Scaling Limit of a Limit Order Book via the Regenerative Characterization of Lévy Trees

High Frequency Asymptotics for the Limit Order Book Peter Lakner and Josh Reed New York University Stern School of Business Sasha Stoikov Cornell University Financial Engineering Manhattan January 19, 2016 Abstract We study the one-sided limit order book corresponding to limit sell orders and model it as a measure-valued process. Limit orders arrive to the book according to a Poisson process and are placed on the book according to a distribution which varies depending on the current best price. Market orders to buy periodically arrive to the book according to a second, independent Poisson process and remove from the book the order corresponding to the current best price. We consider the above described limit order book in a high frequency regime in which the rate of incoming limit and market orders is large and traders place their limit sell orders close to the current best price. Our first set of results provide weak limits for the unscaled price process and the properly scaled measure-valued limit order book process in the high frequency regime. In particular, we characterize the limiting measure-valued limit order book process as the solution to a measure-valued stochastic differential equation. We then provide an analysis of both the transient and long-run behavior of the limiting limit order book process. 1 Introduction The limit order book has become the standard mechanism for trading in modern electronic markets due to its ability to provide a simple and transparent set of rules for the matching of buyers and sellers of securities. Worldwide, according to Russell and Kim [21], order driven markets now account for almost half of the current stock exchanges. However, the widespread adoption of this mechanism for trading and the emergence of High Frequency Traders in these venues have not been without controversy. In particular, events such as the infamous May 6th, 2010 Flash Crash have prompted market observers to question the stability of such exchanges. See Figure 1 below for a 3 minute graph of the S&P500 E-mini futures on the day of the Flash Crash at 2:45 pm. Moreover, as documented in Golub et al. [15], the occurrence of both up and down Mini Flash Crashes, i.e. fast and significantly large one-directional moves in the price of assets, are now also quite common. It is therefore of a practical as well as a theoretical interest to better understand how both the structure and the fundamental parameters of the order book affect the dynamics of price formation. In the present paper, we study the dynamics of the order book over a time scale which is an order of magnitude larger than that of individual order driven events. We refer to this time scale as the high frequency regime and a mathematically precise definition of this regime is given in Definition 3.1 of Section 3 below. Roughly speaking, the high frequency regime which we consider is such that the arrival rate of both limit orders and market orders to the book is large and such that these arrival rates are closely matched to one another. Moreover, we also assume that traders 1 Figure 1: A three minute flashcrash place their individual limit orders in close proximity to the prevailing best price. This regime bears a close resemblance to the conventional heavy traffic regime commonly encountered in the queueing theory literature and indeed many of the tools and techniques of the present paper have been borrowed from the theory of queues. As it turns out, several authors in the past have also adopted techniques from queueing theory in order to study the limit order book. Some of the earliest work in this regard is that of Mendelson [25] who models a market clearing house assuming Poisson arrivals of buy and sell ordres. Cohen et al. [7] consider a continuous time auction where buyers may only place limit orders at two prices, the best bid and the best ask. This work is then extended in Domowitz and Wang [12] to multiple price levels. Smith et al. [30] provide a mean field theory of the limit order book. Recently, Cont et al. [10] have modeled the limit order book as a finite-dimensional continuous time Markov chain. They then show how to use Laplace transform methods in order to calculate several quantities associated with the book such as the direction of the next price move and the probability of executing an order before a change in price. Cont and de Larrard [8] consider a simplified Markovian model of the limit order book in which several key quantities may be computed directly in closed form. A very closely related paper to ours is that of Simatos [28] which elegantly treats a limit order book model similar to our own. In particular, in Simatos [28] a coupling is constructed between the limit order book process and a certain branching random walk. This coupling is then taken advantage of in order to derive necessary and sufficient conditions for the price process to drift to either 0 or +∞. In [24], Maglaras et al. study a fragmented market in which traders may route their orders to one of several competing exchanges. Interestingly, it is shown in Maglaras et al. [24] that on a fluid scale a state-state collapse occurs between the states of the market in each of the various exchanges. Horst and Kreher [18] provide a weak law of large numbers result for the scaling limit of a one-sided limit order book model with fully state dependent dynamics. Finally, we mention the work of Avellaneda et al. [3] which studies the effect of hidden liquidity on the future direction of price movement. Unfortunately, one difficulty commonly encountered in modeling the limit order book is the high 2 dimensionality of its state space. Multiple traders may place their limit orders at any price they wish and as a result in order to keep track of each order on the book at each moment in time one must resort to an infinite-dimensional state space. In the present paper, we confront this difficulty head on by modeling the order book as a measure-valued process. A closely related model has been proposed recently by Abergel and Jedidi [1], who study sequences of multi-dimensional Markov chains and show that the limiting price process converges to a Brownian motion. Our approach in the present paper, though similar in spirit, is limited to modeling one side of the book, which we take to be the sell side (the buy side may treated analogously). Though the framework of Abergel and Jedidi may be most appropriate in describing the normal behavior of a limit order book, we believe that our approach may be useful in understanding situations such as Mini Flash Crashes, where most of the activity happens on one side of the book. We now provide a succinct description of the general setup of our model. A more precise description is given in Section 2. At each point in time t ≥ 0, we let µ(t) ∈ MF (R) represent the state of the order book at time t, where MF (R) denotes the set of finite, non-negative measures on R. Specifically, for each A ∈ B(R), we let µ(t)(A) be the number of limit sell orders on the book at time t whose associated prices lie in the set A. Then, since we are only considering the sell side of the book, the best price at each point in time is given by the lowest price on the book associated with a limit sell order. Mathematically speaking, the best price at time t ≥ 0 is defined by setting p(t) = sup{x ≥ 0 : µ(t)((−∞, x)) = 0}. We next assume that market buy orders arrive to the book according to a Poisson process and upon arrival remove from the book the order corresponding to current best price. On the other hand, limit sell orders also arrive to the book according to a second, independent Poisson process. A limit sell order arriving to the book at time t ≥ 0 will place its order at the price X · p(t−), where X is a positive random variable with distribution G. Finally, we assume the existence of a market maker who never allows the total number of orders on the book to drop below the level a > 0. Our first set of results for the model described above provide limit theorems on the behavior of the unscaled price process as well as the properly scaled order book process in the high frequency regime. These results may be found in Sections 5 and 6, respectively, and are obtained under the crucial assumption that E[ln X] > 0. Note that since we are dealing with the sell side of the book, this assumption implies that traders would on average prefer to sell at a higher price than the current best price. This then places upward pressure on the price and, indeed, as is shown is Theorem 5.1 below, the limiting price process which we obtain is an increasing process. In the terminology of Golub et al. [15], we refer to this situation as an up mini flash crash. Also note that our assumption that E[ln X] > 0 does not immediately imply an increasing price process. In particular, in Simatos [28] it was shown that in the pre-limit it is possible under the same conditions for the price to converge to 0. In Theorem 6.1 below, we provide a limit theorem for the properly scaled measure-valued order book process. In particular, we show that in the limit of the high frequency regime, the majority of the mass of the order book is concentrated at the prevailing best price. Theorem 6.1 may be viewed as providing a link between the limiting price process itself and the limiting distribution of the orders on the book. One of the interesting features of our limiting price process is that its rate of increase is inversely related to the total number of order on the book. This result is in line with the recent empirical findings of Cont et al. [9] which studied the relationship between order flow imbalance and changes in price. In particular, in Cont et al. [9] it is empirically observed that over short intervals of time, 3 changes in price are linear with respect to changes in net order flow and inversely proportional with respect to market depth. Similarly, Hasbrouck and Saar [17] have observed a statistically significant negative relationship between price volatility and order book depth in the vicinity of the best price. Our second set of results in the paper provide a transient as well as a long-term analysis of the limiting price process and the limiting properly scaled order book process. Our transient analysis may be found in Section 7 and essentially relies upon a modification of the Feynman-Kac formula to account for semi-martingales with a singular component to them. In particular, in Section 7 we show how to calculate the expectation of various functionals of the limiting price process and the limiting properly scaled order book process at each point in time. In Section 8, we conduct our long term analysis. As it turns out, the long term behavior of the the limiting price process and the limiting properly scaled order book process depends critically upon whether the incoming rate of limit orders is less than, greater than, or equal to the incoming rate of market orders. In Section 8, we provide the almost sure long-term asymptotics of our limiting processes in each of these three cases. The proofs of many of our results of the paper may be found in the Appendix. 1.1 Notation The following notation will be used throughout this paper. We assume that all random variables are defined on a common probability space (Ω, F, P ). Letting S be a separable and complete metric space, we define D([0, ∞), S) to be the Skorokhod space of all functions with domain [0, ∞) that are right-continuous on [0, ∞), with left limits everywhere in (0, ∞), and taking values in S. We equip D([0, ∞), S) with the standard Skorokhod J1 -topology (see, for instance, Chapter 16 of Billingsley [5]) and its associated Borel σ-algebra. All stochastic processes in this paper are viewed as measurable maps from (Ω, F, P ) to D([0, ∞), S), with S being set equal to either R or MF (R), the set of all finite, non-negative measures on R. The product of a finite number of metric spaces is endowed with the product topology. We endow MF (R) with the topology of weak convergence. That is, a sequence of elements {νn , n ≥ 1} in MF (R) weakly converges to an element ν ∈ MF (R) if and only if ⟨νn , φ⟩ → ⟨ν, φ⟩ as n → ∞, for each φ ∈ Cb (R), where here we have adopted the notation ∫ φ(u)dν(u). ⟨ν, φ⟩ = R For each ξ, ν ∈ MF (R), we recall from (6.11) of Billingsley [5], the definition of the Prohorov metric π, π(ξ, ν) = inf{ε > 0|ξ(A) ≤ ν(Aε ) + ε and ν(A) ≤ ξ(Aε ) + ε for all A ∈ B(R)}. We also recall the well known fact that (MF (R), π) is a separable and complete metric space and moreover (see, for instance, Theorem 6.8 of Billingsley [5]), that π induces the topology of weak convergence on MF (R). The main results in this paper are concerned with the weak convergence of R or MF (R) valued, RCLL (right continuous with left limits) processes. Finally, we denote by e = {t, t ≥ 0} the identity process on R+ . 4 2 The Model In this section, we provide the model which we consider for the remainder of the paper. As mentioned in the Introduction, our approach is to model the limit order book as a measure-valued process. For simplicity, we have chosen to only model one side of the book, which we take to be the sell side of the book. The case of the buy side of the book may be treated analogously. The details of our model are as follows. We assume that limit sell orders arrive to the book according to a Poisson process NL = {NL (t), t ≥ 0} with rate λL . For each i ≥ 1, we denote by τiL = inf{t ≥ 0 : NL (t) ≥ i} the time at which the ith limit sell order arrives to the book. Each time that a limit sell order arrives to the book, it places a single order on the book. The price associated with this order is a random quantity which may be greater than, less than or equal to the current best price. In particular, for each t ≥ 0, let p(t) denote the current best price. p(t) represents the smallest price associated with a limit sell order which is present on the book at time t. Then, a limit sell order arriving to the book at time t will place its order at the price X · p(t−), where X is a strictly positive random variable with distribution G. Beyond requiring that X be a strictly positive random variable, the only other assumption which we place on the distribution G is that there exists a constant 1 < C < ∞ such that P (1/C < X < C) = 1. (1) Note that the above assumptions on the distribution G do not preclude G from being a continuous distribution. In this case our model effectively has a tick size of zero, which one should consider as a theoretical approximation to a market with relatively small tick sizes. In addition, if one restricts the support of G to a set of the form {k n , N1 ≤ n ≤ N2 }, where k > 1 and N1 and N2 integers such that N1 ≤ N2 , then taking the natural log of the price process one ends up with a model with a tick size equal to ln k. Figures 2 through 4 illustrate the three possible transitions of the order book when a limit sell order arrives. They correspond, respectively, to the cases of X > 1, X < 1 and X = 1. p(t−) p(t) Figure 2: A limit sell order is placed on the book above the best price. The price does not change. We next assume that market buy orders arrive to the book according to second Poisson process NM = {NM (t), t ≥ 0} with rate λM . We assume, for convenience, that NM is independent of NL . For each i ≥ 1, we denote by τiM = inf{t ≥ 0 : NM (t) ≥ i} 5 p(t−) p(t) Figure 3: A limit sell order is placed on the book below the best price. The price decreases. p(t−) p(t) Figure 4: A limit sell order is placed on the book at the best price. The price does not change. the time at which the the ith market buy order arrives to the book. Each time that a market buy order arrives to the book, it removes a single limit sell order corresponding the current best price. Figures 5 and 6 illustrate the two possible transitions of the order book when a market buy order arrives. The price may either increase or stay the same. p(t−) p(t) Figure 5: A market buy order arrives to the book and removes one of the limit sell orders corresponding to the best price. Since there were more than one limit sell orders at the best price, the price does not change. We also assume the existence of a “market maker”. The role of the market maker in our model is to never allow the total number of orders on the book to drop below some predetermined level a > 0. In particular, suppose that the total number of orders on the book is at the level a and that a market buy order arrives and removes one of the limit sell orders corresponding to the current best price. The market maker will then instantaneously place a new limit sell order on the book at the same price at which the previous limit sell order was removed. Thus, the total number of orders on the book will never drop below the level a. We let L = {L(t), t ≥ 0} denote the non-decreasing process which keeps track of the points in time at which the market maker places limit sell orders on the book. In other words, L(t) represents the cumulative number of limit orders that the market maker has placed on the book by time t. Before proceeding with a discussion of the full dynamics of the measure-valued order book, we 6 p(t−) p(t) Figure 6: A market buy order arrives to the book and removes the limit sell order corresponding to the best price. This was the last remaining limit sell order at the best price and so the price increases. next provide the details of the simpler process which keeps track of the total number of orders on the book. For each t ≥ 0, let S(t) denote the total number of orders on the book at time t and let S = {S(t), t ≥ 0} be the process keeping track of the total number of orders on the book. Then, according to the discussion in the preceding three paragraphs, it is straightforward to see that one has the identity S(t) = S(0) + NL (t) − NM (t) + L(t), t ≥ 0, (2) where S(0) denotes the total number of orders on the book at time 0. Moreover, using results from the standard theory of reflected random walks (see, for instance, Chapter 2 of Asmussen [2]), one may express L in terms of S(0), NL , NM and a. In particular, we have that L(t) = sup {− (S(0) + NL (s) − NM (s) − a)} ∨ 0, t ≥ 0, (3) 0≤s≤t which also implies the identity ∫ ∞ 1{S(s) > a}dL(s) = 0. 0 Moreover, we note that by (2) and (3) one has that the dynamics of S are identical to those of an M/M/1 queue with an arrival rate of λL and a service rate of λM , which has been adjusted upwards by the amount a. At several points in the paper, and in particular in the proof of Theorem 5.1, it will be helpful to view S as a semi-martingale. Hence, for each t ≥ 0, let M (t) = (NL (t) − λL t) − (NM (t) − λM t) and set M = {M (t), t ≥ 0}. Then, since NM and NL are both Poisson processes, with rates λM and λL , respectively, one immediately sees that the process M is a martingale. Moreover, by (2) we may write S(t) = S(0) + M (t) + B(t), t ≥ 0, (4) where B(t) = (λL − λM )t + L(t) and we set B = {B(t), t ≥ 0}. Since B is of bounded variation, one then obtains from (4) a semi-martingale decomposition of S. This decomposition will be called upon repeatedly in our proofs. We are now in a position to describe the measure-valued order book in detail. For each t ≥ 0, let µ(t) ∈ MF (R) be a counting measure on R which represents the state of the order book at time t. In particular, we assume that for each A ∈ B(R), the quantity µ(t)(A) represents the number of 7 limit sell orders on the book at time t whose prices lie in the set A. We also set µ = {µ(t), t ≥ 0} to be the measure-valued order book process and note that since NL and NM are assumed to be independent Poisson processes, it may be rigorously shown that µ is in fact a Markov process. Next, for each t ≥ 0, let p(t) = sup{x ≥ 0 : µ(t)((−∞, x)) = 0}, (5) denote the leftmost point of the support of µ(t). It then follows that p(t) is the smallest price associated with a limit sell order on the book at time t and hence p(t) represents the current best price. Also note that since the market maker never allows the total number of order on the book to fall below the level a > 0, it follows that the price p(t) is always well-defined. We set p = {p(t), t ≥ 0} to be the price process. The dynamics of the measure-valued order book may now be described as follows. For each x ∈ R, let δx denote the Dirac measure at x and let {Xi , i ≥ 1} be an i.i.d. sequence of random variables, independent of NL and NM , and with common distribution G. Next, let µ(0) ∈ MF (R) be a (random) counting measure representing the state of the book at time t = 0. We then have that, according to the preceding discussion, the state of order book at each point in time t ≥ 0 is given by NL (t) µ(t) = µ(0) + ∑ i=1 δp(τ L −)Xi − i ∫ t 0 δp(s−) dNM (s) + ∫ t 0 δp(s−) dL(s). (6) Note that the assumption that G(0) = 0 ensures that the price always stays positive and that since µ(0)(R) = S(0), we must assume that µ(0)(R) ≥ a, a.s. Next, note that using the identity (2), one has that (6) may be expressed in the more parsimonious form NL (t) µ(t) = µ(0) + ∑ i=1 (δp(τ L −)Xi − δp(τ L −) ) + i i ∫ t 0 δp(s−) dS(s). (7) The representation (7) serves as the starting point for many of our proofs that follow. We will also often find it more convenient to analyze the measure-valued limit order book process by evaluating it using test functions φ in Cb (R). Recall from Section 1.1 that for an arbitrary measure ν ∈ MF (R) and function φ ∈ Cb (R), we denote the inner product of ν and φ by setting ∫ φ(x)dν(x). (8) ⟨ν, φ⟩ = R It then follows upon integrating both sides of (7) that for each φ ∈ Cb (R) and t ≥ 0, we have that NL (t) ⟨µ(t), φ⟩ = ⟨µ(0), φ⟩ + ∑ i=1 (φ(p(τiL −)Xi ) − φ(p(τiL −))) The representation (9) will also be helpful in our proofs that follow. 8 + ∫ t φ(p(s−))dS(s). 0 (9) 3 High Frequency Regime In this paper, we are interested in obtaining asymptotic approximations to both the price process and the measure-valued limit order book process. The asymptotic regime which we will consider in order to obtain our approximations is such that the arrival rate of both limit orders and market orders to the order book is high, while, at the same time, traders place their limit sell orders relatively close to the current best price. We refer to this regime as the “high frequency” regime. The detailed mathematical setup of the high frequency regime is as follows. We consider a sequence of limit order books indexed by n ≥ 1, where n may be roughly thought of as the rate at which market and limit orders arrive to the nth system (see Item 1 of Definition 3.1 below). For each fixed n ≥ 1, the system dynamics are the same as those defined in Section 2, and all quantities associated with the nth order book are denoted by a superscript n. In particular, limit sell orders arrive to the the nth order book according to the Poisson process NLn = {NLn (t), t ≥ 0}, which has rate λnL . For each i ≥ 1, we denote by τiL,n the time at which the ith limit order arrives to the nth order book. Each time that a limit sell order arrives to the nth order book, it places a single order on the book. In particular, letting pn (t) denote the best price at time t ≥ 0 in the nth limit order book, then, a limit sell order arriving to the nth limit order book at time τiL,n will place its order at the price Xn,i pn (τiL,n −). Our assumption on the distribution of Xn,i is given by Item 3 of Definition 3.1 below. Market buy orders arrive to the nth order book according to the Poisson n = {N n (t), t ≥ 0} which has rate λn > 0, and, for each i ≥ 1, we denote by τ M,n the process NM M M i time at which the ith market order arrives to the nth limit order book. Each market order arriving to the book removes a single order corresponding to the current best price. Finally, we assume that in the nth limit order book, there exists a market maker who never allows the total number of orders on the book fall below the level an > 0. The high frequency regime is now defined by the following set of four assumptions. Definition 3.1 (High Frequency Regime). The high frequency regime is defined by assuming that 1. n−1 λnL → λ > 0 and n−1 λnM → λ > 0 as n → ∞, √ 2. n(n−1 λnL − n−1 λnM ) → θ ∈ R as n → ∞, 1/n1/2 for i ≥ 1, where {Xi , i ≥ 1} is a sequence of i.i.d. random 3. For each n ≥ 1, Xn,i = Xi n. variable with common distribution G, independent of NLn and NM 4. n−1/2 an → ã > 0 as n → ∞. We now comment on Items 1 through 4 in Definition 3.1 above. First note that from Item 1 it is evident that the incoming rate of limit sell and market buy orders is on the order of n in the nth limit order book, hence the term “high frequency”. Moreover, note that Item 1 also implies that these rates are closely matched with one another as n grows large. In particular, Item 2 of Definition 3.1 provides a refined asymptotic on the mismatch between the supply of and demand for limit orders in the high frequency regime. This mismatch is characterized through the scalar θ. In general, θ may be positive, negative, or equal to zero. In Section 8, we study the effects of the sign of θ on the long term behavior of the limiting price process. Item 3 of Definition 3.1 implies that as n tends ∞, the distribution of the i.i.d. sequence {Xn,i , i ≥ 1} weakly converges to the constant 1. This in turn implies that in the high frequency regime, the majority of traders place their limit sell in close proximity to the current best price. This type of behavior has been observed 9 empirically by Biais et al. [4] whose studied order flow dynamics at the Paris Bourse and concluded that the majority of order flow occurs close to the best quote. Moreover, note that performing a first order Taylor expansion, one has that for each n ≥ 1, 1/n1/2 Xn,i = Xi = 1+ ln(Xi ) + o(n−1/2 ), i ≥ 1. n1/2 (10) Thus, the expansion (10) suggests that the sign of ln(X) plays an important role in the direction of the price movement and indeed, as mentioned in the Introduction, our results in this paper require the assumption that E[ln X] > 0. Finally, letting ãn = n−1/2 an for each n ≥ 1, note that Item 4 of Definition 3.1 implies that the market maker never allows the total number of orders on the nth √ limit order book to fall too far below the level nã. This can be attributed to the fact that as a result of Item 2 of Definition 3.1, the fluctuations of the total number of orders on the book are on the order of n1/2 . One could also consider a high-frequency regime in which Item 1 of Definition 3.1 is relaxed by only requiring that n−1 λnL → λL > 0 and n−1 λnM → λM > 0, with no specific relationship between λL and λM . In such a regime Item 2 of Definition 3.1 would have to be removed and Item 4 replaced by n−1 an → ā > 0. This regime corresponds to one in which the total number of orders on the book is of order n and a law of large numbers scaling is used for the total number of orders on the book. Although interesting, we do not pursue such a regime in the present paper. We now complete this section by describing our choice of scaling for all relevant processes associated with the nth limit order book. For each n ≥ 1, let S n (t) denote the total number of orders on the nth limit order book at time t ≥ 0. We then define S̃ n (t) = n−1/2 S n (t) to be the diffusion scaled total number of orders on the book and we define the process S̃ n = {S̃ n (t), t ≥ 0}. Note that making use of the semi-martingale decomposition of S n provided by (4) of Section 2, one may also write S̃ n (t) = S̃ n (0) + M̃ n (t) + B̃ n (t), t ≥ 0, (11) n (t) − λnM t), t ≥ 0, M̃ n (t) = n−1/2 (NLn (t) − λnL t) − n−1/2 (NM (12) B̃ n (t) = n1/2 (n−1 λnL − n−1 λnM )t + L̃n (t), t ≥ 0, (13) L̃n (t) = n−1/2 Ln (t), t ≥ 0. (14) where and and We also define the resulting processes M̃ n = {M̃ n (t), t ≥ 0}, B̃ n = {B̃ n (t), t ≥ 0} and L̃n = {L̃n (t), t ≥ 0}, and note that in the sequel it will also be convenient to set n n N̄Ln (t) = n−1 NLn (t) and N̄M (t) = n−1 NM (t), t ≥ 0, n = {N̄ n (t), t ≥ 0}. Next, for and to define the resulting processes N̄Ln = {N̄Ln (t), t ≥ 0} and N̄M M n each n ≥ 1, let µ (t) ∈ MF (R) be the state of the nth limit order book at time t ≥ 0. As it turns out, we will require two assumptions on the initial state of the nth measure-valued order book. These assumptions may be summarized as follows. 10 Assumption 3.2 (Initial state of the order book). We shall assume that for each n ≥ 1, the initial state of the nth order book, µn (0), satisfies the following two conditions: 1. ] 1 [ n 4 E (S (0)) < ∞, 2 n≥1 n sup 2. If x < y are consecutive orders on the nth order book at time t = 0, that is, µn (0)({x}) > 0, µn (0)({y}) > 0, and µn (0)((x, y)) = 0, then y/x ≤ C √1 n . Note that Item 2 of Assumption 3.2 is consistent with our earlier assumption (1) and Item 3 of Definition 3.1. We next define µ̃n (t) = n−1/2 µn (t) to be the state of the diffusion scaled order book at time t ≥ 0. In other words, we set µ̃n (t)(A) = n−1/2 µn (t)(A) for each A ∈ B(R). Substituting the above scalings into the system dynamics equation (9) of Section 2, one now obtains that for each n ≥ 1 and φ ∈ Cb (R), n NL (t) ( ( ) ( ) ( ( ))) 1 ∑ 1/n1/2 n n ⟨µ̃ (t), φ⟩ = ⟨µ̃ (0), φ⟩ + √ φ pn τiL,n − Xi − φ pn τiL,n − n i=1 ∫ t φ(pn (s−))dS̃ n (s), t ≥ 0. + (15) 0 Equation (15) will be helpful as we proceed throughout our proofs in the Appendix. It is also instructive to point out that clearly one must have that ⟨µ̃n (t), 1⟩ = S̃ n (t) for each t ≥ 0. Finally, we note in passing that the price process is not scaled in any way in the high frequency regime. Indeed, in Theorem 5.1 of Section 5 below, we obtain a non-trivial weak limit for the sequence of unscaled price processes in the high frequency regime. 4 Weak Convergence of the Total Number of Orders on the Book In this section, we provide a result on the weak convergence of the sequence of processes {S̃ n , n ≥ 1} defined in Section 3. Recall that these processes keep track of the total number of orders on the book, properly scaled. Our main result is to show that {S̃ n , n ≥ 1} weakly converges in the high frequency regime to what is referred to as a “reflected Brownian motion”. Although this result is to some extent straightforward, it is important for the remainder of the paper and so we state it here with a short proof. We begin by providing a rigorous definition of reflected Brownian motion. Although this definition can take many forms, we feel that the following is most relevant for the remainder of the paper. Definition 4.1. Let X̃ = {X̃(t), t ≥ 0} be a standard Brownian motion and let Z̃(0) be a random variable, independent of X̃. Moreover, let a, θ, σ ∈ ℜ. Then, we say that Z̃ = {Z̃(t), t ≥ 0} is a Brownian motion, reflected at the level a, with constant drift θ and infinitesimal variance σ 2 , started at Z̃(0), if the following three conditions are satisfied almost surely, 11 1. Z̃(t) = Z̃(0) + θt + σ X̃(t) + Ỹ (t) ≥ a, t ≥ 0, 2. Ỹ = {Ỹ (t), t ≥ 0} is a non-decreasing process with Ỹ (0) = 0, 3. The pair (Z̃, Ỹ ) satisfies the complementarity condition ∫ ∞ 1{Z̃(s) > a}dỸ (s) = 0. 0 Moreover, we refer to Ỹ as the local time of Z̃ at a. Note that by conditions (1)-(3) above, one has that Z̃ behaves like a Brownain motion with drift θ and infinitesimal variance σ 2 in the open set (a, ∞), and is instantaneously reflected back into (a, ∞) upon hitting the boundary a. For a further discussion on reflected Brownian motion, one may consult the excellent text of Harrison [16]. The following is now the main result of this section. Proposition 4.2. If S̃ n (0) ⇒ S̃(0) as n → ∞, and Items 1-4 of Definition 3.1 hold, then the pair (S̃ n , L̃n ) ⇒ (S̃, L̃) in D2 ([0, ∞), R) as n → ∞, where S̃ is a Brownian motion, reflected at ã > 0, with constant drift θ and infinitesimal variance 2λ, started at S̃(0), and with local time L̃ at ã. Proof. First note that by (11)-(14) above, we have that we may write S̃ n (t) = S̃ n (0) + M̃ n (t) + n1/2 (n−1 λnL − n−1 λnM )t + L̃n (t), for each t ≥ 0. Next, note that by the discussion in Section 2, it is clear that the pair (S̃ n − ãn , L̃n ) almost surely satisfy conditions (6.11)-(6.13) of Chen and Yao [6]. Hence, by Theorem 6.1 of Chen and Yao [6], we may write (S̃ n − ãn , L̃n ) = (Φ, Ψ)((S̃ n (0) − ãn ) + n1/2 (n−1 λnL − n−1 λnM )e + M̃ n ), where the map (Φ, Ψ) : D([0, ∞), R) 7→ D2 ([0, ∞), R) is continuous at x ∈ C([0, ∞), R). Moreover, Items 1-2 of Definition 3.1 and the martingale central limit theorem n n 1/2 −1 n (see, for instance, Theorem 7.1.4 of Ethier √ and Kurtz [13]) imply that (S̃ (0) − ã ) + n (n λL − n −1 n n λM )e + M̃ ⇒ (S̃(0) − ã) + θe + 2λX̃ as n → ∞, where X̃ is a standard Brownian motion. It therefore follows by Item 4 of Definition 3.1, the representation (S̃ n − ãn , L̃n ) = (Φ, Ψ)((S̃ n (0) − ãn ) + n1/2 (n−1 λnL − n−1 λnM )e + M̃ n ), the almost sure continuity of Brownian motion (see, for instance, Definition 2.1.1 of Karatzas and Shreve [22]), and the continuous mapping theorem (see, for instance, Theorem 3.4.3 of Whitt [31]), that (S̃ n − ãn , L̃n ) ⇒ (S̃ − ã, L̃) = (Φ, Ψ)((S̃(0) − ã) + √ θe + 2λX̃) as n → ∞. However, it is clear by conditions (6.11)-(6.13) of Chen and Yao [6] that this then implies that S̃ is a Brownian motion, reflected at the level ã > 0, with constant drift θ and infinitesimal variance 2λ, started at S̃(0), and with local time L̃ at ã. This completes the proof. Note that Definition 4.1 and Proposition 4.2 together imply that we may write √ S̃(t) = S̃(0) + 2λB̃(t) + θt + L̃(t) ≥ ã, t ≥ 0, (16) where B̃ = {B̃(t), t ≥ 0} is a standard Brownian motion, and L̃ is a non-decreasing process such that ∫ ∞ 1{S̃(s) > ã}dL̃(s) = 0. (17) 0 The representation (16)-(17) will be useful as we proceed. 12 5 Weak Convergence of the Price Process In this section, we provide our main result on the weak convergence of the sequence of price processes {pn , n ≥ 1} in the high frequency regime described in Section 3, see Theorem 5.1 below. Then, in Section 6, we show how Theorem 5.1 may be used to prove that the sequence of diffusion scaled limit order books, {µ̃n , n ≥ 1}, weakly converges in the high frequency regime. The supporting lemmas and propositions for Theorem 5.1 may be found in Sections 10.1 and 10.2 of the Appendix. Before providing the statement of Theorem 5.1, we must first setup the following notation. Let ln = ln(x) be the natural log function on (0, ∞) and, for each n ≥ 1, let ∆µ̃l,n (0) = ⟨µ̃n (0), ln⟩ − S̃ n (0) ln(pn (0)). (18) We then have the following. Theorem 5.1. If E[ln X] > 0 and E[(∆µ̃l,n (0))2 ] → 0 and (pn (0), S̃ n (0)) ⇒ (p(0), S̃(0)) as n → ∞, and if Items 1-4 of Definition 3.1 hold, then pn ⇒ p as n → ∞, where the process p = {p(t), t ≥ 0} is given by (∫ t ) E[ln X] p(t) = p(0) exp d(λs) , for t ≥ 0, (19) S̃(s) 0 where S̃ = {S̃(t), t ≥ 0} is a Brownian motion, reflected at ã > 0, with constant drift θ and infinitesimal variance 2λ, started at S̃(0). Proof. First note that in the notation of Section 10.1, we may write p = exp(pl ). The proof is then a direct consequence of the fact that exp is a continuous map on (−∞, ∞), the continuous mapping theorem (see, for instance, Theorem 3.4.3 of Whitt [31]) and Proposition 10.1 of Section 10.1 of the Appendix. Note that Theorem 5.1 implies that in the high frequency regime, the limiting price process p is an increasing process. This, of course, is a consequence of our assumption that E[ln X] > 0. Moreover, recall by Proposition 4.2 of Section 4 that the reflected Brownian motion S̃ = {S̃(t), t ≥ 0} appearing in Theorem 5.1 keeps track of the total number of orders on the order book. Thus, the fact that S̃(t) appears in the denominator in (78) implies that in the limit the instantaneous rate of increase of the limiting price process is inversely related to the total number of orders on the book. Intuitively speaking, this can be explained by the fact that in order for the price to increase to a new level, one must first remove all of those orders from the book which lie below that level. Moreover, as is shown in Theorem 6.1 below, for n sufficiently large most of the orders on the book lie in a close vicinity of the best price. We also recall that by Item 4 of Definition 3.1, we have that S̃(t) ≥ ã > 0 for each t ≥ 0, and so the integral in (78) is well defined. In Section 7, we study the transient behavior of the limiting price process p given in Theorem 5.1 and in Section 8, we study the long term behavior of p. 6 Weak Convergence of the Order Book Process In this section, we state Theorem 6.1, which provides a weak limit for the sequence {µ̃n , n ≥ 1} of diffusion scaled, measure-valued order book processes in the high frequency regime. Recall first 13 that in the high frequency regime described in Section 3, the distance at which traders place their limit orders from the current best price weakly converges to zero. Hence, it is natural to expect that when n is large, all of the orders on the book will be contained in a small neighborhood of the current best price. Indeed, under appropriate initial conditions, it turns out that this is the case, as we now show. The supporting lemmas and propositions for Theorem 6.1 may be found in Section 10.3. We have the following. Theorem 6.1. If E[ln X] > 0 and E[(∆µ̃nl (0))2 ] → 0 and (pn (0), S̃ n (0)) ⇒ (p(0), S̃(0)) as n → ∞, and if Items 1-4 of Definition 3.1 hold, then µ̃n ⇒ µ̃ as n → ∞, where µ̃ = {µ̃(t), t ≥ 0} is given by µ̃(t) = S̃(t)δp(t) , t ≥ 0, (20) where p = {p(t), t ≥ 0} and S̃ = {S̃(t), t ≥ 0} are as given in Theorem 5.1. Proof. First recall that by Proposition 10.14 of Section 10.3, the sequence {µ̃n , n ≥ 1} is tight in D(R+ , MF (R)). Hence, in order to complete the proof, it suffices by Theorem 13.1 of Billingsley [5] to show that the finite dimensional distributions of {µ̃n , n ≥ 1} converge to those of µ̃ as given by (79). Let t1 , t2 ≥ 0. We will show that (µ̃n (t1 ), µ̃n (t2 )) ⇒ (µ̃(t1 ), µ̃(t2 )) in MF (R) × MF (R) as n → ∞. The proof for an arbitrary number of dimensions follows similarly. Let BL+ (R) denote the set of bounded and non-negative Lipschitz functions on R. Next, for each φ1 , φ2 ∈ BL+ (R), let Fφ1 ,φ2 ∈ Cb (MF (R) × MF (R)) be defined by setting Fφ1 ,φ2 (ν1 , ν2 ) = exp (−⟨ν1 , φ1 ⟩ − ⟨ν2 , φ2 ⟩) and set V = { n ∑ i=1 + } λi Fφi,1 ,φi,2 ; n ∈ N, λ1 , ..., λn ∈ R, φ1,1 , φ1,2 , ..., φn,1 , φn,2 ∈ BL (R) . Then, V ⊂ Cb (MF (R)×MF (R)) and it is straightforward to check that V is an algebra. Moreover, since F = {⟨ · , φ⟩, φ ∈ BL+ (R+ )} separates points on MF (R), it is straightforward to show that V separates points on MF (R) × MF (R). Now let F ∈ V . By Proposition 10.15 and the continuous mapping theorem (see, for instance, Theorem 3.4.3 of Whitt [31]), we have that F (µ̃n (t1 ), µ̃n (t2 )) ⇒ F (µ̃(t1 ), µ̃(t2 )) as n → ∞, where µ̃ is as given in (79). Hence, since F is a bounded function it, follows that E[F (µ̃n (t1 ), µ̃n (t2 ))] → E[F (µ̃(t1 ), µ̃(t2 ))] as n → ∞. It therefore follows by the tightness of the sequence {(µ̃n (t1 ), µ̃n (t2 )), n ≥ 1} and Lemma 3.4.3 and Theorem 3.4.5 of Ethier and Kurtz [13], that (µ̃n (t1 ), µ̃n (t2 )) ⇒ (µ̃(t1 ), µ̃(t2 )) in MF (R) × MF (R) as n → ∞. The proof for n > 2 and t1 , ..., tn ≥ 0 follows similarly. This completes the proof. Note that Theorem 6.1 effectively says that in the limit of the high frequency regime, the entire mass of the order book is contained at the best price. Moreover, recall from Theorem 5.1 of Section 5 that the dynamics of the limiting price process are intricately related to process keeping track 14 of the total number of orders on the book. Thus, taken together, Theorems 5.1 and 6.1 imply interesting behavior for various functionals of the limiting order book. In particular, let ξ(x) = x be the identity function on R and note that ⟨µn (t), ξ⟩ represents the total value of all of the orders on the book at time t. Although the function ξ is clearly not a member of Cb (R), it is still natural to conjecture from Theorem 6.1 that ⟨µ̃n , ξ⟩ ⇒ ⟨µ̃, ξ⟩ as n → ∞. In particular, by Theorem 5.1, this would imply that for each t ≥ 0, one has the convergence (∫ t ) E[ln X] n ⟨µ̃ (t), ξ⟩ ⇒ S̃(t)p(0) exp d(λs) as n → ∞. (21) S̃(s) 0 In Section 7, we provide a method for calculating the expected value of the quantity appearing on the righthand side of (21) at arbitrary points in time. Then, in Section 8, we analyze the long term behavior of the quantity appearing on the righthand side of (21). We now conclude this section by providing a characterization of the limit process µ̃ in Theorem 6.1 as the solution to a measure-valued stochastic differential equation. We have the following. Proposition 6.2. The limiting measure-valued order book process µ̃ given in Theorem 6.1 may be characterized as a solution to the measure-valued SDE, ∫ t ∫ t ′ φ(p(s))dS̃(s) + λE[ln X] φ (p(s))p(s)ds, t ≥ 0, (22) ⟨µ̃(t), φ⟩ = ⟨µ̃(0), φ⟩ + 0 0 for each φ ∈ Cb2 (R), where S̃ = {S̃(t), t ≥ 0} is a Brownian motion, reflected at ã > 0, with constant drift θ and infinitesimal variance 2λ, started at ⟨µ̃(0), 1⟩, and p(t) = sup{x ≥ 0 : µ̃(t)((−∞, x)) = 0}. Proof. The proof follows by (79) of Theorem 6.1 and a direct application of Ito’s formula (see, for instance, Theorem 3.3.3 of Karatzas and Shreve [22]). In general, we believe that the measure-valued SDE (22) characterizes the limit process µ̃ under more relaxed assumptions on the initial conditions of the order book than those contained in the statements of Theorems 5.1 and 6.1. However, under these more relaxed initial conditions, the proofs of Theorems 5.1 and 6.1, as well as the solution to (22), become significantly more complicated while adding little additional insight. We have therefore restricted ourselves in the present paper to the limiting initial condition µ̃(0) = S̃(0)δp(0) . 7 Transient Behavior of the Limiting Price Process In this section, we study the transient behavior of the limiting price process p given in Theorem 5.1, as well as the transient behavior of the limiting order book process µ̃ given in Theorem 6.1. In particular, for each point in time t ≥ 0, we show how to calculate the expected value of the price, the expected total value of the book, and we also show how to calculate the expected value of the total amount of money which the market maker must outlay up until time t, properly discounted. We begin with the easiest case of calculating the expected value of the limiting price process at a fixed point in time. 15 First note that it is quite straightforward to calculate the expected value of the natural log of the limiting price process at each point in time. In particular, letting pl (t) = ln p(t) for each t ≥ 0, one has by Theorem 5.1 that for each t ≥ 0, ] ∫ t [ λE[ln X] l l l l E[p (t) | S̃(0) = x, p (0) = p ] = p + E | S̃(0) = x ds, (23) S̃(s) 0 where S̃ = {S̃(t), t ≥ 0} is a Brownian motion, reflected at the level ã > 0, with constant drift θ and infinitesimal variance 2λ . The transient distribution of reflected Brownian motion is known in closed form (see (4) of Section 1.9 of Harrison [16]) and so in principle the calculations of the expectations in (23) may be preformed. The situation changes when one attempts to calculate the expected value of the actual price itself. In this case, by Theorem 5.1, one is confronted with the calculation ) ] [ (∫ t λE[ln X] ds | S̃(0) = x . (24) E[p(t) | S̃(0) = x, p(0) = p] = pE exp S̃(s) 0 The calculation of the expectation on the righthand side above is in general difficult. Therefore, in order to ease ourselves into the situation, we first consider the case in which S̃ has zero drift, √ i.e. θ = 0. In this case, it is well known that S̃ is equivalent in law to ã + 2λ|X̃|, where X̃ is a standard Brownian motion. We therefore have the equality ] ) [ (∫ t ) ] [ (∫ t √ λE[ln X] λE[ln X] √ ds | 2λX̃(0) = x − ã . E exp ds | S̃(0) = x = E exp S̃(s) 0 0 ã + 2λ|X̃(s)| The expectation on the righthand side above may now be calculated via an application of the Feynman-Kac formula (see, for instance, Theorem 4.4.2 of Karatzas and Shreve [22]). In general, however, S̃ has a singular component to it and so it does not fit neatly into the usual context of the Feynman-Kac formula and the same maneuver cannot be applied. Nevertheless, S̃ is still a semimartingale and so the formula may be appropriately modified in order to yield a partial differential equation whose solution is the desired expected value. In particular, let k : [ã, ∞) 7→ (0, λE[ln X]/ã] be the function defined by k(x) = λE[ln X] , x (25) and note that k is bounded. We then have the following. Theorem 7.1. Suppose that v ∈ C 1,2 is an R-valued function on [0, T ) × [ã, ∞) satisfying the growth condition 2 max |v(t, x)| ≤ Kebx , x ∈ [ã, ∞), 0≤t≤T (26) for some constants K > 0 and b > 0, and that v satisfies − ∂v ∂2v ∂v − kv = λ 2 + θ , on [0, T ) × [ã, ∞), ∂t ∂x ∂x (27) v(T, x) = 1, x ∈ [ã, ∞), (28) subject to 16 ∂v(t, ã) ∂x = 0, t ∈ [0, T ). (29) Then, such a v is unique, and [ v(t, x) = E exp (∫ T −t 0 λE[ln X] ds S̃(s) ) ] | S̃(0) = x . Note, of course, that from Theorem 5.1 one obtains that E[p(T − t)|S̃(0) = x, p(0) = p] = pv(t, x). The proof of Theorem 7.1 is as follows. For the most part, it follows the proof of Theorem 4.4.2 of of Karatzas and Shreve [22], with the proper adjustments necessary. Proof of Theorem 7.1. Recall first that from (16)-(17) we have that S̃ = {S̃(t), t ≥ 0} has the semi-martingale decomposition √ (30) S̃(t) = S̃(0) + 2λB̃(t) + θt + L̃(t), t ≥ 0, where B̃ = {B̃(t), t ≥ 0} is a standard Brownian motion, and L̃ = {L̃(t), t ≥ 0} is a non-decreasing process such that ∫ ∞ 1{S̃(s) > ã}dL̃(s) = 0. (31) 0 Hence, applying Ito’s rule (see, for instance, Theorem 3.3.3 of Karatzas and Shreve [22]) and making note of (27), one obtains that [ d v(t + β, S̃(β)) exp (∫ β k(S̃(s))ds 0 )] = exp (∫ β ) k(S̃(s))ds vx (t + β, S̃(β))dL̃(β) ) (∫ β √ k(S̃(s))ds vx (t + β, S̃(β))dB̃(β). + 2λ 0 0 Also note that by (29) and (31), exp (∫ β ) k(S̃(s))ds vx (t + β, S̃(β))dL̃(β) = 0. 0 (32) Now let τn = inf{t ≥ 0 : |S̃t | ≥ n}. Letting 0 < r < T − t and integrating on [0, r ∧ τn ], the stochastic integral above has zero expectation, and so we obtain that ) ] [ (∫ τ n k(S̃(s))ds 1{τn ≤ r} | S̃(0) = x (33) v(t, x) = E v(t + τn , S̃(τn )) exp [ (∫ 0r ) ] +E v(t + r, S̃(r)) exp k(S̃(s))ds 1{τn > r} | S̃(0) = x . (34) 0 The remainder of the proof now follows similarly to the proof of Theorem 4.4.2 of Karatzas and Shreve [22]. We treat each of the terms in (33) and (34) separately. 17 Regarding (33), note that since k is bounded, we have by (26) that (33) is bounded by ( ) [ ] λE[ln X] exp (T − t) E |v(t + τn , S̃(τn ))|1{τn ≤ r} | S̃(0) = x ã 2 ≤ Kebn P (τn ≤ T |S̃(0) = x). However, since by Lemma 13.5.1 of Whitt [31] there exists a constant C > 0 such that sup |S̃(t)| ≤ C sup |S̃(0) + θt + B̃(t)|, 0≤t≤T 0≤t≤T 2 it follows as in the proof of Theorem 4.4.2 of Karatzas and Shreve [22] that ebn P (τn ≤ T |S̃(0) = x) → 0 as n → ∞. Finally, letting n → ∞ and r → T − t, it follows by the dominated convergence theorem (see, for instance, Theorem 1.8 of Lieb and Loss [23]) and (28) that ) ] [ (∫ r k(S̃(s))ds 1{τn > r} | S̃(0) = x E v(t + r, S̃(r)) exp 0 [ (∫ T −t ) ] λE[ln X] → E exp ds | S̃(0) = x , S̃(s) 0 which completes the proof. We next show how to informally calculate the expected total value of the book at an arbitrary point in time. Note that this is not as easy as simply calculating the expected size of the book and multiplying it by the expected price. In particular, as was already noted in Section 5, the rate of increase of the price is inversely related to the size of the book and so the price itself is determined by the sample path of the size of the book. Now let ξ(x) = x be the identity function. Then, given the form of the limiting order book from Theorem 6.1, we now, informally, must calculate ] [ ] [ E ⟨µ̃(t), ξ⟩ | S̃(0) = x, p(0) = p = E S̃(t)p(t) | S̃(0) = x, p(0) = p . However, by Theorem 5.1, one has that [ E S̃(t)p(t) | S̃(0) = x, p(0) = p ] [ = pE S̃(t) exp (∫ t 0 λE[ln X] ds S̃(s) ) ] | S̃(0) = x . (35) We now proceed to calculate the expectation on the righthand side of (35). The proof of the following result is similar to the proof of Theorem 7.1 and so we omit it. Theorem 7.2. Suppose that v ∈ C 1,2 is a R-valued function on [0, T ) × [ã, ∞) satisfying the growth condition 2 max |v(t, x)| ≤ Kebx , x ∈ [ã, ∞), 0≤t≤T (36) for some constants K > 0 and b > 0, and that v satisfies − ∂2v ∂v ∂v − kv = λ 2 + θ , on [0, T ) × [ã, ∞), ∂t ∂x ∂x 18 (37) subject to v(T, x) = x, x ∈ [ã, ∞), ∂v(t, ã) = 0, t ∈ [0, T ). ∂x Then, such a v is unique, and [ v(t, x) = E S̃(T − t) exp (∫ T −t 0 λE[ln X] ds S̃(s) ) ] | S̃(0) = x . Now suppose that the market maker nets a fixed proportion 0 < α ≤ 1 of the price each time that he or she is asked to place a limit sell order on the the book. It is then natural to attempt to calculate the market maker’s expected discounted cash flow over a finite period of time. In order to make the calculation, first recall that in the model of Section 2, the non-decreasing process L = {L(t), t ≥ 0} keeps track of the points in time at which the market maker places sell orders on the book. Moreover, recall by Proposition 4.2 that L̃n ⇒ L̃ as n → ∞, where L̃ is the local time of S̃ as ã. Hence, assuming that the discount rate is given by γ > 0, then, in the limit, the natural approximation to the expected discounted cash of the market maker is given by ] [∫ T −γt e p(t)dL̃(t) | p(0) = p, S̃(0) = x . (38) αE 0 Now recalling the precise form of the limiting price process p given by Theorem 5.1, we then see that calculating this expectation is equivalent to determining the quantity ) ) ] (∫ t ( [∫ T λE[ln X] − γ ds dL̃(t) | S̃(0) = x . (39) exp αpE S̃(s) 0 0 The following result now provides a partial differential equation whose solution yields the expected value in (39). Its proof is similar in nature to that of Theorem 7.1 and so we omit it. Theorem 7.3. Suppose that v ∈ C 1,2 is a R-valued function on [0, T ) × [ã, ∞) satisfying the growth condition 2 max |v(t, x)| ≤ Kebx , x ∈ [ã, ∞), 0≤t≤T (40) for some constants K > 0 and b > 0, and that v satisfies − ∂v ∂2v ∂v − (k − γ)v = λ 2 + θ , on [0, T ) × [ã, ∞), ∂t ∂x ∂x subject to v(T, x) = 0, x ∈ [ã, ∞), ∂v(t, ã) = −1, t ∈ [0, T ). ∂x Then, such a v is unique, and [∫ v(t, x) = E T −t 0 exp (∫ 0 s( ) ) ] λE[ln X] − γ du dL̃(s) | S̃(0) = x . S̃(u) 19 (41) Note, of course, that if v satisfies the partial differential equation given by Theorem 7.3, then, by (38) and (39), the market maker’s expected discounted payoff is given by ] [∫ T −t −γs e p(s)dL̃(s) | p(0) = p, S̃(0) = x = αpv(t, x). αE 0 This completes our analysis of the transient behavior of p and µ̃. 8 Long Term Behavior of the Limiting Price and Order Book Processes While the results of Section 7 may be thought of as describing the behavior of p and µ̃ over short to medium periods of time, in the current section we focus on the long term behavior of the limiting price process p of Theorem 5.1, and the limiting order book process µ̃ of Theorem 6.1. Our results will be divided into three cases, depending on whether θ < 0, θ > 0 or θ = 0. Note that by (2), these three cases loosely correspond to the arrival rate of limit orders being less than, greater than, or equal to the arrival rate of market orders, respectively. Interestingly, as we will see, it turns out that the long term behavior of p is very different in these three cases. We begin with the case of θ < 0. 8.1 Long Term Behavior for θ < 0 The case of θ < 0 corresponds, loosely speaking, to the arrival rate of limit orders being strictly less than the arrival rate of market orders. Specifically, we have that λnL < λnM for sufficiently large n. In this case, for sufficiently large n, the process S n which keeps track of the total number of orders on the book is identical in distribution to an + Qn , where Qn is the number in system process for an M/M/1 queue with a utilization of ρn = λnL /λnM < 1. One therefore has that the total number of orders on the book never grows too large without returning back down its minimum level an . This fact will turn out to play a critical role in determining the long term behavior of p and distinguishing it from the subsequent cases of θ > 0 and θ = 0. Our first result is to describe the long term behavior pl = ln p, the natural log of the limiting price process. Note that is straightforward to see from Theorem 5.1 that the precise form of pl is given by ∫ t E[ln X] l l d(λs), t ≥ 0, (42) p (t) = p (0) + S̃(s) 0 where S̃ = {S̃(t), t ≥ 0} is a Brownian motion, reflected at ã > 0, with constant drift θ and infinitesimal variance 2λ, started at S̃(0). Now let ∫ ∞ 1 −y e dy, (43) E1 (x) = y x denote the exponential integral function. We then have the following result. Proposition 8.1. If θ < 0, then for each T > 0 and ε > 0, ( ) ) pl (νt) ( −θã/λ lim P sup − −θE[ln X]e E1 (−θã/λ) t > ε = 0. ν→∞ ν 0≤t≤T 20 Proof. Let τ0down = inf{t ≥ 0 : S̃(t) = ã} and τ1up = inf{t ≥ τ0down : S̃(t) ≥ ã + 1} and τ1down = inf{t ≥ τ0up : S̃(t) = ã}. Then, similar to as in Section 5.4 of Harrison [16], one has that if θ < 0, then the process S̃ = {S̃(t), t ≥ 0} is a regenerative process (see Asmussen [2]) where the lengths of the regeneration cycles are distributed according to τ1down − τ0down . Hence, 1/S̃ = {1/S̃(t), t ≥ 0} is also a regenerative process where the legnths of the regeneration cycles are distributed according to τ1down − τ0down . In the case of θ < 0, it is also well known that S̃(t) ⇒ S̃(∞) as t → ∞, where S̃(∞) is equal in distribution to ã + exp(−θ/λ), where exp(−θ/λ) is an exponential random variable with rate −θ/λ. Thus, by the continuous mapping theorem (see, for instance, Theorem 3.4.3 of Whitt [31]), 1/S̃(t) ⇒ 1/S̃(∞). Moreover, by Proposition 3.1 of Hu et al. [19] and 3.5.12 of Karatzas and Shreve [22], we have that tP (τ1down − τ0down > t) → 0 as t → ∞. Thus, since 0 < 1/S̃(t) ≤ ã, the result follows by Theorem 6.1.2 of Asmussen [2] and Thereom 1 of Glynn and Whitt [14]. We now make a few remarks in regard to Proposition 8.1. First note that the proposition implies that with high probability when θ < 0, pl (t) = pl (0) − θE[ln X]e−θã/λ E1 (−θã/λ)t + o(t), which, since the actual limiting price p(t) = exp(pl (t)), implies that ( ) p(t) = p(0) exp −θE[ln X]e−θã/λ E1 (−θã/λ)t exp(o(t)). (44) (45) Next, we comment on the role that the market maker plays in our model. On an intuitive level, in the presence of a market maker who allows the size of the book to drop to low levels, or, in the extreme case, in the absence of any market maker at all, it stands to reason that the number of limit sell orders on the book will drop to low levels and, consequently, incoming market buy orders will tend to push the price upwards at a high rate. The market maker therefore plays an important role in moderating the rate of change of the price. It turns out that one may actually quantify the impact of the market maker in our model by first recalling the well-known asymptotic, lim x→0+ e−2E1 (x) x2 = e2γ , (46) where γ = 0.57721... is the Euler-Mascheroni constant. By Proposition 8.1 above, (46) then implies that almost surely, ( )) ( pl (t) θã 1 lim + exp(−θã/λ) ln − lim = −γ. (47) ã→0 −θE[ln X] t→∞ t λ Hence, one sees that the rate at which the rate of increase of the natural log of the price process increases with respect to the minimum market maker level ã is on the order of ln (−θã/λ). We now move on towards obtaining second order refinements to (44) and (45). Let Pπ denote the probability measure under which S̃(0) is distributed according to the limiting distribution π of S̃, namely ã + exp(−θ/λ), where exp(−θ/λ) is an exponential random variable with rate −θ/λ. And, for t ≥ 0, let Covπ (1/S̃(0), 1/S̃(t)) denote the covariance of 1/S̃(0) and 1/S̃(t) under Pπ . Finally, set ∫ ∞ 2 2 2 σ = 2λ (E[ln X]) Covπ (1/S̃(0), 1/S̃(u))du < ∞. 0 21 We then have the following result. Its proof is similar to that of Proposition 8.1 and therefore has been omitted. Proposition 8.2. If θ < 0, then pl (ν·) − (−θE[ln X]e−θã/λ E1 (−θã/λ)ν·) ν 1/2 ⇒ σB, as ν → ∞, where B is a standard Brownian motion. Note that as far as the actual price process itself is concerned, Proposition 8.2 implies that we may write ( ) p(t) = p(0) exp σBt − θE[ln X]e−θã/λ E1 (−θã/λ)t + o(t1/2 ) . Hence, we see that the limiting price process p may be approximated by a geometric Brownian motion. This concludes our discussion of the case θ < 0, we now move on to the case of θ > 0. 8.2 Long Term Behavior for θ > 0 In the case of θ > 0, one has that for sufficiently large n, the arrival rate of limit orders to the book is larger than the arrival rate of market orders to the book. Thus, in this case, for sufficiently large n, the process S n which keeps track of the total number of orders on the book is identical in distribution to an + Qn , where Qn is the number in system process for an M/M/1 queue with a utilization of ρn = λnL /λnM > 1. However, since an M/M/1 queue with a utilization of greater than one is transient and, since by Theorem 5.1 the rate of increase of the limiting prices process p is inversely related to the size of the book, this implies that the limiting price process will grow more slowly in the case of θ > 0 than in the case of θ < 0. Indeed, the following result bears this point out. Proposition 8.3. If θ > 0, then P ( pl (t) λE[ln X] lim = t→∞ ln t θ ) = 1. Proof. Recall from (42) that we may write l l p (t) = p (0) + ∫ t 0 λE[ln X] ds, t ≥ 0. S̃(s) (48) However, by Ito’s Lemma (see, for instance, Theorem 3.3.3 of Karatzas and Shreve [22]) and by (16), ∫ t ∫ t ∫ t √ 1 1 θ ln(S̃(t)) = ln(S̃(0)) + d( 2λB̃(s)) + ds + dL̃(s) 0 S̃(s) 0 S̃(s) 0 S̃(s) ∫ t λ ds, − 2 0 (S̃(s)) and hence, ∫ t 0 θ ds = ln(S̃(t)) − ln(S̃(0)) − S̃(s) ∫ 22 t 0 √ 1 d( 2λB̃(s)) − S̃(s) ∫ t 0 1 dL̃(s) S̃(s) (49) + ∫ t 0 λ ds. (S̃(s))2 We therefore obtain from (48) and (49) that pl (t) θ λE[ln X] ln(t) pl (0) ln(S̃(t)) ln(S̃(0)) θ 1 = + − − λE[ln X] ln(t) ln(t) ln(t) ln(t) ∫ t ∫ t 1 1 1 λ − ds. dL̃(s) + ln(t) 0 S̃(s) ln(t) 0 (S̃(s))2 ∫ t 0 √ 1 d( 2λB̃(s)) S̃(s) (50) Next recall that if θ > 0, then P (L̃(∞) < ∞ | S̃(0) = x) = 1, for all x ≥ ã. Hence, by the strong law of large numbers for Brownian motion (see, for instance, problem 2.9.3 of Karatzas and Shreve [22]), it follows that given that S̃(0) = x ≥ ã, one has that almost surely, ( ) L̃(t) √ B̃(t) x S̃(t) = lim +θ+ + 2λ = θ > 0. (51) lim t→∞ t→∞ t t t t By (51), one has that almost surely both pl (0) ln(S̃(t)) ln(S̃(0)) θ + − λE[ln X] ln(t) ln(t) ln(t) → 1 (52) and 1 ln(t) ∫ t 0 λ ds → 0, (S̃(s))2 as t → ∞. Next, recall that P (L(∞) < ∞) = 1, and so ∫ t 1 1 dL̃(s) → 0, ln(t) 0 S̃(s) (53) (54) almost surely. For the completion of the proof, by (50), (52), (53), and (54) it is sufficient to show that ∫ t 1 lim dB̃(u) (55) t→∞ 0 S̃(u) almost surely exists and is finite. Note that {∫ t } √ 1 d( 2λB̃(s)), t ≥ 0 0 S̃(s) is a martingale with quadratic variation process } {∫ t 1 d(2λs), t ≥ 0 . 2 0 (S̃(s)) By Revuz and Yor [27], Chapter V, Proposition 1.8, the limit in the expression (55) exists and is finite if and only if ∫ ∞ 1 du < ∞, P-a.s. (56) (S̃(u))2 0 However, (56) follows from (51). This completes the proof. 23 Recalling that pl (t) = ln p(t) for t ≥ 0, and comparing Propositions 8.1 and 8.3 side-by-side, one may now clearly see that the rate of increase of the limiting price process p is faster when θ < 0, than when θ > 0. In fact, p grows like exp(t) when θ < 0, while only growing according to tλE[ln X]/θ when θ > 0. However, Proposition 8.3 does not identify the specific factor in front of the O(tλE[ln X]/θ ) growth term when θ > 0. Nevertheless, upon closer inspection, one may actually see that the proof of Proposition 8.3 yields the following sharper result on the long term behavior of the limiting price process itself when θ > 0. Proposition 8.4. If θ > 0, then ( P lim p(t) t→∞ tλE[ln X]/θ = p(0) ( θ S̃(0) )λE[ln X]/θ R ) = 1, where the random variable R is given by (∫ ∞ )) ( ∫ ∞ ∫ ∞ √ 1 1 λ λE[ln X] ds + d( 2λB̃(s)) − dL̃(s) . R = exp − θ (S̃(s))2 S̃(s) S̃(s) 0 0 0 Note that in above result, there could be some dependence between R and the factor lying in front it. Proof. Recalling that p(t) = exp(pl (t)) and using (50) one obtains that )λE[ln X]/θ ( S̃(t) p(t) = p(0) S̃(0) ( )) (∫ t ∫ t ∫ t √ λE[ln X] 1 λ 1 exp − d( 2λB̃(s)) − dL̃(s) . ds + 2 θ 0 S̃(s) 0 (S̃(s)) 0 S̃(s) Dividing the above by tλE[ln X]/θ , recalling from (51) that S̃(t)/t → θ almost surely and proceeding as in the proof of Proposition 8.3 yields the result. For our final result of this section, we characterize the long term behavior of the total value of the limiting order book. In particular, recall that in the limit, the total value of the book is given by p(t)S̃(t). Hence, essentially the same proof as that of Proposition 8.4 yields the following. Proposition 8.5. If θ > 0, then ( ( )λE[ln X]/θ ) θ p(t)S̃(t) R = 1, P lim λE[ln X]/θ+1 = p(0)θ t→∞ t S̃(0) where R is as given in Proposition 8.4. 8.3 Long Term Behavior for θ = 0 In the case when θ = 0, it is more difficult to obtain the exact asymptotics of the long term behavior of the limiting price process p, as compared to the previous two cases of θ < 0 and θ > 0. However, we are still able to obtain tight upper and lower bounds on the rate of growth of the limiting price process. The proof of our main result in this subsection relies on the following generalization of the l’Hopital rule. The proof of this generalization is quite similar to the proof of the l’Hopital rule itself and so we omit the details. 24 Lemma 8.6. Let f, g ∈ C 1 (R) be such that for every t ∈ (0, ∞), we have that g(t) ̸= 0 and g ′ (t) ̸= 0 and such that limt→∞ |f (t)| = limt→∞ |g(t)| = ∞. Then lim inf t→∞ f ′ (t) f (t) f (t) f ′ (t) ≤ lim inf ≤ lim sup ≤ lim sup . ′ t→∞ g(t) g ′ (t) t→∞ g(t) t→∞ g (t) The following is now the main result of this subsection. Proposition 8.7. If θ = 0, then for every γ > 0, ( ) pl (t) P lim 1 =0 = 1 t→∞ t 2 +γ (57) and P ( lim t→∞ pl (t) =∞ 1 t 2 −γ ) = 1. (58) Proof. We start with proving (58). For each t ≥ 0, let √ l(t) = 2t ln ln t. Then, by Lemma 8.6, (42), and the Law of Iterated Logarithm (see, for instance, Theorem 2.9.23 of Karatzas and Shreve [22]), we have that P -a.s., lim inf t→∞ pl (t) 1 t 2 −γ 1 1 l(t) λE[ln X] t 2 +γ λE[ln X] t 2 +γ lim inf ≥ lim inf 1 ≥ × lim inf = ∞. 1 t→∞ t→∞ S̃(t) t→∞ l(t) 2 − γ S̃(t) 2 −γ This completes our proof of (58). ′′ Next, we proceed to prove (57). Let f (x) = x ((ln x) − 1), and hence f ′ (x) = ln x and f (x) = 1/x. Then, recalling the representation of S̃(t) given in (16), substituting θ = 0 and using Ito’s rule (see, for instance, Theorem 3.3.3 of Karatzas and Shreve [22]), we have that for each t ≥ 0, ∫ t ∫ t ∫ t√ 1 2λ ln S̃(u)dB̃(u) + λ f (S̃(t)) = f (S̃(0)) + ln S̃(u)dL̃(u) + du. 0 S̃(u) 0 0 Taking (17) into consideration, this becomes f (S̃(t)) = f (S̃(0)) + (ln ã)L̃(t) + ∫ t√ 2λ ln S̃(u)dB̃(u) + λ 0 ∫ t 0 1 du, S̃(u) which may be rearranged to yield ∫ t 0 1 du = S̃(u) ] 1[ f (S̃(t)) − f (S̃(0)) − (ln ã)L̃(t) − λ √ ∫ t 2 ln S̃(u)dB̃(u). λ 0 Formulas (42) and (59) now imply that ] [ √ ∫ t l l ln S̃(u)dB̃(u) . p (t) = p (0) + E[ln X] f (S̃(t)) − f (S̃(0)) − (ln ã)L̃(t) − 2λ 0 25 (59) (60) Now, let w = {w(t), t ≥ 0} be (on a possibly different probability space) a standard Brownian motion starting at S̃(0) − ã √ w(0) = > 0. 2λ Let L = {L(t), t ≥ 0} be the local time at zero corresponding to this Brownian motion, and set ∫ t B̄(t) = sign (w(s)) dw(s), t ≥ 0. 0 Then, notice that the three dimensional processes ) } {( S̃(t) − ã L̃(t) √ , √ , B̃(t) , t ≥ 0 and 2λ 2λ {( ) } |w(t)|, L(t), B̄(t) , t ≥ 0 (61) have the same law. In order to see this, we cast the representation of S̃(t) in (16) in the form S̃(t) − ã √ 2λ = 1 S̃(0) − ã √ + B̃(t) + √ L̃(t) 2λ 2λ (62) and also consider the Tanaka formula (see, for instance, Proposition 3.6.8 of Karatzas and Shreve [22]) |w(t)| = S̃(0) − ã √ + B̄(t) + L(t), t ≥ 0. 2λ (63) The identity of the laws of the two processes in (61) follows from (62), (63), and Lemma 3.6.14 of Karatzas and Shreve [22]. Thus {pl (t), t ≥ 0} has the same law as {p̄l (t), t ≥ 0}, where ( √ √ l l p̄ (t) = p (0) + E[ln X] f ( 2λ|w(t)| + ã) − f (S̃(0)) − (ln ã) 2λL(t) − (64) ) √ ∫ t √ ln( 2λ|w(u)| + ã)sign (w(u)) dw(u) , t ≥ 0. 2λ 0 For brevity, we now introduce the notations √ U (t) = 2λ|w(u)| + ã, t ≥ 0, and M (t) = ∫ t 0 √ ln( 2λ|w(u)| + ã)sign (w(u)) dw(u), t ≥ 0, (65) and we note that the process M is a continuous local martingale with quadratic variation ∫ t (ln U (s))2 ds, t ≥ 0. ⟨M ⟩t = 0 Using (64), in order to prove (57) it is sufficient to show that P -a.s., lim L(t) t→∞ t1/2+γ 26 = 0, (66) f (U (t)) t1/2+γ lim t→∞ = 0 (67) = 0. (68) and M (t) t→∞ t1/2+γ lim We proceed in order. Formula (66) follows from (63) and the Law of Iterated Logarithm (see, for instance, Theorem 2.9.23 of Karatzas and Shreve [22]). In order to show (67), first we show that P -a.s., ln U (t) ln l(t) lim sup t→∞ ≤ 1. (69) Indeed, by the Law of Iterated Logarithm (see, for instance, Theorem 2.9.23 of Karatzas and Shreve [22]), √ U (t) lim sup = 2λ l(t) t→∞ and hence lim sup ln l(t) t→∞ ( ) √ ln U (t) −1 = ln 2λ, ln l(t) which implies that lim sup t→∞ ln U (t) ln l(t) ≤ 1. (70) Inequality (70) and the obvious lim inf t→∞ ln(U (t))/ ln l(t) = 0 now imply (69). Now we show (67). By the definition of f (x), we have for each t ≥ 0, ( ) |ln U (t)| |ln l(t)| |f (U (t))| U (t) |ln U (t) − 1| U (t) 1 (71) = (γ+1)/2 ≤ (γ+1)/2 × γ/2 + γ/2 . |ln l(t)| tγ+1/2 t tγ/2 t t t However, by the Law of Iterated Logarithm (see, for instance, Theorem 2.9.23 of Karatzas and Shreve [22]), we have that P -a.s., U (t) lim (γ+1)/2 = 0. t→∞ t Together with (69), this now implies that the right-hand side of (71) converges to zero as t → ∞, and so (67) now follows. In order to complete the proof, it now remains to show that (68) holds. Let h(t) = ∫ t (ln l(s))2 ds, 3 t ≥ 0. By Exercise 1.15 of Chapter V of Revuz and Yor [26], Lemma 8.6 and (69), we have that for every δ > 0, lim sup t→∞ |M (t)| (h(t)) δ+ 12 = lim sup t→∞ |M (t)| l (⟨M ⟩t ) l (⟨M ⟩t ) ≤ lim sup × 1 1 δ+ l (⟨M ⟩t ) (h(t)) 2 t→∞ (h(t))δ+ 2 27 ≤ √ ( ⟨M ⟩t 2 lim sup h(t) t→∞ ) 1 +δ 2 ≤ √ ( (ln U (t))2 2 lim sup t→∞ (ln l(t))2 ) 1 +δ 2 ≤ √ 2. (72) Then, note that for each t ≥ 0, we may write |M (t)| t where ϵ = γ γ+1 . 1 +γ 2 = |M (t)| (h(t)) (1+γ) 2 × ( h(t) t1+ϵ ) 1+γ 2 , (73) Moreover, using the l’Hopital rule we have that (ln l(t))2 h(t) = lim t→∞ (1 + ϵ)tϵ t→∞ t1+ϵ lim = 0. (74) (68) now follows from (72), (73) and (74). This completes the proof of (57), which completes the proof. 9 Extensions In this section we provide some brief discussion on extensions of the model considered so far in the paper. The two extensions we discuss are the case of a 1-sided limit order book with limit order cancelations and the case of a 2-sided limit order book. 9.1 A 1-sided Limit Order Book with Cancelations It has been empirically observed [xx] that a significant proportion of all limit orders placed on the book are eventually canceled before being executed. It is therefore important to discuss how to extend the results of the paper to a model which incorporates order cancelations. We therefore consider the following model of a 1-sided order book with order cancelations. Consider the 1-sided order book model of Section 2 but with the additional feature that for each i ≥ 1 the ith limit sell order to arrive to the book has a patience time di associated with it. If the ith limit sell order is not matched with a market buy order within di units of time after being placed on the book, then it is canceled and removed from the book. For convenience we assume that the sequence {di , i ≥ 1} of patience times is i.i.d. exponentially distributed with mean 1/γ. We now demonstrate how the system equations of Section 2 may be modified in order to account for order cancelations. As usual, we consider a one-sided order book which we take to be the sell side of the book. For each t ≥ 0, we let µ(t) ∈ MF (R) be a finite, non-negative measure on R, where for each A ∈ B(R) we have that µ(t)(A) represents the number of limit sell orders on the book whose prices lie in the set A. We also denote by p(t) = sup{x ≥ 0 : µ(t)((−∞, x)) = 0} the best ask at time t. Now for each t ≥ 0 and x ≥ 0 let W (t, x) = inf{τ ≥ 0 : p(t + τ ) > x} 28 (75) be the amount of time after time t that the best ask first crosses the threshold x. Note that W (t, x) also represents the amount of time that an infinitely patient limit sell order placed at time t at the price x would have to wait before being executed. We then have that NL (t) ∑ NC (t) = i=1 1{di ≤ W (τiL , p(τiL −)Xi )}1{τiL + di ≤ t} represents the total number of orders canceled by time t. Moreover, assuming a market maker level of a > 0 we have that the total number of orders on the book at time t is given by S(t) = S(0) + NL (t) − NM (t) − NC (t) + L(t), where L(t) = sup {− (S(0) + NL (s) − NM (s) − NC (s) − a)} ∨ 0. (76) 0≤s≤t We then have that the state of the order book at time t ≥ 0 is given by NL (t) µ(t) = µ(0) + ∑ i=1 NL (t) − ∑ i=1 δp(τ L −)Xi − 1{di ≤ i ∫ t 0 δp(s−) dNM (s) W (τiL , p(τiL −)Xi )}1{τiL (77) + di ≤ t}δp(τ L −)Xi + i ∫ t 0 δp(s−) dL(s). It may be shown that there exists a unique order book process µ = {µ(t), t ≥ 0} satisfying (77) and the previous definitions. We now provide a modification of Definition 3.1 of the high-frequency regime in order to account for order cancelations. Definition 9.1 (High Frequency Regime with Order Cancelations). The high frequency regime with order cancelations is defined by assuming that 1. n−1 λnL → λ > 0 and n−1 λnM → λ > 0 as n → ∞, √ 2. n(n−1 λnL − n−1 λnM ) → θ ∈ R as n → ∞, 1/n1/2 for i ≥ 1, where {Xi , i ≥ 1} is a sequence of i.i.d. random 3. For each n ≥ 1, Xn,i = Xi n. variable with common distribution G, independent of NLn and NM 4. n−1/2 an → ã > 0 as n → ∞. 5. γ n = γ > 0 for n ≥ 1. Note that Items 1 through 4 of Definition 9.1 are identical to Items 1 through 4 of Definition 3.1 for the high-frequency regime without order cancelations. However, now Item 5 has been added to Definition 9.1, which states that the mean patience times of orders remains the same even as the rate at which orders arrive grows large. In order to provide our main asymptotic results for the 1-sided limit order book with order cancelations, we must first provide the definition of a reflected Ornstein-Uhlenbeck process. 29 Definition 9.2. Let X̃ = {X̃(t), t ≥ 0} be a standard Brownian motion and let Z̃(0) be a random variable, independent of X̃. Moreover, let a, θ, σ, γ ∈ ℜ. Then, we say that Z̃ = {Z̃(t), t ≥ 0} is an Ornstein-Uhlenbeck process, reflected at the level a, with constant drift θ, linear drift coefficient γ and infinitesimal variance σ 2 , started at Z̃(0), if the following three conditions are satisfied almost surely, ∫t 1. Z̃(t) = Z̃(0) + θt − γ 0 Z̃(s)ds + σ X̃(t) + Ỹ (t) ≥ a, t ≥ 0, 2. Ỹ = {Ỹ (t), t ≥ 0} is a non-decreasing process with Ỹ (0) = 0, 3. The pair (Z̃, Ỹ ) satisfies the complementarity condition ∫ ∞ 1{Z̃(s) > a}dỸ (s) = 0. 0 Moreover, we refer to Ỹ as the local time of Z̃ at a. Note that Definition 9.2 of a reflected Ornstein-Uhleneck process is similar to Definition 4.1 of a reflected Brownian motion. The main difference is that the equation in Item 1 of the definition now incorporates a linear drift coefficient. This extra term turns out to represent orders being canceled from the book. We are now ready to state our main weak convergence results for the 1-sided order book with order cancelations. Our first result provides weak convergence of the sequence {S̃ n , n ≥ 1} of properly scaled processes keeping track of the number of orders on the book in the high-frequency regime with order cancelations. Proposition 9.3. If S̃ n (0) ⇒ S̃(0) as n → ∞, and Items 1-5 of Definition 9.1 hold, then the pair (S̃ n , L̃n ) ⇒ (S̃, L̃) in D2 ([0, ∞), R) as n → ∞, where S̃ is an Ornstein-Uhlenbeck process, reflected at ã > 0, with constant drift θ, linear drift coefficient γ and infinitesimal variance 2λ, started at S̃(0), and with local time L̃ at ã. Next, we have the following result on the weak convergence of the sequence {pn , n ≥ 1} of price processes in the high-frequency regime with order cancelations. Theorem 9.4. If E[ln X] > 0 and E[(∆µ̃l,n (0))2 ] → 0 and (pn (0), S̃ n (0)) ⇒ (p(0), S̃(0)) as n → ∞, and if Items 1-5 of Definition 9.1 hold, then pn ⇒ p as n → ∞, where the process p = {p(t), t ≥ 0} is given by (∫ t ) E[ln X] p(t) = p(0) exp d(λs) , for t ≥ 0, (78) S̃(s) 0 where S̃ = {S̃(t), t ≥ 0} is an Ornstein-Uhlenbeck process, reflected at ã > 0, with constant drift θ, linear drift coefficient γ and infinitesimal variance 2λ, started at S̃(0). Finally, we state our main result on the sequence of properly scaled order book processes ≥ 1} in the high-frequency regime with order cancelations. {µ̃n , n 30 Theorem 9.5. If E[ln X] > 0 and E[(∆µ̃nl (0))2 ] → 0 and (pn (0), S̃ n (0)) ⇒ (p(0), S̃(0)) as n → ∞, and if Items 1-5 of Definition 9.1 hold, then µ̃n ⇒ µ̃ as n → ∞, where µ̃ = {µ̃(t), t ≥ 0} is given by µ̃(t) = S̃(t)δp(t) , t ≥ 0, (79) where p = {p(t), t ≥ 0} and S̃ = {S̃(t), t ≥ 0} are as given in Theorem 9.4. Note that Proposition 9.3 and Theorems 9.4 and 9.5 are analogous to Proposition 4.2 and Theorems 5.1 and 6.1, respectively, in the case of no order cancelations. The main difference is that the process S̃ has now been replaced with a reflected Ornstein-Uhlenbeck process. Moreover, similar results to those in Sections 7 and 8 may be obtained in the case of order cancelations. In particular, so long as γ > 0 the process S̃ is positive recurrent and so only the results of the type found in Section 8.1 need apply. 9.2 A 2-sided Limit Order Book In this section we extend the model of the paper to a 2-sided order book as suggested by Simatos [29]. Consider a 2-sided order book where for each t ≥ 0 the state of the order book is given by a pair (µB (t), µS (t)) ∈ MF (R) × MF (R) of finite, non-negative measures on R. For each t ≥ 0 and A ∈ B(R) we have that µB (t)(A) represents the number of limit buy orders on the book whose prices lie in the set A, and µS (t)(A) represents the number of limit sell orders on the book whose prices lie in the set A. Moreover, we denote by pB (t) = inf{x ≥ 0 : µB (t)((x, ∞)) = 0}, (80) pS (t) = sup{x ≥ 0 : µS (t)((−∞, x)) = 0}, (81) and the best bid and best ask, respectively, at time t ≥ 0. Now let NL,B = {NL,B (t), t ≥ 0} be a counting process tracking the cumulative number of limit buy orders which have been placed on the buy side of the book and let NL,S = {NL,S (t), t ≥ 0} be a counting process tracking the cumulative number of limit sell orders which have been placed on the sell side of the book. Moreover, for each i ≥ 1 we define τiL,B = inf{t ≥ 0 : NL,B (t) ≥ i} and τiL,S = inf{t ≥ 0 : NL,S (t) ≥ i} to be the times at which the ith limit buy order or limit sell order, respectively, is placed on their corresponding side of the book. We now assume that the ith limit buy order to arrive to the book will place its order on the buy side of the book at the price XiB pS (τiL,B −), where {XiB , i ≥ 1} is a sequence of i.i.d. random variables with P (0 < XiB < 1) = 1. Similarly, the ith limit sell order to arrive to the book will place its order on the sell side of the book at the price XiS pB (τiL,S −), where {XiS , i ≥ 1} is a sequence of 31 i.i.d. random variables with P (XiS > 1) = 1. Note that the assumptions that P (0 < XiB < 1) = 1 and P (XiS > 1) = 1 ensure that limit buy and sell orders remain on their respective sides of the book and that the bid-ask spread is at all times well-defined and in fact positive. Next let NM,B = {NM,B (t), t ≥ 0} and NM,S = {NM,S (t), t ≥ 0} be counting processes tracking the cumulative number of market buy and sell orders, respectively, that have arrived to the book. Each time that a market buy order arrives it removes the limit sell order corresponding to the best ask. Similarly, each time that a market sell order arrives it removes the limit buy order corresponding to the best bid. Finally, we assume the existence of a market maker who never allows the total number of limit buy orders on the buy side of the book to fall below the level aB > 0, and never allows the total number of limit sell orders on the sell side of the book to fall below the level aS > 0. Specifically, if the number of limit buy orders on the book is at the level aB and a market sell order arrives, then the limit buy order corresponding to the best bid is removed from the book by the market sell order but it is then instantaneously replaced by an equivalent limit buy order by the market maker. A similar dynamic occurs on the sell side of the book such that the total number of limit sell orders never falls below the level aS . Now note that according to the dynamics described above, the total number of limit buy orders on the book at time t ≥ 0 is given by SB (t) = SB (0) + NL,B (t) − NM,S (t) + LB (t), where SB (0) = µB (0)(R) and LB (t) = sup {− (SB (0) + NL,B (s) − NM,S (s) − aB )} ∨ 0, t ≥ 0. 0≤s≤t Similarly, the total number of limit sell orders on the book at time t ≥ 0 is given by SS (t) = SS (0) + NL,S (t) − NM,B (t) + LS (t), where SS (0) = µS (0)(R) and LS (t) = sup {− (SS (0) + NL,S (s) − NM,B (s) − aS )} ∨ 0, t ≥ 0. 0≤s≤t Note also that the processes LB = {LB (t), t ≥ 0} and LS = {LS (t), t ≥ 0} keep track of the cumulative number of limit orders that the market maker has placed on the buy and sell sides of the order book, respectively. Similar to (6) of Section 2 we now have that the order book process (µB , µS ) = {(µB (t), µS (t)), t ≥ 0} is given by the unique solution to the pair of coupled equations NL,B (t) µB (t) = µB (0) + ∑ δp i=1 and L,B −)XiB S (τi − NL,S (t) µS (t) = µS (0) + ∑ i=1 δp L,S −)XiS B (τi − ∫ ∫ t 0 δpB (s−) dNM,B (s) + t 0 δpS (s−) dNM,S (s) + ∫ ∫ t 0 δpB (s−) dLB (s) t 0 δpS (s−) dLS (s), where pB (t) and pS (t) are given by (80) and (81), respectively. One can now define a high frequency regime for the 2-sided order book in a similar manner to Definition 3.1 for the 1-sided order book. However it remains an interesting open problem to derive the scaling limit of the order book in this regime. 32 10 Appendix In the Appendix, we provide the proofs of several key results needed for the proofs of Theorems 5.1 and 6.1 found in the main body of the paper. In particular, in Subsection 10.1 we provide the proof of Proposition 10.1, which is then used in the proof of Theorem 5.1 of Section 5. Next, in Subsection 10.2, we provide the proof of Lemma 10.2, which is used as a supporting lemma in the proof of Proposition 10.1. Finally, in Subsection 10.3, we provide the proof of Proposition 10.14, which is then used in the proof of Theorem 6.1 of Section 6. As it turns out, we will find it convenient at many points throughout the Appendix to work with the natural log of the measure-valued order book process and the natural log of the price process. Specifically, for each t ≥ 0, let µl (t) ∈ MF (R) be the measure defined by setting µl (t)((−∞, b]) = µ(t)((−∞, eb ]), b ∈ R+ . We then define µl = {µl (t), t ≥ 0} to be the natural log of the measure-valued order book process. In a similar manner, setting pl (t) = sup{x ∈ R : µl (t)((−∞, x)) = 0}, t ≥ 0, (82) we define pl = {pl (t), t ≥ 0} to be the natural log of the price process. It is then straightforward using (9) of Section 2 to show that for each φ ∈ Cb (R), one has that µl and pl obey the dynamics NL (t) l l ⟨µ (t), φ⟩ = ⟨µ (0), φ⟩ + ∑ (φ(p i=1 l (τiL −) + ln(Xi )) − φ(p l (τiL −))) + ∫ t φ(pl (s−))dS(s),(83) 0 for t ≥ 0. The system equation (83) will be heavily relied upon throughout the Appendix. We also note that all quantities associated with µl and pl in the nth system of the high frequency regime will be denoted by a superscript n. For instance, pl,n (0) represents the natural log of the price at time t = 0 in the nth system. Moreover, the diffusion scaled natural log of the limit order book in the nth system at time t ≥ 0 is given by µ̃l,n (t)(A) = n−1/2 µl,n (t)(A) for each A ∈ B(R). 10.1 Proof of Proposition 10.1 As mentioned above, our main goal in this subsection is prove Proposition 10.1 below. In order to state this proposition, and also in preparation for our supporting lemmas in this Subsection, we first need to setup some additional notation. First recall from Section 7 the definition of ξ(x) = x as the identity function on R. Then, for each n ≥ 1, set ∆µ̃l,n (t) = ⟨µ̃l,n (t), ξ⟩ − S̃ n (t)pl,n (t), t ≥ 0, (84) and define the process ∆µ̃l,n = {∆µ̃l,n (t), t ≥ 0}. Note, in particular, that ∆µ̃l,n (0) as defined in (84) above coincides with ∆µ̃l,n (0) as defined in (18) of Section 5. We then have the following result. Proposition 10.1. If E[ln X] > 0 and E[(∆µ̃l,n (0))2 ] → 0 and (pl,n (0), S̃ n (0)) ⇒ (pl (0), S̃(0)) as n → ∞, and if Items 1-4 of Definition 3.1 hold, then pl,n ⇒ pl as n → ∞, where the process pl = {pl (t), t ≥ 0} is given by ∫ t E[ln X] l l d(λs), t ≥ 0, (85) p (t) = p (0) + S̃(s) 0 33 and S̃ = {S̃(t), t ≥ 0} is a Brownian motion, reflected at ã > 0, with constant drift θ and infinitesimal variance 2λ, started at S̃(0). We now begin to prove Proposition 10.1 by stating a lemma which is of critical importance. For the remainder of this section, until otherwise noted, we assume that T ≥ 0 is fixed. Then, for each n ≥ 1 and ε > 0, define νεn = inf{t ≥ 0 : |pl,n (t) − pl,n (0)| ≥ ε} ∧ T (86) to be the first time that the absolute value of the difference between the natural log of the price at time t ≥ 0 and the natural log of the price at time t = 0 is at least as great as ε (or T , whichever is smaller). Note that νεn is a stopping time with respect to the price process and that almost surely n one has that νεn ≤ νε+δ for each δ ≥ 0. We now have the following result. Its proof is rather long and technical and consequently has been deferred until Subsection 10.2. Lemma 10.2. If E[ln X] > 0 and E[(∆µ̃l,n (0))2 ] → 0 as n → ∞, and if Items 1-4 of Definition 3.1 hold, then, for each ε > 0, ] [ E sup (∆µ̃l,n (t))2 0≤t≤νεn → 0 as n → ∞. Proof. See Subsection 10.2 of the Appendix. We next provide a quick result which is needed in preparation for the proof of the upcoming Proposition 10.5. We have the following. Lemma 10.3. If Items 1-4 of Definition 3.1 hold, then, for each T ≥ 0, 2    n (t) NL ∑ 1 (ln(Xi ) − E[ln X])  → 0 as n → ∞. E  sup  n 0≤t≤T i=1 Proof. First note that the process   n L (t)  1 N∑  (ln(Xi ) − E[ln X]), t ≥ 0 n  i=1 is a martingale. Hence, by Doob’s martingale inequality (see, for instance, Theorem 1.3.8 of Karatzas and Shreve [22]), we have that for each T ≥ 0,  2  2    n (t) n (T ) NL NL ∑ ∑ 1 1 E  sup  (ln(Xi ) − E[ln X])  ≤ 4E  (ln(Xi ) − E[ln X])  . n n 0≤t≤T i=1 i=1 However, since the sequence {ln Xi , i ≥ 1} is by assumption i.i.d. with finite mean, and independent of NLn , it follows that 2   n (T ) NL ∑ 1 n 1 N (T )Var(ln X). (ln(Xi ) − E[ln X]) |NLn (T ) = E  n n2 L i=1 34 This then implies that  2  ∑ 1 (ln(Xi ) − E[ln X])  E  n n (T ) NL i=1 = 1 n λ T Var(ln X) n2 L → 0 as n → ∞, where the final convergence follows from Item 1 of Definition 3.1. This complete the proof. Now, for each n ≥ 1, define the centered natural log of the price by setting ) ( ∫ t E[ln X] n l,n l,n l,n dN̄L (s) , t ≥ 0, ∆p (t) = p (t) − p (0) + n 0 S̃ (s−) (87) and define the process ∆pl,n = {∆pl,n (t), t ≥ 0}. Proposition 10.5 below essentially shows that for each ε > 0, up until the stopping time νεn , the process ∆pl,n weakly converges to 0 as n tends to ∞. The significance of this convergence can be attributed to the following result. Proposition 10.4. If (pl,n (0), S̃ n (0)) ⇒ (pl (0), S̃(0)) as n → ∞, and if Items 1-4 of Definition 3.1 hold, then ( ) ∫ e E[ln X] l,n n n p (0) + dN̄L (s), S̃ ⇒ (pl , S̃) in D2 ([0, ∞), R) as n → ∞, n (s−) S̃ 0 where S̃ = {S̃(t), t ≥ 0} is a Brownian motion, reflected at ã > 0, with constant drift θ and infinitesimal variance 2λ, started at S̃(0). Proof. First note that for each n ≥ 1, we may write (pl,n (0), S̃ n − ãn ) = Υ(pl,n (0), (S̃ n (0) − ãn ) + n1/2 (n−1 λnL − n−1 λnM )e + M̃ n ), n ≥ 1, (88) where, for (x, z) ∈ R × D([0, ∞), R), the map Υ : R × D([0, ∞), R) 7→ R × D([0, ∞), R) is defined by setting Υ(x, z) = (x, Φ(z)), where the map Φ is as defined in Theorem 6.1 of Chen and Yao [6]. Now recall that the map Φ is continuous at z ∈ C([0, ∞), R). Hence, it follows that Υ is a continuous map at (x, z) ∈ R × C([0, ∞), R). Next, note that since M̃ n is independent of (pl,n (0), S̃ n (0)), it follows as in the proof of Proposition 4.2 that (pl,n (0), (S̃ n (0) − ãn ) + n1/2 (n−1 λnL − n−1 λnM )e + M̃ n ) ⇒ √ l (p (0), (S̃(0) − ã) + θe + 2λX̃) as n → ∞, where X̃ is a standard Brownian motion. Hence, by (88) and the continuous mapping theorem (see, for instance,√Theorem 3.4.3 of Whitt [31]), we now obtain that (pl,n (0), S̃ n − ãn ) ⇒ (pl (0), Φ((S̃(0) − ã) + θe + 2λX̃)) as n → ∞. However, since by Item 4 of Definition 3.1 we have that ãn → ã as n → ∞, this now implies that (pl,n (0), S̃ n ) ⇒ (pl (0), S̃) in R × D([0, ∞), R) as n → ∞, (89) where S̃ = {S̃(t), t ≥ 0} is a Brownian motion, reflected at ã > 0, with constant drift θ and infinitesimal variance 2λ, started at S̃(0). Next, note that by Theorem 5.10 of Chen and Yao [6], the random time change theorem (see, for instance, page 151 of Billingsley [5]) and Item 1 of Definition 3.1, it follows that N̄Ln ⇒ λe as n → ∞. Hence, by (89), we now have that (pl,n (0), S̃ n , N̄Ln ) ⇒ (pl (0), S̃, λe) in R × D2 ([0, ∞), R) as n → ∞. 35 Now, by the Skorokhod representation theorem (see, for instance, Theorem 6.7 of Billingsley [5]), there exists a sequence of processes {(p̂l,n (0), Ŝ n , N̂Ln ), n ≥ 1}, defined on an alternative probability space (Ω̂, F̂, P̂ ), such that d (pl,n (0), S̃ n , N̄Ln ) = (p̂l,n (0), Ŝ n , N̂Ln ), n ≥ 1, (90) and (p̂l,n (0), Ŝ n , N̂Ln ) → (p̂l (0), Ŝ, λe) as n → ∞, P̂ −a.s., (91) where d (p̂l (0), Ŝ, λe) = (pl (0), S̃, λe). (92) Next, recall that for each n ≥ 1, we have that S̃ n (t) ≥ ãn > 0 for each t ≥ 0, and, that by Item 4 of Definition 3.1, we have that ãn → ã > 0 as n → ∞. Hence, by (91) and Lemma 8.3 of Dai and Dai [11], it follows that ( ) ( ) ∫ e ∫ e E[ln X] E[ln X] l,n n n l p̂ (0) + dN̂L (s), Ŝ → p̂ (0) + d(λs), Ŝ , P̂ −a.s., Ŝ(s) 0 Ŝ n (s−) 0 as n → ∞. However, by (90), ( l,n p (0) + ∫ e 0 E[ln X] dN̄Ln (s), S̃ n n S̃ (s−) ) ( d = l,n p̂ (0) + ∫ e E[ln X] Ŝ n (s−) 0 dN̂Ln (s), Ŝ n ) , n ≥ 1, and, similarly, by (92), ( pl (0) + ∫ e 0 E[ln X] d(λs), S̃ S̃(s) ) d = ( p̂l (0) + ∫ e 0 E[ln X] Ŝ(s) d(λs), Ŝ ) . This completes the proof. We are now in a position to prove Proposition 10.5, which actually turns out to be very close to our main result of this Subsection, Proposition 10.1. We have the following. Proposition 10.5. If E[ln X] > 0 and E[(∆µ̃l,n (0))2 ] → 0 as n → ∞, and if Items 1-4 of Definition 3.1 hold, then, for each ε > 0, ] [ E sup (∆pl,n (t))2 0≤t≤νεn → 0 as n → ∞. Proof. First note that by (83), one has that for each n ≥ 1, n NL (t) ∫ t 1 ∑ l,n l,n pl,n (s−)dS̃ n (s), t ≥ 0. ln(Xi ) + ⟨µ̃ (t), ξ⟩ = ⟨µ̃ (0), ξ⟩ + n 0 i=1 36 (93) Next, note that integrating-by-parts, one may write ) ∫ t ∫ t (∫ s− E[ln X] E[ln X] S̃ n (t) dN̄Ln (s) = E[ln X]N̄Ln (t) + dN̄Ln (u) dS̃ n (s) n S̃ n (u−) 0 S̃ (s−) 0 0 ∫ t E[ln X] 1 + 1/2 dN̄Ln (s). n n 0 S̃ (s−) (94) Now subtracting (94) from (93) and using the definition of ∆µ̃l,n (t) in (84) and ∆pl,n (t) in (87), one may then verify after some algebra the remarkably simple relation n NL (t) 1 ∑ S̃ (t)∆p (t) = ∆µ̃ (0) − ∆µ̃ (t) + (ln(Xi ) − E[ln X]) n i=1 ∫ t ∫ t 1 E[ln X] n − 1/2 dN̄L (s) + ∆pl,n (s−)dS̃ n (s). n n 0 S̃ (s−) 0 n l,n l,n l,n (95) Now, for each n ≥ 1, let Z̃ n (t) = (96)  n (t) NL 1 ∑ 1 1  l,n ∆µ̃ (0) − ∆µ̃l,n (t) + (ln(Xi ) − E[ln X]) − 1/2 n n S̃ n (t) i=1 ∫ t 0 for t ≥ 0, and note that by (95) we may write ∆pl,n (t) = Z̃ n (t) + 1 S̃ n (t) ∫ t  E[ln X] dN̄Ln (s) , S̃ n (s−) ∆pl,n (s−)dS̃ n (s). (97) 0 We now use (97) in order to prove our main result. At a high level, our approach is to use an iterative scheme which is fairly standard. Nevertheless, we provide the details below. We begin by defining a sequence of stopping times which will be useful in several localization arguments throughout the proof. Recall by (11) that for each n ≥ 1, we have the decomposition S̃ n (t) = S̃ n (0) + M̃ n (t) + B̃ n (t), t ≥ 0. (98) Next, let ||B̃ n ||T V (t) denote the total variation of the process B̃ n = {B̃ n (t), t ≥ 0} over the interval [0, t], and note that by (13) we may write √ −1 n ||B̃ n ||T V (t) = n|n λL − n−1 λnM |t + L̃n (t). Now, for each δ > 0, we introduce the stopping time n τδn = inf{t ≥ 0 : ||B̃ n ||T V (t) + N̄Ln (t) + N̄M (t)) ≥ δ} ∧ T, and note that since ||B̃ n ||T V (t) − ||B̃ n ||T V (t−) ≤ n−1/2 , and n n (N̄Ln (t) + N̄M (t)) − (N̄Ln (t−) + N̄M (t−)) ≤ n−1 , 37 it follows that 2 n ||B̃ n ||T V (τδn ) + (N̄Ln (τδn ) + N̄M (τδn )) − δ ≤ √ . n (99) The inequality (99) will be useful as we proceed. Let ε > 0 be fixed for the remainder of the proof. We now proceed to show that there exists a δ0 > 0, which is independent of both n and ε, and such that for each 0 ≤ δ < δ0 , we have the convergence [ ] E sup 0≤t≤τδn ∧νεn ∆(pl,n (t))2 → 0 as n → ∞. (100) The convergence (100) will then be used as the base case in an induction argument in order to prove that for each k ≥ 1, we have for 0 ≤ δ < δ0 , the convergence ] [ E sup n ∧ν n 0≤t≤τkδ ε ∆(pl,n (t))2 → 0 as n → ∞. (101) We proceed as follows. First note that since for each n ≥ 1 and t ≥ 0, we have the inequality S̃ n (t) ≥ ãn > 0, it follows by (96) and the identity (x1 + ... + xm )2 ≤ m(x21 + ... + x2m ), m ≥ 1, that (Z̃ n (t))2 ≤   2 n (t) NL ∑ 1 4  (∆µ̃l,n (0))2 + (∆µ̃l,n (t))2 +  (ln(Xi ) − E[ln X]) (ãn )2 n i=1 ( )2 ) ∫ t 1 E[ln X] + dN̄ n (s) . n1/2 0 S̃ n (s−) L (102) Now let δ ≥ 0 be arbitrary. It then follows by the assumption that E[(∆µ̃l,n (0))2 ] → 0 as n → ∞, Lemmas 10.2 and 10.3 above, Proposition 10.4, and Item 4 of Definition 3.1, that ] [ E sup 0≤t≤τδn ∧νεn (Z̃ n (t))2 → 0 as n → ∞. (103) Next, let γ > 0 be arbitrary and note that by (97), (103), the identity (x1 + ... + xm )2 ≤ m(x21 + ... + x2m ), m ≥ 1, and Item 4 of Definition 3.1, it follows that there exists an nγ ≥ 1, which is independent of ε and δ, and such that for n ≥ nγ , one may write ] [ [ (∫ t )2 ] 2 ≤ γ+ 2E E sup (∆pl,n (t))2 sup . (104) ∆pl,n (s−)dS̃ n (s) ãn 0≤t≤τδn ∧νεn 0≤t≤τδn ∧νεn 0 Moreover, using the identity (x1 + ... + xm )2 ≤ m(x21 + ... + x2m ) and the semi-martingale decomposition (11), one has that [ )2 ] (∫ t E sup 0≤t≤τδn ∧νεn ∆pl,n (s−)dS̃ n (s) 0 38 (105) = E [ ≤ 2E sup 0≤t≤τδn ∧νεn [ +2E (∫ sup 0≤t≤τδn ∧νεn [ t sup n n ∆p (s−)d(M̃ (s) + B̃ (s)) 0 (∫ 0≤t≤τδn ∧νεn l,n t l,n n ∆p (s−)dM̃ (s) 0 (∫ t )2 ] )2 ] ∆pl,n (s−)dB̃ n (s) 0 (106) )2 ] . (107) We now proceed to bound the terms in (106) and (107). We begin with (106). Recall that by the Burkholder-Davis-Gundy inequality (see, for instance, Theorem 3.3.28 of Karatzas and Shreve [22]), there exists a constant K1 ≥ 0, which is independent of the martingale M̃ n and the stopping time τδn ∧ νεn , and uniform across all n ≥ 1, such that [ ] )2 ] [∫ n n (∫ t E sup 0≤t≤τδn ∧νεn ∆pl,n (s−)dM̃ n (s) τδ ∧νε ≤ K1 E 0 (∆pl,n (s−))2 d[M̃ n , M̃ n ](s) .(108) 0 However, note from (12) that n [M̃ n , M̃ n ](t) = N̄Ln (t) + N̄M (t), t ≥ 0, and so from the inequality (99) and (108) we obtain that [ )2 ] (∫ t l,n n ∆p (s−)dM̃ (s) E sup 0≤t≤τδn ∧νεn ≤ K1 E ≤ K1 ( [∫ τδn ∧νεn 0 2 δ+√ n (109) 0 l,n 2 (∆p (s−)) d ) [ E sup 0≤t≤τδn ∧νεn ( N̄Ln (s) + l,n (∆p (t)) n N̄M (s) 2 ] ) ] . Next, we proceed to (107). Note first by (99) that for each n ≥ 1 and t ≥ 0, we have that )2 (∫ τ n ∧νεn (∫ )2 t sup 0≤t≤τδn ∧νεn δ ∆pl,n (s−)dB̃ n (s) ≤ 0 ≤ 0 ( 2 δ+√ n |∆pl,n (s−)|d||B̃ n ||T V (s) )2 sup 0≤t≤τδn ∧νεn (∆pl,n (t))2 . Hence, E [ sup 0≤t≤τδn ∧νεn (∫ t 0 ∆pl,n (s−)dB̃ n (s) )2 ] ≤ ( 2 δ+√ n )2 E [ sup 0≤t≤τδn ∧νεn ] (∆pl,n (t))2 . (110) We now obtain from (104),(105), (109), (110), and Item 4 of Definition 3.1, that there exists a constant K2 ≥ 0, which is independent of n, ε and δ, and such that for each n ≥ nγ , [ ] (( ] ) ( )2 ) [ l,n 2 2 2 . ≤ γ + K2 δ + √n + δ + √n E sup0≤t≤τδn ∧νεn ∆(pl,n (t))2 (111) ∆(p (t)) E sup 0≤t≤τδn ∧νεn 39 Next, note that there exists a δ0 > 0, which is independent ε, and such that if 0 ≤ δ < δ0 , then there exists an nδ ≥ 1 such that (( ) ( ) ) 1 2 2 2 < , n ≥ nδ . K2 + δ+√ (112) δ+√ 2 n n Hence, for such 0 ≤ δ < δ0 and n ≥ nδ ∧ nγ , we obtain from (111), after some algebra, that ] [ E sup 0≤t≤τδn ∧νεn ∆(pl,n (t))2 ≤ 2γ. However, since the choice of γ > 0 was arbitrary, this implies that for 0 ≤ δ < δ0 , ] [ E sup 0≤t≤τδn ∧νεn ∆(pl,n (t))2 → 0 as n → ∞, (113) thus proving (100). We now proceed to prove (101) for k ≥ 1. As mentioned at the outset, our proof is by induction on k. It is clear by (100) that (101) holds for the base case of k = 1. Suppose now that (101) holds for k − 1 for some k ≥ 2. That is, suppose that for 0 ≤ δ < δ0 , ] [ E sup n 0≤t≤τ(k−1)δ ∧νεn ∆(pl,n (t))2 → 0 as n → ∞. (114) We now show that (114) holds for k as well. n Note first by (97) that for each t ≥ τ(k−1)δ , we may write l,n ∆p (t) = n Z̃(k−1)δ (t) 1 + n S̃ (t) ∫ t ∆pl,n (s−)dS̃ n (s), (115) ∆pl,n (s−)dS̃ n (s). (116) n τ(k−1)δ ∧νεn where n Z̃(k−1)δ (t) 1 = Z̃ (t) + n S̃ (t) n ∫ n ∧νεn τ(k−1)δ 0 We now claim that E [ n (Z̃(k−1)δ (t))2 n sup n ∧ν 0≤t≤τkδ ε ] → 0 as n → ∞. (117) In order to prove (117), first note that by the basic identity (x1 +...+xm )2 ≤ m(x21 +...+x2m ), m ≥ 1, and the fact that for each n ≥ 1, we have that S̃ n (t) ≥ ãn > 0 for t ≥ 0, it follows by (116) that n (Z̃(k−1)δ (t))2 2 ≤ 2(Z̃ (t)) + n 2 (ã ) n 2 40 (∫ n ∧νεn τ(k−1)δ 0 l,n n ∆p (s−)dS̃ (s) )2 . (118) Next, by (102), the assumption that E[(∆µ̃l,n (0))2 ] → 0 as n → ∞, Lemmas 10.2 and 10.3 above, Proposition 10.4, and Item 4 of Definition 3.1, one has that [ ] E sup n ∧ν n 0≤t≤τkδ ε (Z̃ n (t))2 → 0 as n → ∞. (119) Hence, in order to show (117), it suffices by (118), (119) and Item 4 of Definition 3.1, to show that [ (∫ n )2 ] τ(k−1)δ ∧νεn → 0 as n → ∞. ∆pl,n (s−)dS̃ n (s) E 0 However, proceeding in a manner similar to earlier in the proof, one may show that there exists a constant K3 ≥ 0, which is independent of n, ε and δ, and such that [ [(∫ n ] )2 ] n τ(k−1)δ ∧νε E ∆pl,n (s−)dS̃ n (s) 0 ≤ K3 · E sup n 0≤t≤τ(k−1)δ ∧νεn (∆pl,n (t))2 . (120) The details of the proof of (120) are omitted. Moreover, by the induction hypothesis, we have that ] [ E sup n 0≤t≤τ(k−1)δ ∧νεn (∆pl,n (t))2 → 0 as n → ∞. (121) Hence, (116), (119), (120) and (121) now imply (117). We now use (115) and (117) in order to prove (114) with k − 1 replaced by k. Note first that by (115) and the basic identity (x1 + ... + xm )2 ≤ m(x21 + ... + x2m ), m ≥ 1, we have that for each n n ≥ 1 and t ≥ τ(k−1)δ , (∆pl,n (t))2 2 n ≤ 2(Z̃(k−1)δ (t))2 + n 2 ˜ (a ) (∫ t n τ(k−1)δ ∧νεn ∆pl,n (s−)dS̃ n (s) )2 . (122) Now proceeding in a manner similar to the proof of the base case k = 1, one may show that there exists a constant K4 ≥ 0 such that  )2  (∫ t 2 (123) E ∆pl,n (s−)dS̃ n (s)  sup n n n n n (ãn )2 τ(k−1)δ ∧νε <t≤τkδ ∧νε τ(k−1)δ (( ] ) ( ) ) [ 2 2 2 l,n 2 ≤ K4 δ+√ (∆p (t)) . + δ+√ E sup n n ∧ν n n n τ(k−1)δ ∧νεn <t≤τkδ ε In particular, the constant K4 in (123) is the same constant K2 used in the proof of (111) above. The details of the proof of (123) are omitted. Hence, using the same δ0 and nδ as in (112) above, we obtain that for 0 ≤ δ < δ0 , we have the bound (( ) ( ) ) 2 2 1 2 K4 + δ+√ < , n ≥ nδ , δ+√ 2 n n 41 and so from (122), after some algebra, it follows that for such 0 < δ < δ0 and n ≥ nδ , ] [ ] [ E sup n n ∧ν n τ(k−1)δ ∧νεn <t≤τkδ ε However, by (117), [ E sup n n ∧ν n τ(k−1)δ ∧νεn <t≤τkδ ε (∆pl,n (t))2 n (Z̃(k−1)δ (t))2 ] ≤ 4E ≤ E [ sup n n ∧ν n τ(k−1)δ ∧νεn <t≤τkδ ε sup n ∧ν n 0≤t≤τkδ ε n (Z̃(k−1)δ (t))2 n (t))2 . (Z̃(k−1)δ ] (124) → 0 as n → ∞,(125) and so by (124), E [ sup n n n τ(k−1)δ∧ν n <t≤τkδ ∧νε (∆pl,n (t))2 ε ] → 0 as n → ∞. (126) Thus, by the induction hypothesis and (126), we obtain that [ ] E ≤ E sup n ∧ν n 0≤t≤τkδ ε [ (∆pl,n (t))2 sup l,n n 0≤t≤τ(k−1)δ ∧νεn (∆p (t)) 2 ] +E [ (127) l,n sup n n ∧ν n τ(k−1)δ ∧νεn <t≤τkδ ε (∆p (t)) 2 ] → 0 as n → ∞, as desired. This proves (101). n in (127). First note that for We now complete the proof by removing the stopping times τkδ each n ≥ 1, k ≥ 1 and 0 ≤ δ < δ0 , we may write ] [ sup (∆pl,n (t))2 E 0≤t≤νεn [ n = E 1{τkδ ≥ T} sup n ∧ν n 0≤t≤τkδ ε ] [ ] n (∆pl,n (t))2 + E 1{τkδ < T } sup (∆pl,n (t))2 . 0≤t≤νεn (128) Now note that in order to show that the right-hand side above converges to 0 as n → ∞, it is sufficient to show that the first term in (128) converges to 0 for every fixed k ≥ 1 as n → ∞, and that the second term converges to 0 as k → ∞, uniformly in n. By (127), for each k ≥ 1, ] ] [ [ n E 1{τkδ ≥ T} sup n ∧ν n 0≤t≤τkδ ε (∆pl,n (t))2 ≤ E sup n ∧ν n 0≤t≤τkδ ε (∆pl,n (t))2 → 0, (129) as n → ∞. Thus, in order to complete the proof, it remains to consider the second term in (128). Note that by (1), Item 3 of Definition 3.1, and Item 2 of Assumption 3.2 ln C |pn,l (t) − pn,l (t−)| ≤ √ , n t ≥ 0, thus by the definition of νεn in (86) we have that ln C |pl,n (t)| ≤ pl,n (0) + ε + √ . n 42 (130) From inequality (130), the definition of ∆pl,n (t) in (87), the basic identity (x1 + ... + xm )2 ≤ m(x21 + ... + x2m ), m ≥ 1, the fact that for each n ≥ 1, we have that S̃ n (t) ≥ ãn > 0 for t ≥ 0, and the fact that E[ln X] > 0, it follows that ( ( ))2 ∫ t E[ln X] l,n 2 l,n l,n n sup sup (∆p (t)) = p (t) − p (0) + (131) dN̄L (s) n 0≤t≤νεn 0≤t≤νεn 0 S̃ (s−) ) ( ≤ 2 sup (pl,n (t))2 0≤t≤νεn )2 ) E[ln X] +2 sup dN̄Ln (s) n 0≤t≤νεn 0 S̃ (s) ) (∫ T )2 ( E[ln X] ln C 2 n l,n +2 dN̄L (s) ≤ 2 |p (0)| + ε + √ n S̃ n (s) 0 ) ( )2 ( E[ln X] n ln C 2 +2 ( N̄ (T )) ≤ 2 |pl,n (0)| + ε + √ L ãn n ) ) ( ( E[ln X] 2 ln C 2 l,n √ +2 (N̄Ln (T ))2 . ≤ 2 |p (0)| + ε + ãn n ( (∫ t It therefore follows by (131) and the Cauchy-Schwarz inequality that ( [ ])2 n E 1{τkδ < T } sup (∆pl,n (t))2 0≤t≤νεn (132)  ( )2  )2 ( )2 ( E[ln X] ln C n + (N̄Ln (T ))2  |pl,n (0)| + ε + √ ≤ P (τkδ < T ) · E 4 ãn n [( ] ( ) ) ln C 4 E[ln X] 4 l,n n n 4 |p (0)| + ε + √ ≤ 8P (τkδ < T ) · E + (N̄L (T )) . ãn n Now, using the properties of the Poisson distribution, it is straightforward to show that there exists some K5 > 0 such that ] [( )4 ] 1 1 [ n 4 E (N (T )) ≤ 4 K5 (1 + λnL T )4 → K5 (λT )4 as n → ∞, E N̄Ln (T ) = L 4 n n where the final convergence follows by Item 1 of Definition 3.1. Thus, using (1) and Item 4 of Definition 3.1, it follows that [( ] )4 ( )4 E[ln X] ln C + sup E |pl,n (0)| + ε + √ (N̄Ln (T ))4 < ∞. (133) ãn n n≥1 On the other hand, note that we may write ( ) n n P (τkδ < T ) ≤ P ||B̃ n ||T V (T ) + N̄Ln (T ) + N̄M (T ) ≥ kδ (134) ) ( ) ( ) ( kδ kδ kδ n (T ) ≥ + P N̄Ln (T ) ≥ + P N̄M . ≤ P ||B̃ n ∥|T V (T ) ≥ 3 3 3 43 However, by Proposition 4.2 we have that L̃n (T ) ⇒ L̃(T ) as n → ∞, and by Item 1 of Definition n (T ) ⇒ λT as n → ∞. It therefore follows that the 3.1 it follows that both N̄Ln (T ) ⇒ λT and N̄M n (T ), n ≥ 1} are tight. n sequences of random variables {L̃ (T ), n ≥ 1}, {N̄Ln (T ), n ≥ 1}, and {N̄M Hence, the expression in (134) converges to 0 as k → ∞ uniformly in n, which implies that n lim sup P (τkδ < T ) = 0. (135) k→∞ n≥1 From (132), (133), and (135) it now follows that [ n lim sup E 1{τkδ < T } sup (∆pl,n (t))2 k→∞ n≥1 0≤t≤νεn ] = 0, (136) which completes the proof. Now note that given Proposition 10.5 above, the only remaining obstacle in our way to proving Proposition 10.1 is to show that νεn weakly converges to T for ε sufficiently large. However, in this regard, we have the following result. Proposition 10.6. If E[ln X] > 0 and (pl,n (0), S̃ n (0)) ⇒ (pl (0), S̃(0)) as n → ∞, and if Items 1-4 of Definition 3.1 hold, then, for each δ > 0, there exists an ε > 0 such that lim sup P (νεn < T ) < δ. (137) n≥1 Proof. The proof is by contradiction. Suppose that (137) does not hold. Then, for some δ > 0 and all ε > 0, lim sup P (νεn < T ) ≥ δ. (138) n≥1 We now show that (138) will result in a contradiction with Proposition 10.5 above, thereby yielding (137). For each n ≥ 1, let ∫ t E[ln X] n dN̄Ln (s), t ≥ 0. (139) Ã (t) = n 0 S̃ (s) We first show that if ε > 0 is sufficiently large, then lim P (Ãn (T ) > ε/2) = 0. (140) n→∞ Indeed, note that since by assumption E[ln X] > 0, and since by Item 4 of Definition 3.1, we have that for each t ≥ 0, S̃ n (t) ≥ ãn → ã > 0 as n → ∞, it follows that for n sufficiently large, (∫ T E[ln X] ε dN̄Ln (s) > P (Ã (T ) > ε/2) = P n 2 S̃ (s) 0 ( ) E[ln X] n ε ≤ P . N̄L (T ) > ã/2 2 n 44 ) However, since by Theorem 5.10 of Chen and Yao [6], the random time change theorem (see, for instance, page 151 of Billnigsley [5]) and Item 1 of Definition 3.1, we have that ÑLn ⇒ λe as n → ∞, it follows that the final expression on the righthand side above converges to 0 as n → ∞ whenever E[ln X] λT ã/2 < ε . 2 Thus, (140) is proven. Moreover, note that for fixed n, the quantity P (Ãn (T ) > ε/2) is decreasing in ε, thus for each δ > 0, we have that for sufficiently large ε and n, P (Ãn (T ) > ε/2) < δ . 2 (141) Next we show that on the event {νεn < T, Ãn (T ) ≤ ε/2}, we have that ( )2 sup pl,n (t) − Ãn (t) ≥ κ 0≤t≤νεn where κ = max whenever {( ε 2 + ln c1 )2 , (−ε + ln C1 ) 2 } , ε > max {−2 ln c1 , ln C1 } . First note that sup 0≤t≤νεn ( pl,n (t) − Ãn (t) )2 ≥ ( )2 pl,n (νεn ) − Ãn (νεn ) . (142) (143) (144) (145) By the definition of νεn in (86) on the event {νεn < T, Ãn (T ) ≤ ε/2} we have either pl,n (νεn ) − pl,n (0) ≥ ε or pl,n (νεn ) − pl,n (0) ≤ −ε. By (144) and Item 2 of Assumption 3.2 on the event {νεn < T, Ãn (T ) ≤ ε/2} we have ( )2 ( ε )2 pl,n (νεn ) − Ãn (νεn ) ≥ + ln c1 , 2 and in the second case ( pl,n (νεn ) − Ãn (νεn ) )2 ≥ (−ε + ln C1 )2 , and now (142) follows. Hence, by (138) and (141), it follows for infinitely many n, [ ] ( )2 E sup pl,n (t) − Ãn (t) 0≤t≤νεn ≥ κP (νεn < T, Ãn (T ) ≤ ε/2) = κ(P (νεn < T ) − P (νεn < T, Ãn (T ) > ε/2)) ( ) ≥ κ P (νεn < T ) − P (Ãn (T ) > ε/2) ≥ δ · κ. 2 However, (146) contradicts Proposition 10.5, which completes the proof of (137). 45 (146) Given Propositions 10.4, 10.5 and 10.6 above, we are now in a position to present the proof of Proposition 10.1. We have the following. Proof of Proposition 10.1. Note that by (87) and Proposition 10.4, in order to complete the proof it suffices to show that ∆pl,n ⇒ 0 as n → ∞. In particular, it suffices to show that for each fixed T > 0, sup |∆pl,n (t)| → 0 in probability as n → ∞. (147) 0≤t≤T Let δ > 0 be arbitrary and note that by Proposition 10.6, we may select an ε > 0 such that for sufficiently large n we have that P (νεn < T ) ≤ δ. Now let K > 0 be arbitrary and note that by Proposition 10.5, we have that for sufficiently large n, ( ) sup |∆pl,n (t)| > K P (148) 0≤t≤T = P ( ( )2 sup ∆pl,n (t) > K 2 0≤t≤T ) ) ( ) ( )2 ( )2 K2 K2 n l,n n l,n +P sup ∆p (t) 1{νε = T } > ≤ P sup ∆p (t) 1{νε < T } > 2 2 0≤t≤νεn 0≤t≤T ( ) ( )2 K 2 ≤ P (0 ≤ νεn < T ) + P sup ∆pl,n (t) > 2 0≤t≤νεn ( < 2δ. (148) now implies (147), which completes the proof. 10.2 Proof of Lemma 10.2 In this subsection, we provide the proof of Lemma 10.2 from Section 10.1. The proof is rather long technical and consequently, we divide the subsection into three stages. First, in Subsubsection 10.2.1, we provide stochastic bounds on certain key quantities which will be used later in the proof of Lemma 10.2. Next, in Subsubsection 10.2.2, we prove two lemmas which will be useful in our proof of Lemma 10.2. Finally, in Subsubsection 10.2.3, we provide the proof of Lemma 10.2. 10.2.1 Stochastic Bounds In this subsubsection, we provide two stochastic bounds which will be useful in the proof of Lemma 10.2. We begin with the following. Lemma 10.7. If Items 1-4 of Definition 3.1 hold, then, for each T ≥ 0, [ ] [ ] sup E (L̃n (T ))4 < ∞ and sup E sup (S̃ n (t))4 < ∞. n≥1 n≥1 46 0≤t≤T Proof. Let T ≥ 0. We first prove that if Items 1,2 and 4 of Definition 3.1 hold, then, [ ] sup E (L̃n (T ))4 < ∞. (149) n≥1 In order to see that this is the case, note that by (3) of Section 2 and the basic inequality (x1 + ... + xm )4 ≤ m3 (x41 + ... + x4m ), m ≥ 1, one has that for each n ≥ 1, n (L (T )) 4 = ≤ ( ( n sup {− (S (0) + 0≤t≤T NLn (t) − n NM (t) )4 n − a )} ∨ 0 n S n (0) + an + |λnL − λnM |T + sup |(NLn (t) − λnL t) − (NM (t) − λnM t)| ≤ 43 0≤t≤T ( )4 n (S n (0))4 + (an )4 + (λnL − λnM )4 T 4 + sup ((NLn (t) − λnL t) − (NM (t) − λnM t))4 0≤t≤T ) . Now dividing the above inequality through by n2 , it follows that √ 4−3 (L̃n (T ))4 ≤ (S̃ n (0))4 + (ãn )4 + ( n|n−1 λnL − n−1 λnM |)4 T 4 + sup (M̃ n (t))4 . 0≤t≤T However, by Items 2 and 4 of Definition 3.1, as well as Item 1 of Assumption 3.2, one has that √ (150) sup E[(S̃ n (0))4 + (ãn )4 + ( n|n−1 λnL − n−1 λnM |)4 T 4 ] < ∞. n≥1 Hence, in order to prove (149), it remains to show that [ ] sup E n≥1 sup (M̃ n (t))4 0≤t≤T < ∞. (151) We will use Doob’s martingale inequality (see, for instance, Theorem 1.3.8 of Karatzas and Shreve [22]). In particular, recall from the discussion in Section 2 that the process M̃ n = {M̃ n (t), t ≥ 0} is a martingale. Moreover, using the basic identity (x1 + ... + xm )4 ≤ m3 (x41 + ... + x4m ), m ≥ 1, we have that [ ] n E (M̃ n (T ))4 ≤ 23 n−2 (E[(NLn (T ) − λnL T )4 ] + E[(NM (T ) − λnM T )4 ]) (152) = 23 n−2 (λnL (1 + 3λnL ) + λnM (1 + 3λnM )) → 24 (3)(λ2 ) as n → ∞, where the final convergence follows from Item 1 of Definition 3.1. Thus, the result (151) now follows as a result of (152) and Doob’s martingale inequality (see, for instance, Theorem 1.3.8 of Karatzas and Shreve [22]). This proves (149). We next prove that [ ] sup E n≥1 sup (S̃ n (t))4 0≤t≤T 47 < ∞. (153) Recall from (11) and (13) of Section 3 that for each 0 ≤ t ≤ T , we may write √ S̃ n (t) = S̃ n (0) + M̃ n (t) + n(n−1 λnL − n−1 λnM )t + L̃n (t). However, since {L̃n (t), t ≥ 0} is a non-decreasing process, it follows by the basic identity (x1 + ... + xm )4 ≤ m3 (x41 + ... + x4m ), m ≥ 1, that √ 4−3 sup (S̃ n (t))4 ≤ (S̃ n (0))4 + sup (M̃ n (t))4 + ( n(n−1 λnL − n−1 λnM ))4 T 4 + (L̃n (T ))4 , 0≤t≤T 0≤t≤T which implies that [ 4−3 E sup (S̃ n (t))4 0≤t≤T ] [ ] ≤ E (S̃ n (0))4 + E [ [ ] √ sup (M̃ n (t))4 + ( n(n−1 λnL − n−1 λnM ))4 T 4 0≤t≤T ] +E (L̃n (T ))4 . (153) now follows as a result of the inequality above and (149), (150) and (151). This completes the proof. Next, we have the following result which is our final result of this subsubsection. Let φ ∈ Cb2 (R) be an arbitrary function with bounded first and second derivatives and for each n ≥ 1 and t ≥ 0, define ∆φl,n (t) = n (t) NL { 1 ∑ √ n i=1 (154) ( ) ( ( ) ( ))} 1/n1/2 − λE[ln X] φ pl,n τiL,n − + ln Xi − φ pl,n τiL,n − ∫ t ′ φ 0 ( ) pl,n (s−) ds. Also, define the process ∆φl,n = {∆φl,n (t), t ≥ 0}. We then have the following. Lemma 10.8. If Items 1-4 of Definition 3.1 hold, then, for each φ ∈ Cb2 (R) and T ≥ 0, [ ] E sup |∆φl,n (t)| → 0 as n → ∞. 0≤t≤T Proof. First note by (154) that for each n ≥ 1, t ≥ 0 and φ ∈ Cb2 (R), we have the decomposition ∆φl,n (t) = n NL (t){ ( ( ) ( ))} ) ( ( )) ln(X ) ′ ( 1 ∑ 1/n1/2 i L,n L,n L,n l,n l,n l,n √ φ p (155) τi − + ln Xi τi − −φ p τi − − √ φ p n n i=1 n NL (t) { ( ))} ( 1 ∑ ′ + ln (Xi ) − E[ln X])φ pl,n τiL,n − n i=1 ∫ t ( ) ( ) ′ + E[ln X] φ pl,n (s−) d N̄Ln (s) − λs , 0 48 (156) (157) where N̄Ln (t) is as defined in Section 3 above. Let T ≥ 0. We now treat each of the terms on the righthand side above separately, showing that the expected value supremum of their absolute value over the interval [0, T ] converges to zero in expectation as n goes to ∞. This will then complete the proof. We begin with (155). First note that since φ ∈ Cb2 (R), we have by Taylor’s Theorem that we may write ( ) ln(Xi ) ′ l,n L,n 1 1/n1/2 l,n L,n l,n L,n √ φ (p (τi −)) + O ) − φ(p (τi −)) = φ(p (τi −) + ln Xi , n n ( ) ( ) where, by (1) and φ ∈ Cb2 (R), the O n1 term above is uniformly bounded, i.e. |O n1 | ≤ K1 /n for some constant K1 > 0. We therefore obtain that for each t ≥ 0, we may write n NL (t) { ( ) ))} ( ( ) ( )) ln(X ) ′ ( ( 1 ∑ 1/n1/2 i L,n L,n L,n l,n l,n l,n √ φ p τi − + ln Xi τi − −φ p τi − − √ φ p n n i=1 ≤ K K 1 √ NLn (t) 1 = √ 1 N̄Ln (t). n n n Thus, since {N̄Ln (t), t ≥ 0} is a non-decreasing process, it follows that   n (t) { } NL ∑ ln(Xi ) ′ 1 1/n1/2 ) − φ(pl,n (τiL,n −)) − √ φ (pl,n (τiL,n −))  φ(pl,n (τiL,n −) + ln Xi E  sup √ n n 0≤t≤T i=1 K √ 1 E[N̄Ln (T )] n K1 T n = λ n3/2 L → 0 as n → ∞, ≤ (158) where the final convergence follows by Item 1 of Definition 3.1. This completes our treatment of (155). The expected value of the supremum over the interval [0, T ] of the absolute value of (156) converges to 0 as n → ∞ by Lemma 10.3 and the boundedness of φ. We now complete the proof by showing that the supremum of (157) over the interval [0, T ] converges to zero in expectation as n goes to ∞. First note that for each t ≥ 0, we may write ∫ t ′ E[ln X] φ (pl,n (s−))d(N̄Ln (s) − λs) 0 ∫ t ′ = E[ln X] φ (pl,n (s−))d(N̄Ln (s) − n−1 λnL s) (159) 0 ∫ t ′ φ (pl,n (s−))d((n−1 λnL − λ)s). (160) +E[ln X] 0 We now treat each of the terms (159) and (160) separately. We begin with (159). First note that the process } { ∫ t ′ l,n n −1 n φ (p (s−))d(N̄L (s) − n λL s), t ≥ 0 E[ln X] 0 49 is a martingale. Hence, by Doob’s martingale inequality (see, for instance, Theorem 1.3.8 of Karatzas and Shreve [22]), we obtain that [ ( )2 ] ∫ t ′ l,n n −1 n E sup E[ln X] (161) φ (p (s−))d(N̄L (s) − n λL s) 0≤t≤T 0 [( ∫ 2 ≤ 4(E[ln X]) E T ′ l,n φ (p 0 (s−))d(N̄Ln (s) − n−1 λnL s) )2 ] . However, note that E [( ∫ T 0 [∫ = 1 E n ≤ n−2 λnL T 0 ′ φ (pl,n (s−))d(N̄Ln (s) − n−1 λnL s) ′ (φ (pl,n (s−)))2 d(n−1 λnL s) ′ · T · sup |φ (x)| )2 ] (162) ] x∈R → 0 as n → ∞, where the final convergence follows by Item 1 of Definition 3.1. Thus, combining (161) and (162) together, we obtain that [ )2 ] ( ∫ t ′ φ (pl,n (s−))d(N̄Ln (s) − n−1 λnL s) → 0 as n → ∞, (163) E sup E[ln X] 0≤t≤T 0 which completes our treatment of (159). Next, regarding (160), note that [ ] ∫ t ′ l,n −1 n φ (p (s−))d((n λL − λ)s) E sup E[ln X] 0≤t≤T 0 [∫ T ′ l,n ≤ |n−1 λnL ≤ |n−1 λnL − λ| · E[ln X] · T · sup |φ (x)| − λ| · E[ln X] · E 0 |φ (p (s−))|ds ′ ] x∈R → 0 as n → ∞, where the final convergence follows by Item 1 of Definition 3.1. This completes our treatment of (160), which completes the proof. 10.2.2 Two Lemmas In this subsubsection, we provide two lemmas, both of which play a useful and important role in the proofs in Section 10.2.3 below. Each of these lemmas are of a sample path nature. We begin with the following result. Lemma 10.9. For each n ≥ 1, if 0 ≤ t1 ≤ t2 and pl,n (t2 ) < pl,n (t1 ), then for all pl,n (t2 ) ≤ x1 < x2 ≤ pl,n (t1 ), 1 µ̃l,n (t2 ) ([x1 , x2 ]) + √ ≥ (x2 − x1 )/C. n 50 Proof. Let and { } y1 = max pl,n (t) : t ∈ [t1 , t2 ] , { } T1 = sup t ∈ [t1 , t2 ] : pl,n (t−) = y1 { } S1 = inf t ∈ [t1 , T1 ) : pl,n (·) has no jump in [t, T1 ) . Hence [S1 , T1 ) is the interval on which pl,n (·) achieves its maximum y1 , and if there are several such intervals, then [S1 , T1 ) is the largest of those intervals. We shall now define the intervals [Si , Ti ) recursively. If i ≥ 2 and Ti−1 < t2 , then { } yi = max pl,n (t) : t ∈ [Ti−1 , t2 ] , and { } Ti = max t ∈ [Ti−1 , t2 ] : pl,n (t−) = yi { } Si = inf t ∈ [Ti−i , Ti ) : pl,n (·) has no jump in [t, Ti ) . Let k be the first index such that Tk = t2 ; we stop the recursion at this point. A particular scenario using the above notation is shown in Figure 7. We now have t1 ≤ S1 < T1 ≤ S2 < T2 · · · ≤ Sk < Tk = t2 and y1 > y2 > · · · > yk = pl,n (t2 ), and y1 ≥ pl,n (t1 ). All the points y1 , . . . , yk are in the book at time t2 , i.e., µl,n (t2 ) ({yi }) > 0 for every i = 1, . . . , k. Indeed, the only way an order yi could have been removed from the book is if a market order arrived at the time when yi was the prevailing price, but this could not have happened because that would imply an upward jump of the price at that time. Next we note that NLn (t2 ) − NLn (t1 ) > 0 because pl,n (·) decreased between times t1 and t2 , and so we can define the non-empty set J = {NLn (t1 ) + 1, . . . , NLn (t2 )} . We shall now show that yi+1 − yi ≤ max { 1 √ | ln Xj | : j ∈ J n } . (164) By the definition of Ti+1 , we must have pl,n (Ti ) ≤ yi+1 , which implies that yi − yi+1 ≤ yi − pl,n (Ti ) = pl,n (Ti −) −{pl,n (Ti ). This is a } downward jump of pl,n (·) between times t1 and t2 , hence pl,n (Ti −) − pl,n (Ti ) ∈ √1n | ln Xj | : j ∈ J , which implies (164). All the y’s are in the book at time t2 , and hence { } ( ) 1 l,n µ (t2 ) ([x1 , x2 ]) + 1 max √ | ln Xj | : j ∈ J ≥ x2 − x1 , n and the statement of the lemma follows. Our next lemma completes this section. We have the following. Lemma 10.10. For each n ≥ 1, if 0 ≤ s ≤ t, then ∫ t 1{pl,n (u−) < pl,n (s)}dLn (u) = 0. s 51 y1 y2 y3 p(t) y4 t1 T1 S1 T2 S2 T 3 S3 t 2 = T 4 Figure 7: The shaded circles represent those limit orders remaining on the book at time t2 . They are also the successive maxima of the reversed price process. Proof. First note that it suffices to prove that µ̃l,n (u)([pl,n (s), ∞)) ≥ ãn{for all u ≥ s such that } pl,n (u−) < pl,n (s). Indeed, let y = sup{pl,n (t) : t ∈ [s, u]} and T = inf t ∈ [s, u] : pl,n (t) = y . Clearly s ≤ T < u (this holds even if pl,n has a jump at u). It follows that every order that was on the book at time T is still on the book at time u. So we have ãn ≤ µ̃l,n (T )([pl,n (T ), ∞)) ≤ µ̃l,n (u)([pl,n (T ), ∞)) ≤ µ̃l,n (u)([pl,n (s), ∞)). This completes the proof. 10.2.3 Proof of Lemma 10.2 In this subsubsection, we use the results of Sections 10.2.1 and 10.2.2 above in order to prove Lemma 10.2. We begin by setting up some additional notation. For each ε > 0 and T ≥ 0, let ϱnε = inf{t ≥ 0 : (pl,n (t) − pl,n (0)) ≥ ε} ∧ T (165) be the first time that the natural log of the price is greater than or equal to pl,n (0) + ε (or T , whichever is smaller). Note also the subtle difference between ϱnε as defined in (165) above and νεn as defined in (86) in Section 5. Figure 8 will orientate the reader for the following Proposition. Proposition 10.11. If E[ln X] > 0 and if Items 1-4 of Definition 3.1 hold, then, for each ε, δ, ϑ > 0, ( ) P inf n ϱn ε ≤u≤ϱε+δ (pl,n (u) − pl,n (0)) ≤ ε − ϑ → 0 as n → ∞. (166) Proof. We first show that without any loss of generality, we can assume in this proof that pl,n (0) = 0, a.s., for all n ≥ 1. Define sequence { the } of limit order books corresponding to the sequence of l,n measure-valued processes µ̂ , n ≥ 1 with the (random) initial measure ( ) µ̂l,n (0) ({x}) = µl,n (0) {x + pl,n (0)} , x ∈ R, 52 n ) the same. Next, let µ̂l,n (t) be the random keeping all other random quantities (Xn,i , NLn , NM measure corresponding to the nth book at time t ≥ 0 with the above initial measure, and let { } p̂l,n (t) = sup x ∈ R : µ̂l,n (t) ((−∞, x)) = 0 . (167) Then, clearly we have that p̂l,n (t) = pl,n (t) − pl,n (0) (168) and ( ) µ̂l,n (t) ({x}) = µl,n (t) {x + pl,n (0)} , In particular, x ∈ R. (169) p̂l,n (0) = 0. (170) ϱ̂nε = inf{t ≥ 0 : p̂l,n (t) ≥ ε} ∧ T, (171) Now, for each ε > 0, let so that we have that ϱ̂nε = ϱnε . It then follows from (168) that for each 0 < ϑ < ε, ( ) ( ) ( ) P n inf n (pl,n (u) − pl,n (0)) ≤ ε − ϑ = P n inf n p̂l,n (u) ≤ ε − ϑ . ϱε ≤u≤ϱε+δ ϱ̂ε ≤u≤ϱ̂ε+δ Hence, in order to complete the proof, it suffices to show that the right-hand side of the above equality converges to zero as n goes to ∞. However, by (167), (169) and (170), this implies that for the remainder of the proof it is sufficient to assume that pl,n (0) = 0, a.s., for all n ≥ 1. We proceed as follows. The probability in the statement of the lemma is a non-increasing function of ϑ > 0, so without any loss of generality we can and shall assume that 0 < ϑ < ε. Next, for each n ≥ 1, set ζϑε = inf{t ≥ ϱnε : pl,n (t) ≤ ε − ϑ} ∧ T (172) to be the first time after crossing the threshold ε that the price process drops back down below the level ε − ϑ. Note that in the above we have suppressed the dependence of ζϑε on n. Next, note that clearly ζϑε is a stopping time. Moreover, by definition, ϱnε ≤ ζϑε . Now using (83) along with the definition of ∆φl,n in (154), one obtains that for arbitrary φ ∈ Cb2 (R) and δ > 0, ⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φ⟩ = ⟨µ̃l,n (ϱnε ), φ⟩ + ∆φl,n (ζϑε ∧ ϱnε+δ ) − ∆φl,n (ϱnε ) ∫ ζ ε ∧ϱn ε+δ ϑ ′ +λE[ln X] φ (pl,n (s−))ds ϱn ε + ∫ ζϑε ∧ϱn ε+δ ϱn ε 53 φ(pl,n (s−))dS̃ n (s). (173) pl,n (t) − pl,n (0) ε+δ ε ρnε ρnε+δ t Figure 8: Stopping times ρnε and ρnε+δ . Moreover, since ϱnε and ζϑε ∧ ϱnε+δ are both stopping times, it follows by the semi-martingale decomposition (11) that [∫ ε n ] [∫ ε n ] E ζϑ ∧ϱε+δ φ(pl,n (s−))dS̃ n (s) ζϑ ∧ϱε+δ = E ϱn ε φ(pl,n (s−))dB̃ n (s) . (174) ϱn ε Thus, taking expectations in (173), one obtains that E[⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φ⟩] = E[⟨µ̃l,n (ϱnε ), φ⟩] + E[∆φl,n (ζϑε ∧ ϱnε+δ ) − ∆φl,n (ϱnε )] [∫ ε n ] ζϑ ∧ϱε+δ ′ +λE[ln X]E φ (pl,n (s−))ds (175) ϱn ε +E [∫ ζϑε ∧ϱn ε+δ ϱn ε ] φ(pl,n (s−))dB̃ n (s) . Our next step is to substitute a class of bounded, non-negative functions {φηε ∈ Cb2 (R), η > 0} into (175) which also satisfy the following additional conditions: φηε = (x − ε)2 , x ∈ R, η > 0, 1 1 ′ and sup(φηε ) (x) < , η > 0. sup φηε (x) < η η x≥ε x≥ε (176) (177) We will then use the equality (175) together with our specially chosen functions and the fact that E[ln X] > 0 in order to ultimately prove (166). In order to be concrete, we define our functions φηε ∈ Cb2 (R), η > 0, as follows. For each η > 0, we let φηε be the non-negative function defined by   if x ≤ 0,  c1 arctan (− (ϵ + c2 x)) + c3 2 η (x(− ε) ( φε (x) = ) ) if 0 ≤ x ≤ ε,   (x−ε) 2 12 − + η12 e−η(x−ε) if ε ≤ x, η η 54 where we set 1 + ϵ2 and c3 = 4ϵ3 (arctan ϵ) + ϵ2 . 2ϵ2 We now show how the sequence of functions {φηε , η > 0} may be used in order to obtain a sequence of upper bounds on the probability P (ζϑε < ϱnε+δ ). Specifically, note that by the definition of ϱnε in (165) and of ζϑε in (172), we have that in order to complete the proof, it suffices to show that for each ε, δ > 0 and 0 < ϑ < ε, we have that c1 = 4ϵ3 , c2 = P (ζϑε < ϱnε+δ ) → 0 as n → ∞. (178) We proceed as follows. First note that for each η > 0, we have the decomposition E[⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φηε ⟩] = E[⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φηε ⟩1{ζϑε ∧ ϱnε+δ = ζϑε < T }] +E[⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φηε ⟩1{ζϑε ∧ ϱnε+δ = ϱnε+δ < T }] +E[⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φηε ⟩1{ζϑε ∧ ϱnε+δ = T }]. Next, note that on the event {ζϑε ∧ ϱnε+δ = ζϑε < T }, one has that pl,n (ζϑε ) < pl,n (ϱnε ) and that, also on the event {ζϑε ∧ ϱnε+δ = ζϑε < T }, it follows by (165) and (172) that pl,n (ζϑε ) ≤ ε − ϑ < ε ≤ pl,n (ϱnε ). Hence, by Lemma 10.9 of Section 10.2.2, the fact that by assumption 0 < ϑ < ε, and the fact that φηε is decreasing on [0, ε], we have that E[⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φηε ⟩1{ζϑε ∧ ϱnε+δ = ζϑε < T }] = E[⟨µ̃l,n (ζϑε ), φηε ⟩1{ζϑε ∧ ϱnε+δ = ζϑε < T }] ≥ E[φηε (ε − ϑ/2)µ̃l,n (ζϑε )([ε − ϑ, ε − ϑ/2])1{ζϑε ∧ ϱnε+δ = ζϑε < T }] = ≥ = ϑ2 E[µ̃l,n (ζϑε )([ε − ϑ, ε − ϑ/2])1{ζϑε ∧ ϱnε+δ = ζϑε < T }] 4 ( ) 1 ϑ ϑ2 E[1{ζϑε ∧ ϱnε+δ = ζϑε < T }] −√ 4 2C n ( ) ϑ2 ϑ 1 P (ζϑε < ϱnε+δ ). −√ 4 2C n Moreover, note that for sufficiently large n we have that ( ) ϑ2 ϑ3 1 ϑ P (ζϑε < ϱnε+δ ) ≥ −√ P (ζϑε < ϱnε+δ ), 4 2C 16C n and that by the non-negativity of φηε , it follows that both E[⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φηε ⟩1{ζϑε ∧ ϱnε+δ = ϱnε+δ < T }] ≥ 0 and E[⟨µ̃l,n (ζϑε ∧ ϱnε+δ ), φηε ⟩1{ζϑε ∧ ϱnε+δ = T }] ≥ 0. 55 Hence, by (175) with φ = φηε , one obtains that for sufficiently large n, (ϑ3 /16C)P (ζϑε < ϱnε+δ ) ≤ E[⟨µ̃l,n (ϱnε ), φηε ⟩] + E[∆(φηε )l,n (ζϑε ∧ ϱnε+δ ) − ∆(φηε )l,n (ϱnε )] (179) [∫ ε n ] ζϑ ∧ϱε+δ η ′ l,n +λE[ln X]E (φε ) (p (s−))ds ϱn ε +E [∫ ζϑε ∧ϱn ε+δ ϱn ε ] φηε (pl,n (s−))dB̃ n (s) , which provides our first desired upper bound on P (ζϑε < ϱnε+δ ). We next proceed to bound the terms on the righthand side of (179). In particular, note that we may write [∫ ε n ] [∫ ε n ] ζϑ ∧ϱε+δ λE[ln X]E ϱn ε = λE[ln X]E +E [∫ [∫ +E ϱn ε ζϑε ∧ϱn ε+δ ϱn ε +λE[ln X]E [∫ ζϑε ∧ϱn ε+δ ϱn ε ζϑ ∧ϱε+δ ϱn ε ′ 1{pl,n (s−) < ε}(φηε ) (pl,n (s−))ds 1{pl,n (s−) < ε}φηε (pl,n (s−))dB̃ n (s) [∫ ζϑε ∧ϱn ε+δ ′ (φηε ) (pl,n (s−))ds + E ζϑε ∧ϱn ε+δ ϱn ε φηε (pl,n (s−))dB̃ n (s) ] (180) ] ′ 1{pl,n (s−) ≥ ε}(φηε ) (pl,n (s−))ds l,n 1{p (s−) ≥ ε}φηε (pl,n (s−))dB̃ n (s) ] ] (181) . Therefore, our next step is to analyze the terms (180) and (181), providing convenient upper bounds for each. We begin with (180). First note that by the definition of B̃ n in (13) and Item 2 of Definition 3.1, we have that for sufficiently large n, [∫ ε n ] ζϑ ∧ϱε+δ λE[ln X]E ϱn ε +E ≤ E [∫ [∫ +E ζϑε ∧ϱn ε+δ 1{p (s−) < ζϑε ∧ϱn ε+δ ϱn ε l,n 1{p (s−) < ϱn ε [∫ l,n ϱn ε ζϑε ∧ϱn ε+δ ′ 1{pl,n (s−) < ε}(φηε ) (pl,n (s−))ds ε}φηε (pl,n (s−))dB̃ n (s) (182) ] ′ ε}(λE[ln X](φηε ) (pl,n (s−)) + 2|θ|φηε (pl,n (s−)))ds ] ] 1{pl,n (s−) < ε}φηε (pl,n (s−))dL̃n (s) . Next, note that since λ > 0 and E[ln X] > 0, it follows by the definition of φηε above that for ϑ sufficiently small (and independent of η of and n) and ε − ϑ ≤ x ≤ ε, ′ λE[ln X](φηε ) (x) + 2|θ|φηε (x) ≤ 0. 56 Hence, ′ 1{pl,n (s−) < ε}(λE[ln X](φηε ) (pl,n (s−)) + 2|θ|φηε (pl,n (s−))) ≤ 0 (183) for ϱnε ≤ s < ζϑε ∧ ϱnε+δ and ϑ sufficiently small. Moreover, recall by Lemma 10.10 of Section 10.2.2 that ∫ ζ ε ∧ϱn ϑ ε+δ 1{pl,n (s−) < ε}dL̃n (s) = 0. (184) ϱn ε Now combining (183) and (184), we obtain from (182) that for n sufficiently large and ϑ sufficiently small (how small ϑ should be does not depend on n), ] [∫ ε n ζϑ ∧ϱε+δ λE[ln X]E ϱn ε +E ≤ 0. [∫ ζϑε ∧ϱn ε+δ ′ 1{pl,n (s−) < ε}(φηε ) (pl,n (s−))ds l,n 1{p (s−) < ϱn ε ε}φηε (pl,n (s−))dB̃ n (s) (185) ] This provides our upper bound for (180) We next proceed to bound (181). First recall by (177) that for η sufficiently large we have that 1 η ′ 1{pl,n (s−) ≥ ε}(φηε ) (pl,n (s−)) < and 1{pl,n (s−) ≥ ε}φηε (pl,n (s−)) < 1 . η Hence, for η sufficiently large, it follows that for n sufficiently large (independent of η) we have that [∫ ε n ] ζϑ ∧ϱε+δ λE[ln X]E ϱn ε +E [∫ ζϑε ∧ϱn ε+δ ϱn ε ′ 1{pl,n (s−) ≥ ε}(φηε ) (pl,n (s−))ds 1{pl,n (s−) ≥ ε}φηε (pl,n (s−))dB̃ n (s) (186) ] ≤ (1/η)((λE[ln X] + 2|θ|)T + E[L̃n (T )]) ≤ K1 /η, where the constant K1 > 0 follows by the fact that by (1) we have that 0 < E[ln X] < ∞ and since by Lemma 10.7 of Section 10.2.1, we have that the sequence {(λE[ln(X1 )] + 2|θ|)T + E[L̃n (T )], n ≥ 1} is bounded. (186) now provides our desired upper bound for (181). Finally, note that by Lemma 10.8 of Section 10.2.1, we have that [ E[∆(φηε )l,n (ζϑε ∧ ϱnε+δ ) − ∆(φηε )l,n (ϱnε )] ≤ 2E 57 sup |∆(φηε )l,n (t)| 0≤t≤T ] → 0, (187) as n → ∞. Moreover, note that since pl,n (ϱnε ) ≥ ε, by (177) we have that for η sufficiently large (and independent of n), [ ] 1 1 K2 E[⟨µ̃l,n (ϱnε ), φηε ⟩] ≤ E[S̃ n (ϱnε )] ≤ E sup S̃ n (t) ≤ , (188) η η η 0≤t≤T where [ the constant K ] 2 > 0 follows since by Lemma 10.7 of Section 10.2.1, we have that the sequence n {E sup0≤t≤T S̃ (t) , n ≥ 1} is bounded. Thus, from (179), (185), (186), (187) and (188), we obtain that for each η > 0 sufficiently large and ϑ > 0 sufficiently small and ς > 0, we have that for sufficiently large n, (ϑ3 /16C)P (ζϑε < ϱnε+δ ) ≤ (K1 + K2 )/η + ς. This then implies that P (ζϑε < ϱnε+δ ) ≤ (16C/ϑ3 )((K1 + K2 )/η + ς). However, since η may be made arbitrarily large and ς arbitrarily small, the above inequality implies (178), which completes proof. We now show that Proposition 10.11 implies the following somewhat stronger result. Proposition 10.12. If E[ln X] > 0 and if Items 1-4 of Definition 3.1 hold, then, for each ε > 0, ( )4   → 0 as n → ∞. E  sup (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) 0≤t≤ϱn ε 0≤s≤t Proof. Let ε > 0 and, for each 0 < δ < ε, let {[ti , ti+1 ), i = 0, ..., ⌈2ε/δ⌉ − 1} be a partition of [0, δ⌈2ε/δ⌉), where ti = iδ. Note of course that δ⌈2ε/δ⌉ > ε. Also note that by the definition of ϱnε in (165), one has that ϱnti ≤ ϱnti+1 for i = 0, ..., ⌈2ε/δ⌉−1, and hence {[ϱnti , ϱnti+1 ), i = 0, ..., ⌈2ε/δ⌉−1} forms a natural partition of [0, ϱnδ⌈2ε/δ⌉ ). In the above partition, if ϱnti = ϱnti+1 , then [ϱnti , ϱnti+1 ) = ∅ and we may simply ignore that interval. Now note that we may write sup 0≤t<ϱn ε (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) ≤ 0≤t<ϱn δ⌈2ε/δ⌉ = Uεn (δ), (189) 0≤s≤t (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) sup 0≤s≤t where Uεn (δ) = max sup n i=0,...,⌈2ε/δ⌉−1 ϱn t ≤t<ϱt i i+1 (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) . 0≤s≤t Our goal then is to show that for each 0 < δ < ε, we have that ] [ lim sup E (Uεn (δ))4 ≤ δ 4 , n≥1 58 (190) which, by (189), will complete the proof. First note that by the definition of ϱnε in (165), one has that for ϱnti ≤ t < ϱnti+1 , iδ ≤ sup (pl,n (s) − pl,n (0)) ≤ (i + 1)δ. 0≤s≤t Moreover, by Proposition 10.11, for each ϑ > 0, ( ) l,n l,n n n P iδ − ϑ ≤ (p (t) − p (0)) ≤ (i + 1)δ for ϱti ≤ t < ϱti+1 → 1 as n → ∞. Hence, for each ϑ > 0,  P sup n ϱn ti ≤t<ϱti+1  (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) ≤ δ + ϑ → 1 as n → ∞.(191) 0≤s≤t Next, note we may write the identity ] ] [ ] [ [ E (Uεn (δ))4 = E 1{Uεn (δ) ≤ δ + ϑ}(Uεn (δ))4 + E 1{Uεn (δ) > δ + ϑ}(Uεn (δ))4 . (192) Moreover, clearly [ ] E 1{Uεn (δ) ≤ δ + ϑ}(Uεn (δ))4 ≤ (δ + ϑ)4 . (193) On the other hand, by the definition of ρnε in (165), one has that Uεn (δ) ≤ δ⌈2ε/δ⌉, and so by (190) and (191), it follows that ] [ E 1{Uεn (δ) > δ + ϑ}(Uεn (δ))4 ≤ δ 4 (⌈2ε/δ⌉)4 P (Uεn (δ) > δ + ϑ) (194) → 0 as n → ∞. Thus, since ϑ > 0 was arbitrary, it follows by (192), (193) and (194) that [ ] lim sup E (Uεn (δ))4 ≤ δ 4 , n≥1 which completes the proof. Now note that using the definitions (86) and (165) of νεn and ϱnε , respectively, it follows immediately that νεn ≤ ϱnε . Hence, we obtain that sup 0≤t≤νεn (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) 0≤s≤t ≤ sup 0≤t≤ϱn ε (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) . 0≤s≤t This then immediately implies the following as a result of Proposition 10.12. Proposition 10.13. If E[ln X] > 0 and if Items 1-4 of Definition 3.1 hold, then, for each ε > 0, ( )4   → 0 as n → ∞. E  sup (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) 0≤t≤νεn 0≤s≤t 59 We are now in a position to prove Lemma 10.2. We have the following. Proof of Lemma 10.2. Note that by the definition of ∆µ̃l,n (t) in (84), we may write the simple identity ∆µ̃l,n (t) = ⟨µ̃l,n (t), ξ⟩ − S̃ n (t) sup pl,n (s) + S̃ n (t) sup (pl,n (s) − pl,n (t)), 0≤s≤t 0≤s≤t Thus l,n (∆µ̃ (t)) 2 ( l,n n l,n ≤ 2 ⟨µ̃ (t), ξ⟩ − S̃ (t) sup p (s) 0≤s≤t )2 . + 2 S̃ (t) sup (p (s) − p (t)) (195) )2 ( n l,n 0≤s≤t Let ε > 0. We now show that [ ( )2 ] l,n n l,n E sup ⟨µ̃ (t), ξ⟩ − S̃ (t) sup p (s) →0 0≤t≤νεn l,n (196) 0≤s≤t and E [ sup 0≤t≤νεn ( n l,n l,n S̃ (t) sup (p (s) − p (t)) 0≤s≤t )2 ] → 0, (197) as n → ∞, which, by (195), will complete the proof. We proceed in order. We begin with (196). First note that after some straightforward calculations, it may be shown that ⟨µ̃l,n (t), ξ⟩ − S̃ n (t) sup pl,n (s) (198) 0≤s≤t ≤ S̃ n (t) maxn i=1,...,NL (t) 1/n1/2 ln Xi + ⟨µ̃l,n (0), ξ⟩ − S̃ n (0)pl,n (0). Next, since ⟨µ̃l,n (t), ξ⟩ ≥ S̃ n (t)pl,n (t), it also follows that we may write ( ) l,n n l,n n l,n l,n ⟨µ̃ (t), ξ⟩ − S̃ (t) sup p (s) ≥ S̃ (t) p (t) − sup p (s) (199) 0≤s≤t 0≤s≤t ( ) n l,n l,n l,n l,n = S̃ (t) (p (t) − p (0)) − sup (p (s) − p (0)) . 0≤s≤t Thus, using the basic identity (x1 + ... + xm )2 ≤ m(x21 + ... + x2m ), m ≥ 1, we have that (198) and (199) imply that [ ( )2 ] (1/3)E sup ⟨µ̃l,n (t), ξ⟩ − S̃ n (t) sup pl,n (s) 0≤t≤νεn ≤ E [ sup (S̃ n (t) 0≤t≤νεn 0≤s≤t max n (t) i=1,...,NL 1/n1/2 ln Xi +E[(⟨µ̃l,n (0), ξ⟩ − S̃ n (0)pl,n (0))2 ] 60 )2 ] (200) (201) +E [ sup 0≤t≤νεn ( ( n l,n l,n l,n l,n S̃ (t) (p (t) − p (0)) − sup (p (s) − p (0)) 0≤s≤t ))2 ] . (202) We now show that each of (200) through (202) converge to 0 as n tends to ∞, which will complete our treatment of (196). Regrading (200), note that since P (Xi ≤ C) = 1, we have that ] [ [ ] 1/2 ln C 1/n )2 ≤ E sup (S̃ n (t) maxn ln Xi E sup (S̃ n (t))2 → 0 as n → ∞. n i=1,...,NL (t) 0≤t≤νεn 0≤t≤T Next, for (201), recall that by assumption we have that E[(⟨µ̃l,n (0), ξ⟩ − S̃ n (0)pl,n (0))2 ] → 0 as n → ∞. Finally, regarding (202), note that by the Cauchy Schwarz inequality, Proposition 10.13 and Lemma 10.7 of Section 10.2.1, we have that [ ( ))2 ] ( n l,n l,n l,n l,n E sup S̃ (t) (p (t) − p (0)) − sup (p (s) − p (0)) (203) 0≤t≤νεn ≤ E [ 0≤s≤t n sup (S̃ (t)) 4 0≤t≤T ] E [ sup 0≤t≤νεn → 0 as n → ∞. ( l,n l,n l,n l,n (p (t) − p (0)) − sup (p (s) − p (0)) 0≤s≤t )4 ] This completes our treatment of (196). Finally, note that (197) follows by (203) upon noting that [ ( )2 ] E sup S̃ n (t)( sup pl,n (s) − pl,n (t)) 0≤t<νεn = E [ sup 0≤t≤νεn 0≤s≤t ( ( S̃ n (t) (pl,n (t) − pl,n (0)) − sup (pl,n (s) − pl,n (0)) 0≤s≤t ))2 ] . This completes the proof. 10.3 Proof of Proposition 10.14 In this section, we provide the statement and proof of Proposition 10.14 concerning tightness of the sequence of diffusion scaled, measure-valued order book processes in the high frequency regime. In particular, we prove the following. Proposition 10.14. If E[ln X] > 0 and E[(∆µ̃l,n (0))2 ] → 0 and (pn (0), S̃ n (0)) ⇒ (p(0), S̃(0)) as n → ∞, and if Items 1-4 of Definition 3.1 hold, then the sequence {µ̃n , n ≥ 1} is tight in D(R+ , MF (R)). Proposition 10.14 is used in Section 6 in the proof of Theorem 6.1 which is our main result concerning the weak convergence of the sequence of diffusion scaled, measure-valued order book processes in the high-frequency regime. However, before providing the proof of Proposition 10.14, we first need the following supporting result. Let Lc (R) be the set of Lipschitz continuous functions with compact support on R. We then have the following. 61 Proposition 10.15. If E[ln X] > 0 and E[(∆µ̃l,n (0))2 ] → 0 and (pn (0), S̃ n (0)) ⇒ (p(0), S̃(0)) as n → ∞, and if Items 1-4 of Definition 3.1 hold, then, for each m ≥ 1 and φ1 , ..., φm ∈ Lc (R), we have the joint convergence (⟨µ̃n , φ1 ⟩, ..., ⟨µ̃n , φm ⟩) ⇒ (S̃φ1 (p), ..., S̃φm (p)) in Dm ([0, ∞), R) as n → ∞. (204) Proof. First note that for each n ≥ 1 and φ ∈ Lc (R), we may write ⟨µ̃n , φ⟩ = ⟨µ̃l,n , φ̂⟩, where φ̂ ∈ Lc (R) is given by φ̂(x) = φ(ex ) for x ∈ R. Hence, in order to show (204), it suffices to show that for each m ≥ 1 and φ1 , ..., φm ∈ Lc (R), (⟨µ̃l,n , φ1 ⟩, ..., ⟨µ̃l,n , φm ⟩) ⇒ (S̃φ1 (pl ), ..., S̃φm (pl )) in Dm ([0, ∞), R) as n → ∞. (205) Recall first from (139) the definition Ãn (t) = ∫ t 0 E[ln X] dN̄Ln (s), t ≥ 0, S̃ n (s) and let Ãn = {Ãn (t), t ≥ 0}. Next, let m ≥ 1 and let φ1 , ..., φm ∈ BL(R). It then follows that for each t ≥ 0, we may write (⟨µ̃l,n (t), φ1 ⟩, ..., ⟨µ̃l,n (t), φm ⟩) = (S̃ n (t)φ1 (Ãn (t)), ..., S̃ n (t)φm (Ãn (t))) + ε̃n (t), (206) ε̃n (t) = (⟨µ̃l,n (t), φ1 ⟩ − S̃ n (t)φ1 (Ãn (t)), ..., ⟨µ̃l,n (t), φm ⟩ − S̃ n (t)φm (Ãn (t))). (207) where Moreover, using similar arguments as in the proof of Proposition 10.4 of Section 10.1 above, it is straightforward to show that we have the convergence (S̃ n φ1 (Ãn ), ..., S̃ n φm (Ãn )) ⇒ (S̃φ1 (pl ), ..., S̃φm (pl )) in Dm ([0, ∞), R) as n → ∞. Hence, letting ε̃n = {ε̃n (t), t ≥ 0}, in order to show (205) and complete the proof, it suffices by (206) to show that ε̃n ⇒ 0 as n → ∞. First note that for each i = 1, ..., m, we may write ⟨µ̃l,n , φi ⟩ − S̃ n φi (Ãn ) = (⟨µ̃l,n , φi ⟩ − S̃ n φi (pl,n )) + (S̃ n (φi (pl,n ) − φi (Ãn ))). (208) We now show that both (⟨µ̃l,n , φi ⟩ − S̃ n φi (pl,n )) ⇒ 0 as n → ∞ (209) (S̃ n (φi (pl,n ) − φi (Ãn ))) ⇒ 0 as n → ∞. (210) and By (208), this will then show that ⟨µ̃l,n , φi ⟩ − S̃ n φi (Ãn ) ⇒ 0 as n → ∞, which, by (207), will imply that ε̃n ⇒ 0 as n → ∞ and will compete the proof. We begin with (209). First note that for each t ≥ 0, the measure µ̃l,n (t) is a linear combination of Dirac measures and so we may write n µ̃l,n (t) = S (t) 1 ∑ √ δxnk (t) . n k=1 62 Note also that in the above it is possible that xnj (t) = xnk (t) for j ̸= k. It now follows from the above that for each t ≥ 0 we have that n l,n n l,n ⟨µ̃ (t), φi ⟩ − S̃ (t)φi (p (t)) = S (t) 1 ∑ √ φi (xnk (t)) − S̃ n (t)φi (pl,n (t)) n k=1 S n (t) = 1 ∑ √ (φi (xnk (t)) − φi (pl,n (t))). n k=1 Now let T ≥ 0 and ϵ > 0. We then have from the above that we may write sup ⟨µ̃l,n (t), φi ⟩ − S̃ n (t)φi (pl,n (t)) 0≤t≤T n = sup 0≤t≤T S (t) 1 ∑ √ (φi (xnk (t)) − φi (pl,n (t))) 1{T ≤ νϵn } n (211) k=1 n + sup 0≤t≤T S (t) 1 ∑ √ (φi (xnk (t)) − S̃ n (t)φi (pl,n (t))) 1{T > νϵn }, n (212) k=1 where we recall the definition of νεn from (86) of Section 10.1 above. Hence, in order to show (209), it now suffices to show that both (211) and (212) weakly converge to 0 as n tends to ∞. We first show that (211) converges to 0 in L2 (P ) as n tends to ∞, which implies weak convergence to 0. Let K be the Lipschitz constant associated with φi . We then have that   2 S n (t) ∑ 1 E  sup √ (φi (xnk (t)) − φi (pl,n (t))) 1{T ≤ νϵn } n 0≤t≤T k=1   2 S n (t) ( ) ∑ 1 1{T ≤ νϵn } xnk (t) − pl,n (t) ≤ K 2 E  sup √ n 0≤t≤T k=1 [ ] ( )2 = K 2 E sup ∆µ̃l,n (t) 1{T ≤ νϵn } 0≤t≤T → 0 as n → ∞, where the final convergence above follows by Lemma 10.2 of Section 10.1. Thus, (211) weakly converges to 0 as n tends to ∞. Next, we show that (212) converges to zero in probability as n tends ∞, which implies weak convergence to 0. First note that by Proposition 10.6 of Section 10.1, we have that for every δ > 0, there exists an ϵ > 0 and an N = N (δ, ϵ) such that for n ≥ N , we have P (νϵn < T ) < δ. This then implies for every γ > 0,   S n (t) ∑ 1 P  sup √ (φi (xnk (t)) − S̃ n (t)φi (pl,n (t))) 1{T > νϵn } > γ  ≤ P (νϵn < T ) < δ. n 0≤t≤T k=1 Hence, (212) converges to zero in probability as n tends ∞, which completes the proof of (209). 63 We next proceed to (210). First note that by (147) of Theorem 10.1, we have that pl,n − Ãn ⇒ 0 as n → ∞. (213) Hence, it follows since the function φi ∈ Lc (R) that φi (pl,n ) − φi (Ãn ) ⇒ 0 as n → ∞. Thus, since S̃ n ⇒ S̃ as n → ∞, it is immediate that (210) holds. This completes the proof. We are now in a position to provide the proof of Proposition 10.14. We have the following. Proof of Proposition 10.14. It suffices to check that {µ̃n , n ≥ 1} satisfies the following two conditions of Jakubowski [20] 1. For each T > 0 and η > 0, there exists a compact set KT,η ⊂ MF (R) such that lim inf P (µ̃n (t) ∈ KT,η for all t ∈ [0, T ]) > 1 − η. n→∞ 2. There exists a family F of real, continuous functions F in MF (R) that separates points on MF (R) and is closed under addition such that the sequence {F (µ̃n ), n ≥ 1} is tight in D(R+ , R) for each F ∈ F. We begin with Condition 1. Let T > 0 and η > 0. Next, recall from Theorem 15.7.5 of Kallenberg [21] that a set M ⊂ MF (R) is relatively compact if and only if supξ∈M ξ(R) < ∞ and 1 be such that supξ∈M ξ((−∞, −n) ∪ (n, ∞)) → 0 as n → ∞. Now let KT,η/2 sup P n≥1 ( 1 sup S̃ n (t) > KT,η/2 0≤t≤T ) ≤ η/2. (214) 1 2 exists by Lemma 10.7 of Section 10.2.1. Next, let KT,η/2 Note that such a KT,η/2 be such that sup P n≥1 ( n sup (p (t) + C) > 0≤t≤T 2 KT,η/2 ) ≤ η/2. (215) 2 exists by Theorem 5.1 of Section 5. We now define the set MT,η ⊂ MF (R) Note that such a KT,η/2 by setting 1 MT,η = {ξ ∈ MF (R) : ξ(R) ≤ KT,η/2 and ξ((−∞, 0) ∪ (KT,η/2 , ∞)) = 0}. It is then clear by Theorem 15.7.5 of Kallenberg [21] that MT,η is relatively compact and, moreover, by (214), (215), system equation (6) and the assumption that E[(∆µ̃l,n (0))2 ] → 0 as n → ∞, we have that lim inf P (µ̃n (t) ∈ MT,η for all t ∈ [0, T ]) > 1 − η. n→∞ Thus, the sequence {µ̃n , n ≥ 1} satisfies Condition 1 above. Next, we note that Condition 2 follows from Proposition 10.15 above since the family of continuous functions F = {⟨ · , φ⟩, φ ∈ BL(R)} separates points on MF (R) and is closed under addition. This completes the proof. 64 References [1] F. Abergel and A. Jedidi. A mathematical approach to order book modeling. International Journal of Theoretical and Applied Finance, 16(05), 2013. [2] S. Asmussen. Applied probability and queues, volume 51. Springer Verlag, 2003. [3] M. Avellaneda, J. Reed, and S. Stoikov. 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