Numerical Algebra, Control and Optimization

doi:10.3934/naco.2024012

ON CONTINUOUS AND DISCRETE SAMPLING FOR

PARAMETER ESTIMATION IN MARKOVIAN SWITCHING
DIFFUSIONS

∗
1 1
Yuhang Zhen and Fubao Xi
1 School of Mathematics and Statistics

Beijing Institute of Technology, Beijing 100081, China

Abstract. Parameter estimation of stochastic differential equation has re-

cently been discussed by many authors. The aim of this paper is to study the
rates of convergence of approximate maximum likelihood estimator. More pre-
cisely, for Markovian switching diffusions, we first show the convergence rates
of the continuous maximum likelihood estimator under the Lipschitz condi-
tions. Then we also discuss the probabilistic bounds on |θn,T − θT | under the
non-Lipschitz conditions.

1. Introduction. Let (Ω, F, (Ft )t≥0 , P) be a complete filtered probability space

satisfying the usual conditions (i.e., it is increasing and right continuous and F0
contains all P-null sets). Let N be a positive integer, and put S = {1, 2, . . . , N }
with 2 ≤ N < ∞, the totality of nonnegative integers. The stochastic processes
in this paper are all defined on (Ω, F, (Ft )t≥0 , P). In the whole paper, θ ∈ Θ is
an unknown parameter whose true value is written as θ0 ∈ Θ, where Θ is an open
convex bounded set of Rd .
The asymptotic approach to parameter estimation is frequently adopted because
of its general applicability and relative simplicity. Note that the observation of
diffusion can be continuous or discrete. Continuous observation of a diffusion is a
mathematical idealization and has a very rich theory, for example Itô stochastic cal-
culus, stochastic filtering, inference for continuously observed diffusions and much
more behind it. But the path of the diffusion process changes all the time and no
measuring device can follow a diffusion trajectory continuously. Hence the obser-
vation is always discrete in practice. Research on discretely observed diffusions is
growing recently with a powerful theory of simulation schemes and numerical anal-
ysis of stochastic differential equation (SDE) behind it. In this paper we study the
asymptotic behaviour of maximum likelihood estimator (MLE) and approximate
maximum likelihood estimator (AMLE) of the unknown parameter appearing in
the drift coefficient based on observations of the diffusion process (X(t))t≥0 on a
time interval [0, T ].

2020 Mathematics Subject Classification. Primary: 60H10; Secondary: 34F05.

Key words and phrases. Maximum likelihood estimator; convergence; stochastic differential
equation; Markovian switching.
This work was supported by the National Natural Science Foundation of China (Grant No.
12071031).
This paper is handled by Chao Zhu as guest editor.
∗ Corresponding author: Fubao Xi.

1
2 YUHANG ZHEN AND FUBAO XI

The maximum likelihood method has extensive applications in many fields includ-
ing data processing, error estimation and the parameter estimation of a diffusion
processes. For a linear homogeneous SDE, MLE was studied in [1, 5], and for a
nonlinear homogeneous SDE, MLE was studied in [2, 3, 12], for a nonlinear nonho-
mogeneous SDE, MLE was studied in [7, 8, 11]. Breton [14] studied the convergence
of the estimator θn,T to θT as n → ∞ and T remains fixed. In particular, he showed
2
that |θn,T −θT | = OP ( Tn )1/2 . Rao [13] proved the asymptotic normality and asymp-
totic efficiency of the estimator θn,T . In [4], Jaya mainly concentrated on accuracy
of approximation of the log-likelihood function for in homogeneous diffusions, i.e.,
the following SDE
dX(t) = θX(t)dt + dB(t), t ≥ 0
where (B(t))t≥0 is a standard Brownian motion, θ ∈ Θ ⊆ R is the unknown param-
eter which is to be estimated on the basis of observations of the process X at times
0 = t0 < t1 < ... < tn = T with ti = i Tn , i = 0, 1, ..., n.
It is well known that SDE with Markovian switching has received considerable
attention in the past decades. On the one hand, there are some results in the
theoretical part. We know SDE with Markovian switching have a unique solution
under the Lipschitz and the linear growth conditions (see [10]). And the ergodicity
for the diffusion processes with Markovian switching was established by Mao and
Yuan [10] and the asymptotic stability was proved in [9] and the references therein.
On the other hand, they also have a wide range of applications in some fields, such
as scientific engineering applications [15], structural analysis [14], control systems
and filtering [17], economics and finance [18], and ecological and biological modeling
[19] etc.
Due to the utilities of SDE with Markovian switching in a wide range of appli-
cations, it is natural and important to study its parameter estimation. However,
while parameter estimation for the usual SDE have been well studied, there are few
results for the SDE with Markovian switching. As far as we know, Zhen and Xi [21]
studied least squares estimators for SDE with Markovian switching, Zhen [20] giv-
en a result that approximation of the likelihood function for SDE with Markovian
switching. The goal of this paper is to fill this gap in literature. In this paper, let
(X(t), α(t)) is a right continuous process with left-hand limits on Rd × S, in which
the first component (X(·)) takes values in Rd and the second component (α(t)),
called Markovian switching process. More precisely, X(t) satisfies the following
SDE
dX(t) = b(θ, X(t), α(t))dt + σ(X(t), α(t))dB(t), (1)
where (B(t))t≥0 is standard Brownian motion, θ ∈ Θ ⊆ R is the unknown parame-
ter.
In this paper, we want showed that rate of convergence as ∆ → 0 when the
diffusion is ergodic. However, there are some difficulties to overcome. To solve these
difficulties, we prove some propositions in this paper. More precisely, we consider
that the rates of convergence of AMLE to the continuous MLE of the parameter of
SDE with markovian switching when T remains fixed. Compared with the former
work, all the models discussed only had one random factor, which restricts the
applicability of their models and results. Adding the switching processes to our
work, the coefficients in the model are not same, but vary with the switching process.
Therefore, such research has more practical significance. However, a natural and
important difficulty is that to estimate the parameter of the given system by the
 PARAMETER ESTIMATION IN MARKOVIAN SWITCHING DIFFUSIONS 3

maximum likelihood method, which not only depend on discrete-time observation

values of the state X(t) but also lying in the switching process (α(t))t≥0 . Hence,
there exists an essential difference in the processing method. Heuristically, (α(t))t≥0
is a jumping process, which yields that α(t−) and α(t+) may be at a different state
of S. As a consequence, it is useless to expect that the effect caused by α∆ (t) is
similar to that caused by α(t) no matter how small ∆ is.
The paper is organized as follows. In Section 2, we state problem formulation
and some lemmas. In Section 3, we will give the main result and relate proposition
under the Lipschitz conditions. In Section 4, we give main result and main proof
under the non-Lipschitz conditions. Let C represent positive constants whose values
may change from one place to another.

2. Problem formulation and preliminaries. In this section, we state the SDE

with Markovian switching model and the construction of maximum likelihood func-
tion. The process is observed over [0, T ] and X(t) is distributed according to the
stationary distribution of the process.
In this paper, we study a one-dimensional stationary ergodic diffusion process
described by the following class of SDE:
(
dX(t) = θb(X(t), α(t))dt + dB(t),
0 ≤ t ≤ T, (2)
X(0) = x, α(0) = l,

where α(·) be a right-continuous Markov chain on the probability space taking

values in a finite state space S = {1, 2, ..., N } with generator Q = (qlk )N ×N and
step size ∆ given by
(
qlk ∆ + o(∆), if k 6= l,
P{α(t + ∆) = k|α(t) = l} = (3)
1 + qll ∆ + o(∆), if k = l.

where ∆ ↓ 0. Here qlk 6= 0 is the transition rate from l to k if l = k while

X
qll = − qlk .
k6=l
P
Let H = maxl∈S ql < ∞, where ql = k6=l qlk for l ∈ S. Herein, (X(t), α(t)) is a
two-component process such that X(·) takes values in R, b(·, ·) : R×S → R, θ ∈ Θ is
an unknown parameter whose true value is written as θ0 ∈ Θ, Θ is an open convex
bounded set of R; and B(t) is standard Brownian motion on a filtered probability
space (Ω, F, (Ft )t≥0 , P). We assume that the Markov chain α(t) is Ft -adapted but
independent of the Brownian motion B(·).
Let PθT be the probability measure generated by the process X[0,T ] = {X(t), 0 ≤
t ≤ T } on the measurable space (CT , BT ) of continuous functions on [0, T ] with the
associated Borel σ-algebra BT under the supremum norm. PBT be the probability
measure induced by the standard Brownian process. Under satisfactory condition,
the measures Pθ and PB are equivalent and the Radon-Nikodym derivative of Pθ
w.r.t.PB is given by
( Z )
T
θ2 T 2
Z
dPθ
(X[0,T ] ) = exp θ b(X(t), α(t))dX(t) − b (X(t), α(t))dt , (4)
dPB 0 2 0
4 YUHANG ZHEN AND FUBAO XI

along the sample path X[0,T ] [6]. According to the Girsanov theorem, the likelihood
function has the following expression:
Z T
θ2 T 2
Z
dPθ
ΦT (θ) = log =θ b(X(t), α(t))dX(t) − b (X(t), α(t))dt. (5)
dPB 0 2 0
The MLE of θ is defined as
RT
0
b(X(t), α(t))dX(t)
θT := arg max ΦT (θ) = RT .
θ∈Θ b 2 (X(t), α(t))dt
0
We study the approximation of the MLE θT when the process X is observed at
the points 0 = t0 < t1 < ... < tn = T with ∆ = |ti − ti−1 |, i = 0, 1, 2, ..., n as
n → ∞. In ΦT (θ) one can use Itô type approximation for the stochastic integral
and rectangular approximation for the ordinary integral and obtain an approximate
log-likelihood function
n
X
Φn,T (θ) =θ b(X(ti−1 ), α(ti−1 ))(X(ti ) − X(ti−1 ))
i=1
n
θ2 X 2
− b (X(ti−1) , α(ti−1 ))(ti − ti−1 ). (6)
2 i=1
The AMLE of θ is defined as
Pn
b(X(t ), α(ti−1 ))(X(ti ) − X(ti−1 ))
θn,T Pn 2 i−1
:= arg max ΦT (θ) = i=1 .
θ∈Θ i=1 b (X(ti−1 ), α(ti−1 ))(ti − ti−1 )
In order to prove main results, we cite the following results as lemmas.
Lemma 2.1. Assume that b satisfy the Lipschitz condition, namely there is constant
C > 0 such that
|b(x, l) − b(y, l)| ≤ C|x − y|, x, y ∈ R, l ∈ S. (7)
The Lipschitz condition is used to guarantee the existence of a unique nonexplosive
strong solution of (2) and (3) . And the Lipschitz condition implies the linear growth
condition:
|b(x, l)| ≤ C(1 + |x|), ∀(x, l) ∈ R × S. (8)
Then there is a constant C, which depends only on T, x such that the exact solution
to (2) and (3) have the property that
E sup |X(t)|2 ≤ C,
 
0≤t≤T

and let 0 ≤ s < t < T , the linear growth condition also implies
E[|X(t) − X(s)|2 ] ≤ C|t − s|.
Since the proof is standard, we omit it here.
Lemma 2.2. [4] Let Qn , Rn , Q and R be random variables on the same probability
space (Ω, F, (Ft )t≥0 , P) with P(R > 0) > 0. Suppose |Qn − Q| = OP (δ1n ) and
|Rn − R| = OP (δ2n ) where δ1n , δ2n → 0 as n → ∞. Then
Qn Rn
− = OP (δ1n ∨ δ2n ),
Q R
 PARAMETER ESTIMATION IN MARKOVIAN SWITCHING DIFFUSIONS 5

Xn
where OP means Xn = OP (bn ) : bn is stochastically bounded, i.e.,
Xn
lim sup P{| | > A} = 0.
A→0 n bn
3. Convergence under the Lipschitz conditions. In this section, we main-
ly discuss the convergence of SDE with Markovian switching under the Lipschitz
conditions, and give the probabilistic bounds on |θn,T − θT |.
Let us now present one more useful proposition that will play an important role
in the following sections. Introduce the following notations:
Xn Z T
Qn,T := b(X(ti−1 ), α(ti−1 ))[X(ti ) − X(ti−1 )], QT := b(X(t), α(t))dX(t),
i=1 0
n
X Z T
Rn,T := b2 (X(ti−1 ), α(ti−1 ))[ti − ti−1 ], RT := b2 (X(t), α(t))dt.
i=1 0

Proposition 3.1. Under the assumptions (7) and (8), we have

(a) E|Qn,T − QT |2 = O(∆),
(b) E|Rn,T − RT |2 = O(∆2 ).
Proof. Using (2) and the fact that
Z ti
X(ti ) − X(ti−1 ) = θb(X(s), α(s))ds + B(ti ) − B(ti−1 ),
ti−1

by the definition of Qn,T and QT , we obtain

E|Qn,T − QT |2
n 2
X Z T
=E b(X(ti−1 ), α(ti−1 ))[X(ti ) − X(ti−1 )] − b(X(t), α(t))dX(t)
i=1 0
n Z
X ti
=E θb(X(ti−1 ), α(ti−1 ))b(X(s), α(s))ds
i=1 ti−1

Xn
+ b(X(ti−1 ), α(ti−1 ))[B(ti ) − B(ti−1 )]
i=1
2
Z T Z T
2
− θb (X(t), α(t))dt − b(X(t), α(t))dB(t) (by (2))
0 0

n 2
X Z T
≤2E b(X(ti−1 ), α(ti−1 ))[B(ti ) − B(ti−1 )] − b(X(t), α(t))dB(t)
i=1 0

n Z 2
X ti
2
+ 2θ E b(X(s), α(s))[b(X(ti−1 ), α(ti−1 )) − b(X(s), α(s))]ds
i=1 ti−1

=:N1 + N2 . (9)
Firstly, using the Hölder inequality and Itô isometry , we have,
n Z T 2
X
N1 =E b(X(ti−1 ), α(ti−1 ))[B(ti ) − B(ti−1 )] − b(X(t), α(t))dB(t)
i=1 0
6 YUHANG ZHEN AND FUBAO XI

2
Z T Z T
=E b(X(ti−1 ), α(ti−1 ))dB(t) − b(X(t), α(t))dB(t)
0 0
2
Z T
=E [b(X(ti−1 ), α(ti−1 )) − b(X(t), α(t))]dB(t)
0
Z T
=C E|b(X(ti−1 ), α(ti−1 )) − b(X(t), α(t))|2 dt. (10)
0
By the Lipschtiz condition and Lemma 2.1, we can get
Z T
T 2E|b(X(ti−1 ), α(t)) − b(X(t), α(t))|2 dt
0
n Z
X ti
≤CT E|X(s) − X(ti−1 )|2 ds
i=1 ti−1
n Z
X ti
≤CT |s − ti−1 |ds ≤ C∆. (11)
i=1 ti−1

Let n = [T /∆], the integer part of T /∆. Then,

Z T
2E|b(X(ti−1 ), α(ti−1 )) − b(X(ti−1 ), α(t))|2 dt
0
n Z
X ti
= E[|b(X(ti−1 ), α(ti−1 ))|2 + |b(X(ti−1 ), α(s))|2 ]I{α(ti−1 )6=α(s)} ds.
i=1 ti−1

Noting that X(ti−1 ) and I{α(ti−1 )6=α(s)} are conditionally independent with respect
to the σ-algebra generated by α(ti−1 ), and since the state of the Markov chain is
finite, the Lipschitz condition implies the linear growth condition. We obtain that
Z ti
E[|b(X(ti−1 ), α(ti−1 ))|2 + |b(X(ti−1 ), α(s))|2 ]I{α(ti−1 )6=α(s)} ds
ti−1
Z ti
≤ EC[1 + |X(ti−1 )|2 ]I{α(ti−1 )6=α(s)} ds
ti−1
Z ti
≤ E[E[(1 + |X(ti−1 )|2 )I{α(ti−1 )6=α(s)} |α(ti−1 )]]ds
ti−1
Z ti
= E[E[(1 + |X(ti−1 )|2 )|α(ti−1 )]E[I{α(ti−1 )6=α(s)} |α(ti−1 )]]ds. (12)
ti−1

Here we use the technique from [16]. Observe that by the Markov property and (3),
for 0 ≤ ti−1 ≤ s ≤ ti ≤ T,
E[I{α(ti−1 )6=α(s)} |α(ti−1 )]
X
= I{α(ti−1 )=l} P (α(s) 6= l|α(ti−1 ) = l)
l∈S
X X
= I{α(ti−1 )=l} (qlk (s − ti−1 ) + o(s − ti−1 ))
l∈S k6=l
 
≤ max (−qll )((ti − ti−1 ) + o(ti − ti−1 )) ≤ C∆. (13)
1≤l≤N
 PARAMETER ESTIMATION IN MARKOVIAN SWITCHING DIFFUSIONS 7

Then, we can get that by the boundedness of E|X(t)|2

n 2
X Z T
N1 =2E b(X(ti−1 ), α(ti−1 ))[B(ti ) − B(ti−1 )] − b(X(t), α(t))dB(t)
i=1 0
Z T
≤C 2E|b(X(ti−1 ), α(ti−1 )) − b(X(ti−1 ), α(t))|2 dt
0
Z T
+C 2E|b(X(ti−1 ), α(t)) − b(X(t), α(t))|2 dt
0
≤C∆. (14)
To estimate N2 , let ψi (t) = b(X(t), α(t))(b(X(ti−1 ), α(ti−1 )) − b(X(t), α(t))) for
ti−1 ≤ s ≤ ti , i = 1, 2, ..., n. Then
n Z 2
X ti
E ψi (s)ds
i=1 ti−1

n 2 n
"Z #
X Z ti X ti Z tj
= E ψi (s)ds +2 E ψi (s)ds ψj (s)ds
i=1 ti−1 i=j=1,i<j ti−1 tj−1

=:M1 + M2 . (15)
By the linear growth condition, and using the boundedness of E|X(t)|2 , we can get
n 2 n
X Z ti X Z ti
E ψi (s)ds ≤ (ti − ti−1 ) E(ψi (s))2 ds
i=1 ti−1 i=1 ti−1
n
X Z ti
≤ (ti − ti−1 ) E[|b(X(s), α(s))|2 |b(X(ti−1 ), α(ti−1 )) − b(X(s), α(s))|2 ]ds
i=1 ti−1
n
X Z ti
≤ (ti − ti−1 ) 2CE(1 + |X(s)|2 )[|b(X(ti−1 ), α(s)) − b(X(s), α(s))|2
i=1 ti−1

+ |b(X(ti−1 ), α(ti−1 )) − b(X(ti−1 ), α(s))|2 ]ds

Xn Z ti
≤ (ti − ti−1 ) 2CE(1 + |X(s)|2 )[|X(ti−1 ) − X(s)|2
i=1 ti−1

+ (|b(X(ti−1 ), α(ti−1 ))|2 + |b(X(ti−1 ), α(s))|2 )I{α(ti−1 )6=α(s)} ]ds

Xn Z ti 
≤ (ti − ti−1 ) 2CE(1 + |X(ti−1 )|2 + |X(s) − X(ti−1 )|2 )E|X(ti−1 ) − X(s)|2
i=1 ti−1

+ C(1 + |X(s)|2 )(1 + |X(ti−1 )|2 )I{α(ti−1 )6=α(s)} ds
n
X Z ti 
≤ (ti − ti−1 ) 2CE(1 + |X(ti−1 )|2 )|s − ti−1 | + |s − ti−1 |2
i=1 ti−1

+ C(1 + |X(ti−1 )|4 ) + (1 + |X(ti−1 )|2 )|X(s) − X(ti−1 )|2 I{α(ti−1 )6=α(s)} ]ds
n
(Z
X ti
≤C (ti − ti−1 ) |s − ti−1 |ds
i=1 ti−1
8 YUHANG ZHEN AND FUBAO XI

Z ti 
+ E[E(1 + |X(ti−1 )|2 )I{α(ti−1 )6=α(s)} |α(ti−1 )]
ti−1
 )
+ E|X(s) − X(ti−1 )|2 E(1 + |X(ti−1 )|2 )I{α(ti−1 )6=α(s)} |α(ti−1 ) ds

n
(Z
X ti Z ti
≤C (ti − ti−1 ) |ti − ti−1 |ds + E[E[(1 + |X(ti−1 )|2 )|α(ti−1 )]
i=1 ti−1 ti−1
)
· E[I{α(ti−1 )6=α(s)} |α(ti−1 )]ds

n
(Z )
X ti Z ti
≤C (ti − ti−1 ) |ti − ti−1 |ds + |ti − ti−1 |ds (by (13))
i=1 ti−1 ti−1

≤C∆. (16)
Based on the point of view above, we estimate M2 ,
n
"Z #
X ti Z tj
E ψj (s)ds ψj (s)ds
i=j=1,i<j ti−1 tj−1

n
( ! !)1/2
X Z ti Z tj
2 2
≤ (ti − ti−1 ) E[ψj (s)] ds (tj − tj−1 ) E[ψj (s)] ds
i=j=1,i<j ti−1 tj−1

n
( ! !)1/2
X Z ti Z tj
≤ (ti − ti−1 ) |ti − ti−1 |ds (tj − tj−1 ) |tj − tj−1 |ds
i=j=1,i<j ti−1 tj−1

Xn
≤ (|ti − ti−1 |3 |tj − tj−1 |3 )1/2 ≤ C∆. (17)
i=j=1,i<j

Hence, N2 is O(∆). Combining N1 and N2 completes the proof of (a). We next

prove (b). Let ϕi (t) := b2 (X(ti−1 ), α(ti−1 )) − b2 (X(t), α(t)) for ti−1 ≤ t ≤ ti , i =
1, 2, ..., n. Then
n Z T 2
X
2 2 2
E|Rn,T − RT | =E b (X(ti−1 ), α(ti−1 ))[ti − ti−1 ] − b (X(t), α(t))dt
i=1 0

n Z 2
X ti
2 2
=E [b (X(ti−1 ), α(ti−1 )) − b (X(s), α(s))]ds
i=1 ti−1

n Z 2
X ti
=E ϕi (s)ds
i=1 ti−1

n 2 n
"Z #
X Z ti X ti Z tj
= E ϕi (s)ds +2 E ϕi (s)ds ϕj (s)ds
i=1 ti−1 i=j=1,i<j ti−1 tj−1

=:B1 + B2 . (18)
Firstly,
n 2
X Z ti
B1 = E ϕi (s)ds
i=1 ti−1
 PARAMETER ESTIMATION IN MARKOVIAN SWITCHING DIFFUSIONS 9

n
X Z ti
≤ (ti − ti−1 ) E|ϕi (s)|2 ds
i=1 ti−1
n
X Z ti
= (ti − ti−1 ) E|b2 (X(ti−1 ), α(ti−1 )) − b2 (X(s), α(s))|2 ds
i=1 ti−1
n
X Z ti
= (ti − ti−1 ) E|(b(X(ti−1 ), α(ti−1 )) − b(X(s), α(s)))2
i=1 ti−1

· (b(X(ti−1 ), α(ti−1 )) + b(X(s), α(s)))2 |ds

n
X
≤C |ti − ti−1 |3 ≤ C∆2 . (by (10) and the boundedness of the second term)
i=1
(19)
Now proceeding similar to the estimation of M2 it is easy to see that
B2 ≤ C∆2 .
Combining B1 and B2 , (b) follows.
Theorem 3.2. For some constant C > 0 and for all θ ∈ Θ such that under the
assumptions (7) and (8), we have
|θn,T − θT | = OP (∆). (20)
Proof. Note that
Qn,T QT
θn,T − θT =
− .
Rn,T RT
From Lemma 2.2 it follows that |Qn,T − QT | = OP (∆) and |Rn,T − RT | = OP (∆2 ).
Now the theorem follows easily from the Proposition 3.1.

4. Convergence under the non-Lipschitz conditions. For the above results,

most of the existing convergence results on numerical methods require the coeffi-
cients of SDE to be Lipschitz. However, the global Lipschitz condition is often not
met by many systems of interest. Many researchers began to reduce restrictions
on the coefficient functions, such as the conditions for the local Lipschitz condition,
one-sided Lipschitz conditions or polynomial growth conditions and so on, and have
made some valuable achievements. We observed that there are not many works for
Markovian switching model, and therefore in this section, we develop the MLE and
AMLE for SDE with Markovian switching under the non-Lipschitz conditions.
Assumption 4.1. We let b be bounded. There exist a nondecreasing, differentiable
and concave function ρ : [0, ∞) → [0, ∞) satisfying
Z
dr
=∞ (21)
0+ ρ(r)
such that for all l ∈ S and x, y ∈ R,
|b(x, l) − b(y, l)|2 ≤ ρ(|x − y|2 ).
Remark 4.2. Examples of functions satisfying (21) include ρ(r) = r and con-
cave and increasing functions such as ρ(r) = r log(1/r), ρ(r) = r log(log(1/r)), and
ρ(r) = r log(1/r) log(log(1/r)) for r ∈ (0, ε∗ ) with ε∗ > 0 small enough. When
ρ(r) = r, Assumption 4.1 is just the usual Lipschitz conditions.
10 YUHANG ZHEN AND FUBAO XI

Theorem 4.3. For some constant C > 0 and for all θ ∈ Θ such that under the
assumption 4.1, we have

|θn,T − θT | = OP (∆). (22)

Proof. This proof is similar to Theorem 3.2, so we just give the difference. In fact,
we only need to prove the convergence rate of N1 and N2 in equation (9). Note
that using the Hölder inequality and Itô isometry, we have,
n 2
X Z T
N1 =E b(X(ti−1 ), α(ti−1 ))[B(ti ) − B(ti−1 )] − b(X(t), α(t))dB(t)
i=1 0
2
Z T Z T
=E b(X(ti−1 ), α(ti−1 ))dB(t) − b(X(t), α(t))dB(t)
0 0
2
Z T
=E [b(X(ti−1 ), α(ti−1 )) − b(X(t), α(t))]dB(t)
0
Z T
=C E|b(X(ti−1 ), α(ti−1 )) − b(X(t), α(t))|2 dt, (23)
0

By the non-Lipschtiz condition, Lemma 2.1 and the Jensen inequality, we can get
Z T
T 2E|b(X(ti−1 ), α(t)) − b(X(t), α(t))|2 dt
0
n Z
X ti
≤CT E|ρ(|X(s) − X(ti−1 )|2 )|ds
i=1 ti−1
n Z
X ti
≤CT |ρ(E|X(s) − X(ti−1 )|2 )|ds
i=1 ti−1
n Z
X ti
≤CT |ρ(C|s − ti−1 |)|ds
i=1 ti−1
Xn Z ti
≤CT |ρ(C∆)|ds
i=1 ti−1

=Cρ(∆) → 0. (as ∆ → 0) (24)

The second part of N1 is the same as the previous (12) and (13), we can get that
by the boundedness of E|X(t)|2
n 2
X Z T
N1 =2E b(X(ti−1 ), α(ti−1 ))[B(ti ) − B(ti−1 )] − b(X(t), α(t))dB(t)
i=1 0
Z T
≤T 2E|b(X(ti−1 ), α(t)) − b(X(t), α(t))|2 dt
0
Z T
+T 2E|b(X(ti−1 ), α(ti−1 )) − b(X(ti−1 ), α(t))|2 dt
0
n Z
X ti
≤CT |ρ(E|X(ti−1 ) − X(s)|2 )|ds
i=1 ti−1
 PARAMETER ESTIMATION IN MARKOVIAN SWITCHING DIFFUSIONS 11

n Z
X ti
+ CT E[E[(1 + |X(ti−1 )|2 )|α(ti−1 )]E[I{α(ti−1 )6=α(s)} |α(ti−1 )]]ds
i=1 ti−1
n Z
X ti n Z
X ti
≤CT |ρ(|s − ti−1 |2 )|ds + CT (ti − ti−1 )ds ≤ C∆. (25)
i=1 ti−1 i=1 ti−1

To estimate N2 , let ψi (t) = b(X(t), α(t))(b(X(ti−1 ), α(ti−1 )) − b(X(t), α(t))) for

ti−1 ≤ s ≤ ti , i = 1, 2, ..., n. Then
n Z ti 2
X
E ψi (s)ds
i=1 ti−1

n 2 n
"Z #
X Z ti X ti Z tj
= E ψi (s)ds +2 E ψi (s)ds ψj (s)ds
i=1 ti−1 i=j=1,i<j ti−1 tj−1

=:M1 + M2 . (26)
2
By the linear growth condition, and using the boundedness of E|X(t)| , we can get
n Z ti 2 n Z ti
X X
E ψi (s)ds ≤ (ti − ti−1 ) E(ψi (s))2 ds
i=1 ti−1 i=1 ti−1

Xn Z ti
≤ (ti − ti−1 ) E[|b(X(s), α(s))|2 |b(X(ti−1 ), α(ti−1 )) − b(X(s), α(s))|2 ]ds
i=1 ti−1
n
X Z ti
≤ (ti − ti−1 ) 2CE(1 + |X(s)|2 )[|b(X(ti−1 ), α(s)) − b(X(s), α(s))|2
i=1 ti−1

+ |b(X(ti−1 ), α(ti−1 )) − b(X(ti−1 ), α(s))|2 I{α(ti−1 )6=α(s)} |α(ti−1 )]ds

Xn Z ti 
≤ (ti − ti−1 ) 2CE(1 + |X(s)|2 ) ρ(|X(ti−1 ) − X(s)|2 )
i=1 ti−1

2 2
+ 2(|b(X(ti−1 ), α(ti−1 ))| + |b(X(ti−1 ), α(s))| )I{α(ti−1 )6=α(s)} |α(ti−1 ) ds
n
X Z ti
≤C (ti − ti−1 ) {E(1 + |X(s)|4 )}1/2 {Eρ(|s − ti−1 |4 )}1/2
i=1 ti−1

+ C(1 + |X(s)| )(1 + |X(ti−1 )|2 )I{α(ti−1 )6=α(s)} ds

2

n
(Z )
X ti Z ti
≤C (ti − ti−1 ) ρ(|s − ti−1 |)ds + |s − ti−1 |ds (by (13))
i=1 ti−1 ti−1

≤C∆. (27)
Similar to (28), we estimate M2 ,
n
"Z #
X ti Z tj
E ψi (s)ds ψi (s)ds
i=j=1,i<j ti−1 tj−1

X n
≤C ∆ρ(∆) → 0. (28)
i=j=1,i<j

Hence, N2 is O(∆). Combining N1 and N2 completes the proof of (a). We next

prove (b). Let ϕi (t) := b2 (X(ti−1 ), α(ti−1 )) − b2 (X(t), α(t)) for ti−1 ≤ t ≤ ti , i =
12 YUHANG ZHEN AND FUBAO XI

1, 2, ..., n. Then
E|Rn,T − RT |2
n 2
X Z T
=E b2 (X(ti−1 ), α(ti−1 ))[ti − ti−1 ] − b2 (X(t), α(t))dt
i=1 0

n Z ti 2
X
=E [b2 (X(ti−1 ), α(ti−1 )) − b2 (X(s), α(s))]ds
i=1 ti−1

n Z ti 2
X
=E ϕi (s)ds
i=1 ti−1

n 2 n
"Z #
X Z ti X ti Z tj
= E ϕi (s)ds +2 E ϕi (s)ds ϕj (s)ds
i=1 ti−1 i=j=1,i<j ti−1 tj−1

=:B1 + B2 . (29)
Firstly,
n 2 n
X Z ti X Z ti
B1 = E ϕi (s)ds ≤ (ti − ti−1 ) E|ϕi (s)|2 ds
i=1 ti−1 i=1 ti−1
n
X Z ti
= (ti − ti−1 ) E|b2 (X(ti−1 ), α(ti−1 )) − b2 (X(s), α(s))|2 ds
i=1 ti−1
n
X Z ti
= (ti − ti−1 ) E|(b(X(ti−1 ), α(ti−1 )) − b(X(s), α(s)))2
i=1 ti−1

· (b(X(ti−1 ), α(ti−1 )) + b(X(s), α(s)))2 |ds

n
X
≤C (ti − ti−1 )3 ≤ C∆2 . (by (10) and the boundedness of the second term)
i=1
(30)
Now proceeding similar to the estimation of M2 , it is easy to see that
B2 ≤ C∆2 .
Combining B1 and B2 , (b) follows.

5. Conclusion. In this paper an attempt has been made to provide the founda-
tions of a general result to maximum likelihood estimation for Markovian switching
processes. The result has also helped to produce some new results for maximum
likelihood estimation in diffusions. It is also apparent that there remain areas in
which extensions and consolidation of the ideas presented here can be made.
We do some comparisons of our results with the case of other models. Result of
previous work studied asymptotic theory for the MLE of diffusion processes, however
we consider it of Markovian switching diffusion processes, which greatly enhances
the applicability of our results to many practical models. This result means that for
continuous sampling, the error bound is of order OP (∆) for approximation of the
maximum likelihood estimate of θ. The significance of this result is that it provides
a theoretical basis for approximate computations. The scheme of the proofs for
 PARAMETER ESTIMATION IN MARKOVIAN SWITCHING DIFFUSIONS 13

theorems is classical. Nevertheless, in the proofs is that we have to deal with

the new terms coming from the jump part of the Markovian switching processes.
Therefore, it is easier and less time consuming to make observations in a specific
model.

Acknowledgments. The authors are very deeply grateful to the referees and edi-
tors for their helpful suggestions, which improve the quality of this paper.
REFERENCES
[1] B. Bellach, Consistency asymptotic normality and asymptotic efficiency of the maximum
likelihood estimator in linear stochastic differential equations, Statist., 11 (1980), 227-266.
[2] A. Bose(Basu), Asymptotic theory of estimation in non-linear stochastic differential equations
for the multiparameter case, Thirteenth Conference on Stochastic Processes, 45 (1983), 56-65.
[3] M. Barczy and G. Pap, Asymptotic behavior of maximum likelihood estimator for timeinho-
mogeneous diffusion processes, J. Stat. Plan. Infer., 140 (2010), 1576-1593.
[4] Jaya P. N. Bishwal, Parameter Estimation in Stochastic Differential Equations, Springer
Berlin Heidelberh New York, 1923.
[5] P. D. Feigin, Maximum likelihood estimation for continuous time stochastic processes, J.
Appl. Probab., 8 (1976), 712-736.
[6] P. D. Feigin and A. N. Shiryayev, Statistices of Random Processes I, General Theory Springer-
Verlag, Berlin, 1977.
[7] Y. A. Kutoyantsx, Estimation of a parameter of a diffusion process, Theor. Probab. Appl., 23
(1978), 641-649.
[8] Y. A. Kutoyantsx, Parameter Estimation for Stochastic Processes, Research and Expositions
in Mathematics, 1984.
[9] X. R. Mao, Asymptotic stability for stochastic differential equations with Markovian switch-
ing, Wseas Trans. Circuits and Syst., 1 (2002), 68-73.
[10] X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markov Switching, Imperial
College Press, London, 2006.
[11] M. N. Mishra and P. Rao, Asymptotic study of maximum likelihood estmation for nonhomo-
geneous diffusion processes, Statist., 3 (1985), 193-203.
[12] P. Rao, On Bayes estimation for diffusion fields, applications and new directions, Proc. ISI
Golden Jubilee Conferences,(1981), 504-511.
[13] P. Rao, Asymptotic theory for non-linear least squares estimator for diffusion processes, Ser.
Statist., 14 (1983), 195-209.
[14] L. Shen, On the detectability and observability of continuous stochastic Markov jump linear
systems, J. Math. Anal. Appl., 424 (2015), 878-891.
[15] A. Stoica and I. Yaesh, Jump Markovian based control of wing deployment for an uncrewed
air vehicle, J. Guid. Control. Dyanm., 25 (2002), 407-411.
[16] C. G. Yuan and X. R. Mao, Convergence of the Euler-Maruyama method for stochastic
differential equations wih Markovian switching, Math. Comput. Simulat., 64 (2004), 223-235.
[17] Q. Zhang, Nonlinear filtering and control of a switching diffusion with small observation noise,
J. Control Optim., 36 (1998), 1738-1768.
[18] Q. Zhang, Stock trading: An optimal selling rule, J. Control Optim., 40 (2001), 64-87.
[19] Q. Zhang and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Analysis:
Theory, Methods and Applications, 71 (2009), 1370-1379.
[20] Y. H. Zhen, Approximation of the likelihood function for stochastic differential equations with
Markovian switching, Markov Processes Relat. Fields, 28 (2022), 527-546.
[21] Y. H. Zhen and F. B. Xi, Least squares estimators for stochastic differential equations with
Markovian switching, Discrete. Cont. Dyn-B , 28 (2023), 4068-4086.

Received December 2023; 1st revision February 2024; final revised February 2024;
early access March 2024.