Published by De Gruyter May 24, 2019

A third-order weak approximation of multidimensional Itô stochastic differential equations

Riu Naito and Toshihiro Yamada

From the journal Monte Carlo Methods and Applications

https://doi.org/10.1515/mcma-2019-2036

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Abstract

This paper proposes a new third-order discretization algorithm for multidimensional Itô stochastic differential equations driven by Brownian motions. The scheme is constructed by the Euler–Maruyama scheme with a stochastic weight given by polynomials of Brownian motions, which is simply implemented by a Monte Carlo method. The method of Watanabe distributions on Wiener space is effectively applied in the computation of the polynomial weight of Brownian motions. Numerical examples are shown to confirm the accuracy of the scheme.

Keywords: Stochastic differential equation; weak approximation; third-order method

MSC 2010: 60H35; 65C05; 65C30; 91G60

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 16K13773

Funding statement: This work is supported by JSPS KAKENHI (grant number 16K13773) from MEXT, Japan and a research fund from Tokio Marine Kagami Memorial Foundation.

A Appendix

A.1 Proof of Lemma 5.1

Based on the expansion of the heat kernel p⁢(t,x,y)=E⁢[δy⁢(Xtx)] in [11, formula (A.3)], we have
E⁢[f⁢(Xtx)]=E⁢[f⁢(X¯tx)]+∑j=1ν∑k=1j∑(βi)i=1k⊂ℕ∖{1},s.t.⁢∑i=1kβi=j+k∑(i1,…,ik)∈{1,…,N}k1k!×E[f(X¯tx)H(i1,…,ik)(X¯tx,∏l=1k∑α=(α1,…,αr)∈{0,1,…,d}r,r∈ℕ⁢s.t.⁢∥α∥=βlLα1⋯Lαr-1Vαril(x)×∫0<t1<⋯<tr<tdWt1α1⋯dWtrαr)]+r(t,x),
where r⁢(t,x) is the residual satisfying
supx∈ℝN⁡|r⁢(t,x)|≤C⁢∥f∥∞⁢t(ν+1)/2.
Take ν=7 to get the assertion. ∎

A.2 Proof of Lemma 5.2

First we note that
E⁢[f⁢(X¯tx)⁢H(i1,…,ik)⁢(X¯tx,I(α1,…,αr)⁢(t))]=E⁢[∂i1⁡⋯⁢∂ik⁡f⁢(X¯tx)⁢∫0<t1<⋯<tr<td⁢Wt1α1⁢⋯⁢d⁢Wtrαr].
If αr≠0, we can see that
E⁢[∂i1⁡⋯⁢∂ik⁡f⁢(X¯tx)⁢∫0<t1<⋯<tr<td⁢Wt1α1⁢⋯⁢d⁢Wtrαr]=E⁢[∫0tDαr,tr⁢∂i1⁡⋯⁢∂ik⁡f⁢(X¯tx)⁢∫0<t1<⋯<tr-1<trd⁢Wt1α1⁢⋯⁢d⁢Wtr-1αr-1⁢d⁢tr]=E⁢[∫0t∑ik+1=1N∂i1⁡⋯⁢∂ik⁡∂ik+1⁡f⁢(X¯tx)⁢Dαr,tr⁢X¯tx,ik+1⁢∫0<t1<⋯<tr-1<trd⁢Wt1α1⁢⋯⁢d⁢Wtr-1αr-1⁢d⁢tr]=∑ik+1=1NVαrik+1⁢(x)⁢∫0tE⁢[∂i1⁡⋯⁢∂ik⁡∂ik+1⁡f⁢(X¯tx)⁢∫0<t1<⋯<tr-1<trd⁢Wt1α1⁢⋯⁢d⁢Wtr-1αr-1]⁢d⁢tr.
Also, if αr=0, we have
E⁢[∂i1⁡⋯⁢∂ik⁡f⁢(X¯tx)⁢∫0<t1<⋯<tr<td⁢Wt1α1⁢⋯⁢d⁢Wtrαr]=∫0tE⁢[∂i1⁡⋯⁢∂ik⁡f⁢(X¯tx)⁢∫0<t1<⋯<tr-1<trd⁢Wt1α1⁢⋯⁢d⁢Wtr-1αr-1]⁢d⁢tr.
Iterating this procedure, if r≥4, there exist an integer L=L⁢(k,m)≥1, multi-indices αi, i=1,…,L, and bounded functions Ci:(0,T]×ℝN such that
E⁢[f⁢(X¯tx)⁢H(i1,…,ik)⁢(X¯tx,I(α1,…,αr)⁢(t))]=∑ik+1,…,ik+m=1NE⁢[∂i1⁡⋯⁢∂ik+m⁡f⁢(X¯tx)⁢∏i=1mVανiik+i⁢(x)⁢∫0<t1<⋯<tr<td⁢t1⁢⋯⁢d⁢tr]=t4⁢∑i=1LE⁢[∂αi⁡f⁢(X¯tx)]⁢Ci⁢(t,x).∎

A.3 Proof of Lemma 5.7

Let Jl∈{0,1,…,d}nl, nl≥2, l=1,…,k∈ℕ with ∑l=1knl>6. By applying the Itô formula, we obviously have
∏l=1kIJl⁢(t)=∑αe∈{0,1,…,d}me,e≤LIαe⁢(t)
for some L∈ℕ and multi-indices αe, me∈ℕ, e=1,…,L, which are determined by Jl, l=1,…,k. To show the assertion, it suffices to show that the minimum of me, e=1,…,L. is 4. Let us define
(J1,…,Jk)↦M⁢(J1,…,Jk):-min⁡{|αe|=me;∏l=1kIJl⁢(t)=∑αe∈{0,1,…,d}me,e≤LIαe⁢(t)}.
If |Jl|≥4 for some l, it is obvious that the minimal length M⁢(J1,…,Jk) will be 4 or larger than 4. Then we only need to evaluate the cases (|J1|,|J2|,|J3|)=(2,2,3) and (|J1|,|J2|,|J3|,|J4|)=(2,2,2,2) because if
M⁢(J1,…,J3)≥4when⁢(|J1|,|J2|,|J3|)=(2,2,3),
M⁢(J1,…,J4)≥4when⁢(|J1|,|J2|,|J3|,|J4|)=(2,2,2,2),
then the minimum length M⁢(J1,…,Jk) of any (|J1|,…,|Jk|) such that |Jl|≥2 must be larger than 4.
The case (|J1|,|J2|,|J3|)=(2,2,3). We already see that the minimal length M⁢(J1,J2) when (|J1|,|J2|)=(2,2) is 2, which comes from the term I(0,0) in the right-hand side in Lemma 5.3 with J1=((i11,i21)), J2=((i12,i22)). Further, the minimal length M⁢((0,0),J3) with J3=(i13,i23), i.e., the minimal length of the case I(0,0)⁢I(α1,α2,α3), is 4, which is checked by Lemma 5.4.
The case (|J1|,|J2|,|J3|,|J4|)=(2,2,2,2). We already see that the minimal length M⁢(J1,J2) is 2 when (|J1|,|J2|)=(2,2), which comes from the term I(0,0) of Lemma 5.5. Moreover, M⁢((0,0),(0,0)), i.e., the minimal length of the case I(0,0)⁢I0,0, is 4, which is obtained by I(0,0)⁢(t)⁢I(0,0)⁢(t)=6⁢I(0,0,0,0)⁢(t).
Therefore, we obtain the result. ∎

A.4 Proof of Lemma 5.10

Under the uniformly elliptic condition, the Euler–Maruyama process {X¯t(T/n),x}t∈[0,T] can be regarded as an elliptic Itô process
X¯t(T/n),x=x+∫0tV0⁢(X¯ϕ⁢(s)(T/n),x)⁢d⁢s+∑i=1d∫0tVi⁢(X¯ϕ⁢(s)(T/n),x)⁢d⁢Wsi,t≥0,
where ϕ(s)=sup{kT/n;kT/n≤s}. Then, by combining the method of Watanabe distributions and the integration by parts for elliptic Itô processes of [5], we obtain
〈∂α⁡f⁢(X¯k⁢T/n(T/n),x),G〉=〈f⁢(X¯k⁢T/n(T/n),x),Hα⁢(X¯k⁢T/n(T/n),x,G)〉,f∈𝒮′⁢(ℝN),for⁢k=1,…,n.
When f is a bounded Borel function, we have the following result using the estimate of the Malliavin weight:
|E⁢[f⁢(X¯k⁢T/n(T/n),x)⁢Hα⁢(X¯k⁢T/n(T/n),x,G)]|≤C⁢∥f∥∞⁢∥(det⁡σX¯k⁢T/n(T/n),x)-1∥qγ1⁢∥D⁢X¯k⁢T/n(T/n),x∥l,mγ2⁢∥G∥s,p
for some constants C, γ1, γ2, l,m, p,q>0. By [5], we have ∥D⁢X¯k⁢T/n(T/n),x∥l,m≤C⁢(T) for some non-decreasing function C⁢(⋅)>0. Then we obtain the assertion. ∎

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Received: 2019-02-25

Revised: 2019-04-11

Accepted: 2019-04-19

Published Online: 2019-05-24

Published in Print: 2019-06-01