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![ljwang@ucas.ac.cn](http://html.rhhz.net/ZGKXYDXXB/images/REemail.gif)
中国科学院大学数学科学学院, 北京 100049
摘要: 通过将It?型随机微分方程转换为等价的Stratonovich型和向后It?型随机微分方程,构造出分别与Stratonovich型和向后It?型随机微分方程相容的两种全隐式随机数值格式.这两种数值格式应用于带一个噪声的随机微分方程,且均为均方1阶收敛的.
关键词: 随机微分方程全隐式数值格式It?型随机微分方程Stratonovich型随机微分方程
Consider the numerical approximations for stochastic differential equations (SDEs) in the It? sense as follows
$\left\{ \begin{align} & dX=a\left( t,X \right)dt+b\left( t,X \right)dW\left( t \right), \\ & X\left( {{t}_{0}} \right)={{X}_{0}}, \\ \end{align} \right.$ | (1) |
As is known in Refs.[3-6],there are many kinds of numerical schemes for the SDE (1). However,the construction of these approximations are usually based on the SDEs in the It? sense. Hence,it is meaningful to investigate whether we could construct the methods by using their equivalent SDEs in the sense of Stratonovich and Backward-It?.
1 Full implicit schemes1.1 Equivalentequations in the sense of Stratonovich and Backward-It?
As is known in Refs.[1-2, 5-6],the equivalent equation with respect to the SDE (1) in the sense of Stratonovich is as follows:
$dX=\left( \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( t,X \right) \right)dt+b\left( t,X \right){}^\circ dW\left( t \right)$ | (2) |
Based on the relationship between the It? integral and the Stratonovich integral,we can obtain the SDE (2). Analogously,we can define another stochastic integral,Backward-It? integral,and get the equivalent equation of SDE (1) in the sense of Backward-It?. Let (Ω,F) be a measure space with the probability P and Ft be the σ-algebra generated by the n-dimensional Brownian motions Bi(s)(1≤i≤n,0≤s≤t). We denote by V(S,T) the class of functions
f(t,ω):[0,∞)×Ω→R,
such that
1) (t,ω)→f(t,ω) is B×F-measurable,where B denotes the Borel σ-algebra on [0,∞);
2) f(t,ω) is Ft-adapted and
Furthermore,let L2(P) be a Hilbert space with the following inner product
(X,Y)L2(P):=E[X·Y];X,Y∈L2(P).
Definition 1.1 Suppose that f∈V(0,T) and t→f(t,ω) is continuous for a.e.ω. Then the Backward-It? integral of f is defined by
$\int\limits_{0}^{T}{f\left( t,\omega \right)*dW\left( t \right)}=\underset{h\to 0}{\mathop{\lim }}\,\sum\limits_{j}{f}\left( {{t}_{j+1}},\omega \right)\Delta {{W}_{j}}$ |
Note that this kind of integral has been introduced in Ref.[3]. For convenience,we give it the name Backward-It? integral. According to the definition of the Backward-It? integral,we can easily get the relationship between the It? integral and the Backward-It? integral as follows:
$\int\limits_{0}^{T}{f\left( t,\omega \right)*dW\left( t \right)}=\int\limits_{0}^{T}{f\left( t \right)dW\left( t \right)}+\int\limits_{0}^{T}{\frac{\partial f\left( t \right)}{\partial W}}dt,$ | (3) |
$dX=\left( \left( a-\frac{\partial b}{\partial x}b \right)\left( t,X \right) \right)dt+b\left( t,X \right)*dW\left( t \right),$ | (4) |
1.2 The convergence theorem on mean-square methods from Ref.[4]Theorem 1.1 See Ref.[4]. Suppose that the one-step approximation X(t+h;t,x) has the order of accuracy p1 for the mathematical expectation of the deviation and order of accuracy p2 for the mean-square deviation. More precisely,for arbitrary t0≤t≤t0+T-h,x∈Rn the following inequalities hold:
$\begin{align} & \left| E\left( \left( X-\bar{X} \right)\left( t+h;t,x \right) \right) \right|\le K{{\left( 1+{{\left| x \right|}^{2}} \right)}^{\frac{1}{2}}}{{h}^{{{p}_{1}}}} \\ & {{\left[ E{{\left| \left( X-\bar{X} \right)\left( t+h;t,x \right) \right|}^{2}} \right]}^{\frac{1}{2}}}\le K{{\left( 1+{{\left| x \right|}^{2}} \right)}^{\frac{1}{2}}}{{h}^{{{p}_{2}}}} \\ \end{align}$ | (5) |
p2≥12,p1≥p2+12.
Then for any N and k=0,…,N the following inequality holds:
$\begin{align} & {{\left[ E{{\left| \left( X-\bar{X} \right)\left( {{t}_{k}};{{t}_{0}},{{X}_{0}} \right) \right|}^{2}} \right]}^{\frac{1}{2}}} \\ & \le K{{\left( 1+{{\left| {{X}_{0}} \right|}^{2}} \right)}^{\frac{1}{2}}}{{h}^{{{p}_{2}}-\frac{1}{2}}}, \\ \end{align}$ | (6) |
Theorem 1.2 See Refs.[5-6]. Let the one-step approximation X(t+h;t,x) satisfy the condition of Theorem 2.1. Suppose that
$\begin{align} & \left| E \right.\left( \bar{X}\left( t+h;t,x \right)-\tilde{X}\left( t+h;t,x \right) \right)=O\left( {{h}^{{{p}_{1}}}} \right), \\ & {{\left[ E{{\left| \bar{X}\left( t+h;t,x \right)-\tilde{X}\left( t+h;t,x \right) \right|}^{2}} \right]}^{\frac{1}{2}}}=O\left( {{h}^{{{p}_{2}}}} \right), \\ \end{align}$ | (7) |
Besides,the authors of Refs.[4-6] have mentioned that the increments of Wiener processes should be substituted by truncated random variables in implicit schemes. In detail,
${{\zeta }_{h}}=\left\{ \begin{align} & \xi ,if\left| \xi \right|\le {{A}_{h}}, \\ & {{A}_{h}},if\xi >{{A}_{h}}, \\ & -{{A}_{h}},if\xi <{{A}_{h}}, \\ \end{align} \right.$ | (8) |
1.3 Fullimplicit schemes
First,we consider the SDE (2). As is known,the exact solution is
$\begin{align} & X\left( t+h;t,x \right)=x+\int\limits_{t}^{t+h}{b\left( s,X\left( s \right) \right)}\circ dW\left( s \right)+ \\ & \int\limits_{t}^{t+h}{\left[ \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( s,X\left( s \right) \right) \right]}ds, \\ \end{align}$ | (9) |
$\begin{align} & {{X}_{k+1}}={{X}_{k}}+b\left( {{t}_{k}}+\frac{h}{2},\frac{{{X}_{k}}+{{X}_{k+1}}}{2} \right){{\zeta }_{h}}\sqrt{h}+ \\ & \left[ \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( {{t}_{k}}+\frac{h}{2},\frac{{{X}_{k}}+{{X}_{k+1}}}{2} \right) \right]h. \\ \end{align}$ | (10) |
$\begin{align} & X\left( t+h;t,x \right)=x+\int\limits_{t}^{t+h}{b\left( s,X\left( s \right) \right)}*dW\left( s \right)+ \\ & \int\limits_{t}^{t+h}{\left[ \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( s,X\left( s \right) \right) \right]}ds, \\ \end{align}$ | (11) |
${{X}_{k+1}}={{X}_{k}}+b\left( {{t}_{k}}+h,{{X}_{k+1}} \right){{\zeta }_{h}}\sqrt{h}+\left[ \left( a-\frac{\partial b}{\partial x}b \right)\left( {{t}_{k}}+h,{{X}_{k+1}} \right) \right]h,$ |
$\begin{align} & {{X}_{k+1}}={{X}_{k}}+\left[ \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( {{t}_{k+1}},{{X}_{k+1}} \right) \right]h+ \\ & \left[ \frac{1}{2}\frac{\partial b}{\partial x}b\left( {{t}_{k+1}},{{X}_{k+1}} \right) \right]\zeta _{h}^{2}h+ \\ & b\left( {{t}_{k}},{{X}_{k}} \right){{\zeta }_{h}}\sqrt{h}, \\ \end{align}$ | (12) |
2 Mean-square order of the full implicit schemesTheorem 2.1 The numerical method (10) with
Proof According to an analog of Taylor expansion of the solution (9),the one-step approximation X(t+h;t,x) as follows
$\begin{align} & \bar{X}=x+\left[ \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( t+h,x \right) \right]h+ \\ & b\left( t,x \right)\Delta W\left( h \right)+\frac{1}{2}\frac{\partial b}{\partial x}b\left( t,x \right){{\left( \Delta W\left( h \right) \right)}^{2}}. \\ \end{align}$ | (13) |
$\begin{align} & \tilde{X}=x+\left[ \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( t+\frac{h}{2},\frac{x+\tilde{X}}{2} \right) \right]h+ \\ & b\left( t+\frac{h}{2},\frac{x+\tilde{X}}{2} \right){{\zeta }_{h}}\sqrt{h}. \\ \end{align}$ | (14) |
$\begin{align} & \tilde{X}=x+\left[ \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( t+\frac{h}{2},x \right) \right]h+ \\ & b\left( t,x \right){{\zeta }_{h}}\sqrt{h}+\frac{1}{2}\frac{\partial b}{\partial x}\left( t,x \right)b\left( t,x \right)\zeta _{h}^{2}{{\rho }_{1}}. \\ \end{align}$ | (15) |
$\begin{align} & \bar{X}-\tilde{X}=b\left( t,x \right)\left( \Delta W\left( h \right)-{{\zeta }_{h}}\sqrt{h} \right)+ \\ & \frac{1}{2}\frac{\partial b}{\partial x}\left( t,x \right)b\left( t,x \right)\left( {{\xi }^{2-}}\zeta _{h}^{2} \right)h-{{\rho }_{1}} \\ \end{align}$ | (16) |
$\begin{align} & E{{\left( \bar{X}-\tilde{X} \right)}^{2}}\le KE{{\left( \xi -{{\zeta }_{h}} \right)}^{2}}+ \\ & K{{h}^{2}}E{{\left( {{\xi }^{2}}-\zeta _{h}^{2} \right)}^{2}}+O\left( {{h}^{3}} \right), \\ \end{align}$ | (17) |
Theorem 2.2 The numerical method (12) with
Proof Considering the numerical method (12),we expand the right side of the solution (11) and obtain
$\begin{align} & X\left( t+h;t,x \right) \\ & =x+\int\limits_{t}^{t+h}{b\left( s,X\left( s \right) \right)}*dW\left( s \right)+ \\ & \int\limits_{t}^{t+h}{\left[ \left( a-\frac{\partial b}{\partial x}b \right)\left( s,X\left( s \right) \right) \right]}ds, \\ & =x+\left[ \left( a-\frac{\partial b}{\partial x}b \right)\left( t,x \right) \right]h+b\left( t,x \right)\Delta W\left( h \right)+ \\ & \int\limits_{t}^{t+h}{\int_{t}^{s}{Lb}\left( \theta ,X\left( \theta \right) \right)}d\theta *dW\left( s \right)+ \\ & \int\limits_{t}^{t+h}{\int_{t}^{s}{\frac{\partial b}{\partial x}}b\left( \theta ,X\left( \theta \right) \right)}dW\theta *dW\left( s \right)+{{\rho }^{1}} \\ & =x+\left[ \left( a-\frac{\partial b}{\partial x}b \right)\left( t,x \right) \right]h+b\left( t,x \right)\Delta W\left( h \right)+ \\ & \int\limits_{t}^{t+h}{\int_{t}^{s}{\frac{\partial b}{\partial x}}b\left( t,x \right)}dW\theta *dW\left( s \right)+{{\rho }^{2}}, \\ \end{align}$ | (18) |
$\begin{align} & X\left( t+h;t,x \right) \\ & =x+\left[ \left( a-\frac{\partial b}{\partial x}b \right)\left( t,x \right) \right]h+ \\ & \frac{1}{2}\frac{\partial b}{\partial x}b\left( t,x \right){{\left( \Delta W\left( h \right) \right)}^{2}}+\frac{1}{2}\frac{\partial b}{\partial x}b\left( t,x \right)h+ \\ & b\left( t,x \right)\Delta W\left( h \right)+{{\rho }^{3}} \\ & =x+\left[ \left( a-\frac{1}{2}\frac{\partial b}{\partial x}b \right)\left( t+h,X \right) \right]h+ \\ & \frac{1}{2}\frac{\partial b}{\partial x}b\left( t+h,X \right){{\left( \Delta W\left( h \right) \right)}^{2}}+ \\ & b\left( t,x \right)\Delta W\left( h \right){{\rho }^{4}} \\ \end{align}$ | (19) |
3 Numerical testExample Consider the stochastic differential equation
$dX=tX\left( t \right)dW\left( t \right),X\left( 0 \right)={{X}_{0}}$ | (20) |
$X\left( t \right)={{X}_{0}}\exp \left( \frac{1}{2}\int\limits_{0}^{t}{{{s}^{2}}}ds+\int\limits_{0}^{t}{sdW\left( s \right)} \right).$ |
${{X}_{n+1}}={{X}_{n}}+{{t}_{n}}{{X}_{n}}\Delta {{W}_{n}},$ | (21) |
$\begin{align} & {{X}_{n+1}}={{X}_{n}}-\frac{h}{4}\left( {{t}_{n}}+\frac{h}{2} \right)\left( {{X}_{n}}+{{X}_{n+1}} \right)+ \\ & \frac{1}{2}\left( {{t}_{n}}+\frac{h}{2} \right)\left( {{X}_{n}}+{{X}_{n+1}} \right){{\zeta }_{h}}\sqrt{h} \\ \end{align}$ | (22) |
$\begin{align} & {{X}_{n+1}}={{X}_{n}}-\frac{h}{2}{{\left( {{t}_{n}}+h \right)}^{2}}{{X}_{n+1}}+ \\ & {{t}_{n}}{{X}_{n}}{{\zeta }_{h}}\sqrt{h}+\frac{1}{2}{{\left( {{t}_{n}}+h \right)}^{2}}{{X}_{n+1}}\zeta _{h}^{2}, \\ \end{align}$ | (23) |
The numerical tests examine the behaviors of the numerical methods in two aspects: the sample trajectories produced by the numerical methods and the true solution shown in Fig. 1(d) and the convergence rates of the numerical methods shown in Fig. 1(a),1(b),and 1(c).
Fig. 1
![]() | Download: JPG larger image |
Fig. 1 Numerical test of the example |
Figures 1(a),1(b),and 1(c) show that,comparing with the reference line,the Euler method is of mean-square order 0.5 and the numerical methods (22) and (23) are of the first mean square order. In our experiments we take t=1,X0=10, and h=[0.01,0.02,0.025,0.05,0.1].
Figure 1(d) presents that the numerical approximations (22) and (23) are much closer to the exact solution than the Euler method for X0=2 and h=0.04 within the time interval 0≤t≤1.2.
References
[1] | Mao X R. Stochastic differential equations and applications[M].2nd ed. Horwood Publishing Limited, 2007. |
[2] | ?ksendal B. Stochastic differential equations[M].6th ed. Springer, 2003. |
[3] | Kloeden P E, Platen E. Numerical solution of stochastic differential equations[M].Berlin Heidelberg: Springer-Verlag, 1992. |
[4] | Milstein G N, Tretyakov M V. Stochastic numerics for mathematical physics[M].Berlin Heidelberg: Springer-Verlag, 2004. |
[5] | Milstein G N, Repin Y M, Tretyakov M V. Symplectic integration of Hamiltonian systems with additive noise[J].SIAM J Numer Anal, 2002, 39:2066–2088.DOI:10.1137/S0036142901387440 |
[6] | Milstein G N, Repin Y M, Tretyakov M V. Numerical methods for stochastic systems preserving symplectic structure[J].SIAM J Numer Anal, 2002, 40:1583–1604.DOI:10.1137/S0036142901395588 |
[7] | Deng J, Anton C A, Wong Y S. High-order symplectic schemes for stochastic Hamiltonian systems[J].Commun Comput Phys, 2014, 16(1):169–200.DOI:10.4208/cicp.311012.191113a |