Approximate integration of stochastic differential equations

G. N. Mil'shtein

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 3, pp. 583-588

Résumé

For the Ito equation $$ dX=a(t,X)\,dt+\sigma(t,X)\,dw,\quad X(t_0)=x,\quad t_0\le t\le t_0+T $$ ($w(t)$ is a standard Wiener process) the following approximation is proposed: \begin{gather*} \overline X(t_0)=X(t_0),\quad\overline X(t_0+(k+1)h)= \\ =\overline X(t_0+kh)+\overline\sigma w_{k+1}+\biggl(\overline a-\frac12\overline\sigma\frac{\overline\partial\sigma}{\partial x}\biggr)h+\frac12\overline\sigma\frac{\overline\partial\sigma}{\partial x}w_{k+1}^2 \end{gather*} where $h=T/m$; $k=0,1,\dots,m-1$; $w_1,\dots,w_m$ are independent normal $N(0,h)$ variables. Here the stroke means that the corresponding function is computed at point $(t_0+kh,X(t_0+kh))$. It is shown that $\mathbf M(X(t_0+T)-\overline X(t_0+T))^2=O(h^2)$. The results are generalized to systems of stochastic differential equations. Possibilities of improving the accuracy of the approximation are discussed.