Geodesic
A method of second order accuracy integration of stochastic differential equations
Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 414-419

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For the stochastic differential equation $$ dX=a(t,X)\,dt+\sigma(t,X)\,dw,\qquad X(t_0)=x,\ t_0\le t\le t_0+T, $$ the problem of approximate calculation of the expectation $\mathbf Mf(X_{t_0,x}(t_0+T))$ is considered. Rather a simple method is proposed for recursive modeling of random variables $$ \overline X_{t_0,x}(t_k);\quad k=0,1,\dots;\quad t_k=t_0+kh;\quad h=\frac{T}{m}; $$ such that $$ \mathbf Mf(X_{t_0,x}(t_0+T))=\mathbf Mf(\overline X_{t_0,x}(t_0+T))+O(h^2). $$ 
@article{TVP_1978_23_2_a17,
     author = {G. N. Mil'\v{s}teǐn},
     title = {A method of second order accuracy integration of stochastic differential equations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {414--419},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1978_23_2_a17/}
}
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G. N. Mil'šteǐn. A method of second order accuracy integration of stochastic differential equations. Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 414-419. http://geodesic.mathdoc.fr/item/TVP_1978_23_2_a17/