Approximate integration of stochastic differential equations

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.