# Stochastic differential equations in the weak sense

There are some good discussions in http://mathematica.stackexchange.com regarding the numerical solution of SDEs of the form $dX(t)=a(t,X)dt+b(t,X)dW_t$, wherein $X(0)=X_{0},$ $W_t$ is the Wiener process, such as this one

However, there is no question and answer regarding the weak solutions. I wish to ask a question so as to extend the nice previous discussions.

How could we efficiently implement the following stochastic Runge-Kutta method (in the weak sense) $$\begin{array}{lll} X_{n+1}&=&X_n+\frac{1}{2}b\Delta W_n+\frac{1}{2}b(t_n+\Delta t_n,X_n+a\Delta t_n+b\Delta W_n)\Delta W_n\\ &&+\frac{1}{2}a\Delta t_n+\frac{1}{2}a(t_n+\Delta t_n,X_n+a\Delta t_n+b\Delta W_n)\Delta t_n-\frac{1}{2}b\frac{\partial b}{\partial X}\Delta t_n. \end{array}$$ to find the weak solution of the following nonlinear SDE $$\begin{array}{lll} dX(t)&=&\left(\frac{1}{3}X_t^\frac{1}{3}+6X_t^\frac{2}{3}\right)dt+X_t^\frac{2}{3}dW_t, \ \ 0\leq t\leq T=1, \\[1.5mm] X(0)&=&1, \end{array}$$ and compare the results with the exact solution $X_t=\left(2t+1+\frac{1}{3}W_t\right)^3$.

In fact, we wish to obtain the mean error of the estimate of the expected value $\mathbb{E}[X(T)]$ and plot it, since we are comparing weak convergence of the methods. Besides, what would be the concept of realization here?

Please consider solving this question in Mathematica 8.

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Have you tried implementing it yourself? –  sebhofer Aug 31 at 14:24
I tried to do, but I failed. However, if SRK is complicated please just obtain and compare the results with the well-known method of Euler-Maruyama, or any numerical method you think to do so. –  Fazlollah Soleymani Aug 31 at 14:30
Then you should show what you tried... –  sebhofer Aug 31 at 14:39
Is there however a way to find the weak solution? I know that Euler-Maruyama is doable. Can anyone code it for such a purpose. –  Fazlollah Soleymani Sep 3 at 9:28
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