Dear Arif Ullah,

If v1 and v2 are the white noises, the stochastic equation is not written quite correctly. The equation should be written in the following form

dX= a(X) dt + b(X) v1(t)dt + c(X)v2(t)dt.

The white noises are modeled as

v1(ti)=z1i*Sqrt(N1/dt);

v2(ti)=z2i*Sqrt(N2/dt),

where z1i and z2i are the values of correlated random variables v1 and v2 in the i-th point in time.

Dear Sergiy Prykhodko, Thanks for your answer. in my equation, v1 and v2 are two correlated random variables generated through circulant embedding method with the desired correlation matrix. Now i know how to use Runge Kutta method for Stochastic equation with white noise like dX=a dt +b dW dt, here dW/dt is white noise.

But here in my case i have three terms . second terms and third terms are stochastic terms and here noise is not white noise but we have here two colored noises v1 & v2. 

Dear Arif Ullah,

dW is not white noise, dW/dt is white noise.

If the noise is not white noise, then the noise (color noise) is modeled by another   stochastic equation.

I don't know if this helps: If your noise is not white, then you need additional information. Assuming the simplest case, namely that your colored noise is red noise, then there is a way to write this red noise as white noise plus an additional deterministic part. (see e.g. Uhlenbeck, Ornstein "On the theory of Brownian motion" Phy. Rev. 36, 823 (1930)) Then your system should turn into something with 2 white noise terms and 3 deterministic parts on the right hand side of your equation.

Now applying 2nd Order Runge-Kutta: (I use it myself and I know it under the name of "stochastic Heun scheme" or "improved Euler") I have usually only one noise term, but since it is additive noise, you should get simply two noise terms.

The advantage of this algorithm is, that it integrates the white noise term exactly like the ordinary stochastic Euler scheme and the deterministic part is integrated like deterministic differential equations are integrated according to 2nd order Runge-Kutta.

Here is the scheme for a SDE in the Stratonovich form:

x. = A(x) + B(x) \xi

with \xi being white noise. (Your system should turn into that, just with A(x) consiting of 3 additive parts and having two noise terms of the kind B(x)\xi)

then using constant time steps dt

x_Euler = x(t) + A(x(t)) dt + B(x(t)) \sqrt(dt)

x(t+dt) = x(t) + ( A(x_Euler) + A(x(t)) ) dt/2 + ( B(x_Euler) + B(x(t)) ) \sqrt(dt)/2

(i.e. one takes the arithmetic average of the predicted instant velocity by Euler and the current velocity) just that you should get an additional noise term (i.e. C(x) \xi_2).

This should be correct in my oppinion, as long as your noise is red. If it is something more complicated, then I don't know what to do.

thank you all for response. I have figured it out , how to do this. After generating the correlated noise with Circulant embedding method, the multi dimensional Runge Kutta method (Kloeden, Platen Chapter 15) can be used. 

Dear @Daniil Yurchenko ·, i know , all high order methods for stochastic equations do not show the same order of convergence as they show for ordinary differential equations.