Re: Runge Kunta on SDE

On Thu, 16 Feb 2006, David Langevin wrote:

Dear readers,

I want to apply the Runge Kunta fourth order scheme on the following SDE:

dX = aXdt + bXdW

where a,b are real, dW = sqrt(dt)N(0,1) is a wiener process.

When b=0 in my program all is fine, the solution is the same as in the deterministic case (Hopefully !), the problem I have is in the stochastic part, the scheme doesn't converge to the true solution:

X = X0exp(((a-b^2/2)t+bW)

where X0 is real.

The Runge-Kutta approximation converges to the solution in Stratonovich
sense. If you first convert the Ito SDE to the equivalent Stratonovich
SDE and then apply the Runge-Kutta scheme, it should converge to the right
solution. Try with the following:

dX = a X dt - 1/2 b^2 X dt + b X dW

--
- Simo

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