I have a two dimensional stochastic equation like this
dX=AXdt+BXdV1+CXdV2
Where A B and C are matrices, dV1 and dV2 are correlated with each other and called color noises. My question is , if i want to use Range Kutta scheme, do i need to normalize the color noise and make it of length 1/sqrt(dt) like White noise or we can use Platen's and Kloeden's Range Kutta scheme directly for color noise case.
If g(t) is random, then you have a mixed RODE with a stochastic differential equation. Runge-Kutta methods which were not derived specifically for this will not have order of accuracy higher than 0.5 and will have awful stability regions. You should make sure you use methods specifically designed for random and stochastic situations.
Thanks for your response. can you mention some good references regarding mixed SDEs, this thing is new for me.
I'm not sure of a good reference. But it would probably take a lot of work to beat a straight Euler-Maruyama-type discretization, which would be of strong order min(0.5,alpha) where alpha is the a.s. Holder continuity limit for g(t).
Thanks @ Marc R Roussel, I have these mentioned references but i am confused about color noise.
I have a two dimensional stochastic equation like this
dX=AXdt+BXdV1+CXdV2
Where A B and C are matrices, dV1 and dV2 are correlated with each other and called color noises. My question is , if i want to use Range Kutta scheme, do i need to normalize the color noise and make it of length sqrt(dt) like White noise or we can use Platen's and Kloeden's Range Kutta scheme directly for color noise case.
I see there were some updates. Correlated noise is much easier than a random g(t) with no extra information, so now we can get more specific :).
The Honeycutt papers assume additive noise in their derivation so they don't apply. Plus the "order" they speak of is a very weak sense of order, i.e. not strong order, so it's somewhat misleading.
Kloden and Platen (with Schwartz, commonly abbreviated as KPS) cannot be used on this directly since those algorithms require white noise as part of the derivation for the Wiktorsson approximations. But I wouldn't recommend that anyways: those methods are outdated and there's much more efficient methods. Instead, you should be using the Rossler methods these days:
http://epubs.siam.org/doi/abs/10.1137/09076636X
Once again these assume white noise, so we have to get to there. Essentially what you can do is reverse-engineer correlated noise back to non-diagonal white noise. It's done like this. A correlated Wiener processes can be generated by multiplying by the matrix of SVD factors. If you have a covariance matrix Γ, then for Γ=USV^(-1) the SVD, U*sqrt(S)*sqrt(dt)*eta where eta is a vector of Gaussians gives you a vector of Gaussians with the correlation Γ. Thus defining a new process in terms of these parts from the SVD factorization, we have that
dX=AXdt + X*K*dW_t
where K = D*U*sqrt(S), with D the diagonal matrix with entries [B,C]. That process is equivalent to the one you wrote down, but now the Wiener processes are uncorrelated so SRK methods can be applied. Since K is non-diagonal (and maybe non-commutative? Check it) you'll need to use one of the Order 1.0 methods in the Rossler paper. Or you can use the derivative-free Order 1.0 method from KPS.
If you're lazy the 1.0 derivative-free KPS method is implemented as RKMil in DifferentialEquations.jl and you can follow this tutorial for defining a non-diagonal noise problem and then just make sure to pass the RKMil() algorithm solve.
http://docs.juliadiffeq.org/latest/tutorials/sde_example.html#Example-3:-Systems-of-SDEs-with-Non-Diagonal-Noise-1
But the implementation of that algorithm isn't difficult at all so this would only be worth the effort if you need a fast and efficient implementation.
Thanks @Chris Rackauckas & Marc R Roussel, ,Correlated noise can be generated by Circulant embedding method as you mentioned through SVD factors,
"Helgason (2011), Fast and exact synthesis of stationary multivariate Gaussian time series using circulant embedding" and
Adam M. Sykulski, Donald B. Percival, "EXACT SIMULATION OF NONCIRCULAR OR IMPROPER COMPLEX-VALUED STATIONARY GAUSSIAN PROCESSES USING CIRCULANT EMBEDDING "
also some one can see Wood and Chen papers
I have generated V1 and V2 with the given correlation matrix 'C', Now the point here is that the standard deviation of the noise is not 1/sqrt(dt) as like for weiner process. as i am using weak SDEs schemes, so the simplest weak scheme of order one is the Euler scheme, Euler scheme is converging very well , but when it comes to Range kutta scheme , all range kutta schemes are not converging.
the range kutta scheme suggested by Andreas Rößler in his paper "Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48(3), 922–952."
is different from Platen and Kloeden range kutta scheme just by drift term A, in Andreas Rößler scheme drift term is calculated on three points instead on two points. Andreas Rößler scheme has an advantage but for large number of white noises.
You need to not generate the noises as correlated, and instead write the decomposition parts into the noise term of the SDE so that way you have a white noise SDE with just a more complicated diffusion term. That's what I described above. Once you have the white noise process, then you can just apply the SRK methods.
If you look at the Rossler and derivative-free KPS schemes you'll see that the sqrt(dt) is part of the schemes itself.
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