I have a problem about whether one can allow some dependence when using Richardson extrapolation method.Consider the Euler scheme for the numerical solution of a SDE with the Richardson extrapolation: Let [$]X_{T}(h)[$] be the numerical approximation of [$]X_{T}[$], the solution of a SDE at time [$]T[$], with the step size [$]h[$], Richardson extrapolation here typically means the scheme [$]2E(f(X_{T}(h)))-E(f(X_{T}(2h)))[$] to approximate [$]E(f(X_{T}))[$], where [$]f(.)[$] is a payoff function. Suppose [$]N_{h}^{1}[$],....,[$]N_{h}^{T/h}[$] are required standard normal random variables from the simulation to get one path [$]X_{T}(h)[$]. To get [$]X_{T}(2h)[$], can we just collect [$]T/(2h)[$] number of standard normal random variables from [$]N_{h}^{1}[$],....,[$]N_{h}^{T/h}[$], instead of simulating new ones, please? This means [$]X_{T}(h)[$] and [$]X_{T}(2h)[$] are then dependent, instead of independent. I assume this treatment would improve the computational efficiency, but I find no such a discussion in the literature... Thanks for your comments.

Kloeden and Platen discuss RE (and Romberg extrapolation). The original source is Talay and Tubaro 1990. "Expansion of the global error for the numerical schemes solving Stochastic Differential Equation".It's a bit more computational and you have to be careful with the random numbers on the rough dt mesh.rn(dt) = (rn1(dt/2) + rn2(dt/2) /sqrt(2).In general, RE is a good all-round scheme and it improves Euler. Caveat: RE is less effective for discontinuities.In many cases RE can be faster because you don't need to take too small of a dt. So it has an upside.Here is some code to show the idea

Still a question: can we use the random samples, such as the normal rv, from the fine dt mesh for the rough dt mesh ? It seems in the literature that the simulation on the fine dt mesh and on the rough dt mesh are completely independent.

QuoteOriginally posted by: fmfreshmanStill a question: can we use the random samples, such as the normal rv, from the fine dt mesh for the rough dt mesh ? It seems in the literature that the simulation on the fine dt mesh and on the rough dt mesh are completely independent.How do they do it? I have not been able to find a source.As mentioned, I usern(dt) = (rn1(dt/2) + rn2(dt/2) /sqrt(2)and I got it from somewhere I can remember. Maybe Kloeden or on Wilmott(?) Gut feeling is that the random numbers should _not_ be independent otherwise you are approximating a different path on two meshes..

Gilles Pages, Vincent Lemaire have written an article or two on this subject not to long ago. with some explanation of how to reuse the fine dt mesh.