Is it possible to apply Richardson extrapolation with Euler-Maruyama scheme to improve strong rate of convergence of stochastic differential equations?


From what I know all of the research has focused on weak convergence via extrapolation. The book by Kloden, Platen, and Schurz has a good discussion on them.

There are a few problems one has to solve to do "good" Richardson extrapolation with SDEs. For one, the reason why it was interesting was because the error estimates ended up being in the square powers $h^{2n}$ and that won't necessarily be the case with SDEs, especially since it would have to be based on a strong order Runge-Kutta type method and there's no obvious "higher-order choice" like the midpoint method was for ODEs (there is for weak order, which is why this was investigated). This means that its strongest point seems to be diminished. Then you have to come up with a good scheme from interpolating Brownian motion since during the calculation you do a lot of "backfilling" at lower $h$'s, and so you'd have to do that properly to not change the sampling distribution. Lastly, extrapolation methods seemed to work really well for smooth and stable problems, but tend to have problems with more unstable or discontinuous problems. The regularity of SDEs is always pretty weak, and problems seem to stem more from instability (since the noise term also changes the stability conditions A LOT!) than error (which can usually be made lower than the standard deviation of the noise anyways).

For those reasons, no one has taken a good look at extrapolation methods for strong approximation. That doesn't mean no one should (I might in the near future), but suggests that getting comfortable with the Runge-Kutta type methods might have a higher payoff. As always in numerical computing, YMMV.

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