How to do double stochastic integrals?

[crowsonkb]

Is it possible to use or modify this code to do double stochastic integrals in a reasonable way, in order to support higher order SDE solvers? I am using your BrownianTree implementation with my own custom solvers right now and would like to try a higher order solver.

Thank you,
Katherine Crowson

[patrick-kidger]

Right now torchsde is capable of providing both Davie and Davie-Foster approximations to Levy area (which is the double stochastic integral that usually matters). This can be done with e.g. BrownianInterval(..., levy_area_approximation="foster").

You can then access this at call-time by doing bm(..., return_A=True).

Note that this isn't the usual Wiktorrson approximation. That is typically used to give higher-order strong convergence, but is quite computationally expensive. The Davie[-Foster] approximations instead only give higher-order Wasserstein convergence, but are computationally much cheaper.

Let me know if you need more details. This topic is a bit of a can of worms, and the mathematics is not well-documented either here or in the academic literature.