frankschae
commented
4 years ago One step ahead could possibly be done by implementing this (complicated-looking) procedure:
https://epubs.siam.org/doi/10.1137/S0036139992235706
for non-commutative noise based on two Wiener processes. Since it doesn't rely on additional random numbers, we should be able to see the first order convergence.
In another paper of Ryden and Wiktorsson (https://www.sciencedirect.com/science/article/pii/S0304414900000533) they say about the above procedure:
The method by Gaines and Lyons differs from all the above ones in that it is exact and based on inversion of the joint characteristic function of rho2 and the Levy stochastic area integral Aij(2) (see (1.3)) rather than on a probabilistic representation and analysis of the iterated Itoˆ integral. It is certainly also the fastest method; Gaines and Lyons report that simulation of two Wiener increments and one iterated Itoˆ integral takes about the same time as the simulation of approximately 14 standard normal variables. However, the method is also by far the most complicated one to implement and indeed sometimes – although seldomly – requires on-line numerical Fourier inversion of the characteristic distribution. Hence code for this operation must be included in the simulation package.
@frankschae added very strong validation for the non-commutative noise handling, except we're missing the final nail in the coffin which is an actual convergence test on a non-commutative noise SDE showing 1.0. Instead, we have a bunch of unit tests that show the iterated integrals and closures have the right distributions, and we have something that shows, on a commutative SDE which we know the analytical solution, a 1.0 convergence rate. However, we do not have a non-commutative SDE with an analytical solution. When using analytic-less convergence testing, this normally assumes that the random numbers are all shared so the distribution is exactly kept between the different dts, but since we do not handle the extra terms of the iterated integrals for that, the analytic-less convergence testing won't necessarily be calculating the same iterated integral as dt shrinks. This means we don't have the final piece of proof, but are reasonably confident that it's working to put it out there, but it would be good to finish up this final proof someday.