Documentation of ImplicitRKMil - StochasticDiffEq.jl

My goal is to undertsand the actual implementation of the ImplicitRKMil algorithm here. Are there any notes or book sections directly relating to the code here?

The perspective of my question is this: I have a bunch of stiff SDEs where ImplicitRKMil with theta=1 (i.e. backward Euler) works best by far. However, it’s still not super stable, so I’m asking myself whether it would be possible to have higher-order implicit methods (in the sense that Euler is the simplest RK method)?

SDELab: A package for solving stochastic differential equations in MATLAB - ScienceDirect describes it.

With theta=1 it’s not implicit Euler, it’s still an implicit Milstein method.

Going to Order 1.5 is very expensive in terms of the Levy area calculations. For details see [1801.00784] Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise

Thanks for your answer, Chris.

I will check this out - would it be welcome if I add a docstring based on this? I think it’s nice to have some maths there relating to the code.

Yeah, you’re right. I have additive noise though, in which case this makes no difference, does it?

Well, it sure looks involved… Feels like this is too far out of my expertise to try something myself, unfortunately.

Yes that would be fine.

That does make a difference to an extent. SKenCarp should be better there, but yeah there aren’t that many methods focused on that domain and it would be nice to make some more.