Here is a nicely balanced book on SDE by two top numerical analysts
https://epubs.siam.org/doi/10.1137/1.9781611976434
i.e. continuous space to discrete space (as it should be).
Most books and articles on SDE give numerical methods short shrift...
In fact, most books only tell half the story. That's why SDE can be so intimidating.
Paper models don't crash.
I recommend it to students, in combination with C++ and Python.
It is true that there is nothing in a stochastic differential equation that is not in a Fokker-Planck equation, but the stochastic differential equation is so much easier to write down and manipulate that only an excessively zealous purist would try to eschew the technique.
C.W. Gardiner (2004) Handbook of Stochastic Methods, for Physics, Chemistry and the Natural Sciences, Springer.
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FPE (Fokker Planck) PDE Remarks
Daniel J. Duffy
Attention Points on current thinking
. Watertight mathematical formulation replaced by (ad-hoc) heuristics.
. Somewhat outdated finite difference schemes used.
Of course, extensive numerical experimental testing should ensure that it works in practice. But how does it work in theory?
Some possible Remedies
1. Domain transformation of PDE to unit interval/square (no ad-hoc domain truncation!).
2. Seamless Fichera/Feller/Green formula boundary conditions (mathematically robust). Well-posed PDE now.
3. Morph FPE into a conservative/self-adjoint PDE, which subsumes the pesky convective term and allowing elegant numerical schemes.
4. ADI/Craig Sneyd/Crank Nicolson are a bit alt-modisch; we have about 5 improvements in Duffy (2022).
5. No issues with Dirac BCs; integral evaluates to 1.
6. I use ADE, Marchuk, Strang, Yanenko schemes. Even Method of Lines (MOL).
Duffy, D.J. (2022) Numerical Methods in Computational Finance Wiley.