By analogy with differential equations, which use vanilla calculus to define deterministic dynamics, we can define stochastic differential equations, which use stochastic calculus to define random dynamics.
SDEs are time-indexed, causal stochastic processes which notionally integrate an ordinary differential equation over some driving noise. Stochastic partial differential equations are to SDEs as PDEs are to ODEs.
Useful in state filters, optimal control, financial mathematics, etc.
Usually, we talk about ‘differential’ equations, but usually SDEs are more conveniently defined through integral equations rather than differential equations, in the sense that the driving noise process is integrated; the derivative is not necessarily well-defined without extra machinery.. When you do differentiate the noise process, it leads, AFAICT, to Malliavin calculus, or something? I am not sure about that theory.
Useful tools: infinitesimal generators, martingales, Dale Robert’s cheat sheet, Itô-Taylor expansions…
Warning: beware the terminology problem that some references take SDEs to be synonymous with Itô processes, whose driving noise is Brownian. Some writers, when they really want to be clear that they are not assuming Brownian motion. Some SDEs driven by Lévy noise, use the term sparse stochastic processes.
1 Pathwise solutions
See pathwise solutions of stochastic differential equations.
2 Infinite dimensional
3 Incoming
4 References
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