A Complete Decomposition of Stochastic Differential Equations

Samuel Duffield

Keywords: Stochastic differential equations, diffusion models

TL;DR: An SDE has prescribed marginals p(x,t) iff it decomposes into a unique scalar field and some symmetric and skew-symmetric matrix fields.

Abstract: We show that any stochastic differential equation with prescribed time-dependent marginal distributions admits a decomposition into three components: a unique scalar field governing marginal evolution, a symmetric positive-semidefinite diffusion matrix field and a skew-symmetric matrix field.

Submission Number: 5