Absolute continuity for some one-dimensional processes

TL;DR

This paper presents a simple Fourier-based method to prove the absolute continuity of time marginals for various one-dimensional stochastic processes, including those with irregular coefficients and driven by Lévy noise.

Contribution

It introduces an elementary comparison technique using Fourier transforms, extending absolute continuity results to processes with non-smooth, random, and path-dependent coefficients.

Findings

Established absolute continuity for SDEs with Hölder continuous coefficients

Extended results to Lévy-driven SDEs and some SPDEs

Applicable even when Malliavin calculus is not feasible

Abstract

We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with H\"{o}lder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations and to some L\'{e}vy-driven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavin differentiable.