Variational calculus for diffusions

TL;DR

This paper extends the variational formulation of certain expectations to diffusions, providing new insights into invertibility and attainability within stochastic calculus.

Contribution

It generalizes the classic variational approach to include diffusion-dependent functions and relaxes integrability conditions, offering new theoretical tools.

Findings

Extended variational formulation to diffusion-dependent functions

Provided entropic characterization of diffusion perturbation invertibility

Discussed conditions for attaining the infimum in the variational problem

Abstract

We expand the classic variational formulation of $-\log\mathbb{E}\left[e^{-f}\right]$ to the case where f depends on a diffusion, and not only a on Brownian motion, while decreasing the integrability hypothesis on f. We also give an entropic characterisation of the invertibility of a perturbation of a diffusion and discuss the attainability of the infimum in the aforementioned variational formulation.