Optimal bounds for the densities of solutions of SDEs with measurable and path dependent drift coefficients

TL;DR

This paper derives explicit, optimal bounds for the probability density functions of solutions to SDEs with irregular, path-dependent, and time-inhomogeneous drifts, extending to broader diffusion classes.

Contribution

It provides the first explicit, optimal density bounds for SDE solutions with irregular, path-dependent drifts and generalizes these results to wider diffusion coefficient classes.

Findings

Derived explicit, optimal density bounds for SDE solutions.

Identified the worst-case SDE for bound optimality.

Extended bounds to a broader class of diffusion coefficients.

Abstract

We consider a process given as the solution of a stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Explicit and optimal bounds for the Lebesgue density of that process at any given time are derived. The bounds and their optimality is shown by identifying the worst case stochastic differential equation. Then we generalise our findings to a larger class of diffusion coefficients.