Decaying derivative estimates for functions of solutions to non-autonomous SDEs

TL;DR

This paper establishes uniform and decaying bounds over time for derivatives of solutions to the backward Kolmogorov equation related to non-autonomous SDEs, under specific integrability and stability assumptions.

Contribution

It introduces new derivative estimates for solutions to non-autonomous SDEs' Kolmogorov equations, including examples and applications to McKean-Vlasov processes.

Findings

Derived uniform and decaying bounds for derivatives

Provided examples satisfying the assumptions

Applied results to McKean-Vlasov equations

Abstract

We produce uniform and decaying bounds in time for derivatives of the solution to the backwards Kolmogorov equation associated to a stochastic processes governed by a time dependent dynamics. These hold under assumptions over the integrability properties in finite time of the derivatives of the transition density associated to the process, together with the assumption of remaining close over all $[0,\infty)$, or decaying in time, to some static measure. We moreover provide examples which satisfy such a set of assumptions. Finally, the results are interpreted in the McKean-Vlasov context for monotonic coefficients by introducing an auxiliary non-autonomous stochastic process.