Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions

TL;DR

This paper extends regularity results for Kolmogorov equations to cases where coefficients are not globally Lipschitz but satisfy Lyapunov-type growth conditions, broadening the class of stochastic processes with well-behaved solutions.

Contribution

It introduces new Lyapunov-based conditions on coefficients that ensure regularity preservation in Kolmogorov equations, generalizing previous Lipschitz assumptions.

Findings

Regularity of solutions is maintained under sub-logarithmic growth conditions.

Weak convergence rates of numerical schemes are established under these conditions.

Counterexamples of non-Lipschitz coefficients are addressed with new Lyapunov criteria.

Abstract

Given global Lipschitz continuity and differentiability of high enough order on the coefficients in It\^{o}'s equation, differentiability of associated semigroups, existence of twice differentiable solutions to Kolmogorov equations and weak convergence rates of numerical approximations are known results. In this work and against the counterexamples of Hairer et al.(2015), the drift and diffusion coefficients having Lipschitz constants that are $o(\log V)$ and $o(\sqrt{\log V})$ respectively for a function $V$ satisfying $(\partial_t + L)V\leq CV$ is shown to be a generalizing condition in place of global Lipschitz continuity for the above.