On the existence and regularity of local times

TL;DR

This paper establishes general conditions for the existence and regularity of local times in multi-dimensional stochastic processes, addressing open problems in stochastic differential equations driven by fractional Brownian motion and extending results to other processes.

Contribution

It provides a unified framework for analyzing local times, including new insights into fractional Brownian motion solutions and extensions to Rosenblatt processes and Gaussian quasi-helices.

Findings

Established conditions for local time existence and regularity.

Included solutions to SDEs driven by fractional Brownian motion.

Extended results to Rosenblatt process and Gaussian quasi-helices.

Abstract

We study the existence and regularity of local times for general $d$-dimensional stochastic processes. We give a general condition for their existence and regularity properties. To emphasize the contribution of our results, we show that they include various prominent examples, among others solutions to stochastic differential equations driven by fractional Brownian motion, where the behavior of the local time was not fully understood up to now and remained as an open problem in the stochastic analysis literature. In particular this completes the picture regarding the local time behavior of such equations, above all includes high dimensions and both large and small Hurst parameters. As other main examples, we also show that by using our general approach, one can quite easily cover and extend some recently obtained results on the local times of the Rosenblatt process and Gaussian quasi-helices.