Sample Path Properties of Volterra Processes

TL;DR

This paper investigates the regularity and sample path properties of Volterra processes, revealing how the process's regularity is influenced by the underlying semimartingale and the kernel function, with applications to fractional Lévy processes.

Contribution

It provides new insights into the regularity of Volterra processes, especially at jump times, and determines the optimal Hölder exponent for fractional Lévy processes.

Findings

Sample path regularity depends on the kernel function F.

Processes exhibit worst regularity at jump times of X.

Derived the optimal Hölder exponent for fractional Lévy processes.

Abstract

We consider the regularity of sample paths of Volterra processes. These processes are defined as stochastic integrals $$ M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, $$ where $X$ is a semimartingale and $F$ is a deterministic real-valued function. We derive the information on the modulus of continuity for these processes under regularity assumptions on the function $F$ and show that $M(t)$ has "worst" regularity properties at times of jumps of $X(t)$. We apply our results to obtain the optimal H\"older exponent for fractional L\'{e}vy processes.