Sample Path Properties of Volterra 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.