On the Supremum of Certain Families of Stochastic Processes

TL;DR

This paper investigates conditions for the supremum of a family of stochastic processes to tend to zero as a parameter approaches zero, extending classical continuity criteria and applying results to stochastic integrals with Poisson measures.

Contribution

It provides new conditions under which the supremum of stochastic processes converges to zero, generalizing Kolmogorov's continuity criteria and applying to processes driven by Poisson measures.

Findings

Established criteria for supremum convergence as   

Compared new conditions with Kolmogorov's classical criteria

Applied results to stochastic integrals with Poisson random measures

Abstract

We consider a family of stochastic processes $\{X_t^\epsilon, t \in T\}$ on a metric space $T$, with a parameter $\epsilon \downarrow 0$. We study the conditions under which \lim_{\e \to 0} \P \Big(\sup_{t \in T} |X_t^\e| < \delta \Big) =1 when one has the \textit{a priori} estimate on the modulus of continuity and the value at one point. We compare our problem to the celebrated Kolmogorov continuity criteria for stochastic processes, and finally give an application of our main result for stochastic intergrals with respect to compound Poisson random measures with infinite intensity measures.