Large jumps of $q$-Ornstein-Uhlenbeck processes

TL;DR

This paper studies the sample path behavior of $q$-Ornstein-Uhlenbeck processes, revealing that they exhibit large jumps across the domain with positive probability, and the count of such jumps follows a Poisson distribution asymptotically.

Contribution

It demonstrates the existence of large jumps in $q$-Ornstein-Uhlenbeck processes and characterizes their count distribution as Poisson, extending understanding of their path properties.

Findings

Large jumps occur with positive probability for all $q\in(-1,1)$.

Number of jumps in an enlarged window converges to a Poisson distribution.

The process crosses from one domain boundary to the other via these jumps.

Abstract

We continue the investigation of sample paths of $q$-Ornstein-Uhlenbeck process. We show that for all $q\in(-1,1)$, the process has big jumps crossing from near one end point of the domain to the other with positive probability. Moreover, the number of such jumps in an appropriately enlarged window converges weakly to a Poisson random variable.