On certain bounds for first-crossing-time probabilities of a jump-diffusion process

TL;DR

This paper derives explicit lower bounds for the first-crossing-time probabilities of a jump-diffusion process with constant boundaries, using sample-path analysis and stochastic order comparisons, with special focus on Poisson-driven jumps.

Contribution

It provides new explicit bounds for first-crossing-time distributions of jump-diffusions, enhancing understanding of boundary crossing probabilities in stochastic processes.

Findings

Explicit lower bounds for crossing-time density and distribution functions.

Improved bounds using stochastic process comparison methods.

Detailed analysis of Poisson-driven jump processes.

Abstract

We consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit lower bounds for the first-crossing-time density and for the first-crossing-time distribution function. In the case of the distribution function, the bound is improved by use of processes comparison based on the usual stochastic order. The special case of constant jumps driven by a Poisson process is thoroughly discussed.