Nonlinear boundary crossing probabilities of Brownian motion with random jumps

TL;DR

This paper develops explicit formulas for boundary crossing probabilities of Brownian motion with jumps, enabling approximation for nonlinear boundaries with flexible jump processes, and provides an easy-to-implement numerical algorithm.

Contribution

It introduces a general approach for calculating boundary crossing probabilities of Brownian motion with complex jump processes, including correlated and non-identically distributed jumps.

Findings

Explicit formulas derived for crossing probabilities with jumps.

Numerical algorithm demonstrated with numerical examples.

Applicable to a wide class of jump processes and boundaries.

Abstract

We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.