On approximation rates for boundary crossing probabilities for the multivariate Brownian motion process

TL;DR

This paper investigates how small changes in the boundary affect the crossing probabilities of multivariate Brownian motion, providing bounds under broad conditions relevant to mathematical finance.

Contribution

It extends univariate boundary crossing probability bounds to the multivariate case under broad assumptions, including Lipschitz conditions.

Findings

Derived bounds for boundary crossing probabilities of multivariate Brownian motion.

Established stability results for boundary crossing probabilities under boundary perturbations.

Provided upper bounds for first boundary crossing time densities.

Abstract

Motivated by an approximation problem from mathematical finance, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad assumptions on the nature of the boundary, including the Lipschitz condition (in a Hausdorff-type metric) on its time cross-sections, we obtain an analogue of the Borovkov and Novikov (2005) upper bound for the difference between boundary hitting probabilities for "close boundaries" in the univariate case. We also obtained upper bounds for the first boundary crossing time densities.