On Markov chain approximations for computing boundary crossing probabilities of diffusion processes

TL;DR

This paper introduces a Markov chain approximation method with Brownian bridge correction to accurately compute boundary crossing probabilities for diffusion processes, with proven convergence and promising numerical results.

Contribution

It develops a new discrete approximation method with convergence proof for boundary crossing probabilities of diffusions, improving accuracy with Brownian bridge correction.

Findings

Convergence of the approximation is proven for broad classes of boundaries and diffusions.

Numerical results show an $O(n^{-2})$ convergence rate for smooth boundaries.

The method effectively computes boundary crossing probabilities with high accuracy.

Abstract

We propose a discrete time discrete space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the proposed approximation with the Brownian bridge correction is $O(n^{-2})$ in the case of $C^2$-boundaries and a uniform time grid with $n$ steps.