Boundary crossing probabilities for diffusions with piecewise linear drifts

TL;DR

This paper introduces an approximation method for calculating boundary crossing probabilities of one-dimensional diffusion processes with piecewise linear drifts, providing explicit formulas and convergence analysis, with applications in reliability modeling.

Contribution

It presents a novel approximation scheme for boundary crossing probabilities using explicit Laplace transform expressions for diffusions with piecewise linear drifts.

Findings

The approximation scheme converges at a quantifiable rate.

Explicit Laplace transform expressions enable practical computation.

Application to reliability models extends standard Wiener process analysis.

Abstract

We propose an approach to approximate the boundary crossing probabilities for general one-dimensional diffusion processes, and derive the convergence rate for this approximation scheme. There results are based on the explicit expression of the Laplace transforms of the first passage densities for diffusions with piecewise linear drifts. The proposed method is applied to a reliability problem where the standard degradation model based on Wiener process is extended to diffusion processes with piecewise linear drifts.