Analytical Survival Analysis of the Ornstein-Uhlenbeck Process

TL;DR

This paper derives an analytical expression for the survival probability of an Ornstein-Uhlenbeck process using asymptotic methods, providing a comprehensive solution that aligns well with numerical results and accounts for boundary leakage.

Contribution

It introduces a novel asymptotic approach to obtain a uniform analytical solution for the survival probability of Ornstein-Uhlenbeck processes over broad domains.

Findings

Analytical solution matches numerical results closely.

Solution accounts for probability leakage at short times.

Broad applicability across natural and engineering sciences.

Abstract

We use asymptotic methods from the theory of differential equations to obtain an analytical expression for the survival probability of an Ornstein-Uhlenbeck process with a potential defined over a broad domain. We form a uniformly continuous analytical solution covering the entire domain by asymptotically matching approximate solutions in an interior region, centered around the origin, to those in boundary layers, near the lateral boundaries of the domain. The analytic solution agrees extremely well with the numerical solution and takes into account the non-negligible leakage of probability that occurs at short times when the stochastic process begins close to one of the boundaries. Given the range of applications of Ornstein-Uhlenbeck processes, the analytic solution is of broad relevance across many fields of natural and engineering science.