Exact Asymptotics for the Exit Time Probabilities of Scalar Ornstein-Uhlenbeck Bridges

TL;DR

This paper derives precise asymptotic formulas for the probabilities that scalar Ornstein-Uhlenbeck bridges exit a domain, providing tools for accurate probability estimation in stochastic processes with small noise.

Contribution

It introduces an asymptotic series expansion for exit time probabilities of OU bridges, valid in specific regions, enhancing accuracy in stochastic process analysis.

Findings

Asymptotic series accurately estimates exit probabilities

Series validity in regions with smooth functions

Enables precise probability evaluation for OU processes

Abstract

This paper aims to derive accurate asymptotic estimates for the exit time probabilities of scalar Ornstein-Uhlenbeck (OU) bridges. The exit time probabilities are expressed as an asymptotic series in powers of a small parameter that characterizes the intensity of the noise inputs. It is shown that the series is valid in certain regions where all its terms are smooth functions. The results enable an accurate evaluation of the probability for a corresponding OU process to escape from a domain before a specified time, provided its initial and terminal states are known.