Analytical Survival Analysis of the Non-autonomous Ornstein-Uhlenbeck Process

TL;DR

This paper derives analytical expressions for the survival probability of a periodic non-autonomous Ornstein-Uhlenbeck process using asymptotic and integral methods, with the asymptotic approach shown to be more accurate.

Contribution

It introduces two analytical methods for calculating survival probabilities in non-autonomous Ornstein-Uhlenbeck processes, comparing their effectiveness.

Findings

Asymptotic method outperforms the integral approach in accuracy.

Derived explicit formulas for survival probability and exit rate.

Validated analytical solutions through numerical comparison.

Abstract

The survival probability for a periodic non-autonomous Ornstein-Uhlenbeck process is calculated analytically using two different methods. The first uses an asymptotic approach. We treat the associated Kolmogorov Backward Equation with an absorbing boundary by dividing the domain into an interior region, centered around the origin, and a "boundary layer" near the absorbing boundary. In each region we determine the leading-order analytical solutions, and construct a uniformly valid solution over the entire domain using asymptotic matching. In the second method we examine the integral relationship between the probability density function and the mean first passage time probability density function. These allow us to determine approximate analytical forms for the exit rate. The validity of the solutions derived from both methods is assessed numerically, and we find the asymptotic method to be superior.