Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift

TL;DR

This paper analyzes the survival probability and first-passage-time statistics of a Wiener process with an exponential time-dependent drift, providing explicit solutions and series expansions to understand its behavior.

Contribution

It introduces a series expansion approach for solving the first-passage-time problem of a Wiener process with exponential time-dependent drift, including explicit second-order solutions.

Findings

Explicit second-order solutions for survival probability

Series expansion method for time-inhomogeneous drift

Good agreement with numerical simulations

Abstract

The survival probability and the first-passage-time statistics are important quantities in different fields. The Wiener process is the simplest stochastic processwith continuous variables, and important results can be explicitly found from it. The presence of a constant drift does not modify its simplicity; however, when the process has a time-dependent component the analysis becomes difficult. In this work we analyze the statistical properties of the Wiener process with an absorbing boundary, under the effect of an exponential time-dependent drift. Based on the backward Fokker-Planck formalism we set the time-inhomogeneous equation and conditions that rule the diffusion of the corresponding survival probability.We propose as the solution an expansion series in terms of the intensity of the exponential drift, resulting in a set of recurrence equations. We explicitly solve the expansion up to second order and comment on higher-order solutions. The first-passage-time density function arises naturally from the survival probability and preserves the proposed expansion. Explicit results, related properties, and limit behaviors are analyzed and extensively compared to numerical simulations.