Statistics of the first passage area functional for an Ornstein–Uhlenbeck process

Michael J Kearney,Richard J Martin

doi:10.1088/1751-8121/abd677

Abstract

We consider the area functional defined by the integral of an Ornstein–Uhlenbeck process which starts from a given value and ends at the time it first reaches zero (its equilibrium level). Exact results are presented for the mean, variance, skewness and kurtosis of the underlying area probability distribution, together with the covariance and correlation between the area and the first passage time. Among other things, the analysis demonstrates that the area distribution is asymptotically normal in the weak noise limit, which stands in contrast to the first passage time distribution. Various applications are indicated.

Highlights

First passage problems play a special role in the theory of stochastic processes [1]
We are interested in the first passage area functional associated with the relaxation of an Ornstein-Uhlenbeck process to zero
The area functional A is defined by the time integral of the process starting from a given initial value up to the first passage time T to reach zero

Summary

Introduction

First passage problems play a special role in the theory of stochastic processes [1]. A topical example, which involves the study of integrated Ornstein-Uhlenbeck processes and consideration of functionals of Brownian motion, relates to so-called diffusing diffusivity models These have been invoked to explain the behaviour of diffusion-type processes which have a linear time dependence of the mean-square displacement, but accompanied by a non-Gaussian displacement distribution, possibly crossing over to Gaussian on long timescales. Describing the fluctuating diffusivity through subordination highlights the importance of the probability density of the integrated diffusivity (a functional), which lends itself to even more generalised treatments, including making full use of spectral concepts [35, 36] The link between this and the current work is only indirect, but the broader contextual setting is interesting, and the results and techniques developed here may find wider use.

General theory

The first passage time statistics

Discussion