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

TL;DR

This paper derives exact statistical properties of the area under an Ornstein-Uhlenbeck process until it hits zero, revealing asymptotic normality in the weak noise limit and contrasting with the first passage time distribution.

Contribution

It provides explicit formulas for moments and correlations of the area functional, and analyzes its asymptotic behavior, advancing understanding of stochastic process functionals.

Findings

Exact mean, variance, skewness, kurtosis of the area distribution

Asymptotic normality of the area in the weak noise limit

Contrasts with the distribution of first passage times

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. Amongst 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.