On Sharp Large Deviations for the bridge of a general Diffusion

TL;DR

This paper derives precise large deviation estimates for the exit probability of a diffusion process bridge as the conditioning time approaches zero, highlighting the role of the drift under certain conditions.

Contribution

It provides sharp asymptotics for exit probabilities of diffusion bridges, revealing when the drift influences these probabilities and when it does not.

Findings

Exit probability asymptotics are independent of drift in 1D under certain conditions.

Drift can significantly affect exit probabilities if the conditions are not met.

Results are motivated by applications in numerical simulation.

Abstract

We provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a $d$-dimensional general diffusion process $X$, as the conditioning time tends to $0$. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift $b$ of $X$. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided $b$ enjoyes a simple condition that is always satisfied in dimension $1$. On the other hand, we show that the drift can be influential if this assumption is not satisfied. }