First-exit-time problems for two-dimensional Wiener and Ornstein–Uhlenbeck processes through time-varying ellipses

Abstract

We study the first-exit-time problem for the two-dimensional Wiener and Ornstein–Uhlenbeck processes through time-varying ellipses which run according to specific rules depending on the processes. We obtain the Laplace Transform of the first-exit-time probability density function and the corresponding moments. For both processes, some computational results on the first-exit-time densities are provided by means of the numerical inversion of the relevant Laplace Transforms. Moreover, we also investigate the asymptotic behaviour of the first-exit-time moments when the ellipse grows. In particular, an asymptotic exponential trend holds for the first-exit-time density of the mean-reverting Ornstein–Uhlenbeck process.

Keywords:

• Transition density
• first-passage-time densities
• first-exit-time moments
• asymptotic behaviour

Mathematics Subject Classification (2020):

• 60J60
• 60J70

Disclosure statement

The authors declare that they have no known potential conflict of interest.

Additional information

Funding

This research is partially funded by the ‘ European Union – Next Generation EU’ through MUR-PRIN 2022 project 2022XZSAFN ‘Anomalous Phenomena on Regular and Irregular Domains: Approximating Complexity for the Applied Sciences’, and by the research group GNCS of INdAM (Istituto Nazionale di Alta Matematica).