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 behavior 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.
First-exit-time problems for two-dimensional Wiener and Ornstein-Uhlenbeck processes through time-varying ellipses
Di Crescenzo, Antonio;Giorno, Virginia;Nobile, Amelia Giuseppina;Spina, Serena
2024-01-01
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 behavior 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.