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
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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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4855112
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