For $a,b >0,$ we consider a temporally homogeneous, one-dimensional diffusion process $X(t)$ defined over $I = (-b, a),$ with infinitesimal parameters depending on the sign of $X(t).$ We suppose that, when $X(t)$ reaches the position $0,$ it is reflected rightward to $\delta$ with probability $p >0$ and leftward to $- \delta$ with probability $1-p,$ where $\delta >0.$ Closed analytical expressions are found for the mean exit time from the interval $(-b,a),$ and for the probability of exit through the right end $a,$ in the limit $\delta \rightarrow 0 ^+,$ generalizing the results of Lefebvre, holding for asymmetric Wiener process. Moreover, in alternative to the heavy analytical calculations, a numerical method is presented to estimate approximately the quantities above. Furthermore, on the analogy of skew Brownian motion, the notion of skew diffusion process is introduced. Some examples and numerical results are also reported.

Abundo, M. (2009). First-Passage Problems for Asymmetric Diffusions and Skew-diffusion Processes. OPEN SYSTEMS & INFORMATION DYNAMICS, 16(4), 325-350 [10.1142/S1230161209000256].

First-Passage Problems for Asymmetric Diffusions and Skew-diffusion Processes

Abstract

For $a,b >0,$ we consider a temporally homogeneous, one-dimensional diffusion process $X(t)$ defined over $I = (-b, a),$ with infinitesimal parameters depending on the sign of $X(t).$ We suppose that, when $X(t)$ reaches the position $0,$ it is reflected rightward to $\delta$ with probability $p >0$ and leftward to $- \delta$ with probability $1-p,$ where $\delta >0.$ Closed analytical expressions are found for the mean exit time from the interval $(-b,a),$ and for the probability of exit through the right end $a,$ in the limit $\delta \rightarrow 0 ^+,$ generalizing the results of Lefebvre, holding for asymmetric Wiener process. Moreover, in alternative to the heavy analytical calculations, a numerical method is presented to estimate approximately the quantities above. Furthermore, on the analogy of skew Brownian motion, the notion of skew diffusion process is introduced. Some examples and numerical results are also reported.
Scheda breve Scheda completa Scheda completa (DC)
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/06 - Probabilita' e Statistica Matematica
English
Con Impact Factor ISI
Asymmetric diffusion, Brownian motion, First-exit time
Abundo, M. (2009). First-Passage Problems for Asymmetric Diffusions and Skew-diffusion Processes. OPEN SYSTEMS & INFORMATION DYNAMICS, 16(4), 325-350 [10.1142/S1230161209000256].
Abundo, Mr
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/8578
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