Stopped diffusion processes: Boundary corrections and overshoot

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:120

Article

Gobet, Emmanuel^1,2  Menozzi, Stephane³ 

[1] Univ Grenoble, Lab Jean Kuntzmann, F-38041 Grenoble 9, France

[2] CNRS, F-38041 Grenoble 9, France

[3] Univ Paris 07, Lab Probabil & Modeles Aleatoires, F-75013 Paris, France

关键词: Stopped diffusion;    Time-dependent domain;    Brownian overshoot;    Boundary sensitivity;   

DOI  :  10.1016/j.spa.2009.09.014

来源: Elsevier

【 摘 要 】

For a stopped diffusion process in a multidimensional time-dependent domain D, we propose and analyse a new procedure consisting in Simulating the process with an Euler scheme with step size Delta and stopping it at discrete times (i Delta)(i is an element of N*) in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal n (t, x) at any point (t, x) on the parabolic boundary of D, and its amplitude is equal to 0.5826(...)vertical bar n*sigma vertical bar(t, x)root Delta where sigma stands for the diffusion coefficient of the process. The procedure is thus extremely easy to use. In addition, we prove that the rate of convergence w.r.t. Delta for the associated weak error is higher than without shifting, generalizing the previous results by Broadie et al. (1997) [6] obtained for the one-dimensional Brownian motion. For this, we establish in full generality the asymptotics of the triplet exit time/exit position/overshoot for the discretely stopped Euler scheme. Here, the overshoot means the distance to the boundary of the process when it exits the domain. Numerical experiments support these results. (C) 2009 Elsevier B.V. All rights reserved.

【 授权许可】

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