Stopped diffusion processes: boundary corrections and overshoot

Abstract : 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 $(i\Delta)_{i\in\N^*}$ 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 $0.5826 (...) |n^*\sigma|(t,x)\sqrt \Delta$ 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, generalizin g previous results by \cite{broa:glas:kou:97} 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.
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Stochastic Processes and their Applications, Elsevier, 2010, 120 (2), pp.130-162. 〈10.1016/j.spa.2009.09.014〉
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Emmanuel Gobet, Stéphane Menozzi. Stopped diffusion processes: boundary corrections and overshoot. Stochastic Processes and their Applications, Elsevier, 2010, 120 (2), pp.130-162. 〈10.1016/j.spa.2009.09.014〉. 〈hal-00157975v3〉

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