Stopped diffusion processes: Boundary corrections and overshoot

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 Δ and stopping it at discrete times (i Δ)_{i ∈ 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^* σ | (t, x) sqrt(Δ) where σ 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. Δ 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.