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 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 ) 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.