Stopped diffusion processes: boundary corrections and overshoot

TL;DR

This paper introduces a boundary correction method for simulating stopped diffusion processes in multidimensional domains, improving convergence rates and providing theoretical and numerical validation for the approach.

Contribution

It proposes a simple boundary shift procedure for Euler scheme simulations of diffusion processes, enhancing convergence rates and extending previous one-dimensional results to higher dimensions.

Findings

The boundary shift improves the weak convergence rate of the Euler scheme.

The asymptotics of exit time, position, and overshoot are fully characterized.

Numerical experiments confirm theoretical improvements.

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.