Higher-Order Approximation of Exit Functionals in Sampling-Based Stochastic Model Predictive Control

Abstract

Safety evaluation in sampling-based stochastic model predictive control often requires numerical estimation of exit functionals. The approximation of first-exit times and exit indicators is therefore a key numerical bottleneck, and discretization error in these quantities directly affects the resulting controller. This paper studies how existing higher-order methods for strong approximation of exit times can be brought into safe control. Two cases are highlighted. For general noncommutative dynamics, an adaptive order-1 Milstein discretization is used together with Lévy-area simulation via Wiktorsson's method. For commutative dynamics, an adaptive order-1.5 construction achieves a stronger exit-time rate. Under a local anti-concentration condition on the exit-time law, we show that strong exit-time approximation transfers to strong approximation of the failure indicator. The methods are then studied in the context of chance-constrained path integral control, which provides an exact continuous-time representation of safety through exit events. Numerical experiments compare the two cases in terms of strong exit-time error, failure-indicator error, and closed-loop constraint satisfaction, showing improvement over Euler-Maruyama and thereby enabling existing and future techniques whose applicability depends on improved strong approximation.