Tight estimates of exit and containment probabilities for nonlinear stochastic systems

TL;DR

This paper develops a method to derive tight bounds on the containment probabilities of nonlinear stochastic systems by comparing their pull strengths to those of well-understood processes like Ornstein-Uhlenbeck, with implications for control problems.

Contribution

It introduces a dominance relationship based on pull strength to bound containment probabilities of nonlinear stochastic systems, linking to contraction theory.

Findings

Derived tight bounds for nonlinear system containment probabilities.

Established a dominance relationship that implies bounds on probabilities.

Linked the approach with contraction theory and provided application examples.

Abstract

Tight estimates of exit/containment probabilities are of particular importance in many control problems. Yet, estimating the exit/containment probabilities is non-trivial: even for linear systems (Ornstein-Uhlenbeck processes), the containment probability can be computed exactly for only some particular values of the system parameters. In this paper, we derive tight bounds on the containment probability for a class of nonlinear stochastic systems. The core idea is to compare the "pull strength" (how hard the deterministic part of the system dynamics pulls towards the origin) experienced by the nonlinear system at hand with that of a well-chosen process for which tight estimates of the containment probability are known or can be numerically obtained (e.g. an Ornstein-Uhlenbeck process). Specifically, the main technical contribution of this paper is to define a suitable dominance relationship between the pull strengths of two systems and to prove that this dominance relationship implies an order relationship between their containment probabilities. We also discuss the link with contraction theory and highlight some examples of applications.