Grid-Free Computation of Probabilistic Safety with Malliavin Calculus

TL;DR

This paper introduces a grid-free method using Malliavin Calculus to compute probabilistic safety regions for stochastic differential equations, avoiding the need for state space discretization.

Contribution

It presents a novel approach that leverages Malliavin Calculus to directly compute safety regions without state space gridding, improving efficiency and scalability.

Findings

Enables computation of safety regions without discretizing the state space

Uses a functional minimization approach at the safety region boundary

Provides a scalable alternative to existing grid-based methods

Abstract

This work concerns continuous-time, continuous-space stochastic dynamical systems described by stochastic differential equations (SDE). It presents a new approach to compute probabilistic safety regions, namely sets of initial conditions of the SDE associated to trajectories that are safe with a probability larger than a given threshold. The approach introduces a functional that is minimised at the border of the probabilistic safety region, then solves an optimisation problem using techniques from Malliavin Calculus, which computes such region. Unlike existing results in the literature, the new approach allows one to compute probabilistic safety regions without gridding the state space of the SDE.