Grid-Free Computation of Probabilistic Safety with Malliavin Calculus

We work with continuous-time, continuous-space stochastic dynamical systems described by stochastic differential equations (SDE). We present 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. We introduce a functional that is minimised at the border of the probabilistic safety region, then solve an optimisation problem using techniques from Malliavin Calculus that computes such region. Unlike existing results in the literature, this new approach allows one to compute probabilistic safety regions without gridding the state space of the SDE.