Error estimates for second order Hamilton-Jacobi-Bellman equations.Approximation of probabilistic reachable sets

Mohamed Assellaou; Olivier Bokanowski; Hasnaa Zidani

doi:10.3934/dcds.2015.35.3933

Article Contents

2015, Volume 35, Issue 9: 3933-3964. Doi: 10.3934/dcds.2015.35.3933

This issue Previous Article Ergodicity conditions for zero-sum games Next Article Large deviations for some fast stochastic volatility models by viscosity methods

Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets

1.
Unité des mathématiques appliquées (UMA), ENSTA ParisTech, 828 Bd Maréchaux, 91120 Palaiseau
2.
Laboratoire Jacques-Louis Lions, UMR 7598, Université Paris-Diderot (Paris 7), UFR de Mathématiques - 5 rue Thomas Mann, 75205 Paris CEDEX 13

Received: April 30, 2014

Revised: October 31, 2014

Published: May 2017

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Abstract

This work deals with numerical approximations of unbounded and discontinuous value functions associated to some stochastic control problems. We derive error estimates for monotone schemes based on a Semi-Lagrangian method (or more generally in the form of a Markov chain approximation). A motivation of this study consists in approximating chance-constrained reachability sets. The latters will be characterized as level sets of a discontinuous value function associated to an adequate stochastic control problem. A precise analysis of the level-set approach is carried out and some numerical simulations are given to illustrate the approach.

Keywords:

Error estimates for HJB equations,
numerical approximation,
unbounded and discontinuous value function,
chance constraints,
level-set approach,
stochastic reachability analysis.

Mathematics Subject Classification: Primary: 65M15, 93E20; Secondary: 49L25.

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