Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

Olivier Bokanowski (1, 2) , Athena Picarelli (3, 4) , Hasnaa Zidani (3, 4)

1 LJLL - Laboratoire Jacques-Louis Lions
2 OC - Optimisation et commande
3 UMA - Unité de Mathématiques Appliquées
4 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems

This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton- Jacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.

Mots clés: sadco; maximum cost; lookback options; viscosity notion; reachable sets under state constraints; Hamilton-Jacobi equations; oblique Neuman boundary condition; error estimate

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Olivier Bokanowski, Athena Picarelli, Hasnaa Zidani. Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost. Applied Mathematics and Optimization, 2015, 71 (1), pp.125--163. ⟨10.1007/s00245-014-9255-3⟩. ⟨hal-00931025v2⟩