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

Journal Article · Sun Feb 15 04:00:00 EST 2015 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-014-9255-3· OSTI ID:22470060

Bokanowski, Olivier ^[1]; Picarelli, Athena ^[2]; Zidani, Hasnaa ^[3]

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1. Laboratoire Jacques-Louis Lions, Université Paris-Diderot (Paris 7) UFR de Mathématiques - Bât. Sophie Germain (France)
2. Projet Commands, INRIA Saclay & ENSTA ParisTech (France)
3. Unité de Mathématiques appliquées (UMA), ENSTA ParisTech (France)

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.

• OSTI ID: 22470060
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 1 Vol. 71; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
97 MATHEMATICS AND COMPUTING
BOUNDARY CONDITIONS
COMPUTERIZED SIMULATION
CONVERGENCE
DIFFERENTIAL EQUATIONS
DYNAMIC PROGRAMMING
LAGRANGIAN FUNCTION
MATHEMATICAL SOLUTIONS
OPTIMAL CONTROL
STOCHASTIC PROCESSES
VISCOSITY