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

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

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