# 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.

- Authors:

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

- Publication Date:

- OSTI Identifier:
- 22470060

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Applied Mathematics and Optimization; Journal Volume: 71; Journal Issue: 1; Other Information: Copyright (c) 2015 Springer Science+Business Media New York; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; BOUNDARY CONDITIONS; COMPUTERIZED SIMULATION; CONVERGENCE; DIFFERENTIAL EQUATIONS; DYNAMIC PROGRAMMING; LAGRANGIAN FUNCTION; MATHEMATICAL SOLUTIONS; OPTIMAL CONTROL; STOCHASTIC PROCESSES; VISCOSITY

### Citation Formats

```
Bokanowski, Olivier, E-mail: boka@math.jussieu.fr, Picarelli, Athena, E-mail: athena.picarelli@inria.fr, and Zidani, Hasnaa, E-mail: hasnaa.zidani@ensta.fr.
```*Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost*. United States: N. p., 2015.
Web. doi:10.1007/S00245-014-9255-3.

```
Bokanowski, Olivier, E-mail: boka@math.jussieu.fr, Picarelli, Athena, E-mail: athena.picarelli@inria.fr, & Zidani, Hasnaa, E-mail: hasnaa.zidani@ensta.fr.
```*Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost*. United States. doi:10.1007/S00245-014-9255-3.

```
Bokanowski, Olivier, E-mail: boka@math.jussieu.fr, Picarelli, Athena, E-mail: athena.picarelli@inria.fr, and Zidani, Hasnaa, E-mail: hasnaa.zidani@ensta.fr. Sun .
"Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost". United States.
doi:10.1007/S00245-014-9255-3.
```

```
@article{osti_22470060,
```

title = {Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost},

author = {Bokanowski, Olivier, E-mail: boka@math.jussieu.fr and Picarelli, Athena, E-mail: athena.picarelli@inria.fr and Zidani, Hasnaa, E-mail: hasnaa.zidani@ensta.fr},

abstractNote = {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.},

doi = {10.1007/S00245-014-9255-3},

journal = {Applied Mathematics and Optimization},

number = 1,

volume = 71,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}