Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60F10 Large deviations
- 93E20 Optimal stochastic control

**Résumé:** We consider a control problem of an ergodic process
where the objective is to maximize over long term the
probability to overperform a given level. This is formulated as
a large deviations control problem for which the standard dynamic programming
methods may not be applied directly. We solve this problem by adopting
a duality approach leading to a risk-sensitive ergodic control problem.
In a continuous-time diffusion setting, we state a verification theorem
in terms of partial differential equations for this dual problem.
We then turn back to the primal problem by means
of large deviations techniques. We derive the optimal rate function
and nearly optimal controls for the large deviations optimization problem.
Finally, explicit solutions are provided in a linear-quadratic case.

**Mots Clés:** *Large deviations ; risk-sensitive control ; controlled diffusion ; Bellman equation ; duality
*

**Date:** 2002-02-06

**Prépublication numéro:** *PMA-707*