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Article Dans Une Revue SIAM Journal on Control and Optimization Année : 2009

Stochastic Target Problems with Controlled Loss

Résumé

We consider the problem of finding the minimal initial data of a controlled process which guarantees to reach a controlled target with a given probability of success or, more generally, with a given level of expected loss. By suitably increasing the state space and the controls, we show that this problem can be converted into a stochastic target problem, i.e. find the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one. Unlike the existing literature on stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls, together with the appropriate boundary conditions. These results are applied to the problem of quantile hedging in financial mathematics, and are shown to recover the explicit solution of Föllmer and Leukert (1999).
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Dates et versions

hal-00323383 , version 1 (22-09-2008)

Identifiants

  • HAL Id : hal-00323383 , version 1

Citer

Bruno Bouchard, Nizar Touzi, Romuald Elie. Stochastic Target Problems with Controlled Loss. SIAM Journal on Control and Optimization, 2009, 48 (5), pp.3123-3150. ⟨hal-00323383⟩
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