Stochastic target problems with controlled loss
Bouchard, Bruno; Elie, Romuald; Touzi, Nizar (2009), Stochastic target problems with controlled loss, SIAM Journal on Control and Optimization, 48, 5, p. 3123-3150. http://dx.doi.org/10.1137/08073593X
TypeArticle accepté pour publication ou publié
Journal nameSIAM Journal on Control and Optimization
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Abstract (EN)We consider the problem of ﬁnding the minimal initial data of a controlled processwhich 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, weshow that this problem can be converted into a stochastic target problem, i.e., ﬁnding the minimalinitial data of a controlled process which guarantees to reach a controlled target with probability one.Unlike in the existing literature on stochastic target problems, our increased controls are valued in anunbounded set. In this paper, we provide a new derivation of the dynamic programming equation forgeneral stochastic target problems with unbounded controls, together with the appropriate boundaryconditions. These results are applied to the problem of quantile hedging in ﬁnancial mathematicsand are shown to recover the explicit solution of F¨ ollmer and Leukert [Finance Stoch., 3 (1999), pp.251–273].
Subjects / KeywordsDiscontinuous Viscosity Solutions; Stochastic target problem; Quantile Hedging
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Generalized stochastic target problems for pricing and partial hedging under loss constraints - Application in optimal book liquidation Bouchard, Bruno; Dang, Ngoc Minh (2013) Article accepté pour publication ou publié