Abstracts

We study a stochastic target problem with a controlled probability of success on a set of deterministic dates. Proceeding as in [1] we can suitably increase the state space and the controls to reduce the problem to a more standard stochastic target one. More precisely we can reduce the problem to a problem of super-replication of a Bermudean option. However the increased controls are then valued in an unbounded set. Nevertheless we can deduce the related dynamic programming equation. We then apply our results to the so-called quantile hedging example of [2]. We can then extend their result to the case where the constraint holds on a set of deterministic dates and find a pseudo-explicit solution.

[1] Bouchard B., Elie R. and Touzi N., (2009). Stochastic target problems with controlled loss. SIAM Journal on Control and Optimization, 48 (5), pp. 3123-3150.
[2] Follmer H. and Leukert P., (1999). Quantile Hedging. Finance and Stochastics, 3, pp. 251-273.