Portfolio Optimization under a Quantile Hedging Constraint

We consider a class of Markovian optimal stochastic control problems in which two controlled processes have to meet a probabilistic shortfall constraint at some terminal date. Following the arguments of Bouchard, Elie and Imbert (2010) we convert this initial problem into a state constraint one where the constraint is defined via an auxiliary value function characterizing the corresponding viability domain. Therefore, the viability domain is not given a priori but is naturally integrated into an auxiliary value function which solves, in a viscosity sense, a nonlinear parabolic PDE. Proceeding as in Bouchard, Elie and Imbert (2010) we can derive, in the interior of the domain, a Hamilton-Jacobi-Bellman characterization of the original value function. However, the auxiliary value function involves an additional controlled state variable coming from the diffusion of the probability of reaching the target and belonging to the compact set [0,1]. This leads to non-trivial boundaries for the original value function that need to be discussed.