Portfolio Optimization Under A Quantile Hedging Constraint

We study a problem of portfolio optimization under a European quantile hedging constraint. More precisely, we consider a class of Markovian optimal stochastic control problems in which two controlled processes must meet a probabilistic shortfall constraint at some terminal date. We denote by V the corresponding value function. Following the arguments introduced in the literature on stochastic target problems, we convert this problem into a state constraint one in which the constraint is defined by means of an auxiliary value function v characterizing the reachable set. This set is therefore not given a priori but is naturally integrated in v solving, in a viscosity sense, a nonlinear parabolic partial differential equation (PDE). Relying on the existing literature, we derive, in the interior of the domain, a Hamilton–Jacobi–Bellman characterization of V. However, v 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 nontrivial boundaries for V that must be discussed. Our main result is thus the characterization of V at those boundaries. We also provide examples for which comparison results exist for the PDE solved by V on the interior of the domain.