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 [Formula: see text] 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 [Formula: see text] characterizing the reachable set. This set is therefore not given a priori but is naturally integrated in [Formula: see text] 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 [Formula: see text]. However, [Formula: see text] involves an additional controlled state variable coming from the diffusion of the probability of reaching the target and belonging to the compact set [Formula: see text]. This leads to nontrivial boundaries for [Formula: see text] that must be discussed. Our main result is thus the characterization of [Formula: see text] at those boundaries. We also provide examples for which comparison results exist for the PDE solved by [Formula: see text] on the interior of the domain.