Summary: 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.