Optimal Control under Stochastic Target Constraints

We study a class of Markovian optimal stochastic control problems in which the controlled process Zν is constrained to satisfy an almost sure constraint Zν(T)∈ Gᵈ⁺¹ P-a.s. at some final time $T>0$. When the set is of the form G:=\(x,y)ᵈ:g(x,y)≥0\, with g nondecreasing in y, we provide a Hamilton–Jacobi–Bellman characterization of the associated value function. It gives rise to a state constraint problem, where the constraint can be expressed in terms of an auxiliary value function w which characterizes the set D:=\(t,Zν(t))∈[0,T]ᵈ⁺¹:Zν(T)∈ G a.s. for some ν\. Contrary to standard state constraint problems, the domain D is not given a priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function w, which itself is a viscosity solution of a nonlinear parabolic PDE. Applying ideas recently developed in Bouchard, Elie, and Touzi [SIAM J. Control Optim., 48 (2009), pp. 3123–3150], our general result also allows us to consider optimal control problems with moment constraints of the form E[g(Zν(T))]≥0 or P[g(Zν(T))≥0]≥ p.