Optimal Control under Stochastic Target Constraints

Abstract

We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^{\nu}$ is constrained to satisfy an almost sure constraint $Z^{\nu}(T)\in G\subset\mathbb{R}^{d+1}$ $\mathbb{P}$-a.s. at some final time $T>0$. When the set is of the form $G:=\{(x,y)\in\mathbb{R}^d\times\mathbb{R}:g(x,y)\geq0\}$, 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 $D:=\{(t,Z^{\nu}(t))\in[0,T]\times\mathbb{R}^{d+1}:Z^{\nu}(T)\in G$ a.s. for some $\nu\}$. 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 $\mathbb{E}[g(Z^{\nu}(T))]\geq0$ or $\mathbb{P}[g(Z^{\nu}(T))\geq0]\geq p$.