Variational inequalities and smooth-fit principle for singular stochastic control problems in Hilbert spaces

We consider a class of infinite-dimensional singular stochastic control problems. These can be thought of as spatial monotone follower problems and find applications in spatial models of production and climate transition. Let (D,ℳ,μ) be a finite measure space and consider the Hilbert space H := L2(D,ℳ,μ;ℝ). Let then X be an H-valued stochastic process on a suitable complete probability space, whose evolution is determined through an SPDE driven by a self-adjoint linear operator 𝒜 and affected by a cylin- drical Brownian motion. The evolution of X is controlled linearly via an H-valued control consisting of the direction and the intensity of action, a real-valued nondecreasing right-continuous stochastic process, adapted to the underlying filtration. The goal is to minimize a discounted convex cost- functional over an infinite time-horizon. By combining properties of semi- concave functions and techniques from viscosity theory, we first show that the value function of the problem V is a C1,Lip(H)-viscosity solution to the corresponding dynamic programming equation, which here takes the form of a variational inequality with gradient constraint. Then, by allowing the deci- sion maker to choose only the intensity of the control and requiring that the given control directionˆ n is an eigenvector of the linear operator 𝒜, we estab- lish that the directional derivative Vˆ n is of class C1(H), hence a second-order smooth-fit principle in the controlled direction holds for V . This result is ob- tained by exploiting a connection to optimal stopping and combining results and techniques from convex analysis and viscosity theory.