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, ) be a finite measure space and consider the Hilbert space H: =L² (D, M, ; 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 cylindrical 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 semiconcave functions and techniques from viscosity theory, we first show that the value function of the problem V is a C^1, 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 decision maker to choose only the intensity of the control and requiring that the given control direction n is an eigenvector of the linear operator A, we establish that the directional derivative V₍ is of class C¹ (H), hence a second-order smooth-fit principle in the controlled direction holds for V. This result is obtained by exploiting a connection to optimal stopping and combining results and techniques from convex analysis and viscosity theory.