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 $(D,\mathcal{M},μ)$ be a finite measure space and consider the Hilbert space $H:=L^2(D,\mathcal{M},μ; \mathbb{R})$. 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 $\mathcal{A}$ 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 {$C^{1,\mathrm{Lip}}(H)$}-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 $\hat{n}$ is an eigenvector of the linear operator $\mathcal{A}$, we establish that the directional derivative $V_{\hat{n}}$ is of class $C^1(H)$, 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.

💡 Research Summary

The paper studies a class of infinite‑dimensional singular stochastic control problems formulated in a Hilbert space (H:=L^{2}(D,\mathcal{M},\mu;\mathbb{R})). The state process (X) evolves according to a stochastic partial differential equation (SPDE) driven by a self‑adjoint linear operator (\mathcal{A}) and a cylindrical Wiener process. Control enters linearly through an (H)-valued, non‑decreasing, right‑continuous process (I); the control can be decomposed into a direction (\hat n\in H) and an intensity (a real‑valued non‑decreasing process). The objective is to minimize a discounted convex cost functional over an infinite horizon: \