## Second-order smooth-fit for a class of singular stochastic control problems in infinite dimension

We prove a second-order smooth-fit principle for 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 $(D,\mathcal{M},\mu)$ be a measure space and consider the Hilbert space $H:=L^2(D,\mathcal{M},\mu; \mathbb{R})$. 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 linear operator $\mathcal{A}$ and affected by a cylindrical Brownian motion. The evolution of $X$ is controlled linearly via a vector-valued control consisting of the direction and the intensity of action. The former is a unitary vector in $H$, while the latter is a real-valued nondecreasing right-continuous stochastic process, adapted to the underlying filtration. The goal is to minimize an infinite time-horizon, discounted convex cost-functional. 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, allowing the decision maker to choose only the intensity of the control, and requiring that the given direction of control $\hat{n}$ 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.

Area: IS14 - Stochastic Control and Game-theoretic Models in Economics and Finance (Giorgio Ferrari)

Keywords: Singular stochastic control, Variational Inequalities in infinite dimension, Stochastic Partial Differential Equations, Smooth-fit principle

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