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

Federico Salvatore, Università di Bologna
Ferrari Giorgio, Center for Mathematical Economics (IMW), Bielefeld University
Riedel Frank, University of Bielefeld
Roeckner Michael, University of Bielefeld

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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