Variational Inequalities and Smooth-Fit Principle for Singular Stochastic Control Problems in Hilbert Spaces

Salvatore Federico, Giorgio Ferrari, Frank Riedel and Michael Röckner
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Salvatore Federico: Center for Mathematical Economics, Bielefeld University
Giorgio Ferrari: Center for Mathematical Economics, Bielefeld University
Michael Röckner: Center for Mathematical Economics, Bielefeld University

No 692, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

Abstract: 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 $(D,\mathcal{M},\mu)$ be a finite 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 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 $C^{1,Lip}(H)$-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.

Keywords: infinite-dimensional singular stochastic control; viscosity solution; variational inequality; infinite-dimensional optimal stopping; smooth-fit principle (search for similar items in EconPapers)
Pages: 37
Date: 2024-06-11
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