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Erscheinungsjahr: 
2024
Schriftenreihe/Nr.: 
Center for Mathematical Economics Working Papers No. 692
Verlag: 
Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld
Zusammenfassung: 
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,μ) be a finite measure space and consider the Hilbert space H:=L2(D,M,μ;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 C1,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 nis an eigenvector of the linear operator A , we establish that the directional derivative Vnis 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.
Schlagwörter: 
infinite-dimensional singular stochastic control
viscosity solution
variational inequality
infinite-dimensional optimal stopping
smooth-fit principle
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