# On a Class of Infinite-Dimensional Singular Stochastic Control Problems

## TL;DR

This paper investigates a class of infinite-dimensional singular stochastic control problems, providing a rigorous formulation, deriving optimality conditions, and explicitly solving a specific model using advanced mathematical techniques.

## Contribution

It introduces a rigorous formulation for infinite-dimensional control problems and derives explicit solutions using first-order optimality conditions.

## Key findings

- Explicit optimal control expression in a specific model
- Necessary and sufficient first-order optimality conditions
- Application of semigroup theory and convex analysis

## Abstract

We study a class of infinite-dimensional singular stochastic control problems with applications in economic theory and finance. The control process linearly affects an abstract evolution equation on a suitable partially-ordered infinite-dimensional space X, it takes values in the positive cone of X, and it has right-continuous and nondecreasing paths. We first provide a rigorous formulation of the problem by properly defining the controlled dynamics and integrals with respect to the control process. We then exploit the concave structure of our problem and derive necessary and sufficient first-order conditions for optimality. The latter are finally exploited in a specification of the model where we find an explicit expression of the optimal control. The techniques used are those of semigroup theory, vector-valued integration, convex analysis, and general theory of stochastic processes.