We study a class of infinite-dimensional singular stochastic control problems that might find 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. Our main contribution is to provide a rigorous formulation of the problem by properly defining the controlled dynamics and integrals with respect to the control process, and then to derive necessary and sufficient first-order conditions for optimality. The latter are finally exploited in a specification of the model where we determine an optimal control. The techniques used are those of semigroup theory, vector-valued integration, convex analysis, and general theory of stochastic processes.
On a class of infinite-dimensional singular stochastic control problems
Federico S.;
2021-01-01
Abstract
We study a class of infinite-dimensional singular stochastic control problems that might find 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. Our main contribution is to provide a rigorous formulation of the problem by properly defining the controlled dynamics and integrals with respect to the control process, and then to derive necessary and sufficient first-order conditions for optimality. The latter are finally exploited in a specification of the model where we determine an optimal control. The techniques used are those of semigroup theory, vector-valued integration, convex analysis, and general theory of stochastic processes.
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