On a Class of Infinite-Dimensional Singular Stochastic Control Problems
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.
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Center for Mathematical Economics
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