Title: Parameter-dependent Stochastic Optimal Control in Finite Discrete Time,
Abstract: We prove a general existence result in stochastic optimal control in discrete
time where controls take values in conditional metric spaces, and depend on the
current state and the information of past decisions through the evolution of a
recursively defined forward process. The generality of the problem lies beyond
the scope of standard techniques in stochastic control theory such as random
sets, normal integrands and measurable selection theory. The main novelty is a
formalization in conditional metric space and the use of techniques in
conditional analysis. We illustrate the existence result by several examples
including wealth-dependent utility maximization under risk constraints with
bounded and unbounded wealth-dependent control sets, utility maximization with
a measurable dimension, and dynamic risk sharing. Finally, we discuss how
conditional analysis relates to random set theory.