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.

A transfer principle for second-order arithmetic, and applications

“…The following minimum theorem was proved in [10,Theorem 4.4] in finite dimensions, and in full generality in Jamneshan and Zapata [38, Theorem 5.13], and applied in e.g. Cheridito et al [8] and Jamneshan et al [37] successfully to stochastic control.…”

“…Proof By proposition 4.6, one can identify X with a conditionally compact set in L 0 (R) n . From a result in Jamneshan et al [37,Section 5], one can identify f with a sequentially lower semi-continuous 8 function L 0 (R) n → L 0 (R). Theorem 4.8 proves the claim.…”

“…19 Conditional metric spaces were introduced in full generality in Drapeau et al [15]. Conditional metric spaces were applied in Jamneshan et al [37] to stochastic control. It was argued there that they are a viable alternative to the established measurable selection techniques.…”

“…There is a practical interest in such model-theoretic results since the transfer principle is a highly efficient tool which replaces the tedious work of proving by hand conditional versions of classical results which are relevant in applications. Existing areas of application include probability (Jamneshan et al [36]), mathematical economics (Backhoff and Horst [3], Bielecki et al [5], Cheridito et al [8], Drapeau and Jamneshan [14], Filipovic et al [20], Frittelli and Maggis [23,24], Hansen and Richard [31], and Kabanov and Stricker [39]), stochastic optimization (Cheridito and Hu [9], Cheridito and Stadje [11], and Jamneshan et al [37]), set-valued analysis and measurable selection theory (Jamneshan et al [37] and Jamneshan and Zapata [38]), Lebesgue-Bochner spaces and vector duality (Drapeau et al [17] and Grad and Jamneshan [26]), and probabilistic analysis (Guo [27,28], Guo et al [29], Guo and Zhang [30], and Jamneshan and Zapata [38]).…”

Measures and Integrals in Conditional Set Theory

“…See [6,11,12,15,20,23] for further results in conditional analysis and conditional set theory. For applications of conditional analysis, we refer to [2,3,5,7,8,10,13,14,17,19]. See [4,16] for other related results.…”