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, taking values in conditional metric spaces, depend on the current information and past decisions. The general form of the problem lies beyond the scope of standard techniques in stochastic control theory, the main novelty is a formalization in conditional metric space and the use of conditional analysis. We illustrate the existence result by several examples such as wealth-dependent utility maximization under risk constraints and utility maximization with a conditional dimension. We also provide a discussion as to how our methods compare to techniques based on random sets.

Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations

“…In the case X = R d (d ∈ N), Kabanov and M. Safarian ([19], Proposition 5.4.3) proved that a stable set is sequentially closed if and only if it is the set of measurable selectors of a random closed set. This was generalized in [20], Theorem 5.1 to an arbitrary Polish space X ; see also ([16], Theorem 5.4.1). We complement this result by providing the corresponding Boolean valued representation.…”

“…It is well known that stable compact sets are represented by compact sets in the Boolean valued model; see, e.g., [21,22], which amount to the equivalence (1) ⇔ (2) below in the present context. In conditional set theory, it is used the terminology conditional compactness; see [3,4,20]. In particular, it was proven in ([23], Theorem 5.12) and ([16], Theorem 5.4.2) that, in the case that X = R (d ∈ N), a set is stably compact if and only it is the set of measurable selectors of a random compact set.…”

Random optimization on random sets

“…The same approach is also used in the setting of set-valued risk measures in presence of proportional transaction costs, [7,6], where the random orders defined by the solvency sets play a crucial role. More recently, a conditional analysis approach is considered to solve stochastic optimal problems in discrete-time [8] with applications in Finance and Economics. Similarly, a backward approach is implemented in [1].…”

Notions, Definitions, and Models

“…With the deep development of the theory of random conjugate spaces, Guo [15] earlier found that some basic theorems involving w * -compactness and weak compactness for normed spaces are no longer valid for general complete random normed modules under random w * -topology and random weak topology. On the other hand, to provide a simplifying proof of noarbitrage criteria, Kabanov and Stricker [35] proved the randomized Bolzano-Weierstrass theorem, which states that every almost surely bounded sequence of random variables with values in the Euclidean space admits an almost surely convergent randomized subsequence although it does not admit any almost surely convergent subsequence, which also motivates the subsequent development of the theory of random sequential compactness [21,33]. Motivated by the work [14,15,35,5], Drapeau, et.al [2] presented the notions of conditional sets and conditional topology with an attempt to provide the theory of conditional compactness suitable for the further development of random functional analysis or more general conditional analysis.…”

“…Contrasted with this, the theory of random metric spaces as a random generalization of ordinary metric spaces had not made much substantive progress for a quite long time although random metric spaces were earliest presented in random functional analysis. With the notion of d-σ-stability for subsets of a random metric space presented in [33], such a situation was beginning to change, a series of deep developments on random metric spaces were achieved [23]. Although similar to σ-stability, d-σ-stability depends on random metric structure rather than an L 0 -module structure.…”