L⁰-convex compactness and its applications to random convex optimization and random variational inequalities

Abstract

First, this paper introduces the notion of L 0 -convex compactness for a special class of closed convex subsets-closed L 0 -convex subsets of a Hausdorff topological module over the topological algebra L 0 (F, K), where L 0 (F, K) is the algebra of equivalence classes of random variables from a probability space (Ω, F, P ) to the scalar field K of real numbers or complex numbers, endowed with the topology of convergence in probability. Then, this paper continues to develop the theory of L 0convex compactness by establishing various kinds of characterization theorems on L 0 -convex compactness for L 0 -convex subsets of a class of important topological modules-complete random normed modules, in particular, we make full use of the theory of random conjugate spaces to establish the characterization theorem of James type on L 0 -convex compactness for a closed L 0 -convex subset of a complete random normed module, which also surprisingly implies that our notion of L 0convex compactness coincides with GordanŽitković's notion of convex compactness in the context of a closed L 0 -convex subset of a complete random normed module. As the first application of our results, we give a fundamental theorem on random convex optimization (or, L 0 -convex optimization), which includes Hansen and Richard's famous result as a special case. As the second application, we give an existence theorem of solutions of random variational inequalities, which generalizes H.Brezis' classical result from a reflexive Banach space to a random reflexive complete random normed module. It should be emphasized that a new method, namely the L 0convex compactness method, is presented for the second application since the usual weak compactness method is no longer applicable in the present case. Besides, our fundamental theorem on random convex optimization can be also applied in the study of optimization problems of conditional convex risk measures, which will be given in our future papers.

No-arbitrage concepts in topological vector lattices

“…Remark 5. 13 If the convex subset B is closed, then it is bounded in probability (which means that K 1 satisfies NUPBR) if and only if it is convexly compact; see [45] for further details. Now, we consider K 0 and (K α ) α>0 together.…”

“…Remark 6. 13 Let us have a closer look at the situation U = L ∞ . Suppose that the convex cone K 0 satisfies NFLVR.…”

“…Definition 7.12 A strategy δ ∈ (S) and the corresponding portfolio S δ are called self-financing for S if S δ = S δ 0 + δ • S. Definition 7. 13 We denote by sf (S) the set of all self-financing strategies for S.…”

“…However, the space (L 0 , ≤) is an example of a topological vector lattice; indeed it is even a so-called Fréchet lattice. The properties of the space L 0 and in particular its positive cone L 0 + have already been studied in the literature, often with a focus to applications in finance; see, for example [3,11,13,22,[24][25][26][27]30,43,45]. We also mention the related paper [4], where the equivalence between economic viability and no-arbitrage in the presence of Knigthian uncertainty has been studied, and the recent paper [31], where a reverse approach to model uncertainty is presented.…”

“…13 By Lemma 3.11 the following statements are equivalent:(i) K 1 satisfies NUPBR. (ii) B α is topologically bounded for all α > 0.…”