Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

Guo, Tiexin; Zhang, Erxin; Wang, Yachao; Guo, Zichen

doi:10.1016/j.jmaa.2019.123644

Cited by 23 publications

(26 citation statements)

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“…Besides, stimulated by the study of some problems in Banach space theory or financial applications, RN modules were also independently introduced and studied by Haydon, et al in [36] as a tool for the study of ultrapowers of Lebesgue-Bochner function spaces, by Hansen and Richard in [37] where a class of special random inner product modules (called conditional L 2 -spaces) was introduced and a conditional version of Riesz representation theorem was established in order to represent a dynamic equilibrium pricing function, and by Filipović, et al in [13] where the locally L 0 -convex topology was introduced in order to establish random convex analysis as a tool for the study of conditional convex risk measures. Now, the theory of RN modules has grown into a coherent whole-random functional analysis, see [22,29,30] for a detailed historical survey.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…What is more is that L 0 (F , X) is neither compact in general under the topology of convergence in probability, where X is a compact convex subset of a normed space B, but Proposition 6.3.3 of [6] or Corollary 2.10 of this paper shows that L 0 (F , X) is a T ε,λ -closed, L 0convex and sequentially compact subset of the σ-stable RN module L 0 (F , B). Since T ε,λ -closed L 0 -convex subsets involved in the main results of this paper are random sequentially compact but not compact in general, one can not count on that every continuous selfmapping on them has a fixed point, the fixed point theorems [10,29,2] previously established for RN modules or random operators motivate us to introduce the following σ-stable continuous mappings as follows.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this paper, we continue to prove that Proposition 3.1 of [10] is still true for a σ-stable T c -continuous mapping, see Lemma 3.6 of this paper. In [29,Theorem 4.1], for the study of dynamic portfolio selection problem, Guo, et al proved that a problem of solving a class of backward stochastic equations with a terminal condition in the RN module L p F0 (F T ) amounts to one of studying the fixed points of the associated operator on L p F0 (F T ), where the classical metric fixed point theorems were successfully generalized to RN modules. In particular, it was also pointed out in [29] that it was still an open problem to establish Schauder or Krasnoselskii-type fixed point theorem in RN modules, even at that time how to formulate such fixed point theorems was a problem.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In [29,Theorem 4.1], for the study of dynamic portfolio selection problem, Guo, et al proved that a problem of solving a class of backward stochastic equations with a terminal condition in the RN module L p F0 (F T ) amounts to one of studying the fixed points of the associated operator on L p F0 (F T ), where the classical metric fixed point theorems were successfully generalized to RN modules. In particular, it was also pointed out in [29] that it was still an open problem to establish Schauder or Krasnoselskii-type fixed point theorem in RN modules, even at that time how to formulate such fixed point theorems was a problem. In fact, the earliest random generalization of the classical Schauder theorem is Theorem 10 of [2], which was given for a random operator for the study of random integral equations and random equilibrium price vectors.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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The noncompact Schauder fixed point theorem in random normed modules and its applications

Guo¹,

Wang²,

Xu³

et al. 2021

Preprint

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Random normed modules (briefly, RN modules) are a random generalization of ordinary normed spaces, whose L 0 -norm induces two kinds of most useful topologies (called the (ε, λ)-topology and the locally L 0 -convex topology). The purpose of this paper is to generalize the classical Schauder fixed point theorem to RN modules under the two kinds of topologies. Motivated by the randomized version of the classical Bolzano-Weierstrass theorem, we first systematically and deeply study the random sequential compactness under the (ε, λ)-topology and random total boundedness under the locally L 0convex topology for a σ-stable subset of a σ-stable RN module, establishing the Hausdorff theorem on their equivalence, which allows us to construct the well defined random Schauder projection and countably many decompositions of the mapping in question so that we can prove Schauder fixed point theorem in a σ-stable RN module, namely every σ-stable continuous mapping (under either of the two topologies) of a random sequentially compact closed L 0 -convex subset into itself has a fixed point. The new fixed point theorem both unifies all the random generalizations currently available of the classical Brouwer or Schauder fixed point theorem and meets the need of the future applications of RN modules to stochastic analysis and stochastic finance.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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The noncompact Schauder fixed point theorem in random normed modules and its applications

Guo¹,

Wang²,

Xu³

et al. 2021

Preprint

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“…By the way, we also naturally consider applications of the theory of L 0 -convex compactness to the optimization problem of conditional convex risk measures, we will specially study it in [28] since conditional convex risk measures are not strictly L 0 -convex and coercive. The theory of L 0 -convex compactness can be also used to establish the fixed point theorem for nonexpansive mappings in complete random normed modules [27]. The remainder of this paper is organized as follows: Section 2 first recapitulates some known basic notions and facts and then introduces the concept of L 0 -convex compactness and develop its theory with a series of characterization theorems.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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.

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An exponential representation for random C₀–semigroups

Ming

Dai

Zhang

2022

Math Methods in App Sciences

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Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

Cited by 23 publications

References 43 publications

The noncompact Schauder fixed point theorem in random normed modules and its applications

The noncompact Schauder fixed point theorem in random normed modules and its applications

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

An exponential representation for random C₀–semigroups

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Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

Cited by 23 publications

References 43 publications

The noncompact Schauder fixed point theorem in random normed modules and its applications

The noncompact Schauder fixed point theorem in random normed modules and its applications

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

An exponential representation for random C0–semigroups

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Product

Resources

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

An exponential representation for random C₀–semigroups