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Tiexin Guo

Publications and source records attributed to Tiexin Guo.

At least 19 recordsLinked to original sources
arXiv2026

Survey of Metric fixed point theory in random functional analysis

Based on the idea of randomizing the traditional space theory of functional analysis, random functional analysis has been developed as functional analysis over random metric spaces, random normed modules and random locally convex modules. Since these random frameworks have much more complicated algebraic, topological and geometric structures than their prototypes, the development of fixed point theory in random functional analysis had been almost stagnant before 2010. Unexpectedly, with the deep development of stable set theory fixed point theory in random functional analysis, including both its metric and topological fixed point theory, has made considerable progress in the recent 15 years. The purpose of this paper is to survey the important progress in metric fixed point theory in random functional analysis, including the random Banach contraction mapping principle and Caristi fixed point theorem on complete random metric spaces, and fixed point theorems for random nonexpansive and asymptotically nonexpansive mappings in complete random normed modules. Besides, the connections among the topics surveyed, random equations and random fixed point theorems for random operators are also briefly mentioned.

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arXiv2026

Anticipated backward stochastic Volterra integral equations and their applications to nonzero-sum stochastic differential games

In [J. Wen, Y. Shi, Stat. Probab. Lett. 156 (2020) 108599] the authors first introduced a kind of anticipated backward stochastic Volterra integral equations (anticipated BSVIEs, for short). By virtue of the duality principle, it is found in this paper that the anticipated BSVIEs can be applied to the study of stochastic differential games. Naturally, in order to develop the related theories and applications of BSVIEs, in this paper we deeply investigate a more general class of anticipated BSVIEs whose generator includes both pointwise and average time-advanced functions. In theory, the well-posedness and the comparison theorem of anticipated BSVIEs are established, and some regularity results of adapted M-solutions are proved by applying Malliavin calculus, which cover the previous results for BSVIEs. Further, using linear anticipated BSVIEs as the adjoint equation, we present the maximum principle for the nonzero-sum differential game system of stochastic delay Volterra integral equations (SDVIEs, for short) for the first time. As one of the applications of the principle, a Nash equilibrium point of the linear-quadratic differential game problem of SDVIEs is obtained.

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arXiv2026

A random demiclosedness principle for random asymptotically nonexpansive mappings

By making full use of the inherent connection between the theory of random conjugate spaces and the theory of classical conjugate spaces, in this paper we establish a random demiclosedness principle for a random asymptotically nonexpansive mapping, which generalizes Xu's classical demiclosedness principle from a uniformly convex Banach space to a complete random uniformly convex random normed module: let $(E,\|\cdot\|)$ be a complete random uniformly convex random normed module, $E^{*}$ the random conjugate space of $E$, $G$ an almost surely bounded closed $L^{0}$-convex subset of $E$ and $f: G \rightarrow G$ a random asymptotically nonexpansive mapping, then $(I-f)$ is random demiclosed at $θ$, namely, for each sequence $\{x_{n}, n\in \mathbb{N}\}$ in $G$, if $\{x_{n}, n\in \mathbb{N}\}$ converges in $σ(E, E^{*})$ to $x$ and $\{(I-f)x_{n}, n\in \mathbb{N}\}$ converges to $θ$, then $(I-f)x=θ$, where $I$ denotes the identity operator on $E$ and $σ(E, E^{*})$ the random weak topology on $E$.

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arXiv2025

A new complete proof of the random Brouwer fixed point theorem and its implied consequences of unification

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of $L^{0}$-simplicial subdivisions of $L^{0}$-simplexes. Based on this, we are able to achieve a new complete proof of the random Brouwer fixed theorem in random Euclidean spaces, which can provide a solid foundation for various contemporary applications of interest. Afterward, we unify the works currently available and closely related to the random Brouwer fixed theorem: we first prove that the stochastic Brouwer fixed point theorem occurring elsewhere in stochastic analysis is equivalent to a special case of our random Brouwer fixed theorem, and then prove a general random Borsuk theorem and its equivalence with the random Brouwer fixed theorem. Finally, we conclude this paper with commentaries on recent state of study of the famous Schauder conjecture.

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arXiv2025

The random Kakutani fixed point theorem in random normed modules

Based on the recently developed theory of random sequential compactness, we prove the random Kakutani fixed point theorem in random normed modules: if G is a random sequentially compact L0-convex subset of a random normed module, then every -stable Tc-upper semicontinuous mapping F:G to 2G such that F(x) is closed and L0-convex for each x in G, has a fixed point. This is the first fixed point theorem for set-valued mappings in random normed modules, providing a random generalization of the classical Kakutani fixed point theorem as well as a set-valued extension of the noncompact Schauder fixed point theorem established in Math. Ann. 391(3), 3863--3911 (2025).

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arXiv2024

Common fixed point theorems for a commutative family of nonexpansive mappings in complete random normed modules

In this paper, we first introduce and study the notion of random Chebyshev centers. Further, based on the recently developed theory of stable sets, we introduce the notion of random complete normal structure so that we can prove the two deeper theorems: one of which states that random complete normal structure is equivalent to random normal structure for an $L^0$-convexly compact set in a complete random normed module; the other of which states that if $G$ is an $L^0$-convexly compact subset with random normal structure of a complete random normed module, then every commutative family of nonexpansive mappings from $G$ to $G$ has a common fixed point. We also consider the fixed point problems for isometric mappings in complete random normed modules. Finally, as applications of the fixed point theorems established in random normed modules, when the measurable selection theorems fail to work, we can still prove that a family of strong random nonexpansive operators from $(Ω,\mathcal{F},P)\times C$ to $C$ has a common random fixed point, where $(Ω,\mathcal{F},P)$ is a probability space and $C$ is a weakly compact convex subset with normal structure of a Banach space.

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arXiv2024

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

Motivated by the randomized version of the classical Bolzano--Weierstrass theorem, in this paper we first introduce the notion of a random sequentially compact set in a random normed module and develop the related theory systematically. From these developments, we prove the corresponding Schauder fixed point theorem: let $E$ be a random normed module and $G$ a random sequentially compact $L^0$--convex set of $E$, then every $σ$--stable continuous mapping from $G$ to $G$ has a fixed point, which unifies all the previous random generalizations of the Schauder fixed point theorem. As one of the applications of the theorem, we prove the existence of Nash equilibrium points in the context of conditional information. It should be pointed out that the main challenge in this paper lies in overcoming noncompactness since a random sequentially compact set is generally noncompact.

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arXiv2024

The relations among the notions of various kinds of stability and their applications

First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application of which, it is easy to see that the notion of $d$-$σ$-stability introduced for a nonempty subset of a random metric space can be regarded as a special case of the notion of $σ$-stability introduced for a nonempty subset of a random normed module, as another application we give the final version of the characterization for a $d$-$σ$-stable random metric space to be stably compact. Second, we prove that an $L^{\infty}$-module is an $L^{p}$-normed $L^{\infty}$-module iff it is generated by a complete random normed module, from which it is easily seen that the gluing property of an $L^{p}$-normed $L^{\infty}$-module can be derived from the $σ$-stability of the generating random normed module, as applications the known and new basic facts of module duals for $L^{p}$-normed $L^{\infty}$-modules can be obtained, in a simple and direct way, from the theory of random conjugate spaces of random normed modules. Third, we prove that a random normed space is order complete iff it is complete with respect to the $(\varepsilon,λ)$-topology, as an application it is proved that the $d$-decomposability of an order complete random normed space is exactly its $d$-$σ$-stability. Finally, we prove that an equivalence relation on the product space $X\times B$ of a nonempty set $X$ and a complete Boolean algebra $B$ is regular iff it can be induced by a $B$-valued Boolean metric $d$ on $X$, as an application it is proved that a nonempty subset of a Boolean set $(X,d)$ is universally complete iff it is a $B$-stable set defined by a regular equivalence relation.

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arXiv2023

Well-posedness and regularity of mean-field backward doubly stochastic Volterra integral equations and applications to dynamic risk measures

In this paper, the theory of mean-field backward doubly stochastic Volterra integral equations (MF-BDSVIEs) is studied. First, we derive the well-posedness of M-solutions to MFBDSVIEs, and prove the comparison theorem for such a type of equations. Furthermore, the regularity result of the M-solution for MF-BDSVIEs is established by virtue of Malliavin calculus. Finally, as an application of the comparison theorem, we obtain the properties of dynamic risk measures governed by MF-BDSVIEs.

math.PR↗
arXiv2023

Adaptive design of experiment via normalizing flows for failure probability estimation

Failure probability estimation problem is an crucial task in engineering. In this work we consider this problem in the situation that the underlying computer models are extremely expensive, which often arises in the practice, and in this setting, reducing the calls of computer model is of essential importance. We formulate the problem of estimating the failure probability with expensive computer models as an sequential experimental design for the limit state (i.e., the failure boundary) and propose a series of efficient adaptive design criteria to solve the design of experiment (DOE). In particular, the proposed method employs the deep neural network (DNN) as the surrogate of limit state function for efficiently reducing the calls of expensive computer experiment. A map from the Gaussian distribution to the posterior approximation of the limit state is learned by the normalizing flows for the ease of experimental design. Three normalizing-flows-based design criteria are proposed in this work for deciding the design locations based on the different assumption of generalization error. The accuracy and performance of the proposed method is demonstrated by both theory and practical examples.

stat.ME↗
arXiv2022

Some nonlinear characterizations of reflexive Banach spaces

It is well known that in the calculus of variations and in optimization there exist many formulations of the fundamental propositions on the attainment of the infima of sequentially weakly lower semicontinuous coercive functions on reflexive Banach spaces. By either some constructive skills or the regularization skill by inf--convolutions we show in this paper that all these formulations together with their important variants are equivalent to each other and equivalent to the reflexivity of the underlying space. Motivated by this research, we also give a characterization for a normed space to be finite dimensional: a normed space is finite dimensional iff every continuous real--valued function defined on each bounded closed subset of this space can obtain its minimum, namely the converse of the classical Weierstrass theorem also holds true.

math.FA↗
arXiv2021

The fundamental theorem of affine geometry in regular $L^0$-modules

Let $(Ω,{\mathcal F},P)$ be a probability space and $L^0({\mathcal F})$ the algebra of equivalence classes of real-valued random variables defined on $(Ω,{\mathcal F},P)$. A left module $M$ over the algebra $L^0({\mathcal F})$(briefly, an $L^0({\mathcal F})$-module) is said to be regular if $x=y$ for any given two elements $x$ and $y$ in $M$ such that there exists a countable partition $\{A_n,n\in \mathbb N\}$ of $Ω$ to $\mathcal F$ such that ${\tilde I}_{A_n}\cdot x={\tilde I}_{A_n}\cdot y$ for each $n\in \mathbb N$, where $I_{A_n}$ is the characteristic function of $A_n$ and ${\tilde I}_{A_n}$ its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular $L^0({\mathcal F})$-modules: let $V$ and $V^\prime$ be two regular $L^0({\mathcal F})$-modules such that $V$ contains a free $L^0({\mathcal F})$-submodule of rank $2$, if $T:V\to V^\prime$ is stable and invertible and maps each $L^0$-line segment to an $L^0$-line segment, then $T$ must be $L^0$-affine.

math.FA↗
arXiv2020

The Hille-Yosida generation theorem for almost surely bounded $C_{0}$--semigroups of continuous module homomorphisms$^1$

In this paper, we first study some properties peculiar to $C_{0}$--semigroups of continuous module homomorphisms and give a characterization for such a $C_{0}$--semigroup to be almost surely bounded. Then, based on these, we establish the Hille-Yosida generation theorem for almost surely bounded $C_{0}$--semigroups of continuous module homomorphisms, which generalizes some known results. Moreover, the counterexample constructed in this paper also shows that it is necessary to require the almost sure boundedness for such $C_{0}$--semigroups.

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arXiv2019

Optimization of conditional convex risk measures

Optimization of conditional convex risk measure is a central theme in dynamic portfolio selection theory, which has not yet systematically studied in the previous literature perhaps since conditional convex risk measures are neither random strictly convex nor random coercive. The purpose of this paper is to give some basic results on the existence and uniqueness on this theme, in particular our results for conditional monotone mean--variance and conditional entropic risk measures are complete and deep. As the basis for the work of this paper, this paper first begins with a brief introduction to random functional analysis, including the historical backgrounds for its birth and some important advances. This paper then further surveys some recent progress in random convex analysis and its applications to conditional convex risk measures. Finally, based on these, we establish a concise sufficient and necessary condition for a return to be a solution to the optimization problem of conditional monotone mean--variance. We also make use of the recently developed theory of $L^0$--convex compactness to establish the existence of the optimization problem of conditional entropic risk measure when the conditional mean of returns is given and the returns fall within a random closed ball. Besides, the related uniqueness problems are also solved.

math.OC↗
arXiv2019

$L^0$--convex compactness and its applications to random convex optimization and random variational inequalities

In 2010, Gordan Žitković introduced the notion of convex compactness for a convex subset of a linear topological space and gave some important applications to both nonlinear analysis and mathematical economics in [ Gordan Žitković, Convex compactness and its applications, Math. Finance Econom. 3(1) (2010) 1--12 ]. Motivated by Gordan Žitković's idea, in this paper we introduce the notion of $L^0$--convex compactness for an $L^0$--convex subset of a topological module over the topological algebra $L^0(\mathcal{F},K)$, where $L^0(\mathcal{F},K)$ is the algebra of equivalence classes of random variables from a probability space $(Ω,\mathcal{F},P)$ to the scalar field $K$ of real numbers or complex numbers, endowed with the topology of convergence in probability. This paper continues to develop the theory of $L^0$--convex compactness by establishing various kinds of characterization theorems for $L^0$--convex subsets of a class of important topological modules--complete random normed modules, in particular, we make use of the theory of random conjugate spaces to give a characterization theorem of James type for a closed $L^0$--convex subset of a complete random normed module. As applications, we successfully generalize some basic theorems of classical convex optimization and variational inequalities from a convex function on a reflexive Banach space to an $L^0$--convex function on a random reflexive random normed module. Since the usual weak compactness method fails in the random setting of this paper and in particular, since the difficulties caused by the partial order structure of the range of an $L^0$--valued function also frequently occurs in the study of problems involved in this paper, we are forced to discover a series of new skills to meet the needs of this paper.

math.FA↗
arXiv2019

On $d$--$σ$--stability in random metric spaces and its applications

In 2010, the first author of this paper introduced the notion of $σ$--stability for a nonempty subset of an $L^0(\mathcal{F},K)$--module in [T.X. Guo, Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal. 258(2010), 3024--3047], this kind of $σ$--stability is purely algebraic and leads to a series of deep developments of random normed modules and random locally convex modules. Motivated by this, A. Jamneshan, M. Kupper and J. M. Zapata recently introduced another kind of $σ$--stability for a nonempty subset of a random metric space $(E,d)$, called $d$--$σ$--stability since it depends on the random metric $d$. $d$--$σ$--stability coincides with the previous $σ$--stability in the case of random normed modules, which motivates us in this paper to generalize the precise form of Ekeland's variational principle from a complete random normed module to a complete $d$--$σ$--stable random metric space. Besides, this paper also utilize $d$--$σ$--stability to generalize Nadler's fixed point theorem for a multivalued contraction mapping from a complete metric space to a complete random metric space. To our surprise, our simple fixed point theorem, however, can derive the known basic fixed point theorems of contraction type for both random operators and $σ$--stable mappings on a complete random normed module. A lot of examples shows the study of random metric spaces is more complicated than that of random normed modules.

math.FA↗
arXiv2019

$L^0$--convex compactness and random normal structure in $L^0(\mathcal{F},B)$

Let $(B,\|\cdot\|)$ be a Banach space, $(Ω,\mathcal{F},P)$ a probability space and $L^0(\mathcal{F},B)$ the set of equivalence classes of strong random elements (or strongly measurable functions) from $(Ω,\mathcal{F},P)$ to $(B,\|\cdot\|)$. It is well known that $L^0(\mathcal{F},B)$ becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue--Bochner function spaces and random functional analysis. Let $V$ be a closed convex subset of $B$ and $L^0(\mathcal{F},V)$ the set of equivalence classes of strong random elements from $(Ω,\mathcal{F},P)$ to $(B,\|\cdot\|)$, the central purpose of this paper is to prove the following two results: (1). $L^0(\mathcal{F},V)$ is $L^0$--convexly compact if and only if $V$ is weakly compact; (2). $L^0(\mathcal{F},V)$ has random normal structure if $V$ is weakly compact and has normal structure. As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules and Banach space techniques, can be applied in the other related aspects.

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