In J. Wen, Y. Shi (2020) [39] 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.

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,·) 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 → G a random asymptotically nonexpansive mapping, then (I-f) is random demiclosed at θ, namely, for each sequence {x_n, n∈ℕ} in G, if {x_n, n∈ℕ} converges in σ(E, E^*) to x and {(I-f)x_n, n∈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.

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.

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.

Based on both the fundamental theorem of affine geometry in regular L0-modules and the recent progress in random convex analysis, this paper characterizes the stable and fully order preserving and order reversing operators acting on the class of proper lower semicontinuous L0-convex functions 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 commutative family of strong random nonexpansive operators from (Ω ,ℱ,P)× C to C has a common random fixed point, where (Ω ,ℱ,P) is a probability space and C is a weakly compact convex subset with normal structure of a Banach space.

In this paper, we first establish the following fixed point theorem for a random asymptotically nonexpansive mapping, which can be regarded as a random generalization of the classical Goebel-Kirk fixed point theorem: let ( E, & Vert;& sdot;& Vert;) be a complete random uniformly convex random normed module and G be an almost surely bounded closed L0-convex subset of E , then every random asymptotically nonexpansive mapping f from G to G has a fixed point in G . Second, we prove that the set Y of fixed points off is closed and L0-convex. Finally, we show that every eventually random asymptotically nonexpansive mapping f also has a fixed point. Since the classical method used to prove the Goebel-Kirk fixed point theorem for an asymptotically nonexpansive mapping does not work directly for the current random setting, we are forced to make use of the connection between the random uniform convexity of the complete random normed module ( E, & Vert;& sdot;& Vert;) and the uniform convexity of the abstract L p ( E )-space generated by E , where p is a given positive number with 1 < p < +infinity . Specifically, we decompose a random asymptotically nonexpansive operator on G into a sequence of smaller operators on a bounded closed convex subset of Lp(E) such that each smaller operator is a classical eventually asymptotically nonexpansive mapping on the corresponding bounded closed convex subset. Consequently, by using the a-stability off and G , we can establish a precise relation between the fixed point set off and the fixed point sets of these smaller operators, which makes us finally complete the proofs of the above mentioned main results.

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 L-0-convex subset of a random normed module, then every sigma-stable T-c-upper semicontinuous mapping F : G -> 2(G) \ {empty set} such that F(x) is closed and L-0-convex for each x is an element of 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 [Guo et al., Math. Ann. 391(3), 3863-3911 (2025)].

Motivated by the stochastic differential game problem, in this paper we introduce and study a more general class of anticipated backward stochastic Volterra integral equations (anticipated BSVIEs, for short) whose generator includes both pointwise time-advanced functions and average time-advanced functions. 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. Moreover, by virtue of the duality principle, 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 theorem, a Nash equilibrium point of the linear-quadratic differential game problem of SDVIEs is obtained. It should be pointed out that optimal controls of stochastic Volterra integral equations can be regarded as a special case of our game problem.

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 relevant 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.

Based on the recently developed theory of sigma-stable sets and stable compactness, we first establish the random Markov-Kakutani fixed point theorem in a random locally convex module: let (E,P) be a random locally convex module and G be a nonempty stably compact L-0-convex subset of E, then every commutative family of T-c(P-cc)-continuous L-0-affine mappings from G to G has a common fixed point, where P-cc is the sigma-stable hull of P and T-c(P-cc) is the locally L-0-convex topology induced by P-cc. Second, we prove that the random Markov-Kakutani fixed point theorem implies the algebraic form of the known random Hahn-Banach theorem. Finally, we establish a more general strict separation theorem in a random locally convex module, which provides not only a more general geometric form of the random Hahn-Banach theorem but also another proof for the random Markov-Kakutani fixed point theorem. Therefore, as a byproduct, the work of this paper also shows that the algebraic and geometric forms of the random Hahn-Banach theorem are equivalent. It should be pointed out that the main challenge in this paper lies in overcoming noncompactness since a stably compact set is generally noncompact.

By making full use of the theory of random sequential compactness in random normed modules, in this paper we establish a noncompact Dotson fixed point theorem: if C is a σ–stable random sequentially compact L0–star–shaped subset of a random normed module, then every random nonexpansive mapping T:C→C has a fixed point. Furthermore, we obtain an existence result for best approximations in random normed modules: let E be a random normed module, T:E→E a random nonexpansive mapping with a fixed point u and C a closed, σ–stable and T–invariant subset of E such that T(C)‾ is random sequentially compact, then the set of best approximations of u in C is nonempty, which generalizes the classical result of Smoluk. In addition, we also get an existence result for invariant approximations in random normed modules. A significant distinction between the proofs of our results in random normed modules and the corresponding classical results in normed spaces is that the σ–stability of both the sets and mappings involved in the random setting plays a prominent part in the proofs of the main results of this paper.

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 systematically develop the related theory. Based on 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 $\sigma$--stable continuous mapping from $G$ to $G$ has a fixed point, which unifies all the previous random generalizations of Schauder fixed point theorem. As one of 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 difficulty of our whole paper lies in overcoming noncompactness since a random sequentially compact set is very often noncompact.

First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of d- σ -stability in a random metric space can be regarded as a special case of the notion of σ -stability in 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^p -normed L^∞ -module is exactly generated by a complete random normed module so that the gluing property of an L^p -normed L^∞ -module can be derived from the σ -stability of the generating random normed module, as applications the direct relation between module duals and random conjugate spaces are given. Third, we prove that a random normed space is order complete iff it is (ε ,λ ) -complete, 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 of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric 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.

Failure probability (FP) estimation problem is a 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). Considering the remarkable achievements of neural networks, we aim to leverage this powerful tool for surrogate modeling and sampling purposes. 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. The relative error of FP estimation achieved by the proposed methods is consistently below ten percent.

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.

本文首先深入研究随机局部凸模中的稳定紧集,证明它关于（ε,λ）-拓扑T ε,λ 是完备的,并给出它的一个简明的特征,即一个σ-稳定集是稳定紧的当且仅当它的每个具有有限交性质的由σ-稳定的T ε,λ -闭子集组成的σ-稳定族必有非空交.在此基础上,对定义在稳定紧集上的σ-稳定的、真的、下半连续的■-值函数给出相应的Weierstrass定理,并由此证明一个稳定紧的L~0-凸集必为L~0-凸紧的.然后,对L~0-凸集引进L~0-端点的概念并对L~0-凸紧集证明相应的Krein-Milman定理,同时给出这个推广的Krein-Milman定理与经典的Krein-Milman定理的某些有趣的比较与联系.最后,作为应用,证明定义在L~0-凸紧集上的真下半连续L~0-拟凸函数f必达到最小值.进一步地,如果f还是L~0-仿射的,那么f的最小值也可以在L~0-端点达到.

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 attain its minimum, namely the converse of the classical Weierstrass theorem also holds true.

Let (Omega, F, P) be a probability space, R the scalar field of real numbers, L degrees(F, R) the equivalence classes of R-valued F-measurable random variables on Omega, (E, || . ||) and (F, || . ||) two complete random normed modules over R with base (Omega, F, P). The main theorem of this paper is the following approximation result for random delta-nearsurjective epsilon-isometries between random normed modules: if f : E -> F is a stable random delta-nearsurjective epsilon-isometry with f(0) = 0, where epsilon, delta is an element of L degrees (F, R) and epsilon, delta >= 0, then there exists a surjective L degrees -linear random isometry U between E and F such that || f(x) - U(x)|| <= 4e for all x is an element of E. Furthermore, making use of the above result and the relations between random normed modules and classical normed spaces, we give the approximation result for sample-continuous random operators: let (X,|| . ||) and (Y,|| . ||) be two real separable Banach spaces, epsilon degrees and delta degrees two nonnegative random variables and f : Omega x X -> Y a random operator such that f(w,.) : X -> Y is a continuous delta degrees (w)-nearsurjective epsilon degrees (w)-isometry and f(w, 0) = 0 for any w epsilon Omega, then there exist a sample-linear and almost everywhere (briefly, a.e.) isometric random operator U : Omega X -> Y and Omega(0) is an element of F with P(Omega(0)) = 1 such that || f (w, x) - U(w, x) I < 4 epsilon degrees (w),for all(w, x) E Omega(0) x X. It is the first time that sample-linear and a.e. isometric random operators are used to approximate sample-continuous nonlinear random operators.

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.