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In this article we present the multivariate approximation of time splitting random functions defined on a box or R N , N ∈ N , by neural network operators of quasi-interpolation type. We achieve these approximations by obtaining quantitative-type Jackson inequalities engaging the multivariate modulus of continuity of a related random function or its partial high-order derivatives. We use density functions to define our operators. These derive from the logistic and hyperbolic tangent sigmoid activation functions. Our convergences are both point-wise and uniform. The engaged feed-forward neural networks possess one hidden layer. We finish the article with a great variety of applications. [ABSTRACT FROM AUTHOR]

In this article we present the multivariate approximation of time splitting random functions defined on a box or RN,N∈N, by neural network operators of quasi-interpolation type. We achieve these approximations by obtaining quantitative-type Jackson inequalities engaging the multivariate modulus of continuity of a related random function or its partial high-order derivatives. We use density functions to define our operators. These derive from the logistic and hyperbolic tangent sigmoid activation functions. Our convergences are both point-wise and uniform. The engaged feed-forward neural networks possess one hidden layer. We finish the article with a great variety of applications.

Abstract Approximations are derived for both nonlinear heat equations and singularly perturbed nonlinear wave equations with highly oscillating random force on boundary and strong interaction. By a diffusion approximation method, if the interaction is large and the singular perturbation is small enough, the approximation of the nonlinear wave equation is an one dimensional stochastic ordinary differential equation with white noise from the boundary which is exactly the same as that of the nonlinear heat equation. [ABSTRACT FROM AUTHOR]

One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish sharp weighted Trudinger–Moser inequalities (Theorems 1.1, 1.2 and 1.3) and Caffarelli–Kohn–Nirenberg inequalities in the borderline case p = N (Theorems 1.5, 1.6 and 1.7) with best constants. Existence of extremal functions is also investigated for both the weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities. Radial symmetry of extremal functions for the weighted Trudinger–Moser inequalities are established (Theorem 1.4). Moreover, the sharp constants and the forms of the optimizers for the Caffarelli–Kohn–Nirenberg inequalities in some particular families of parameters in the borderline case p = N will be computed explicitly. Symmetrization arguments do not work in dealing with these weighted inequalities because of the presence of weights and the failure of the Polyá – Szegö inequality with weights. We will thus use a quasi-conformal mapping type transform and the corresponding symmetrization lemma to overcome this difficulty and carry out proofs of these results. As an application of the Caffarelli–Kohn–Nirenberg inequality, we also establish a weighted Moser–Onofri type inequality on the entire Euclidean space R 2 (see Theorem 1.8). [ABSTRACT FROM AUTHOR]

This paper deals with the L 2 − L ∞ filtering problem for stochastic systems with time-varying delay. First, based on the Bessel–Legendre inequality, a new stochastic integral inequality is established, which is called Bessel–Legendre stochastic inequality. Then, a new Lyapunov–Krasovskii functional is constructed for a stochastic time-varying delay system by utilizing Legende polynomials. With the help of the Bessel–Legendre stochastic inequality and the new Lyapunov–krasovskii functional, an L 2 − L ∞ filter is developed, which can guarantee the filtering error system to be asymptotically mean-square stable with a prescribed L 2 − L ∞ performance level. Finally, numerical examples are given to illustrate the effectiveness of the proposed filtering approach. The example results show that the proposed approach is less conservative than existing ones in designing the L 2 − L ∞ filter for the stochastic time-delay system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the topological structure of solution sets for stochastic evolution inclusions in Hilbert spaces when the semigroup is compact as well as noncompact. It is shown that the solution set is nonempty, compact, and anRδ-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As applications of the obtained results, an example is given. [ABSTRACT FROM PUBLISHER]

For any nonzero elements x; y in a normed space X, the angular and skew-angular distance is respectively defined byα [x; y] = ∥ x/∥x∥;-y∥y∣∥ avd β[x,y] =∥ x/∥x∥;-y∥y∣∥ . Also inequality α≤fiβcharacterizes inner product spaces. Operator ver- sion of αhas been studied by Pečarić, Rajić, and Saito, Tominaga, and Zou et al. In this paper, we study the operator version of βby using Douglas' lemma. We also prove that the operator version of inequality α≥ β holds for commutating normal operators. Some examples are presented to show essentiality of these conditions. [ABSTRACT FROM AUTHOR]

We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is defined over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem, we effectively convert stability of the SQVIP into that of a one stage stochastic program with stochastic cone constraints. Under some moderate conditions, we derive Hölder outer semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter. The stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with SQVIP constraints. [ABSTRACT FROM AUTHOR]

Variational inequality modeling, analysis and computations are important for many applications, but much of the subject has been developed in a deterministic setting with no uncertainty in a problem's data. In recent years research has proceeded on a track to incorporate stochasticity in one way or another. However, the main focus has been on rather limited ideas of what a stochastic variational inequality might be. Because variational inequalities are especially tuned to capturing conditions for optimality and equilibrium, stochastic variational inequalities ought to provide such service for problems of optimization and equilibrium in a stochastic setting. Therefore they ought to be able to deal with multistage decision processes involving actions that respond to increasing levels of information. Critical for that, as discovered in stochastic programming, is introducing nonanticipativity as an explicit constraint on responses along with an associated 'multiplier' element which captures the 'price of information' and provides a means of decomposition as a tool in algorithmic developments. That idea is extended here to a framework which supports multistage optimization and equilibrium models while also clarifying the single-stage picture. [ABSTRACT FROM AUTHOR]

We present a scalable computing framework for the solution stochastic multiobjective optimization problems. The proposed framework uses a nested conditional value-at-risk (nCVaR) metric to find compromise solutions among conflicting random objectives. We prove that the associated nCVaR minimization problem can be cast as a standard stochastic programming problem with expected value (linking) constraints. We also show that these problems can be implemented in a modular and compact manner using PLASMO (a Julia -based structured modeling framework) and can be solved efficiently using PIPS-NLP (a parallel nonlinear solver). We apply the framework to a CHP design study in which we seek to find compromise solutions that trade-off cost, water, and emissions in the face of uncertainty in electricity and water demands. [ABSTRACT FROM AUTHOR]

Stochastic convexity and its applications are very important in mathematics and probability (Aequationes Mathematicae 20:184-197, 1980). There are two well-known inequalities for convex stochastic processes: Jensen's inequality and Hermite-Hadamard's inequality. Recently, Hafiz (Stoch Anal Appl 22:507-523, 2004) has provided fractional calculus for some stochastic processes. The problem is how to formulate these inequalities for stochastic processes in the class of fractional calculus and that is what is done in this paper. Our results generalize the corresponding ones in the literature. [ABSTRACT FROM AUTHOR]

We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by the total variation flow, respectively, by a singular p -Laplace-type operator. We impose nonlinear zero Neumann boundary conditions and precisely investigate their connection with the coefficient fields of the noise. This solves an open problem posed in Barbu et al. (2013) [7] and Barbu and Röckner (2015) [10] . [ABSTRACT FROM AUTHOR]

Recently, in the class of convex stochastic processes, Kotrys (Aequat Math 83:143-151, ; Aequat Math 86:91-98, ) proposed upper and lower bounds of mean-square stochastic integrals by using Hermite-Hadamard inequality. This paper shows that these bounds can be refined. Our results extend and refine the corresponding ones in the literature. Finally, an open problem for further investigations is given. [ABSTRACT FROM AUTHOR]

In this paper we apply modified Newton method based on sample average approximation (SAA) to solve stochastic variational inequality with stochastic second-order cone constraints (SSOCCVI). Under some moderate conditions, the SAA solution converges to its true counterpart with probability approaching one at exponential rate as sample size increases. We apply the theoretical results for solving a class of stochastic second order cone complementarity problems and stochastic programming problems with stochastic second order cone constraints. Some illustrative examples are given to show how the globally convergent method works and the comparison results between our method and other methods. Furthermore, we apply this method to portfolio optimization with loss risk constraints problems. [ABSTRACT FROM AUTHOR]

A stochastic variational inequality is proposed to model an elasto-plastic oscillator excited by a filtered white noise. We prove the ergodic property of the process and characterize the corresponding invariant measure. This extends Bensoussan--Turi's method (2008, Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators. Appl. Math. Optim., 58, 1-27) with a significant additional difficulty of increasing the dimension. Two points boundary value problem in dimension 1 is replaced by elliptic equations in dimension 2. In the present context, Khasminskii's method (1980, Stochastic Stability of Differential Equations. The Netherlands: Sijthoff and Noordhof) leads to the study of degenerate Dirichlet problems with partial differential equations and non-local boundary conditions. [ABSTRACT FROM AUTHOR]

Lutwak and Zhang proposed the concept of $L_{p}$-centroid bodies. Then Feng and Wang gave the notion of general $L_{p}$-centroid bodies. In this article, based on the $L_{p}$-dual affine surface area, we address Shephard type problems for the general $L_{p}$-centroid bodies. [ABSTRACT FROM AUTHOR]

We describe the one-dimensional Čebyšëv subspaces of a JBW-triple M by showing that for a non-zero element x in M, $\mathbb{C}x$ is a Čebyšëv subspace of M if and only if x is a Brown-Pedersen quasi-invertible element in M. We study the Čebyšëv JBW-subtriples of a JBW-triple M. We prove that for each non-zero Čebyšëv JBW-subtriple N of M, exactly one of the following statements holds: We also provide new examples of Čebyšëv subspaces of classic Banach spaces in connection with ternary rings of operators. [ABSTRACT FROM AUTHOR]

In this paper, we study an Nicholson-type delay system with delays. New criteria for the boundedness and exponential stability of positive solutions of Nicholson-type system with time-varying delays are established by applying the fundamental solution matrix, inequality techniques and Lyapunov method. Two examples with their computer simulations are presented to illustrate the effectiveness of the theoretical findings. Our results are new and supplement some previously known ones. [ABSTRACT FROM AUTHOR]

This paper is concerned with solving a stochastic variational inequality problem (for short, SVIP) from a viewpoint of minimization of mixed conditional value-at-risk (CVaR). The regularized gap function for SVIP is used to define a loss function for the SVIP and mixed CVaR to measure the loss. In this setting, SVIP can be reformulated as a deterministic minimization problem. We show that the reformulation is a convex program for a huge class of SVIP under suitable conditions. Since mixed CVaR involves the plus function and mathematical expectation, the neural network smoothing function and Monte Carlo method are employed to get an approximation problem of the minimization reformulation. Finally, we consider the convergence of optimal solutions and stationary points of the approximation. [ABSTRACT FROM AUTHOR]

We consider the one dimensional symmetric simple exclusion process with additional births and deaths restricted to a subset of configurations where there is a leftmost hole and a rightmost particle. At a fixed rate birth of particles occur at the position of the leftmost hole and at the same rate, independently, the rightmost particle dies. We prove convergence to a hydrodynamic limit and discuss its relation with a free boundary problem. [ABSTRACT FROM AUTHOR]

We derive a new inequality involving either the F or Gamma distribution and its quantiles and call it either the F-Inequality or G-Inequality. The stochastic representation of the new inequality involves α, p ∈ (0, 1) such that if p > α and k > 1 with W being a random variable with an F(v1, v2) or Gamma(τ, θ) distribution, then 1/pP(W < Wp/k) > 1/αP(W < Wα/k), where for any γ between 0 and 1, Wγ is defined by α = P(W < Wα). The inequality reverses for k ∈ (0, 1); it becomes equality for k = 1 and, trivially, for k = ∞. This inequality seems to hold for a larger class of distributions that include for example the F, Gamma, Cauchy, and special cases of the Beta distribution and perhaps others. However, we provide rigorous proofs only for the F and Gamma distributions. [ABSTRACT FROM AUTHOR]

The notions of association and dependence of random variables, rearrangements, and heterogeneity via majorization ordering have proven to be most useful for deriving stochastic inequalities. In this survey article we first show that these notions are closely related to three basic inequalities in classical mathematical analysis: Chebyshev’s inequality, the Hardy-Littlewood-Pólya rearrangement inequality and Schur functions. We then provide a brief review of some of the recent results in this area. An overall objective is to illustrate that classical mathematical inequalities of this type play a central role in the developments of stochastic inequalities.

An introduction is presented in which the editor discusses articles in the issue on topics including the development of stochastic variational inequalities, the use of Lipschitzian maps, and the natural gas transmission process.

Abstract: In this paper, we compare the largest order statistics arising from independent heterogeneous gamma random variables based on the likelihood ratio order. Let be independent gamma random variables with having shape parameter and scale parameter , , and let denote the corresponding largest order statistic. Let denote the largest order statistic arising from independent and identically distributed gamma random variables with having shape parameter and scale parameter , the arithmetic mean of ’s. It is shown here that is stochastically greater than in terms of the likelihood ratio order. The result established here answers an open problem posed by Balakrishnan and Zhao (2013), and strengthens and generalizes some of the results known in the literature. Numerical examples are also provided to illustrate the main result established here. [Copyright &y& Elsevier]

Some new weakly singular Henry-Gronwall type integral inequalities with "maxima" are established in this paper. Applications to Caputo fractional differential equations with "maxima" are also presented. [ABSTRACT FROM AUTHOR]

Abstract: This paper aims to study stochastic variational inequalities with special emphasis to incorporate uncertainty in transportation models. The main contribution of this paper is two fold. First, we introduce the so-called elliptic regularization technique in the context of stochastic variational inequalities. Our motivation to study regularization is due to the fact that network problems lead naturally to monotone variational inequalities. Therefore, to employ computational methods which are designed for strongly monotone variational inequalities, we resort to the regularization. Second, we perform a thorough comparison of our approach which is a rigorous approach, with a commonly studied sample-path approach for stochastic variational inequalities. Two small scale network equilibrium problems are analyzed in detail to better illustrate the conceptual difference between the two approaches as well as the commonly used computational methods. [Copyright &y& Elsevier]

In this paper, we characterize the solution of a nonlinear reflected Backward Stochastic Differential Equations (BSDE) as the unique solution of a Stochastic Variational Inequality (SVI). This approach leads to a priori estimate for the increment of the predictable component of the solution of the reflected BSDE. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider stochastic affine variational inequality problems with nonlinear perturbation (for short, SVIPP). Firstly, we formulate SVIPP as an ERM problem that minimizes the expected residual of the so-called gap function. Furthermore, we study some properties of the ERM problem under some suitable conditions. By means of quasi-Monte Carlo method, we obtain a discrete approximation of ERM problem. Finally, we consider the convergence of optimal solutions and stationary points of the approximation problem as sample size increases. [Copyright &y& Elsevier]

The article discusses the criteria for the commutativity of non-negative operators and projections in terms of operator inequality in von Neumann algebra. It states that characterization of traces in all positive normal functional in von Neuman algebra was obtained by using the inequalities of the operator. It mentions that the continues function in von Neuman algebra were derived from Weierstrass' theorem on the uniform approximation of a continues polynomial interval in functional calculus. It adds that the replacement of the projection in operator inequalities using a non-negative operator to preserve its commutativity is impossible in several cases.

The mean square stability for invariant manifolds of nonlinear stochastic differential equations is considered. The stochastic stability analysis is reduced to the estimation of the spectral radius of some positive operator. For the important case of manifolds with codimension one, a constructive spectral analysis of this operator is carried out. On the basis of this spectral technique, parametrical criteria of the stochastic stability of limit cycle and 2-torus are developed. [ABSTRACT FROM AUTHOR]

Motivated by an investigation of finding necessary and sufficient conditions for the Kolmogorov strong law of large numbers for the general Gini's mean difference, new versions of the classical Lévy, Ottaviani, and Hoffmann-J\ooTrgensen inequalities are obtained for a sequence of Banach space valued random variables. No geometric conditions are imposed on the Banach space. An application to the general Gini's mean difference in a Banach space setting is presented. [ABSTRACT FROM PUBLISHER]

Spontaneous disease extinction can occur due to a rare stochastic fluctuation. We explore this process, both numerically and theoretically, in two minimal models of stochastic viral infection dynamics. We propose a method that reduces the complexity in models of viral infections so that the remaining dynamics can be studied by previously developed techniques for analyzing epidemiological models. Using this technique, we obtain an expression for the infection clearance time as a function of kinetic parameters. We apply our theoretical results to study stochastic infection clearance for specific stages of HIV and HCV dynamics. Our results show that the typical time for stochastic clearance of a viral infection increases exponentially with the size of the population, but infection still can be cleared spontaneously within a reasonable time interval in a certain population of cells. We also show that the clearance time is exponentially sensitive to the viral decay rate and viral infectivity but only linearly dependent on the lifetime of an infected cell. This suggests that if standard drug therapy fails to clear an infection then intensifying therapy by adding a drug that reduces the rate of cell infection rather than immune modulators that hasten infected cell death may be more useful in ultimately clearing remaining pockets of infection. [ABSTRACT FROM AUTHOR]

In this paper, an efficient approach is used to evaluate the stochastic variations of the bulk permittivity for complex periodic composite material. This method is based on the stochastic collation method together with a periodic 3-D finite-difference approach. Based on the proposed method, the influence of the input uncertainties (e.g., mixture inclusion permittivity, conductivity, volume fraction, and host cell periodicity) on the complex bulk permittivity is discussed. [ABSTRACT FROM AUTHOR]

We propose in this work an original estimator of the conditional intensity of a marker-dependent counting process, that is, a counting process with covariates. We use model selection methods and provide a nonasymptotic bound for the risk of our estimator on a compact set. We show that our estimator reaches automatically a convergence rate over a functional class with a given (unknown) anisotropic regularity. Then, we prove a lower bound which establishes that this rate is optimal. Lastly, we provide a short illustration of the way the estimator works in the context of conditional hazard estimation. [ABSTRACT FROM AUTHOR]

In this article notions of exponential dominance and uncertainty for weighted and unweighted distributions are explored and used to compare values of the informational energy function and the differential entropy. Stochastic inequalities and bounds for cross-discrimination and uncertainty measures in weighted and unweighted residual life distribution functions and related reliability measures are presented. Moment-type inequalities for the comparisons of weighted and unweighted residual life distributions are also presented. [ABSTRACT FROM PUBLISHER]

For stochastic differential equations of pure jumps, though the Poincaré inequality does not hold in general, we show that W1 H transportation inequalities hold for its invariant probability measure and for its process-level law on right continuous paths space in the L¹ -metric or in uniform metrics, under the dissipative condition. Several applications to concentration inequalities are given. [ABSTRACT FROM AUTHOR]

Abstract: The consideration of uncertainties plays an increasing role in the design of geotechnical structures. An important procedure in the uncertainty analysis is the reliability assessment whereby the required statistical quantities and distributions are assumed to be known exactly. Due to small sample observations and missing information this is not the case in practical applications and uncertainties in the stochastic parameters themselves have to be considered. In our study we estimate the parameter variation of soil properties by statistical procedures and perform an extended reliability analysis of a shallow foundation considering these uncertainties. We propose a novel approach based on a Taylor series expansion which enables the estimation of the failure probability variation very efficiently. [Copyright &y& Elsevier]

Two mathematical crystallization models describing structure formations in instability zones are proposed and justified. The first model, based on a phase field system, describes crystallization processes in binary alloys. The second model, based on a modified Biot model of a porous medium and the convective Cahn–Hilliard model, governs oriented crystallization. Physical interpretation and numerical analysis are discussed. Bibliography: 23 titles. Illustration: 5 figures. [ABSTRACT FROM AUTHOR]

In this paper stochastic relations and closure results for weighted and transformed distributions including the proportional hazards model are presented. Exponential approximations to the class of increasing failure rate (IFR) and decreasing failure rate (DFR) weighted distributions including transformed distributions with monotone weight functions are obtained. These include approximations via the proportional hazards and length-biased exponential distributions. Also, bounds and moment-type inequalities for weighted life distributions including proportional hazards models and some applications are presented. [ABSTRACT FROM AUTHOR]