Generalized stochastic processes in algebras of generalized functions

Stochastic processes with paths in a generalized function algebra are defined and it is shown that there exists an embedding of generalized functional stochastic processes into such ones. Gaussian stochastic processes with paths in an algebra of generalized functions are characterized by their first and second moments and an application to stochastic differential equations is given.     

Generalized stochastic evolution equations

An approach to generalized stochastic evolution equations is presented which is based on a generalized Ito formula. This allows the consideration of interesting examples which are stochastic generalizations of evolution equations of mixed type or second order in time hyperbolic equations. It includes more standard material involving a Gelfand triple of spaces as a special case. Several examples are given which illustrate the use of the abstract theory presented.     

Sampling and reconstruction for shift-invariant stochastic processes

In this paper, combining stochastic processes with shift-invariant spaces, we introduce shift-invariant stochastic processes. It is a general case of the classical band-limited stochastic processes and a kind of non-band-limited stochastic processes. Two sampling theorems are obtained for the shift-invariant stochastic processes. The results for band-limited stochastic processes and shift-invariant spaces are generalized by our new results.     

Global stability and stabilization of more general stochastic nonlinear systems

This paper considers the global stability and stabilization of more general stochastic nonlinear systems. Due to the absence of the conventional assumptions (e.g., Lipschitz condition), the stochastic nonlinear systems under investigation may have more than one weak solution. However, the most associated results are only applicable to the stochastic systems having a unique strong solution, and therefore, it is meaningful to refine and extend the relevant concepts and methods to the more general case. In this paper, the concepts of stochastic stability in the more general sense are first introduced to cover the stochastic nonlinear systems having more than one weak solution. Then, the generalized stochastic Barbashin–Krasovskii theorem and LaSalle theorem are established, which present the criterions of stochastic stability for more general stochastic nonlinear systems. As one of the main contributions in this paper, we rigorously prove the generalized stochastic Barbashin–Krasovskii theorem. Moreover, based on the generalized theorems, the output-feedback and state-feedback stabilization are accomplished respectively for two classes of high-order stochastic nonlinear systems under rather weaker assumptions comparing to the existing literature.     

Generalized Brownian functionals and stochastic integrals

The purpose of this paper is two-fold; i) a new class of generalized Brownian functionals, in fact generalized linear functionals, is introduced and ii) generalized stochastic integrals based on creation operators are discussed. These topics are in line with the causal calculus of Brownian functionals.Communicated by H. H. Kuo     

Nonlinear stochastic differential equations containing generalized delta processes

In this paper, some classes of nonlinear stochastic differential equations involving the generalized delta process and its derivatives are considered in the framework of the Colombeau generalized stochastic process theory. All initial problems considered are proven to have a unique unbiased solution. Several examples are exposed that provide more insight into the assumptions made in the theorems.     

Moderate deviations for estimators under exponentially stochastic differentiability conditions

We introduce two exponentially stochastic differentiability conditions to study moderate deviations for M-estimators. Under a generalized exponentially stochastic differentiability condition, a moderate deviation principle is established. Some sufficient conditions of the exponentially stochastic differentiability and examples are also given.     

L-shaped decomposition of two-stage stochastic programs with integer recourse

We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.     

Strong solutions for stochastic partial differential equations of gradient type

Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the “distance” defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction–diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained.     

Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise

We introduce and study generalized stochastic derivatives on Kondratiev-type spaces of nonregular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to properties of stochastic derivatives in the Gaussian analysis. As an example, we calculate the generalized stochastic derivative of a solution of a stochastic equation with Wick-type nonlinearity. Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 737–758, June, 2008.     

About the complexity of two-stage stochastic IPs

Mathematical Programming - We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer...     

Three types of exact solutions to Wick-type generalized stochastic Korteweg-de Vries equation

A Wick-type generalized stochastic Korteweg-de Vries equation is researched. By means of Hermite transformation, white noise theory and Riccati equation mapping method, three types of exact solutions to the generalized stochastic Korteweg-de Vries equation, which include the functional solutions of hyperbolic-exponential type, trigonometric-exponential type and exponential type, are derived.     

Numerical analysis of stochastic oscillators on supercomputers

The problem of numerical analysis of stochastic differential equations (SDEs) with oscillating solutions is investigated. The expectation and variance of SDE numerical solutions are shown as functions of the mesh size of integrating the generalized Euler method. Results of some numerical experiments on the simulation of linear and nonlinear stochastic oscillators on the supercomputer of the Siberian Supercomputer Center are presented.     

On a class of stochastic differential equations arising from the stochastic approximation theory

The paper dealt with generalized stochastic approximation procedures of Robbins-Monro type. We consider these procedures as strong solutions of some stochastic differential equations with respect to semimartingales and investigate their almost sure convergence and mean square convergence     

Harnack inequality and strong Feller property for stochastic fast-diffusion equations

As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333-1350], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, this paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.     

Dynamics of a non-autonomous stochastic Gilpin-Ayala model

This paper is concerned with a stochastic non-autonomous Gilpin-Ayala model. First, it is shown that this model has a global positive solution. Then sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence of the solution are established. The critical number between weak persistence and extinction is obtained. Finally, the lower- and upper-growth rate of the solution are investigated. Several numerical figures are introduced to illustrate the results. Some recent results are improved and generalized.     

On the generalized 2-D stochastic Ginzburg-Landau equation

This paper proves the existence and uniqueness of solutions in a Banach space for the generalized stochastic Ginzburg-Landau equation with a multiplicative noise in two spatial dimensions. The noise is white in time and correlated in spatial variables. The condition on the parameters is the same as in the deterministic case. The Banach contraction principle and stochastic estimates in Banach spaces are used as the main tool.     

Stability analysis and optimal control of stochastic singular systems

In this paper, problems of stability and optimal control for a class of stochastic singular systems are studied. Firstly, under some appropriate assumptions, some new results about mean-square admissibility are developed and the corresponding LMI sufficient condition is given. Secondly, finite-time horizon and infinite-time horizon linear quadratic (LQ) control problems for the stochastic singular system are investigated, in which the coefficients are allowed to be random in control input and quadratic criterion. Some results involving new stochastic generalized Riccati equation are discussed as well. Finally, the proposed LQ control model for stochastic singular systems provides an appropriate and effective framework to study the portfolio selection problem in light of the recent development on general stochastic LQ problems.     

Doubly stochastic operators and rearrangement theorems

This paper characterizes doubly stochastic operators between two L¹ spaces of random variables in terms of convex functions on the real line. This characterization is then applied to proving some Hardy-Littlewood-Pólya-type rearrangement theorems. The conditional form of Jensen's inequality is also derived and a condition for equality obtained. Moreover, some known results concerning doubly stochastic operators are also generalized.     

On a Lie-theoretic approach to generalized doubly stochastic matrices and applications

In this article, we study generalized doubly stochastic matrices using the theory of Lie groups and Lie algebras. Applications to the inverse eigenvalue problem for symmetric doubly stochastic matrices are presented.