Nonlinear Stochastic Difference Equations Driven by Martingales

Abstract

We provide in this paper a systematic development of nonlinear stochastic difference equations driven by martingales (that depend on a spatial parameter); three such equations are considered. We begin with the existence and uniqueness of solutions and continue with the study of stochastic properties, such as the martingale and Markov properties, along with ? irreducibility and recurrence. We discuss in the final section the discrete-time flow and asymptotic flow properties of the solution process.     

Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

General Solution of Systems of Nonlinear Difference Equations with Continuous Argument

We investigate the structure of the general solution of a system of nonlinear difference equations with continuous argument in the neighborhood of an equilibrium state.     

Evolving Systems of Stochastic Differential Equations

Journal of Theoretical Probability - We introduce Evolving Systems of Stochastic Differential Equations. This model generalizes the well-known stochastic differential equations with Markovian...     

Instability of Certain Nonlinear Stochastic Differential Equations

We prove existence and uniqueness of the solution X^ε_t of the SDE, X^ε_t = εB_t + ∫^t₀u^{q −1} _ε(s, X^ε_t) ds, where X^ε_t is a one-dimensional process and u_ε(t, x) the density of X^ε_t (ε > 0, q > 1). We show that the closure of (X^ε_t; 0 ≤ t ≤ 1) with respect to Hölder norm, when ε goes to 0, is a.s. equal to an explicit family of continuous functions. We obtain similar results, considering SDE′s where the drift coefficient is equal to ± sgn(x) u(t, x).     

Asymptotic Stability of Nonlinear Stochastic Evolution Equations

Some results on the pathwise asymptotic stability of solutions to stochastic partial differential equations are proved. Special attention is paid in proving sufficient conditions ensuring almost sure asymptotic stability with a non-exponential decay rate. The situation containing some hereditary characteristics is also treated. The results are illustrated with several examples.     

Adaptive Stabilization of Nonlinear Stochastic Systems

The purpose of this paper is to study the problem of asymptotic stabilization in probability of nonlinear stochastic differential systems with unknown parameters. With this aim, we introduce the concept of an adaptive control Lyapunov function for stochastic systems and we use the stochastic version of Artstein's theorem to design an adaptive stabilizer. In this framework the problem of adaptive stabilization of a nonlinear stochastic system is reduced to the problem of asymptotic stabilization in probability of a modified system. The design of an adaptive control Lyapunov function is illustrated by the example of adaptively quadratically stabilizable in probability stochastic differential systems. Accepted 9 December 1996     

Stochastic Convolution-Type Heat Equations with Nonlinear Drift

Abstract

In this article, we study the solution of a class of stochastic convolution-type heat equations with nonlinear drift. For general initial condition and coefficients, we prove existence and uniqueness by using the characterization theorem and Banach's fixed-point theorem. We also give an implicit solution, which is a well-defined generalized stochastic process in a suitable distribution space. Finally, we investigate the continuous dependence of the solution on the initial data as well as the dependence on the coefficient.     

Homoclinic Orbits of Nonlinear Functional Difference Equations

In this paper, by using the critical point theory, we obtain the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of nontrivial homoclinic orbits which decay exponentially at infinity is obtained.      

一类非线性时滞差分方程的稳定性

得到了一类线性非自治时滞差分方程的零解的一致稳定、一致渐近稳定和全局渐近稳定的充分条件.     

非线性差分方程的多重周期解

用变分方法得到一类非线性差分方程多重周期解的存在性．我们的结果推广了Cai,Yu和Guo[Comput．Math．Appl．,52（2006）,1630-1647]的结果,并且这里给出的证明显著地简化了．     

Multitime Methods for Systems of Difference Equations

Systems of difference equations containing small parameters are studied by a constructive perturbation scheme analogous to the one developed by the authors for the study of differential equations. The method results in an averaging procedure for difference equations, and it is particularly well suited to certain highly oscillatory, nonlinear systems. The method is applied to problems from population genetics, pattern recognition, and the numerical analysis of stiff differential equations     

Sequential Systems of Linear Equations Algorithm for Nonlinear Optimization Problems with General Constraints

In Ref. 1, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints was proposed. At each iteration, this new algorithm only needs to solve four systems of linear equations having the same coefficient matrix, which is much less than the amount of computation required for existing SQP algorithms. Moreover, unlike the quadratic programming subproblems of the SQP algorithms (which may not have a solution), the subproblems of the SSLE algorithm are always solvable. In Ref. 2, it is shown that the new algorithm can also be used to deal with nonlinear optimization problems having both equality and inequality constraints, by solving an auxiliary problem. But the algorithm of Ref. 2 has to perform a pivoting operation to adjust the penalty parameter per iteration. In this paper, we improve the work of Ref. 2 and present a new algorithm of sequential systems of linear equations for general nonlinear optimization problems. This new algorithm preserves the advantages of the SSLE algorithms, while at the same time overcoming the aforementioned shortcomings. Some numerical results are also reported.     

Ergodicity for Nonlinear Stochastic Equations in Variational Formulation

This paper is concerned with nonlinear partial differential equations of the calculus of variation (see [13]) perturbed by noise. Well-posedness of the problem was proved by Pardoux in the seventies (see [14]), using monotonicity methods. The aim of the present work is to investigate the asymptotic behaviour of the corresponding transition semigroup P_t. We show existence and, under suitable assumptions, uniqueness of an ergodic invariant measure ν. Moreover, we solve the Kolmogorov equation and prove the so-called "identite du carre du champs". This will be used to study the Sobolev space W^1,2(H,ν) and to obtain information on the domain of the infinitesimal generator of P_t.