Second order properties of distribution tails and estimation of tail exponents in random difference equations

According to a celebrated result of Kesten (Acta Math 131:207–248, 1973), random difference equations have a power-law distribution tail in the asymptotic sense. Empirical evidence shows that classical estimators of tail exponent of random difference equations, such as Hill estimator, are extremely biased for larger values of tail exponents. It is argued in this work that the bias occurs because the power-tail region is too far in the tail from a practical perspective. This is supported by analysis of a few examples where a stationary distribution of random difference equation is known explicitly, and by proving a weaker form of the so-called second order regular variation of distribution tails of random difference equations, which measures deviations from the asymptotic power tail. The latter, in particular, suggests a specific second order term for a distribution tail. Estimation of tail exponents can be adapted by taking this second order term into account. One such method available in the literature is examined, and a new, simple, regression type estimator is proposed. Simulation study shows that the proposed estimator works very well. ARCH models of interest in Finance and multiplicative cascades used in Physics are considered as motivating examples throughout the work. Extension to multidimensional random difference equations with nonnegative entries is also considered.     

Solution of partial differential equations by a modified random walk

A new Monte Carlo technique is applied to solve difference equations of elliptic and parabolic partial differential equations with given boundary values. Fixed random walk is extended to modified random walk, whereby a random walk is made on a maximum square. The average number of steps and the computational time in a modified random walk is much less than in a fixed random walk. Numerical examples support the utility of this method.     

Mean square convergent three points finite difference scheme for random partial differential equations

In this paper, the random finite difference method with three points is used in solving random partial differential equations problems mainly: random parabolic, elliptic and hyperbolic partial differential equations. The conditions of the mean square convergence of the numerical solutions are studied. The numerical solutions are computed through some numerical case studies.     

Relationship between spectral and coefficient criteria of mean-square stability for systems of linear stochastic differential and difference equations

We establish the relationship (equivalence) between the spectral and algebraic (coefficient) criteria (the latter is represented in terms of the Sylvester matrix algebraic equation) of mean-square asymptotic stability for three classes of systems of linear equations with varying random perturbations of coefficients, namely, the ltô differential stochastic equations, difference stochastic equations with discrete time, and difference stochastic equations with continuous time.     

Green's functions for elliptic and parabolic equations with random coefficients II

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

     

Investigation of the Solutions of Linear Systems of Difference Equations with Random Coefficients

The aim of the present paper is to give an equation which defines the moments equations for the solutions of linear systems of difference equations with random coefficients. Further we consider the cases when the coefficients depend on or do not depend on a Markov chain. Moreover, we investigate the stability of the null solutions for a linear system of difference equations in the mean and in the mean square.     

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.     

The correlation matrix of random solutions of a dynamical system with Markov coefficients

For dynamical systems which are described by systems of differential or difference equations dependent on a finite-valued Markov process, we suggest a new form of equations for moments of their random solution. We derive equations for a correlation matrix of random solutions. Kiev Economic Institute, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 338–348, March, 1999.     

A probability wave description of stochastic variables whose means satisfy a system of difference equations

Probability wave theory is used to study the behavior of stochastic vectors whose means satisfy ordinary first-order difference equations. Difference-differential equations are given for the probability waves corresponding to the difference model for the means. Analogues of the Liouville and Ehrenfest theorems are proved. A first-order difference equation for the evolution of the component dispersion of the random vector is obtained. An algorithm for solving the wave equations is proposed. The results from analyzing some solutions to the probability wave equations are presented. The relationship of the finite-difference method to the manifestation of the particle-wave dualism is discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115. No. 1, pp. 56–76. April. 1998.     

ASYMPTOTICALLY MEAN ALMOST PERIODIC SOLUTIONS TO RANDOM DIFFERENCE EQUATIONS

In this paper,we first investigate some basic properties of asymptotically mean almost periodic random sequences on Z + and then show some properties of asymptotically mean almost periodic solutions to random difference equations.     

On the asymptotic optimality of orthoregressional estimators

It is shown that the orthoregressional (STLS) parameter estimators in linear algebraic systems (including autonomous difference equations with matrix coefficients) converge to the maximum likelihood estimators and thus become asymptotically best in the limit case of large variances of the random coordinates on the variety of solutions to the system observed with additive random perturbations.     

Numerical solutions of random mean square Fisher-KPP models with advection

This paper deals with the construction of numerical stable solutions of random mean square Fisher-Kolmogorov-Petrosky-Piskunov (Fisher-KPP) models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear inhomogeneous system of random differential equations. Then, by extending to the random framework, the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity, the results are illustrated by comparing the results with a test problem where the exact solution is known.     

Asymptotic Behavior of Solutions of Some Difference Equations Defined by Weakly Dependent Random Vectors

In this article, we consider not only stochastic differential equations driven by the Wiener process but also by processes with stationary increments from the view points of time series analysis for mathematical finance. Corresponding to Black-Scholes type stochastic differential equations, we consider difference equations defined by weakly dependent sequence of random vectors and examine the asymptotic behavior of their solutions.     

A Monte Carlo method for finite difference equations of elliptic type in a multiregion domain

This paper describes a Monte Carlo procedure for the solution of elliptic difference equations requiring a previous subdivision of the integation domain into convex regions. Using Green's function for these regions, it is shown how it is possible to avoid processing random walks inside each region.     

模糊随机有限元平衡方程的摄动解法*

对模糊随机有限元平衡方程作λ水平截集,得随机区间平衡方程,然后基于平衡方程中有关力学量之间的关系,将随机区间平衡方程转化为两类普通随机平衡方程求解,利用小参数摄动理论导得求随机区间位移的递归方程组.文中还详细推导了计算模糊随机位移、模糊随机应变和模糊随机应力数字特征的计算公式.     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

On the distribution of the first exit time and overshoot in a two-sided boundary crossing problem

We consider a random walk generated by a sequence of independent identically distributed random variables. We assume that the distribution function of a jump of the random walk equals an exponential polynomial on the negative half-axis. For double transforms of the joint distribution of the first exit time from an interval and overshoot, we obtain explicit expressions depending on finitely many parameters that, in turn, we can derive from the system of linear equations. The principal difference of the present article from similar results in this direction is the rejection of using factorization components and projection operators connected with them.     

Exponential Stability for Discrete Time Linear Equations Defined by Positive Operators

In this paper the problem of exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear positive operators acting on an ordered finite dimensional Hilbert space is investigated. The class of linear equations considered in this paper contains as particular cases linear equations described by Lyapunov operators or symmetric Stein operators as well as nonsymmetric Stein operators. Such equations occur in connection with the problem of mean square exponential stability for a class of difference stochastic equations affected by independent random perturbations and Markovian jumping as well us in connection with some iterative procedures which allow us to compute global solutions of discrete time generalized symmetric or nonsymmetric Riccati equations. The exponential stability is characterized in terms of the existence of some globally defined and bounded solutions of some suitable backward affine equations (inequalities) or forward affine equations (inequalities).     

Difference equations with random delay

We present a first step towards a general theory of difference and differential equations incorporating unbounded random delays. The main technical tool relies on recent work of Lian and Lu, which generalizes the multiplicative ergodic theorem by Oseledets to Banach spaces.     

The role of coefficients of a general SPDE on the stability and convergence of a finite difference method

In this paper for the approximate solution of stochastic partial differential equations (SPDEs) of Itô-type, the stability and application of a class of finite difference method with regard to the coefficients in the equations is analyzed. The finite difference methods discussed here will be either explicit or implicit and a comparison between them will be reported. We prove the consistency and stability of these methods and investigate the influence of the multiplier (particularly multiplier of the random noise) in mean square stability. From stochastic version of Lax-Richtmyer the convergence of these methods under some conditions are established. Numerical experiments are included to show the efficiency of the methods.