一类随机脉冲微分系统的稳定性

本文给出了随机脉冲微分系统零解的最终稳定性的定义,利用Liapunov函数,得到了非线性随机脉冲微分系统零解一致最终稳定性及一致最终渐进稳定性和最终不稳定的充分条件.     

Adaptive multi-element polynomial chaos with discrete measure: Algorithms and application to SPDEs

We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve partial differential equations with random parameters. The method is based on numerical construction of orthogonal polynomial bases in terms of a discrete probability measure. To this end, we compare the accuracy and efficiency of five different constructions. We develop an adaptive procedure for decomposition of the parametric space using the local variance criterion. We then couple the ME-PCM with sparse grids to study the Korteweg–de Vries (KdV) equation subject to random excitation, where the random parameters are associated with either a discrete or a continuous probability measure. Numerical experiments demonstrate that the proposed algorithms lead to high accuracy and efficiency for hybrid (discrete–continuous) random inputs.     

Implicit Stochastic Runge–Kutta Methods for Stochastic Differential Equations

In this paper we construct implicit stochastic Runge–Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

Sparse Data Representation of Random Fields

Mathematical models with uncertainties are often described by stochastic partial differential equations (SPDEs) with multiplicative noise. The coefficients, the right-hand side, the boundary conditions are modelled by random fields. As a result the solution is also a random field. We offer to use the Karhunen-Loève expansion (KLE) to compute a sparse data format for the fast generation and representation of these random fields. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of both, the sparse hierarchical matrix format as well as the low-rank Kronecker tensor format. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anticipative calculus for Lévy processes and stochastic differential equations*

We develop an anticipative calculus for Lévy processes with finite second moment for analysing anticipating stochastic differential equations. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals with respect to the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and present anticipative stochastic integration, the Wick product and the so-called 𝒮-transform. These concepts serve as tools for studying general Wick type stochastic differential equations with anticipative initial conditions. We apply the 𝒮-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.     

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.     

An algebra of Stein operators

We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are kth degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for the density of the product of k independent symmetric variance-gamma distributed random variables.     

A random differential transform method: Theory and applications

In this article, a Differential Transform Method (DTM) based on the mean fourth calculus is developed to solve random differential equations. An analytical mean fourth convergent series solution is found for a nonlinear random Riccati differential equation by using the random DTM. Besides obtaining the series solution of the Riccati equation, we provide approximations of the main statistical functions of the stochastic solution process such as the mean and variance. These approximations are compared to those obtained by the Euler and Monte Carlo methods. It is shown that this method applied to the random Riccati differential equation is more efficient than the two above mentioned methods.     

Mean periodic solutions of a linear inhomogeneous first-order differential equation with random coefficients

We obtain deterministic first-order linear differential equations with ordinary and variational derivatives and deterministic initial conditions for the expectation and the second moment function of the solution of an ordinary scalar first-order linear inhomogeneous differential equation whose coefficients are random processes. We derive existence conditions for mean periodic solutions. In particular, we consider Gaussian and uniformly distributed random coefficients.     

The parametrix method for parabolic SPDEs

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Itô–Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.     

On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument

We consider the application of the Krylov-Bogolyubov-Mitropol’skii asymptotic method and Runge-Kutta methods to the investigation of oscillating solutions of quasilinear second-order differential equations with random deviations of argument. For specific equations, we obtain approximate numerical solutions and characteristics of random oscillations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1433–1441, October, 1999.     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

A random Euler scheme for Carathéodory differential equations

We study a random Euler scheme for the approximation of Carathéodory differential equations and give a precise error analysis. In particular, we show that under weak assumptions, this approximation scheme obtains the same rate of convergence as the classical Monte–Carlo method for integration problems.     

On the Existence of Periodic Solutions for Certain Classes of Systems of Differential Equations with Random Pulse Influence

We establish conditions for the existence of periodic solutions for systems of differential equations with random right-hand side and random pulse influence at fixed times. We consider the case of small pulse perturbation and weakly nonlinear systems.     

On Markov Chain Approximations to Semilinear Partial Differential Equations Driven by Poisson Measure Noise

We consider the stochastic model of water pollution, which mathematically can be written with a stochastic partial differential equation driven by Poisson measure noise. We use a stochastic particle Markov chain method to produce an implementable approximate solution. Our main result is the annealed law of large numbers establishing convergence in probability of our Markov chains to the solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains.     

On the statistical estimation of the initial probability distribution based on the observations of dynamics at the end of an interval

We consider the problem of estimation of density of a random variable playing the role of initial value for a certain dynamics. The dynamics is defined by a differential equation whose solution is observable at the end of an interval. This problem is called the problem of estimation according to indirect observations. We propose a procedure for the estimation of density based on the method of transformation of measure along the integral curve in combination with kernel estimates.     

Equations with Random Gaussian Operators with Unbounded Mean Value

We consider an equation in a Hilbert space with a random operator represented as a sum of a deterministic, closed, densely defined operator and a Gaussian strong random operator. We represent a solution of an equation with random right-hand side in terms of stochastic derivatives of solutions of an equation with deterministic right-hand side. We consider applications of this representation to the anticipating Cauchy problem for a stochastic partial differential equation.     

A general stochastic fixed-point theorem for continuous random operators on stochastic domains

We prove that under certain conditions a continuous random operator with stochastic domain has a random fixed point provided that each realization has a (deterministic) fixed point. As a by-product we obtain a selection theorem for the interior of certain convex-valued measurable correspondences. We apply our results to obtain stochastic Krasnoselski- and Rothe-type theorems and existence results for random differential and integral equations.     

Théorème limite pour une équation différentielle à coefficient aléatoire à mémoire longue

We consider an ordinary differential equation depending on a small parameter and with a long-range random coefficient. We establish that the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion. The index of the fractional Brownian motion depends on the asymptotic behavior of the covariance function of the random coefficient. The proof of the convergence uses the T. Lyons theory of “rough paths”. To cite this article: R. Marty, C. R. Acad. Sci. Paris, Ser. I 338 (2004).