Irreducibility and strong Feller property for non-linear SPDEs

Under some non-degeneracy condition, the strong Feller property and irreducibility are studied for non-linear stochastic partial differential equations driven by multiplicative noise within the framework called ‘variational approach’. Our result for irreducibility can be applied to equations with locally monotone coefficients. In some special cases, we discuss the Hölder continuity of the associated Markov semigroups. The main results are applied to several examples such as stochastic Burgers equation, stochastic porous media equation and stochastic fast diffusion equation.     

Averaging principle for multiscale stochastic reaction‐diffusion‐advection equations

Stochastic averaging principle is a powerful tool for studying qualitative analysis of multiscale stochastic dynamical systems. In this paper, we will establish an averaging principle for stochastic reaction‐diffusion‐advection equations with slow and fast time scales. Under suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

Basket options valuation for a local volatility jump–diffusion model with the asymptotic expansion method

In this paper we discuss the basket options valuation for a jump–diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expansion method to approximate the conditional expectation of the stochastic variance associated with the basket value process. The numerical tests show that the suggested method is fast and accurate in comparison with the Monte Carlo and other methods in most cases.     

Bogoliubov averaging principle of stochastic reaction–diffusion equation

The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation.     

Wong–Zakai approximations and attractors for stochastic reaction–diffusion equations on unbounded domains

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction–diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong–Zakai approximations of stochastic reaction–diffusion equation. Then, we show that the attractors of Wong–Zakai approximations converges to the attractor of the stochastic reaction–diffusion equation for both additive and multiplicative noise.     

Basket options valuation for a local volatility jump-diffusion model with the asymptotic expansion method

In this paper we discuss the basket options valuation for a jump-diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expansion method to approximate the conditional expectation of the stochastic variance associated with the basket value process. The numerical tests show that the suggested method is fast and accurate in comparison with the Monte Carlo and other methods in most cases.     

On a random scaled porous media equation

It is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well-posed in L^∞. In the fast diffusion case we show existence in Lp.     

Fast-diffusion limit with large noise for systems of stochastic reaction-diffusion equations

We consider a class of a stochastic reaction-diffusion equations with additive noise. In the limit of fast diffusion, one can approximate solutions of the stochastic reaction–diffusion equations by the solution of a suitable system of ordinary differential equation only describing the reactions, but due to nonlinear interaction of large diffusion and fluctuations in the limit new effective reaction terms appear. We focus on systems with polynomial nonlinearities and illustrate the result by applying it to a predator-prey system and a cubic auto-catalytic reaction between two chemicals.     

Weak Euler Approximation for Itô Diffusion and Jump Processes

This article studies the rate of convergence of the weak Euler approximation for Itô diffusion and jump processes with Hölder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backward Kolmogorov equation in Hölder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.     

Plongement stochastique des systèmes lagrangiens

We define an operator which extends classical differentiation from smooth deterministic functions to certain stochastic processes. Based on this operator, we define a procedure which associates a stochastic analog to standard differential operators and ordinary differential equations. We call this procedure stochastic embedding. By embedding Lagrangian systems, we obtain a stochastic Euler–Lagrange equation which, in the case of natural Lagrangian systems, is called the embedded Newton equation. This equation contains the stochastic Newton equation introduced by Nelson in his dynamical theory of Brownian diffusions. Finally, we consider a diffusion with a gradient drift, a constant diffusion coefficient and having a probability density function. We prove that a necessary condition for this diffusion to solve the embedded Newton equation is that its density be the square of the modulus of a wave function solution of a linear Schrödinger equation. To cite this article: J. Cresson, S. Darses, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.     

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large reaction term: The almost periodic framework

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large nonlinear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic nonlinear convection–diffusion equation which we explicitly derive in terms of appropriate functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of $\Sigma$-convergence for stochastic processes.     

Cahn-Hilliard stochastic equation: Strict positivity of the density

This paper deals with the Cahn-Hilliard stochastic equation driven by a space-time white noise with a non-linear diffusion coefficient. Using new lower estimate of the kernel, we prove the "local" existence of the density without non-degeneracy condition in a case of Hölder continuous trajectories, and we show that the density of any vector is lower bounded by a strictly positive continuous function under a non-degeneracy condition.     

零漂移广义双曲线扩散过程构建和最优鞅估计函数

由于广义双曲线分布在资产收益率分布拟合中的优异表现,以规模测度和速度测度为工具,构建出边际分布服从广义双曲线分布的扩散过程.利用1.5阶强泰勒近似法离散化随机微分方程,以二次最优鞅估计函数法得到模型参数估计量,结果显示:鞅估计函数法能够快速、准确地对正态逆高斯扩散过程作出参数估计,并且能获得高精度渐近协方差矩阵.     

与年龄相关的模糊随机种群扩散系统的指数稳定性

讨论了一类与年龄相关的模糊随机种群扩散系统,系统受两种不确定性因素的影响,即随机和模糊.在有界和Lipschitz条件下,利用Ito公式和Gronwall引理,建立了均方意义下与年龄相关的模糊随机种群扩散系统指数稳定性的判定准则并通过数值例子对所给出的结论进行了验证.     

与年龄相关的模糊随机种群扩散系统的均方散逸性

讨论了一类与年龄相关的模糊随机种群扩散系统,该系统受随机和模糊两种不确定性因素的影响.在有界的条件(弱于线性增长条件)和Lipschitz条件下,利用It公式和Bellman-Gronwall-Type引理,建立了均方意义下与年龄相关的模糊随机种群扩散系统均方散逸性的判定准则.并通过数值例子对所给出的结论进行了验证.     

Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u_t= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.     

Weak martingale solutions to the stochastic Landau–Lifshitz–Gilbert equation with multi-dimensional noise via a convergent finite-element scheme

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. By using the Doss–Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent $\theta$-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.     

Lower bounds for densities of uniformly elliptic random variables on Wiener space

 In this article, we generalize the lower bound estimates for uniformly elliptic diffusion processes obtained by Kusuoka and Stroock. We define the concept of uniform elliptic random variable on Wiener space and show that with this definition one can prove a lower bound estimate of Gaussian type for its density. We apply our results to the case of the stochastic heat equation under the hypothesis of unifom ellipticity of the diffusion coefficient. Received: 6 November 2001 / Revised version: 27 February 2003 / Published online: 12 May 2003 Key words or phrases: Malliavin Calculus – Density estimates – Aronson estimates