The dynamic programming equation for stochastic optical control in hilbert spaces: a variational approach

The global existence of a point wise solution to the Hamilton-Jacobi equation for totally observed controlled diffusions in Hilbert spaces is proved by studying the corresponding control problem. The optimality principle for the control problem leads to local results, whilst an a priori bound is achieved by introducing a secondary minimization problem.     

Finite Difference Approximations for Stochastic Control Systems with Delay

Abstract

This article considers the computation issues of the infinite dimensional HJB equation arising from the finite horizon optimal control problem of a general system of stochastic functional differential equations with a bounded memory treated in [2 Chang , M.H. , Pang , T. , and Pemy , M. accepted. Optimal control of functional stochastic differential equations with a bounded memory. Stochastics 80 ( 1 ): 69 – 96 . [Google Scholar]]. The finite difference scheme, using the result in [1 Barles , G. , and Souganidis , P.E. 1991 . Convergence of approximative schemes for fully nonlinear second order equations . J. Asymptotic Analysis 4 : 557 – 579 . [Google Scholar]], is obtained to approximate the viscosity solution of the infinite dimensional HJB equation. The convergence of the scheme is proved using the Banach fixed point theorem. The computational algorithm also is provided based on the scheme obtained.     

A Strong Approximation Theorem for Stochastic Recursive Algorithms

The constant stepsize analog of Gelfand–Mitter type discrete-time stochastic recursive algorithms is shown to track an associated stochastic differential equation in the strong sense, i.e., with respect to an appropriate divergence measure.     

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     

巴拿哈空间中随机年龄结构种群方程的数值分析

本文研究年龄结构随机种群方程的离散误差,在空间离散中用到Galerkin公式,时间离散中用到显式欧拉公式.     

Almost Sure Asymptotic Stability of Stochastic Volterra Integro-Differential Equations with Fading Perturbations

Abstract

In this note, we address the question of how large a stochastic perturbation an asymptotically stable linear functional differential system can tolerate without losing the property of being pathwise asymptotically stable. In particular, we investigate noise perturbations that are either independent of the state or influenced by the current and past states. For perturbations independent of the state, we prove that the assumed rate of fading for the noise is optimal.     

A formula for the lyapunov numbers of a stochastic flow. application to a perturbation theorem

We present a formula for the Lyapunov exponents of the flow of a nonlinear stochastic system. (These exponents characterise the asymptotic behaviour of the derivative flow, and negative exponents are associated with clustering of the flow). This formula is analogous to that of Khas'minskii, who deals with a linear system. We use this fojoruila to show that if we have an ordinary dynamical system which is Lyapunov stable (i.e. all the exponents are negative) then so are certain stochastic perturbations of it.     

Monte-Carlo estimation of time-dependent statistical characteristics of random dynamical systems

The problem of estimating the time-dependent statistical characteristics of a random dynamical system is studied under two different settings. In the first, the system dynamics is governed by a differential equation parameterized by a random parameter, while in the second, this is governed by a differential equation with an underlying parameter sequence characterized by a continuous time Markov chain. We propose, for the first time in the literature, stochastic approximation algorithms for estimating various time-dependent process characteristics of the system. In particular, we provide efficient estimators for quantities such as the mean, variance and distribution of the process at any given time as well as the joint distribution and the autocorrelation coefficient at different times.     

Implicit Scheme for Stochastic Parabolic Partial Diferential Equations Driven by Space-Time White Noise

We extend Rothe's method of solving linear parabolic PDEs to the case of nonlinear SPDEs driven by space-time white noise. When the nonlinear terms are Lipschitz functions we prove almost sure convergence of the approximations uniformly in time and space. When the nonlinear drift term is only measurable we obtain the convergence in probability, by using Malliavin calculus.     

Analysis of a stochastic predator-prey system with foraging arena scheme

ABSTRACT

This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results.     

On some elliptic optimal control problems with state constraints

In this paper optimality conditions will be derived for elliptic optimal control problems with a restriction on the state or on the gradient of the state. Essential tools are the method of transposition and generalized trace theorems and green's formulas from the theory of elliptic differential equations.     

Remarks on non-linear noise excitability of some stochastic heat equations

We consider nonlinear parabolic SPDEs of the form ${\partial }_{t}u=\Delta u+\lambda \sigma \left(u\right)\dot{w}$ on the interval $(0,L)$, where $\dot{w}$ denotes space–time white noise, $\sigma$ is Lipschitz continuous. Under Dirichlet boundary conditions and a linear growth condition on $\sigma$, we show that the expected $L^2$-energy is of order $exp[\text{const}\times {\lambda }^4]$ as $\lambda \to \infty$. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result in Khoshnevisan and Kim.