Auto-Bcklund transformation and exact solutions of Wick-type stochastic Burgers equation

Burgers equation in random environment is studied. In order to give the exact solutions of random Burgers equation, we only consider the Wick-type stochastic Burgers equation which is the perturbation of the Burgers equation with variable coefficients by white noise W（t）=Bt, where Bt is a Brown motion. The auto-Baecklund transformation and stochastic soliton solutions of the Wick-type stochastic Burgers equation are shown by the homogeneous balance and Hermite transform. The generalization of the Wick-type stochastic Burgers equation is also studied.

Stochastic Viscosity Solutions for SPDEs with Discontinuous Coefficients

In this paper, a class of nonlinear stochastic partial differential equations with discontinuous coefficients is investigated. This study is motivated by some research on stochastic viscosity solutions under non-Lipschitz conditions recently. By studying the solutions of backward doubly stochastic differential equations with discontinuous coefficients and constructing a new approximation function fn to the coefficient f, we get the existence of stochastic viscosity sub-solutions (or super-solutions).The results of this paper can be seen as the extension and application of related articles.

Extended Jacobi Elliptic Function Rational Expansion Method and Its Application to （2＋1）-Dimensional Stochastic Dispersive Long Wave System

In this work, by means of a generalized method and symbolic computation, we extend the Jacobi elliptic function rational expansion method to uniformly construct a series of stochastic wave solutions for stochastic evolution equations. To illustrate the effectiveness of our method, we take the （2＋ 1）-dimensional stochastic dispersive long wave system as an example. We not only have obtained some known solutions, but also have constructed some new rational formal stochastic Jacobi elliptic function solutions.

UNIQUENESS OF THE MILD SOLUTION OF SEMILINEAR STOCHASTIC EVOLUTION EQUATION IN HILBERT SPACE

In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: [GRAPHICS] where W(t) is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is an infinitesimal generator of a strongly continuous semigroup s(t) on Y, f(t, y): [0, T] x Y --> Y, and G(t, y): [0, T] X Y --> L(H, Y), y0: OMEGA --> Y is a ramdom variable of square integrable. We apply theory of the semigroup and obtain two conclusions of uniqueness of the mild solution of (1) which include the corresponding results in [4].

THE SOLUTION OF RANDOM EIGENVALUE PROBLEM WITH SMALL STOCHASTIC PROCESSES

This paper considers an eigenvalue problem containing small stochastic processes. For every fixed is, we can use the Prufer substitution to prove the existence of the random solutions lambda(n) and u(n) in the meaning of large probability. These solutions can be expanded in epsilon regularly, and their correction terms can be obtained by solving some random linear differential equations.

THE PATHWISE SOLUTION FOR A CLASS OF QUASILINEAR STOCHASTIC EQUATIONS OF EVOLUTION IN BANACH SPACE Ⅲ

This is the third part of the papers with the same title. We will discuss the problem of convergence of the semi-implicit difference scheme for a class of quasilinear SEE, which generalize the Crandall's work to the stochastic case.

Backward Doubly Stochastic Differential Equations with Stochastic Non-Lipschitz Coefficients

In this paper,we prove an existence and uniqueness theorem for backward doubly stochastic differential equations under a new kind of stochastic non-Lipschitz condition which involves stochastic and timedependent condition.As an application,we use the result to obtain the existence of stochastic viscosity solution for some nonlinear stochastic partial differential equations under stochastic non-Lipschitz conditions.

Dynamics of a prey-predator system under Poisson white noise excitation

The classical Lotka-Volterra （LV） model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural environment has been modeled as Poisson white noise, is in- vestigated by using the stochastic averaging method. The averaged generalized It6 stochastic differential equation and Fokkerlanck-Kolmogorov （FPK） equation are derived for prey-predator ecosystem driven by Poisson white noise. Approximate stationary solution for the averaged generalized FPK equation is obtained by using the perturbation method. The effect of prey self-competition parameter e2s on ecosystem behavior is evaluated. The analytical result is confirmed by corresponding Monte Carlo （MC） simulation.

Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are surveyed.The existence and uniqueness of invariant measures are established for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white noises.Some further developments,problems,and challenges in this direction are also discussed.

Benefits of Stochastic Optimization for Scheduling Energy Storage in Wholesale Electricity Markets

We propose a two-stage stochastic model for optimizing the operation of energy storage. The model captures two important features: uncertain real-time prices when day-ahead operational commitments are made;and the price impact of charging and discharging energy storage. We demonstrate that if energy storage has full flexibility to make real-time adjustments to its day-ahead commitment and market prices do not respond to charging and discharging decisions, there is no value in using a stochastic modeling framework, i.e., the value of stochastic solution is always zero. This is because in such a case the energy storage behaves purely as a financial arbitrageur day ahead, which can be captured using a deterministic model.We show also that prices responding to its operation can make it profitable for energy storage to "waste" energy, for instance by charging and discharging simultaneously, which is normally sub-optimal. We demonstrate our model and how to calibrate the price-response functions from historical data with a practical case study.

AN APPROXIMATION SCHEME FOR BLACK-SCHOLES EQUATIONS WITH DELAYS

This paper addresses a finite difference approximation for an infinite dimensional Black-Scholesequation obtained by Chang and Youree (2007).The equation arises from a consideration ofan European option pricing problem in a market in which stock prices and the riskless asset prices havehereditary structures.Under a general condition on the payoff function of the option,it is shown thatthe pricing function is the unique viscosity solution of the infinite dimensional Black-Scholes equation.In addition,a finite difference approximation of the viscosity solution is provided and the convergenceresults are proved.