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.     

SPDE in Hilbert space with locally monotone coefficients

The aim of this paper is to extend the usual framework of SPDE with monotone coefficients to include a large class of cases with merely locally monotone coefficients. This new framework is conceptually not more involved than the classical one, but includes many more fundamental examples not included previously. Thus our main result can be applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic p-Laplace equation and stochastic porous media equation with non-monotone perturbations.     

Full-Discrete Finite Element Method for Stochastic Hyperbolic Equation

This paper is concerned with the finite element method for the stochastic wave equation and the stochastic elastic equation driven by space-time white noise. For simplicity, we rewrite the two types of stochastic hyperbolic equations into a unified form. We convert the stochastic hyperbolic equation into a regularized equation by discretizing the white noise and then consider the full-discrete finite element method for the regularized equation. We derive the modeling error by using "Green's method" and the finite element approximation error by using the error estimates of the deterministic equation. Some numerical examples are presented to verify the theoretical results.     

Optimal control for two-dimensional stochastic second grade fluids

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional. We show that the Gâteaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.     

A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions

Nonlinear partial differential equation with random Neumann boundary conditions are considered. A stochastic Taylor expansion method is derived to simulate these stochastic systems numerically. As examples, a nonlinear parabolic equation (the real Ginzburg-Landau equation) and a nonlinear hyperbolic equation (the sine-Gordon equation) with random Neumann boundary conditions are solved numerically using a stochastic Taylor expansion method. The impact of boundary noise on the system evolution is also discussed.     

Stochastic stability of Burgers equation

The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space. Motivated by the numerical results of Hairer and Voss, we prove that the Burgers equation is stochastic stable in the sense that statistically steady regimes of fluid flows of stochastic Burgers equation converge to that of determinstic Burgers equation as noise tends to zero.     

利率均值回复模型下标的资产期权定价

在利率均值回复金融市场中 ,给出了财富贴现过程的随机微分方程 ;证明了与之联系的倒向随机微分方程解的存在唯一性 .最后 ,从倒向随机微分方程的解出发 ,得到了欧式期权定价的条件期望定价公式 .     

Existence theorem for a stochastic wave equation with discontinuous unbounded coefficients

We prove an existence theorem for stochastic hyperbolic equations with measurable locally bounded coefficients. A solution of a stochastic hyperbolic equation is understood as a martingale solution of the stochastic inclusion corresponding to the equation.     

Mean–variance portfolio selection under a constant elasticity of variance model

This paper discusses a mean–variance portfolio selection problem under a constant elasticity of variance model. A backward stochastic Riccati equation is first considered. Then we relate the solution of the associated stochastic control problem to that of the backward stochastic Riccati equation. Finally, explicit expressions of the optimal portfolio strategy, the value function and the efficient frontier of the mean–variance problem are expressed in terms of the solution of the backward stochastic Riccati equation.     

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.     

Gaussian density estimates for the solution of singular stochastic Riccati equations

Stochastic Riccati equation is a backward stochastic differential equation with singular generator which arises naturally in the study of stochastic linear-quadratic optimal control problems. In this paper, we obtain Gaussian density estimates for the solutions to this equation.     

Pathwise Stability of Degenerate Stochastic Evolutions

For linear stochastic evolution equations with linear multiplicative noise, a new method is presented for estimating the pathwise Lyapunov exponent. The method consists of finding a suitable (quadratic) Lyapunov function by means of solving an operator inequality. One of the appealing features of this approach is the possibility to show stabilizing effects of degenerate noise. The results are illustrated by applying them to the examples of a stochastic partial differential equation and a stochastic differential equation with delay. In the case of a stochastic delay differential equation our results improve upon earlier results.     

Large deviations and approximations for slow-fast stochastic reaction-diffusion equations

A large deviation principle is derived for a class of stochastic reaction-diffusion partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.     

由L\'{e}vy过程驱动的仿射方程关联的无限时区的最优二次控制

讨论线性二次最优控制问题, 其随机系统是由 L\'{e}vy 过程驱动的具有随机系数而且还具有仿射项的线性随机微分方程. 伴随方程具有无界系数, 其可解性不是显然的. 利用 $\mathscr{B}\mathscr{M}\mathscr{O}$ 鞅理论, 证明伴随方程在有限 时区解的存在唯一性. 在稳定性条件下, 无限时区的倒向随机 Riccati 微分方程和伴随倒向随机方程的解的存在性是通过对应有限 时区的方程的解来逼近的. 利用这些解能够合成最优控制.     

Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

Stochastic elastic equation driven by fractional Brownian motion

In this paper, we consider the stochastic elastic equation driven by a cylindrical fractional Brownian motion. The regularities of the solution to the linear stochastic problem corresponding to the stochastic elastic equation are proved. Then, we obtain the existence of the solution using the Picard iteration.     

What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?

We prove that a quantum stochastic differential equation is the interaction representation of the Cauchy problem for the Schrödinger equation with Hamiltonian given by a certain operator restricted by a boundary condition. If the deficiency index of the boundary-value problem is trivial, then the corresponding quantum stochastic differential equation has a unique unitary solution. Therefore, by the deficiency index of a quantum stochastic differential equation we mean the deficiency index of the related symmetric boundary-value problem.In this paper, conditions sufficient for the essential self-adjointness of the symmetric boundary-value problem are obtained. These conditions are closely related to nonexplosion conditions for the pair of master Markov equations that we canonically assign to the quantum stochastic differential equation.     

Maximal regularity for non-autonomous stochastic evolution equations in UMD Banach spaces

A non-autonomous stochastic linear evolution equation in UMD Banach spaces of type 2 is considered. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to a stochastic partial differential equation.