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1.
The stochastic solute dispersion model studied in the previous article, can be applied to more realistic velocity variations by approximating them as piecewise constant. This requires treatment by a boundary value formulation, which raises problems connected with entropy considerations. A method is developed to deal with these by the introduction of a specially designed compensator function into the boundary value probability integral for calculating solute concentration. Applying this even for a single velocity step yields an intractable integration, but a suitable approximation is constructed that allows it to be evaluated in analytical form. The result is that a Gaussian solute plume impinging on a velocity step is transmitted as a modulated and compressed or dilated quasi-Gaussian. Plume dispersion is encapsulated in an enhancement factor F that multiplies the diffusive, linear time, dispersion. F is also time dependent; at the time of step penetration it equals kinematical dilation, but anneals away non-linearly so that a length scale can be established over which downstream effects of a velocity step on the dispersion extends. 相似文献
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Xicheng Zhang 《Journal of Functional Analysis》2010,258(4):1361-1425
In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier-Stokes equations are also investigated. 相似文献
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《Stochastic Processes and their Applications》2014,124(5):1974-2002
In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE. 相似文献
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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. 相似文献
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《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(3-4):203-256
Using the decomposition of solution of SDE, we consider the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process (nonanticipativity equality constraint). It is shown that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman stochastic partial differential equation (HJB SPDE). This appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of the Wiener process is used. Our approach extends the Wong-Zakai type results [20] from SDE to the stochastic dynamic programming equation by showing how this arises as average of the limit of a sequence of deterministic dynamic programming equations. The stochastic characteristics method of Kunita [13] is used to represent the value function. By choosing the Lagrange multiplier equal to its nonanticipative constraint value the usual stochastic (nonanticipative) optimal control and optimal cost are recovered. This suggests a method for solving the anticipative control problems by almost sure deterministic optimal control. We obtain a PDE for the “cost of perfect information” the difference between the cost function of the nonanticipative control problem and the cost of the anticipative problem which satisfies a nonlinear backward HJB SPDE. Poisson bracket conditions are found ensuring this has a global solution. The cost of perfect information is shown to be zero when a Lagrangian submanifold is invariant for the stochastic characteristics. The LQG problem and a nonlinear anticipative control problem are considered as examples in this framework 相似文献
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Franco Flandoli Francesco Russo 《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(1-2):11-54
In this note we prove a precise asymptotic estimate for Laplace type functionals for a parabolic SPDE. We use a large deviation principle, the stochastic Taylor expansion, some exponential inequalities and support theorems for our stochastic partial differential equation 相似文献
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In this paper, we consider a risk process with stochastic return on investments. The basic risk process is the classical risk process while the return on the investment generating process is a compound Poisson process plus a Brownian motion with positive drift. We obtain an integral equation for the ultimate ruin probability which is twice continuously differentiable under certain conditions. We then derive explicit expressions for the lower bound for the ruin probability. We also study a joint distribution related to exponential functionals of Brownian motion which is required in the derivations of the explicit expressions for the lower bound. 相似文献
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Benjamin Gess 《Journal of Functional Analysis》2012,263(8):2355-2383
Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the “distance” defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction–diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained. 相似文献
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We develop a dynamic structural model for the wealth of individual mortgagors in a mortgage pool. We model the process of default and prepayment and, by taking a limit as the pool size goes to infinity, derive a stochastic partial differential equation (SPDE) which can be used to describe the evolution of the loss process from the pool. We prove existence and uniqueness of solutions to this SPDE and show how our model is able to capture, in a flexible way, the prices of credit risky tranches of mortgage-backed securities under different market conditions. 相似文献
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Dejun Luo 《Bulletin des Sciences Mathématiques》2009,133(3):205-228
We consider the stochastic flow generated by Stratonovich stochastic differential equations with non-Lipschitz drift coefficients. Based on the author's previous works, we show that if the generalized divergence of the drift is bounded, then the Lebesgue measure on Rd is quasi-invariant under the action of the stochastic flow, and the explicit expression of the Radon-Nikodym derivative is also presented. Finally we show in a special case that the unique solution of the corresponding Fokker-Planck equation is given by the density of the stochastic flow. 相似文献
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《Stochastic Processes and their Applications》2020,130(10):5865-5887
We give a new definition of a Lévy driven CARMA random field, defining it as a generalized solution of a stochastic partial differential equation (SPDE). Furthermore, we give sufficient conditions for the existence of a mild solution of our SPDE. Our model unifies all known definitions of CARMA random fields, and in particular for dimension 1 we obtain the classical CARMA process. 相似文献
14.
Wei Liu 《Applied Mathematics and Optimization》2010,61(1):27-56
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. 相似文献
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Hyek Yoo. 《Mathematics of Computation》2000,69(230):653-666
The paper concerns finite-difference scheme for the approximation of partial differential equations in , with additional stochastic noise. By replacing the space derivatives in the original stochastic partial differential equation (SPDE, for short) with difference quotients, we obtain a system of stochastic ordinary differential equations. We study the difference between the solution of the original SPDE and the solution to the corresponding equation obtained by discretizing the space variable. The need to approximate the solution in with functions of compact support requires us to introduce a scale of weighted Sobolev spaces. Employing the weighted -theory of SPDE, a sup-norm error estimate is derived and the rate of convergence is given.
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The authors consider a stochastic heat equation in dimension d=1 driven by an additive space time white noise and having a mild nonlinearity.It is proved that the functional law of its solution is absolutely continuous and possesses a smooth density with respect to the functional law of the corresponding linear SPDE. 相似文献
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In this paper, we extend Walsh’s stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous
spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be
equivalent to Dalang’s one. Then we study existence and regularity of the density of the probability law for the real-valued
mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply
the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension
and the stochastic wave equation in space dimension d=1,2,3. Moreover, for these particular examples, known results in the literature have been improved.
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18.
A numerical scheme for a stochastic partial differential equation of heat equation type is considered where the drift is locally bounded and the dispersion may be state dependent. Uniform convergence in probability is obtained.
Roger Pettersson: Partially supported by the EU grant ref. ERBF MRX CT96 0057A. 相似文献
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In this paper we study a stochastic partial differential equation (SPDE) with Hölder continuous coefficient driven by an -stable colored noise. The pathwise uniqueness is proved by using a backward doubly stochastic differential equation backward (SDE) to take care of the Laplacian. The existence of solution is shown by considering the weak limit of a sequence of SDE system which is obtained by replacing the Laplacian operator in the SPDE by its discrete version. We also study an SDE system driven by Poisson random measures. 相似文献
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《Stochastic Processes and their Applications》2020,130(8):4968-5005
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided. 相似文献