3,783 research outputs found
Weak order for the discretization of the stochastic heat equation driven by impulsive noise
Considering a linear parabolic stochastic partial differential equation
driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H,
t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an
impulsive cylindrical process and Q describes the spatial covariance structure
of the noise; Tr(A^{-\alpha})0 and A^\beta Q is bounded
for some \beta\in(\alpha-1,\alpha]. A discretization
(X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space
(parameter h>0) and a \theta-method in time (parameter \Delta t=T/N). For
\phi\in C^2_b(H;R) we show an integral representation for the error
|E\phi(X^N_h)-E\phi(X_T)| and prove that
|E\phi(X^N_h)-E\phi(X_T)|=O(h^{2\gamma}+(\Delta t)^{\gamma}) where
\gamma<1-\alpha+\beta.Comment: 29 pages; Section 1 extended, new results in Appendix
Simulation of stochastic Volterra equations driven by space--time L\'evy noise
In this paper we investigate two numerical schemes for the simulation of
stochastic Volterra equations driven by space--time L\'evy noise of pure-jump
type. The first one is based on truncating the small jumps of the noise, while
the second one relies on series representation techniques for infinitely
divisible random variables. Under reasonable assumptions, we prove for both
methods - and almost sure convergence of the approximations to the true
solution of the Volterra equation. We give explicit convergence rates in terms
of the Volterra kernel and the characteristics of the noise. A simulation study
visualizes the most important path properties of the investigated processes
Large Deviations for Stochastic Partial Differential Equations Driven by a Poisson Random Measure
Stochastic partial differential equations driven by Poisson random measures
(PRM) have been proposed as models for many different physical systems, where
they are viewed as a refinement of a corresponding noiseless partial
differential equations (PDE). A systematic framework for the study of
probabilities of deviations of the stochastic PDE from the deterministic PDE is
through the theory of large deviations. The goal of this work is to develop the
large deviation theory for small Poisson noise perturbations of a general class
of deterministic infinite dimensional models. Although the analogous questions
for finite dimensional systems have been well studied, there are currently no
general results in the infinite dimensional setting. This is in part due to the
fact that in this setting solutions may have little spatial regularity, and
thus classical approximation methods for large deviation analysis become
intractable. The approach taken here, which is based on a variational
representation for nonnegative functionals of general PRM, reduces the proof of
the large deviation principle to establishing basic qualitative properties for
controlled analogues of the underlying stochastic system. As an illustration of
the general theory, we consider a particular system that models the spread of a
pollutant in a waterway.Comment: To appear in Stochastic Process and Their Application
Stochastic Minimum Principle for Partially Observed Systems Subject to Continuous and Jump Diffusion Processes and Driven by Relaxed Controls
In this paper we consider non convex control problems of stochastic
differential equations driven by relaxed controls. We present existence of
optimal controls and then develop necessary conditions of optimality. We cover
both continuous diffusion and Jump processes.Comment: Pages 23, Submitted to SIAM Journal on Control and Optimizatio
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