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Register a new userStrong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation
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In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process ${u(t)}_{tin [0,T]}$ satisfying a stochastic linear evolution equation with a positive-type memory term driven by an additive Gaussian noise. The equation can be written in an abstract form as $$ dd u + (int_0^t b(t-s) Au(s) , dd s), dd t = dd W^{_Q}, tin (0,T]; quad u(0)=u_0 in H, $$ where $W^{_Q}$ is a $Q$-Wiener process on $H=L^2({mathcal D})$ and where the main example of $b$ we consider is given by $$ b(t) = t^{beta-1}/Gamma(beta), quad 0 < beta <1. $$ We let $A$ be an unbounded linear self-adjoint positive operator on $H$ and we further assume that there exist $alpha >0$ such that $A^{-alpha}$ has finite trace and that $Q$ is bounded from $H$ into $D(A^kappa)$ for some real $kappa$ with $alpha-frac{1}{beta+1}<kappa leq alpha$. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter $Delta t =T/n$), and a standard continuous finite element method in space (parameter $h$). Let $u_{n,h}$ be the discrete solution at $T=nDelta t$. We show that $$ (E | u_{n,h} - u(T)|^2)^{1/2}={mathcal O}(h^{ u} + Delta t^gamma), $$ for any $gamma< (1 - (beta+1)(alpha - kappa))/2 $ and $ u leq frac{1}{beta+1}-alpha+kappa$.
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In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract It^o form as $$ dd X(t) + left (int_0^t b(t-s) A X(s) , dd s right) , dd t = dd W^{_Q}(t), tin (0,T]; ~ X(0) =X_0in H, $$ oindent where $W^Q$ is a $Q$-Wiener process on the Hilbert space $H$ and where the time kernel $b$ is the locally integrable potential $t^{rho-2}$, $rho in (1,2)$, or slightly more general. The operator $A$ is unbounded, linear, self-adjoint, and positive on $H$. Our main assumption concerning the noise term is that $A^{( u- 1/rho)/2} Q^{1/2}$ is a Hilbert-Schmidt operator on $H$ for some $ u in [0,1/rho]$. The numerical approximation is achieved via a standard continuous finite element method in space (parameter $h$) and an implicit Euler scheme and a Laplace convolution quadrature in time (parameter $Delta t=T/N$). %Let $X_h^N$ be the discrete solution at time $T$. Eventually let $varphi : Hrightarrow R$ is such that $D^2varphi$ is bounded on $H$ but not necessarily bounded and suppose in addition that either its first derivative is bounded on $H$ and $X_0 in L^1(Omega)$ or $varphi = | cdot |^2$ and $X_0 in L^2(Omega)$. We show that for $varphi : Hrightarrow R$ twice continuously differentiable test function with bounded second derivative, $$ | E varphi(X^N_h) - E varphi(X(T)) | leq C ln left(frac{T}{h^{2/rho} + Delta t} right) (Delta t^{rho u} + h^{2 u}), $$ oindent for any $0leq u leq 1/rho$. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.
We provide an exhaustive treatment of Linear-Quadratic control problems for a class of stochastic Volterra equations of convolution type, whose kernels are Laplace transforms of certain signed matrix measures which are not necessarily finite. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than $1/2$ as a special case. We establish the correspondence of the initial problem with a possibly infinite dimensional Markovian one in a Banach space, which allows us to identify the Markovian controlled state variables. Using a refined martingale verification argument combined with a squares completion technique, we prove that the value function is of linear quadratic form in these state variables with a linear optimal feedback control, depending on non-standard Banach space valued Riccati equations. Furthermore, we show that the value function of the stochastic Volterra optimization problem can be approximated by that of conventional finite dimensional Markovian Linear--Quadratic problems, which is of crucial importance for numerical implementation.
We study discrete-time simulation schemes for stochastic Volterra equations, namely the Euler and Milstein schemes, and the corresponding Multi-Level Monte-Carlo method. By using and adapting some results from Zhang [22], together with the Garsia-Rodemich-Rumsey lemma, we obtain the convergence rates of the Euler scheme and Milstein scheme under the supremum norm. We then apply these schemes to approximate the expectation of functionals of such Volterra equations by the (Multi-Level) Monte-Carlo method, and compute their complexity.
We consider the dynamics of an elastic beam which is clamped at its left end to a vibrating support and which can move freely at its right end between two rigid obstacles (the stops). We model the contact with Signorinis complementary conditions between the displacement and the shear stress. For this infinite dimensional contact problem, we propose a family of fully discretized approximations and their convergence is proved. Moreover some examples of implementation are presented.
In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centered at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.
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