Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise

We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean-$p$ Holder continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space-time (Sobolev-Holder) regularity of the solution together with a maximal type bound on the spatial Sobolev norm. As one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family.

Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term

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.

Higher order Grunwald approximations of fractional derivatives and fractional powers of operators

We give stability and consistency results for higher order Grunwald-type formulae used in the approximation of solutions to fractional-in-space partial differential equations. We use a new Carlson-type inequality for periodic Fourier multipliers to gain regularity and stability results. We then generalise the theory to the case where the first derivative operator is replaced by the generator of a bounded group on an arbitrary Banach space.

Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation

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$.