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

作者：Kovacs Mihaly^*; Printems Jacques

来源：Journal of Mathematical Analysis and Applications, 2014, 413(2): 939-952.

DOI：10.1016/j.jmaa.2013.12.034

• Voltorra equations
• Euler scheme
• Stochastic partial differential equations
• Finite element method
• Weak convergence
• Convolution quadrature
• Fractional differential equations

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 Ito form as %26lt;br%26gt;dX(t) + (integral(t)(0) b(t - s)AX(s)ds) dt = dW(Q)(t), t is an element of (0, T]; X(0) = X-0 is an element of H, %26lt;br%26gt;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 is an element of (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((nu-1/rho)/2)Q(1/2) is a Hilbert-Schmidt operator on H for some nu is an element of [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). We show that for phi : H -%26gt; R twice continuously differentiable test function with bounded second derivative, %26lt;br%26gt;vertical bar E phi(X-h(N)) - E phi(X(T))vertical bar %26lt;= C ln(T/h(2/rho) +Delta t) (Delta t(rho nu) + h(2 nu)), %26lt;br%26gt;for any 0 %26lt;= nu %26lt;= 1/rho. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.

• 出版日期2014-5-15