Upper bounds on the rate of convergence of truncated stochastic infinite-dimensional differential systems with H-regular noise

摘要

The rate of H-convergence of truncations of stochastic infinite-dimensional systems du = [Au + B(u)] dt + G (u) dW, u (0, .) = u(0) epsilon H with nonrandom, local Lipschitz-continuous operators A, B and G acting on a separable Hilbert space H, where u = u (t, x): [0, T I x D --> R-d (D C Rd) is studied. For this purpose, some new kind of monotonicity conditions on those operators and an existing H-series expansion of the Wiener process W are exploited. The rate of convergence is expressed in terms of the converging series-remainder h(N) = Sigma(+infinity)(k=N+1) alpha(n), where alpha(n) epsilon R-+(1) are the eigenvalues of the covariance operator Q of W. An application to the approximation of semilinear stochastic partial differential equations with cubic-type of nonlinearity is given too. (C) 2006 Elsevier B.V. All rights reserved.