Upper Bounds on the Rate of Convergence of Truncated Stochastic Infinite-Dimensional Differential Systems with H-Regular Noise

The rate of H-convergence of truncations of stochastic infinite-dimensional systems du = [Au + B(u)]dt + G(u)dW, u(0, ·) = u0 ∈ 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 ] × ID → IRd (ID ⊂ IRd) is studied. For this purpose, some new kind of monotonicity conditions on those operators and an existing H-series expansion of the space-time noise W are exploited. The rate of convergence is expressed in terms of the converging series-remainder h(N) = ∑+∞ k=N+1 α 2 n belonging to the trace of related covariance operator Q of W with eigenvalues αn ∈ IR1 of Q. An application to the approximation of semilinear stochastic partial differential equations with cubic-type of nonlinearity is given too.