Upper bounds on the rate of convergence of truncated stochastic infinite-dimensional differential systems with [formula omitted]-regular noise

Abstract

The rate of H-convergence of truncations of stochastic infinite-dimensional systems d u = [ Au + B ( u ) ] d t + G ( u ) d W , u ( 0 , · ) = u 0 ∈ 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 ] × D → R d ( D ⊂ R d ) 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 ) = ∑ k = N + 1 + ∞ α n , where α n ∈ 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.