Long-term convergence and stability of linear-implicit theta methods for moment-dissipative stochastic differential equations with multiplicative noise

Strong and mean square convergence of linear drift-implicit Theta methods Y to X governed by systems of non-linear stochastic differential equations (SDEs) with monotone drift and degenerate state-dependent noise driven by standard Wiener processes are studied. For this purpose, we begin with collecting basic properties of strong solutions X and systematically investigate uniform boundedness and asymptotic stability of moments (Lyapunov-type functionals) of their both solutions X and approximations Y . Under some polynomial growth conditions, the one-sided Lipschitz-continuity of the drift with non-increasing nonlinearities and uniform Lipschitz-continuity of diffusion coefficients, we can verify uniform boundedness of moments, ergodicity, exponential p -th mean stability, contractivity and rates of mean square convergence of Y to X with implicitness parameters θ ≥ 0.5 θ ≥ 0.5 , even on infinite time-intervals. While applying a fairly general convergence theorem (2006) on Hilbert spaces, we justify the global rate rg = 0.5 r g = 0.5 of mean square convergence of those methods for systems with state-dependent noise on infinite time-horizons. Three simple examples with power-law non-linearities support our findings (among them, a simplified 1D Ginzburg–Landau model from field theory, systems with dissipative power-laws and non-linear stochastic heat equations), demonstrating some potential applicability of our findings. Our Theta methods with θ ≥ 0.5 θ ≥ 0.5 can control the long-term behavior of numerical dynamics by their moment-dissipativity and are fairly easily implementable.