Stability issues in the numerical solution of stochastic differential equations

The aim of this talk is the analysis of various stability issues for numerical methods designed to solve stochastic differential equations. We first aim to consider a nonlinear system of Ito stochastic differential equation (SDE) dX(t)=f(X(t))dt+g(X(t))dW(t), t>0. Under suitable regularity conditions, exponential mean-square stability holds, i.e. any two solutions $X(t)$ and $Y(t)$ of the SDE with $\mathbb{E}|X_0|^2<\infty$ and $\mathbb{E}|Y_0|^2<\infty$ satisfy $$ \mathbb{E}|X(t)-Y(t)|^2\leq \mathbb{E}|X_0-Y_0|^2 e^{\alpha t}, $$ with $\alpha<0$. We aim to investigate its numerical counterpart when trajectories are generated by stochastic linear multistep methods, in order to provide stepsize restrictions ensuring analogous exponential mean-square stability properties also numerically. We next move to the following second order stochastic differential equation $$ \ddot{x}=f(x)-\eta s^2(x)\dot x+\varepsilon s(x)\xi(y), $$describing the position of a particle subject to the deterministic forcing $f(x)$ and a random forcing $\xi(t)$ of amplitude $\varepsilon$. The dynamics exhibits damped oscillations, with damping parameter $\eta$. We aim to analyze asymptotic mean-square stability properties for partitioned Runge-Kutta methods and multistep linear methods, thought as applied to the system equivalent to $$ \left\{ \begin{array}{rl} dX(t))&=V(t)dt,\\ dV(t)&=-\eta s^2(X(t))V(t) dt+f(X(t))dt+\varepsilon s(X(t))dW(t). \end{array} \right. $$ This is a joint work with E. Buckwar (Univ. of Linz), M. Moccaldi and B. Paternoster (Univ. of Salerno). [1] Buckwar, R. D'Ambrosio, Exponential mean-square stability of linear multistep methods, submitted. [2] K. Burrage, G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput. 29(1), 245--264 (2007).