Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations
The authors consider drift-implicit two-step schemes of the form \[ \begin{aligned}\alpha_2 X_{i+1}+ \alpha_1 X_i+ \alpha_0 X_{i-1} &= h[\beta_2 f(t_{i+1}, X_{i+1})+ \beta_1 f(t_i, X_i)+ \beta_0 f(t_{i-1}, X_{i-1})]\\ &\quad +\gamma_1 G(t_i, X_i)\Delta W_i+ \gamma_0 G(t_{i-1}, X_{i-1})\Delta W_{i-1}\end{aligned} \] with given initial values \(X_0\), \(X_1\) for the system of SDEs \[ dX(t)= f(t, X(t))\,dt+ G(t, X(t))\,dW(t). \] They investigate when the numerical scheme shares asymptotic properties of the system choosing as a test equation the linear scalar SDE \[ dX(t)=\lambda X(t)\,dt+\mu X(t)\,dW(t) \] with complex coefficients \(\lambda\), \(\mu\).
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