Un schéma multipas d'approximation de l'équation de Langevin. (A multistep approximation method for the Langevin equation) (Q808100)

Consider, under suitable assumptions, the second-order stochastic differential (Langevin) equation \[ dX_ t=V_ tdt,\quad dV_ t=- \alpha (t)V_ tdt+b(t,X_ t)dt+\sum^{n}_{i=1}\sigma_ i(t,X_ t)dW^ i_ t+\int_{U}c(t,Y_{t-},u)q(dt,du), \] with initial conditions \(X_ 0=\xi\) and \(V_ 0=\eta\), where \((W^ 1_ t,W^ 2_ t,...,W^ n_ t)\) is an n-dimensional standard Wiener process, \(q(\omega;ds,du)=p(\omega;ds,du)-ds\otimes F(du),\) and p is a stationary Poisson measure on \({\mathbb{R}}_+\times U\) (with \(\sigma\)-finite characteristic measure F) independent of the Wiener process. A multistep method for the approximation of the solution is proposed and studied; it allows a faster convergence than the Euler-Maruyama scheme for the usual non-degenerate equations. Such method is easy to implement and allows convenient numerical simulations for such processes.