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Un schéma multipas d’approximation de l’équation de Langevin. (A multistep approximation method for the Langevin equation). (French) Zbl 0731.60051
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.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65C99 Probabilistic methods, stochastic differential equations 65C05 Monte Carlo methods 68U20 Simulation (MSC2010)
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