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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.

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