×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gard, T.C., Introduction to stochastic differential equations, (1987), Dekker New York
[2] Gihman, I.I.; Skorohod, A.V., The theory of stochastic processes, Vol. 3, (1979), Springer New York · Zbl 0404.60061
[3] Maruyama, G., Continuous Markov processes and stochastic equations, Rend. circ. mat. Palermo, 4, 48-90, (1955) · Zbl 0053.40901
[4] McShane, E.J., Stochastic calculus and stochastic models, (1974), Academic Press New York · Zbl 0292.60090
[5] Milstein, G.N., Approximate integration of stochastic differential equations, Theory probab. appl., 19, 557-562, (1974) · Zbl 0314.60039
[6] Nelson, E., Dynamical theories of Brownian motion, (1967), Princeton Univ. Press Princeton, NY · Zbl 0165.58502
[7] Pardoux, E.; Talay, D., Discretization and simulation of stochastic differential equations, Acta appl. math., 3, 23-47, (1985) · Zbl 0554.60062
[8] Platen, E., An approximation method for a class of Itô processes with jump component, Liet. mat. rink., 22, 124-136, (1982) · Zbl 0497.60057
[9] Rümelin, W., Numerical treatment of stochastic differential equations, SIAM J. num. anal., 19, 604-613, (1982) · Zbl 0496.65038
[10] Talay, D., Résolution trajectorielle et analyse numérique des équations différentielles stochastiques, Stochastics, 9, 275-306, (1983) · Zbl 0512.60041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.