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An adaptive scheme for the approximation of dissipative systems

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  • Lemaire, Vincent

Abstract

We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, a regular explicit Euler scheme-with constant or decreasing step-may explode and implicit Euler schemes are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems.

Suggested Citation

  • Lemaire, Vincent, 2007. "An adaptive scheme for the approximation of dissipative systems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1491-1518, October.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:10:p:1491-1518
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    References listed on IDEAS

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    1. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
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    Cited by:

    1. Pagès, Gilles & Panloup, Fabien, 2014. "A mixed-step algorithm for the approximation of the stationary regime of a diffusion," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 522-565.
    2. Cohen, Serge & Panloup, Fabien, 2011. "Approximation of stationary solutions of Gaussian driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2776-2801.
    3. Gadat, Sébastien & Panloup, Fabien & Saadane, Sofiane, 2016. "Stochastic Heavy Ball," TSE Working Papers 16-712, Toulouse School of Economics (TSE).
    4. Cohen, Serge & Panloup, Fabien & Tindel, Samy, 2014. "Approximation of stationary solutions to SDEs driven by multiplicative fractional noise," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1197-1225.
    5. Chen, Peng & Deng, Chang-Song & Schilling, René L. & Xu, Lihu, 2023. "Approximation of the invariant measure of stable SDEs by an Euler–Maruyama scheme," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 136-167.
    6. Panloup, Fabien, 2008. "Computation of the invariant measure for a Lévy driven SDE: Rate of convergence," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1351-1384, August.
    7. Gilles Pagès & Clément Rey, 2023. "Discretization of the Ergodic Functional Central Limit Theorem," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-44, March.
    8. Laruelle Sophie & Pagès Gilles, 2012. "Stochastic approximation with averaging innovation applied to Finance," Monte Carlo Methods and Applications, De Gruyter, vol. 18(1), pages 1-51, January.
    9. Pagès, Gilles & Rey, Clément, 2020. "Recursive computation of invariant distributions of Feller processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 328-365.
    10. Pagès Gilles & Rey Clément, 2019. "Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications," Monte Carlo Methods and Applications, De Gruyter, vol. 25(1), pages 1-36, March.

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