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A mixed-step algorithm for the approximation of the stationary regime of a diffusion

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  • Pagès, Gilles
  • Panloup, Fabien

Abstract

In some recent papers, some procedures based on some weighted empirical measures related to decreasing-step Euler schemes have been investigated to approximate the stationary regime of a diffusion (possibly with jumps) for a class of functionals of the process. This method is efficient but needs the computation of the function at each step. To reduce the complexity of the procedure (especially for functionals), we propose in this paper to study a new scheme, called the mixed-step scheme, where we only keep some regularly time-spaced values of the Euler scheme. Our main result is that, when the coefficients of the diffusion are smooth enough, this alternative does not change the order of the rate of convergence of the procedure. We also investigate a Richardson–Romberg method to speed up the convergence and show that the variance of the original algorithm can be preserved under a uniqueness assumption for the invariant distribution of the “duplicated” diffusion, condition which is extensively discussed in the paper. Finally, we conclude by giving sufficient “asymptotic confluence” conditions for the existence of a smooth solution to a discrete version of the associated Poisson equation, condition which is required to ensure the rate of convergence results.

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  • 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.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:522-565
    DOI: 10.1016/j.spa.2013.07.011
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    References listed on IDEAS

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    1. 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.
    2. Gilles Pag`es & Fabien Panloup, 2007. "Approximation of the distribution of a stationary Markov process with application to option pricing," Papers 0704.0335, arXiv.org, revised Sep 2009.
    3. Bernard Lapeyre & Emmanuel Temam, 2001. "Competitive Monte Carlo methods for the pricing of Asian options," Post-Print hal-01667057, HAL.
    4. 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.
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    Cited by:

    1. Ganguly, Arnab & Sundar, P., 2021. "Inhomogeneous functionals and approximations of invariant distributions of ergodic diffusions: Central limit theorem and moderate deviation asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 74-110.
    2. Gadat, Sébastien & Panloup, Fabien & Pellegrini, C., 2020. "On the cost of Bayesian posterior mean strategy for log-concave models," TSE Working Papers 20-1155, Toulouse School of Economics (TSE), revised Feb 2022.

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