IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v36y2023i1d10.1007_s10959-023-01237-w.html
   My bibliography  Save this article

Discretization of the Ergodic Functional Central Limit Theorem

Author

Listed:
  • Gilles Pagès

    (LPSM, Sorbonne Université)

  • Clément Rey

    (Institut Polytechnique de Paris)

Abstract

In this paper, we study the discretization of the ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see Bhattacharya in Z Wahrscheinlichkeitstheorie Verwandte Geb 60:185–201, 1982) which states the following: Given a stationary and ergodic Markov process $$(X_t)_{t \geqslant 0}$$ ( X t ) t ⩾ 0 with unique invariant measure $$\nu $$ ν and infinitesimal generator A, then, for every smooth enough function f, $$(n^{1/2} \frac{1}{n}\int _0^{nt} Af(X_s){\textrm{d}}s)_{t \geqslant 0}$$ ( n 1 / 2 1 n ∫ 0 nt A f ( X s ) d s ) t ⩾ 0 converges in distribution towards the distribution of the process $$(\sqrt{-2 \langle f, Af \rangle _{\nu }} W_{t})_{t \geqslant 0}$$ ( - 2 ⟨ f , A f ⟩ ν W t ) t ⩾ 0 with $$(W_{t})_{t \geqslant 0}$$ ( W t ) t ⩾ 0 a Wiener process. In particular, we consider the marginal distribution at fixed $$t=1$$ t = 1 , and we show that when $$\int _0^{n} Af(X_s)ds$$ ∫ 0 n A f ( X s ) d s is replaced by a well chosen discretization of the time integral with order q (e.g. Riemann discretization in the case $$q=1$$ q = 1 ), then the CLT still holds but with rate $$n^{q/(2q+1)}$$ n q / ( 2 q + 1 ) instead of $$n^{1/2}$$ n 1 / 2 . Moreover, our results remain valid when $$(X_t)_{t \geqslant 0}$$ ( X t ) t ⩾ 0 is replaced by a q-weak order approximation (not necessarily stationary). This paper presents both the discretization method of order q for the time integral and the q-order ergodic CLT we derive from them. We finally propose applications concerning the first order CLT for the approximation of Markov Brownian diffusion stationary regimes with Euler scheme (where we recover existing results from the literature) and the second order CLT for the approximation of Brownian diffusion stationary regimes using Talay’s scheme (Talay in Stoch Stoch Rep 29:13–36, 1990) of weak order two.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-023-01237-w
    DOI: 10.1007/s10959-023-01237-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-023-01237-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10959-023-01237-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. Ganidis, H. & Roynette, B. & Simonot, F., 1999. "Convergence rate of some semi-groups to their invariant probability," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 243-263, February.
    4. Mei, Hongwei & Yin, George, 2015. "Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic differential equations with Markov switching," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3104-3125.
    5. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    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. Basak, Gopal K. & Hu, Inchi & Wei, Ching-Zong, 1997. "Weak convergence of recursions," Stochastic Processes and their Applications, Elsevier, vol. 68(1), pages 65-82, May.
    8. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Honoré, Igor, 2020. "Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2127-2158.
    7. Fabian Dickmann & Nikolaus Schweizer, 2014. "Faster Comparison of Stopping Times by Nested Conditional Monte Carlo," Papers 1402.0243, arXiv.org.
    8. Yi Chen & Jing Dong & Hao Ni, 2021. "ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 559-594, May.
    9. 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.
    10. Jian Wang & Xiang Gao & Zhili Sun, 2021. "A Multilevel Simulation Method for Time-Variant Reliability Analysis," Sustainability, MDPI, vol. 13(7), pages 1-16, March.
    11. Ahmed Kebaier & J'er^ome Lelong, 2015. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Papers 1510.03590, arXiv.org, revised Jul 2017.
    12. Stéphane Crépey & Noufel Frikha & Azar Louzi & Gilles Pagès, 2023. "Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-04304985, HAL.
    13. Wei Fang & Zhenru Wang & Michael B. Giles & Chris H. Jackson & Nicky J. Welton & Christophe Andrieu & Howard Thom, 2022. "Multilevel and Quasi Monte Carlo Methods for the Calculation of the Expected Value of Partial Perfect Information," Medical Decision Making, , vol. 42(2), pages 168-181, February.
    14. Lay Harold A. & Colgin Zane & Reshniak Viktor & Khaliq Abdul Q. M., 2018. "On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters," Monte Carlo Methods and Applications, De Gruyter, vol. 24(4), pages 309-321, December.
    15. Hideyuki Tanaka & Toshihiro Yamada, 2012. "Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method," Papers 1210.0670, arXiv.org, revised Nov 2013.
    16. Nagy Shady Ahmed & El-Beltagy Mohamed A. & Wafa Mohamed, 2020. "Multilevel Monte Carlo by using the Halton sequence," Monte Carlo Methods and Applications, De Gruyter, vol. 26(3), pages 193-203, September.
    17. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Post-Print hal-02430430, HAL.
    18. Alexander Veretennikov, 2023. "Polynomial Recurrence for SDEs with a Gradient-Type Drift, Revisited," Mathematics, MDPI, vol. 11(14), pages 1-16, July.
    19. Lokman A. Abbas-Turki & Stéphane Crépey & Babacar Diallo, 2018. "Xva Principles, Nested Monte Carlo Strategies, And Gpu Optimizations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(06), pages 1-40, September.
    20. Simonella, Roberta & Vázquez, Carlos, 2023. "XVA in a multi-currency setting with stochastic foreign exchange rates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 59-79.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-023-01237-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.