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Limit theorems for weighted and regular Multilevel estimators

Author

Listed:
  • Giorgi DaphnĂ©

    (Laboratoire de Probabilités et ModÚles Aléatoires, UMR 7599, UPMC Paris 6 (Sorbonne Université), France)

  • Lemaire Vincent

    (Laboratoire de Probabilités et ModÚles Aléatoires, UMR 7599, UPMC Paris 6 (Sorbonne Université), France)

  • PagĂšs Gilles

    (Laboratoire de Probabilités et ModÚles Aléatoires, UMR 7599, UPMC Paris 6 (Sorbonne Université), France)

Abstract

We aim at analyzing in terms of a.s. convergence and weak rate the performances of the Multilevel Monte Carlo estimator (MLMC) introduced in [7] and of its weighted version, the Multilevel Richardson–Romberg estimator (ML2R), introduced in [12]. These two estimators permit to compute a very accurate approximation of I0=đ”Œâą[Y0]${I_{0}=\mathbb{E}[Y_{0}]}$ by a Monte Carlo-type estimator when the (non-degenerate) random variable Y0∈L2⁹(ℙ)${Y_{0}\in L^{2}(\mathbb{P})}$ cannot be simulated (exactly) at a reasonable computational cost whereas a family of simulatable approximations (Yh)h∈ℋ${(Y_{h})_{h\in\operatorname{\mathcal{H}}}}$ is available. We will carry out these investigations in an abstract framework before applying our results, mainly a Strong Law of Large Numbers and a Central Limit Theorem, to some typical fields of applications: discretization schemes of diffusions and nested Monte Carlo.

Suggested Citation

  • Giorgi DaphnĂ© & Lemaire Vincent & PagĂšs Gilles, 2017. "Limit theorems for weighted and regular Multilevel estimators," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 43-70, March.
    • Handle: RePEc:bpj:mcmeap:v:23:y:2017:i:1:p:43-70:n:4
    DOI: 10.1515/mcma-2017-0102
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    References listed on IDEAS

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    1. Al Gerbi Anis & Jourdain Benjamin & ClĂ©ment Emmanuelle, 2016. "Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators," Monte Carlo Methods and Applications, De Gruyter, vol. 22(3), pages 197-228, September.
    2. K. Bujok & B. M. Hambly & C. Reisinger, 2015. "Multilevel Simulation of Functionals of Bernoulli Random Variables with Application to Basket Credit Derivatives," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 579-604, September.
    3. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
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