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Central Limit Theorem for the $σ$-antithetic multilevel Monte Carlo method

Mohamed Ben Alaya, Ahmed Kebaier, Thi Bao Tram Ngo

Abstract

In this paper, we introduce the $σ$-antithetic multilevel Monte Carlo (MLMC) estimator for a multi-dimensional diffusion which is an extended version of the original antithetic MLMC one introduced by Giles and Szpruch \cite{a}. Our aim is to study the asymptotic behavior of the weak errors involved in this new algorithm. Among the obtained results, we prove that the error between on the one hand the average of the Milstein scheme without Lévy area and its $σ$-antithetic version build on the finer grid and on the other hand the coarse approximation stably converges in distribution with a rate of order 1. We also prove that the error between the Milstein scheme without Lévy area and its $σ$-antithetic version stably converges in distribution with a rate of order $1/2$. More precisely, we have a functional limit theorem on the asymptotic behavior of the joined distribution of these errors based on a triangular array approach (see e.g. Jacod \cite{c}). Thanks to this result, we establish a central limit theorem of Lindeberg-Feller type for the $σ$-antithetic MLMC estimator. The time complexity of the algorithm is carried out.

Central Limit Theorem for the $σ$-antithetic multilevel Monte Carlo method

Abstract

In this paper, we introduce the -antithetic multilevel Monte Carlo (MLMC) estimator for a multi-dimensional diffusion which is an extended version of the original antithetic MLMC one introduced by Giles and Szpruch \cite{a}. Our aim is to study the asymptotic behavior of the weak errors involved in this new algorithm. Among the obtained results, we prove that the error between on the one hand the average of the Milstein scheme without Lévy area and its -antithetic version build on the finer grid and on the other hand the coarse approximation stably converges in distribution with a rate of order 1. We also prove that the error between the Milstein scheme without Lévy area and its -antithetic version stably converges in distribution with a rate of order . More precisely, we have a functional limit theorem on the asymptotic behavior of the joined distribution of these errors based on a triangular array approach (see e.g. Jacod \cite{c}). Thanks to this result, we establish a central limit theorem of Lindeberg-Feller type for the -antithetic MLMC estimator. The time complexity of the algorithm is carried out.

Paper Structure

This paper contains 42 sections, 25 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. General framework
  3. Milstein scheme without Lévy area
  4. Notations
  5. Settings and some standard results
  6. Main results
  7. The $\sigma$-antithetic scheme
  8. The $\sigma$-antithetic MLMC method
  9. Functional limit theorem for the errors
  10. Central limit theorem
  11. The time complexity
  12. Expanding analysis of $\sigma$-antithetic scheme
  13. Expansion of the error $X^{nm,\tilde{\sigma}}_{\frac{i}{n}}-X^{nm,\tilde{\sigma}}_{\frac{i-1}{n}}$, with $\tilde{\sigma}\in\{\rm Id,\sigma\}$
  14. $\bullet$ The term $N^{nm,\tilde{\sigma}}_{\frac{i-1}{n}}$:
  15. $\bullet$ The term $M^{nm,\tilde{\sigma},1}_{\frac{i-1}{n}}$:
  16. ...and 27 more sections

Key Result

Lemma 2.1

Under (Assume), for $p\geq2$ there exists a constant $C_p$, independent of $n$, such that

Theorems & Definitions (47)

  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • ...and 37 more