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Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

Chenxu Pang, Xiaojie Wang

TL;DR

The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, which helps to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC.

Abstract

In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with non-commutative diffusion coefficients in the globally Lipschitz setting, a kind of one-half order truncated Milstein-type scheme without Lévy areas was recently introduced by Giles and Szpruch (2014), which combined with the antithetic multilevel Monte Carlo (MLMC) gives the optimal overall computational cost $\mathcal{O}(ε^{-2})$ for the required target accuracy $ε$. Nevertheless, many nonlinear SDEs in applications have non-globally Lipschitz continuous coefficients and the corresponding theoretical guarantees for antithetic MLMC are absent in the literature. In the present work, we aim to fill the gap and analyze antithetic MLMC in a non-globally Lipschitz setting. First, we propose a family of modified Milstein-type schemes without Lévy areas to approximate SDEs with non-globally Lipschitz continuous coefficients. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where even the diffusion coefficients are allowed to grow superlinearly. This then helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. Since getting rid of the Lévy areas destroys the martingale properties of the scheme, the analysis of both the convergence rate and the desired variance becomes highly non-trivial in the non-globally Lipschitz setting. By introducing an auxiliary approximation process, we develop non-standard arguments to overcome the essential difficulties. Numerical experiments are provided to confirm the theoretical findings.

Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

TL;DR

The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, which helps to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC.

Abstract

In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with non-commutative diffusion coefficients in the globally Lipschitz setting, a kind of one-half order truncated Milstein-type scheme without Lévy areas was recently introduced by Giles and Szpruch (2014), which combined with the antithetic multilevel Monte Carlo (MLMC) gives the optimal overall computational cost for the required target accuracy . Nevertheless, many nonlinear SDEs in applications have non-globally Lipschitz continuous coefficients and the corresponding theoretical guarantees for antithetic MLMC are absent in the literature. In the present work, we aim to fill the gap and analyze antithetic MLMC in a non-globally Lipschitz setting. First, we propose a family of modified Milstein-type schemes without Lévy areas to approximate SDEs with non-globally Lipschitz continuous coefficients. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where even the diffusion coefficients are allowed to grow superlinearly. This then helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. Since getting rid of the Lévy areas destroys the martingale properties of the scheme, the analysis of both the convergence rate and the desired variance becomes highly non-trivial in the non-globally Lipschitz setting. By introducing an auxiliary approximation process, we develop non-standard arguments to overcome the essential difficulties. Numerical experiments are provided to confirm the theoretical findings.
Paper Structure (19 sections, 17 theorems, 213 equations, 5 figures)

This paper contains 19 sections, 17 theorems, 213 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The antithetic MLMC revisited
  3. Antithetic MLMC in a non-globally Lipschitz setting
  4. Settings and the modified Milstein scheme
  5. The antithetic MLMC method and main result
  6. Strong convergence order of the MM scheme
  7. Variance analysis for multilevel estimators
  8. Numerical examples
  9. Conclusion and future research
  10. Proof of Theorem \ref{['theorem:moments-bound-of-MM']}
  11. Proof of lemmas in Section \ref{['section:strong-convergence-order-of-the-MM']}
  12. Proof of Lemma \ref{['lemma:strong-convergence-rate-of-sde-and-auxiliary-process']}
  13. Proof of Lemma \ref{['lemma:holder-continuity-of-MM']}
  14. Proof of Lemma \ref{['lemma:errors-between-sde-and-auxiliary-process-in-continuous-version']}
  15. Proof of Lemma \ref{['lemma:strong-convergence-rate-of-MM-and-auxiliary-process']}
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $P$, $Z_{\ell}$ and $Z$ be defined as def:functional-P-in-mlmc, definition:estimator-of-multilevel-estimator and equation:final-multilevel-estimator, respectively. Let $\hat{P}_{\ell}$be the corresponding level $\ell$ numerical approximation. If there exist positive constants $\alpha$, $\beta$, where $\mathcal{C}_{\ell}$ is the computational complexity of $Z_{\ell}$, then there exists a posit

Figures (5)

  • Figure 1: Strong convergence rates of the Tamed Milstein scheme (TMS1) for the generalized stochastic FHN model \ref{['equation:fhn-model']} in mean-squred sense (Left) and in quadratic sense (Right)
  • Figure 2: Convergence rates of variance (Left) and computational cost (Right) for a smooth payoff function $P = 2X_{1}(T)+\sin(X_{2}(T))$
  • Figure 3: Strong convergence rates of the projected Milstein scheme (PMS) for the 3/2 Heston model \ref{['equation:heston-model']} in mean-squred sense (Left) and in quadratic sense (Right)
  • Figure 4: Convergence rates of variance (Left) and computational cost (Right) for a smooth payoff function $P = S_{T}$
  • Figure 5: Convergence rates of variance (Left) and computational cost (Right) for a call option $P = e^{-rT} \max\{ S_{T} -K,0 \}$

Theorems & Definitions (28)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Theorem 4.6
  • Theorem 5.1
  • ...and 18 more