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

Author

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  • Pang, Chenxu
  • Wang, Xiaojie

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 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.

Suggested Citation

  • Pang, Chenxu & Wang, Xiaojie, 2024. "Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 178(C).
  • Handle: RePEc:eee:spapps:v:178:y:2024:i:c:s030441492400173x
    DOI: 10.1016/j.spa.2024.104467
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    References listed on IDEAS

    as
    1. Michael B. Giles & Lukasz Szpruch, 2012. "Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation," Papers 1202.6283, arXiv.org, revised May 2014.
    2. Iro Ren'e Kouarfate & Michael A. Kouritzin & Anne MacKay, 2020. "Explicit solution simulation method for the 3/2 model," Papers 2009.09058, arXiv.org, revised Jan 2021.
    3. Desmond J. Higham, 2015. "An Introduction to Multilevel Monte Carlo for Option Valuation," Papers 1505.00965, arXiv.org.
    4. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    Full references (including those not matched with items on IDEAS)

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