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A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

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  • Buckwar, Evelyn
  • Sickenberger, Thorsten

Abstract

In this article we compare the mean-square stability properties of the θ-Maruyama and θ-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the θ-Milstein method and thus, for some choices of θ, the conditions on the step-size, are much more restrictive than those for the θ-Maruyama method; (ii) the precise stability region of the θ-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partial implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter σ. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.

Suggested Citation

  • Buckwar, Evelyn & Sickenberger, Thorsten, 2011. "A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(6), pages 1110-1127.
    • Handle: RePEc:eee:matcom:v:81:y:2011:i:6:p:1110-1127
    DOI: 10.1016/j.matcom.2010.09.015
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    Cited by:

    1. Vargas, Alessandro N. & Caruntu, Constantin F. & Ishihara, João Y. & Bouzahir, Hassane, 2022. "Stochastic stability of switching linear systems with application to an automotive powertrain model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 191(C), pages 278-287.
    2. de la Cruz, H. & Jimenez, J. C, 2020. "Exact pathwise simulation of multi-dimensional Ornstein–Uhlenbeck processes," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    3. Rathinasamy, Anandaraman & Nair, Priya, 2018. "Asymptotic mean-square stability of weak second-order balanced stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential systems," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 276-303.
    4. Tocino, A. & Zeghdane, R. & Senosiaín, M.J., 2021. "On the MS-stability of predictor–corrector schemes for stochastic differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 289-305.
    5. Michael B. Giles & Christoph Reisinger, 2012. "Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance," Papers 1204.1442, arXiv.org.

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