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Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motion
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SYSNO ASEP 0557426 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motion Author(s) Tubikanec, I. (AT)
Tamborrino, M. (GB)
Lánský, Petr (FGU-C) RID, ORCID
Buckwar, E. (AT)Number of authors 4 Article number 113951 Source Title Journal of Computational and Applied Mathematics. - : Elsevier - ISSN 0377-0427
Roč. 406, May 1 (2022)Number of pages 29 s. Language eng - English Country NL - Netherlands Keywords GARCH model ; Feller's boundary classification ; numerical splitting schemes ; log-ODE method ; boundary preservation ; moment preservation OECD category Statistics and probability R&D Projects GF20-21030L GA ČR - Czech Science Foundation (CSF) Method of publishing Open access Institutional support FGU-C - RVO:67985823 UT WOS 000789740200019 EID SCOPUS 85121879507 DOI https://doi.org/10.1016/j.cam.2021.113951 Annotation We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller's boundary classification. We compare the frequently used Euler-Maruyama and Milstein methods, two Lie-Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong-Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler-Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we prove that the limit distribution of the splitting and ODE methods exists for all stepsize values and parameters. Third, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler-Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie-Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features. Workplace Institute of Physiology Contact Lucie Trajhanová, lucie.trajhanova@fgu.cas.cz, Tel.: 241 062 400 Year of Publishing 2023 Electronic address https://doi.org/10.1016/j.cam.2021.113951
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