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Algebraic structure of vector fields in financial diffusion models and its applications

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  • Yusuke Morimoto
  • Makiko Sasada

Abstract

High-order discretization schemes of SDEs using free Lie algebra-valued random variables are introduced by Kusuoka [Adv. Math. Econ., 2004, 5, 69–83], [Adv. Math. Econ., 2013, 17, 71–120], Lyons–Victoir [Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 2004, 460, 169–198], Ninomiya–Victoir [Appl. Math. Finance, 2008, 15, 107–121] and Ninomiya–Ninomiya [Finance Stochast., 2009, 13, 415–443]. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical methods. The authors have found a special Lie algebraic structure on the vector fields in the major financial diffusion models. Using this structure, we can solve the flows associated with vector fields analytically and efficiently. Numerical examples show that our method reduces the computation time drastically.

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  • Yusuke Morimoto & Makiko Sasada, 2017. "Algebraic structure of vector fields in financial diffusion models and its applications," Quantitative Finance, Taylor & Francis Journals, vol. 17(7), pages 1105-1117, July.
    • Handle: RePEc:taf:quantf:v:17:y:2017:i:7:p:1105-1117
    DOI: 10.1080/14697688.2016.1264618
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    Cited by:

    1. Melike Bildirici & Yasemen Ucan & Sérgio Lousada, 2022. "Interest Rate Based on The Lie Group SO(3) in the Evidence of Chaos," Mathematics, MDPI, vol. 10(21), pages 1-9, October.

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