Evaluation of integrals with fractional Brownian motion for different Hurst indices

Fei Gao, Shuaiqiang Liu, Cornelis W. Oosterlee*, Nico M. Temme

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

In this paper, we will evaluate integrals that define the conditional expectation, variance and characteristic function of stochastic processes with respect to fractional Brownian motion (fBm) for all relevant Hurst indices, i.e. (Formula presented.). Particularly, the fractional Ornstein–Uhlenbeck (fOU) process gives rise to highly nontrivial integration formulas that need careful analysis when considering the whole range of Hurst indices. We will show that the classical technique of analytic continuation, from complex analysis, provides a way of extending the domain of validity of an integral from (Formula presented.) to the larger domain (Formula presented.). Numerical experiments for different Hurst indices confirm the robustness and efficiency of the integral formulations presented. Moreover, we provide accurate and highly efficient financial option pricing results for processes that are related to the fOU process, with the help of Fourier cosine expansions.

Original languageEnglish
Pages (from-to)847-866
Number of pages20
JournalInternational Journal of Computer Mathematics
Volume100
Issue number4
DOIs
Publication statusPublished - 2023

Bibliographical note

Funding Information:
F. Gao would like to thank the China Scholarship Council [CSC, grant number 202006280439] for the financial support. N. M. Temme acknowledges financial support from Spanish Ministry of Science and Innovation (Ministerio de Ciencia e Innovación), Spain, project MTM2012-11686. N. M. Temme thanks CWI, Amsterdam, for general support.

Publisher Copyright:
© 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

Funding

F. Gao would like to thank the China Scholarship Council [CSC, grant number 202006280439] for the financial support. N. M. Temme acknowledges financial support from Spanish Ministry of Science and Innovation (Ministerio de Ciencia e Innovación), Spain, project MTM2012-11686. N. M. Temme thanks CWI, Amsterdam, for general support.

Keywords

  • 60G22
  • 65D30
  • 91G20
  • 91G60
  • analytic continuation
  • conditional density function
  • COS method option pricing
  • Fractional Brownian motion
  • fractional Ornstein–Uhlenbeck process

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