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ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations

Author

Listed:
  • Yi Chen

    (Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208)

  • Jing Dong

    (Graduate School of Business, Columbia University, New York, New York 10027)

  • Hao Ni

    (Department of Mathematics, University College London, London WC1E 6BT, United Kingdom)

Abstract

Consider a fractional Brownian motion (fBM) B H = { B H ( t ) : t ∈ [ 0 , 1 ] } with Hurst index H ∈ ( 0 , 1 ) . We construct a probability space supporting both B H and a fully simulatable process B ^ ∈ H such that sup t ∈ [ 0 , 1 ] | B H ( t ) − B ^ ∈ H ( t ) | ≤ ∈ with probability one for any user-specified error bound ∈ > 0 . When H > 1 / 2 , we further enhance our error guarantee to the α-Hölder norm for any α ∈ ( 1 / 2 , H ) . This enables us to extend our algorithm to the simulation of fBM-driven stochastic differential equations Y = { Y ( t ) : t ∈ [ 0 , 1 ] } . Under mild regularity conditions on the drift and diffusion coefficients of Y , we construct a probability space supporting both Y and a fully simulatable process Y ^ ∈ such that sup t ∈ [ 0 , 1 ] | Y ( t ) − Y ^ ∈ ( t ) | ≤ ∈ with probability one. Our algorithms enjoy the tolerance-enforcement feature, under which the error bounds can be updated sequentially in an efficient way. Thus, the algorithms can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently.

Suggested Citation

  • Yi Chen & Jing Dong & Hao Ni, 2021. "ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 559-594, May.
  • Handle: RePEc:inm:ormoor:v:46:y:2021:i:2:p:559-594
    DOI: 10.1287/moor.2020.1078
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    References listed on IDEAS

    as
    1. Laskin, Nick, 2000. "Fractional market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 482-492.
    2. Nan Chen & Zhengyu Huang, 2013. "Localization and Exact Simulation of Brownian Motion-Driven Stochastic Differential Equations," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 591-616, August.
    3. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
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