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

Yi Chen (), Jing Dong () and Hao Ni ()
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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

Mathematics of Operations Research, 2021, vol. 46, issue 2, 559-594

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.

Keywords: Primary: 65C05, 60G22, Primary: Simulation: system dynamics; analysis of algorithms: computational complexity, fractional Brownian motion, stochastic differential equation, Monte Carlo simulation (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:46:y:2021:i:2:p:559-594

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