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

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.