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

Consider a fractional Brownian motion (fBM) B-H = {B-H(t) : t is an element of [0, 1]} with Hurst index H is an element of (0, 1). We construct a probability space supporting both B-H and a fully simulatable process (B) over cap (H)(epsilon) such that sup(t is an element of[0,1])vertical bar B-H(t) - (B) over cap (H)(epsilon)(t)vertical bar <= epsilon with probability one for any user-specified error bound epsilon > 0. When H > 1/2, we further enhance our error guarantee to the alpha-Holder norm for any alpha is an element of (1/2, H). This enables us to extend our algorithm to the simulation of fBM-driven stochastic differential equations Y = {Y(t) : t is an element of [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) over cap (epsilon) such that sup(t is an element of[0,1])vertical bar Y(t) - (Y) over cap (epsilon)(t)vertical bar <= epsilon 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.

Automated summary

The article introduces an algorithm for simulating fractional Brownian motion, supporting user-specified error bounds and providing different error guarantees for different Hurst indices. Building on this, the algorithm is extended to simulate stochastic differential equations driven by fBM, allowing for efficient estimation of function expectations.