L^p uniform random walk-type approximation for fractional Brownian motion with Hurst exponent 0 < H < (1)/(2)

2020/07/29 by Alberto Ohashi, Ohashi, Alberto, Francys Andrews de Souza +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2007.15472

openalex publication_date 2020/07/29 · openalex created_date 2020/08/03 · openalex updated_date 2026/07/28

Abstract

In this note, we prove an L^p uniform approximation of the fractional Brownian motion with Hurst exponent 0 &lt; H &lt; (1)/(2) by means of a family of continuous-time random walks imbedded on a given Brownian motion. The approximation is constructed via a pathwise representation of the fractional Brownian motion in terms of a standard Brownian motion. For an arbitrary choice ε_k for the size of the jumps of the family of random walks, the rate of convergence of the approximation scheme is O(ε_k^{p(1-2λ)+ 2(δ-1)}) whenever max\0,1-(pH)/(2)\&lt; δ&lt; 1, λ∈ ((1-H)/(2), (1)/(2) + (δ-1)/(p)).

Related

• A random walk approximation to fractional Brownian motion
• A Generalized Correlated Random Walk Converging to Fractional Brownian Motion
• Level crossings of fractional Brownian motion
• A strong uniform approximation of sub-fractional Brownian motion
• Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion