Exact Results for the Roughness of a Finite Size Random Walk

TL;DR

This paper derives exact results showing that finite size effects significantly increase the effective Hurst exponent in random walks, especially relevant for high-frequency stock price analysis, and highlights slow convergence to the theoretical value.

Contribution

It provides exact finite size results for the Hurst exponent using Spitzer's identity, clarifying how finite size impacts empirical estimates in financial data.

Findings

Finite size effects strongly increase the estimated Hurst exponent.

Convergence to the asymptotic H=1/2 is slow.

Implications for high-frequency financial data analysis.

Abstract

We consider the role of finite size effects on the value of the effective Hurst exponent H. This problem is motivated by the properties of the high frequency daily stock-prices. For a finite size random walk we derive some exact results based on Spitzer's identity. The conclusion is that finite size effects strongly enhance the value of H and the convergency to the asymptotic value (H=1/2) is rather slow. This result has a series of conceptual and practical implication which we discuss.