Results on the Supremum of Fractional Brownian Motion

TL;DR

This paper investigates the distribution and bounds of the supremum of reflected fractional Brownian motion with Hurst parameter greater than 1/2, establishing relations to hitting times and applications in financial markets.

Contribution

It introduces new relationships between the supremum distribution and hitting times for fractional Brownian motion with H>1/2, including bounds and applications.

Findings

Distribution of supremum related to hitting time for H>1/2.

Upper bounds on supremum distribution using Jensen's and Markov's inequalities.

Sharper bounds derived from Gamma distribution properties.

Abstract

We show that the distribution of the square of the supremum of reflected fractional Brownian motion up to time a, with Hurst parameter-H greater than 1/2, is related to the distribution of its hitting time to level $1,$ using the self similarity property of fractional Brownian motion. It is also proven that second moment of supremum of reflected fractional Brownian motion up to time $a$ is bounded above by $a^{2H}.$ Similar relations are obtained for the supremum of fractional Brownian motion with Hurst parameter greater than 1/2, and its hitting time to level $1.$ What is more, we obtain an upper bound on the complementary probability distribution of the supremum of fractional Brownian motion and reflected fractional Brownian motion up to time a, using Jensen's and Markov's inequalities. A sharper bound is observed on the distribution of the supremum of fractional Brownian motion by the properties of Gamma distribution. Finally, applications of the given results to the financial markets are investigated and partial results are provided.