Bounds on the Expected Value of Maximum Loss of Fractional Brownian Motion

Abstract

It has been theoretically proven through present study that the expected value of maximum loss of fractional Brownian motion up to fixed time t with Hurst parameter [1/2, 1) is bounded above by tH??/2 and below by tH/2. These new bounds provide improvement on those bounds which have been previously derived in the literature. In order to search for closer bounds, numerical study is also performed through discretization method and multivariate Gaussian variables have been examined. The simulated values of the expected value of maximum loss of fractional Brownian motion have been provided through the use of Cholesky decomposition. As a consequence of the simulation study, it has been observed that as the Hurst parameter increases, the values of the expected maximum loss of fractional Brownian motion decreases.