Two-Step Variations for Processes Driven by Fractional Brownian Motion With Application in Testing for Jumps From the High Frequency Data

In this dissertation we introduce the realized two-step variation of stochastic processes and develop its asymptotic theory for processes based on fractional Brownian motion and on more general Gaussian processes with stationary increments. The realized two-step variation is analogous to the realized 1, 1-order bipower variation introduced by Barndorff-Nielsen and Shephard [8] but mathematically is simpler to deal with. The powerful techniques of Wiener/Itˆo/Malliavin calculus for establishing limit laws play a key rule in our proofs. We include some stochastic simulations as an illustration of our theory. As a result of our study, we provide test statistics for testing for jumps in high frequency data and establish their consistency and asymptotic normality under the null hypothesis that there are no jumps. Testing for jumps from high frequency data has important applications in Financial Mathematics.