Long-range dependent completely correlated mixed fractional Brownian motion

Abstract: In this paper we introduce the long-range dependent completely correlated mixed fractional Brownian motion (ccmfBm). This is a process that is driven by a mixture of Brownian motion (Bm) and a long-range dependent completely correlated fractional Brownian motion (fBm, ccfBm) that is constructed from the Brownian motion via the Molchan--Golosov representation. Thus, there is a single Bm driving the mixed process. In the short time-scales the ccmfBm behaves like the Bm (it has Brownian H\"older index and quadratic variation). However, in the long time-scales it behaves like the fBm (it has long-range dependence governed by the fBm's Hurst index). We provide a transfer principle for the ccmfBm and use it to construct the Cameron--Martin--Girsanov--Hitsuda theorem and prediction formulas. Finally, we illustrate the ccmfBm by simulations.