  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.
