Long-range dependent completely correlated mixed fractional Brownian motion

Josephine Dufitinema, Foad Shokrollahi, Tommi Sottinen and Lauri Viitasaari

Stochastic Processes and their Applications, 2024, vol. 170, issue C

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 Hölder index and quadratic variation). However, in the long time-scales it behaves like the fBm (it has long-range dependence governed by the fBms 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.

Keywords: Cameron–Martin–Girsanov–Hitsuda theorem; Fractional Brownian motion; Mixed fractional Brownian motion; Prediction; Transfer principle (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2023.104289

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