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Long-range dependent completely correlated mixed fractional Brownian motion

Author

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  • Dufitinema, Josephine
  • Shokrollahi, Foad
  • Sottinen, Tommi
  • Viitasaari, Lauri

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.

Suggested Citation

  • Dufitinema, Josephine & Shokrollahi, Foad & Sottinen, Tommi & Viitasaari, Lauri, 2024. "Long-range dependent completely correlated mixed fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 170(C).
    • Handle: RePEc:eee:spapps:v:170:y:2024:i:c:s0304414923002612
    DOI: 10.1016/j.spa.2023.104289
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