We proposed a novel class of Gaussian processes, the multi-mixed sub-fractional Brownian motion (mmsfBm) and its Ornstein-Uhlenbeck counterpart. The mmsfBm is an infinite linear combination of independent sub-fractional Brownian motions, a construction that enables it to capture a continuum of scaling properties and provides a significant mathematical advantage over finite-sum models. We rigorously proved that the local roughness of these processes is defined by the infimum of their Hurst exponents. We further showed that both processes are non-semimartingales and possess the conditional full support (CFS) property. The preservation of these unique regularity properties under the Ornstein-Uhlenbeck transformation is a key finding, confirming the robustness of this new framework for modeling complex, multi-scale systems in finance and other fields.
Citation: Foad Shokrollahi, Tommi Sottinen, Mounir Zili. Multi-mixed sub-fractional Brownian motion and Ornstein–Uhlenbeck processes[J]. AIMS Mathematics, 2026, 11(2): 3464-3498. doi: 10.3934/math.2026141
We proposed a novel class of Gaussian processes, the multi-mixed sub-fractional Brownian motion (mmsfBm) and its Ornstein-Uhlenbeck counterpart. The mmsfBm is an infinite linear combination of independent sub-fractional Brownian motions, a construction that enables it to capture a continuum of scaling properties and provides a significant mathematical advantage over finite-sum models. We rigorously proved that the local roughness of these processes is defined by the infimum of their Hurst exponents. We further showed that both processes are non-semimartingales and possess the conditional full support (CFS) property. The preservation of these unique regularity properties under the Ornstein-Uhlenbeck transformation is a key finding, confirming the robustness of this new framework for modeling complex, multi-scale systems in finance and other fields.
| [1] | P. Abry, F. Sellan, The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation, Appl. Comput. Harmonic Anal., 3 (1996), 377–383. |
| [2] |
E. Azmoodeh, T. Sottinen, L. Viitasaari, A. Yazigi, Necessary and sufficient conditions for Hölder continuity of Gaussian processes, Stat. Probab. Lett., 94 (2014), 230–235. https://doi.org/10.1016/j.spl.2014.07.030 doi: 10.1016/j.spl.2014.07.030
|
| [3] |
C. Bender, T. Sottinen, E. Valkeila, Pricing by hedging and no-arbitrage beyond semimartingales, Finance Stoch., 12 (2008), 441–468. https://doi.org/10.1007/s00780-008-0074-8 doi: 10.1007/s00780-008-0074-8
|
| [4] |
T. Bojdecki, L. G. Gorostiza, A. Talarczyk, Fractional Brownian density process and its self-intersection local time of order $k$, J. Theor. Probab., 17 (2004), 717–739. https://doi.org/10.1023/B:JOTP.0000040296.95910.e1 doi: 10.1023/B:JOTP.0000040296.95910.e1
|
| [5] |
T. Bojdecki, L. G. Gorostiza, A. Talarczyk, Sub-fractional Brownian motion and its relation to occupation times, Stat. Probab. Lett., 69 (2004), 405–419. https://doi.org/10.1016/j.spl.2004.06.035 doi: 10.1016/j.spl.2004.06.035
|
| [6] |
E. N. Charles, Z. Mounir, On the sub-mixed fractional Brownian motion, Appl. Math. J. Chin. Univ., 30 (2015), 27–43. https://doi.org/10.1007/s11766-015-3198-6 doi: 10.1007/s11766-015-3198-6
|
| [7] |
P. Cheridito, Arbitrage in fractional Brownian motion models, Finance Stoch., 7 (2003), 533–553. https://doi.org/10.1007/s007800300101 doi: 10.1007/s007800300101
|
| [8] | A. Cherny, Brownian moving averages have conditional full support, Ann. Appl. Probab., 18 (2008), 1825–1830. |
| [9] |
C. R. Dietrich, G. N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput., 18 (1997), 1088–1107. https://doi.org/10.1137/S1064827592240555 doi: 10.1137/S1064827592240555
|
| [10] |
D. Gasbarra, T. Sottinen, H. van Zanten, Conditional full support of Gaussian processes with stationary increments, J. Appl. Probab., 48 (2011), 561–568. https://doi.org/10.1239/jap/1308662644 doi: 10.1239/jap/1308662644
|
| [11] |
P. Guasoni, M. Rásonyi, W. Schachermayer, Consistent price systems and face-lifting pricing under transaction costs, Ann. Appl. Probab., 18 (2008), 491–520. https://doi.org/10.1214/07-AAP461 doi: 10.1214/07-AAP461
|
| [12] | Y. Mishura, M. Zili, Stochastic analysis of mixed fractional Gaussian processes, ISTE Press, Elsevier, 2018. https://doi.org/10.1016/C2017-0-00186-6 |
| [13] | D. Revuz, M. Yor, Continuous martingales and Brownian motion, Grundlehren der mathematischen Wissenschaften, Vol. 293, Springer-Verlag, 1991. https://doi.org/10.1007/978-3-662-06400-9 |
| [14] |
C. Tudor, Some properties of the sub-fractional Brownian motion, Stochastics, 79 (2007), 431–448. https://doi.org/10.1080/17442500601100331 doi: 10.1080/17442500601100331
|
| [15] |
M. Zili, Mixed sub-fractional Brownian motion, Random Oper. Stochastic Equ., 22 (2014), 163–178. https://doi.org/10.1515/rose-2014-0017 doi: 10.1515/rose-2014-0017
|
| [16] | M. Zili, An optimal series expansion of sub-mixed fractional Brownian motion, J. Numer. Math. Stochastics, 5 (2013), 93–105. |
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Foad Shokrollahi, Tommi Sottinen, Mounir Zili. Multi-mixed sub-fractional Brownian motion and Ornstein–Uhlenbeck processes[J]. AIMS Mathematics, 2026, 11(2): 3464-3498. doi: 10.3934/math.2026141
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