Stochastic Calculus with a Special Generalized Fractional Brownian Motion

Résumé

This work is a first step toward developing a stochastic calculus theory with respect to the generalized fractional Brownian motion, which a recently introduced Gaussian process is extending both fractional and sub-fractional Brownian motions. A Malliavin divergence operator and a stochastic symmetric integral with respect to this process are defined, and sufficient integrability conditions are provided. Moreover, corresponding Ito formulas are established, then applied to introduce a generalized version of the fractional Black–Scholes option pricing model.

Keywords: Fractional, Sub-frational, Brownian motion, Malliavin Calculus, Stochastic, Symmetric integral, Black-Scholes equation

MSC: 60G15, 60G22, 60H05.

REFERENCES

[1] Aloes, E., Mazet, O., & Nualart, D. (2001). Stochastic calculus with respect to Gaussian processes. The Annals of Probability, 29(2), 766-801.. Search in Google Scholar . Digital Object Identifier
[2] Aloes, E., & Nualart, D. (2003). Stochastic integration with respect to the fractional Brownian motion. Stochastics and Stochastic Reports, 75(3), 129-152.‏. Search in Google Scholar. Digital Object Identifier
[3] Bojdecki, T., Gorostiza, L. G., & Talarczyk, A. (2004). Sub-fractional Brownian motion and its relation to occupation times. Statistics & Probability Letters, 69(4), 405-419. Search in Google Scholar . Digital Object Identifier

[4] E. Nouty, C. & Zili, M. (2015). On the sub-mixed fractional Brownian motion. Applied Mathematics-A Journal of Chinese Universities, 30, 27-43.. Search in Google Scholar.   Digital Object Identifier

[5] Houdré, C., & Villa, J. (2003). An example of infinite dimensional quasi-helix. Contemporary Mathematics, 336, 195-202.‏. Search in Google Scholar  MR
[6] Mandelbrot, B. B., & Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM review, 10(4), 422-437. Search in Google Scholar  Digital Object Identifier
[7] Mishura, Y. (2008). Stochastic calculus for fractional Brownian motion and related processes (Vol. 1929). Springer Science & Business Media.‏. Search in Google Scholar.   Digital Object Identifier
[8] Mishura, Y., & Zili, M. (2018). Stochastic analysis of mixed fractional Gaussian processes. Elsevier. Search in Google Scholar .

[9] Nourdin, I. (2012). Selected aspects of fractional Brownian motion (Vol. 4). Milan: Springer. Search in Google Scholar     
[10] Nualart, D. (2006). The Malliavin calculus and related topics (Vol. 1995, p. 317). Berlin: Springer. Search in Google Scholar  Digital Object Identifier
[11] Peltier, R. F., & Véhel, J. L. (1995). Multifractional Brownian motion: definition and preliminary results (Doctoral dissertation, INRIA). Search in Google Scholar     Inria view
[12] Russo, F., & Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probability theory and related fields, 97, 403-421. Search in Google Scholar.     Digital Object Identifier
[13] Sghir, A. (2013). The generalized sub-fractional Brownian motion. Communications on Stochastic Analysis, 7(3), 2. Search in Google Scholar  . Digital Object Identifier
[14] Sottinen, T., & Valkeila, E. (2003). On arbitrage and replication in the fractional Black–Scholes pricing model. Statistics & Decisions, 21(2), 93-108. Search in Google Scholar       Digital Object Identifier
[15] Tudor, C. (2007). Some properties of the sub-fractional Brownian motion. Stochastics An International Journal of Probability and Stochastic Processes, 79(5), 431-448. Search in Google Scholar      Digital Object Identifier  
[16] Tudor, C. (2008). Some aspects of stochastic calculus for the sub-fractional Brownian motion. Ann. Univ. Bucuresti, Mathematica, 199-230. . Search in Google Scholar      Article 
[17] Yan, L., Shen, G., & He, K. (2011). Itô's formula for a sub-fractional Brownian motion. Communications on Stochastic Analysis, 5(1), 9. Search in Google Scholar       Digital Object Identifier
[18] Zili, M. (2017). Generalized fractional Brownian motion. Modern Stochastics: Theory and Applications, 4(1), 15-24. Search in Google Scholar      Digital Object Identifier
[19] Zili, M. (2018). On the generalized fractional Brownian motion. Mathematical Models and Computer Simulations, 10(6), 759-769. Search in Google Scholar     Digital Object Identifier
[20] Zili, M. (2006). On the mixed fractional Brownian motion. International Journal of stochastic analysis, 2006. Search in Google Scholar      Digital Object Identifier
[21] Zili, M. (2014). Mixed sub-fractional Brownian motion. Random Operators and Stochastic Equations, 22(3), 163-178. Search in Google Scholar   Digital Object Identifier

Communicated Editor: Mezerdi Brahim
Manuscript received Dec 03, 2023; revised Feb 02, 2024; accepted Feb 02, 2024; published May 11, 2024