## Electronic Journal of Probability

### The FBM Itô's Formula Through Analytic Continuation

#### Abstract

The Fractional Brownian Motion can be extended to complex values of the parameter $\alpha$ for $\Re\alpha \gt {1\over 2}$. This is a useful tool. Indeed, the obtained process depends holomorphically on the parameter, so that many formulas, as Itô formula, can be extended by analytic continuation. For large values of $\Re\alpha$, the stochastic calculus reduces to a deterministic one, so that formulas are very easy to prove. Hence they hold by analytic continuation for $\Re\alpha \le 1$, containing the classical case $\alpha =1$.

#### Article information

Source
Electron. J. Probab., Volume 6 (2001), paper no. 26, 22 pp.

Dates
Accepted: 1 October 2001
First available in Project Euclid: 19 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1461097656

Digital Object Identifier
doi:10.1214/EJP.v6-99

Mathematical Reviews number (MathSciNet)
MR1873303

Zentralblatt MATH identifier
1008.60074

Subjects
Primary: 60H05: Stochastic integrals
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Feyel, D.; de La Pradelle, A. The FBM Itô's Formula Through Analytic Continuation. Electron. J. Probab. 6 (2001), paper no. 26, 22 pp. doi:10.1214/EJP.v6-99. https://projecteuclid.org/euclid.ejp/1461097656

#### References

• E. Alòs and D. Nualart, An extension of Itô's formula for anticipating processes. In Journal of Theoretical Probability, 11, 493-514 (1998)
• E. Alòs, 0. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes. In Annals of Probability, 29, 76-801 (2001)
• E. Alòs, 0. Mazet and D. Nualart, Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 1/2. In Stoch. processes and applications, 86, 121-139, (2000)
• E. Alòs, J.A. León and D. Nualart, Stochastic stochastic calculus for fractional Brownian motion with Hurst parameter lesser than 1/2. In Taiwanesse J. of Math. 5, no 3, 609-632, (2001)
• A. Ayache, S. Léger, M. Pontier, Drap brownien fractionnaire. To appear in Potential Analysis
• P. Carmona and L. Coutin, Stochastic integration with respect to fractional Brownian motion. In CRAS, Paris, Série I, t.330, 231-236, (2000)
• W. Dai and C. C. Heyde, Itô's formula with respect to fractional Brownian motion and its application. In Journal of Appl. Math. and Stoch. An., 9, 439-448 (1996)
• L. Decreusefond, Regularity properties of some stochastic Volterra integrals with singular kernels. Preprint, 2000
• L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion. In Potential Analysis, 10, 177-214 (1998)
• T.E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion. I. Theory. In SIAM J. Control Optim. 38 (2000), no 2, 582-612
• D. Feyel and A. de la Pradelle, Espaces de Sobolev gaussiens. In Ann.Inst.Fourier, t.39, fasc.4, p.875-908, (1989)
• D. Feyel and A. de la Pradelle, Capacités gaussiennes. In Ann. Inst. Fourier, t.41, f.1, p.49-76, (1991)
• D. Feyel and A. de la Pradelle, Opérateurs linéaires gaussiens. In Proc. ICPT91, Ed. Bertin, Potential Analysis, vol 3,1, p.89-106 (1994)
• D. Feyel and A. de la Pradelle, Fractional Integrals and Brownian Processes. In Publication de l'Université d'Evry Val d'Essonne, (1996)
• D. Feyel and A. de la Pradelle, On fractional Brownian processes. In Potential Analysis, 10, 273-288 (1999)
• D. Feyel, Polynômes harmoniques et inégalités de Nelson. In Publication de l'Université d'Evry Val d'Essonne, 1995.
• B. Gaveau and P. Trauber, L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel. In J. Functional Analysis, 46, 230-238 (1982)
• M. L. Kleptsyna, P. E. Kloeden and V. V. Anh, Existence and uniqueness theorems for stochastic differential equations with fractal Brownian motion. Problems Inform. Transmission 34 (1999), no 4, 332-341
• S. J. Lin, Stochastic analysis of fractional Brownian motions. In Stochastics and Stoch. Reports, 55, 121-140 (1995)
• B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications. In SIAM Review, 10(4), 422-437 (1968)
• J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier. In Studia Mathematica, 8, 139-40 (1939)
• D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands. In Prob. Theory Rel. Fields, 78, 535-581 (1988)
• I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motion. In Bernoulli, 5, 571-587 (1999)
• N. Privault, Skorohod stochastic integration with respect to a non-adapted process on Wiener space. In Stochastics and Stochastics Reports, 65, 13-39, (1998)
• A. V. Skorohod, On a generalization of a stochastic integral. In Theory Probab. Appl., 20, 219-233 (1975)
• A. S. Üstünel, The Itô formula for anticipative processes with non-monotonous time scale via the Malliavin calculus. In Probability Theory and Related Fields 79, 249-269 (1988)
• M. Zähle, Integration with respect to fractal functions and stochastic calculus, I. In Probability Theory and Related Fields 111, 333-374, (1998)