An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter

Stochastic Processes and their Applications

Volumn 104, Issue 1, 2003, Pages 81-106

  Bender, Christian ^a  

^a UNIVERSITY OF KONSTANZ   (Germany)

Author keywords

Fractional Brownian motion; Fractional white noise; It formula; Local time; Tanaka formula; Unified treatment for arbitrary Hurst parameter

Indexed keywords

EID: 0037361234     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(02)00212-0     Document Type: Article

References (31)

1. Alòs, E., Mazet, O., Nualart, D., 2000. Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 1/2 . Stochastic Process. Appl. 86, 121-139.
2. Alòs, E., Mazet, O., Nualart, D., 2001. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766-801.
3. Alòs, E., Nualart, D., 2000. Stochastic integration with respect to the fractional Brownian motion. Preprint, Barcelona.
4. Berman, S.M., 1991. Self-intersections and local nondeterminism of Gaussian processes. Ann. Probab. 19, 160-191.
5. Carmona, P., Coutin, L., Montseny, G., 2000. Stochastic integration with respect to fractional Brownian motion. Preprint.
6. Coutin, L., Nualart, D., Tudor, C.A., 2001. Tanaka formula for the fractional Brownian motion. Preprint, Barcelona.
7. Decreusefond, L., Üstünel, A.S., 1998. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177-214.
8. Duncan, T.E., Hu, Y., Pasik-Duncan, B., 2000. Stochastic calculus for fractional Brownian motion. I. Theory. SICON 38, 582-612.
9. Elliott, R.J., van der Hoek, J., 2001. Fractional Brownian motion and financial modelling. In: Kohlmann, M., Tang, S. (Eds.), Proceedings of a Workshop on Mathematical Finance, Konstanz Oct. 5-7, 2000, Birkhäuser, Basel, pp. 140-150.
10. Gel'fand, I.M., Vilenkin, N.Y., 1964. Generalized Functions, Vol. 4, Academic Press, New York.
11. Hida, T., 1980. Brownian Motion. Springer, Berlin.
12. Hille, E., Phillips, R.S., 1957. Functional Analysis and Semigroups. American Mathematical Society, Providence, RI.
13. Holden, H., Øksendal, B., Ubøe, J., Zhang, T., 1996. Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Birkhäuser, Boston.
14. Hu, Y., Øksendal, B., 1999. Fractional white noise calculus and applications to finance. Preprint, Oslo.
15. Hu, Y., Øksendal, B., 2000. Chaos expansion of local time of fractional Brownian motions. Preprint, Oslo.
16. Hu, Y., Øksendal, B., 2001. Weighted local time for fractional Brownian motion and applications to finance. Preprint, Oslo.
17. Karatzas, I., Shreve, S.E., 1991. Brownian Motion and Stochastic Calculus. Springer, New York.
18. Kubo, I., 1983. Itô formula for generalized Brownian functionals. In: Kallianpur, G. (Ed.), Lecture Notes in Control and Information Sciences, Vol. 49. Springer, Berlin, pp. 155-166.
19. Kuo, H.-H., 1996. White Noise Distribution Theory. CRC Press, Boca Raton, FL.
20. Lin, S.J., 1995. Stochastic analysis of fractional Brownian motion. Stoch. Stoch. Rep. 55, 121-140.
21. Lindstrøm, T., 1993. Fractional Brownian fields as integrals of white noise. Bull. London Math. Soc. 25, 83-88.
22. Ma, J., Protter, P., Yong, Y., 1994. Solving forward-backward stochastic differential equations explicitly-a four step scheme. Probab. Theory Related Fields 98, 339-359.
23. Mandelbrot, B., Van Ness, J., 1968. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422-437.
24. Nualart, D., 1995. The Malliavin Calculus and Related Topics. Springer, New York.
25. Pipiras, V., Taqqu, M.S., 2000. Integral questions related to fractional Brownian motion. Probab. Theory Related Fields 118, 251-291.
26. Privault, N., 1998. Skorohod stochastic integration with respect to non-adapted processes on Wiener space. Stoch. Stoch. Rep. 65, 13-39.
27. Rogers, L.C.G., 1997. Arbitrage with fractional Brownian motion. Math. Finance 7, 95-105.
28. Samko, S.G., Kilbas, A.A., Marichev, O.I., 1993. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon.
29. Samorodnitsky, G., Taqqu, M.S., 1994. Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance. Chapman & Hall, London.
30. Taylor, M.E., 1996. Partial Differential Equations I: Basic Theory. Springer, New York.
31. Widder, D.V., 1944. Positive temperatures on an infinite rod. Trans. Amer. Math. Soc. 55, 85-95.