A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

• Autorzy: Alpay D. , Kipnis A.
• Języki publikacji: EN

Abstrakty

EN

Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.

Słowa kluczowe

EN

stochastic integral; white noise space; fractional Brownian motion

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2013
• Tom: Vol. 33, no. 3
• Strony: 395--417
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Alpay D.

• Ben Gurion University of the Negev Department of Electrical Engineering P.O.B. 653, Be’er Sheva 84105, Israel

autor

Kipnis A.

• Ben Gurion University of the Negev Department of Electrical Engineering P.O.B. 653, Be’er Sheva 84105, Israel

Bibliografia

• [1] D. Alpay, H. Attia, D. Levanony, On the characteristics of a class of Gaussian processes within the white noise space setting, Stochastic Process. Appl. 120 (2010), 1074–1104.
• [2] D. Alpay, H. Attia, D. Levanony, White noise based stochastic calculus associated with a class of Gaussian processes, Opuscula Math. 32 (2012) 3, 401–422.
• [3] D. Alpay, P. Jorgensen, D. Levanony, A class of Gaussian processes with fractional spectral measures, J. Funct. Anal. 261 (2011) 2, 507–541.
• [4] C. Bender, An S-transform approach to integration with respect to a fractional Brownian motion, Bernoulli 9 (2003) 6, 955–983.
• [5] F. Biagini, B. Øksendal, A. Sulem, N. Wallner, An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion, stochastic analysis with applications to mathematical finance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 2041, 347–372.
• [6] F. Biagini, Y. Hu, B. Øksendal, T. Zhang, Stochastic calculus for fractional Brownian motion and applications. Probability and its Applications (New York), Springer-Verlag London Ltd., London, 2008.
• [7] R.M. Blumenthal, R.K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, vol. 29, Academic Press, New York, 1968.
• [8] T.E. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion. I. Theory, SIAM J. Control Optim. 38 (2000) 2, 582–612 (electronic).
• [9] R.J. Elliott, J. van der Hoek, A general fractional white noise theory and applications to finance, Math. Finance 13 (2003) 2, 301–330.
• [10] T. Hida, H. Kuo, J. Potthoff, L. Streit, White noise, vol. 253 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1993. An infinite-dimensional calculus.
• [11] E. Hille, R.S. Phillips, Functional analysis and semi-groups, American Mathematical Society, Providence, R.I., 1974. Third printing of the revised edition of 1957, American Mathematical Society Colloquium Publications, vol. XXXI.
• [12] H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic Partial Differential Equations, Probability and its Applications, Birkhäuser Boston Inc., Boston, MA, 1996.
• [13] Y. Hu, B. Øksendal, Fractional white noise calculus and applications to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003) 1, 1–32.
• [14] M.G. Krein, On the problem of continuation of helical arcs in Hilbert space, C.R. (Doklady) Acad. Sci. URSS (N.S.) 45 (1944), 139–142.
• [15] H.-H. Kuo, White noise distribution theory, Probability and Stochastics Series, CRC Press, Boca Raton, FL, 1996.
• [16] J. von Neumann, I.J. Schoenberg, Fourier integrals and metric geometry, Trans. Amer.Math. Soc. 50 (1941), 226–251.
• [17] L. Schwartz, Analyse. III, vol. 44 of Collection Enseignement des Sciences [Collection: The Teaching of Science], Hermann, Paris, 1998, Calcul intégral.
• [18] V.I. Bogacev, Gaussian Measures, Mathematical Surveys and Monographs, American Math. Soc. 1998.