We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation Lévy process with a Volterra-type kernel. This class of processes contains, for example, fractional Lévy processes as studied by Marquardt [Bernoulli 12 (2006) 1090–1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an Itô formula which separates the different contributions from the memory due to the convolution and from the jumps.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-textAlternatively, the document is available for a cost of $15. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
References
[1] Albeverio, S., Daletsky, Y., Kondratiev, Y.G. and Streit, L. (1996). Non-Gaussian infinite dimensional analysis. J. Funct. Anal. 138 311–350.
[2] Alòs, E., Mazet, O. and Nualart, D. (2001). Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 766–801.
[3] Bender, C. (2003). An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stochastic Process. Appl. 104 81–106.
[4] Bender, C. (2003). An S-transform approach to integration with respect to a fractional Brownian motion. Bernoulli 9 955–983.
[5] Biagini, F., Øksendal, B., Sulem, A. and Wallner, N. (2004). An introduction to the white noise theory and Malliavin calculus for fractional Brownian motion. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 347–372.
[6] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Boca Raton, FL: Chapman and Hall.
[7] Decreusefond, L. and Savy, N. (2006). Anticipative calculus for filtered Poisson processes. Ann. Inst. H. Poincaré Probab. Statist. 42 343–372.
[8] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Berlin: Springer.
[9] Kubo, I. (1983). Itô formula for generalized Brownian functionals. Lecture Notes in Control and Inform. Sci. 49 (G. Kallianpur, ed.) 155–166. Berlin: Springer.
[10] Lee, Y. and Shih, H. (2000). Itô formula for generalized Lévy functionals. In Quantum Information II (T. Hida and K. Saitô, eds.) 87–105. River Edge, NJ: World Scientific.
[11] Lee, Y. and Shih, H. (2004). The product formula for multiple Lévy–Itô integrals. Bull. Inst. Math. Acad. Sin. (N.S.) 32 71–95.
[12] Løkka, A. and Proske, F. (2006). Infinite dimensional analysis for pure jump Lévy processes on the Poisson space. Math. Scand. 98 237–261.
[13] Marquardt, T. (2006). Fractional Lévy processes, CARMA processes and related topics. Ph.D. thesis, Munich Univ. Technology.
[14] Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12 1090–1126.
[15] Nualart, D. (2003). Stochastic calculus with respect to the fractional Brownian motion and applications. Contemp. Math. 336 3–39.
[16] Nualart, D. and Vives, J. (1990). Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités XXIV 154–165. Lecture Notes in Math. 1426. Berlin: Springer.
[17] Øksendal, B. and Proske, F. (2004). White noise for Poisson random measures. Potential Anal. 21 375–403.
[18] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge Univ. Press.
