Bernoulli

  • Bernoulli
  • Volume 14, Number 2 (2008), 499-518.

Stochastic calculus for convoluted Lévy processes

Christian Bender and Tina Marquardt

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Abstract

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.

Article information

Source
Bernoulli, Volume 14, Number 2 (2008), 499-518.

Dates
First available in Project Euclid: 22 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.bj/1208872115

Digital Object Identifier
doi:10.3150/07-BEJ115

Mathematical Reviews number (MathSciNet)
MR2544099

Zentralblatt MATH identifier
1173.60017

Keywords
convoluted Lévy process fractional Lévy process Itô formula Skorokhod integration

Citation

Bender, Christian; Marquardt, Tina. Stochastic calculus for convoluted Lévy processes. Bernoulli 14 (2008), no. 2, 499--518. doi:10.3150/07-BEJ115. https://projecteuclid.org/euclid.bj/1208872115


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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.