Itô Formula for Integral Processes Related to Space-Time Lévy Noise


In this article, we give a new proof of the Itô formula for some integral processes related to the space-time Lévy noise introduced in [1] [2] as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Lévy noise with finite variance: a maximal inequality for the p-th moment of the stochastic integral, and the Itô representation theorem leading to a chaos expansion similar to the Gaussian case.

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Balan, R. and Ndongo, C. (2015) Itô Formula for Integral Processes Related to Space-Time Lévy Noise. Applied Mathematics, 6, 1755-1768. doi: 10.4236/am.2015.610156.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Balan, R.M. (2015) Integration with Respect to Lévy Colored Noise, with Applications to SPDEs. Stochastics, 87, 363- 381.
[2] Balan, R.M. (2014) SPDEs with α-Stable Lévy Noise: A Random Field Approach. International Journal of Stochastic Analysis, 2014, Article ID: 793275.
[3] Walsh, J.B. (1989) An Introduction to Stochastic Partial Differential Equations. Ecole d’Eté de Probabilités de Saint- Flour XIV. Lecture Notes in Math, 1180, 265-439.
[4] Rajput, B.S. and Rosinski, J. (1989) Spectral Representations of Infinitely Divisible Processes. Probability Theory and Related Fields, 82, 451-487.
[5] Samorodnitsky, G. and Taqqu, M.S. (1994) Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
[6] Applebaum, D. (2009) Lévy Processes and Stochastic Calculus. 2nd Edition, Cambridge University Press, Cambridge.
[7] Kunita, H. (2004) Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms. In: Rao, M.M., Ed., Real and Stochastic Analysis, New Perspectives, Birkhaüser, Boston, 305-375.
[8] Resnick, S.I. (2007) Heavy Tail Phenomena: Probabilistic and Statistical Modelling. Springer, New York.
[9] Balan, R.M. and Ndongo, C.B. (2015) Intermittency for the Wave Equation with Lévy White Noise. Arxiv: 1505.04167.
[10] Di Nunno, G., Oksendal, B. and Proske, F. (2009) Malliavin Calculus for Lévy Processes with Applications to Finance. Springer-Verlag, Berlin.

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