Itô Formula for Integral Processes Related to Space-Time Lévy Noise
ABSTRACT
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.
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.
KEYWORDS
Lévy Processes, Poisson Random Measure, Stochastic Integral, ItÔ Formula, ItÔ Representation Theorem
Lévy Processes, Poisson Random Measure, Stochastic Integral, ItÔ Formula, ItÔ Representation Theorem
Cite this paper
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.
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.
References
[1] Balan, R.M. (2015) Integration with Respect to Lévy Colored Noise, with Applications to SPDEs. Stochastics, 87, 363- 381.
http://dx.doi.org/10.1080/17442508.2014.956103 [2] Balan, R.M. (2014) SPDEs with α-Stable Lévy Noise: A Random Field Approach. International Journal of Stochastic Analysis, 2014, Article ID: 793275.
http://dx.doi.org/10.1155/2014/793275 [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.
http://dx.doi.org/10.1007/BFb0074920 [4] Rajput, B.S. and Rosinski, J. (1989) Spectral Representations of Infinitely Divisible Processes. Probability Theory and Related Fields, 82, 451-487.
http://dx.doi.org/10.1007/BF00339998 [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.
http://dx.doi.org/10.1017/CBO9780511809781 [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.
http://dx.doi.org/10.1007/978-1-4612-2054-1_6 [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.
http://dx.doi.org/10.1007/978-3-540-78572-9
[1] Balan, R.M. (2015) Integration with Respect to Lévy Colored Noise, with Applications to SPDEs. Stochastics, 87, 363- 381.
http://dx.doi.org/10.1080/17442508.2014.956103 [2] Balan, R.M. (2014) SPDEs with α-Stable Lévy Noise: A Random Field Approach. International Journal of Stochastic Analysis, 2014, Article ID: 793275.
http://dx.doi.org/10.1155/2014/793275 [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.
http://dx.doi.org/10.1007/BFb0074920 [4] Rajput, B.S. and Rosinski, J. (1989) Spectral Representations of Infinitely Divisible Processes. Probability Theory and Related Fields, 82, 451-487.
http://dx.doi.org/10.1007/BF00339998 [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.
http://dx.doi.org/10.1017/CBO9780511809781 [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.
http://dx.doi.org/10.1007/978-1-4612-2054-1_6 [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.
http://dx.doi.org/10.1007/978-3-540-78572-9