Stochastic partial differential equations driven by multi-parameter white noise of Lévy processes

Øksendal, Bernt

doi:10.1090/S0033-569X-08-01090-5

Author: Bernt Øksendal
Journal: Quart. Appl. Math. 66 (2008), 521-537
MSC (2000): Primary 60H15, 60H40; Secondary 60G51, 60G57, 60G60, 35R60
DOI: https://doi.org/10.1090/S0033-569X-08-01090-5
Published electronically: July 3, 2008
MathSciNet review: 2445527
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a short introduction to the white noise theory for multiparameter Lévy processes and its application to stochastic partial differential equations driven by such processes. Examples include temperature distribution with a Lévy white noise heat source, and heat propagation with a multiplicative Lévy white noise heat source.

References

David Applebaum, Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics, vol. 93, Cambridge University Press, Cambridge, 2004. MR 2072890
Fred Espen Benth, Integrals in the Hida distribution space $({\scr S})^*$, Stochastic analysis and related topics (Oslo, 1992) Stochastics Monogr., vol. 8, Gordon and Breach, Montreux, 1993, pp. 89–99. MR 1268006
Fred Espen Benth and Arne Løkka, Anticipative calculus for Lévy processes and stochastic differential equations, Stoch. Stoch. Rep. 76 (2004), no. 3, 191–211. MR 2072380, DOI https://doi.org/10.1080/10451120410001716880
Rama Cont and Peter Tankov, Financial modelling with jump processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2042661
Giulia Di Nunno, Bernt Øksendal, and Frank Proske, White noise analysis for Lévy processes, J. Funct. Anal. 206 (2004), no. 1, 109–148. MR 2024348, DOI https://doi.org/10.1016/S0022-1236%2803%2900184-8

G. Di Nunno, B. Øksendal and F. Proske: Malliavin Calculus for Lévy Processes and Applications to Finance. Forthcoming book, to be published by Springer.

Takeyuki Hida, Brownian motion, Applications of Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1980. Translated from the Japanese by the author and T. P. Speed. MR 562914

Takeyuki Hida, Hui-Hsiung Kuo, Jürgen Potthoff, and Ludwig Streit, White noise, Mathematics and its Applications, vol. 253, Kluwer Academic Publishers Group, Dordrecht, 1993. An infinite-dimensional calculus. MR 1244577

Helge Holden, Bernt Øksendal, Jan Ubøe, and Tusheng Zhang, Stochastic partial differential equations, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1996. A modeling, white noise functional approach. MR 1408433

Kiyosi Itô, Multiple Wiener integral, J. Math. Soc. Japan 3 (1951), 157–169. MR 44064, DOI https://doi.org/10.2969/jmsj/00310157

Ju. M. Kabanov, A generalized Itô formula for an extended stochastic integral with respect to Poisson random measure, Uspehi Mat. Nauk 29 (1974), no. 4(178), 167–168 (Russian). MR 0397876

Ju. M. Kabanov, Extended stochastic integrals, Teor. Verojatnost. i Primenen. 20 (1975), no. 4, 725–737 (Russian, with English summary). MR 0397877

Alberto Lanconelli and Frank Proske, On explicit strong solution of Itô-SDE’s and the Donsker delta function of a diffusion, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004), no. 3, 437–447. MR 2085642, DOI https://doi.org/10.1142/S0219025704001670

Arne Løkka, Bernt Øksendal, and Frank Proske, Stochastic partial differential equations driven by Lévy space-time white noise, Ann. Appl. Probab. 14 (2004), no. 3, 1506–1528. MR 2071432, DOI https://doi.org/10.1214/105051604000000413

Tom Lindstrøm, Bernt Øksendal, and Jan Ubøe, Wick multiplication and Itô-Skorohod stochastic differential equations, Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988) Cambridge Univ. Press, Cambridge, 1992, pp. 183–206. MR 1190500

Arne Løkka and Frank Norbert Proske, Infinite dimensional analysis of pure jump Lévy processes on the Poisson space, Math. Scand. 98 (2006), no. 2, 237–261. MR 2243705, DOI https://doi.org/10.7146/math.scand.a-14994

David Nualart and Wim Schoutens, Chaotic and predictable representations for Lévy processes, Stochastic Process. Appl. 90 (2000), no. 1, 109–122. MR 1787127, DOI https://doi.org/10.1016/S0304-4149%2800%2900035-1

Bernt Øksendal and Frank Proske, White noise of Poisson random measures, Potential Anal. 21 (2004), no. 4, 375–403. MR 2081145, DOI https://doi.org/10.1023/B%3APOTA.0000034329.34647.fd

Bernt Øksendal, Frank Proske, and Mikael Signahl, The Cauchy problem for the wave equation with Lévy noise initial data, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 2, 249–270. MR 2235547, DOI https://doi.org/10.1142/S0219025706002330

Bernt Øksendal and Agnès Sulem, Applied stochastic control of jump diffusions, 2nd ed., Universitext, Springer, Berlin, 2007. MR 2322248

Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR 1739520

John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI https://doi.org/10.1007/BFb0074920