Stochastic partial differential equations driven by multi-parameter white noise of Lévy processes
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
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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
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References
- D. Applebaum: Lévy Processes and Stochastic Calculus. Cambridge Univ. Press, 2004. MR 2072890 (2005h:60003)
- F. E. Benth: Integrals in the Hida distribution space $(\mathcal {S})^\ast$. In T. Lindstrøm, B. Øksendal and A. S. Üstunel (editors): Stochastic Analysis and Related Topics. Gordon and Breach, 1993, pp. 89–99. MR 1268006 (95e:60052)
- F. E. Benth and A. Løkka: Anticipative calculus for Lévy processes and stochastic differential equations. Stochastics and Stochastics Reports 76 (2004), 191–211. MR 2072380 (2006g:60081)
- R. Cont and P. Tankov: Financial Modelling with Jump Processes. Chapman & Hall/CRL 2004. MR 2042661 (2004m:91004)
- G. Di Nunno, B. Øksendal and F. Proske: White noise analysis for Lévy processes. J. Funct. Anal. 206 (2004), 109–148. MR 2024348 (2004j:60148)
- G. Di Nunno, B. Øksendal and F. Proske: Malliavin Calculus for Lévy Processes and Applications to Finance. Forthcoming book, to be published by Springer.
- T. Hida: Brownian Motion. Springer, 1980. MR 562914 (81a:60089)
- T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: White Noise. Kluwer, 1993. MR 1244577 (95f:60046)
- H. Holden, B. Øksendal, J. Ubøe and T. Zhang: Stochastic Partial Differential Equations. Birkhäuser, 1996. Second Edition to appear in 2008/2009. MR 1408433 (98f:60124)
- K. Itô: Multiple Wiener Integral. J. Math. Soc. Japan 3 (1951), 157–169. MR 0044064 (13:364a)
- Y. Kabanov: A generalized Itô formula for an extended stochastic integral with respect to a Poisson random measure. (In Russian). Usp. Mat. Nauk 29 (1974), 167–168. MR 0397876 (53:1732)
- Y. Kabanov: On extended stochastic integrals. Theory Probab. Appl. 20 (1975), 710–722. MR 0397877 (53:1733)
- A. Lanconelli and F. Proske: On explicit strong solution of Itô-SDEs and the Donsker delta function of a diffusion. Inf. Dim. Anal. Quant. Probab. 7 (2004), 437–447. MR 2085642 (2005h:60186)
- A. Løkka, B. Øksendal and F. Proske: Stochastic partial differential equations driven by Lévy space-time white noise. Annals Appl. Probab. 14 (2004), 1506–1528. MR 2071432 (2005d:60078)
- T. Lindstrøm, B. Øksendal and J. Ubøe: Wick multiplication and Itô-Skorohod stochastic differential equations. In S. Albeverio et al. (editors): “Ideas and Methods in Mathematical Analysis, Stochastics and Applications”. Cambridge Univ. Press, 1992, pp. 183–206. MR 1190500 (94d:60084)
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- D. Nualart and W. Schoutens: Chaotic and predictable representations for Lévy processes. Stoch. Proc. Appl. 90 (2000), 109–122. MR 1787127 (2001j:60090)
- B. Øksendal and F. Proske: White noise of Poisson random measures. Potential Analysis 21 (2004), 375–403. MR 2081145 (2005i:60134)
- B. Øksendal, F. Proske and M. Signahl: The Cauchy problem for the wave equation with Lévy noise initial data. Inf. Dim. Anal. Quantum Probab. Rel. Topics 9 (2006) 249–270. MR 2235547 (2007e:60059)
- B. Øksendal and A. Sulem: Applied Stochastic Control of Jump Diffusions. Springer. Second Edition 2007. MR 2322248
- K.-I. Sato: Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, 1999. MR 1739520 (2003b:60064)
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Additional Information
Bernt Øksendal
Affiliation:
Center of Mathematics for Applications (CMA), Department of Mathematics, University of Oslo, P. O. Box 1053 Blindern, N–0316 Oslo, Norway, Norwegian School of Economics and Business Administration (NHH), N–5014 Bergen, Norway
Email:
oksendal@math.uio.no
Received by editor(s):
May 15, 2007
Published electronically:
July 3, 2008
Additional Notes:
I want to thank Yaozhong Hu and Jiang Lun Wu for valuable communications.
Article copyright:
© Copyright 2008
Brown University
The copyright for this article reverts to public domain 28 years after publication.