The Annals of Applied Probability

Stochastic partial differential equations driven by Lévy space-time white noise

Arne Løkka, Bernt Øksendal, and Frank Proske

Full-text: Open access

Enhanced PDF (179 KB)

Abstract

In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Lé vy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d≤3, then this solution can be represented as a classical random field in L2(μ), where μ is the probability law of the L évy process. The starting point of our theory is a chaos expansion in terms of generalized Charlier polynomials. Based on this expansion we define Kondratiev spaces and the Lévy Hermite transform.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 3 (2004), 1506-1528.

Dates
First available in Project Euclid: 13 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1089736294

Digital Object Identifier
doi:10.1214/105051604000000413

Mathematical Reviews number (MathSciNet)
MR2071432

Zentralblatt MATH identifier
1053.60069

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60H40: White noise theory 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
Lévy processes white noise analysis stochastic partial differential equations

Citation

Løkka, Arne; Øksendal, Bernt; Proske, Frank. Stochastic partial differential equations driven by Lévy space-time white noise. Ann. Appl. Probab. 14 (2004), no. 3, 1506--1528. doi:10.1214/105051604000000413. https://projecteuclid.org/euclid.aoap/1089736294


Export citation

References

  • Aase, K., Øksendal, B., Privault, N. and Ubøe, J. (2000). White noise generalizations of the Clark--Haussmann--Ocone theorem with application to mathematical finance. Finance Stoch. 4 465--496.
    Mathematical Reviews (MathSciNet): MR1779589
  • Albeverio, S., Hida, T., Potthoff, J. and Streit, L. (1989). The vacuum of the Høegh-Krohn model as a generalized white noise functional. Phys. Lett. B 217 511--514.
    Mathematical Reviews (MathSciNet): MR981544
    Digital Object Identifier: doi:10.1016/0370-2693(89)90087-7
  • Albeverio, S. and Høegh-Krohn, R. (1974). The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space time. J. Funct. Anal. 16 39--82.
    Mathematical Reviews (MathSciNet): MR356761
    Digital Object Identifier: doi:10.1016/0022-1236(74)90070-6
    Zentralblatt MATH: 0279.60095
  • Albeverio, S., Kondratiev, Y. G. and Streit, L. (1993). How to generalize white noise analysis to non-Gaussian spaces. In Dynamics of Complex and Irregular Systems (Ph. Blanchard, L. Streit, M. Sirugue-Collin and D. Testard, eds.). 120--130. World Scientific, Singapore.
    Mathematical Reviews (MathSciNet): MR1340427
  • Applebaum, D. and Wu, J.-L. (1998). Stochastic partial differential equations driven by Lévy space-time white noise. Preprint, Univ. Bochum, Germany.
    Mathematical Reviews (MathSciNet): MR1796675
  • Benth, F. E. and Løkka, A. (2002). Anticipative calculus for Lévy processes and stochastic differential equations. Unpublished manuscript.
  • Bers, L., John, F. and Schechter, M. (1964). Partial Differential Equations. Interscience.
    Mathematical Reviews (MathSciNet): MR163043
    Zentralblatt MATH: 0126.00207
  • Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
    Mathematical Reviews (MathSciNet): MR1406564
  • Cramer, H. (1946). Mathematical Methods in Statistics. Princeton Univ. Press.
    Mathematical Reviews (MathSciNet): MR16588
    Zentralblatt MATH: 0063.01014
  • Di Nunno, G., Øksendal, B. and Proske, F. (2004). White noise analysis for Lévy processes. J. Func. Anal. To appear.
    Mathematical Reviews (MathSciNet): MR2024348
    Digital Object Identifier: doi:10.1016/S0022-1236(03)00184-8
  • Dobrushin, R. L. and Minlos, R. L. (1977). Polynomials in linear random functions. Russian Math. Surveys 32 71--127.
  • Elliott, R. and van der Hoek, J. (2000). A general fractional white noise theory and applications to finance. Preprint.
  • Gelfand, I. M. and Vilenkin, N. J. (1968). Generalized Functions 4. Academic Press, San Diego.
    Zentralblatt MATH: 0159.18301
  • Hida, T. (1982). White noise analysis and its applications. In Proc. Int. Mathematical Conf. (L. H. Y. Chen, ed.) 43--48. North-Holland, Amsterdam.
  • Hida, T. and Ikeda, N. (1965). Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral. Proc. Fifth Berkeley Symp. Math. Stat. Probab. II 117--143. Univ. California Press, Berkeley.
    Mathematical Reviews (MathSciNet): MR219131
    Zentralblatt MATH: 0212.19803
  • Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L. (1993). White Noise. Kluwer, Dordrecht.
    Mathematical Reviews (MathSciNet): MR1244577
  • Holden, H., Øksendal, B., Ubøe, J. and Zhang, T.-S. (1996). Stochastic Partial Differential Equations---A Modeling, White Noise Functional Approach. Birkhäuser, Boston.
    Mathematical Reviews (MathSciNet): MR1408433
    Zentralblatt MATH: 0860.60045
  • Hu, Y. and Øksendal, B. (2003). Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 1--32.
    Mathematical Reviews (MathSciNet): MR1976868
    Digital Object Identifier: doi:10.1142/S0219025703001110
    Zentralblatt MATH: 1045.60072
  • Itô, K. (1956). Spectral type of the shift transformation of differential processes and stationary increments. Trans. Amer. Math. Soc. 81 253--263.
    Mathematical Reviews (MathSciNet): MR77017
  • Itô, Y. and Kubo, I. (1988). Calculus on Gaussian and Poisson white noises. Nagoya Math. J. 111 41--84.
    Mathematical Reviews (MathSciNet): MR961216
  • Kachanowsky, N. A. (1997). On an analog of stochastic integral and Wick calculus in non-Gaussian infinite dimensional analysis. Methods Funct. Anal. Topology 3 1--12.
    Mathematical Reviews (MathSciNet): MR1768162
  • Kondratiev, Y. (1978). Generalized functions in problems of infinitedimensional analysis. Ph.D. thesis, Kiev Univ.
  • Kondratiev, Y., Da Silva, J. L. and Streit, L. (1997). Generalized Appell systems. Methods Funct. Anal. Topology 3 28--61.
    Mathematical Reviews (MathSciNet): MR1768164
  • Kondratiev, Y., Da Silva, J. L., Streit, L. and Us, G. (1998). Analysis on Poisson and gamma spaces. Infin. Dim. Anal. Quantum Probab. Relat. Top. 1 91--117.
    Mathematical Reviews (MathSciNet): MR1611915
    Digital Object Identifier: doi:10.1142/S0219025798000089
    Zentralblatt MATH: 0924.60018
  • Kondratiev, Y., Leukert, P. and Streit, L. (1994). Wick calculus in Gaussian analysis. Unpublished manuscript, Univ. Bielefeld, Germany.
  • Kuo, H. H. (1996). White Noise Distribution Theory. CRC Press, Boca Raton, FL.
    Mathematical Reviews (MathSciNet): MR1387829
    Zentralblatt MATH: 0853.60001
  • Lindstrøm, T., Øksendal, B. and Ubøe, J. (1991). Stochastic differential equations involving positive noise. In Stochastic Analysis (M. Barlow and N. Bingham, eds.) 261--303. Cambridge Univ. Press.
    Mathematical Reviews (MathSciNet): MR1166414
  • Løkka, A. and Proske, F. (2002). Infinite dimensional analysis of pure jump Lévy processes on the Poisson space. Preprint, Univ. Oslo.
  • Mueller, C. (1998). The heat equation with Lévy noise. Stochastic Process. Appl. 74 67--82.
    Mathematical Reviews (MathSciNet): MR1624088
    Digital Object Identifier: doi:10.1016/S0304-4149(97)00120-8
  • Nualart, D. and Schoutens, W. (2000). Chaotic and predictable representations for Lévy processes. Stochastic Process. Appl. 90 109--122.
    Mathematical Reviews (MathSciNet): MR1787127
    Digital Object Identifier: doi:10.1016/S0304-4149(00)00035-1
  • Nualart, D. and Zakai, M. (1986). Generalized stochastic integrals and the Malliavin calculus. Probab. Theory Related Fields 73 255--280.
    Mathematical Reviews (MathSciNet): MR855226
    Digital Object Identifier: doi:10.1007/BF00339940
    Zentralblatt MATH: 0601.60053
  • Obata, N. (1994). White Noise Calculus and Fock Space. Lecture Notes in Math. 1577. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1301775
    Zentralblatt MATH: 0814.60058
  • Øksendal, B. (1996). An introduction to Malliavin calculus with applications to economics. Working paper No 3/96, Norwegian School of Economics and Business Administration.
  • Øksendal, B. and Proske, F. (2004). White noise of Poisson random measures. Potential Anal. To appear.
    Mathematical Reviews (MathSciNet): MR2081145
    Digital Object Identifier: doi:10.1023/B:POTA.0000034329.34647.fd
    Zentralblatt MATH: 1060.60069
  • Potthoff, J. and Timpel, M. (1995). On a dual pair of smooth and generalized random variables. Potential Anal. 4 637--654.
    Mathematical Reviews (MathSciNet): MR1361381
    Digital Object Identifier: doi:10.1007/BF02345829
    Zentralblatt MATH: 0839.46035
  • Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
    Mathematical Reviews (MathSciNet): MR1739520
  • Thangavelu, S. (1993). Lectures on Hermite and Laguerre Expansions. Princeton Univ. Press.
    Mathematical Reviews (MathSciNet): MR1215939
    Zentralblatt MATH: 0791.41030
  • Us, G. F. (1995). Dual Appell systems in Poissonian analysis. Methods Funct. Anal. Topology 1 93--108.
    Mathematical Reviews (MathSciNet): MR1402159
  • Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Ecole d'Été de Probabilités de St. Flour XIV. Lecture Notes in Math. 1180 266--439. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR876085
    Zentralblatt MATH: 0608.60060
  • Wick, G. C. (1950). The evaluation of the collinear matrix. Phys. Rev. 80 268--272.
    Mathematical Reviews (MathSciNet): MR38281
    Digital Object Identifier: doi:10.1103/PhysRev.80.268
    Zentralblatt MATH: 0040.13006

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more