The Annals of Applied Probability

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

Arne Løkka, Bernt Øksendal, and Frank Proske

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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 d3, 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.

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

First available in Project Euclid: 13 July 2004

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

Lévy processes white noise analysis stochastic partial differential equations


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.

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  • 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.
  • 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.
  • 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.
  • 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.
  • Applebaum, D. and Wu, J.-L. (1998). Stochastic partial differential equations driven by Lévy space-time white noise. Preprint, Univ. Bochum, Germany.
  • 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.
  • Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • Cramer, H. (1946). Mathematical Methods in Statistics. Princeton Univ. Press.
  • Di Nunno, G., Øksendal, B. and Proske, F. (2004). White noise analysis for Lévy processes. J. Func. Anal. To appear.
  • 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.
  • 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.
  • Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L. (1993). White Noise. Kluwer, Dordrecht.
  • Holden, H., Øksendal, B., Ubøe, J. and Zhang, T.-S. (1996). Stochastic Partial Differential Equations---A Modeling, White Noise Functional Approach. Birkhäuser, Boston.
  • Hu, Y. and Øksendal, B. (2003). Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 1--32.
  • Itô, K. (1956). Spectral type of the shift transformation of differential processes and stationary increments. Trans. Amer. Math. Soc. 81 253--263.
  • Itô, Y. and Kubo, I. (1988). Calculus on Gaussian and Poisson white noises. Nagoya Math. J. 111 41--84.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Nualart, D. and Schoutens, W. (2000). Chaotic and predictable representations for Lévy processes. Stochastic Process. Appl. 90 109--122.
  • Nualart, D. and Zakai, M. (1986). Generalized stochastic integrals and the Malliavin calculus. Probab. Theory Related Fields 73 255--280.
  • Obata, N. (1994). White Noise Calculus and Fock Space. Lecture Notes in Math. 1577. Springer, Berlin.
  • 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.
  • Potthoff, J. and Timpel, M. (1995). On a dual pair of smooth and generalized random variables. Potential Anal. 4 637--654.
  • Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • Thangavelu, S. (1993). Lectures on Hermite and Laguerre Expansions. Princeton Univ. Press.
  • Us, G. F. (1995). Dual Appell systems in Poissonian analysis. Methods Funct. Anal. Topology 1 93--108.
  • 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.
  • Wick, G. C. (1950). The evaluation of the collinear matrix. Phys. Rev. 80 268--272.