On the relationship between Wick calculus and stochastic integration in the Lévy white noise analysis

  • Maria M. Frei
  • Nikolai A. KachanovskyEmail author
Research Article


We deal with Lytvynov’s generalization of the chaotic representation property and the corresponding spaces of regular generalized functions in the Lévy white noise analysis. Our goal is to find a relationship between the Wick calculus and stochastic integration on these spaces. In particular, we consider the Wick multiplication (a natural multiplication on spaces of generalized functions) under the sign of the stochastic integral, and construct a formal representation of the extended stochastic integral via the Pettis integral, using the Wick product. As application of our results, we consider some stochastic equations with Wick type nonlinearities.


Lévy process Extended stochastic integral Wick product 

Mathematics Subject Classification

46F25 60G51 60H05 



The authors are very grateful to the referee for helpful advices.


  1. 1.
    Benth, F.E., Di Nunno, G., Løkka, A., Øksendal, B., Proske, F.: Explicit representation of the minimal variance portfolio in markets driven by Lévy processes. Math. Finance 13(1), 55–72 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berezansky, Yu.M., Kondratiev, Yu.G.: Spectral Methods in Infinite-Dimensional Analysis. Mathematical Physics and Applied Mathematics, vol. 12. Kluwer, Dordrecht (1995)Google Scholar
  3. 3.
    Berezansky, Yu.M., Sheftel, Z.G., Us, G.F.: Functional Analysis, Vol. II. Operator Theory: Advances and Applications, vol. 86. Birkhäuser, Basel (1996)Google Scholar
  4. 4.
    Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)Google Scholar
  5. 5.
    Di Nunno, G., Øksendal, B., Proske, F.: Malliavin Calculus for Lévy Processes with Applications to Finance. Universitext. Springer, Berlin (2009)CrossRefGoogle Scholar
  6. 6.
    Di Nunno, G., Øksendal, B., Proske, F.: White noise analysis for Lévy processes. J. Funct. Anal. 206(1), 109–148 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dyriv, M.M., Kachanovsky, N.A.: On operators of stochastic differentiation on spaces of regular test and generalized functions of Lévy white noise analysis. Carpathian Math. Publ. 6(2), 212–229 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frei, M.M.: Wick calculus on spaces of regular generalized functions of Lévy white noise analysis. Carpathian Math. Publ. 10(1), 82–104 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frei, M.M., Kachanovsky, N.A.: Some remarks on operators of stochastic differentiation in the Lévy white noise analysis. Methods Funct. Anal. Topology 23(4), 320–345 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hida, T.: Analysis of Brownian Functionals. Carleton Mathematical Lecture Notes, vol. 13. Carleton University, Ottawa (1975)Google Scholar
  11. 11.
    Holden, H., Øksendal, B., Ubøe, J., Zhang, T.-S.: Stochastic Partial Differential Equations. Probability and its Applications. Birkhäuser, Boston (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Itô, K.: Spectral type of the shift transformation of differential processes with stationary increments. Trans. Amer. Math. Soc. 81(2), 253–263 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kabanov, Yu.M., Skorohod, A.V.: Extended stochastic integrals. In: Proceedings of the School-Seminar on the Theory of Random Processes, part 1, pp. 123–167. Inst. Fiz. i Mat. Akad. Nauk Litovsk. SSR, Vilnius (1975) (in Russian)Google Scholar
  14. 14.
    Kachanovsky, N.A.: An extended stochastic integral and a Wick calculus on parametrized Kondratiev-type spaces of Meixner white noise. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11(4), 541–564 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kachanovsky, N.A.: Extended stochastic integrals with respect to a Lévy process on spaces of generalized functions. Mathematical Bulletin of Taras Shevchenko Scientific Society 10, 169–188 (2013)Google Scholar
  16. 16.
    Kachanovsky, N.A.: On extended stochastic integrals with respect to Lévy processes. Carpathian Math. Publ. 5(2), 256–278 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kachanovsky, N.A.: Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis. Methods Funct. Anal. Topology 21(4), 336–360 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kachanovsky, N.A., Tesko, V.A.: Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space. Ukrainian Math. J. 61(6), 873–907 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kondratiov, Yu.G.: Generalized Functions in Problems of Infinite-Dimensional Analysis. Ph.D. Thesis, Kyiv (1978)Google Scholar
  20. 20.
    Lytvynov, E.: Orthogonal decompositions for Lévy processes with an application to the gamma, Pascal, and Meixner processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6(6), 73–102 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nualart, D., Schoutens, W.: Chaotic and predictable representations for Lévy processes. Stochastic Process. Appl. 90(1), 109–122 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schoutens, W.: Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics, vol. 146. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  23. 23.
    Skorohod, A.V.: Integration in Hilbert Space. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 79. Springer, Berlin (1974)Google Scholar
  24. 24.
    Skorohod, A.V.: On a generalization of a stochastic integral. Theory Probab. Appl. 20(2), 219–233 (1976)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Solé, J.L., Utzet, F., Vives, J.: Chaos expansions and Malliavin calculus for Lévy processes. In: Benth, E.F., et al. (eds.) Stochastic Analysis and Applications. Abel Symposia, vol. 2, pp. 595–612. Springer, Berlin (2007)CrossRefGoogle Scholar
  26. 26.
    Surgailis, D.: On \(L^2\) and non-\(L^2\) multiple stochastic integration. In: Arató, M., Vermes, D., Balakrishnan, A.V. (eds.) Stochastic Differential Systems. Lecture Notes in Control and Information Sciences, vol. 36, pp. 212–226. Springer, Berlin (1981)Google Scholar
  27. 27.
    Vershik, A.M., Tsilevich, N.V.: Fock factorizations and decompositions of the \(L^2\) spaces over general Lévy processes. Russian Math. Surveys 58(3), 427–472 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Precarpathian National UniversityIvano-Frankivs’kUkraine
  2. 2.Institute of MathematicsNASUKievUkraine

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