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Multiple Wiener-Ito integrals possessing a continuous extension
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  • Published: March 1990

Multiple Wiener-Ito integrals possessing a continuous extension

  • David Nualart1 &
  • Moshe Zakai2 

Probability Theory and Related Fields volume 85, pages 131–145 (1990)Cite this article

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  • 12 Citations

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Summary

LetF(W) be a Wiener functional defined byF(W)=I n(f) whereI n(f) denotes the multiple Wiener-Ito integral of ordern of the symmetricL 2([0, 1]n) kernelf. We show that a necessary and sufficient condition for the existence of a continuous extension ofF, i.e. the existence of a function ø(·) from the continuous functions on [0, 1] which are zero at zero to ℝ which is continuous in the supremum norms and for which ø(W)=F(W) a.s, is that there exists a multimeasure μ(dt 1,...,dt n ) on [0, 1]n such thatf(t 1, ...,t n ) = μ((t 1, 1]), ..., (t n , 1]) a.e. Lebesgue on [0, 1]n. Recall that a multimeasure μ(A 1,...,A n ) is for every fixedi and every fixedA i,...,Ai-1, Ai+1,...,An a signed measure inA i and there exists multimeasures which are not measures. It is, furthermore, shown that iff(t 1,t 2, ...,t n ) = μ((t 1, 1], ..., (t n , 1]) then all the tracesf (k),\(k \leqq \left[ {\frac{n}{2}} \right]\) off exist, eachf(k) induces ann−2k multimeasure denoted by μ(k), the following relation holds

$$I_n (f) = \sum\limits_{k = 0}^{[n/2]} {\left( { - \frac{1}{2}} \right)^k \frac{{n!}}{{k!(n - 2k)!}}\int\limits_{[0,1]^{n - 2k} } {W_{t_1 } } \cdot \cdot \cdot W_{t_{n - 2k} } \cdot \mu ^{(k)} (dt_1 ,...,dt_{n - 2k} )} $$

and each of the integrals in the above expression equals the multiple Stratonovich or Ogawa type integral of the tracef(k), namely

$$\int\limits_{[0, 1]^{n - 2k} } {W_{t_1 } } \cdot \cdot \cdot W_{t_{n - 2k} } \mu ^{(k)} (dt_1 , . . . , dt_{n - 2k} ) = I_{n - 2k} \circ (f^{(k)} ).$$

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References

  1. Blei, R.C.: Fractional dimensions and bounded fractional forms. Mem. Am. Math. Soc.5, number 331, 1985

    Google Scholar 

  2. Blei, R.C.: Multilinear measure theory and the Grothendieck factorization theorem. Proc. Lond. Math. Soc.56, 529–546 (1988)

    Google Scholar 

  3. Clarckson, J.A., Adams, C.K.: On definitions of bounded variation for functions of two variables. Trans. Am. Math. Soc.35, 824–854 (1933)

    Google Scholar 

  4. Fréchet, M.: Sure les functionnelles bilinéaires. Trans. Am. Math. Soc.16, 215–234 (1915)

    Google Scholar 

  5. Gilbert, J.E., Ito, T., Schreiber, B.M.: Bimeasure algebras on locally compact groups, J. Funct. Anal.64, 134–162 (1985)

    Google Scholar 

  6. Gross, L.: Abstract Wiener Space. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability II, part I, pp. 31–42, Berkeley: University of California Press, 1967

    Google Scholar 

  7. Horowitz, J.: Une remarque sur les bimeasures. In: Dellacherie, C., Meyer, P.-A., Weil, M. (eds.) Seminaire de Probabilites XI. (Lect. Notes Math., vol. 581, pp. 58–64) Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  8. Hu, Y.Z., Meyer, P.A.: Sur les intégrales multiples de Stratonovich. In: Azéma, J., Meyer P.A., Yor, M. (eds) Séminaire de Probabilités XXII (Lect. Notes Math., vol. 1321, pp. 72–81) Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  9. Johnson, G.W., Kallianpur, G.: Some remarks on Hu and Meyer's paper and infinite dimensional calculus on finitely additive cannonical Hilbert space. Theor Probab. Appl. (to appear)

  10. Nualart, D., Zakai, M.: Generalized multiple stochastic integrals and the representation of Wiener functionals. Stochastics23, 311–330 (1988)

    Google Scholar 

  11. Nualart, D., Zakai, M.: On the relation between the Stratonovich and Ogawa integrals. Ann. Probab.17, 1536–1540 (1989)

    Google Scholar 

  12. Sole, J.L., Utzet, F.: Stochastic integral and trace. Stochastics (to appear)

  13. Sugita, H., Hu-Meyer's multiple Stratonovich integral and essential continuity of multiple Wiener integral. Bull. Soc. Math., 2e Série,113, 463–474 (1989)

    Google Scholar 

  14. Watanabe, S.: Lectures on stochastic differential equations and Malliavin Calculus. Tata Institute of Fundamental Research. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  15. Wiener, N.: The homogeneous chaos. Am. J. Math.,55, 897–936 (1938)

    Google Scholar 

  16. Wiener, N.: Nonlinear problems in random theory. Cambridge: M.I.T. Press 1958

    Google Scholar 

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Author information

Authors and Affiliations

  1. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, E-08007, Barcelona, Spain

    David Nualart

  2. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Moshe Zakai

Authors
  1. David Nualart
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  2. Moshe Zakai
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Partially supported by CICYT, grant PB86-0238

Partially supported by fund for promotion of research at the Technion

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Nualart, D., Zakai, M. Multiple Wiener-Ito integrals possessing a continuous extension. Probab. Th. Rel. Fields 85, 131–145 (1990). https://doi.org/10.1007/BF01377634

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  • Issue Date: March 1990

  • DOI: https://doi.org/10.1007/BF01377634

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Keywords

  • Continuous Function
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Signed Measure
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