Multiple Wiener-Ito integrals possessing a continuous extension View Full Text


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Article Info

DATE

1990-03

AUTHORS

David Nualart, Moshe Zakai

ABSTRACT

LetF(W) be a Wiener functional defined byF(W)=In(f) whereIn(f) denotes the multiple Wiener-Ito integral of ordern of the symmetricL2([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(dt1,...,dtn) on [0, 1]n such thatf(t1, ...,tn) = μ((t1, 1]), ..., (tn, 1]) a.e. Lebesgue on [0, 1]n. Recall that a multimeasure μ(A1,...,An) is for every fixedi and every fixedAi,...,Ai-1, Ai+1,...,An a signed measure inAi and there exists multimeasures which are not measures. It is, furthermore, shown that iff(t1,t2, ...,tn) = μ((t1, 1], ..., (tn, 1]) then all the tracesf(k), off exist, eachf(k) induces ann−2k multimeasure denoted by μ(k), the following relation holds and each of the integrals in the above expression equals the multiple Stratonovich or Ogawa type integral of the tracef(k), namely More... »

PAGES

131-145

References to SciGraph publications

  • 1988. Sur les integrales multiples de Stratonovitch in SÉMINAIRE DE PROBABILITÉS XXII
  • 1977. Une remarque sur les bimesures in SÉMINAIRE DE PROBABILITÉS XI
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    http://scigraph.springernature.com/pub.10.1007/bf01377634

    DOI

    http://dx.doi.org/10.1007/bf01377634

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