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
and each of the integrals in the above expression equals the multiple Stratonovich or Ogawa type integral of the tracef(k), namely
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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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DOI: https://doi.org/10.1007/BF01377634