A characterization of the kernels associated with the multiple integral representation of some functionals of the Wiener process
نویسندگان
چکیده
In this paper we present a characterization of those Wiener functionals that are the likelihood ratio for a 'signal plus independent noise' model. The characterization is expressed in terms of the representation of such functionals in a series of multiple Wiener integrals. Let {ys, 0 <s =z r} be a random process with measurable sample functions satisfying ] y, ] G K as. Let { Ws, 0 <s < T} be a Wiener process which is independent of the {v,. 0 G s < T) process. Let A,=exp Then A, is the unique solution to the integral equation A, = 1 + J 'Asys dI+', 0 and admits the series representation A,=limin q.m. i u,(t) N-m n=O where uO(t) = 1 and u,(t) = /,,'u,-,(s)J., dW,. Let 3 w denote the u-field generated by { W,, 0 < B =z t}. Since convergence in quadratic mean commutes with conditional expectation, we have (as was observed in [l] and [3])-+,I% ") = $i +,(dI%W) n=O where and the integrals are iterated Ito integrals. The functional g(W) = E(A,] ??r!) is a nonnegative functional of the Brownian motion, its Wiener-It0 representation is given by (1) with the kernels u,(.. .) satisfying (2); that is, the n-th order kernel u,(t , ,. .. , t,) is the n-th order moment of a process which is independent of 94
منابع مشابه
A Representation for Characteristic Functionals of Stable Random Measures with Values in Sazonov Spaces
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