3 Decomposition of a counting process associated with

and let Fz denote the a-algebra generated by the following collections of events:

{Ai < ai,...,Aj < aj}n{m = j}(l{Z(T) < y}; (6)

Theorem 1. The process N(z) can be written as the sum

N(z)= i Y(y)dA(y) + M(z), (7) o where M(z) is a martingale with respect to the filtration (Fz \ 0 < z < zo) and

26 V. Bagdonavicius, A. Bikelis, V. Kazakevicius and M. Nikulin Proof. Fix y < z, j > 1 and denote by X = 1{A1<a1..,Aj<aj}• Then

Xl{m=j} (N(z) - N(y)) = E[X 1{S.+a3y<T<s3+A,z}]

(because 1{z(t)<y'}Y(x) = 0 for all x > y). Since the union of collections (5) and (6) is closed with respect to finite intersections of events, obtained equalities mean, by the Monotone Class Theorem, that

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