By N. Balakrishnan
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Extra resources for Advances on Theoretical and Methodological Aspects of Probability and Statistics
So let be a tame function and has the representation where is C∞-bounded and s1<…
38) Now applying the Ito formula to and hence and Rt, we get . 42) above to interchange the integral signs we have used Fubini’s theorem which is justified because of the following. s. , since P=Q. 42) is justified. t. 14)]. 1 Let Wt, 0ՅtՅT, be a Wiener martingale with respect to ( t), 0ՅtՅT. Let As be -progressively measurable such that is an ( t)-martingale. Assume that (As) satisfy the following further conditions. s. s. MANDAL We apply the lemma on , defined on the product space (⍀×⍀, ⊗ , µ) and .
We denote a typical point in Ω×Ω by (ω, ω′). For any F-measurable function ζ on Ω, ζ† will denote the function on Ω×Ω defined by which clearly is an F⊗Fmeasurable function. Also, ζ′ will denote the function on Ω×Ω given by ζ′(ω, ω′)=ζ(ω′). Thus, and . So, under µ, is an independent copy of . Also, with this notation for (ω, ω′) ∈ Ω×Ω. 28) for (ω, ω′) ∈ Ω×Ω. Since (Xt) is independent of (Yt) under (Q, ( ) is independent of ( ) under µ. Also, since, under µ, ( ) has the same distribution as the joint distribution of ( ) and ( ), under µ, is the same as that of and ( ).
Advances on Theoretical and Methodological Aspects of Probability and Statistics by N. Balakrishnan