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The second theorem called "Girsanov II" is indeed a special case of the general "Girsanov I" from above with $$Y_t=W_t,$$$$X_t=-\int_0^t\Theta_udW_u$$. We can show that $$[Y,X]=-\int_0^t\Theta_udu$$ using general Stochastic Calculus rules (e.g. see p.37, 6.6 here): $$[Y,X]=[W_t,-\int_0^t\Theta_udW_u]=-\int_0^t\Theta_ud[W_u,W_u]=-\int_0^t\Theta_udu$$ since ...



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