By Norihiko Kazamaki

In 3 chapters on Exponential Martingales, BMO-martingales, and Exponential of BMO, this ebook explains intimately the gorgeous homes of continuing exponential martingales that play an important position in a variety of questions in regards to the absolute continuity of likelihood legislation of stochastic strategies. the second one and crucial target is to supply an entire record at the interesting effects on BMO within the concept of exponential martingales. The reader is believed to be conversant in the overall idea of continuing martingales.

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**Sample text**

Which completes the proof. 15 : [] Let T be an innovation time such that P(T > O) > O, f~' = {T > 0}, fi; =`T, la,, 1 d R ' - p(fy)Ia,dP. To, so that if X is a martingale, then XIa, is a martingale under P' and in addition we have (XIa,) = (X)Ia,. Tt) under P such that X ' = XIa,. T[). 9 it follows that on the probability system ( f l ' , . T~)) there exists a bounded martingale M ' for which exp(-~(M')oo) LI(p1). Let now M = M'Ia,. Then it is a bounded martingale over (`T,) such that e x P ( s ( M ) ~ o ) ~ LI(P).

L. THE REVERSE HOLDER INEQUALITY 55 Therefore, it follows that E[$(M)oo : g(M)oo > A] _ E [ E ( M ) ~ : T < ec] <_ E[E(M)T : T < ec] <_ AP(T < oc) 2pA < 2 p _ ~ l P ( g ( M ) o o > 5~). 5). Secondly, let T be any fixed stopping time. TT+t, M~ = MT+t - MT (O < t < ec). T[) is a martingale with respect to dPq Note that E(M')~ = E(M)T+t/g(M)T. An elementary calculation shows that IIM'llBMOr(p,) <_ IIMllBMOr(p) for every r > 1. Thus above, we get IIM'llSMo~(e,) < (I,(p). 5). Namely, we have E[{$(M)~o/$(M)T}P : A] <_ Kp,M,P(A).

This is one of the most important results in the theory of H1. On the other hand, in 1972 R. K. Getoor and M. J. Sharpe ([21]) introduced the concept of a conformal martingale and by using conformal martingales they established the duality of//1 and B M O in the probabilistic setting. 1) II M IlSMO, = sup E [ I M ~ - MT_IPlJrTy p oo T where the supremum is taken over all stopping times T. The class {M : [[MllBMO, < ec} is denoted by B M O v , and observe that II II~MO, is a norm on this space. From the HSlder inequality it follows at once that for p < q, B M O q C B M O p .