By Norihiko Kazamaki
ISBN-10: 0387580425
ISBN-13: 9780387580425
ISBN-10: 3540580425
ISBN-13: 9783540580423
In 3 chapters on Exponential Martingales, BMO-martingales, and Exponential of BMO, this publication explains intimately the attractive homes of continuing exponential martingales that play a necessary position in quite a few questions about the absolute continuity of chance legislation of stochastic tactics. the second one and crucial target is to supply an entire document at the interesting effects on BMO within the thought of exponential martingales. The reader is believed to be conversant in the overall conception of constant martingales.
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Extra resources for Continuous Exponential Martingales and BMO
Example text
On the other hand, one can easily see that (Bt^I) ~ H a \ Loo. Observe now that the martingale E[B~IZ't] is not in B M O . Incidentally, if M, N E B M O and if f is a Lipschitz function on R 2, that is, [f(x,y) - f(x',y')J < c(]x x' I + [y - y ' ] ) for all (x,y), (x',y') E R 2, with a constant c > 0, then the martingale (0 _< t < e~) is also in B M O . TT] + E[IN~ - NTIWT]) <_ 2c(l[MllsMo ~ + IINII~MO~). ~. ] belong to the class B M O . 2) where the supremum is taken over all stopping times T satisfying P ( T < ec) > 0.
Doob we get I]~(M(k)) -~(M)Hm -- [IE(M'k))- E(M)I]/_/~ _< (M(k/) - <: 2s g ( M ( k ) ) o o - g ( M ) ~ where r -1 + s -1 = 1. This completes the proof. 3, the generalization to the right continuous BMO-martingales is impossible. Corollary 3. 1. into Hi. If M C B M O , then x ~-~ ~ ( x M ) is a continuous mapping of R 1 It is remarkable that, even if M E Hp for all p > 0, g(M) does not necessarily belong to the class H1. 1. Tt) be a one dimensional Brownian motion starting at 0 and let T be the stopping time defined by T=inf{t:Bt <_t-l).
Then we have p(p@lM) < q and so 1 a(-~M) (p~)
1. Tz] <_ C2. J and E[exp(-Mo~)l~] have (A2), so that p(M) <_ 2. Finally we shall show that p(M) - 1 < - a---~ 1 if p(M) < ec. For that, it is enough to verify that if p - 1 > ~(-~), then p >_p(M). If p > 2, it is trivial, because p(M) <_ 2. So we let 1 < p < 2 a n d p - l > ~(-~). p) is valid. T. ] satisfies (Ap). ] satisfies (Av). Therefore, we find that p(M) <_p. [] Let M E B M O . ] satisfy all (Ap).
Continuous Exponential Martingales and BMO by Norihiko Kazamaki
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