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I try to understand Theorem 10 on page 107 of Protter's Stochastic integration and differential equations. The proof is really long, and for now, I just want to get an intuition.

Here is the theorem:

Let $Z$ be a cadlag supermartingale with $Z_0=0$ of class $D$, and such that all jumps of $Z$ occur at totally inaccessible stopping times. Then there exists a unique increasing, continuous, adapted process $A$ with $A_0=0$ such that $M_t = Z_t+A_t$ is a uniformly integrable martingale.

Having worked out the discrete version, the conclusion of this more general theorem is not surprising, except for the part that "totally inaccessible jumps" leads to the existence of "continuous" FV component $A$. Can someone please provide some intuition here? Perhaps using some simple examples or under some stronger assumptions? I don't care so much about rigorousness here, just some intuitions. Thanks.

Jay.H
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1 Answers1

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In the basic Doob-Meyer decomposition, we have $Z=M-A$ with $M$ a martingale and $A$ a predictable integrable increasing process. Comparing left limits $Z_-=M_--A_-$ with predictable projections ${}^pZ={}^pM-A$ and using the fact that ${}^pM=M_-$ because $M$ is a martingale, we have $Z_--{}^pZ=A-A_-$. Thus $A$ is continuous if and only if $Z_-$ is indistinguishable from ${}^pZ$. Let $T$ be a predictable time. Then on $\{T<\infty\}$, we have by hypothesis $Z_T=Z_{T-}$, so $$ {}^pZ_T=\Bbb E[Z_T|\mathcal F_{T-}]=\Bbb E[Z_{T-}|\mathcal F_{T-}]=Z_{T-}. $$ It follows that ${}^pZ=Z_-$, and so $A$ is continuous.

John Dawkins
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