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I have a question concerning the paper, Lemma 6.3, which is based on Prop. 5. Actually this proposition does not state what is used, but I think the following is done:

Say, we have a local Martingale $M$, a predictable process $\lambda$, a local martingale $L$ which is orthogonal to $M$ in the sense that $ML$ is a local martingale such that

$$ Z=1-\int Z \lambda dM+L $$

is a strictly positive local martingale. Can you always rewrite $Z$ to $Z=\mathcal{E}(-\lambda\cdot M + \tilde{L})$ with $d\tilde{L}_t=\frac{1}{Z_{t-}}dL_t$ so $\tilde{L}$ is still orthogonal to $M$?

One gets

$$ d(M\tilde{L})=\tilde{L}dM+\frac{M}{Z}dL+\frac{1}{Z}d[M,L] $$

Now, if $\tilde{L}$, $\frac{M}{Z}$ and $\frac{1}{Z}$ are well behaved, e.g. locally bounded everything is fine. Otherwise the integrals might not be local martingales. But that is not obvious, is it? I've seen the same trick a couple of times, always without going into details.

Thank you!

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