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I am looking for a martingale representation theorem for positive semimartingales. Using the answer to this question: Martingale representation theorem for Levy processes

My best guess is (subject to integrability condition, in one dimension for simplicity): $$ M_t = M_0 + \int_0^t M_s v_s dWs + \int_0^t \int_R M_s u_s(x) \tilde{N}(ds, dx)$$

where $\tilde{N}(ds, dx)$ is the compensated measure of the underlying Lévy process, but as I said its just a guess. Is it correct? Do I need any conditions for $u_s(x)$?

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  • $\begingroup$ Hi, What do you mean exactly by "martingale representation theorem for positive semimartingales" ? What is the filtration with respect to which you have martngales ? If it is a filtration generated by a semimartingale with jumps, you can take a look at theorem 204 p.177 of the book Theory of SDEs with jumps and applications, by Rong Situ. There's also in the same book the martingale representation property with respect to a (brownian and poisson point process)-filtration on p.68. $\endgroup$
    – user20368
    Commented Jan 5, 2012 at 10:49

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The correct answer seems to be exponential martingales. E.g. for Levy exponential martingales we have the representation: $$ M_t = M_0 + \int_0^t M_s v_s dWs + \int_0^t \int_R M_s (e^x-1) \tilde{N}(ds, dx) $$

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