## Positive martingale representation with jumps

I am looking for a martingale representation theorem for positive semimartingales. Using the answer to this question: http://mathoverflow.net/questions/70981/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 $u_s(x)$?

-

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