Sorry, Gang. I did not mean to hide anything. For Azema martingales, Emery is the expert on this. The reference paper "On the Azema martingale" gave a very nice explantation on this. Basically, they can be charaterized as a process with poisson with time change. Philip also explain the Azema martingale in his book from another point of view.(It is closely related to Brownian excursion, try JeanBlanc on google, who did some nice work, the reference can be find in Philip's another paper on credit). In terms of analytic tractability, the model is Markov and the infinitesimal generator is given in Emery's paper. So in principle, if you can work with Levy processes, so can you do it here. (of course, Alan is the expert on this here )About the predictable representation theorem if the model is Azema martingale, it is not given in very detail in this paper, but you can find the answer in another paper by Philip. Same place, #6. Of course, it is not written there explicitly for Azema, but it is for Markov processes (which include Azema). Now, back to my data cleaning and coffee drinking job....

A diffusion process can never have jumps along the path; only the boundary conditions can have jumps. Therefore, the jump-diffusion process doesn’t literally mean to put jumps in the diffusion, but rather integrate the two processes. The option price can have jumps but not the diffusion component.

QuoteOriginally posted by: Vincentjump-diffusion model can be considered as a Brownian motion plus a poisson process. Not just that, the following paper go one step further. And it has the esential juice to price the options.Xue, Xing Xiong Martingale representation for a class of processes with independent increments and its applications. Applied stochastic analysis (New Brunswick, NJ, 1991), 279--311, Lecture Notes in Control and Inform. Sci., 177, Springer, Berlin, 1992.

lyuping,What did Xing Xiong Xue's SDE look like? I hope this isn't another yoga-toga...

I haven't seen the martingale representation in Jeanblanc & pontier's, which is super-hedging, if I understand correctly.Protter's is great in the completeness sense. Xue's model is still in what Vincent said a Levy model , but as I said it has some juice. It claims that every F^Y local martingale M can be represented as M(t)= M_0+\int_0^t (H(s), U(s)) \times (dW, dN)where W is a Brownain motion, N is a compenstaed Poisson process, the filtration F^Y is generated by Y, which is driven by(W,N) in a complicted SDE: dY= \beta(t,Y) dt + \gama(t,Y) dW + compensated jump terms of Y. Though Xue didn't go further to give an explict option valuation formula, it did give a market completeness result based on this main result.newton, I don't know what is yoga-toga, so ...

QuoteOriginally posted by: SimplicioNewton << I've got two good pseudo Greek letters yoga & toga, I'm working on a hedging strategy; something along the lines of technical-technical analysis, essentially a pattern analysis on the utterances of technical analysts. At least it promises to make interesting listening: we can combine QF kung-fu with vanilla yoga, and adjust our net option toga when we observe a candlestick of head and shoulders.Whar are your Yoga & Toga,... Simplicio ?

This thread is going to take a lot of editing! Would one or two contributors please have a go at summarizing...?!P

I suggest transfering this thread to the technical forum and starting over!MJ

From memory a summary of this thread might be:1. Jump-diffusion models can be used to model kurtosis in asset returns2. But they lead to incomplete markets and therefore require a discussion of how to treat risk preferences3. *** Space where discussion of risk preferences should be! ***4. Discussion of Azema martingales which no-one except Martingale can quite visualize! Very frustrating as this process (which incorporates jumps and yet still keeps complete markets) is probably very interesting and useful.5. Fizzles out ...