Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases?



Martingale representation theorem for Levy processes
Hi,


Here is a theorem that might answer your question (it is coming from Chesnay, Jeanblanc-Piqué and Yor's book "Mathematical Methods for Financial Markets"). 


It is theorem (11.2.8.1 page 621) here it is :


(edit note : be carefull as mentioned by G. Lowther there's a typo in the book regarding the domain of integration in the conditions over $\psi$ (defined hereafter) )


Let $X$ be an $R^d$ valued Lévy Process and $F^X$ its natural filtration. Let $M$ be an $F^X$-local Martingale. Then there exist an $R^d$-valued predictable process $\phi$ and an predictable function $\psi : R^+ \times \Omega \times R^d\to R$ such that :


-$\int\_0^t \phi^i(s)^2ds <\infty$ almost surely 


-$\int\_0^t \int\_{|x|> 1} |\psi(s,x)|ds\nu(dx) <\infty$ almost surely 


-$\int\_0^t \int\_{|x|\le 1} \psi(s,x)^2ds\nu(dx) <\infty$ almost surely 


and 


$M\_t=M\_0+ \sum\_{i=0}^d \int\_0^t \phi^i(s)dW^i\_s + \int\_0^t \int\_{R^d} \psi(s,x)\tilde{N}(ds,dx)$


Where $\tilde{N}(ds,dx)$ is the compensated measure of the Lévy process $X$ and $\nu$ the associated Lévy measure.


Moreover if $(M\_t)$ is square integrable martingale then we have :


$E[(\int\_0^t \phi^i(s)dW^i\_s)^2]=E[\int\_0^t \phi^i(s)^2ds]<\infty$


and 


$E[(\int\_0^t \int\_{R^d} \psi(s,x)\tilde{N}(ds,dx))^2]=E[ \int\_0^t ds \int\_{R^d} \psi(s,x)^2\nu(dx)]<\infty$


and $\phi$ and $\psi$ are essentially unique. 


The theorem is not proved in the book but there is a reference to the following parpers :


1/H. Kunita and S. Watanabe. On square integrable martingales. Nagoya J.
Math., 30:209–245, 1967


2/H. Kunita. Representation of martingales with jumps and applications to
mathematical finance. In H. Kunita, S. Watanabe, and Y. Takahashi, editors,
Stochastic Analysis and Related Topics in Kyoto. In honour of Kiyosi Itô,
Advanced studies in Pure mathematics, pages 209–233. Oxford University
Press, 2004.


Regards


