Thanks a lot for the interesting blog. Concerning Theorems 1 and 2, I was looking for references in the literature, but so far I haven’t found any. Can you help here?

Marcus

]]>You define a predictable sigma-algebra on {{\mathbb R}_+\times\Omega}, denoted by {\mathcal{P}} to be the sigma algebra generated by the left-continuous and adapted processes. You also write that a adapted process is one which in turn depends on the sigma algebra i.e that each r.v is measuable. This seems circular to me, do you mean that we take a left contious processes and then generate it’s natural filtration? Or have I gotten something wrong?

]]>When the function f is one dimensional, it is easy to understand what is \Delta f (X_s) = f (X_s + jump ) – f( X_s). But what is the signification of \Delta f (X_s) when f is a two dimensional function for example?

Is it \Delta f (X_s, Y_s) = f(X_s + jump of X_s, Y_s + jump of Y_s) – f ( X_s, Y_s) ?

or \Delta f (X_s, Y_s) = f(X_s + jump of X_s, jump of Y_s) – f ( X_s, Y_s) + (X_s, jump of Y_s) – f ( X_s, Y_s + jump of Y_s).

Thank you !

]]>The following thesis deals with an Ito calculus for Dirichlet processes (using non-anticipative functionals and not only functions)

http://www.math.columbia.edu/~thaddeus/theses/2010/fournie.pdf

Enjoy!! ]]>

I am not sure if you remember, that six months ago, I consulted you this question on Brownian Motions under different filtrations and existence of quadratic covariation as limit of Riemann sum. You mentioned that you had a counter-example where the Riemann sum diverged.

This problem has remained a mystery to me until now. I have so far not been able to construct an counterexample, nor can I prove convergence. I am therefore wondering if you will have time to revisit this problem.

Thanks in advance!

Best,

Ryan