In this talk we will revisit the notion of weak Dirichlet process which is the natural extension of semimartingale with jumps. If X is such a process, then it is the sum of a local martingale M and a martingale orthogonal process A in the sense that [A, N ] = 0 for every continuous local martingale N . We remark that if [A] = 0 then X is a Dirichlet process. The notion of Dirichlet process is not very suitable in the jump case since in this case A is forced to be continuous.
The talk will discuss the following points.
- To provide a (unique) decomposition which is also new for semimartingales with jumps.
- To discuss some new stability theorem for weak Dirichlet processes through C0,1 transformations.
- To discuss various examples of such processes arising from path-dependent martingale problems. This includes path-dependent stochastic differential equations with involving a distributional drift and with jumps.
The talk is based on a joint paper with E. Bandini (Bologna).