Semimartingales with jumps, weak Dirichlet processes and path-dependent martingale problems – Francesco Russo (ENSTA-Paris)

Venue

Aula Seminari

Abstract

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.

1. To provide a (unique) decomposition which is also new for semimartingales with jumps.
2. To discuss some new stability theorem for weak Dirichlet processes through C0,1 transformations.
3. 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).