6. EDS with jumps

Brownian motion can be seen as the most elementary model for describing a random phenomenon whose value varies continuously. More general processes can be obtained, such as the solution of EDS with Brownian motion as input. However, when describing physical phenomena, or in the field of finance and insurance, observed processes may present discontinuities whose location and amplitude are random. The counting of events that cause these discontinuities is classically described by Poisson processes.

First, we'll look at Lévy processes, which are jump processes that generalize Poisson processes. We will then consider the construction of the stochastic integral with respect to random measures involving jumps. Finally, we'll see how Itô's formula extends in this kind of situation. The results are given here without demonstrations, and the emphasis is on providing an intuitive...