Itô-Doeblin lemma for non-continuous semimartingales

Question

On wikipedia there are some results on Itô's lemma applied to non-continuous semimartingales (sometimes called Itô-Doeblin lemma)

I have looked up the books by Malliaris, Oksendal and Doob, but they only mention the case of Poisson jumps, not the more general case.

Where can I find a reference and a demonstration of Itô's lemma for non-continuous semimartingales?

Answer 1

As saz pointed out, there are several books on jump processes which give demonstrations and examples on semimartingales. Some of these books are:

1. Stochastic Differential Equations and Diffusion Processes - Ikeda & Watanabe
2. Stochastic Integration and Differential Equations - Philip E. Protter
3. Limit Theorems for Stochastic Processes - Jacod & Shiryaev
4. Introduction to stochastic calculus with applications - Fima C. Klebaner