Most General Theory of Stochastic Integration

Question

I've learnt continuous stochastic integration using the classical books: - Revuz & Yor, - Karatzas & Shreve and - Oksendal.

Now I want to learn general stochastic integration, i.e. possibly discontinuous semimartingales.

Can anyone recommend some literature, which covers this topics and also provides some exercises?

Answer 1

The monograph Stochastic Integration and Differential Equations by Philip E. Protter covers stochastic integration with respect to semimartingales and provides exercises at the end of each chapter. This is the standard reference, I would say (at least for graduate students).

The first part of Limit Processes for Stochastic Processes by Jean Jacod and Albert Shiryaev also discusses stochastic integration with respect to semimartingales. Although this is not an easy read, you should manage if you are familiar with stochastic processes and integration (with respect to continuous martingales) and if you like rigorous books. However, the book doesn't contain any exercises.

Answer 2

Protters book uses a more modern approach based on functional analysis - whereas Jacod uses the "classical approach". Jacods exposition is very short (but nevertheless rigorous) - the whole theory is developed in less than 60 pages. (Personally i found Jacods book better)

I want to point out a third option. In fact this is perhaps the most rigorous and self contained treatment you can get (as literally every basic property of martingales is proven as well) - however it requires a lot of time and dedication. Its the book "Probabilities and Potential B: Theory of Martingales" from Dellacherie & Meyer.