Good book that contains stochastic integration, martingales and Lévy-processes?

Question

Does anyone know about any good and easy interoductory books which contins information about martingales, sotchastic integration and Lévy-processes?

I have tried reading: http://www.cambridge.org/us/academic/subjects/statistics-probability/probability-theory-and-stochastic-processes/levy-processes-and-stochastic-calculus-2nd-edition and it is very hard, I am not really able to get much out of it. Do you know about any lower level texts you can reccomend please?, which contains stochastic calculus and the theory about Lévy processes?

I would like it to introduce stochastic calculus from scratch, where the only prerequisites are real analysis and measure theory.

Answer 1

It depends a little bit on your interests, but as you might know, stochastic processes and Itô-calculus is excessively used in quantitative finance. I can recommend some books which really explain the basics of stochastic integration and stochastic differential equations. However, these books have a (strong) focus towards financial applications.

1. 'Elementary Stochastic calculus with finance in view' by Thomas Mikosch, is a good book. It starts from scratch and really explains the basics of stochastic integration and stochastic differential equations. Only the last chapter is related to financial applications.

If you want more 'difficult' books then I would suggest:

2. Steven Shreve: 'Stochastic calculus for finance'
3. Michael Steele: 'Stochastic Calculus and Financial Applications'

If you are really not interested in any financial applications at all, then I would recommend 'Introduction to Stochastic Integration' by K. L. Chung and R.J. Williams.

Answer 2

Since Lévy processes are used a lot in finance, there are several books on this topic (that is, Lévy processes and their applications in finance). For example "Stochastic Calculus for Finance II" by S. Shreve introduces stochastic integration and contains some material on Lévy processes. However, if you are less interested in applications, but more in the theory behind it, then this might not be your first choice.

For an introduction to Lévy processes I recommend "Stochastic Processes" by Barndorff-Nielsen & Sato. On $\approx$ 60 pages they present the most important results on Lévy processes and the book is quite readable, I would say.

There are plenty of books on stochastic integration. If you are new to stochastic integration, it might be a good idea to start with stochastic integration with respect to Brownian motion and then have a look at the general theory afterwards. For example, in "Brownian Motion - An Introduction to Stochastic Processes" by Schilling & Partzsch, stochastic integration (with respect to Brownian motion) is introduced (and proved) in such a way that it can be generalized to stochastic integration with respect to martingales without difficulties.