Summary We will now study the stochastic integration of predictable processes against martingale-valued measures. Important examples are the Brownian, Poisson and Lévy-type cases. In the case where the integrand is a sure function, we investigate the associated Wiener–Lévy integrals, particularly the important example of the Ornstein–Uhlenbeck process and its relationship with self-decomposable random variables. In Section 4.4, we establish Itô's formula, which is one of the most important results in this book. Immediate spin-offs from this are Lévy's characterisation of Brownian motion, Burkholder's inequality and estimates for stochastic integrals. We also introduce the Stratonovitch, Marcus and backwards stochastic integrals and indicate the role of local time in extending Itô's formula beyond the class of twice-differentiable functions.

Integrators and integrands

In Section 2.6, we identified the need to develop a theory of integration against martingales that is not based on the usual Stieltjes integral. Given that our aim is to study stochastic differential equations driven by Lévy processes, our experience with Poisson integrals suggests that it might be profitable to integrate against a class of real-valued independently scattered martingale-valued measures M defined on (S, I). Here S = ℝ+ × E, where E ∈ B(ℝd) and I is the ring comprising finite unions of sets of the form I × A where A ∈ B(E) and I is itself a finite union of intervals.