I have recently encountered an excellent lecture note on Brownian motion. However, I could not find the source (textbooks/authors/…) form which this note is taken.
Could you elaborate on the (possible source) of this note?
Below is the beginning of the note.
Chapter 8 Brownian motion and Itô calculus
Brownian motion is a continuous analogue of simple random walks (as described in the previous part), which is very important in many practical applications. This importance has its origin in the universal properties of Brownian motion, which appear as the continuous scaling limit of many simple processes. Moreover, it is also intimately related to martingales and bounded-variation processes in continuous time. Brownian motion is a very rich structure that inherits properties from various fields of mathematics [à compléter].
This chapter presents in a first section the main properties of Brownian as well as various constructions. The second section presents Itô calculus for Brownian motion: this construction is only a particular case of stochastic calculus for semi-martingales; we choose however to present it here for two reasons: many applications do not require the general framework and moreover this general framework is abstract enough, so that a short introduction in a simple case may help for the next chapter.
Before going any further in this chapter, the reader is encouraged to read again the reminder of the properties of normal random variables presented in the first part.
8.1 Definition, construction and properties
8.1.1 Definition and structure
Definition 8.1.1 (Brownian motion). Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. A Brownian motion $\left(B_t\right)_{t \in \mathbb{R}_{+}}$is a process with values in $\mathbb{R}$ such that:
- $B_0=0$,
- the maps $t \mapsto B_t(\omega)$ are almost surely continuous,
- the increments $B_{t+s}-B_t$ are independent from $\sigma\left(B_u, u \leq t\right)$ and are centered normal r.v. with variance s, for all $t, s>0$.
Its existence is postponed to the next subsection and we assume now that it exists. The Brownian motion exhibits many interesting features mentioned in the previous chapter, that relates it to other probabilistic tools.
Best Answer
This is Damien Simon's lecture at the French Polytechnic School (summer school 2012).
http://www.cmap.polytechnique.fr/~ecolemathbio2012/index.php
You would find there many very interesting contributions.