Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 39

DIFFUSION

A diffusion or a diffusion process is a solution to a stochastic differential equation. In this chapter, we will present some results about diffusion, in particular, Itô diffusion.

39.1 Basic Concepts and Facts

Definition 39.1 (Compact Set). A set K ⊆ R is called a compact set if every open cover of K contains a finite subcover; that is, if {V_i}_{i I} is a collection of open sets such that

then there exists a finite subcollection {V_ij}_j=1,2,…,n such that

Definition 39.2 (Closure). Let E ⊆ R. The closure of the set E, written as , is the smallest closed set in R that contains E.

Definition 39.3 (Support). The support of a function f on R is the closure of the set

Definition 39.4 (C^1,2-Function). A function f on R × R is called a C^1,2-function if f(t, x) is continuously differentiable on t (i.e., is continuous) and twice continuously differentiable on x (i.e., is continuous). ...