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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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 KR is called a compact set if every open cover of K contains a finite subcover; that is, if {Vi}i I is a collection of open sets such that

equation

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

equation

Definition 39.2 (Closure). Let ER. 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

equation

Definition 39.4 (C1,2-Function). A function f on R × R is called a C1,2-function if f(t, x) is continuously differentiable on t (i.e., is continuous) and twice continuously differentiable on x (i.e., is continuous). ...

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