ABSTRACT
In this chapter, we provide a brief, ab ovo, review of general stochastically continuous processes with stationary and independent increments also called Lévy processes. Two basic examples of such processes, the Brownian motion and the Poisson process, residing on the opposite ends of the spectrum of Lévy processes, are discussed in the preceding Chapters. Definition 4.1.
Process X(t), t ≥ 0 is said to have stationary (homogeneous) increments if, for any t 1, t 2 ≥ 0, h > 0, the distributions of the increments, https://www.w3.org/1998/Math/MathML"> X ( t 1 + h ) − X ( t 1 ) , and X ( t 2 + h ) − X ( t 2 ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0043-02.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
are identical.
Definition 4.2.Process X(t), t ≥ 0 is said to have independent increments if, for any 0 < t 1 < · · · < tn , the increments https://www.w3.org/1998/Math/MathML"> X ( t 1 ) − X ( 0 ) , X ( t 2 ) − X ( t 1 ) , … , X ( t n ) − X ( t n − 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0043-03.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
are independent random variables.
