# CHAPTER 27

# POISSON PROCESSES

A Poisson process is a continuous-time stochastic process that counts the number of events in a given time interval. In mathematical finance, Poisson processes are used to build jump processes for modeling asset prices. In this chapter, we present the mathematical definition of the Poisson process and relevant results.

# 27.1 Basic Concepts and Facts

**Definition 27.1** (Poisson Process). Let λ > 0. A stochastic process {*N*_{t} : *t* ≥ 0} is said to be Poisson process with parameter λ if it satisfies the following conditions:

*P*{

*N*

_{0}= 0} =

*P*{ω :

*N*

_{0}(ω) = 0} = 1.

*s*<

*t*, the random variable

*N*

_{t}−

*N*

_{s}is a Poisson random variable with parameter λ(

*t*−

*s*):

*N*

_{t}has independent increments, that is, for any 0 ≤

*t*

_{1}<

*t*

_{2}< … <

*t*

_{n}, the random variables

*N*

_{t1},

*N*

_{t2}−

*N*

_{t1},…,

*N*

_{tn}−

*N*

_{tn−1}are independent.

*N*

_{t}:

*t*≥ 0} are right-continuous functions with left-hand limits.

**Definition 27.2** (Poisson Process with Respect to Filtrations). Let {_{t} : *t* ≥ 0} be a filtration, and let λ > 0. A stochastic process {*N*_{t} : *t* ≥ 0} is said to be Poisson process with respect to the filtration {_{t} : *t* ≥ 0} and with parameter λ if it satisfies the following ...

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