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

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:

(a) P{N₀ = 0} = P{ω : N₀(ω) = 0} = 1.

(b) For any 0 ≤ s < t, the random variable N_t − N_s is a Poisson random variable with parameter λ(t − s):

(c) The N_t has independent increments, that is, for any 0 ≤ t₁ < t₂ < … < t_n, the random variables N_t1, N_t2 − N_t1,…, N_tn − N_tn−1 are independent.

(d) Almost all sample paths of {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 ...