Compound Poisson process

A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate $λ > 0$ and jump size distribution G, is a process ${Y (t) : t \geq 0}$ given by

Y (t) = \sum_{i = 1}^{N (t)} D_{i}

where, ${N (t) : t \geq 0}$ is the counting variable of a Poisson process with rate $λ$ , and ${D_{i} : i \geq 1}$ are independent and identically distributed random variables, with distribution function G, which are also independent of ${N (t) : t \geq 0} .$

When $D_{i}$ are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process.

Properties of the compound Poisson process

The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:

E (Y (t)) = E (D_{1} + \dots + D_{N (t)}) = E (N (t)) E (D_{1}) = E (N (t)) E (D) = λ t E (D) .

Making similar use of the law of total variance, the variance can be calculated as:

\begin{aligned} var (Y (t)) & = E (var (Y (t) ∣ N (t))) + var (E (Y (t) ∣ N (t))) \\ = E (N (t) var (D)) + var (N (t) E (D)) \\ = var (D) E (N (t)) + E (D)^{2} var (N (t)) \\ = var (D) λ t + E (D)^{2} λ t \\ = λ t (var (D) + E (D)^{2}) \\ = λ t E (D^{2}) . \end{aligned}

Lastly, using the law of total probability, the moment generating function can be given as follows:

\Pr (Y (t) = i) = \sum_{n} \Pr (Y (t) = i ∣ N (t) = n) \Pr (N (t) = n)

\begin{aligned} E (e^{s Y}) & = \sum_{i} e^{s i} \Pr (Y (t) = i) \\ = \sum_{i} e^{s i} \sum_{n} \Pr (Y (t) = i ∣ N (t) = n) \Pr (N (t) = n) \\ = \sum_{n} \Pr (N (t) = n) \sum_{i} e^{s i} \Pr (Y (t) = i ∣ N (t) = n) \\ = \sum_{n} \Pr (N (t) = n) \sum_{i} e^{s i} \Pr (D_{1} + D_{2} + \dots + D_{n} = i) \\ = \sum_{n} \Pr (N (t) = n) M_{D} (s)^{n} \\ = \sum_{n} \Pr (N (t) = n) e^{n \ln (M_{D} (s))} \\ = M_{N (t)} (\ln (M_{D} (s))) \\ = e^{λ t (M_{D} (s) - 1)} . \end{aligned}

Exponentiation of measures

Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.

μ (A) = \Pr (D \in A) .

Let δ₀ be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure

\exp (λ t (μ - δ_{0}))

where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by

\exp (ν) = \sum_{n = 0}^{\infty} \frac{ν^{* n}}{n!}

and

ν^{* n} = \underset{n factors}{\underset{⏟}{ν * \dots * ν}}

is a convolution of measures, and the series converges weakly.

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