Compound Poisson process

Template:Short description Template:Refimprove 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 <math>\lambda > 0</math> and jump size distribution G, is a process <math>\{\,Y(t) : t \geq 0 \,\}</math> given by

<math>Y(t) = \sum_{i=1}^{N(t)} D_i</math>

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

When <math> D_i </math> are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process. Template:Source needed

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:

<math>\operatorname E(Y(t)) = \operatorname E(D_1 + \cdots + D_{N(t)}) = \operatorname E(N(t))\operatorname E(D_1) = \operatorname E(N(t)) \operatorname E(D) = \lambda t \operatorname E(D).</math>

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

<math>

\begin{align} \operatorname{var}(Y(t)) &= \operatorname E(\operatorname{var}(Y(t)\mid N(t))) + \operatorname{var}(\operatorname E(Y(t)\mid N(t))) \\[5pt] &= \operatorname E(N(t)\operatorname{var}(D)) + \operatorname{var}(N(t) \operatorname E(D)) \\[5pt] &= \operatorname{var}(D) \operatorname E(N(t)) + \operatorname E(D)^2 \operatorname{var}(N(t)) \\[5pt] &= \operatorname{var}(D)\lambda t + \operatorname E(D)^2\lambda t \\[5pt] &= \lambda t(\operatorname{var}(D) + \operatorname E(D)^2) \\[5pt] &= \lambda t \operatorname E(D^2). \end{align} </math>

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

<math>\Pr(Y(t)=i) = \sum_n \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) </math>

<math>

\begin{align} \operatorname E(e^{sY}) & = \sum_i e^{si} \Pr(Y(t)=i) \\[5pt] & = \sum_i e^{si} \sum_{n} \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) \\[5pt] & = \sum_n \Pr(N(t)=n) \sum_i e^{si} \Pr(Y(t)=i\mid N(t)=n) \\[5pt] & = \sum_n \Pr(N(t)=n) \sum_i e^{si}\Pr(D_1 + D_2 + \cdots + D_n=i) \\[5pt] & = \sum_n \Pr(N(t)=n) M_D(s)^n \\[5pt] & = \sum_n \Pr(N(t)=n) e^{n\ln(M_D(s))} \\[5pt] & = M_{N(t)}(\ln(M_D(s))) \\[5pt] & = e^{\lambda t \left( M_D(s) - 1 \right) }. \end{align} </math>

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.

<math>\mu(A) = \Pr(D \in A).\,</math>

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

<math>\exp(\lambda t(\mu - \delta_0))\,</math>

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

<math>\exp(\nu) = \sum_{n=0}^\infty {\nu^{*n} \over n!}</math>

and

<math> \nu^{*n} = \underbrace{\nu * \cdots *\nu}_{n \text{ factors}}</math>

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

See also

• Poisson process
• Poisson distribution
• Compound Poisson distribution
• Non-homogeneous Poisson process
• Campbell's formula for the moment generating function of a compound Poisson process