Compound Poisson process

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A compound Poisson process with rate λ > 0 and jump size distribution G is a continuous-time stochastic process \{\,Y(t) : t \geq 0 \,\} given by

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

where, \{\,N(t) : t \geq 0\,\} is 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\,\}.\,

[edit] Properties of the compound Poisson process

Using conditional expectation, the expected value of a compound Poisson process can be calculated as:

\,E(Y(t)) = E(E(Y(t)|N(t))) = E(N(t)E(D)) = E(N(t))E(D) = \lambda t E(D).

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

\, \operatorname{var}(Y(t)) = E(\operatorname{var}(Y(t)|N(t))) + \operatorname{var}(E(Y(t)|N(t)))
\, = E(N(t)\operatorname{var}(D)) + \operatorname{var}(N(t)E(D))
\, = \operatorname{var}(D)E(N(t)) + E(D)^2 \operatorname{var}(N(t))
\, = \operatorname{var}(D)\lambda t + E(D)^2\lambda t
\, = \lambda t(\operatorname{var}(D) + E(D)^2)
\, = \lambda t E(D^2).

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)
\,E(e^{sY}) = \sum_{i} e^{si} \Pr(Y(t)=i)
\, = \sum_{i} e^{si} \sum_{n} \Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n)
\, = \sum_{n} \Pr(N(t)=n) \sum_{i} e^{si} \Pr(Y(t)=i|N(t)=n)
\, = \sum_{n} \Pr(N(t)=n) \sum_{i} e^{si}\Pr(D_1 + D_2 + \cdots + 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^{\lambda t \left ( M_D(s) - 1\right ) }.

[edit] 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.

\mu(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(\lambda t(\mu - \delta_0))\,

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

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

and

\nu^{*n} = \underbrace{\nu * \cdots *\nu}_{n\ \mathrm{factors}}

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

[edit] See also