Compound Poisson process explained

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 byY(t) = \sum ^{N(t)} _{i =1} 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:

\operatorname E(Y(t)) = \operatorname E(D _{1} + … + D _{N(t)} ) = \operatorname E(N(t))\operatorname E(D _{1)} = \operatorname E(N(t)) \operatorname E(D) = λ t \operatorname E(D).

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

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

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\mid N(t) = n)\Pr(N(t) = n) \begin{align} \operatorname E(e ^{sY} ) & = \sum _{i} e ^{si} \Pr(Y(t) = i) \\[ 5 pt] & = \sum _{i} e ^{si} \sum _{n} \Pr(Y(t) = i\mid N(t) = n)\Pr(N(t) = n) \\[ 5 pt] & = \sum _{n} \Pr(N(t) = n) \sum _{i} e ^{si} \Pr(Y(t) = i\mid N(t) = n) \\[ 5 pt] & = \sum _{n} \Pr(N(t) = n) \sum _{i} e ^{si} \Pr(D _{1} + D _{2} + … + D _{n=i)} \\[ 5 pt] & = \sum _{n} \Pr(N(t) = n) M ^{n} _{D(s)} \\[ 5 pt] & = \sum _{n} \Pr(N(t) = n) e ^{n ln (M _{D(s))}} \\[ 5 pt] & = M _{N(t)} ( ln (M _{D(s)))} \\[ 5 pt] & = e ^{λ t \left( M _{D(s)} -1 \right)} . \end{align}

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( λ t(\mu - \delta _{0))}

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

\exp(\nu) = \sum ^{inf ty} _{n =0} {\nu ^{*n} \over n!}

and

\nu ^{*n} = \underbrace{\nu * … *\nu} _{n}

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