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Concept
Compound Poisson process
Summary
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 and jump size distribution G, is a process given by
where, is the counting variable of a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which are also independent of
When are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process.
The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:
Making similar use of the law of total variance, the variance can be calculated as:
Lastly, using the law of total probability, the moment generating function can be given as follows:
Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.
Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure
where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by
and
is a convolution of measures, and the series converges weakly.
Official source
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