- 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 given by
where, is a Poisson process with rate λ, and are independent and identically distributed random variables, with distribution function G, which are also independent of
Properties of the compound Poisson process
Using conditional expectation, the expected value of a compound Poisson process can be calculated 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:
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.
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.
See also
Categories:- Poisson processes
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