A compound Poisson process is a continuous-time (random) 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. A compound Poisson process, parameterised by a rate λ > 0 and jump size distribution G, is a process { Y ( t ) : t ≥ 0 } given by
Y ( t ) = ∑ i = 1 N ( t ) D i where, { N ( t ) : t ≥ 0 } is a Poisson process with rate λ , and { D i : i ≥ 1 } are independent and identically distributed random variables, with distribution function G, which are also independent of { N ( t ) : t ≥ 0 } .
When D i are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process which has the feature that two or more events occur in a very short time .
Using conditional expectation, the expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:
E ( Y ( t ) ) = E ( E ( Y ( t ) | N ( t ) ) ) = E ( N ( t ) E ( D ) ) = E ( N ( t ) ) E ( D ) = λ t E ( D ) . Making similar use of the law of total variance, the variance can be calculated as:
var ( Y ( t ) ) = E ( var ( Y ( t ) | N ( t ) ) ) + var ( E ( Y ( t ) | N ( t ) ) ) = E ( N ( t ) var ( D ) ) + var ( N ( t ) E ( D ) ) = var ( D ) E ( N ( t ) ) + E ( D ) 2 var ( N ( t ) ) = var ( D ) λ t + E ( D ) 2 λ t = λ t ( var ( D ) + E ( D ) 2 ) = λ t E ( D 2 ) . Lastly, using the law of total probability, the moment generating function can be given as follows:
Pr ( Y ( t ) = i ) = ∑ n Pr ( Y ( t ) = i | N ( t ) = n ) Pr ( N ( t ) = n ) E ( e s Y ) = ∑ i e s i Pr ( Y ( t ) = i ) = ∑ i e s i ∑ n Pr ( Y ( t ) = i | N ( t ) = n ) Pr ( N ( t ) = n ) = ∑ n Pr ( N ( t ) = n ) ∑ i e s i Pr ( Y ( t ) = i | N ( t ) = n ) = ∑ n Pr ( N ( t ) = n ) ∑ i e s i Pr ( D 1 + D 2 + ⋯ + D n = i ) = ∑ n Pr ( N ( t ) = n ) M D ( s ) n = ∑ n Pr ( N ( t ) = n ) e n ln ( M D ( s ) ) = M N ( t ) ( ln ( M D ( s ) ) ) = e λ t ( M D ( s ) − 1 ) . Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.
μ ( A ) = Pr ( D ∈ 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 ( μ − δ 0 ) ) where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by
exp ( ν ) = ∑ n = 0 ∞ ν ∗ n n ! and
ν ∗ n = ν ∗ ⋯ ∗ ν ⏟ n factors is a convolution of measures, and the series converges weakly.
