Let N(t) be a Poisson process and let be a compound Poisson process, where X1,X2,... are iid. First, show that the conditional Laplace transform for Z given N (the number of jumps occurred up to time...

Let N(t) be a Poisson process and let

be a compound Poisson process, where X₁,X₂,... are iid. First, show that the conditional Laplace transform for Z given N (the number of jumps occurred up to time t) is

where LX is the Laplace transform of X. Use this to prove that the Laplace transform of Z is

where LN(c;t) is the Laplace transform of N(t), as given in (11.9).