Consider a compound Poisson process Y (t) = N(t) n=1 ξn, where N(t) is a Poisson process with rate λ and the ξn are i.i.d. and independent of N. Suppose ξ1 has a mean μ, variance σ2 and moment...

Consider a compound Poisson process Y (t) = N(t) n=1 ξn, where N(t) is a Poisson process with rate λ and the ξn are i.i.d. and independent of N. Suppose ξ1 has a mean μ, variance σ2 and moment generating function ψ(α) = E[eαξ1 ]. Show that the following are martingales with respect to Y :