Answer Happy

Claims arrive at an insurance company following a Poisson process with rate 1 > 0. Each time a claim arrives the company pays a random amount S. Assume each claim is independent of the others and that the density of the claims is g(s). Due to inflation and the ability of the insurance company to invest the premiums collected, the longer a claim is delayed, the less it costs the company. If a claim is discounted at rate ß, then show that the company's ultimate liability L:= Sie ßti, where {t;} is an arrival time (not the inter-arrival time) of the Poisson process and S; is the ith random claim, has mean and variance 1 E[L]= B S sg(s)ds Var(L) = 26 S.*s?915)ds. Hint: Note that the pair (Ti,S;) forms a marked Poisson process.