Calculate the mean and variance of S(t) and of U(t), where (S(r)),>0 and (U(r)),>0 are the aggregate claim process and the surplus process of the described model.

An insurance company has an initial surplus of 100 and premium loading factor of 20%. Assume that claims arrive according to a Poisson process with parameter A = 5 and the size of claims Xi are iid random variables with Xi ,-,, exp( 1-0). The time unit is 1 week. Assume that 1 month is 4 weeks.
(a) Calculate the average number of claims on any given day, week and month. Let t’ > 0 be an instance of time. Calculate the probability that at least one claim occurs within 5 days after t’. Calculate also the probability that at least 2 claims occur within 5 days after t’. (b) Let t = 2 months. Calculate the mean and variance of S(t) and of U(t), where (S(r)),>0 and (U(r)),>0 are the aggregate claim process and the surplus process of the described model.
(c) Derive an upper bound for the ultimate ruin probability using Lundberg’s inequality.

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