Suppose that the classical risk model applies. Let $(U(t){)}_{t}0$ denote the surplus process, that is$,$

$U(t)=u+ct-S(t)$

where the aggregate claims process $(S(t){)}_{t}0$ is a compound Poisson process with Poisson parameter $=4.$ Assume that the premium income rate per time unit is $c=16,$ where the time$-$unit is a day, and that $u=50.$ Assume that the individual claims, denoted by $x_i,$iinN, are distributed according to a Gamma distribution $x_i(2,4).$

i$.$ Check if the condition $c>E[x_1]$ on the premium income rate is satisfied. Why is this condition important?

ii$.$ Calculate the mean and the variance of $S(8).$

iii. Calculate $|)>(23.$

iv$.$ What is the average surplus process after $10$ days?

v$.$ Write down an equation for the adjustment coefficient $R$ and calculate the value of $R.$ Using your answer, determine an upper bound for the ultimate probability of ruin using Lundberg's inequality.

vi$.$ Consider a new surplus process hat$(U)$ given by

hat$(U)(t)=50+10t-S(t)$

How does the ultimate ruin probability for hat$(U)$ compare to the one for $U?$ Provide a proof of your answer.