2. (a) [10 marks] State the definition of the following objects from risk theory. Make sure you define all objects you use in the definitions. i. The surplus process (U(t))r20- ii. A compound Poisson process (S(()):20 with Poisson parameter A. ili. The adjustment coefficient R for a classical risk process. iv. The ultimate ruin probability w(u) for a given initial surplus u. (b) [8 marks] Consider the setting of a classical risk process with Poisson parameter A i. State Lundberg's inequality. Make sure you define all objects you use. ii. Prove that an upper bound for the adjustment coefficient R is given by 20m1 RS where o is the premium loading factor and mx - E[X*] for & = 1, 2 and X] is the random variable denoting the first claim amount in the classical risk process. (c) [7 marks] Assume now that the aggregate claim process is a compound Poisson process with Poisson parameter A and individual claims are exponentially distributed with parameter cr. As usual, # denotes the premium loading factor. i. Write down an expression for the average claim size, the expected number of claims in a unit of time, and the ultimate ruin probability (w). ii. How does tu) change as A changes? And as u changes? Briefly motivate your answers. iii. How does the ruin probability in finite time (u, () compare to (u)? How does v(u, t) change as & changes? Briefly motivate your answers