Exercise 2. (Ruin problem) Consider an insurance company earning premiums at a constant rate p per unit of time. Suppose also that there is a Poisson process with intensity > 0 which has jump times T 2 are i.i.d. and distributed by the exponential distribution with density de-de. At Tk the insurance company pays a claim of value YA, where Y, Y2.... are i.i.d. random variables which are independent of the Poisson process and strictly positive, i.e. P[Y > 0) = 1. Suppose that the initial reserve of the company is x > 0. Hence, at time t > 0, the reserve of the company is X = + pt - Y, k where the sum is over all k such that Tk 0: X: 1, rewrite the event Ti 0 no matter how large r is. In fact, show that P[T: e-Kr (1) where K > 0 is independent of x. (We will soon see a type of converse to this inequality.) (c) Let's now get an idea of when ruin is inevitable. Suppose Y, is integrable, i.e. E[Y] 0. Argue that P[T: 0 such that I(61) = 0. ('L' is for Lundberg.) (e) By consider the exponential change of measure' with parameter @u, show Lundberg's inequality: P[Tx 0. (2) Exercise 2. (Ruin problem) Consider an insurance company earning premiums at a constant rate p per unit of time. Suppose also that there is a Poisson process with intensity > 0 which has jump times T 2 are i.i.d. and distributed by the exponential distribution with density de-de. At Tk the insurance company pays a claim of value YA, where Y, Y2.... are i.i.d. random variables which are independent of the Poisson process and strictly positive, i.e. P[Y > 0) = 1. Suppose that the initial reserve of the company is x > 0. Hence, at time t > 0, the reserve of the company is X = + pt - Y, k where the sum is over all k such that Tk 0: X: 1, rewrite the event Ti 0 no matter how large r is. In fact, show that P[T: e-Kr (1) where K > 0 is independent of x. (We will soon see a type of converse to this inequality.) (c) Let's now get an idea of when ruin is inevitable. Suppose Y, is integrable, i.e. E[Y] 0. Argue that P[T: 0 such that I(61) = 0. ('L' is for Lundberg.) (e) By consider the exponential change of measure' with parameter @u, show Lundberg's inequality: P[Tx 0. (2)