Ruin Theory

Hi Jia

You are right that this chapter is fairly formulae heavy :-(. Hopefully by the time you've practised past questions though, many of the formulae will start to embed themselves. This is what I recommend my tutorial students learn:


Definition of a Poisson process: N(t)

Definition of a compound Poisson process: S(t) = X1 + ... + XN(t) where N(t) is Poi(λt) and Xi is the amount of the ith claim. {Xi} are i.i.d.

Definition of the surplus process: U(t) = U + ct - S(t)

Definition of premium: c = (1+θ)λE[X]

Definition of the four ruin probabilities: Ψ(U), Ψ(U,t), Ψh(U), Ψh(U,t)

Know how the factors: U, t, θ, λ, μ, σ^2 affect the ultimate and finite probabilities of ruin

Be able to solve λ +cr = λMX(r) for the adjustment coefficient r

Lundberg’s inequality: Ψ(U) ≤ exp(rU)

Definition of insurer's expected profit = θλE[X]

Be able to solve λ +cnetr = λMY(r) for the adjustment coefficient in the presence of reinsurance

Definition of cnet = (1+θ)E(S) - (1+ξ)E[SR]

Definition of insurer's expected profit in presence of reinsurance = cnet - E[SI]

Derivation of formula for upper bounds on adjustment coefficient: R < 2θm1/m2 where m1 = E[X] and m2 = E[X^2]

Derivation of formula for lower bounds on adjustment coefficient: R > (1/M) ln (c/λm1) where claim amounts are continuously distributed on (0, M)

Distribution of time until the first claim for a Poisson process: T1 is Exp(λ)

Calculate probability of ruin on first claim, P(U(T1) < 0) where U(T1) = U + cT1 - X1


Hope this is a useful list

Anna

 

You're welcome - sorry, couldn't get the formatting right for ages - kept putting a line through my maths! Think I've sorted it now though