Risk Theory

The following sections are included:

• Dedication
• Preface
• Contents

The following sections are included:

• The ruin probability
  • Total claim size
• Classical model with exponential claim distribution
• Ruin probability in classical risk model
• Discount penalty function
• Exercises

In this model the spans between successive occurrence times are not exponential random variables anymore, but follow a common general distribution, denoted by

Let us introduce the notation (A, c, B) for this setup, called Sparre-Andersen risk model…

The following sections are included:

• De Vylder’s approximation
• Beekman-Bowers’ approximation
• Diffusion approximation
• Limit survival approximation
• Two-sided bounds
• Exercises

The following sections are included:

• Limit distribution of the maximum
• Max-stability
• Maximum domain of attraction
• Point processes
• Exercises

The following sections are included:

• Representation theory
• Karamata’s theory
• Closure with respect to convolution
• Exercises

The absence of exponential moment of claims is to be understood as that the claims can take huge values. This is the case for several situations of the insurance practice. The constantly increasing demand for insurance force us to proceed beyond the constraint of Cramér condition and to take into consideration claim distributions with heavy tails.

The following sections are included:

• Weighted sums
• Renewal risk model with interest force
• Convolution equivalence
• Structure of subexponentiality
• The product distribution
• Exercises

We study the random walk, whose increments follow a distribution with heavy tails and negative mean value. The maximum of the random walk is finite almost surely and we are interesting to find the asymptotic expressions of the ruin probability. Taking into account the property of the subexponentiality we establish that the main reason of ruin is that one of the increments becomes too large. This is the single big jump principle.

The following sections are included:

• Compound interest
• Risk model under constant interest rate
  • Extension of Lundberg inequality
• Finite horizon risk model
  • Classical risk model
  • Risk model with Pareto claim
  • Renewal risk model
• Exercises

The loan is the most popular way to face the deficit of the insurance company. The question is how large a loan can the company borrow, so that it can survive and return back the money. As it is to be proved, the company has no chances of survival if the negative surplus exceeds the level −c/δ, where c is a constant premium rate and δ represents the constant interest force for this loan. So, we introduce the following model, under which we re-examine the ruin probability, taking into account the possibility to borrow from some financial institution.

In the renewal risk model (A, c, B) we follow three basic independence assumptions. Firstly, independence among the inter-claim times {θ_k, k ∈ ℕ }. Next, independence among the claim sizes {Z_k, k ∈ ℕ }. The last is the independence between the previous two sequences. However, these independence assumptions reduce their practical value, in cases where there are clear indication of dependence. In particular, the last form of independence between the two sequences, is questionable in all kinds of insurance.

The following sections are included:

• Ruin under investment
• Asymptotic independence
• Asymptotic without moment conditions
• Exercises

The following sections are included:

• Vague convergence
• Convergence to Poisson process
• Breiman’s asymptotic formula
• Exercises

The following sections are included:

• Bibliography
• Index