Discrete-time Risk Process Research Articles

Ruin Probabilities as Recurrence Sequences in a Discrete-Time Risk Process

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support {0,1,…,m+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\{0,1,\\ldots ,m+1\\}}$$\\end{document}, for some integer m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{m\\ge 1}$$\\end{document}. The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.

On the risk of credibility premium rules

ABSTRACT A discrete-time risk process is considered where the full distribution of the claim size X is not completely known to the insurance company. Rather, it assumes that the distribution of X given is where Z is some structural random variable for which a prior is available. The main emphasis of the paper is the unconditional ruin probability in this setting where the premium is either updated according to incoming information about the claim distribution or computed by the expected value principle. This is in turn studied via the conditional ruin probability , for which large deviations estimates are available. Rigorous proofs are given only for the case of the forming a scale parameter family, including the classical case of gamma claims with a gamma prior. However, the analysis readily suggests what should be the behaviour of in different models for the claims.

Ruin probability for discrete risk processes

Abstract We study the discrete time risk process modelled by the skip-free random walk and derive results connected to the ruin probability and crossing a fixed level for this type of process. We use the method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.

Inequalities for the ruin probability in a controlled discrete-time risk process

Ruin probabilities in a controlled discrete-time risk process with a Markov chain interest are studied. To reduce the risk of ruin there is a possibility to reinsure a part or the whole reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a constant stationary policy. The relationships between these inequalities are discussed. To illustrate these results some numerical examples are included.

Bounds for the Ruin Probability of a Discrete-Time Risk Process

We consider a discrete-time risk process driven by proportional reinsurance and an interest rate process. We assume that the interest rate process behaves as a Markov chain. To reduce the risk of ruin, we may reinsure a part or even all of the reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a stationary policy. To illustrate these results, a numerical example is included.

Bounds for the Ruin Probability of a Discrete-Time Risk Process

We consider a discrete-time risk process driven by proportional reinsurance and an interest rate process. We assume that the interest rate process behaves as a Markov chain. To reduce the risk of ruin, we may reinsure a part or even all of the reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a stationary policy. To illustrate these results, a numerical example is included.

The compound binomial model with randomized decisions on paying dividends

Consider a discrete time risk process based on the compound binomial model. The insurer pays a dividend of 1 with a probability q 0 when the surplus is greater than or equal to a non-negative integer x . We will derive recursion formulas and an asymptotic estimate for the ruin probability, the probability function of the surplus prior to the ruin time, and the severity of ruin, etc.

Ruin probabilities with a Markov chain interest model

Finite and infinite time ruin probabilities in a discrete time risk process with a Markov chain interest model are studied. Recursive and integral equations for the ruin probabilities are given. When interest rates are non-negative, generalized Lundberg inequalities for the infinite time ruin probability are derived by inductive and martingale approaches. When interest rates can be negative and loss distributions have regularly varying tails, asymptotic formulas for the finite time ruin probability are given by an inductive approach on the recursive equations.

Time in the red in a two state Markov model

Consider a discrete time risk process based on a two state Markov chain. The premium is assumed to be the reciprocal of an integer, the benefit is unity and the initial surplus is a multiple of the premium. We will derive recursion formulae for the joint distribution of ruin and severity of ruin and for the expected time to surplus zero from a given negative surplus. Further we will derive asymptotic estimates for the ruin probability, the severity of ruin and their joint distribution. In conclusion we will calculate the expected time in the red.

Recursive calculation of finite-time ruin probabilities

We develop a simple algorithm for the numerical calculation of finite-time ruin probabilities in a general discrete-time risk process model. These probabilities can be used for the calculation of approximations for the finite-time ruin probabilities in the classical actuarial risk model.