The finite-time ruin probability under the compound binomial risk model

Abstract

We study the compound binomial ruin model, which is considered to be the discrete analogue of the classical compound Poisson model. Our key result is a simple approach for inverting a generating function whose argument is the discount factor when we know the inverse of the same generating function, which this time has argument that is the solution to Lundberg's equation. The main idea comes from a result in Dickson and Willmot (2005) who discuss the classical model. We are then able to derive the probability distribution of the time to ruin and to go beyond the results in Dickson and Willmot (2005) by deducing the distribution of the first hitting time of a specific level and the duration of the time when the surplus is negative. The paper contains several illustrative examples where specific claim-amount distributions are considered.

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable, dependent claim occurrences

“…In this context, the discrete‐time compound binomial risk model was first proposed by Gerber as a discrete‐time version of the continuous‐time compound Poisson (CP) model in risk theory. Later, new results were obtained by Michel, Shiu, Willmot, Dickson, Cheng et al, Liu et al, Lefèvre and Loisel, Li and Sendova, Eryilmaz, Tuncel and Tank, and Eryilmaz using this model. Cossette et al defined and studied a discrete‐time risk model with I 1 , I 2 ,… as a Markov process and referred to it as the compound Markov binomial model.…”

Parisian ruin for the dual risk process in discrete-time

“…Step 1 For , in Eq. ( 2.6 ), and using the fact that for , we have that where is the classic finite-time survival probability in the compound binomial risk model, which has been extensively studied in the literature, (see [ 19 ] and references therein) and alternatively can be evaluated using Lemma 2 .…”

“…Later, Lefèvre and Loisel [ 18 ] derive a Seal-type formula based on the ballot theorem (see [ 27 ]) and a Picard–Lefèvre-type formula for the corresponding finite-time survival probability, namely . For further results on finite-time probabilities, see [ 19 ] and references therein. The finite-time ruin probabilities, in general, prove difficult to tackle and the literature on the subject remains few.…”

An identity based on the generalised negative binomial distribution with applications in ruin theory

“…In this section we give a simple illustration of the usefulness of equation (3.1). Li & Sendova (2013) show that the probability function of the time of ruin when aggregate claims are geometrically distributed with probability function (1 − q)q x for x = 0, 1, 2, … is…”

“…The ultimate ruin probability for this model can be defined in two different ways. The definition used in this note is that used by Li & Sendova (2013), namely but we could have equally used the definition given by Gerber (1988) and reworked the results we use below from Li & Sendova (2013). The net profit condition for this model is .…”