The compound Poisson risk model with dependence under a multi-layer dividend strategy

On the compound Poisson risk model with dependence and a threshold dividend strategy

The Gerber–Shiu discounted penalty function in a delayed renewal risk model with multi-layer dividend strategy

On a perturbed Sparre Andersen risk model with multi-layer dividend strategy

On the classical risk model with credit and debit interests under absolute ruin

The Discounted Penalty Function with Multi-Layer Dividend Strategy in the Phase-Type Risk Model

On a risk model with debit interest and dividend payments

In this paper, the absolute ruin in the compound Poisson risk model with interest and a constant dividend barrier is investigated. First, integro-differential equations satisfied by the expected discounted dividend payments are derived. The explicit expressions are obtained when the individual claim size is exponential distributed. Second, the moment generating function of the discounted dividends is considered, and integro-differential equations satisfied by the moment generating function of the discounted dividends are derived. Third, by a “differential” argument, the time to recovery to zero from a given negative surplus is considered. Finally, how long it takes for the surplus process to reach the dividend barrier is discussed.

On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula

Exit problems for jump processes with applications to dividend problems

According to the characteristics of risk model with perturbed by diffusion, a new kind of stochastic process— F-B process is proposed following from Fuzzy Markov Skeleton process. Then the ruin probabilities (caused by oscillation or by a claim) are discussed by F-B process. We derive a distribution of the ruin probability and a distribution of the surplus immediately before the time of ruin. Then continuous model and discrete model, risk model with compound Poisson process and risk model with double Poisson process, are used to illustrate the implementation of the model. Next limit theorems of two continuous risk models perturbed by diffusion are discussed. Finally simulation experiment is introduced.

This chapter complements the material of Chap. 4 in the sense that explicit identification of requisite quantities is possible with stronger parametric assumptions about either the (marginal) distribution of the claim sizes or that of the interclaim times. Convenient parametric assumptions involve Erlang (or Coxian) distributions, due to the generality and mathematical tractability inherent in the use of such models. Sect. 5.1 directly generalizes the classical Poisson model of Chap. 3 by essentially assuming a (marginal) Erlang form for the interclaim times. A Lagrange polynomial approach together with Laplace transform arguments is used to explicitly identify all components of the Gerber-Shiu function. The independent model with an arbitrary interclaim time distribution and exponential claim sizes is then considered in Sect. 5.2. The conditioning argument of Sect. 5.2 is then extended in Sect. 5.3 to a model where the (marginal) claim size distribution is assumed to be of Coxian form. In particular, identification of the Gerber-Shiu function is shown to be possible for special choices of the penalty function, and more generally for an arbitrary penalty function of the deficit. Finally, the Coxian assumption of Sect. 5.3 is replaced by a (possibly infinite) mixture of Erlang distributions, resulting in tractable results for ruin probabilities, the deficit at ruin, and related quantities.

On a modification of the classical risk process

In this paper, we propose a discrete-time counterpart of the classical compound Poisson and several applications of a bivariate extension of the Dickson-Waters discretisation. We expose numerous examples, including forth-looking scenario of ruin and surplus models We consider the integro-differential equation m′(u) = λ+δ c1 m(u) − λ c1 ∫ u 0 to expose application of a compound-Poisson risk models in barrier strategies invoking dividend surplus. Keywords— Bivariate risk process, Discretisation, Dividend barrier strategy, Copulas, Capital allocation, Proportional reinsurance

In this thesis, we present the new results found in exploring the asymptotic tail probability of the phase-type scale mixture distributions, and apply a subclass of the phase-type scale mixtures, the class of Erlangized scale mixture distributions, to the approximation of the probability of ruin.We consider the class of phase-type scale mixture distributions, which can be written as MellinnStieltjes convolution P*G of a phase-type distribution G and a nonnegative but otherwise ar-bitrary distribution P. Such a class can also be seen as the class of distributions of the product of two independent random variables: a phase-type random variable Y s G and a nonnegative random variable S s P. We call S the scaling random variable and its corresponding distribution P the scaling distribution. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction.In this thesis, particular focus is on phase-type scale mixture distributions where the scaling distribution has discrete support m such a class has been recently used in risk applications to approximate heavy-tailed distributions. We extend an existing scheme for numerically calculating the probability of ruin of a classical CramernLundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a subclass of the phase-type scale mixtures, which is also a dense class of distributions that we denominate Erlangized scale mixtures (ESM). Such a class corresponds to nonnegative and absolutely continuous distributions which can be written as a MellinnStieltjes convolution P*Gmnof a discrete nonnegative distribution P with an Erlang distribution Gmn. A distinctive feature of such a class is that it contains heavy-tailed distributions when the scaling distributions have unbounded support, and it provides sharp approximations to heavy-tailed distributions. We suggest a simple methodology for constructing a sequence of distributions having the form P*Gmnwith the purpose of approximating the integrated tail distribution of the claim sizes. Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.

Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform