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

In this paper, a compound Poisson risk model with time-dependent claims is studied under a multi-layer dividend strategy. A piecewise integro-differential equation for the Gerber-Shiu function is derived and solved. Asymptotic formulas of the ruin probability are obtained when the claim size distributions are heavy-tailed.

Parisian ruin with a threshold dividend strategy under the dual Lévy risk model

On a risk model with debit interest and dividend payments

In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the claim amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie–Gumbel–Morgenstern copula. In this framework, we derive the integro-differential equation and the Laplace transform (LT) of the Gerber–Shiu discounted penalty function. An explicit expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for claim amounts having an exponential distribution.

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

<abstract><p>In this paper, a classical risk model with liquid reserves and proportional investment is considered, and the expected total discounted dividend before ruin of insurance companies under the threshold dividend strategy is studied. First, the integral differential equations of the expected total discounted dividend before ruin satisfying certain boundary conditions is derived. Second, since the explicit solutions of the equations cannot be obtained, the numerical approximation solutions are obtained by the sinc approximation method. Finally, we discuss the effects of parameters such as risk capital ratio and liquid reserve on the expected total discounted dividend before ruin by some examples.</p></abstract>

We show an analytic method to construct a bivariate distribution function (DF) with given marginal distributions and correlation coefficient. We introduce a convenient mathematical tool, called a copula, to connect two DFs with any prescribed dependence structure. If the correlation of two variables is weak (Pearson's correlation coefficient $|\rho| <1/3 $), the Farlie-Gumbel-Morgenstern (FGM) copula provides an intuitive and natural way for constructing such a bivariate DF. When the linear correlation is stronger, the FGM copula cannot work anymore. In this case, we propose to use a Gaussian copula, which connects two given marginals and directly related to the linear correlation coefficient between two variables. Using the copulas, we constructed the BLFs and discuss its statistical properties. Especially, we focused on the FUV--FIR BLF, since these two luminosities are related to the star formation (SF) activity. Though both the FUV and FIR are related to the SF activity, the univariate LFs have a very different functional form: former is well described by the Schechter function whilst the latter has a much more extended power-law like luminous end. We constructed the FUV-FIR BLFs by the FGM and Gaussian copulas with different strength of correlation, and examined their statistical properties. Then, we discuss some further possible applications of the BLF: the problem of a multiband flux-limited sample selection, the construction of the SF rate (SFR) function, and the construction of the stellar mass of galaxies ($M_*$)--specific SFR ($SFR/M_*$) relation. The copulas turned out to be a very useful tool to investigate all these issues, especially for including the complicated selection effects.

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

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.

People who work in the field of actuarial science pay more and more attention to the risk model with dividend strategy, it has become one of the hot topics in the current actuarial science research. In this paper, we want to study the Gerber-Shiu expected discounted penalty function due to oscillation which written as , bd u , considering a classical compound poisson risk model perturbed by diffusion in the presence of a constant dividend. The integro expression of the Gerber-Shiu function is derivated by the strong markov property and also , bd u is continuous and twice continuously differentiable. then we obtain the integro-differential equation of the Gerber-Shiu function by to formula, It is unique to research the ultimate ruin probability due to oscillation compared with other articles. Finally, we give the explicit expression of solution of integro-differential equation satisfied by , bd u when the claim sizes are exponential distribution.

On a class of renewal risk models with a constant dividend barrier

Exit problems for jump processes with applications to dividend problems

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

研究阈值分红策略下带扩散的二维对偶模型，得到分红现值的期望所满足的积分微分方程组，并用此方程组解得收入服从指数分布时的分红折现的期望具体表达式，应用拉普拉斯变换求得收益服从任意分布时的解，最后解得了此模型下的最优边界。 We study the two-dimensional dual model with diffusion under a threshold dividend strategy. We obtain a group of integro-differential equations satisfied by the expected total sum of discount dividends until ruin. And explicit results when the gains of the two projects are exponentially distributed are derived. By applying the method of the Laplace transform, we solve the case where gains follow general distributions. We also illustrate how to calculate the optimal dividend threshold.

ABSTRACTIn this paper, we discuss an asymmetric extension of Farlie–Gumbel–Morgenstern copulas studied by several authors and obtain the range of the parameter. We derive an expression for regression function and the properties of these copulas are studied in detail. Also, explicit expressions for various measures of association are obtained. These measures are numerically compared for some particular parametric values of the copulas.