Cary Chi-Liang Tsai

A generalized defective renewal equation for the surplus process perturbed by diffusion

Abstract In this paper, we consider the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process. We generalize the defective renewal equation for the expected discounted function of a penalty at the time of ruin in Garber and Landry [Insurance: Math. Econ. 22 (1998) 263]. Then an asymptotic formula for the expected discounted penalty function is proposed. In addition, the associated claim size distribution is studied, and reliability-based class implications for the distribution are given.

On the discounted distribution functions of the surplus process perturbed by diffusion

Abstract In this paper, we derive explicit expressions for the discounted joint and marginal distribution functions of the surplus immediately prior to the time of ruin and the deficit at the time of ruin, and for the discounted distribution function of the amount of the claim causing ruin, based on the surplus process of ruin theory with an independent diffusion process. Furthermore, we show that these distribution functions satisfy defective renewal equations.

On the expectations of the present values of the time of ruin perturbed by diffusion

Abstract In this paper, we consider the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process. A compound geometric distribution and the expectations of the present values of the time of ruin due to oscillation and a claim, respectively, are studied. Recursive formulas and explicit expressions for the moments of, and the asymptotic formulas and the Tijms-type approximations for, the compound geometric distribution and the expectations of the present values of the time of ruin are derived. In addition, explicit analytical solutions to the compound geometric distribution and to these expectations can be obtained if the claim size distribution is a combination of exponentials or a mixture of Erlangs. Finally, a lower bound and an upper bound on the compound geometric distribution are proposed, provided the associated claim size distribution is in some of the reliability-based classes.

On the moments of the surplus process perturbed by diffusion

Abstract In this paper we extend the results of Lin and Willmot [Insurance: Mathematics and Economics 27 (2000) 19–44] to those based on the surplus process perturbed by diffusion. We first derive the expression for the (discounted) moments of deficit at the time of ruin. An upper bound is also given if the claim size distribution function satisfies a certain condition. Next, we show that the joint moment of the penalty function and the time of ruin due to a claim satisfies a defective renewal equation and has an explicit expression; then the joint moment of the deficit at ruin and the time of ruin is just a special case by an appropriate choice of the penalty function. Finally, the moments of the time of ruin due to oscillation and caused by a claim, respectively, are studied. We also find that these two kinds of moments of the time of ruin have the same recursive expressions.

On the ordering of ruin probabilities for the surplus process perturbed by diffusion

In this paper, we study orders of pairs of ruin probabilities resulting from two individual claim size random variables for corresponding continuous time surplus processes perturbed by diffusion with different premium rates, relative security loadings, and variance parameters of the diffusion processes. We show that high frequency and low severity risks yield smaller ruin probabilities than low frequency and high severity risks. These ordering relationships can also be used to obtain upper and/or lower bounds on ruin probabilities. Finally, some examples are given to illustrate the results of the theorems.In this paper, we study orders of pairs of ruin probabilities resulting from two individual claim size random variables for corresponding continuous time surplus processes perturbed by diffusion with different premium rates, relative security loadings, and variance parameters of the diffusion processes. We show that high frequency and low severity risks yield smaller ruin probabilities than low frequency and high severity risks. These ordering relationships can also be used to obtain upper and/or lower bounds on ruin probabilities. Finally, some examples are given to illustrate the results of the theorems.

An effective method for constructing bounds for ruin probabilities for the surplus process perturbed by diffusion

In this paper, we first study orders, valid up to a certain positive initial surplus, between a pair of ruin probabilities resulting from two individual claim size random variables for corresponding continuous time surplus processes perturbed by diffusion. The results are then applied to obtain a smooth upper (lower) bound for the underlying ruin probability; the upper (lower) bound is constructed from exponentially distributed claims, provided that the mean residual lifetime function of the underlying random variable is non-decreasing (non-increasing). Finally, numerical examples are given to illustrate the constructed upper bounds for ruin probabilities with comparisons to some existing ones.

Ruin time and aggregate claim amount up to ruin time for the perturbed risk process

We consider the classical Sparre-Andersen risk process perturbed by a Wiener process, and study the joint distribution of the ruin time and the aggregate claim amounts until ruin by determining its Laplace transform. This is first done when the claim amounts follow respectively an exponential/Phase-type distribution, in which case we also compute the distribution of recovery time and study the case of a barrier dividend. Then the general distribution is considered when ruin occurs by oscillation, in which case a renewal equation is derived.

Incorporating the Bühlmann credibility into mortality models to improve forecasting performances

In this paper, we incorporate the Bühlmann credibility into three mortality models (the Lee–Carter model, the Cairns–Blake–Dowd model, and a linear relational model) to improve their forecasting performances, as measured by the MAPE (mean absolute percentage error), using mortality data for the UK. The results show that the MAPE reduction ratios for the three mortality models with the Bühlmann credibility are all significant. More importantly, the MAPEs under the three mortality models with the Bühlmann credibility are very close to each other for each age and forecast year. Thus, by incorporating the Bühlmann credibility we are able to converge the forecasting MAPEs resulting from the three different mortality models to a lower and more consistent level. Moreover, we provide a credibility interpretation with an individual time trend for age x and a group time trend for all ages. Finally, we apply the forecasted mortality rates both with and without the Bühlmann credibility to the net single premiums of life insurance products, and compare the corresponding MAPEs.