In this paper, the classical compound Poisson model under periodic observation is studied. Different from the random observation assumption widely used in the literature, we suppose that the inter-observation time is a constant. In this model, both the finite-time and infinite-time Gerber-Shiu functions are studied via the Laguerre series expansion method. We show that the expansion coefficients can be recursively determined and also analyze the approximation errors in detail. Numerical results for several claim size density functions are given to demonstrate effectiveness of our method, and the effect of some parameters is also studied.

In this paper, we discuss a generalization of the classical compound Poisson model with claim sizes following a compound distribution. As applications, we consider models involving zero-truncated geometric, zero-truncated negative-binomial and zero-truncated binomial batch-claim arrivals. We also provide some ruin-related quantities under the resulting risk models. Finally, through numerical examples, we visualize the behavior of these quantities.

This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.

In this paper, we consider some dividend problems in the perturbed compound Poisson model under a constant barrier dividend strategy. We approximate the expected present value of dividend payments before ruin and the expected discounted penalty function based on the COS method, and construct some nonparametric estimators by using a random sample on claim number and individual claim sizes. Under a large sample size setting, we perform an error analysis of the estimators. We also provide some simulation results to verify the effectiveness of this estimation method when the sample size is finite.

In this paper, we consider a mixed dividend strategy in a dual risk model. The mixed dividend strategy is the combination of a threshold dividend and a Parisian implementation delays dividend under periodic observation. Given a series of discrete observation points, when the surplus level is larger than the predetermined bonus barrier at observation point, the Parisian implementation delays dividend is immediately carried out, and the threshold dividend is performed continuously during the delayed period. We study the Gerber-Shiu expected discounted penalty function and the expected discounted dividend payments before ruin in such a dual risk model. Numerical illustrations are given to study the influence of relevant parameters on the ruin-related quantities and the selection of the optimal dividend barrier for a given initial surplus level.

In this paper, we extend the optimal dividend and capital injection problem with affine penalty at ruin in (Xu, R. & Woo, J.K. (2020). Insurance: Mathematics and Economics 92: 1–16) to the case with singular dividend payments. The asymptotic relationships between our value function to the one with bounded dividend density are studied, which also help to verify that our value function is a viscosity solution to the associated Hamilton–Jacob–Bellman Quasi-Variational Inequality (HJBQVI). We also show that the value function is the smallest viscosity supersolution within certain functional class. A modified comparison principle is proved to guarantee the uniqueness of the value function as the viscosity solution within the same functional class. Finally, a band-type dividend and capital injection strategy is constructed based on four crucial sets; and the optimality of such band-type strategy is proved by using fixed point argument. Numerical examples of the optimal band-type strategies are provided at the end when the claim size follows exponential and gamma distribution, respectively.

This paper studies the open-loop equilibrium strategies for a class of non-zero-sum reinsurance–investment stochastic differential games between two insurers with a state-dependent mean expectation in the incomplete market. Both insurers are able to purchase proportional reinsurance contracts and invest their wealth in a risk-free asset and a risky asset whose price is modeled by a general stochastic volatility model. The surplus processes of two insurers are driven by two standard Brownian motions. The objective for each insurer is to find the equilibrium investment and reinsurance strategies to balance the expected return and variance of relative terminal wealth. Incorporating the forward backward stochastic differential equations (FBSDEs), we derive the sufficient conditions and obtain the general solutions of equilibrium controls for two insurers. Furthermore, we apply our theoretical results to two special stochastic volatility models (Hull–White model and Heston model). Numerical examples are also provided to illustrate our results.

This paper studies a Pareto-optimal reinsurance problem when the contract is subject to default of the reinsurer. We assume that the reinsurer can invest a share of its wealth in a risky asset and default occurs when the reinsurer's end-of-period wealth is insufficient to cover the indemnity. We show that without the solvency regulation, the optimal indemnity function is of excess-of-loss form, regardless of the investment decision. Under the solvency regulation constraint, by assuming the investment decision remains unchanged, the optimal indemnity function is characterized element-wisely. Partial results are derived when both the indemnity function and investment decision are impacted by the solvency regulation. Numerical examples are provided to illustrate the implications of our results and the sensitivity of solution to the model parameters.

Layer reinsurance treaty is a common form obtained in the problem of optimal reinsurance design. In this paper, we study allocations of policy limits in layer reinsurance treaties with dependent risks. We investigate the effects of orderings and heterogeneity among policy limits on the expected utility functions of the terminal wealth from the viewpoint of risk-averse insurers faced with right tail weakly stochastic arrangement increasing losses. Orderings on optimal allocations are presented for normal layer reinsurance contracts under certain conditions. Parallel studies are also conducted for randomized layer reinsurance contracts. As a special case, the worst allocations of policy limits are also identified when the exact dependence structure among the losses is unknown. Numerical examples are presented to shed light on the theoretical findings.

Network mutual aid platforms is one of the popular risk-sharing models in recent years, and they have almost 200 million members in China. However, current mutual aid platforms does not satisfy the actuarial rules in either the apportionment method or the pricing principle. Hence, a variety of mutual aid models which enable mutual aid members with different risks to exchange their risks in a transparent and actuarial fair way have been proposed in this paper. Besides, the decision-making frameworks for participants choosing between the mutual aid platform and similar insurance products, or choosing no risk sharing are constructed, respectively. Decisions are made based on the principle of maximizing expected utility. Moreover, the optimization problems of maximizing profit and minimizing risk are constructed, respectively. Through the principle of individual fairness and relative fairness, the problem of adverse selection of the platform can also be reduced. Finally, the actual mutual aid plan is compared with similar insurance products to discuss the advantages of the optimized plan.

Two-part framework and the Tweedie generalized linear model (GLM) have traditionally been used to model loss costs for short-term insurance contracts. For most portfolios of insurance claims, there is typically a large proportion of zero claims that leads to imbalances, resulting in lower prediction accuracy of these traditional approaches. In this article, we propose the use of tree-based methods with a hybrid structure that involves a two-step algorithm as an alternative approach. For example, the first step is the construction of a classification tree to build the probability model for claim frequency. The second step is the application of elastic net regression models at each terminal node from the classification tree to build the distribution models for claim severity. This hybrid structure captures the benefits of tuning hyperparameters at each step of the algorithm; this allows for improved prediction accuracy, and tuning can be performed to meet specific business objectives. An obvious major advantage of this hybrid structure is improved model interpretability. We examine and compare the predictive performance of this hybrid structure relative to the traditional Tweedie GLM using both simulated and real datasets. Our empirical results show that these hybrid tree-based methods produce more accurate and informative predictions.

In the context of mortality forecasting, “rotation” refers to the phenomenon that mortality decline accelerates at older ages but decelerates at younger ages. Since rotation is typically subtle, it is difficult to be confirmed and modeled in a statistical, data-driven manner. In this paper, we attempt to overcome this challenge by proposing an alternative modeling approach. The approach encompasses a new model structure, which includes a component that is devoted to measuring rotation. It also features a modeling technique known as ANCOVA, which allows us to statistically detect rotation and extrapolate the phenomenon into the future. Our proposed approach yields plausible mortality forecasts that are similar to those produced by Li et al. [Extending the Lee-Carter method to model the rotation of age patterns of mortality decline for long-term projections. Demography 50 (6), 2037–205, and may be considered more advantageous than the approach of Li et al. in the sense that it is able to generate not only static but also stochastic forecasts.