Spectrally negative Lévy risk model under Erlangized barrier strategy

We find the optimal dividend strategy in two related risk models under an affine penalty for ruin. The first risk model is the classical Cramer–Lundberg risk model, and the second is the so-called dual risk model. Under both models, for exponentially distributed jumps, we show that the optimal dividend strategy is a barrier strategy and obtain the barrier explicitly. Moreover, we prove that the optimal barrier increases with respect to the parameters of the affine penalty, while the penalized value function decreases with respect to the penalty. We also compute the expected time until ruin and show that the expected time of ruin increases with respect to the parameters of the affine penalty. Finally, we present some numerical examples to demonstrate the relationship between the results for the classical and dual risk models. Our main contributions are in comparing the classical and dual risk models side-by-side and in obtaining explicit expressions for the penalized value functions, the optimal barriers, and the expected times of ruin. Also, we contrast the free-boundary condition associated with the barrier strategies in the two models.

We consider the dual of the generalized Erlang(n) risk model with a barrier dividend strategy. We derive integro‐differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin and the moment‐generating function of the discounted dividend payments until ruin, respectively. The results are illustrated by several examples.

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

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 paper, we investigate the optimal dividend problem under Parisian ruin with affine penalty payments at Parisian ruin time. The underlying risk process is assumed to be a spectrally negative Levy risk process. With the help of the dynamic programming principle, we prove that the value function associated to our optimal control problem is the smallest solution with certain characteristics to the corresponding Hamilton–Jacobi–Bellman (HJB) equation. In addition, the form of the performance function under barrier dividend strategy is expressed in terms of various extended scale functions. Then we identify a condition under which the performance function under certain barrier strategy is also a solution to the HJB equation, which in turn illustrates the optimalilty of such barrier dividend strategy among all admissible strategies. Various numerical examples are also given when the underlying risk process is compound Poisson process, Brownian motion with drift and jump-diffusion process.

Fluctuation theory for level-dependent Lévy risk processes

Since De Finetti first introduced the concept of dividend in 1957, an increasing number of scholars have started to study different dividend strategies. With the advent of barrier dividend strategies and band dividend strategies, ratcheting dividend strategies have also become a new research direction in recent years. This type of dividend strategy stipulates that once the level of dividend increases, it will not decline. The problem of determining the optimal dividend strategy can be formulated as a Hamilton-Jacobi-Bellman equation (hereinafter referred to as the HJB equation). This article explores the problem of one-fold optimal ratcheting dividend strategies by solving the HJB equation and leveraging the fluctuation theory of the Lévy processes. The research is conducted separately under the Brownian risk model, the classical risk model, and the classical risk model with drift. Additionally, a proof of the optimality of the ratcheting dividend strategy is provided.

This article explores the application of Lévy risk models in actuarial science, focusing on optimal dividend strategies and ruin problems. We investigate the expected present value of total operating costs until ruin for insurance companies, a concept crucial for analyzing risk management strategies within these firms. Utilizing the COS method, we examine this function under various dividend strategies, including constant barrier dividends, linear and non linear strategies, and strategies related to the maximum process. Our analysis is conducted within the framework of discrete observation points for the surplus process of insurance companies, providing a comprehensive understanding of ruin-related issues under different dividend schemes. The numerical experiments presented in this article demonstrate the effectiveness of our approach.

This paper studies the discounted Gerber-Shiu type function for a perturbed actuarial risk model with interest and periodic dividend strategy. Under the assumption of exponentially distributed inter-observation times, some integro-differential equations of the discounted Gerber-Shiu type function at ruin are presented. For exponential claim amounts, explicit expressions satisfied by the Laplace transforms of ruin time are provided and some numerical examples are investigated to demonstrate our results.

A new approach to solve the continuous-time stochastic inventory problem using the fluctuation theory of Lévy processes is developed. This approach involves the recent developments of the scale function that is capable of expressing many fluctuation identities of spectrally one-sided Lévy processes. For the case with a fixed cost and a general spectrally positive Lévy demand process, we show the optimality of an (s, S)-policy. The optimal policy and the value function are concisely expressed via the scale function. Numerical examples under a Lévy process in the β-family with jumps of infinite activity are provided to confirm the analytical results. Furthermore, the case with no fixed ordering costs is studied.

In this short paper, we show how fluctuation identities for Lévy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Lévy insurance risk model with a dividend barrier.

This study investigates a combined optimal financing, reinsurance and dividend distribution problem for a big insurance portfolio. A manager can control the surplus by buying proportional reinsurance, paying dividends and raising money dynamically. The transaction costs and liquidation values at bankruptcy are included in the risk model. Under the objective of maximising the insurance company's value, we identify the insurer's joint optimal strategies using stochastic control methods. The results reveal that managers should consider financing if and only if the terminal value and the transaction costs are not too high, less reinsurance is bought when the surplus increases or dividends are always distributed using the barrier strategy.

For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (Insur Math Econ 37:324–334, 2005), by considering a general claim amount distribution.

In this paper, we apply the concept of the Parisian implementation delay to dividend payments and assume that the claim amount paid Zi by the insurance company for the ith time follows an exponential distribution. We give the Laplace transformation of the ruin time for a risk model with a Parisian implementation delay and give display expressions using the scale function.

We consider a discrete-time risk model with delayed claims and a dividend payment strategy. It is assumed that every main claim will induce a by-claim which may be delayed for one time period with a certain probability. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. Dividends are paid to the shareholders according to an admissible strategy. The company controls the amount of dividends in order to maximize the cumulative expected discounted dividends prior to ruin minus the expected discounted penalty value at ruin. We obtain some properties of the optimal dividend-payment strategy, and offer high efficiency algorithms for obtaining the optimal strategy, the optimal value function and the expectation of time of ruin under the optimal strategy. Our method is mainly to transform the value function. Numerical examples are presented to illustrate the transformation method and the impact of the penalty for ruin on the optimal strategy, the corresponding expected present value of dividends and the expectation of time of ruin.