Optimal Dividend Problem In The Dual Model

The optimal dividend problem is a stochastic control problem in financial insur-ance and it is also one of the popular research topics in actuarial science,and it has high academic values.De Finetti^[1]solved the optimal dividend problem in a discrete random walk model,since then many research groups began researching into the op-timal dividend problem under more general and more realistic model assumptions and until nowadays this turns out to be a rich and challenging field of research that needs the combination of tools from analysis,probability and stochastic control.Some schol-ars,such as Gerber and Shiu^[2],have studied various improved classical risk models.In order to be more realistic,Jun Cai^[3]proposed a piecewise deterministic compound Poisson risk model in 2009.Unlike the classical risk model,the premium rate of this model depends on the current surplus.The dual risk model might be appropriate for some companies that specializes in inventions and discoveries,which pays costs continuously and has occasional prof-it.Generally,when solving the problem of optimal dividend for such risk model,the derivation of the dynamic programming equation（DPE）is”heuristic”,which is based on the assumption that the value function is smooth enough,and the verification the-orem is established under the additional condition of regularity of the value function.Based on the study of existing literature and further analysis of more practical models,this paper study several optimal dividend problems in the dual risk model of the risk model with surplus-dependent premium by using stochastic control theory,martingale method and measure-valued HJB method.Markov-modulated risk model is more realistic than classical risk model,and it has attracted more and more attention in recent years.Its premiums,claim size and claim counting process are conditionally independent under a given Markov chain.This assumption is too strong in some applications.For this reason,Reinhard and Snooss^[4,5]and other scholars have studied the discrete-time semi-Markov risk model.In order to study the continuous-time semi-Markov risk model,we consider the semi-additive functional of semi-Markov process and its measure-valued Poisson equation in the fifth chapter.This paper carries out the following aspects’.In Chapter 2,Chapter 3 and Chapter 4,the general dividend problem,the opti-mal dividend problem with transaction costs and the restricted dividend problem of the dual risk model of the risk model with surplus-dependent premium are considered respectively.In Chapter 2,a mixed control problem（not limited to continuous dividend-sharing）is studied,in Chapter 3,an impulse control problem is studied,and in Chapter4 studies a classic control problem.In this paper,three control problems are dealt with in a unified framework.we show that a dividend strategy is a Markov strategy if and only if it is a（predictable）additive functional of the controlled surplus process.We derive strictly the dynamic programming equation（DPE）under the Markov strategies by virtue of the theory of measure-valued generators,and we obtain heuristically the measure-valued DPE for three different optimal dividend problems under all admis-sible strategies.The corresponding verification theorem is proved without additional assumptions on the regularity of the value function.Furthermore,we show that the optimal strategy is indeed a stationary Markov strategy.By analyzing the Lebesgue de-composition of the measure-valued DPE,we obtain the corresponding quasi-variational inequalities of the optimal dividend problem and give the partition of the surplus state space corresponding to the optimal strategy and present that the optimal strategy is a Markov strategy with a band structure.In Chapter 5,the semi-additive functional and its expectations for semi-Markov processes are studied.We generalized the concept of additive functionals of an SMPs to the semi-additive functionals of SMPs.Motivated by Jacod and Skorokhod^[6],we characterize semi-additive functionals of an SMP in terms of a c`adl`ag function and a measurable function,and present the necessary and sufficient conditions of being a semimartingale,a local martingale or a special semimartingale for a semi-additive functional of SMP.Furthermore,we also study the expectation of semi-additive func-tionals of SMPs,and prove that it satisfies a measure-valued Poisson equation（because the inter-jump time in SMPs follows a general,not necessarily absolutely continuous distribution）and give the uniqueness condition of the solution to the measure-valued Poisson equation.In some special cases,the measure-valued Poisson equation degen-erates to the usual Poisson equation.