The Optimal Control Problem In A Piecewise-deterministic Compound Poisson Risk Model

As a practical matter, the insurance company thirsts for an optimum state, that is themaximum expected profit and the minimum risk. Risk theory is the most theoretical part ininsurance mathematics which derives from risk management in insurance. It is mainly aboutseveral actuarial variables such as ruin probability, the deficit at ruin etc, which the insur-ance company cared about. The applications of stochastic process, stochastic analysis andthe martingale method make many fruitful achievements in characterizing several actuarialvariables mentioned above have been made. From the perspective of income, the insurancecompany is concerned about the maximum expected the utility of the wealth and the dividendpayments. All of these belong to stochastic control problems. During the nineteen nineties,stochastic control theory has been applied by into the insurance. Asmussen etc.(1997)[1]andBrowne(1995)[2]respectively obtained optimal investment strategy and the optimal dividendstrategy under diffusion model by the Hamilton-Jacobi-Bellman (HJB) equation approach.Their works are the breakthrough of the research for the problems. Later, on the study of theoptimal problem many scholars have achieved valuable fruits from different angle of view.For the risk model, one of the principle focuses of most scholars is the continuous timerisk model which mainly divided into two classes: One is diffusion model, another is com-pound Poisson model. Both of these risk models, the Brownian motion with drift and classicalcompound Poisson process are for the most part basic. Both the two risk models have station-ary independent increments, so many problems are easier to solve. Many academicians usethem to portray the company’s surplus process. But only in the Brownian motion with driftsetting, we obtain better results. The optimal dividend problem in the classical risk modelis always one of the most difficult topics. The main reason is that the optimal equation hasneither the boundary condition nor continuously differentiable solution.Various modified compound Poisson risk models have appeared in the literature, such asGerber and Shiu(2006b)[3]. Many of them are characterized by their sample paths between anytwo consecutive claims. To this Jun Cai(2009)[4]research the piecewise-deterministic com-pound Poisson risk model. Compared with the classical risk model, the aggregate claimsfollow a compound Poisson process, while the premium rate is depending on the surplus pro- cess, which makes it agree to the fact much more. For this model, there are two motivationsto study. From the insurer’s point of view, a higher surplus level allows the insurer to reducepremium to stay competitive. In the lower surplus level, the insurer might charge a higherpremium to avoid the possibility of insufficient funds. From a mathematical point of view, theclass of risk model includes a variety risk models. Such as the classical risk model, risk mod-els with multi-threshold dividend strategy, credit interest or even interest earnings with liquidreserves, etc. These papers more focus on the discussion of the Gerber-Shiu function, such asLin and Pavlova(2006)[5], Lin and Sendova(2008)[6], Cai and Dickson(2002)[7], Cai,Feng andWillmot(2009)[8]. The optimal control problem is still rarely considered about the piecewise-deterministic compound Poisson risk model.Based on the situation above, several optimal control problems in the piecewise-deterministic compound Poisson risk model are mainly researched in this thesis. In the following, I willintroduce the content of every Chapter in detail.In Chapter1, a simply introduction of stochastic control theory which will be used in thefollowing chapters is given. Most of the contents are borrowed from Schmidli(2008)[9]andDavis(1993)[10].In Chapter2, we consider the restricted dividend problem in the piecewise-deterministiccompound Poisson risk model.Dividend is payments that the company gives to the shareholder or the person who pro-vides the initial surplus. In a sense, the total dividend payments represent company’s benefit.Thus, how to pay dividends to maximize the total dividend payments is always one of thehotspot issues in finance and insurance. In the problem of restricted dividend, it is assumedthat the dividend rate be restricted, the purpose of the optimization is finding the optimaldividend strategy to maximize the expected discounted dividend payments before ruin. Therestricted dividend problem is postulated by Jeanblanc-Picque′and Shiryaev (1995)[11]andAsmussen and Taksar (1997)[1]. They considered only dividend strategies with an upperbound for the dividend rate in the Brownian motion model. It was shown that the optimal div-idend strategy is a threshold strategy. Gerber and Shiu (2006a)[12]provided some calculationsfor this model. In the classical risk model, Gerber and Shiu (2006b)[3]examined the analo-gous questions, they showed that the optimal dividend strategy is also a threshold strategy for an exponential claim amount distribution. For the classical risk model with restricted dividendpayments, Schmidli (2008)[9]discussed the optimal dividend strategy for general distributedclaims. It was shown that the optimal policy is formed by a generalized multi-threshold strat-egy.In aforementioned references, the admissible dividend rate under a threshold strategyis bounded by some constant. While, the premium rate is depending on the surplus processin this paper, so the admissible dividend rate is controlled by a locally Lipschitz continuousfunction, and which also depends on the surplus process. We show that the optimal policyis formed by a generalized multi-threshold strategy. By comparing the dividend rate controlfunction and the premium rate function, we discuss in different situation to get the optimalpolicy. In order to be sure that a certain solution to the HJB equation really is the value func-tion, we characterise the value function among all solutions. Moreover, the value function isnot necessarily differentiable. And we show that the density of the value function can onlyhave upward jumps. Finally, we consider an optimal dividend problem for linear premiumrate and exponential claims.In Chapter3, we consider the unrestricted dividend payments in the piecewise-determini-stic compound Poisson risk model.And in relative terms, the dividend rate will not be controlled by the ceiling. Then the op-timal strategy is of singular control type. To determine the value function and optimal policy,we characterise the value function, containing the locally bounded property, the liner growthproperty, the locally Lipschitz continuous property and the convergence. As with the vari-ational approach, we will obtain a Hamilton-Jacobi-Bellman equation, and get the optimalpolicy by discussions. In the optimal strategy, the value function is the minimal solution to theHJB equation and the solution of HJB equation is the value function under certain conditions.In Chapter4, we consider an impulse control problem in a piecewise-deterministic com-pound Poisson risk model. In this chapter, the objective is to maximize the total discountedutility of dividend payments, the utility function is expressed by ()=1(),∈(0,1]. To take the dividend analysis even more realistic, the insurance company is allowed topay out dividends to its shareholders with taking into account fixed transaction costs and a taxrate. These costs prohibit the continuous trade, which leads to consider the dividend quantity while also consider dividend time. It will radically affect the expression of optimal polityand boils down to the impulse control problem. While compared with the problem withouttransaction costs and taxes, it will become difficult. Due to the additional difficulty, this kindof optimal dividend problem is still rarely considered in the insurance literature, especially inthe compound Poisson model.The corresponding optimization problem was considered in Paulsen(2007)[13]for a gen-eral diffusion process, in Jeanblanc-Picque′and Shiryaev(1995)[11]for the constant drift andvolatility case and in Cadenillas etc.(2007)[14]for a mean-reverting diffusion. For the classicalrisk model, Thonhauser and Albrecher(2011)[15]characterize the value function according tothis paper. In the case of=1, Bai and Guo(2010)[16]obtain the analytical solutions ofthe optimal value function and the optimal dividend strategy when claims are exponentiallydistributed.In the chapter, the payments are subject to both fixed and proportional costs. We obtainthat the optimal value function can be characterized as a smallest solution of a set of quasi-variational inequalities (QVI), or equivalently as the smallest fixed point of the first jumpoperator in the domain of the absolutely continuous along trajectories functions. QVI controlis the optimal strategy. And it is shown that the value function and intervention operator arelocally Lipschtz continuity along trajectories. First we discuss the optimal stopping problem.Second, we obtain the main results of impulse control problem are given. Finally we providesome auxiliary results and prove the main results of this paper.In Chapter5, we consider the numerical method and some examples about the problemin a particular case which is involved in chapter4.Base on the result in the last chapter, the chapter focused on a particular case. In the caseof=1, by discussing the corresponding quasi-variational inequality, we provide a methodto numerically find the optimal value function and the optimal dividend strategy (if they exist)when the claims sizes are exponentially distributed. There can be essentially three differentsolutions depending on the model parameters and costs.(i) Whenever surpluses reach a barrierˉ, they are reduced toˉ*through a dividend payment*, and the surplus process continuesuntil ruin.(ii) Whenever surpluses reach a barrierˉ, everything is paid out as dividends, andthe surplus process continues until ruin.(iii) If there is no optimal policy, the value function is approximated by increasing barriers. Finally, two examples are given when the premium ratefunction is constant and linear function respectively.