Research On Risk Models Modulated By Markov Chains

Several risk models modulated by Markov chain are built and examined in this dissertation. Firstly, existence of these models, probabilistic construction and path-depict are proved in the strictly mathematic sense. Secondly, ruin problems of model, and ruin quantities, especially Gerber-Shiu discounted penalty function are probed.The risk model modulated by Markov chain is a generalization of classical risk model. Classical risk model involves only two stochastic processes, which are Poisson counting process of claims and independent identically distributed claim amount process. Meanwhile, these two processes are independent. The risk model modulated by Markov chain, proposed in this dissertation, adds a Markov chain which is a new control process other than two stochastic processes mentioned above. Furthermore, other processes other than these three processes are also included. These processes are not independent but dependent. From the respective of math, the existence of risk models is desired to solve at first. The ruin problems of model, optimal control problems and so on can be solved only after the existence of the model is proved. The existence of classical risk model is undoubted which can be proved in strictly mathematic sense making use of independent product space technique. However, if a risk model involves two or more than two dependent stochastic processes, for example, risk model modulated by Markov chain, simply using independent product space technique cannot prove the existence of risk models. Therefore, many experts tend to neglect this issue and they take it for granted that the existence of model is default which is not scientific and not desirable as well. By contrast, this dissertation proves the existence of newly built risk model modulated by Markov chain at first, and then investigates the probabilistic construction and nature.The innovation points and structure of this dissertation are as follows. I. Innovation PointsThe main findings are originated by the author except Chapter1Introduction, Section2.1and2.4of Chapter2. There are seven innovation points in total.1. Assembly method of constructing dependent random variables by using independent product space is put forward.The assembly method mentioned here is not a big invention but is clarified as a method systematically at the first time in this dissertation. Furthermore, this method is fully applied in this dissertation and is hoped to be very useful in constructing other dependent random variables.2. Several new findings of Markov chain are obtained.Theory about Markov chain has been studied deeply and widely. This dissertation mainly attempts to explore the risk model modulated by Markov chain and meanwhile gets many new findings about Markov chain.(1) A sort of replacement of random time for a Markov chain is investigated. Under a certain hypothesis, chain after replacement of time still has Markov property.There have been many studies about time replacement of Markov process, using some advanced concept such as additive function of the Markov Process. The random time replacement studied in this dissertation is under the elementary framework of Markov chain.(2) The proposition about multidimensional process derived by q process and Markov risk model is time-homogeneous Markov chain is proved. Moreover, Markov property and time-homogeneous property of stochastic process with reward are obtained.3. For Markov risk model, the integral equation, recursive formulas and analytic expression of Gerber-Shiu discounted penalty function are derived.4. Double-Markov risk model is introduced for the first time and survival probability and condition survival probability of it are solved.5. The existence of Markov-modulated risk model is solved. Furthermore, Path-depict and probabilistic construction are defined. The Markov-modulated risk process with tax is given by Path-depict.6. Criterion, necessary conditions and probabilistic construction of the Markov dependent risk model are proposed, which can settle the problems of model existence and model checking.7. The newly introduced semi-Markov dependent risk model reflects the meaning of Semi. A criterion, necessary conditions of model and probabilistic construction are put forward to settle the problems of model existence and model checking. II Structure and ContentsThis dissertation has seven chapters and is divided by two parts.Part I (Chapter1and Chapter2). At first, the basic theory of Markov chain is introduced. Then, several new findings raised by the author are revealed:new results of q process, a sort of replacement of random time for a Markov chain and Markov property and time-homogeneous property of stochastic process with reward. At last, assembly method of constructing dependent random variables by using independent product space is put forward.Part II (Chapter3-7). Five Markov-chain modulated risk models are explored. Each model is studied in each chapter respectively, with different emphasis.1. Markov risk model (Chapter3):The integral equation, recursive formulas and analytic expression of Gerber-Shiu discounted penalty function are derived.2. Double-Markov risk model (Chapter4):survival probability and ruin probability of it are explored. This model is extending the claim paid process to a Markov chain. Because claim paid moment is a jump point of Markov chain and claim paid amount is a Markov chain as well, the model is called double Markov risk model. The integral equation, recursive formulas and analytic expression of survival probability, condition survival probability are derived.3. Markov-modulated risk model (Chapter5):The existence and Path-depict of it are explored. Though this model is introduced by predecessors, the Markov environment cannot stated more clearly in many papers. The strictly mathematic definitions of Markov-modulated risk model U=(A; J,S,X)are given, A=(C,Q,G,F) is characteristic group of model U among these. The premium, accurate mathematic description for dependent Markov environment for claim paid and claim amount, model existence and probabilistic construction are stated. The Path-depict is given for the Markov-modulated risk process with tax.4. Markov dependent risk model (Chapter6):Criterion, necessary conditions and probabilistic construction of it are proposed, which can settle the problems of model existence and model checking. Markov dependent risk model is proposed by predecessors which involves three dependent Stochastic Processes. Meanwhile, the dependent relationship of these three stochastic processes is depicted by a package relation. This chapter answers several questions such as the existence of three processes, the relations of every two processes, how to judge whether the three processes can make up a Markov dependent risk model, is there any criterion to judge and so on.5. Semi-Markov dependent risk model (Chapter7):Criterion, necessary conditions and probabilistic construction of it are proposed, which can settle the problems of model existence and model checking. Albrecher and Boxma (2005) have introduced Markov dependent risk model and mentioned Semi-Markov, but the meaning of Semi isn’t reflected. The newly introduced semi-Markov dependent risk model, as the special case of Markov dependent risk model studied in Chapter6of this dissertation, reflects the meaning of Semi.