| In recent years,more and more actuarial theorists and practitioners pay attention to the consideration of dependent structures in the risk processes.In this paper,two different forms of dependent risks are considered in fully discrete reinsurance models.One is to assume that the risk-free interest rate is dependent on the inter-claim time.The second is to consider investment,in which risk-free interest rates and investment returns are dependent.We study actuarial quantities such as ruin probabilities,deficit at ruin and the surplus before ruin.Using recursive technique and martingale method respectively,some upper and lower bounds are obtained for these quantities.This thesis is divided into three chapters.Chapter 1 is prolegomenon.Firstly,the research situation of discrete-time renewal risk model is introduced.Then we introduce the development of fully discrete-time risk model.Finally we present the main results of this thesis.In chapter 2 we firstly give the reinsurance model with dependent risks.Then the recursive equation satisfied by finite time ruin probability and the integral equation of ultimate ruin probability are obtained by recursive method.As special cases,related results for proportional reinsurance and stop-loss reinsurance are derived.Take the proportional reinsurance as example,we give the upper and lower bounds for ruin probability.We also give the functional upper bounds for tail of deficit distribution.In chapter 3 we study the dependent reinsurance model with risk investment.The model is presented firstly.Then the recursive equation satisfied by ruin probability is obtained by recursive method,and the upper bound estimation for ruin probability is given.By constructing an upper martingale,a different upper bound estimation is obtained.The recursive equation satisfied by the deficit distribution is given.When the claim amounts follow the NBU distribution,the upper bound of the deficit is obtained.An integral equation for the surplus before ruin is also studied. |
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