Study On Dividend Problems Under Some Risk Models

People who work in the field of actuarial science pay more and more attention to the risk model with dividend strategy, it has be-come one of the hot topics in the current actuarial science research. The thesis discusses dividend problems in some risk model, we obtain some results about absolute ruin probability, the present value of all dividend payments and the nth moment of the present value of divi-dend payments. According to the content of the thesis, it is divided into the following chapters:In the first chapter, the history about development of risk theory and the current situation about dividend risk model are briefly re-viewed. Furthermore, we make an outline of the main research work of this thesis.In Chapter2, elementary basis of knowledge is presented briefly.In Chapter3, we study a perturbed Poisson risk model with con-stant interest and a threshold dividend strategy under absolute ruin. We derive integro-differential equations for the expectation of the discounted dividend payments and the moment generating function with certain boundary conditions. Furthermore, we obtain integro-differential equations for the Gerber-Shiu discounted penalty function, then transform them into Volterra integro equations. The explicit ex-pression of the absolute ruin probability for exponential claim is also obtained.Chapter4studies a compound Poisson surplus model with liq-uid reserves and constant dividend barrier under absolute ruin. We first derive the integro-differential equations satisfied by the moment-generating function and moment of the discounted dividend payments until absolute ruin. Then, applying these results, we get the explicit expressions of them for exponential claims and discuss the impact of the model parameters on the expected dividend payments by numer-ical examples.In Chapter5, we consider a perturbed dual risk model with in-terest and a constant dividend barrier. First, the integral-differential equation satisfied by the ruin probability is given, and the closed form for ruin probability is provided in the case where profits follow an ex-ponential distribution. Second, the integral-differential equation satis-fied by the moments of the discounted dividend payments is given, and the closed form for the expected dividend payments is also obtained in the case where profits follow an exponential distribution.In Chapter6, we consider a correlated discrete risk model under stochastic interest rates, in which every main claim may produce a by-claim and the occurrence of the by-claim may be delayed. The company pays dividends to policyholders when the surplus is higher than a constant dividend barrier b. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space, we have obtained the difference equations for the expected present value of total dividend payments prior to ruin. Explicit expressions for the corresponding results are derived in a special case, numerical examples are also given.In Chapter7, we consider the dividend payments with constant dividend barrier in a delayed discrete-time interaction risk model. The interaction comes from the assumption that each main claim in one class induces a by-claim in the other class with a certain probability, Specifically the occurrences of induced claim may be delayed. A sys-tem of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is de-rived and solved. Explicit expressions for the expected present value of total dividend payments are derived when the claim size distribu-tions have finite support, numerical illustrations of the results are also given.In Chapter8, we give a brief summary of the thesis and intro-duction to future work.