Study On Ruin Theory And Dividend Problems For Some Risk Models

We will study some problems in insurance through absolute ruin,Gerber-Shiu expected discounted penalty function(Simply called Gerber-Shiu function) and optimal dividend payments.The risk models that we study can roughly be divided into two kinds:one is the risk models with interests,the other is L(?)vy risk model.1.The risk models with interests:Usually,the study on absolute ruin problems will go on through the study on Gerber-Shiu function.With the knowledges of Stochastic Process and Stochastic Differential Equation we obtain the integro-differential equation and boundary values satisfied by the Gerber-Shiu function.Then the explicit expression of it under exponential claims and the differential equation are obtained.Thus we discuss the effects of credit and debit interests on it through data analysis. Through the research of the mean,the moment generating function and the higher moments of discounted aggregate dividend payments,we discuss the optimal dividend payments problems.We deduce the the integro-differential equations and boundary values satisfied by them through probability way.And we also characterize the optimal value function as the viscosity solution of the associated Hamilton-Jacobi-Bellman(HJB) equation.Furthermore,we study the effects of credit and debit interests on the mean of discounted aggregate dividend payments and optimal dividend barrier through data analysis.2.L(?)vy risk model:If the L(?)vy measureⅡof L(?)vy risk surplus process U has a densityπwhich is log convex,then for q＞0 the scale function W^(q) is convex in the interval (a^*,∞),where a^* is the largest value at which W^(q)′attains its global minimum. As a consequence,the barrier strategy at a^* is an optimal strategy.Organization and outline of this thesis:Chapter 1.We introduce some basic risk models,optimal dividend problems and confluent hypergeometric equation.Chapter 2.We attempt to study the dividend payments in the classical risk model under absolute ruin.Dividends are paid to the shareholders according to a barrier strategy b.At first,we obtain the integro-differential equations satisfied by the moment generating function and moments of the discounted dividend payments and we also prove the continuous property of them at zero.Then,applying these results,we get the explicit expressions of the moment generating function and moments of the discounted dividend payments for exponential claims.Furthermore, we discuss the optimal dividend barrier when the claim sizes have a common exponential distribution.Finally,we give the numerical examples of optimal dividend problem for exponential claims and Erlang(2) claims.In the last section,we considered the Gerber-Shiu function.Part of the results has been published on Applied Stochastic Models in Business and Industry,the other part has been submitted to Acta Mathematicae Applicatae Sinica.Chapter 3.We consider the dividend payments in the classical risk model with credit and debit interests under absolute ruin.We first obtain the integro-differential equations satisfied by the moment generating function and moments of the discounted aggregate dividend payments.Secondly,applying these results, we get the explicit expressions of them for exponential claims.Then,we give the numerical analysis of optimal dividend barrier and the expected discounted aggregate dividend payments which are influenced by the debit and credit interests. Finally,we find the integro-differential equations satisfied by the Laplace transform of absolute ruin time and give its explicit expressions when the claim sizes are exponentially distributed.(submitted to Statistics & Probability Letters.)Chapter 4.We study absolute ruin questions for the perturbed compound Poisson risk process with investment and debit interest.We first consider the stochastic Dirichlet problem and from which we derive a system of integro-differential equations and the boundary conditions satisfied by the function.Secondly,we derive the integral equations and a defective renewal equation under some special cases,then based on the defective renewal equation we give two asymptotic results for the expected discounted penalty function when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims,respectively.Finally, we investigate some explicit solutions and numerical results when claim sizes are exponentially distributed.(Published on Methodology and Computing in Applied Probability).Chapter 5.In this chapter,we consider the perturbed compound Poisson risk model under absolute ruin with debit interest and a constant dividend barrier. Integro-differential equations satisfied by the expectation of discounted dividend payments,the moment generating function and the expected discounted penalty function(Gerber-Shiu function) with certain boundary conditions are obtained. For some special cases,explicit expressions are obtained.(Accepted by Acta Mathematica Scientia).Chapter 6.In this chapter,we consider the optimal dividend problems of the perturbed compound Poisson risk model with investment interest.Our aim is to find an optimal dividend strategy that maximize the cumulative expected discounted dividend payments.We characterize the optimal value function as the viscosity solution of the associated Hamilton-Jacobi-Bellman(HJB) equation and we prove that there exists an optimal barrier strategy which is optimal among all admissible dividend strategy in some special cases.(Submitted to Journal of Computational and Applied Mathematics).Chapter 7.In this chapter,we study the optimal dividend strategy for spectrally negative L(?)vy risk processes.Firstly,we will review some basic results on log-convexity and complete monotonicity of functions that will be needed later on. Then,we discuss the convex solutions for two kinds of integro-differential equations. Finally,we discuss the optimality of barrier strategy under some special cases.(Submitted to Journal of Computational and Applied Mathematics).