Risk Theory For Some Risk Models And Related Problems

Levy processes are stochastic processes with independent and stationary increments, and play a fundamental role in Mathematical Finance. On the other hand, many important risk processes are also special cases of Levy processes, such as Brownian motion, the compound Poisson process, the compound Poisson process perturbed by Brownian motion. In Chapters 2-4 of this thesis, we study ruin problems for a Levy processes with positive and negative jumps and for some Levy processes with investment. In Chapters 5-6, we introduce two Ornstein-Uhlenbeck type risk processes driven by some Levy process. The optimal dividend problem and the moments of the cumulative dividend are studied, respectively.The asymptotic problems of ruin problems are important topics in risk theory. In.Chapters7-8, when the conditions for the exponential estimate are not satisfied, a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk are derived and non-exponential asymptotic forms for solutions of defectiverenewal equations are obtained. All the results are applied to risk models. In last chapter, we consider a jump-diffusion model with a dependent setting, where the claim inter-occurrence times depend on the previous claim size.Organization and outline of this thesisChapter 1: We introduce the definition of and some important theorems related to Levy processes, and the definitions of some light-tailed and heavy-tailed distributions are introduced also. Some notations which will be used throughout the thesis are set down in this chapter.Chapter 2: Garrido and Morales [47] introduced a Levy risk process only with negative jumps, in which they studied the Gerber-Shiu function. The aim of this chapter is to extend their work to a general Levy risk process with positive and negative jumps, that is, the Levy risk process has the formU(t)=u + ct-S(t),t≥0,{S(t)} is a jump Levy process with positive and negative jumps. We prove that the GerberShiu functionΦsatisfies a renewal equation and a infinite series expression ofΦis obtained. Some asymptotic behaviors of the ruin probability as u→∞are discussed.Chapter 3: In this chapter, we assume the surplus of the insurer at time t under some investment assumption is denoted by Ut:U_t=e^Mt(u+∫₀^te^-MsdR_s),t≥0,U₀=u.where R_t=ct-J_t+σB_t is a Levy risk process. The aggregate claims process {Jt,t≥0}is a jump Levy process without drift and diffusion coefficient, started at 0. The process{e^Mt=e^δt+rWt,t≥0}is a geometric Brownian motion,δ> 0 and r are constants, and{W_t,t≥0} is an one-dimensional Brownian motion independent of {Rt, t≥0}. In thischapter, it is shown that the ruin probabilities (by a jump or by oscillation) of the resulting surplus process satisfy certain integro-differential equations.Chapter 4: Let U(t) be the surplus of insurer at time t with the liquid reserve level,. the credit interest force r≥0 and debit interest forceδ> 0. Then the surplus process {U(t), t≥0} satisfies the following stochastic differential equation:where u≥0 is the initial value and c > 0 is a constant premium rate, defined as c = (1 +θ)EZ(1), whereθ≥0 is the security loading factor. {Z(t), t≥0} is a subordinator with zero drift.We study the absolute ruin questions by defining the joint distributed function of the absolute ruin time, the surplus immediately before absolute ruin and the deficit at absolute ruin. Using an approximation scheme, a general expression for the joint distributed function of the risk process driven by a subordinator is obtained.Chapter 5: In this chapter, a controlled Ornstein-Uhlenbeck type model is studied whose reserve R_t is assumed to be governed by the stochastic differential equationdR_t=(μ+ρR_t-l(t))dt+σdW_t,where {W_t,t≥0} is a standard Brownian motion,μ,ρ,σ>0 are constants and l(t) is the rate of dividend payment at time t. It is shown how the optimal return function and the optimal dividend-payment strategy can be calculated when the rate function l(t) is restricted so that the function l(t) varies in [0, M] for some M <∞.Chapter 6: In this chapter, we present a spectrally negative a-stable Ornstein-Uhlenbeck (1 <α≤2) type risk process X := {X_t,t≥0} with dividend barrier, which is the solution to the linear stochastic differential equationdX_t=-λX_tdt+dZ_t-dD_t,with X₀ = x, and the parameterλ> 0, Z := {Z_t,t≥0} be a spectrally negativeα-stableprocess, withα∈(1,2]. D := {D_t, t≥0} be the cumulative dividends process. The momentsof the present value of dividend payments until ruin are provided in terms of the Wright's generalized hypergeometric function 2ψ1.Chapter 7: For a random walk on the real line with negative mean, we obtain a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk when the conditions for the exponential estimate are not satisfied. The results are applied to the Sparre Andersen model, some new results on the probability of ruin are presented.Chapter 8: In this chapter, we derive non-exponential asymptotic forms for solutions of defective renewal equations. Applications of this result is given to the Gerber-Shiu discounted penalty function in the classical risk model.Chapter 9: In this chapter, we consider a jump-diffusion model with a dependent setting, where the claim inter-occurrence times depend on the previous claim size. We study the joint distribution of the time of ruin, the surplus prior to ruin and the deficit at ruin, and an exact analytical expression for the Laplace transform of the the joint distribution is derived. We also study the dividend problem for the above model with a dividend strategy. A system of integro-differential equations with certain boundary conditions satisfied by the expected discounted dividend payments prior to ruin is derived and an infinite series expression of it is obtained.