Two Types Of Risk Model

This dissertation is devoted to the study of two classes of risk models. The foundations of theories for Levy processes were laid down by the study of infinitely divisible distributions in the late 20's. With the development of theories for Markov processes and random walks, the properties of Levy processes were intensively studied. Specially, the Levy processes with only positive or negative jumps have been discussed widely. In Chapter 1, we study the general Levy risk model in terms of theories for Levy processes. It is shown that the expected discounted penalty function satisfies an integro-differential equation. Under some assumptions, we discuss the ruin probability and the joint distributions of several actuarial diagnostics in Levy risk model from another points of view. Theorem 1.2.1 The expected discounted penalty function Φ satisfies the following integro-differential equation, for u > 0,Theoreml.3.1 The expected discounted penalty function Φ satisfies the following integral equation, for u > 0,where ρ is the positive root of Generalized Lundberg's equation;Theorem 1.3.2 Suppose that the Levy measure v is finite and u{dx) = Xaeaxdxfor x < 0, fixed a > 0, fixed A > 0, and satisfies/0+o° uj(u)du < +oo for u > 0, where w(u) = /y1"00 u;(u, a;— u)v(dx),then we have+ OO(u) = ^n=0where2 2c f+°°g{u) = -^e⁷tu;hi(u) = /, 9l(u)=Theorem 1.4.1 If the adjustment coefficient exists, the ruin probability tp satisfies the Lundberg's inequalityip{u) < e^Ru, u > 0.Theoreml.4.2 If the adjustment coefficient exists, the ruin probability ip satisfiese-RuTheoreml.5.1 The ruin probability tp has the following representation ip(u) ="￡ T Q{0,dxi) J Q{dxudx2)---ru r+os/ Q{dxn-2,dxn-i) / Q(dxn-i,dxn).J—oo JuTheorem 1.5.2 The joint distribution of the ruin time T and the the surplusat ruin Ut has the following representation[ Q{0,dXl) [ Q(dxudx2)n=\J-°° J-°°/ Q(dxn-2,dxn-i) /J — oo Jun), u>0,y>0.Theorem 1.5.3 The joint distribution of the ruin time T and the the surplus prior to ruin Ut- has the following representationr+ooF2(u,x)=^Y^ / / v{dz)Q{0,dxQ)+lf r Q{o,dX1) rA n₂ ■/—oo ./-O/■u /-+OO r/ Q(eten₂,<￡En_i.) / /Ju-x Ju JoOO +OO rxn—u+x/where u > 0 and x > 0.Theorem 1.5.4 The joint distribution of the ruin time T, the the surplus priorto ruin Ut- and the surplus at ruin Ut has the following representationF(u, x, y) = [ ? ' / / u(dz)Q(0, dxo)A y Jqn=2/■u ru+y rxn-u+x/ Q(dxn₂,dxn-i) / / v{dz)Q(dxn-i,dxn),Ju-x Ju JOwhere u > 0, x > 0, y > 0.In Chapter 2, we discuss a general renewal risk model, and study the ruin probability of finite time and some distributions of several important actuarial diagnostics by ladder epochs and ladder heights. Theorem 2.2.1 The finite time survival probability has the following represen-tation ? +0° rlip{u,t) = J2iP*n(u^) / P*n{u,t-s)dP2{s)}, u,t>0.Theorem 2.2.2 The multivariate ruin function ip(u,t,x,y) = P{t(u) < t,X+(u) < x,Y+(u) > y) satisfies the integral equationtp(u, t, x, y) =I(u < x)ip(0, t,x — u,y + u)+ru rt/ / ip(u-v,t- s,x,y)dP{v,t), u,t,x,y>Q: Jo Joand its solution isip(u,t,x,y) = / %l>(O,t-s,x-u + v,J{x-u)+ JOJ{x-u)where P(v,s) = Y;i=oP*n(v,s).Theorem 2.2.3 The multivariate ruin function tp(u,t,x,y,z) = P(t(u) y,Z+{u) > z), for all u,t,x,y,z > 0, we havei[>(u, t, x,y, z) =I(u < x)ip(0, t,x — u,y + u,z + n)+ru rtI I tp(u-v,t s,x,y,z)dP(v,t), Jo Joand its solution is ip(u, t, x,y,z) = / / ip{O,t — s,x — u + v,y + u — v,z + u — v)dP(v, s).J(x-u)+ JO...