Study On Ruin Problems For Some Risk Models

Actuarial is a applied mathematics generated from the insurance risk management. Risk theory is one of the most active research areas. Under some practical conditions, we can model the surplus process of the insurance company, analysis and measure the risk, and provide theoretical basis for the insurance, by methods of probability and statistics, stochastic process and integro-differential equation, company. Ruin theory is the core of risk theory, there have been many well-known results for classical risk model and its generalized models.In this thesis, we discuss the ruin problems for some generalized risk models.1. A perturbed MAP risk model under a threshold dividend strategy Consider the following risk model: where c2=C1-d, d (0<d≤c1) is the dividend rate,{N(t),t≥0} is the claim number process,{Xi, i≥1} is the claims,{B(t),t≥0} is a standard Brownian motion,{J(t),t≥0} is the potential environment affecting the insurance company, which is a Markov chain, with state space E={1,2,..., n} and density matrix Do+D1. Here, Do=[D0,ij]i,jn=1, D1=[D1,ij]i,jn=1satisfying: where D0,ij and D1,ij represent respectively:(1) the instantaneous rate of transition from state i to state j (j≠i) in E without an accompanying claim,(2) the instantaneous rate of transition from state i to state j (possibly j=i) in E with an accompanying claim. We call{N(t),J(t),t≥0)a Markovian arrival process(MAP).The corresponding risk model is called MAP risk model. For a transition of{J(t),t≥0)from state i to state j at the time of a claim, we assume that the accompanying claim size has density function fij(x),distribution function Eij(x),Laplace transform fij(s)(?)fij(x)e-sxdx and finite mean μij.Given the initial surplus is u and the initial MAP state is i∈E,define Φij(u,b)=E[e-δTBω(Ub(Tb-),|Ub(Tb)|)I(Tb<∞,J(Tb)=j)|Ub(0)=u,J(0)=i], where i,j=1,2,...,n. For ease of notation, write whereΦ1(u,b))=[Φ1,ij(u,b))]i,jn=l,Φ2(u,b))=[Φ2,ij(u,b))]i,jn=1. First,we can get the integro-differential equations of the Gerber-Shiu function: where C1=c1I,C2=c2I,D=diag(σ12/2,σ22/2,...,σn2/2),f(x)=[fij(x)]i,jnj=1, ω(u)=[ωij(u)]i,jn=1,Φk":(u,b)=[Φk,ij"(u,b)]i,jn=1,Φk’(u,b)=[Φk,ij’(u,b)]i,jn=1,k=1,2. ij(u)=(?)ω(u,u-x)fij(x)dx.They satisfy the boundary condition:Φ1(0,b)=I, and the continuous condition: Φ1(b-,b)=Φ2(b+,b),Φ1’[(b-,b)=Φ2’(b+,b).Then, we haveTheorem1The analytical solution of Gerber-Shiu function Φ(u,b) and Φ2(u,b) can be obtained as follows:(i) the Gerber-Shiu function Φ(u) for the risk process without dividend is: where M1(u)=L-1{[Lc1(s)]-1(Ds+C1)}, N1(u)=L-1{[Lc1(s)]-1D},(ii)when0≤u<b, Gerber-Shiu function Φ1(u, b) is where in which (iii) when u>b, Gerber-Shiu function Φ2(u,b) is where φ2(b,b)=Φ1(b, b), Φ2(b,b)=Φ1’(b,b), M2(u)=L-1{[Lc2(s)]-1(Ds+C2)}, N2(u)=L-1{[Lc2(s)]-1D}, Let D(t) be the cumulative amount of dividends paid out up to time t, define to be the total discounted dividends until time of ruin T&,, where d=c1-c2,δ>0. Its expectation is:Similarly,we can get the analytical solution of Vk(u,b)=(Vk,1(u,b),Vk,2(u,b),...,Vk,n(u, b))(?) where k=1,2.Theorem2The expectation of the total dividend payments until ruin V1(u,b) and V2(u,b) can be obtained as follows:(i) For0≤u<b,(ii) For u≥b, where V2(b,b)=V1(b,b), V2’(b,b)=V1’(b,b).We can also get the matrix renewal equation of the Gerber-Shiu functon:Theorem3The Gerber-Shiu function Φ2(u,b) satisfies the matrix renewal equa-tion: By the above renewal equation,we can get the asymptotic behavior of Φ(u,b) when the claim size distributions are heavy-tailed distributions.Assumption1There exist a matrix H=[Hij]i,jn=1,and a distribution function F with density f such thatTheorem4(1) For δ>0,assume that limu→∞.ωij(u)/Fij(u)=kij>0,if Assumption1holds with f∈(?)d,then(ii)Forδ=0,assume that limu→∞Ωij(u)/Fe,ij(u)=Kij>0,if Assumption1holds with F∈(?)d,then where Ωij(u)=(?)ωij(x)dx,Fe,ij(u)=(?)Fij(x)dx,Ω(u)=|Ωij(u)]i,jn2.The renewal risk process with two classes of claims umder a threshold dividend strategyConsider the following risk model: where u≥0is the initial surplus,c2=c1-d,d is the dividend rate,{S(t),t≥0)is the aggregate claim amount process,which is assumed to be generated from two classes of insurance risks,i.e., where Sk(t) and {Nk(t),t≥0} represent the aggregate claims and claim numbers up to time t from the k-class risk, k=1,2.{Xi,i≤1} are the individual claim amounts from the first class and assumed to be positive i.i.d. r.v. with common distribution function P, density p and Laplace transform p(s)=(?) p(x)e-sxdx, while {Yi,i≥1} are the individual claim amounts from the second class and assumed to be positive i.i.d. r.v with common distribution function Q, density q and Laplace transform q(s)=(?) q(y)e-sydy. Let {Vi,i≥1} and {Wi,i≥1} be the i.i.d. inter-claim times of {Ni(t),t≥0}and {N2(t),t≥0}, with distribution function and density function F, f and G, g, respectively, and assume that F is a PH(α,A,α) distribution and G is a PH(β,B,b(?)) distribution.Since the ruin may be caused by the claim from the first or second class, we can define K to be the cause-of-ruin random variable: K=k if the ruin is caused by a claim of class k, k=1,2, then define to be the Gerber-Shiu function at ruin caused by a claim of class k.By the definition of phase-type distribution, each Vi corresponds to the time to absorption in a terminating continuous Markov chain {It(i), t≥0} with common n transient states {E1,E2,..., En} and an absorbing state EQ. Similarly, we denote by {Jt(i), t≥0} the corresponding terminating continuous Markov chain of Wi with common m transient states {F1,F2,...,Fm}and an absorbing state F0. To analyze the model using Markvian techniques, denote {(I(t),J(t)),t≥0} to be the underlying state process which is denned by We know that {(I(t),J(t)), t≥0} is a two-dimensional Markov chain with state {(E1,F1),...,(En,Fi),(E1,F2),...,(En,F2),...,(E1,Fm)...,(En,Fm)}, initial distri--bution γ=β(?)α, and density matrix D=Im×m(?)A+B(?)In×n+Im×m (?)(α(?)α)+(b(?)β)(?)InGiven initial state (I(0),J(0))=(Ei,Fj), We define where k=1,2, i=1,2,...,n,j=1,2,...,m. For ease of notation write where k=1,2,i=1,2,...,n,j=1,2,...,m. First,we obtain the integro-differential equations of the Gerber-Shiu functions.Then,we can solute them as followsTheorem5The analytical expressions of Gerber-Shiu functions(?)(u,b)and (?)(u,b),k=1,2are obtained as follows:(i)the Gerber-Shiu function (?)for risk process without dividend payments is where (ii) when0≤u<b, the Gerber-Shiu function (?)(u,b) and (?)(u,b) are where in which(iii) when u≥b,the Gerber-Shiu function (?)(u,b) and (?)(u,b) are where (?)(b,b)=(?)(b,b) Let D(t) be the cumulative amount of dividends paid out up to time t, define to be the total discounted dividends until time of ruin Tb. Its expectation isSimilarly, for V(k)(u,b)=(Vij(k)(u,b),i=1,...,n,j=1,..., m)(?), we haveTheorem6The analytical solutions of V(1)(u,b) and V(2)(u,b) are obtained as follows:(i) when0≤u<b,(ii) when u≥b, where V(2)(b,b)=V(1()b,b).3. A two-dimensional perturbed risk model with stochastic premiums Consider the following risk model: where U(t)=(U1(t),U2(t))(?) denotes the two-dimensional surplus process, Yi=(Yu,Y2i)(?) i≥1denote the pairs of claims (Y1i and Y2i represent two different kinds of claims, such as vehicle damages and bodily injuries in an accident), Xi=(X1i,X2i)(?), i≥1denote the premiums paid for different kinds of claims, N1(t) and N2(t) are the total number of premiums and the total number of claims up to time t, respectively, u=(u1,u2)(?) denotes the initial surplus vector,B(t)=(B1(f),B2(f))(?) is a two-dimensional Brown-ian motion with constant correlation coefficient r∈[-1,1],which adds an additional uncertain to the aggregate claims or the premiums incomes, while σ1≥0and σ2≥0denote the diffusion coefficient of B{t). We assume {Xi,i≥1} and {Yi,i≥1} are sequences of i.i.d random vectors, having the same distributions with X={Xx,X2} distributed by F(x1,x2) and Y={Yi,Y2} distributed by G(y1,y2), respectively; De--note the marginal distribution functions of X and Y by Fi(xi), F2{x2) and Gi(yi), G2(y2), respectively, write Fi=1-Fi and Gi=1-Gi for i=1,2.We consider the ruin time Tmax=inf{t>0|max{U1(t),U2(t)}<0}, define the (infinite) time ruin probability and finite time ruin probability as: andFor ease of notation, writemi(si,s2)=E[exp{siX1+s2X2}],m2(s1,s2)=E[exp{siY1+s2Y2}];f(s1,s2)=λ1(m1(-s1,-s2)-1)+λ2(m2(s1,s2)-1)+1/2[σ12s12+2rσ1σ2s1s2+σ22s22];s10=suo{s1|m2(s1,0)<∞}, s20=sup{s2|m2(0,s2)<∞};A0={(si,s2)|s1≥0,s2≥0,m2(si,s2)<∞}\(0,0).Then we haveTheorem7If s10,s20>0, then whereΔ0={(S1,s2)∈A0|f(S1,s2)=0}.When the claims are heavy-tailed distributions, we haveTheorem8Suppose that G∈S, G2∈S, where G1and G2are the marginal distribution functions of Y, then for each fixed time T>0, we have4. An autoregressive risk model with dependent rates of interest Consider the following risk model: or equivalently, the surplus process {Un,n>1} satisfies where Here, U0=u≥0is the initial surplus, Xn denotes the total premiums during nth period with initial value X0=x0, and the parameter a can be interpreted as the proportion of last year’s business, which will remain in this year’s portfolio; Yn denotes the total claims during nth period with initial value Yo=yo, and the parameter b cam be interpreted as the proportion of the old business in the new portfolio; In denotes the interest rate during nth period with initial value I0=i0, and {In,n≥1} has dependent AR(1) structure. We assume that {Wn,n≥1},{Zn,n≥1} and {Rn,n≥1} are three sequences of i.i.d. nonnegative random variables, and they are independent mutually with common distribution functions F(w)=P{W≤ω), G(z)=P[Z1≤z) and H(r)=P(R1≤r) respectively and F(0)=G(0)=H(0)=0. Furthermore, in order to keep stationary property of the model and fit the practical problems in the sense, we assume that0<a<1,0<6<1and0<c<1.Let T=inf{n:Un<0} with inf{(?)}=∞be the time of ruin, UT-1be the surplus immediately before ruin and|UT|be the deficit at ruin, define the Gerber-Shiu function as Let w(x,y)=I{y>d},δ=0, write Φ5(u, x0, y0, i0) as <φ(u,x0,y0,i0,d), then which is called the infinite time severity of ruin, i.e., the probability that ruin occurs and that the deficit at ruin is greater than d>0. Let w(x,y)=1,δ=0, or let d=0in φ(u,x0,y0,i0,d), we get the infinite time ruin probability Denote the Gerber-Shiu function before or at time n by ψδ,n(u,x0,y0,i0), i.e. similarly, the finite time severity of ruin and the finite time ruin probability can be defined as From the definition, we know Φδ,1(u,x0,y0,i0)≤Φδ,2(u,x0,y0,i0)≤Φ,δ3(u,x0,y0,i0)≤...,and For the Gerber-Shiu function, we haveTheorem9If by0-ax0<u, then for all n≥1, we have where that the integral interval does not include u.By the above recursive and integral equations, we can getTheorem10Let B be an NBU distribution and Λ be a nonnegative function. Assume that B and Λ satisfy and them for all u≥0,a≥b(1+ci0+K2),b≤p/(1+p),by0-ax0≤u,we haveTheorem11Let G∈R-λ for some λ>0.Assume that(1+I1)-1has a ppsilive λ-th moment,i.e. the for all n≥1,we have where Cn(i0) is given recursively byLetΥ=inf{n；n>T,Un>0}be the first time at which the risk process Un crosses above O after T,then clearly, denote the duration of the first negative surplus,or the duration of ruin.We write the probability that the duration of ruin is m as Qm(u,x0,y0,i0)(?)P(T=m).We havTheorem12Qm(u,x0,y0,i0)satisfies the following equation where Mk,m(u,x0,y0,i0) can be calculated in the following steps: where Mk,m represents the probability that the duration of ruin is m when the ruin time is k.