Study Of The Bankruptcy Of The Types Of Risk Models

This dissertation is devoted to dealing with the ruin theory for some kinds of risk models which include distributions of extreme values in the classical risk model with constant interest force ,a class of renewal risk process whose claims occur as an Mixed Exponential process and a class of delayed renewal processes.The classical risk model has been generalized in many papers to conform to reality since it was put forward.One of the examples is the classical risk model with constant interest force .There are many correlated papers treating constant interest force such as Cai(2002),Li(2004),Wu(2002),Wu(2005). Among them , Wu(2005)derived the united distribution of the ruin time ,the surplus immediately prior to ruin and the deficit at ruin .Besides,Li(2004)discussed the first hitting-time in the classical risk model with constant interest force.Based on these conclusions, The first chapter of the dissertation is to discuss several important united distributions running in the classical risk model with constant interest force.They include the ruin time,the fist recovery time,the supreme profits before ruin, the minimum profits between ruin and recov-ery,the surplus immediately prior to ruin,and the deficit at ruin . We obtain these expressions mainly by Strong Markov property of the surplus with constant interest force.The main results are as follows:Theorem 1.1 when u > 0, let G₁(u,a) = p^u(sup₀≤t≤T_δ Uδ(t) > a,Tδ < ∞), thenoo?i(u,a) = ips(a)\C -f OU Jpoo ptH / /i(?, ue5t + c eSvdv - a)Jl\n^4^ Jo* /in*(i, ae"' + c / e*wdu - a)d<. /oTheorem 1.2 let M = max{\Us(t)\,Ts < t < T?}, then< oo, \US(T,)\ = y) = ^i Theorem 1.3 when a > w > 0, x, y > 0,let G3(u;a,x,y)dxdy = PU(U5(TS⁾ e dx,\Us(Ts)\ € dy,T, > oo),thenGz(u;a, x,y)/ h(t,ue6t + c e6vdv-a)e5vdv - a)dt] / lit > 5^l In I±l±)Ap(t, a, x) Jo du + cf(x + y)dxdydt.Theorem 1.4 if a > u > 0, b > y > 0, a > x > 0,if let G4(u;x,y,a,b)dxdy= Pu(Us{Ts-)edx,\Us(T5)\edy, sup ￡/,(*)< a,CKKTfinf Us(t) >-bX < oo),06 Mn X C)\p(t,u,x)f{x + y)dtdxdy Jo ou + cMb)z vdv — a)* hn*(t,ae5t/°/8x + cIn-------)\p(t, a, x)f(x + y)dtdxdy}.da + cAt present,the distribution of the claim interval has been generalized to Erlang in many papers such as Dickson(1998) where David CM. Dickson made a research on ruin probability ,deficit distribution ,barrier problem and so on when the claim interval yields Erlang .I'll discuss the similar problems as well supposing the interval yields Mixed Exponential distribution.The main results are as follows:Theorem 2.1.1 If the claim amount Zi yields the exponential distribution with parameter ft , thenwhere A2 =Theorem 2.1.2 S(s) satisfies :^, c2s5(0) + f3i/32mi + c(A\/3i 4- A2(32) — (/?i + B2)c d(s) —-----------■■------------------------------------------------------'■------=—,c2s2 - (Pi + /32)cs + Pif32 - [/?i/?2 cs(AiPi + A2/32)]f(s)where m^s the expectation of Z{, f(s) means the Laplace Transform of /. Theorem 2.2 G(u, y) satisfies :6(0)where (s,y) means the Laplace Transform of fu y g(0,x)dx , = c2s5{Q) + Bifarrn + c^ft + A2/32) - (ft 4- (ft + p2)cs + ftft - [ft ft - (ft (c2rTheorem 2.3 If the claim amount Z-t yields the exponential distribution with parameter j3 , thenx(u b) =ZlLth^In chapter 3,we discuss a special delayed risk process on the condition that the density function of the distribution of the time prior to the first claim satisfies:when /(*) = ae^at,-at roo eayJ\ fj + (1 q)ae. Jo e°yK(y)dyThe main results are as follows:Theorem 3.2.1 The Gerber-Shiu penalty function m^fu) satisfies:mhS(u)kx{Q)E{V) fu1 + 8/ ms{u - y)dHe(y) + exp{-------(u -Jo Ju ck _,f ²-* r m,(t - y)dH(y)dtdx - ^"Jo c \n fc=0{-x)k / exp{----—-t}tn-2^k{l + 9)E{Y)r{t)dtdx, Jx cwherePs(t) = --/o(-l)V / exp{-^-^} / rns(t-y)dHiy)tn'2-kdtdu Ju c J0+ / (-l)V / exp{----^^^}(lJO Ju Cparticularly , when n = 2,mg(u - y)dHe(y) + exp{/o+-----y-—D2exp{------u]ec2c2 whereC | /*OO /*0O C i /*￡exp{-------u} / / exp{---------1} / ms{t — y)dH(y)dtdac Jo Jx c JoX _] /*OO /*OO ￡ 1exp{------u} / / exp{--------t}(l + 6)E(Y)r(t)dtdx,c Jo Jx cr°° r°° 5 + a fl D2 = / / exp{---------1} / ms{t-y)dH{y)dtduJo Ju c Jo/ exp{----—t}(l + O)E(Y)r{t)dtdu. Ju Ccorollary 3.2.2 let 5 = 0,w — 1 then ms(u) = il>(u), then the ruin proba-bility ij)\ (u) satisfies 'u) = mowo r^{u y)dHe{y) +1 T " JO Juf W- y)dH{y)dtdx - [I ^f J^ C Jo c yn Z). fc_Q ■(^x)k / exp{--t}tn-2-kH(t)dtdx, Jx ckn₂where/?OO />CX3 />t^n,0 = / (-l)V / exp{—?} iP(t-y)dH(y)tnJO Ju c Jo+ / {-l)kuk / exp{--t}tn-2-kH(t)dtdu.JO Ju cparticularly , when n = 2-2-kdtduQa2 a f°° f°° a fl— exp{— u) I I exp{----1) / ip(t — y)dH(y)dtdxc Jo Jx c Joa-exp{-n} r f^exp{--t}H{t)dtdx,c Jo Jx cwhereI exp{—t) I ip(t — y)dH{y)dtdu Ju c Jo+ / / exp{----t}^H{t)d.tdu.Jo Ju cIf the claim interval yields generalized Erlang,we get that:Theorem 3.3.1 The Gerber-Shiu penalty function mi)($(?) satisfies :mi,s{u) = ———j.—- / ms(u — y)dHe(y) + / exp{-------(u — t)} ^<sup> (u - v)dHe(y) +1 -r v Jo Jun-l n-1l/>0O O ptexp{^(W-i)} / ij,(t-y)dH(y)dtJu c Jo^(u - t)}H{t)dt - Z fa ft) (a - Pi V c whereut) = -bC Jq C CThe Gerber-Shiu penalty function mii($(u) satisfies :———— / ms{u - y)dHe(y) + exp{------(u-1 +y Jo A c]exp{(—)?} ceXp{(_l^l)(M_t)} / ms(t-y)dH(y)dt c Jo/ &cp{(-^-)(u-t)}(lJ u ^whereWillmot(2004) has deduced the integrate relation of penalty functions between delayed risk processes and normal ones about this model. Particularly,let w = 1 and 5 = 0,we get the relation of the ruin probability between the two renewal ones.In this chapter,if f(t) is changed into Erlang(n) or generaUzd Erlang(n),-we can get the integral relation between the penalty function and the ruin probability of the renewal process.Considering that Erlang(l) is also exponential ,thus we can treat Willmot(2004) as a special case of this chapter.