第10章-随机过程在保险精算中的应用.pdf

1/43kJIkJIGoBackFullScreenCloseQuitAAALLL?III 0)?PoissonL,Xk,k 1p.d?.?5bkES(t)=EN(t)EX1=t?xi$?S?ct ES(t)=(c )t 0,t 0.-c=(1+)(10.2.2)0,SSS?KKK111(relative security load-ing).JL?:9/43kJIkJIGoBackFullScreenCloseQuitdtLk-O9L-C?.?5bct S(t),t 0-OL.udr?limtU(t)=+,a.s.?U3,JLU?K?xi“?”,T=inft:U(t)0,inf =+(10.2.3)10/43kJIkJIGoBackFullScreenCloseQuit?.Lundberg-Cram er?xi?V(u)=P(T|U(0)=u),u 0(10.2.4)?V.?Vd?xiGU?-IPR(u)=1 (u)=PU(t)0|U(0)=u(10.2.5)KLJu?xi?V)Ve?V(u)VR(u)11/43kJIkJIGoBackFullScreenCloseQuit10.2.2 NNN!XXXLundberg?N!X?Vg|N!X?V?.10.2.2 b?g?1X(r)=E(erX)=Z0erxdF(x)=1+rZ0erx1F(x)dx(10.2.6)?3?:?,?S3.gX(r)=1+cr=1+(1+)r(10.2.7)3?)PRN!X12/43kJIkJIGoBackFullScreenCloseQuit555 1:duX(r)3SO?ek?)K7?P RN!X,vcZ0eRx1 F(x)dx=1(10.2.8)5?cZ01 F(x)dx=c=11+1Kc1 F(x),x 0V.?e-f(x)=eRxc1 F(x)Kyf(x)V,)?N!XR?d513/43kJIkJIGoBackFullScreenCloseQuit5552:PV(t)=ct S(t)duV(t),t 0-OL?kV(t)(r)=EerV(t)=?V(t)?0,t 0qV(1)(r)=Eer(cS(1)=ecrEerS(1)=ecrexp(X(r)1)=expX(r)+cr9N!X?V(1)(R)=1l?kV(t)(R)=1,t 014/43kJIkJIGoBackFullScreenCloseQuitoeN!X3KX(r)=1+cre?cZ0erx1 F(x)dx=1V(1)(r)=1.5553:f(10.2.7)k?Xg?kU3115/43kJIkJIGoBackFullScreenCloseQuit 10.2.1?g?XlX?N!X)dKg?X?F(x)=(1 ex,x 0;0,x 0.X?1X(r)=Z+0erxdF(x)=(1 r)1,r 0d?=2X?1X(r)=Z+0erxf(x)dx=(1 r)2,r 0N!XK?V(u)vXe?(u)eRu,u 0(10.2.9)yyy?n(u)L?u)31ng?9c?Vn=0,1,2,8Byn(u)eRu0(u)eRub?n(u)eRu yn+1(u)eRuk1g?u)?b?1g?u)?mmlVete1g?u)?t 03tu|G?7?u+ct?1g?Lu+ct?u)?V Px u+ct=1 F(u+ct)18/43kJIkJIGoBackFullScreenCloseQuit?Lu+ct?21g?u)|G?Ju+ct x?EkU3?e5?u)JLk-Od?VkXe48n+1(u)=Z+0?1 F(u+ct)+Zu+ct0n(u+ct x)dF(x)?etdt?d8Bb?n+1(u)=Z+0?Z+u+ctdF(x)+Zu+ct0n(u+ct x)dF(x)?etdtZ+0?Z+u+cteR(u+ctx)dF(x)+Zu+ct0eR(u+ctx)dF(x)?etdt?x u+ctR(u+ct x)019/43kJIkJIGoBackFullScreenCloseQuit=keR(u+ctx)1d?5?n+1(u)Z+0?Z+0eR(u+ctx)dF(x)?etdt=eRuZ+0eRctZ+0eRxdF(x)etdt=eRuZ+0e(+Rc)tX(R)dt=X(R)eRuZ+0e(+Rc)tdt=X(R)+RceRu?dN!X?VgX(R)=1(1+)R=1+cR20/43kJIkJIGoBackFullScreenCloseQuit?kX(R)+RceRu=1+(1+)R+RceRu=+Rc+RceRu=eRu=kn+1(u)eRu?n(u)eRu?n?k(u)=limnn(u)eRu21/43kJIkJIGoBackFullScreenCloseQuityyy?-Xt=eRU(t)=X0eRVtX0=eRu,RN!XVt=ct S(t)e-Yt=RVt,t 0KYt,t 0k-O?L.EeY1=EeRV1=MV1(R)=1ul6.5.1Xt,t 0K?.|K?n?limtXt=X tPT t=EXT|T tPT t+EXt|T tPT t(10.2.10)5?t TU(t)0Xt=eRU(t)1 ddNnLebesguen3(10.2.10)-t?4?eRu=EXt|T PT +EX|T=PT=qlimtU(t)=+,a.s.?kX=0,a.s.l?23/43kJIkJIGoBackFullScreenCloseQuitkeRu=EXT|T PT dd?(u)=eRuEeRU(t)|T 25?U(T)1d=(u)eRul?Lundberg?y.24/43kJIkJIGoBackFullScreenCloseQuitLundberg?-53uU5u?Ju?YS?K1m?XUd?xib?xiUJ?JuUN=?VKS?K1=uEexp(lnuX)1ln 1U?1+(1+)R=X(R)?R=lnu2dLundberg?(u)eRu,eF?S?K1?u=lnRK?y(u)eRu=eln=.25/43kJIkJIGoBackFullScreenCloseQuit?dLundberg?()=limu(u)=0(10.2.11)?V7LK?k0 (u)eRudYK?0 limu(u)limueRu=0d?)VR()=1 ()=1(10.2.12)26/43kJIkJIGoBackFullScreenCloseQuit10.2.3?VVVvvv?!?V)V?wL 10.2.3 G(u,y),u 0,y 0G(u,y)=PU(T)y,T|U(0)=u=L?Ju?iy?Vd?J30ym?k(u)=limyG(u,y),u 0(10.2.13)kXe(J27/43kJIkJIGoBackFullScreenCloseQuitnnn 10.2.2 G(u,y)vXeG(u,y)u=cG(u,y)cZu0G(u x,y)dF(x)cF(u+y)F(u),u 0(10.2.14)yyy1g?u)?b?1g?u)?mmlVet?xi31g?u)?t?Ju+ct?xv0 x u+ct1g?Ju+ct xd-O5?iy?VG(u+ct x,y)?u+ct x u+ct y1g?iy1g?28/43kJIkJIGoBackFullScreenCloseQuit3u+ct,u+ct+y?VF(u+ct+y)F(u+ct)d?VkG(u,y)=Z+0Zu+ct0G(u+ctx,y)dF(x)+F(u+ct+y)F(u+ct)etdt5C-z=u+ct=t=zuc,dt=dzckG(u,y)=ceucZ+uezcZz0G(zx,y)dF(x)+F(z+y)F(z)dz?G(u,y)uu?G(u,y)u=cG(u,y)+ceucezcZu0G(u x,y)dF(x)+F(u+y)F(u)=?(10.2.14)29/43kJIkJIGoBackFullScreenCloseQuiten?G(0,y)?wLnnn 10.2.3 G(0,y)LG(0,y)=cZy01 F(x)dx,y 0(10.2.15)yyykk 0 G(u,y)(u)eRu?0 G(,y)=limuG(u,y)limueRu=kG(,y)=0Z+0G(u,y)du Z+0eRudu=1R-(y)=R+0G(u,y)duKk0 .10.2.14l0?+uu?G(0,y)=c(y)cZ+0Zu0G(u x,y)dudF(x)+cZ+0F(u+y)F(u)du30/43kJIkJIGoBackFullScreenCloseQuitC-v=u x?G(0,y)=c(y)+cZ+0Z+0G(v,y)dvdF(x)+cZ+0F(u+y)F(u)du=c(y)+cZ+0(y)dF(x)+cZ+0F(u+y)F(u)duR+0dF(x)=1?m?cp-=kG(0,y)=cZy01 F(x)dx31/43kJIkJIGoBackFullScreenCloseQuitnnn 10.2.4 3J0?V(0)=11+(10.2.16)yyy?g?F(x)u?K?C=Z+01 F(x)dxdn10.2.3k(0)=limyG(0,y)=limycZy01 F(x)dx=cZ+01 F(x)dx=c=11+?J0)VR(0)=1+32/43kJIkJIGoBackFullScreenCloseQuit?e5)Vv?nnn 10.2.5)VR(u)vXeR0(u)=cR(y)cZu0R(u x)dF(x),u 0(10.2.17)yyyd10.2.1310.2.14k0(u)=c(u)cZu0(u x)dF(x)c1 F(u)(10.2.18)R(u)=1 (u)?k33/43kJIkJIGoBackFullScreenCloseQuitR0(u)=c1 R(u)cZu01 R(u x)dF(x)Zu01 F(u)=cR(y)+cZu0R(u x)dF(x)F(u)=Ru0dF(x)34/43kJIkJIGoBackFullScreenCloseQuit 10.2.3 eg?l=F(x)=1 ex,x 0)R(u)n10.2.5kR0(u)=cR(y)cZu0R(u x)exdx-y=u xkR0(u)=cR(y)cZu0R(y)euydyR00(u)=(1+)R0(u)35/43kJIkJIGoBackFullScreenCloseQuit=keyu(1+)R0(u)=H1H1=R0(0)=(1+)2?R(u)=11+expu(1+)2+H2?u=0dR(0)=1+?H2=1?kR(u)=1 11+expu(1+)236/43kJIkJIGoBackFullScreenCloseQuit10.2.4 CramerCCCqqq?Feller?#yE|;?euFeller#n?V?CqLnnn 10.2.6 3Lundberg-Cramer;?.k1(0)=11+2Lundberg-Cram erCq3?C?(u)CeRu,u =limu(u)CeRu=1.37/43kJIkJIGoBackFullScreenCloseQuityyy PR(u)=1 (u)=P(U(t)0|U(0)=u),KLJu?xi?V)V.kg?u)?T1g?X1?)VA?V,?R(u)=Z0et?Zu+ct0R(u+ct z)dF(z)?dtCx=u+ct,?R(u)=cecuZuecx?Zx0R(x z)dF(z)?dxddR(u).u?,?R0(u)=cR(u)cZu0R(u z)dF(z)38/43kJIkJIGoBackFullScreenCloseQuit3?R(t)R(0)=cZt0R(u)du+cZt0Zu0R(uz)d1F(z)dud?Ru0R(uz)d1F(z)=R(0)1F(u)R(u)+Ru0R0(uz)1F(z)dz“,=?R(t)R(0)=cR(0)Rt01F(u)du+cRt0Ru0R0(uz)1F(z)dzdue3?-?gS?Rt0Ru0R0(uz)1F(z)dzdu=Rt0(RtzR0(uz)1F(u)du)dz=Rt0R(tz)R(0)1F(z)dz2“?R(t)=R(0)+cZt0R(t z)1 F(z)dz39/43kJIkJIGoBackFullScreenCloseQuitdulimuR(u)=1-t?1=R(0)+cZ01 F(z)dz=R(0)+cuk(0)=1 R(0)=c=11+(1)?y“c,?R(t)=1cR01F(z)dz+cRt0R(tz)1F(z)dz=1cRt1F(z)dzcRt01R(tz)1F(z)dzl?(t)=cZ01 F(z)dz+cZt0(t z)1 F(z)dz40/43kJIkJIGoBackFullScreenCloseQuitducZ01 F(z)dz=c=11+1#.3#eRtRN!X,-A(t)=eRt(t),a(t)=ceRtZt1 F(z)dzf(z)=ceRz1 F(z)=?A(t)=a(t)+Zt0A(t z)f(z)dzqR0f(z)dz=1,l?#.a(t)N4Z0a(t)dt=1R1+=C1.41/43kJIkJIGoBackFullScreenCloseQuitua(t)30,)v#n?.dd#n?limteRt(t)=limtA(t)=C1R0zf(z)dz=CLlimt(t)CeRt=1l?L-C?.Cq(2)?y.42/43kJIkJIGoBackFullScreenCloseQuit 10.2.4?g?lF(x)=(1 ex,x 0;0,x 0.Cq?V)N!XR=(1+)1X(r)=11r?0X(r)=d(11r)dr=(1 r)20X(R)=(1 R)2=(1+)2dCramerCq?kC=(1+)2(1+)=11+43/43kJIkJIGoBackFullScreenCloseQuit=k(u)11+expy(1+),u +?!0?;?Lundberg-Cramer?.Nd?1U?xi?JL*La*L5