考虑一般投资收益和时间相依索赔情形下二维带扰动风险模型的有限时间破产概率渐近估计.pdf

2023,43A(5):15291558http:/m?/e?6x?.?km?VC?O?(fE?611731):kL?6?xx?.,b?xiJ?Ldm4Lx,?mml Sarmanov(?.?u?KCzx,?km?V?C.?L?m4LO?L evy L,Vasicek|?.,Cox-Ingersoll-Ross(CIR)|?.,Heston?.,?A/e?V?C.c:x?.;m;?V.MR(2010)Ka:62E20;62P05a:O211.4zI:A?:1003-3998(2023)05-1529-301Cc5,x?.?V?Ja1,8,10,11,19,21,23.u?xi5,EU?,d?xx?.?JF,?xx?.?xx?.?g,29p?xx?.?Ak-,z37,9,12,1618,20,25,27.XILFO7KF,?xi?xk?5?-?K,?3?xi?VL?-.b?xi?,?=?xi3|,Cz?,nzb?B3?n.37KF?8U,?xi|?U?p?x?,d,?xi?k(5x5.kb?xi?Ld L evy Lx,?xi5,L?d-Lx,X,CIR|?.x?L Markov L,u/K$?5?.x?L$Ud Markov Lx.?y?xi?V?OL$Lp,XJL?xi?k5,)K,b?xi?Ld?m4Lx,?6?xx?.?V?C?O.vF:2021-06-21;?F:2022-07-05E-mail:;78:Ig,7(71271042)!?H“8(2020ZB014)?H:“c8(202201AU070051)Supported by the NSFC(71271042),the Yunnan Normal University(2020ZB014)and the YunnanProvince Science and Technology Department(202201AU070051)1530n?Vol.43 A?m4L(t),t 0?xi?L(0)=0.?B1(t),t 0 B2(t),t 0 IOK$vB2(t)=B1(t)+p1 2B3(t),1,1,B3(t)IOK$B1(t).u,?m t 0?xi?JL(U1(t),U2(t)U1(t)U2(t)=e(t)xy+c1Zt0e(s)dsc2Zt0e(s)ds+1Zt0e(s)dB1(s)2Zt0e(s)dB2(s)N(t)Xi=1Xie(i)N(t)Xj=1Yje(j),(1.1),(x,y)O?7;(c1,c2)?c1,c2 0;i 0,i=1,2,6X;(Xi,Yi),i 1?S?,?mm 1,2,0?C;?m n=nPi=1i,n 1?#t=t?gtL N(t):N(t)=Pi=1Iit,t 0,IAL8 A?5.?.(1.1),?km?V(x,y;t)=P(Tmax t|U1(0)=x,U2(0)=y),Tmax=inf t 0:max(U1(t),U2(t)0?m.?1|e?Xe.1 2!0?y?(J;?LO?L evy L,Vasicek|?.,CIR|?.,Heston?.,31 3!?A/e?V?C;1 4!y?(J.2O?9(J?C oL?,3z?U.u a b,Pa b=maxa,b,a b=mina,b.uvXl1=liminf(x,y)(,)f(x,y)g(x,y)limsup(x,y)(,)f(x,y)g(x,y)=l2,?f(,)g(,),?l2 1,f(x,y).g(x,y);?l1 1,f(x,y)&g(x,y);?l1=l2=1,f(x,y)g(x,y);?0 l1 l2 0.No.5?:m?/e?6x?.?V1531?b?9?mmln?Sarmanov(?,=b?2.1?(X,Y,)S?(Xi,Yi,i),i 1?E,SO30,)?F,G H vP(X dx,Y dy,dz)=(1+121(x)2(y)+131(x)3(z)+232(y)3(z)dF(x)dG(y)dH(z),(2.2),12,13 23?;1(x),2(y)3(z)vE(1(X)=E(2(Y)=E(3(Z)=0,(2.3)limx1(x)=d1,limy2(y)=d2 0,(2.4)1+121(x)2(y)+131(x)3(z)+232(y)3(z)0,x,y,z 0.(2.5)?(2.2)u z 3m 0,)?,(2.3)?P(X dx,Y dy)=(1+121(x)2(y)dF(x)dG(y),(2.6)=(X,Y)l?Sarmanov.aq/,P(X dx,dz)=(1+131(x)3(z)dF(x)dH(z),(2.7)P(Y dy,dz)=(1+232(y)3(z)dG(y)dH(z).(2.8)z26,3 b1,b2 b3?|1(x)|b1,|2(y)|b2,|3(z)|b3,x,y,z 0.(2.9)?e5k?P.d(2.4),s 0,31(s),32(s)33(s),31(s):=1+13d13(s)0,32(s):=1+23d23(s)0,(2.10)33(s):=1+3(s)(13d1+23d2)/eCd 0.(2.11)3()?k.5,3?b31,b32 b33?31()b31,32()b32 33()b33.-b1,b2b3?C,ObH1,bH2bH3,=dbH1(s)=31(s)dH(s),dbH2(s)=32(s)dH(s),dbH3(s)=33(s)dH(s).(2.12)?(X,Y,)(X,Y,)?,(X,Y,)(X,Y,)k?(X,Y,)?mp.,?(Xk,Yk,k),k 1 E?(X,Y,)?S?.u,?!?Ee?#L,bN(m)t=Xk=1I(b(m)kt),b(m)1=bm,b(m)k=bm+kXi=2i,k 2,m=1,2,3;(2.13)bN(4)t=Xk=1I(b(4)kt),b(4)1=b1,b(4)2=b1+b2,b(4)k=b1+b2+kXi=3i,k 3;(2.14)1532n?Vol.43 AbN(5)t=Xk=1I(b(5)kt),b(5)1=b2,b(5)2=b2+b1,b(5)k=b2+b1+kXi=3i,k 3.(2.15)#b(m)t=EbN(m)t=Xk=1P(b(m)k t),m=1,2,3,4,5.(2.16)?b?(Xi,Yi,i),i 1,(Xi,Yi,i),i 1,(t),t 0,(B1(t),B2(t),t 0(b1,b2,b3).n 2.1?xx?.(1.1).?F R,G R,0,0.3b?2.1?e,e3 max+,2,?RT0Ee(s)ds 0 T ,K 0 t T,(x,y;t)F(x)G(y)P(t),P(t)=eCdZt0Ee(+)(u)db(3)u+ZZu+vtEe(u)(u+v)db(5)udv+dbH2(u)(db(1)v dv)+ZZu+vtEe(u)(u+v)db(4)udv+dbH1(u)(db(2)v dv).?L(t),t 0 L evy L,w,k 2.1.2.1?xx?.(1.1).?(t),t 0 L evy L,()(t),t 0?Laplace.3n 2.1?e,K 0 0.?e?.2.2?xx?.(1.1).?(X,Y)l(2.6)?Sarmanov S F,G R,0.?(X,Y)C .3b?2.1?e,e3 max2,2?RT0Ee(s)ds 0 T ,K 0 0,KEeb(u)a(u+v)=?kaemv/2kacosh(kav)+sinh(kav)m/2?2aml/2(bB1(v)1)e(m2ka+b)u/2bB1(v)e2uka+b!2ml/2 exp(r0m 2ka+b24ka+b2bB1(v)e2uka+b 1#),(3.4)ks=m2+2s2/2,bB1(v)=1 4ka+b(2ka+bm)bB2(v)2,bB2(v)=2am+2kacoth(kav).yd(3.1),Eeb(u)a(u+v)=E?e(a+b)Ru0rsdsaRu+vursds?=Ene(a+b)Ru0rsdsE?eaRu+vursds|Fu?o.?y?oNgkO E?eaRu+vursds|Fu?,?O Eeb(u)a(u+v).r(t)d(3.2)(,?r(t)?L,eT t P B(t,eT):=E?eaReTtrsds|Ft?=f(t,rt),f(t,r)uJC t r?.?E?eaRu+vursds|Fu?,k)f(t,r)?wL.z 22,IL“?,?,-dt?u 0”n?1534n?Vol.43 A f(t,r)v?,L)?f(t,r)?wL.0 s t eT,Enea(t)B(t,eT)?Fso=EneaRt0rudu E?eaReTtrsds|Ft?|Fso=EneaReT0rudu|Fso=ea(s)B(s,eT).?ea(t)B(t,eT)=ea(t)f(t,rt)?.?ea(t)f(t,rt)?d?ea(t)f(t,rt)?=f(t,rt)dea(t)+ea(t)df(t,rt)=ea(t)?arf+ft+m(l r)fr+frr 2r2/2?dt+ea(t)frdWt.-dt?u 0?ft(t,r)+m(l r)fr(t,r)+frr(t,r)2r2/2=arf(t,r),(3.5)f(eT,r)=1,?r.(3.6)(1)?=0,=Vasicek?.(3.5)ft(t,r)+m(l r)fr(t,r)+frr(t,r)2/2=arf(t,r).(3.7)z 22,T?)?/f(t,r)=erC1(t,eT)A1(t,eT),C1(t,eT)A1(t,eT)u t 0,eT?.f(t,r)=erC1(t,eT)A1(t,eT)“(3.7)?C01(t,eT)+mC1(t,eT)a?r A01(t,eT)mlC1(t,eT)+2C21(t,eT)2#f(t,r)=0.(3.8)?r,(3.8)o?,?C01(t,eT)+mC1(t,eT)a=0,K,eUC r?,(3.8)o 0.u?C01(t,eT)=mC1(t,eT)a.(3.9)l?,A01(t,eT)=mlC1(t,eT)+2C21(t,eT)/2.(3.10)d(3.6),C1(eT,eT)=A1(eT,eT)=0.?(3.9)(3.10)9?C1(t,eT)=am?1 em(eTt)?,(3.11)A1(t,eT)=?al a222m2?(eT t)+?alma22m3?em(eTt)+a224m3e2m(eTt)?alm3a224m3?.(3.12)u,B(t,eT)?wL B(t,eT)=f(t,rt)=ertC1(t,eT)A1(t,eT),0 t eT,C1(t,eT)A1(t,eT)Od(3.11)(3.12)(.?Eeb(u)a(u+v)=Ene(a+b)(u)B(u,u+v)o=eA1(u,u+v)EneC1(u,u+v)ru(a+b)(u)o.(3.13)?E?eC1(u,u+v)ru(a+b)(u)?,I Eep1rt+p2(t),p1,p2.-D(p1,p2,t)=Eep1rt+p2(t),rtpdC,dz 22,4.4.10 (t)=Rt0rsds=No.5?:m?/e?6x?.?V1535(mlt+r0 rt+Wt)/m pdC,?p1 p2,D(p1,p2,t)k?.ep1rt+p2Rt0rsdsA It o?D(p1,p2,t)=ep1r0+?p1ml+p2122?Zt0D(p1,p2,s)ds+(p2 p1m)Zt0D(p1,p2,s)p1ds,?1?5 D(p1,p2,s)/p1=E(rsep1rs+p2Rs0rvdv).u?D(p1,p2,t)u t?D(p1,p2,t)t+(p1m p2)D(p1,p2,t)p1=?p1ml+p2122?D(p1,p2,t),(3.14)D(p1,p2,0)=ep1r0.?(3.14)?),k?(3.14)?A?).dA?,dp1ds=(p1m p2),dtds=1,p1(0)=c,t(0)=0.(3.15)u,(3.15)?)p1(s)=p2m+?c p2m?ems,t(s)=s.(3.16)?U(s)=D(p1(s),p2,t(s),u(3.14)=ze?,dUds=D(p1,p2,t)tdtds+D(p1,p2,t)p1dp1ds=D(p1,p2,t)t+(p1m p2)D(p1,p2,t)p1.(3.14),KdUds=?p1ml+p2122?U(s),U(0)=D(p1(0),p2,t(0)=D(c,p2,0)=ecr0.u,U(s)=exp?cr0+mlZs0p1(v)dv+22Zs0p21(v)dv?.(3.17)d(3.16),p1 t 5L c s,?“(3.17)?D(p1,p2,t)=exp?A1(p1,p2)+A2(p1,p2)t+A3(p1,p2)emt+A4(p1,p2)e2mt?,(3.18)A1(p1,p2)=p2r0m+?l+p22m2?p1p2m?+?p1p2m?224m,A2(p1,p2)=p2l+p2222m2,A3(p1,p2)=?r0 l p22m2?p1p2m?,A4(p1,p2)=2(p1p2/m)24m.,(3.18),EneC1(u,u+v)ru(a+b)(u)o=E?ea(1emv)mru(a+b)(u)?=D?a?1 emv?/m,(a+b),u?.(3.19)?(3.19)“(3.13)=?(3.3).(2)?=1/2,=CIR?.(3.5)ft(t,r)+m(l r)fr(t,r)+2rfrr(t,r)/2=arf(t,r).(3.20)T?)k/f(t,r)=erC2(t,eT)A2(t,eT),“(3.20)?f(t,r)?C02(t,eT)+mC2(t,eT)+22C22(t,eT)a?r A02(t,eT)mlC2(t,eT)?=0.1536n?Vol.43 Aaq/,?C02(t,eT)+mC2(t,eT)+22C22(t,eT)a=0 A02(t,eT)mlC2(t,eT)=0,=C02(t,eT)=mC2(t,eT)+2C22(t,eT)/2 a A02(t,eT)=mlC2(t,eT).?,Tv C2(eT,eT)=A2(eT,eT)=0.k)C2(t,eT).-Z(t,eT)=C2(t,eT)+(m+2ka)/2,ka=m2+2a2/2.u,Z0(t,eT)=22Z2(t,eT)2kaZ(t,eT)Z(eT,eT)=(m+2ka)/2?|,?Z(t,eT)=?24ka+2h1m+2ka14kaie2ka(eTt)?1.uC2(t,eT)=asinh(ka(eT t)kacosh(ka(eT t)+m2sinh(ka(eT t).(3.21),?)A2(t,eT).A02(t,eT)=mlC2(t,eT)|A2(eT,eT)=0?A2(t,eT)=2mla2ln kaem(eTt)/2kacosh(ka(eT t)+m2sinh(ka(eT t)!.(3.22)u,CIR?.f(t,rt)?w)B(t,eT)=f(t,rt)=ertC2(t,eT)A2(t,eT),0 t eT,C2(t,eT)A2(t,eT)d(3.21)(3.22)(.?E?eb(u)a(u+v)?=Ene(a+b)(u)B(u,u+v)o=eA2(u,u+v)EneC2(u,u+v)ru(a+b)(u)o.(3.23)?EeC2(u,u+v)ru(a+b)(u),?D0(p1,p2,t)=Eep1rt+p2(t),p1 p2.dz 13,rt(t)=Rt0rsds?k1,aqu D(p1,p2,t)k?,?m2 22p2,D0(p1,p2,t)=exp?c(p1,p2)r0+m (p2)mlt22ml2ln(p1,p2)e(p2)t(p1,p2)1?,(3.24)(p2)=pm2 22p2,c(p1,p2)=m (p2)22(p2)2(p1,p2)e(p2)t 1,(p1,p2)=1 2(p2)(p2)m+p12.(3.25),dn 3.1 m2 22(a+b),ud(3.24)k E?eC2(u,u+v)ru(a+b)(u)?,l?(3.23).n?y.n 3.1?6?xx?.(1.1).?(X,Y)?SF R G R,0.?(X,Y)l?Sarmanov,=(2.6).?L(t),t 0 d(3.1)(3.2)?E,(t),t 0,(Xi,Yi),i 1,i,i 1 (B1(t),B2(t),t 0.u,?t 0 k(x,y;t)F(x)G(y)?eCdZt0Ee(+)(u)du+2ZZu+vtEe(u)(u+v)+Ee(u)(u+v)dudvo,Ee(+)(u),Ee(u)(u+v)Ee(u)(u+v)dn 3.1?.No.5?:m?/e?6x?.?V1537y?n?y.ky Vasicek?./.d(3.3)kEe(t)=expneA1()+eA2()t+eA3()emt+eA4()e2mtoeA1()=m(l r0)3224m3,eA2()=?22m2 l?,eA3()=m?r0 l+2m2?,eA4()=224m3.KRt0Ee(s)ds 0,udn 2.1=y?(J.gy CIR?./.d(3.4)?Ee(t)=?emt/2cosh(kt)+m2ksinh(kt)?2ml/2 expn2r0m+2kcoth(kt)o.KRt0Ee(s)ds 0,udn 2.1?(J.n 3.1?y.3.2Au Heston?.1993 c,Steven Heston?a?.,Heston?.14.Black-Scholes?.,Heston?.?.-L?St=e(t),t 0?d Heston?.,=dSt=Stdt+rtStdW(1)t,(3.26),x|;W(1)t,t 0 IOK$;rt,t 0?Ldrt=m(l rt)dt+rtdW(2)t.(3.27)d?,r0 0;l 0 LY;m 0 LE?Y l?;0 N?L,v 2ml 2;W(2)t,t 0 IOK$K$W(1)t,t 0,=dW(1)t=dW(2)t+p1 2dW(3)t,(3.28),1,0 X,W(3)t,t 0 IOK$W(2)t,t 0.n 3.2?L St=e(t),t 0 d(3.26)(3.28).a,b 0,e(2+2m)(a+b)+(a+b)22SF R G R,0,?(X,Y)l?Sarmanov,=(2.6).?L St=e(t),t 0 d(3.26)(3.28)?E,(t),t 0,(Xi,Yi),i 1,i,i 1(B1(t),B2(t),t 0.e3 max+,2 v 2m+(+1)0,(x,y;t)F(x)G(y)?eCdZt0Ee(+)(u)du+2ZZu+vtEe(u)(u+v)+Ee(u)(u+v)dudv?,Ee(+)(u),Ee(u)(u+v)Ee(u)(u+v)(3.30)?.ydn 3.2,3(3.32)(3.34)?L,-b=0,a=,u=0 9 v=t?E?e(t)?=exp?t+mlt+r0+c?,+2(1 2)2m?+f(0,t)?,un 2.1=?(J,n 3.2?y.4(J?y4.1?nkeZy(JI?X.?F u 0,)?,e F R,0 ,K 0 a b 0,K3 CF 1 DF 0,?0 0,d(4.2)?xp=o(F(x),x .(4.3)1540n?Vol.43 A?X Y?C,e X?F R,p KC YvEYp x,Y M)F(x)=E?YIY M?,(4.4),z 15?0,p 0?eP?XY x?Y?CF(x)?pYpIY+IY 0,0.?KC 1 2z?X,Y X.e3 +vEk 0,KP X1 x,Y2 y F(x)G(y)E?12?,(4.7)P X1 x,Y1 y,X2 x=o(1)F(x)G(y).(4.8)yd(4.6)H older?,?0 l kElk(Ek)l/0,?b (1,)a (0,1)v a a,+2 E?lk?Ikb?x,Y2 y?C.,-E1=1 b.u P X1 x,Y2 y-?PX1 x,Y2 y=3Ps=1P X1 x,Y2 y,Es:=3Ps=1Qs.dF R,G R,(4.4)9(4.9),3x1 0,y1 0?x x1,y y1,Q1F?xa?P Y2 y aF(x)G(y)E2 x1,y2 y1?x x2,y y2,Q2=P X1 x,Y2 y,a 1 b,2 x,Y2 y,a 1 b,a 2 b+P X1 x,Y2 y,a 1 b,2 bG?ya?P X1 x+ZbaZbaF(x/u)G(y/v)P(1 du,2 dv)+F?xb?P Y2 y,2 b aF(x)G(y)E1+F(x)G(y)ZbaZbauvP(1 du,2 dv)+bF(x)G(y)E?2I2b?No.5?:m?/e?6x?.?V1541 CF(x)G(y)+F(x)G(y)E?12?.d(4.4),(4.5),(4.10)H older?,3 x3 x2,y3 y2?x x3,y y3kQ3=P X1 x,Y2 y,1 b,2 b+P X1 x,Y2 y,1 b,2 bG?yb?P X1 x,1 b+E?I1b,2bP(X1 x|1,2)P(Y2 y|1,2)?CF(x)G(y)+CF(x)G(y)EnI1b,2b?+1I11+I11?+2I21+I2b+2I2bo CF(x)G(y)+CF(x)G(y)hE?+21I1b?i+2hE?+22I2b?i+2 CF(x)G(y).u,du Q1,Q2 Q3?V P X1 x,Y2 y?C.,yyV P X1 x,Y2 y?Ce.Q2?,3 x4 x3,y4 y3?x x4,y y4,P X1 x,Y2 y P X1 x,Y2 y,a 1 b,a 2 bF(x)G(y)E?12Ia1bIa2b?F(x)G(y)nE?12?E?12I1b?E?12I2b?o.d H older?,(4.9)(4.10)?E?12I1a?hE?+1I1b?+E?12I2b?C.u,0,x x4 y y4,(1 C)F(x)G(y)En12o P X1 x,Y2 y (1+C)F(x)G(y)En12o,=yC(4.7).gy(4.8).0,d(4.2),H older?,(4.9)(4.3),P X1 x,Y1 y,X2 x=Zx/DF0Zy/DG0+Zx/DF0Zy/DG+Zx/DFZy/DG0+Zx/DFZy/DG!F?xv?G?yu?F?xu?P 1 du,2 dv CF(x)G(y)Zx/DF0Zy/DG0(u u+)(v v+)F?xu?P 1 du,2 dv1542n?Vol.43 A+CF(x)Zx/DF0Zy/DG(v v+)?uDGy?+P 1 du,2 dv+CG(y)Zx/DFZy/DG0(u u+)?vDFx?+P 1 du,2 dv+CZx/DFZy/DG?uDGy?+?vDFx?+P 1 du,2 dv CF(x)G(y)En(1+1)(2+2)IX1xo+CF(x)y(+)En(2+2)+1o+Cx(+)G(y)En(1+1)+2o+Cx(+)y(+)En+1+2o=o(1)F(x)G(y).n?y.?n(eXk,eYk,ek),k 1o?S?,E(eX,eY,e).?(eX,eY,e)?SOeF,eG eH,=deF(x)=?1 1(x)b1?dF(x),deG(y)=?1 2(y)b2?dG(y),deH(z)=?1 3(z)b3?dH(z),1(x),2(y)3(z)db?2.1,bi,i=1,2,3,d(2.9).u,eF(x)=Zx(1 1(u)/b1)dF(u)(1 d1/b1)F(x),x ,(4.11)eG(y)=Zy(1 2(v)/b2)dG(v)(1 d2/b2)G(y),y ,(4.12)d1 d2d(2.4).?/,0 ,0?i 0,-i=1+2+i,i=1+2+i,i(t)=e(i)Iit,i(t)=e(i)Iit,bki(t)=e(k)i)I(k)it,k=1,2,3,4,5,(k)id(2.13)(2.15).?q 0 kXn=1nqsn1n(n 1)!esC(sq 1),q=0,1,2,C(sq+1 1),(4.13)q q?.q 0,0 l n 2.1?,H older?,Xn=1nqEln(t)=Xn=1nqEnIntE?el(n)|n?o=Xn=1nqZt0E(el(s)sn1n(n 1)!esds=Zt0E(el(s)Xn=1nqsn1n(n 1)!esds C(tq+1 1)Zt0Eel(s)ds C(T)Zt0?Ee(s)?l/ds C(T)?Zt0Ee(s)ds?l/,0 t T,(4.14),?1?d?.3(4.14)?q=0,0 l 0 t T,KEln(t),?n 1.(4.15)n 4.33n 2.1?e,?i 1,j 1 0 x,Yii(t)y eCdF(x)G(y)Eb+3i(t),eCdd(2.4);(2)?i x,Yjj(t)y F(x)G(y)Enb1i(t)b4j(t)o;(3)?i j,P Xii(t)x,Yjj(t)y F(x)G(y)Enb2j(t)b5i(t)o.yPi(sk,sm;t)=e(1+k1+sk+m1+sm+i)I1+sk+sm+it,1 k m i(4.16)i(sk;t)=e(1+k1+sk+i)I1+k1+sk+it,1 k i.(4.17)1544n?Vol.43 Ad(4.15)deH(w)=(1 3(v)/b3)dH(w),(4.18)?,?n 0 l ,kEnln(en;t)o=EnEhln(en;t)|enio=Zt0E?el(1+u)?deH(u)2Zt0E?el(1+u)?dH(u)=2E?ln(t)?(4.19)Enln(ek,en;t)o=EnEhln(ek,en;t)|ek,enio 4E?ln(t)?x,Yii(t)y eCdF(x)G(y)Eb+3i(t).5?(Xi,Yi,i)l Sarmanov,=b?2.1,u,P Xii(t)x,Yii(t)y-?P Xii(t)x,Yii(t)y=E P(Xii(t)x,Yii(t)y|1,i1).n 4.2,(4.15),(4.19),(4.7),(4.11)(4.12)?P Xii(t)x,Yii(t)y=f0P Xii(t)x,Yii(t)y (f12+f13)PneXii(t)x,Yii(t)yo(f12+f23)PnXii(t)x,eYii(t)yo(f13+f23)PnXii(ei;t)x,Yii(ei;t)yo+f12PneXii(t)x,eYii(t)yo+f13PneXii(ei;t)x,Yii(ei;t)yo+f23PnXii(ei;t)x,eYii(ei;t)yof0F(x)G(y)E+i(t)(f12+f13)(1 d1/b1)F(x)G(y)E+i(t)(f12+f23)(1 d2/b2)F(x)G(y)E+i(t)(f13+f23)F(x)G(y)Eh+i(ei;t)i+f12(1 d1/b1)(1 d2/b2)F(x)G(y)E+i(t)+f13(1 d1/b1)F(x)G(y)Eh+i(ei;t)i+f23(1 d2/b2)F(x)G(y)Eh+i(ei;t)i=eCdF(x)G(y)Zt033(u)Ehe(+)(i1+u)Ii1+utiH(du)=eCdF(x)G(y)Eb+3i(t),(4.21)i(;t)d(4.17).i x,Yjj(t)y.du(Xi,i)l?Sarmanov,=(2.7),?3n 4.2?12=23=0?P Xii(t)x,Yjj(t)y=(1+13b1b3)P?Xii(t)x,Yjj(j;t)y?13b1b3PneXii(t)x,Yjj(j;t)yo 13b1b3PnXii(ei;t)x,Yjj(ei,j;t)yo+13b1b3PneXii(ei;t)x,Yjj(ei,j;t)yoNo.5?:m?/e?6x?.?V1545:=(1+13b1b3)L1 13b1b3L2 13b1b3L3+13b1b3L4,(4.22),j(;t)j(,;t)Od(4.17)(4.16).L1,du(Yj,j)l?Sarmanov,=(2.8),?3n 4.2?12=13=0?L1=(1+23b2b3)P?Xii(t)x,Yjj(t)y?23b2b3PnXii(t)x,eYjj(t)yo 23b2b3PnXii(t)x,Yjj(ej;t)yo+23b2b3PnXii(t)x,eYj(ej;t)yo,(4.23)aqu(4.21)?L1F(x)G(y)Eni(t)b2j(t)o.d(4.20)9aqu L1?L3F(x)G(y)Eni(ei;t)j(ei,b2;t)o,b2d(2.12).aq/,L2?1 d1b1?F(x)G(y)Eni(t)b2j(t)o L4?1 d1b1?F(x)G(y)Eni(ei;t)j(ei,b2;t)o.L1,L2,L3 L4“(4.22),(2.10)9(2.12)(2.14)?i x,Yjj(t)y F(x)G(y)Enb1i(t)b4j(t)o.?i j,aqu i x,Yjj(t)y F(x)G(y)Enb2j(t)b5i(t)o.n?y.n 4.43n 2.1?e,?N 0 x,NXj=1Yjj(t)yNXi=1NXj=1P Xii(t)x,Yjj(t)y.ykyV PnNPi=1Xii(t)x,NPj=1Yjj(t)yo?C.?0 v+2 ,?0 1?(1 )(+)(1 )x!Nm=1Ymm(t)(1 )y!,KPNXi=1Xii(t)x,NXj=1Yjj(t)y=PNXi=1Xii(t)x,NXj=1Yjj(t)y,E EcNXi=1NXj=1P Xii(t)(1 )x,Yjj(t)(1 )y+NXi=1NXj=1X1mN,m6=iP Xii(t)x/N,Yjj(t)y/N,Xmm(t)x/(N 1)+NXi=1NXj=1X1mN,m6=jP Xii(t)x/N,Yjj(t)y/N,Ymm(t)y/(N 1):=K1+K2+K3.(4.24)K1,dn 4.3,F R9 G Rv?x y kK1=NXi=j=1+N1Xj=1NXi=j+1+N1Xi=1NXj=i+1P Xii(t)(1 )x,Yjj(t)(1 )y1546n?Vol.43 A(1 )(+)F(x)G(y)eCdNXi=1E?b+3i(t)?+N1Xj=1NXi=j+1E?b2j(t)b5i(t)?+N1Xi=1NXj=i+1E?b1i(t)b4j(t)?(1 )(+)NXi=1NXj=1P Xii(t)x,Yjj(t)y(1+)NXi=1NXj=1P Xii(t)x,Yjj(t)yNXi=1NXj=1P Xii(t)x,Yjj(t)y+CF(x)G(y).(4.25)K2,kK2=X1mi=jN+X1i=jmN+X1ij=mN+X1j=miN+X1mijN+X1ijmN+X1mjiN+X1jmiN+X1jimN+X1im x/N,Yjj(t)y/N,Xmm(t)x/(N 1):=K20+K21+K22+K29.(Xm,m)l?Sarmanov (2.7),?3n 4.2?12=23=0 kK20(1+13b1b3)N1Xm=1NXi=m+1P Xii(i;t)x/N,Yii(i;t)y/N,Xmm(t)x/(N 1)+13b1b3N1Xm=1NXi=m+1PnXii(em,i;t)x/N,Yii(em,i;t)y/N,eXmm(em;t)x/(N 1)o:=(1+13b1b3)K201+13b1b3K202,i(,;t)m(;t)Od(4.16)(4.17).3 K201,5?(Xi,Yi,i)lSarmanov,=b?2.1,Kdn 4.2,(4.15),(4.19)(4.8)?,v?x y kK201N1Xm=1NXi=m+1?f0P Xii(t)x/N,Yii(t)y/N,Xmm(t)x/(N 1)+f12PneXii(t)x/N,eYii(t)y/N,Xmm(t)x/(N 1)o+f13PneXii(ei;t)x/N,Yii(ei;t)y/N,Xmm(t)x/(N 1)o+f23PnXii(ei;t)x/N,eYii(ei;t)y/N,Xmm(t)x/(N 1)oo CF(x)G(y).d(4.11),(4.19)(4.20),aqu K201 CF(x)G(y)?y?,v?x y k K202 CF(x)G(y).?v?x y,K20 CF(x)G(y).aq/,v?No.5?:m?/e?6x?.?V1547?x y,?k=1,2,9,K2k CF(x)G(y).K2k CF(x)G(y),k=1,2,9,“K2?K2 CF(x)G(y),v?xy.?/,K3 CF(x)G(y)v?x y.gyV PnNPi=1Xii(t)x,NPj=1Yjj(t)yo?Ce.u,P?NXi=1Xii(t)x,NXj=1Yjj(t)y?P?Ni=1(Xii(t)x),Nj=1(Yjj(t)y)?NXi=1NXj=1P Xii(t)x,Yjj(t)yNXi=1NXj=1X1mNi6=mP Xii(t)x,Yjj(t)y,Xmm(t)xNXi=1NXj=1X1mNj6=mP Xii(t)x,Yjj(t)y,Ymm(t)y:=K01 K02 K03.(4.26)aqu K2 CF(x)G(y)?y,k K02 CF(x)G(y)K03 CF(x)G(y)v?x y.(4.24)(4.26)=?(J.n 4.4?y.0 r 1,0 l ,0 t T,d H older?(4.14)?Xn=1nq?Eln(t)?r?Xn=1n(q+1)/rEln(t)?r?Xn=11/n1/(1r)?1r 0+2 ,d H older?,Cr?(4.27),Xi=1Xj=1iqjqEn?i(t)+i(t)?j(t)+j(t)oXi=1Xj=1iqjq?E?i(t)+i(t)?+2+?+2?E?hj(t)+j(t)i+2+?+2 CXi=1Xj=1iqjq(?E?+(+2)i(t)?+2+?Eh+2i(t)i?+2)(?E?+(+2)j(t)?+2+?Eh+2j(t)i?+2),q 0.(4.28)aq/,Xi=1Xj=1iqjqEn?i(t)+i(t)?+j(t)+i(t)hj(t)+j(t)io.(4.29)1548n?Vol.43 An 4.53n 2.1?e,?0 x,Pj=1Yjj(t)y?F(x)G(y)=0,(4.30)limNlimsup(x,y)(,)P?Pi=1Xii(t)x,Pj=NYjj(t)y?F(x)G(y)=0.(4.31)yky(4.30).?M vPi=11i2 x,Xj=1Yjj(t)y?Xi=NXj=1P?Xii(t)x/(i2M),Yjj(t)y/(j2M)?=?XNij?P?Xii(t)x/(i2M),Yjj(t)y/(j2M)?:=L01+L02+L03.L01,(4.22)(4.23)J?-?L01.N5,(Xi,i)l?Sarmanov,=(2.7),?3n 4.2?12=23=0,d L01oL01k,k=1,2,3,4.q(Yj,j)l?Sarmanov,=(2.8),?3n 4.2?12=13=0,d L011o L011k,k=1,2,3,4.?e5(4.2),(4.3),(4.28),(4.29)(4.27)?L0111=XNi x/(i2M),Yjj(t)y/(j2M)?=XNij Zx/(i2MDF)0Zy/(j2MDG)0+Zx/(i2MDF)0Zy/(j2MDG)+Zx/(i2MDF)Zy/(j2MDG)0+Zx/(i2MDF)Zy/(j2MDG)!F?x/(i2Mu)?G?y/(j2Mv)?P?i(t)du,j(t)dv?CF(x)G(y)XNiji2(+)j2(+)Z0Z0(u u+)