支付破产时刻赤字的常利率经典模型的最优分红问题.pdf

下载

下载

8积分

10积分

下载

10积分

VIP价：8积分

河北工业大学硕士学位论文支付破产时刻赤字的常利率经典模型的最优分红问题姓名：闫文娟申请学位级别：硕士专业：应用数学指导教师：刘国欣2010-12 à ó’ŒÆa¬Æ Ø©|G» ž»iff~|Dz;fi.ff`©ù¯KÁ‡fl©ïÄff´|G» ž»iff~|Dz;fi.ff`©ù¯K"3fl©¥§·‚Äkđ åff©ùüѧÙ̇gŽ5 YingFang&RongWu(2007)"ÏLÄ 5y n§·‚ `üÑAT÷vffHJB§"¿ÏL?˜Úffy² `üÑ´˜‡ .üÑ"¿ ¢ ê©Ùž`üÑ÷vff^‡"3ؑ åff©ùüќ¹e§aqGerber,Shiu&Smith£2006¤ff{§·‚?Ø |G» ž»ié`©ùüÑffK"ùÙ¥§Ü6‡Aې§å ­‡ffŁ^"'…c:~|Dz;fi.§|G» ž»i§`©ù§Ä 5y n§Ü6‡Aې§i |G» ž»iff~|Dz;fi.ff`©ù¯,first,weconsidertherestricteddividendstrategy,whichfromYingFangandRongWu’smainideas(2007).Throughthedynamicprogrammingprinciple,,,resemblingGerberShiuSmith’smethod(2006),wediscussedthecoverthedeficitruinhowtoaffecttheoptimaldividendstrategy,especially,:theclassicalmodelwithconstantinterestrate,coverthedeficitruin,optimaldividend,thedynamicprogrammingprinciple,confluenthypergeo-metricequationii |G» ž»iff~|Dz;fi.ff`©ù¯KÎÒ`²(Ω,F,{Ft}t≥0,P)VǘmxЩO 7c ¤Â\Çr~Ł|Çpi©ùüÑN(t) –tžff¢ gêYizgff¢ Œ S(t)\O¢ δòyÇTpi©ùüя žff» žZpiT©ùüя ž» ž|Gff»ipiЌ#NüÑ8dpi©ùüÑpietžff©ùÇt၌©ùÇV(x)Ł¼êVpi(x)8I¼êpi∗k©ùüÑb∗`©ùüÑéAff©ù>.iv à ó’ŒÆa¬Æ Ø©1˜ÙÚó°ŽêÆ´ gé x!7K!Ý]†ãÖ +ffºx?1ýß!©Û!µ Ú+nff˜€Æ‰"3°ŽêÆù‡ ŒffA^êÆNX¥§ºxnØ洃 Ú<'5ff­‡|¤Ü©ƒ˜"3éºxnØz{cffïÄ{§¥§c<®²·‚È\ ´Lff² Ú´affïĤJ"ÐffºxnØ̇´ÏLïÄ xúiff» VÇ!» ž!» »i °Žþ5©Ûúi¡ ffºx" Ł˜«²L¢N§ xúiff À3'%úi$Effºx¯KffÓžÓ '%úiŒ±gC‘5õŒffÂç‘Xéù˜¯K'5fffl\§ xúiff`©ù¯K3 ›­VeŒ“r\ ºxïÄöffÀ" xúi`©ù¯Kffïȱˆ 1957cff1›Ê3IS°Ž¬Æ"3ù‡¬Æþ§DeFinetteBrunoJÑúiAÏé» cÏ"òy©ùffŒŁ±9ƒAff©ùüÑ"¦y² 3b xúiffJ{+1Ú-1üÚff{üfflўmff‘ÅiÄfffi.e§`©ùüÑ´˜‡>.üÑ"=§‡L,˜.ffJ{fiÜŁ©ù©‰úiff À"¦Óžy² `©ùff>.´XÛ (½ff"3dƒ §Ïé`©ùüÑff?Øòff ëYžmffºxfi."Bu¨hlmann(1970)±9Gerber(1969)?Ø ²;fi.ff`©ù¯K"‘X‘Å››nØ A^ ºxn؁`©ù¯KffïÄ"Assumsen&Taksar(1997)ÏLHJB§ïÄ ‘¤£ÙK$Äfi.ff`©ù¯K"y² ©ùÇk~êþ.ž§`©ùüÑ´˜‡ .üѧ= J{‡L,˜>.ž§AT±˜‡Œ± ÉffŒ©ùÇ?1©ù" ©ùÇvk›ž§`©ùüя˜‡>.üÑ"Albrecher&Thonhauser(2008)ïÄ ~|Dz;fi.ff`©ù¯K"Cai,j,Gerber&Yang(2006)ïÄffK´‘/±|ÇffO-Ufi."Avrametal.(2007)ÚLocffer(2008)ïÄfffi.ÊKffLe´vyL§"‘XïÄfffl\§ïÄö‚3ïÄ©ù¯Kž\\ õff›Ú^‡§¦ƒ3bCy¢ffӞA^ 2ff+"Gerber&Shiu(2006)ïÄ ²;fi.©ùÇk›ffœ¹e§y² ¢ Ñlê©Ùž§`©ùüÑ´˜‡ .üÑ"Azcue&Muler£2005¤Ä `©ù9`2 x¯Keff²;fi."L kka&zervos£2007¤?Ø ‘'~ ´¤^ff`©ù¯K"Gerber,Shiu&Smith£2006¤?Ø |G» ž»iff²;fi.ff©ù¯K"Kulenko&Schmidli£2007¤?Ø ‘5]ff²;fi.ff`©ù¯K"nܱþff0 ·‚Œ±w §©ù¯KffïÄ3ØÓfffi.cJeØäffªu õ§bCy¢§Óž5­[!§kud§fl©ïÄfffi.´‘~|Çff²;fi."ù˜fi.´²;fi.ffí2§Äff´žXòúiJ{Ý] ÃX;O½Å ƒaffúxÝ]effJ{L§"dDickson&Waters£2004¤ff*:Ñu§·‚b½ úiOy» ž§ 1 |G» ž»iff~|Dz;fi.ff`©ù¯KÀ¬á=±gk]7|Gúiff» »i§±dŁgfi ÀÂÖff ("ƒé©ù¯K5­ À |ffӞ§·‚±|G» ž»iù˜1rN Àéúi±9Ý <ffÂÖ"ddÚuÑ ˜‡#ff¯Kµ3Ä|G» ž»iœ¹eff`©ù¯K"fl©ff( 9̇SNSüXeµ1 ٧̇0 fl©fffi.b "1nÙ§?Ø|G» ž»iff~|Dz;fi.3©ùÇk›ffœ¹eff`©ù¯K"ÏLÄ 5y n HJB§"dHJB§9Ł¼ê5Ÿ E˜©ùüѧ¿y²düя`§¿é ¢ Ñlê©ùžff`üÑ9Ł¼ê÷vff^‡"1oÙ§?Ø|G» ž»iff~|Dz;fi.3©ùÇvk›ffœ¹eff`©ù¯K"ÄuGerber,Shiu&Smith£2006¤ffgŽ§·‚l,˜‡ Ýé©ùüя>.üÑff|G» ž»iff~|Dz;fi.ff©ù¯K?1?Ø"¿3¢ ÑlAύê©Ùž§* |G» ž»ié©ùüÑffffK"2 à ó’ŒÆa¬Æ Ø©1 Ùºxfi. ‘6ffVǘm(Ω,F,{Ft}t≥0,P)Ę‡Ð©O 7xff xúiffJ{L§§3ØÄ©ùffœ¹e§tžffJ{∫∫ttXtt=xer+cer(t−s)ds−er(t−s)dS(s)00Ù¥∑N(t)S(t)=Yii=1ùp{Xt}´'u{Ft}t≥0·Aff"±c>0½Â ¤Â\ǧ±r≥0½Â˜~Ł|Ç"±S(t)½Â –tžff\O¢ "{s(t),t≥0}´˜‡EÜÑtL§"±N(t)½Â –tžff¢ gê§{N(t),t≥0}´˜‡ëêλ§λ>0ffÑtL§"±Yi½Âzg¢ ff¢ Œ §{Yi,i=1,2,3...}´˜ ÕáÓ©Ùff‘ÅCþ§Ù©Ù¼êP(x),P(0)=0"VǗݼêp(x),E[Yi]=µ<∞"…{Yi}†{N(t)}´ƒpÕáff"y3Ä xúi±©ùüÑpiéúiJ{?1©ù"±DpiL«üÑpietžff\È©tù§b ©ùL§´†ëm4ff"KüÑpietžffJ{L§µ∫∫ttXpi=Xer(t−s)dDpit−s)s=xert+er(tt−(cdS−dS(s)−dDpis)00±TpiL«©ùüяpižff» žµTpi=inf{t≥0;Xpi<0}t±ZpiTL« À3úi\w» ž|Gff» ž»iµpiZpipiT+XpiT=0piK8I¼êµ∫T∫piTpiVpi(x)=E[e−δtdDpi−e−δTpi|XpitT||Xpipi0=x]=E[e−δtdDpi−e−δTpiZpi|XpitTpi0=x]00dªL«ff´3ЩO 7x,©ùüяpiffœ¹e§\È » žffòy©ù†» ž»iòyŁƒ ffêÆÏ""= À|G » ž»iƒ ffÀÂÃyŁffÏ""Ù¥òyÇδ>0§b δ>r§… » u) § ÀØ2©ù"3 |G» ž»iff~|Dz;fi.ff`©ù¯KéAffŁ¼êµV(x)=supVpi(x)pi∈ΠÙ¥Π´3ØÓ^‡e¤kŒ#NüÑff8ܧ=Œ#NüÑ8"eV(x)Œ± §±pi∗L«éAff`©ùüѧkµV(x)=Vpi∗(x)4 à ó’ŒÆa¬Æ Ø©1nّ åff©ù¯K3fl!¥§Ä xúi±˜‡‘ åff©ùÇ?1©ù"±dpiL«©ùüÑpietžtff©ùÇ" 0≤dpi≤ᧁŒ©ùÇα÷v0<α<c"tƒAff§·‚k∫TpiVpi(x)=E[e−δtdpidt−e−δTpiZpitT|Xpipi0=x]0V(x)=Vpi∗(x)§3-1Ł¼ê5Ÿ5Ÿµ¼êV(x)3(0,∞)þ´üNO\ff"… x→∞ž,V(x)→α"δy²µ-piε´éuЩO 7xff˜‡©ùüѧ÷vVpiε(x)≥V(x)−ε"-{dpiε}L«©tùÇ"双u0ff?¿¢ê"éuЩO 7y,y>x½Â˜‡#ff©ùüÑp˜i§-p˜i÷vXe^‡µdpiε,t≤Ttpiεdp˜i=tα,t>TpiεKXp˜i>Xpiεét≤Tttpiε"K∫Tp˜iVt(y)≥Vp˜i(y)=E[e−δdp˜idδTt−e−p˜iZp˜itT|Xp˜ip˜i0=y]0∫Tpiε≥E[e−δtdpiεdt−e−δTpiεZpiε|Xpiε0=x]≥V(x)−εtTpi0εdεff?¿5§ŒV(y)>V(x)"¤±V(x)O¼ê"y3ÄüÑpˆiµdpˆi=α"tKpˆiüÑeff» žµ∫∫ttTtpˆi=inf{t≥0:xer+(c−α)ds−er(t−s)dS(s)<0}00 x→∞žµTpˆi→∞"Kd››Âñ½n§·‚kµlimx→∞E[e−δTpˆi|Xpˆi0=x]=05 |G» ž»iff~|Dz;fi.ff`©ù¯K¤±limx→∞E[e−δTpˆiZpˆiT|Xpˆipˆi0=x]=0K§ x→∞žµ∫Tpˆilimtαdt−e−δTpˆiZpˆiT|Xpˆix→∞Vpˆi(x)=l→imx∞E[e−δpˆi0=x]0∫Tpˆi=l→impˆix∞E[e−δtαdt|X0=x]−lim0x→∞E[e−δTpˆiZpˆiT|Xpˆipˆi0=x]∫Tpˆi=lim[e−δαtαdt|Xpˆi0=x]=x→∞E0δKk∫∞αVpˆi≤V(x)≤E−δt[eαdt|Xpˆi0=x]=0δαlimx→∞V(x)=δ5Ÿµ¼êV(x)´oÊÜ[ëYff"y²µÐ©O 7u§u≥x>0-piu={du}uff?¿Œ#NüÑ"-h=1lnc+rutrc+L«rxJ{L§3Øu)¢ ffœ¹efflx uffžm"-TiL«1ig¢ u)ffžm§uiL«éAff1igff¢ "éxÄüÑpiµ0,t≤h½T=dpi1≤htdpi∧uT1t>ht−h,K©ùu)ffVǏҴ xúiffJ{3Øu)¢ ffœ¹e ˆuffVÇ"dVǏe−λh§Kµ∫TpiV(x)≥Vpi(x)=E[e−δtdpidt−e−δTpiZpiT|Xpipi0=x]t0∫Tpi=P(T1>h)E[e−δtdpidt−e−δTpiZpiT|Ttpi1>h,Xpi0=x]0∫Tpi+P(T1≤h)E[e−δtdpidt−e−δTpiZpiT|Tpi1≤h,Xpi0=x]t0=e−λhe−δhVpiu(u)−(1−e−λh)E[e−δTpiZpipiT|Xpi0=x]∑∞≥e−λhe−δhVpiu(u)−(1−e−λhT)E[e−δixi]i=1=e−(λ+δλµ)hVpiu(u)−(1−e−λh)δ6 à ó’ŒÆa¬Æ Ø© Hpi¥ffŒ§·‚kµV(x)≥e−λµ(λ+δ)hV(u)−(1−e−λh)δqϏ1−e−(λ+δλ+δ)h≤(λ+δ)h≤(u−x)cλ1−e−(λ+δ)h≤λh≤(u−x)cŒ µ0≤V(u)−V(x)≤(1−e−(λ+δλµ)h)V(u)+(1−e−λh)δ(λ+δ)α+λ2µ≤(u−x)c䤱V(x)´oÊÜ[ëYff"§3-2HJB§9`üÑdÄ 5y n§·‚ V(x)÷vffHJB§µ∫∞max′{(c−d+rx)V(x)+d−(λ+δ)V(x)+λV(x−y)p(y)dy}=0()0≤d≤α0Ù¥∫∞∫x∫∞V(x−y)p(y)dy=V(x−y)p(y)dy−(y−x)p(y)dy00xLˆª()ff†>´Œzd(1−′V(X))"du()´‚5ff§K-()ŒzffdŒ± ¤µ′0,V(x)>1′d(x)=∈[0,α],V(x)=1′α,V(x)<1b V(X)´]ff§Kdv(X)ff5ŸŒ§3′b=inf{x:V(x)≤1}kµ0,x<bd(x)=()α,x≥b7 |G» ž»iff~|Dz;fi.ff`©ù¯K=′0,V(x)>1d(x)=′α,V(x)≤1dup(x)´ëYff§Kd()·‚Œ±w V(x)3(0,b)9(b,∞)´ëYŒ‡ff"y3b b>0§-x=b§dHJB§Œ± µ′′(c+rx)V(b−)=α+(c+rx−α)V(b+)=′′′(c+rx)(V(b−)−V(b+))=α(1−V(b+))()3c¡·‚®²b 0<α<c" d()·‚Œ± µ′′V(b−)=V(b+)=1 ′V(x)3(0,∞)þëYŒ‡"½µd()‰½ffüÑpi∗={d∗=d(x)}´¤kŒ#NüÑ¥ff`üÑ"ty²µÄk§pi∗´Œ#NüÑ"-Vpi∗(x)L«ƒAffŁ¼ê"é∧e−δ(tTpi∗)V(Xpi∧∗tT)Api∗^Itoˆúª§·‚kµ∧∫∧tTpi∗e−δ(tTpi∗)V(Xpi∧∗tT)=V(x)+e−δs′[(c−dpi∗s+rXpi∗s−)V(Xpi∗s−)−δV(Xpi∗dspi∗s−)]0∑∗+e−δs[V(Xpi∧s−V(Xpi∗s−))]()∗0<s≤tTpi∗,Xpi∗s−,Xpisy3Ä|G» ž»iffœ¹"-TiL«1ig¢ u)ffžm§YiL«éAff1igff¢ "XJXpi∗T=Xpi∗−YiTi−i≥0§úiج» § Àff|GÂ֏∗0"eXpi∗T=XpiiT−Yi<i−0§KTi=Tpi§úi» § ÀI‡|G» ž»iµZpi∗T=−pi∗(XpiT−Yi)=−(Xpi∗T)i−i=z ¢ u)ž§ ÀA|GµZpi∗T=−inf(Xpi∗iT,0)i¤±V(XT)=V(XiTi−−Yi)+Zpi∗Ti8 à ó’ŒÆa¬Æ Ø©duXpi∗s−,Xpi∗s3¢ u)ž¤á§Kµ∑e−δs[V(Xpi∗s−V(Xpi∗∧s−))]0<s≤tTpi∗,Xpi∗s−,Xpi∗s∑N(t)=e−δTi[V(Xpi∗T−Ypi∗i+Zpi∗T)−V(XiT)]i−i−i=1∑N(t)∑N(t)∗∗=e−δTi[V(XpiT−Yi−i)−V(XpiT)]+e−δTiZpi∗i−Tii=1i=1∑N(t)=e−δTi[V(Xpi∗T−Yi−i)−V−δT(Xpi∗T)]+epi∗Zpi∗i−Tpi∗i=13()ªüà Ï"§·‚kµ∧∫∧tTpi∗′Eδ(tT)[e−pi∗V(Xpi∧∗∗tT)]=V(x)+Ee−δs[(c−dpi∗s+rXpi∗pi∗s−)V(Xpis−)−δV(Xpi∗s−)]ds0∑N(t)∗+e−δTpi∗ZpiT+Ee−δTi[V(Xpi∗−Ypi∗Ti−i)−V(Xpi∗T)]i−i=15¿ ∑N(t)∫∧∫tTpi∗∗∗∞Ee−δTi[V(XpiT−Yi−i)−V(XpiT)]−λe−δs[V(Xpi∗∗s−Y)−V(Xpii−s)]p(y)dydsi−i−00i=1´˜‡Ï"0ff "¤±§·‚Œ± µ∧∫∧tTpi∗E(tT[e−δpi∗)V(Xpi∧∗e−δs[(c−dpi∗s+rXpi∗′s−)V(Xpi∗s−)−(λ+δ)V(Xpi∗tT)]=V(x)+Epi∗s−)0∫∞+λV(Xpi∗Tpi∗s−−y)p(y)dy]ds+e−δZpi∗Tpi∗0dHJB§()§·‚kµ∫∞′(c−dpi∗s+rXpi∗s−)V(Xpi∗s−)−(λ+δ)V(Xpi∗s−)+λV(Xpi∗s−−y)p(y)dy+dpi∗s=00¤±∧∫∧tTpi∗Eδ(tT)[e−pi∗V(Xpi∧∗sdpi∗sds−e−δTpi∗Zpi∗tT)]+E[e−δpi∗T]=V(x)pi∗0-t→∞§dV(x)ff§·‚Œ± µ∫Tpi∗Vpi∗(x)=Es[e−δdpi∗pi∗sdst−e−δTZpi∗T|Xpi∗pi0=x]=V(x)0¤y¤á"9 |G» ž»iff~|Dz;fi.ff`©ù¯K§3-3ê©Ù¢ ff .üÑ©ù¯KÄê¢ ©Ù§=p(x)=βe−βx§3`üÑ()eff©ù"dHJB§()§·‚kµ∫∞′(c+rx)Vb(x)−(λ+δ)Vb(x)+λVb(x−y)p(y)dy=00<x<b()0∫∞′(c−α+rx)Vb(x)−(λ+δ)Vb(x)+λVb(x−y)p(y)dy+α=0x≥b()0A^d(d+β) (),()§·‚Œ±kµx′′′(c+rx)Vb(x)+(β(c+rx)+r−λ−δ)Vb(x)−βδVb(x)=00<x<b()′′′(c−α+rx)Vb(x)+(β(c−α+rx)+r−λ−δ)Vb(x)−βδVb(x)+βα=0x≥b()K()ff)µλ+δ+δcVb(x)=A1(x+cr)re−βxMδλ(+1,+1,β(x+))+rrrcλ+δB1(x+−βx)reUδλ+δc(+1,+1,β(x+))0<x<b()rrrr5¿ α´(),()ff˜‡A)§…¤k©ùüÑVb(x)ج‡Lα"δδK()ff)µc−αλ+δVb(x)=Ax2(x+)re−βUδλ+δc−αα(+1,+1,β(x+))+x≥b()rrrrδÙ¥M(a,b,z)ÚU(a,b,z)©O´1˜aÚ1 aÜ6‡Aې§"Maz(a)2z2(a)zn(a,b,z)=1+++...+n+...b(b)22!(b)nn!UpiM(a,b,z)z1−bM(1+a−b,2−b,z)(a,b,z)={−}sinpibΓ(1+a−b)Γ(b)Γ(a)Γ(2−b)(a)n=a(a+1)(a+2)...(a+n−1),(a)0=1y3I‡(½ëêA1,A2,B1§d·‚I‡é n‡'uA1,A2,B1ff§µdƒcff?Ø·‚Œ± µ′cVb(0+)−(λ+δ)Vb(0+)=0=·‚kµλλ+δβcA1Mδ(+1,+2,)+B1Uδλ+δβc(+1,+2,)=0()λ+δ+rrrrrrr10 à ó’ŒÆa¬Æ Ø©d^‡Vb(b−)=Vb(b+)§·‚kµc−αλ+δλ+δc−αα(b+)rA2Uδ(+1,+1,β(b+))+eβb=rrrrδcλ+δλ+δcδλ+δc(b+)r[A1Mδ(+1,+1,β(b+))+B1U(+1,+1,β(b+))]()rrrrrrréu ˜‡^‡§5¿ µ∫∞∫x∫∞Vb(x−y)p(y)dy=Vb(x−y)p(y)dy−(y−x)p(y)dy00x∫x∫∞=Vb(y)p(x−y)dy−(y−x)p(y)dy0x∫b∫x∫∞=Vb(y)p(x−y)dy+Vb(y)p(x−y)dy−(y−x)p(y)dy0bxKò()9()‘\()§·‚kµ∫x∫c−αλ+δc−αbλA2(y+)rUδλ+δ(+1,+1,β(y+))dy+λVb(y)eβydybrrrr0∫c−αλ+δ−A+12r(x+)rUδλ+δc−αλα∞βx(+1,+2,β(y+))−e−β(y−x)p(y)dy=0rrrrβδx-x↓b,·‚  ˜‡^‡µ∫bcλ+δλ+δcλ(y+)rMδ(+1,+1,β(y+))dyA10rrrr∫bcλ+δλ+δc+λ(y+)rUδ(+1,+1,β(y+))dyB10rrrrc−αλ+δ−r(b++1δλ+δc−α)rU(+1,+2,β(b+))A2rrrr∫λα∞−eβb−β(y−b)e−βbdy=0()ρδbd(),()9()§·‚Œ± A1,A2,B1ffLˆª"Ӟ§2d′V(b∗)=1§·‚Œ± `©ùüÑeéAffb∗ffêŁ)"11 |G» ž»iff~|Dz;fi.ff`©ù¯K1oÙà åff©ù¯Kb ©ùüѱ>.üÑ?1" ©ùüÑpiéAff©ù>.b,­:?Ø>.üÑ©ùœ¹eff|G» ž»ié©ùüÑffK"§4-1Vpi(x)ff5Ÿ∫T∫piTpiVpi(x)=E[e−δtdDpi−e−δTpipi|XtT||Xpipi0=x]=E−δt[edDpi−e−δTpiZpiT|Xpipi0=x]t00∫Tpi=E[e−δtdDpi|Xpit0=x]−E[e−δTpi|XpiT||Xpipi0=x]0V(x)=supVpi(x)pi∈Π-∫TpiWpi(x)=Wb(x)=E[e−δtdDpipi|Xt0=x]0Rpi(x)=Rb(x)=E[e−δTpi|XpiT||Xpipi0=x]=E[e−δTpiZpipiT|Xpi0=x]KVpi(x)=Wpi(x)−Rpi(x)=Wb(x)−Rb(x)dFangYing(2009)Œ§‰xff¼ê§Wpi(x)÷vµ∫x(c+rx)W′b(x)−(λ+δ)Wb(x)+λWb(x−y)p(y)dy=00<x≤b()0…k>.^‡µW′b(b−)=W′b(b+)=1=W′pi(b)=W′b(b)()Чµ∫x′(c+rx)h(x)−(λ+δ)h(x)+λh(x−y)p(y)dy=0()0 h(x)§ffš²…)§KµWh(x)pi(x)=Wb(x)=0<x≤b()′h(b)12 à ó’ŒÆa¬Æ Ø©ddŒ§ h(x)  Łž§Wpi(x)ˆ Œ"-b0=sup{b≥′0:h(b)=min′y≥0h(y)}§Kb0`üÑ" džéAff\È©ùWb0(x)§KµWh(x)b0(x)=(0<x≤b0)′h(b0)W′b0(b0)=1 b0>0§KW′′b0(b0)=0 e5©ÛRb(x)"Rb(x)=Rpi(x)=E[e−δTpi|XpiT||Xpipi0=x]=E[e−δTpiZpiT|Xpipi0=x]5Ÿ:Rb(x)÷vÈ©)‡©§µ∫x∫∞′(c+rx)Rb(x)−(λ+δ)Rb(x)+λRb(x−y)p(y)dy+λ(y−x)p(y)dy=00<x≤b()0x…k>.^‡µ′Rb(b)=0()y²µ-∫τ0<x<b,τ>0§…uerτ+certdt<0"Ä[0,τ]S1˜g¢ u)ffžm09êþ±9¢ ´Ä —» "·‚kµ∫τ∫∫ττ∫xert+certd∫tτ0Rb(x)=e−(δ+λ)τRrtb(xe+certdt)+λe−(δ+λ)tRb(uert+certdt−y)p(y)dydt0000∫τ∫∞∫τ+λe−(δ+λ)τ∫(y−uert−certdt)p(y)dydt()0uert+τcertdt00ò()'uτ¦ §¿-τ=0"·‚kµ∫x∫∞′(c+rx)Rb(x)−(λ+δ)Rb(x)+λRb(x−y)p(y)dy+λ(y−x)p(y)dy=0()0xÄx=b§…?¿ffτ>0"d()§·‚kµ∫τ∫b∫τ∫∞R)τb(b)=e−(δ+λR−(δ+λ)tb(b)+λeRb(b−y)p(y)dydt+λe−(δ+λ)t(y−b)p(y)dydt()000bò()'uτ¦ ¿-τ=0§·‚kµ∫b∫∞−(λ+δ)Rb(b)+λRb(b−y)p(y)dy+λ(y−b)p(y)dy=0()0bò() x=b¿†()' §·‚kµ′Rb(b)=0()13 |G» ž»iff~|Dz;fi.ff`©ù¯K5Ÿ:-R(x)L«ØÄ©ùžffúi» žff» ž»iòyÏ""=R(x)=E[e−δT|xT||X0=x]"K′R(b)Rb(x)=R(x)−h(x)0<x≤b′h(b)y²µ†()ƒq§·‚kµ∫x∫∞′(c+rx)R(x)−(λ+δ)R(x)+λR(x−z)p(z)dz+λ(y−x)p(y)dy=0()0xd()9()·‚ Rb(x)−R(x)´§()ff)§ŒPµRb(x)−R(x)=kh(x)Ù¥kŒd^‡()(½§²LOŽ·‚Œ± µ′R(b)Rb(x)=R(x)−h(x)0<x≤b()′h(b)²L±þff?Ø·‚Œ±w µh(x)′Vpi(x)=Vb(x)=Wb(x)−Rb(x)=(1+R(b))−R(x)0<x≤b()′h(b)d()9()§·‚kµ′Vb(b)=10<x≤bb `üÑ3§Pb∗§Kµ′′Vb∗(b∗)=0§4-2|G» ž»ié©ùüÑffKÄê¢ ©Ùp(y)=βe−βy"A^Žfd(d+β) ()§·‚kµx(c+rx)W′′b(x)+(β(c+rx)+r−λ−δ)W′b(x)−βδWb(x)=00<x≤b()A^Žfd(d+β) ()§·‚kµx′′′(c+rx)Rb(x)+(β(c+rx)+r−λ−δ)Rb(x)−βδRb(x)=00<x≤b()dVb(x)=Wb(x)−Rb(x)§Ó ff§·‚kµ′′′(c+rx)Vb(x)+(β(c+rx)+r−λ−δ)Vb(x)−βδVb(x)=00<x≤b()14 à ó’ŒÆa¬Æ Ø©-x=b0d()§·‚Œ± µWb0r+cb0(b0r−λ−δ)=+()δβδÓ ff§-x=b∗d()§·‚kµ∗r+cr−λ−δVb∗(b∗b)=+()δβδy3·‚5oÑ' ˜eb09b∗§-β=1§KµRb(x)=Lb(x)=E[e−δT|Xpi0=x]d5Ÿ()Œµ∂−∂′′′′(L∂bR(b))h(b)−∂(∂bh(b))R(b)b(x)=h(x)()∂b′(h(b))2ÏLOŽ·‚Œ± µh(b)=C1f1(b)+C2f2(b)cλ+δf1(b)=e−βb(b+)rMδλ+δc(+1,+1,β(b+))rrrrλ+δλ+δcf2(b)=e−βbc(b+)rUδ(+1,+1,β(b+))rrrr1r+δC1=D(βU0r+δ+2−βU1)λ1δC2=D(βM1+(1+)βM2)0rDδ+r)0=β(1+)U1Mβ(δ2+Mrr+δ+1U2λMcλ+δλ+δβck=()rMδ(+k,+k,)rrrrUcλ+δδλ+δβck=()rU(+k,+k,)rrrrλfR1(b)(b)=cβ(M1+(1+δ)Mr2)ò±þffLˆª‘\()§²LOŽ§·‚Œ± µ∂Lb(x)<0∂bK∂∂∂∂Vb(x)=Wb(x)−Lb(x)>Wb(x)∂b∂b∂b∂b15 |G» ž»iff~|Dz;fi.ff`©ù¯K b=b0ž§·‚kµ∂Wb(x)|∂bb=b0=0 b=b∗žµ∂Wb(x)|b=b∗<0∂bKdWb(x)ff5Ÿ·‚Œ± µb0<b∗nܱþff?Ø·‚Œ± µ ¢ ê¢ ©Ù…β=1žµb0<b∗?˜Ú§ β=1ž§d()9()·‚Œ± µWb0(b0)<Vb∗(b∗)=3Ä|G» ž»iffœ¹e§ƒAff`>.©ùüÑff©ù>.‡'ØÄ|G» ž»iœ¹eff©ù>.p§ù˜(Ø´ÎÜ·‚ff˜„@£ff" ЩO 7 uéAff`©ù>.ž§|G» žff»i ff Àff\ÈÀÂÃ'ØÄ|G» ž»iffÂÃp"I‡5¿ff´§ù˜(Ø´†úiffЩO 7ݘ7 ƒéXff"Remarkµ3Gerber,Shiu,Smith(2006)¥?Ø ²;fi.ffÓ ff¯K§ †fl©Ø˜ ff(Ø"3ØÄ~|Çffœ¹e§Wb0(b0)=Vb∗(b∗)"16 à ó’ŒÆa¬Æ Ø©ë©z[1]Abramowitz,M.;Irene,:withFormula,;UnitedStatesDepartmentofCommerce,-fice:Washington,.,1972.[2]Asmussen,;WorldScientific:Singapore,2000.[3]Asmussen,S.;Taksar,-ance:MathematicsandEconomics1997,20,1-15.[4]Bjhlmann,:Springer-Verlag;1970.[5]Cai,J.;Gerber,.;Hailiang,,10(2),94-119"[6]DeFinettiB.”Suun’impostazionealternativadellteoriacollettivadelrischio.”TransactionsoftheXVthInternationalCongressofActuaries,1957,2:433-443[7],Waters,.,,2004,34:49-74[8]Fleming,.;Soner,;Springer-Verlag:NewYork,1993.[9]Fang,YingandWu,Rong(2007)’OptimalDividendStrategyintheCompoundPoissonModelwithConstantInterest’,StochasticModels,23:1,149-166.[10]FangYing(2009)OptimalDividendProblemsfortheRiskModelwithConstantInterestRate[11]Gerber,.;Shiu,.;.[12]Gerber,.;Shiu,,10(2),76-93.[13]Gerber,.;Shiu,.;,.[14]Gerber,-graph8,,Homewood, |G» ž»iff~|Dz;fi.ff`©ù¯K[15]Gerber,HansU.,.”OnOptimalDividends:FromReflectiontoRefrac-tion,”JournalofComputationalandApplieMathematics,toappear.[16]Gerber,.;Shiu,:,8(1),1-20.[17](1979).-tion,Philadelphia.[18](2004).OnoptimaldividendstrategiesinthecompoundPoissonmodel.[19]Jeanblanc-PicquW,M.,.”OptimizationoftheFlowofDividends,”RussianMathematicalSurveys20:257-277.[20],GuojingWang,:MathematicsandEcon-cmics40(2007)104-112[21]Knuth,,Volume1:FundamentalAlgorithms,,MA:Addison-Wesley.[22]Kulenko,N.,Schmidli,H.,OptimaldividendstrategiesinaCrame´:MathematicsandEconomics(2007).[23]Paulsen,J.;Gjessing,:MathematicsandEconomice1997,20,215-22318 à ó’ŒÆa¬Æ Ø©—fi–dØ© ¤ƒS,•Q²‰ƒ·ÏÚ|±ff<‚L«©%ffafiÚª”ff¹¿œÄk§afi·ff “4I! ǜ˜†±5§4P“î>È¢ff£Æ Ýڰæ°ff£Æ° flfla/Ú-yX·"zgÚ4P“ff 6Ñ4·ÉÞf§zgÚ4P“ff 6Ñ´·?Úffå:" ´34P“aaØðff e§·ff;’Y²âk Jp§.’Ø©â ±^| ¤"4P“Ø=3ƒþ´·ff “§3óŁ ÝÚ<) ÝþÓ ´·ÆSff;‰"l4P“fiþÆ ff£Æƒ ڏ<ƒ §ò´·˜)ffãL"/dŬ§•4P“Lˆ·Âpff¹¿Ú©%ffafiœÙg§·‡afi¤kQ² L·ffP“Ú†·˜åóŁLffÓ;§\‚3·¤ffL§¥‰ ·éõff¡¡ffÏ§3·ªu ¤ ûff¦Æƒ´ƒž§/d•\‚¤k<L«afiœafi·ffӀÚQ²ÏL·ff*lÓƧvk\‚ffÏ§·ffa¬)ãò¬(Jéõ§3d©%ff•Œ[L«afiœafi·ff[<"afi¦‚·¤‰ff˜ƒœ„kéõ·Ã{˜˜ ¶ff“Úl<‰·ff ڐϧ3d©%ffL«afi§\‚ff¶i·ò˜†µP3%œ §©%afi3zaƒ¥Äўm"flflØ©ff;[ ǜA©ï "˜"c› ff19