Aspects on asset liability management via stochastic programming
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(35) tzkmghen U#ulac rstvw@ijacdyk ~ dyehtvdfk#ula b>nyehg o n xztnIwhalxacuWgtzn k ghau{rskst 6 U#ay]ghrsaetvw .iGadywOUseau{rsn whalk d wghr#a w"g{dfk##dfe{ al~tvd gtzn k nfogrsa5eacdyxzet dla ea Og U#ehk " ksnfgrsalejdfbsb#ehnmdyu{r ghn ijnalxghrsa etw dy~ ale{w a acr#d ~tzn enyo d5edfghtnyk#dyx]tk~yaw"gnye9tzkmghacye{d gtzks5ehtvw Idfk# ehac dye9 tvwghrs< a U#whaGnfo Ughtxtg"5o Usk#u ghtnyk#wcd wacwu etz>ac qn k ?a Usi(dfksk/dfk pInyeyalkw"gale k "}
(36) grsaleehtvw ijacdyOw Useacw >acwhta~ dfetvdfk#ulatk#u ex U#aw}~ dfex Usa d g etw whal a " "#p/n ehmdf k #u n k#tzghtnyk#dyx ~ dfex Usa d g etw ydyKw U#whacj k#aceww n k` p =d U#ww acecmn whalkdyk#(|e d w ac ~ mdyk#(dfw n mx Uga al~tvd gtzn k`h U#whac6m nyk#ksn.dyk# $ dyi(=d dcd t " #nyejdfxx ghrsaw aMbsenysxali(wcnyk#xz n ksa acultwhtnyk tvw i(dya dfk ghr#a b>nyehg o n xztn6tvw/u n k#whtaceha,ghn >a an ~ alegrsa acultwhtnyk rsnyeet dlnyk " k i U#xgt w"g{df a5b>nyehg o n xztn.b#ehn sxzaci(wlqgrsaMbn e gho nyxtzn tvw k#nfgu n k#w tvaceha ghn >a a n ~yace grsatzk~ acw ghijalkmgrsnyeet dlnyk:m@ Ugtzkw"gacdy5b>aletzntvuldyxzxeal#dyxdyk#u a " alk#ulaygrsanybsght i(dfxqacultwhtnykItvwksn5xnyk#yaled5bn e gho nyxtzn>@ Ug@dQw ghe{d ghacy sacwu etzstks5rsn ghr#a b>nyehg o n xztn w r#n Usxv adfxvdfk#ulac o nye@adyu{r bacehtn.tk ghtija " nyghr ghr#aijacdfk ~ dfetvdfk#ulajdyk#; Ughtxtg" o Usk#u ghtnyk.o edyiGacn e wr#d ~yaGacalk a ghack#a/ghn grsaj i Usxzghtqb>aletzn.uldywhaGmIpIn ww t k f dfk# d dfk#ww n k fs# " kQghrsadfehghtvu xacw u tzghaQdyn ~ ay ghrsabsenysxali(w dfea b Useab>nyxtvu (nybsghtiGet dcdfghtnykb#ehn sxzaci(w w tk#ulaksnGw b>acultdyx b U#ehb>n whanyo9ghrsatzacxswnfo:ghrsatzk~yaw"gijalkmgtvw}w b>acult #a " rsalk5Oe U#kskstks dfk tzkcw Use{dfku a(u n iGbdfky:ghrsa( b U#ehb>n whajnfo ghrsa(tk~yacw ghijack g{wtwghn/>adfsxaGghnIu n ~ aleghr#a b#d ijalkmgw ghn grsa/b>nyxtul rsnyxvace?w " alk#ula/tgtvwjijnyea dybsbsenybsetvd gha5ghn o n ehb i Usxdfgha ghr#a acultwhtnyk bsenysxali d wQdfkdyww algQxztvdf#tzxtg",bseny#xzaci !:p W}rsalea/grsa xtdystxztzghtacw dfea a bsxtultgxz/ulnyk#whtsaleac rsalkIghrsa bn e gho nyxtzntvwu{rsn whalk " dytzk` !]p bsenysxali(wijd >a w tks xza w gdyya n eU i Usxzght w gdyya}bsenysxali(?w "]}rsa #e{w gUb i Usxzght w gdyya iGnacxw o n ehb i Usxvd gha9yOw U#u{r dyw 5r#dfi acew dfk r#dyehksa w sd whOw Usijagrsao UOg UseaghnGab>aleho acu ghx ksn k "Up/nyea dy~ dyk#u aijnsalxvwlyOw U#u{r d w]ghrsa nyksao nye i U#xdfghac@m e{dyxaldyk#Ge{dfks a tk#ulnyebn edfgha ghrsa edyk#n iGk#acwwtksrsalealkmgtzkIb>nyehg o n xztnQwhalxacu ghtnyk " rsalk.edyk#nyijksawhwtwtzk~ nyx~yac` a i U#w ga galeiGtksar#d g}tzg ijacdyk#wo nye}d whnyex Ugtzn k(ghnGao acd w tsxa " ai(d jaltzghr#ale}eh7a 6WUstzea ghrsaGu n k#w"gedytzkmg{w}ghnr#nyxvMtgrIbseny#dystzxtzg" y>nyeeh7a 6 U#tzeaghr#a u n k#w"gedytzkmg{w}ghn5rsnyxvMtgr dMbseaca #ksac bsenydfstxztzg"y]ehacw U#xgtzksMtk d/u{r#dfku a(u n k#w"gedytzk#ac.bsehn sxali w aca5r#dfeksaw dfk# nnyb>al e 6"@}rstvw tk#Inyou n k#w ghe{dftk g{wdyeha tzk#ulmx Uac.tzk ace 7g 5 w f qijnalxUnyod [ Usgu{rMback#whtzn k5o U#k# "
(37) k#a5g"b>a/nfoxztzo a/tk#cw U#edyk#u aMbsen U#uWgn 2>aceha66whnyijaMxto aItk#Ow Use{dfk#ula ulnyijb#dfk#tzaw dyxgrsn Us r.k#nfgIn Useb#dye gksale twdyk.7a 6WUstg" xztk ya.ulnykmghe{dyu gtzghr Udfe{dfkmghaca "G U#u{r dGbn xztvu 5tzxx`b#d (grsai(d t i Usi nyoUdG U#dfe{dfkmghacac5dfijn UskmgdfkQgrsa~ dfex Usanfoqdfk tzka rsalk ghrsaju n kmghe{dyuWgi(d Og U#ehaw t " a " ghrsaGb>nyxtul rsn xaceeacd u{rsacwdQulalehgdftk dyyay n etacw B" ehacksk#dfk dfk# u{r dye g{w c w rsn Jrsn ghr#acwha b>nyxtultzawi(d /a(au nyijb>n whac tzkmghn5ghr#a U#dyedykmghala dfijn Uskmg dyk# dfk n bghtnyk6tgr ghrsa w"geht ya b#ehtvu ajyt~yack grsaQ U#dfe{dfkmghacac dfijn U#k ng " }r#twMau nyijb>n whtgtzn ktvwgrsa.dywhtw5nfo dyk tkm~ acw ghijalkmgMw"gedfghal r#aleaIghr#a u n iGbdfkghetzawUghnealbsxtvuld ga ghrsan bghtnyk "U U#e grsaleijnyeay ehacksk#dykGdyk#(u{r dye g{w Ow Us yacw g ghr#dfgt 2>acehackmgtzk~ acw ghijalkmgw ghe{d ghacytacw grsali(whalx~yacwijd 5a u nykw tvaleacMd wdyww algwtkMghr#a bn e gho nyxtzn(whalxacu ghtnyk bsehn sxaliG" n ac~yacec}nybsghtnykealbsxtvuld gtzn knacw5ksnfgMdfbsb>acdyegn a Ow Ustg{dfsxaIo n egrsa atwhr . . . . . . . . . . . . . . !. . . . #. . . . ". . . . !.
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(42) . ijtzk#tzijtedla xt ,t∈{0,...,T }. wOUs"auWggn. f0 (x0 (ξ0 ), ξ0 ) + +...+.
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(45). P ξ2 |ξ1 [f2 (x2 (ξ2 ), ξ2 ). " cd " l " u. ! . P ξT |ξT −1 [fT (xT (ξT ), ξT )] . . .]],. x0 (ξ0 ) ∈ X0 (ξ0 ), − xt (ξt ) ∈ Xt (→ x t−1 (ξt ), ξt ),. #. ! . #. ! #. t = 1, . . . , T.. aleay ξ ehacbseacwhalkmgwGdfxx}edyk#nyi tko n ehi(d gtzn k ksn kdfg gtzija t sa #ksa n kd w"g{dfk #dfe{(bsetny#dystzxtzg"Gwhb#dyula (Ω, , P ) sdyk#
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(53) Useb#dfehghksaceu n ijb#dfktvw r#ac ghn U#w a@ghrsactze6u Useehackmgdyww algu xvdyww tuldfghtnyk`#r#d ~tzks st 2 alealkmg d whwha g u xvdyww aw dyxgrsn Us r5ulxd whwhacwq aleaksnyg d uWght~yacxz ghe{dysac 6" acwu etz#tzks@ghrsa "n tzkmg b#ehn #dfstxtg" o Usk#uWgtzn k nfowhn/ijdyk.dywwha gwgn d/wd ghtvw o dyuWgnye acyeala(ea 6WUstehawghrsa wu ack#dfetnMghealagn r#d ~ adIxvdfeya(k Usi ace nyo}#edyk#u{rsacwcUi(d tzk#/ghrsa5ehacw U#xgtzks bsehn sxali rdfe{MghnMw n xz~ a " o a r#d ~ aghnn5o al4#edyk#u{rsacwghnMdyuc6u Use{d galx awhuleht>a grsa b#ehn #dfstxtg" o U#k#uWgtzn k` ghrsa5nybgtzijet dcd gtzn k tzxx whala IgnIa bsxnytzg wh b Usetz n U#wtkm~ acw ghijalkmg@n bsbn e Zg Uskstzghtacw r#tu{r xnmn Iynn rsalk ac~ dymx Ud gha n ~ aleghr#aQwhulalk#dyehtn5ghealay] Ug@#d rsalk al~ dymx U#dfgha U#k#ale}grsaghZe Usabsehn #df#tzxtg"(ijadyOw Useh a "] G k : 8 +n agehjgnjulnyk#w ghe{dftk(grsao acdywhtsxza wha gnyo:n Use !:p=bsehn sxaliMtzk nye{aleghnGxtzijtzg grsabnmwhwhtstzxtzg"jghnju{r#dywhawh b Usetn U#wbsen sg{wl r#tzxad gqgrsawhdyijagtzijaksnygqeha U#u tksghr#abace o n ehi(dfku a}nfo>ghrsaijnacxmGw"gtzxxsea g{dftkstzk# dGksadfe n bghti(dfx9whnyex Ughtnyk tzkw tvaghr#ao acdywhtsxzawha ng " w ghnu{r#d w"gtuMb#ehn ye{dfijijtzks #d w a acultwhtnykOw Usbsb>nyehg5w w ghaci i(d >a.whalackdywd #xd u In /ghrsX a Uw ace?w " U#whale@tks b Ug{wrstvw rsale@Ow Us "auWgtz~ aG~tal nfo ghrsa(b#ehn #dfstxtg" stw ghetz@ Ughtnyknyo9o UOg U#ehaal~yackmgwdyk#(etw gnyxale{dfk#uladyw acxzx dywghrsaw"g{d ghanfo9ghr#au n ijb#dfky @b U#w r#acwQd Ug gnykdyk#,n Ug5bn b#wQd sacu tvwhtzn k " #nye d U#w acectzgQtvwksnyg5u xacdyersn Fghrstvw sacu tvwhtzn k/ d wnyg{dftksac`dfkMrsn dfxzghale{d gtzn k#w ghnghrsa tzks@b Ugtxzx d 2 acu gghrsa n Ug b Ung " k dykd ghghalijbgghn ~mtvOw U#dfxmt dcarsn ghrsanybgtzi(dyxwhnyex Ughtnykjo enyi2ghrsaw"gnu{r#dyw ghtvu}bsenyye{dfijijtks ijnacx n Usxv ~ dfe tzghr tks b Usgcq aQw n xz~ ac d/xvdfeya(k Usi ace nfobseny#xzacijw dfk# w ghn eha grsa(eacOw Usxg{.w " tkmghacehb>nyxvd gtzks >a g" alalk grsajehacw Usxzgww ghn eha9` ajijd .ed 2ye{dfbsrwnyo dybsbsen tzi(d galx5r#n grsajacultwhtzn k u{r#dyksyawtzghrIgrsajw"g{d gha nfoqgrsaGulnyijb#dykm " a grsalk eacdyxzet dlagr#d g}grsacwhagdysxzaw}i(d Y a Uw a5d wd b#ehn jo nyen gdytzk#tzks(au tvw tnykw o enyi ghrsa w ghnu{r#d w"gtubseny edyijiGtks iGnacx" }rsany~tn U#w e{d #dyu Gnfo`grstvwdfbsbsen d u{r(tw gr#d g} aksnjxnyks ale yalg}dykQnybsghtijdyx9a ultwhtnyk o nyedyt~yackGw gdfgha}nfo>ghrsaulnyijb#dykm "]}rsacehadyeha rsn al~yaceqdk Usi aceqnfo9dy~ dyk g{df ac?w " . . . . !. . #. !. #. . . . . . . . . .
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(56) .
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(58) }r#tw gZUskstks tvw}b>aleho nyeiGau n i stkstksjalbgr #ew gw adfe{u{rnfo`grsawhulalk#dyehtn@gehacatzghr u{rdfks acwgn tzg gehnu 15 w }o dyw g #d u mo dyw g o nye dye.w a76WUsalk#ultzk#Ibsenfghnulnyxulnyijijnyksx Uw aQo n eksacw gha ack#ale{wau nyijb>n whtgtzn k" . . . . . . . . 8 n+ : !. }r#tw5b#dfb>ale acdyxwQtgrgrsa.dyyyealmd gtzn k6nfo d whwha g{wtkmghn6d whwha g5ulxd whwhacwQtzkw"gnu{rdyw ghtvu #b ehn ye{dfijijtzks !:p ijnalxvw." a5whrsn rsn JgrsaQwhalxacu ghtnyk nfo}grsakUsi >ale nfodywwha gw gnjtzk#ulxmUatzkMdfk nybgtzijtedcdfghtnykQijnacx`d 2 acuWg{w ghrsa 6WU#dyxztzg"(nfo]ghrsawhnyxeUghtnyk5tzk5g"nG}d w r#alk,dfkdybsbsen tzi(d gtzn k nfoghrsaMbsenydfstxztzg" ijacdywOUsea5tvw U#whac " ]te{w"gxz ghr#a 6WU#dyxztzg" nyoUghrsaGw n mx Ugtzn k/twd 2 acu ghacMk#al dfght~yacxz #dywai(d ijd ya@dyxzx]nybgtzi(dfx]nyek#acdfe nybgtzi(dfx whnyex Ugtzn k#wtko acd w tsxa "Iaculnyk#sxz ]n k ghrsab>n whtzght~yaQwhtsay]ghrsa dfy ehac dfghtnyk b#ehac~yalkmg{wghrsa a bsxzn tg{d gtzn k nyo}wh b Usetn U#wbsen #gwacijdyk#d gtzk#5o enyi dMbnn e@dfbsbsen tzi(dfghtnyk nyogrsa(o Usxx b#ehn #dfstxtg"ijacdyOw Usea ":@ Usehghrsacehijnyeay a w r#n .rsn Udfxmtzko n ehi(dfghtnyko ehn ighrsa wu ack#dfetn gehacaijd QY a U#whac5gnG Ustvaghrsa@d whwha gdf yealmd ghtnykQbsehnulac Usea " . . . 8 Bi : !. kghrstvwb#dybaceQ a tehauWgxz,u n iGbdfeaIw"gnu{rdyw ghtvuMxztksacdyebseny edyiGijtks ghn6d b#dfe{dfija galetmdcac L=9& (: 3RQbn xztvu " ehac~tzn U#wu n ijb#dfetwhnykwjtzk,grsa xtgale{d gZUseha/rd ~ya/>alalk tzghr ea nybsghtiGtedla ac ijt bn xztvu tacw?" @Use grsaleijnyeays a@u n k#w gheZU#uWgdjrset b>nyxtulysrsaceha d b#dyedyiGalghacehet dlajbn xztvu (tvw}u nykw"gOe U#u ghajo ehn i d@k Usi ace nyo]w ghnu{r#dyw ghtvuxtksacdyebseny edyi ijtkswhnyex Ugtzn k#.w " a.whrsn Fghr#dfg5 nybgtzijet dltks grsa b#dyedyija ghacewjnyoghrstvw5bn xztvu a whtykstuldykmghx/n Ugbace o n ehi grsa(w n mx Ugtzn k#wo ehn iLd b UseaGw ghnu{r#d w"gtu xztksadfebsenyye{dfijijtks o n ehb i Usxvd ghtnyk ".
(59) v6,:8 . k5dystzghtnykjgnghr#ab#dfb>ale{w yghrsagrsacwhtvw nye r#d wqehawcU#xgacjtk(g"nb#ehn ye{dfi(wUrstvu{r ar#nyb>adfeT a U#wha o Usxo nyenfgrsale}w"gnu{r#dyw ghtvu}bsenyye{dfijijtksehaw adfe{u{rsale{wqdfkjbsed uWgtgtzn ksale{w." }r#a #ew gbseny edyi tvwdfk,a 4(ultzackmgtzijbsxalijalkmg{d ghtnyk,nyogrsa/ksacw gha alk#acew dyxz nyetgrsi grsatzijbsxalijalkmg{d ghtnykQtw}awhulehtatkQ qdfb>ale}h B"}rsaw au nykbseny edyi ksnfg sacwu etz>actzk dykm bdfb>ale( Ug Uw a6tk qdybacewjWqh jdfk# tvw(dfk a galk#whtzn knyodyk dyxz alse{dftvu5iGnacxztks xvdfks U#df ayr#tu{r dfxxn wjtg(gn rdfk#xaIw ghnu{r#dyw ghtvu bsenyye{dfijijtks b#ehn sxzaci(.w "I}rstvw bsehn ye{dfiM s
(60) ! p twGawhulehta tzk acu ghtnyk I>alxn Y " nfgr b#ehn ye{dfi(wdyehao ealacxzQd ~ dytzxvdfsxaghnjghr#aehaw adfe{u{rQu n iGb i Uskstzg" " . . . . !. #. . P
(61)
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(68) P [g (x , ξ , X, P )]. " .
(69). !. . #. ! #. !. x0 ∈X0. 0. 0. 0. ξ1. 1. 0. 1. 0. 0. 0. ξ1. 1. 0. 1. #. n ac~yacecqdyw w"g{d gac bseal~tn U#whxyq a5dfeaksnygGdysxzaQghn dyseacwwgrstvw b#ehn sxzaci tzeacu ghx" kw"gacdyQatvwhulehalghtedlagrsaw b#d u anfo:b>n ww tsxan Ugulnyijacwtk gn d whulalk#dyehtngheala dfk#5w n xz~ a grsa@dfb#bsehn ti(d ghabsenysxali ijtkstzijet dla f (x (ξ )) +
(70) P¯ [g (x , ξ , X, P¯ )], " !. x0 ∈X0. . #.
(71) rsalea P¯ salksnyghacw:ghr#absenydfstxztzg"ijacdywOUsea u n eheacwhb>nyk#tks}gngrsa wu ack#dfetn}gheala"] k@grstw bsenysxaliM mal~yacktzo9ghrsabseny#dystzxtzg" ijacd wcU#eha ¯ b#ehn ~tvacwdfkQd ulu6U#edfghaawhulehtbghtnykQnfo9ghr#a o Usxx9bsehn #df#tzxtg"(ijadyOw Useha P o nye}grsa #ew g w gPdf ae{dfk#snyi nUsgu n ijacwc grsaehawcU#xgtzksGnybsght i(dfx #e{w"g w gdyya w n mx Ugtzn k i(d >a}o dfe o ehn i ghr#a eacdyxnybgtzi(dfxwhnyxeUghtnyk:fto g (x , ξ , X, P¯ ) 1 twd5bnn edybsbsen tzi(d gtzn k/nfo g (x , ξ , X, P ) " sn ea sdfijbsxay9tzo aGr#d ~ 1a i(0dfk/ dywwha gw 1 0 1 ghn tk~yaw"gtk` UgQo al se{dfk#u{rsaw(t k d g5xdfghale5w"g{dfyawjtzk,grsaIgehacay ghr#alea/tzxxa tw g w @b Usehtn Uw}bsehn sgw}tkMghr#a@whulalk#dyehtn gehaca@d gxvd galew gdf acwcsdfk# rsack#u a txzx , ξal1km, g X,P¯tg)rea aIdyk6n ~ ale aw"gtzi(d gtzn k nfo g (x , ξ , X, P ) " k6grstwb#dfb>ale( a5a g1b(xaceh0tij w"gehtvuWgtzn k#wnyk.ghrsajo adywhtz#xza(wha 1g e0albs1xvdyultzk#5grsaGo acd w tsxajwha g X X¯ ⊂ X 6"j}r#aG@b Use bnmw a/tvwjgn ijd ya g (x , ξ , X, ¯ P¯ ) d alg ghace5dybsbsen ti(d gtzn k6nfo g1 (x0 , ξ1 , X, P ) ghr#dyk 1 0 1 # s r v t { u. r z t x ` x t iGb#ehn ~ agrsa #e{w"g w gdyyaw n mx Ugtzn k " g1 (x0 , ξ1 , X, P¯ ). r#alk dyk !]p bsenysxali<tvw:whbau t#ac9fdyk@tiGb>nyehgdykmg]u{rsnytvu a tw]rstvu{r dyww alg]u xvdywwhacw gn tzku ex U#a "}mb#tucdfxxz ghrsa@d whwha g{wdfeadf yeal dfghatk gnuldfghal nyetzawcw Uu{rMd w}o nyealtykMw ghnu nyijaw"gtu>nykswlma g{=u "U o ghnno acdyww alg u xvdyww awdfeaa U#whac9 ghrsack(ghr#aijnsalx#i(d Gbace o n ehi bnn ehx whtk#u aQghrsaQo acdywhtsxza5w algGnacw ksnygGu n kmgdftk dfk ynn w n mx Usghtnyk#?w "
(72) k grsaQnyghrsace r#dfk9Utzk#ulmx Utzk#/ghnnIi(dfk dyww algwxacd swgn u n iG@b Ugdfghtnyk#dyx t 4(.u Usxzghtacw Gd wghr# a "nytkmg bseny#dystzxtzg" tvw"geht Ugtzn knfo grsab>n ww tsxadywwha g talxvsw dyehaealbseacwhalkmgac U#whtzks whulalk#dyehtn wc tzku eacdywhtksQgrsak Usi acenfodyww algwea 6WUsteha` w U#wghn/tk#ulehadywha grsa(tvghr nyogrsawhulalk#dyehtn ghealawl ghr U#wQi(d tzk# ghrsaIb#ehn sxzaci b i U#u{r,xdyeh alne " k,ghr#twQb#dfb>ale ghrsaIeacw ghetvuWghtnykwjnfo ghrsa o acd w tsxa w algwtvwh.u U#whwhac/dyn ~ agd yacwghrsaGw r#dyba nfodf yealmd ghtks(dywwha gwtzkmghn5styyace dyww alg@u xvdyww aw ehalo aleeha/gnMd ww kmghr#a ghtvujdyww algw B"(}rsa(df yealmd ghtnyk tw Ustsac.m. U#dfx tzkso nyeijdfghtnyk o enyi grsa !:p bsenysxaliMqtk6nye{ace@ghn #k#6ehaw"gehtvuWgtzn k#w@n k ghrsa5o acd w tsxa w algw}rstvu{rMehacijn ~yacwgrsaghack#alku gn(u{r#dywhaw b U#ehtn U#w}b#ehn sgwcrstxza@dfgghrsa@wdfijaghtija bacehijtzgw ghr#aMwhnyx~yace@gn #k ynn6d whwha gju n i stk#d gtzn k#?w " a ghe ghrstvwjdfbsb#ehnmdyu{r nyk d #dfea >nyksaw !:p bseny#xzaci sdyk# #k# gr#d ggrsaleatwksn(w tykstucdfkmgdy~ dfkmg{dfyanyo U#whtks ijnyeagr#dfkjg" ndyww alg u n i stk#d gtzn k#wcydywUxnyks@d w ghrsacGdyehaucdfea o Usxxz u{rsn whalk " ao Usehghrsace #k#5ghrd gwhtz kst >uldfkmgb>aleho nyeijdyk#u atvwxn w g}tzo ghrsa@xtvdfstxztzg"Qwhtsatwk#nfgu n k#w tvacehaQrsack ghrsa dyww algwdfeadf yeal dfghatk gndyww algu xvdyww awlactgrsale! U#w tks Udfx`tko nyei(d gtzn k5o enyi ghrsa !:p #nyem df yeal dfghtksjdyww algw U#whtzk#QpIdfe yn tzOg dija grsn9>d Usyijack gac tzghrMghr#a xztvdf#tzxtg"Qrsac tzk#julehatzgnfoUr#dfebadyk#M}tzkmg " . . . . $!. #. . . . . . . !. #. . . $& 3+ D$ 3 H <% *+ 3. H .4;%-0,4, *;: : ;D, < 3. 4,A8 , / 3 D= /# D H 4C3+G ).4C% %6 <B) 1,34 "! # ##$-%'.E*)+,%-,( . . }rsa ulnyk#ulalbgnfow ghnu{r#d w"gtu@bsehn ye{dfijijtzk#(n ~yaledwu alkdfetzn(gehacacwtwnyk#xzMnyk#a nfoi(dykm bnmwhwhtsxza/dybsbsen dyu{r#acw ghr#dfgi(d a Uw a6o n edyww alg(xztvdfstxtg"ijdyk#df alijalkmgn" ksnyghrsace bnmwhwhtsxzadybsbsen dyu{r5tw grsaYU#whanfoUb#dfe{dfija galetmdcacQbn xztvu tacw w aca@p Usx~yal ! 6" oq axznn Qd ggrsabsenysxali " Wdyk#Mdywcw U#iGagr#d ggrsa@~ dfetvdfsxacw}i(d >a t~ta tzkmgn n ksawha g}ealbseacwhalkmghtksgrsaw"g{d ganyo`ghrsaulnyijb#dfk x sdfk#Qnyksaw alg ealb#ehaw ack gtzk#@ghr#a acultwhtnyk ai(d i(d ay y ghrsack ghrsabseny#xzaci i(d aea o n ehb i Usxvd ghaQd w !. . #. ! #. . min f0 (x0 ) + y0. +···+.
(73).
(74). ξ1.
(75). − − min [f1 (→ x 1 (ξ1 ), ξ1 ) + ξ2 |ξ1 [ min f2 (→ x 2 (ξ2 ), ξ2 ) y2 (ξ2 ) → − [ min f ( x (ξ ), ξ )] · · · ]],. y1 (ξ1 ). ξT |ξT −1. yT (ξT ). T. T. . T. T. !. " d . #.
(76) w?"]gn". " y " u " m " a. !. x0 = g(¯ x, y0 ), y 0 ∈ Y0 , xt (ξt ) = g(xt−1 (ξt−1 ), yt (ξt ), ξt ), yt (ξt ) ∈ Yt (xt−1 (ξt−1 ), ξt ),. . !. #. #. !. #. t = 1, . . . , T, t = 1, . . . , T.. !. . #. kIghrstvwo n ehibUsxvd ghtnyk:sghr#a w gd ga nyoqghrsaGu nyijb#dyk5d gghtija t tvwdstzeacu go Usk#u ghtnyk nyoUghrsa w gdfgha nfoqghr#a u n ijb#dfk5dfggtzija dyk#Mgrsa au tvw tnyk/i(d a d gghtija 9dywa bseacww a x (ξ ) = g(x (ξ ), y (ξ ),t −ξ )1" wr#dywacalkMijalkmghtnyksa acdfextzacectgrsa@tiGack#whtzn k nyod tw"gnt u{r#dyw ghtvuQt−1 b#ehn yt−1 e{dfijijtztks t bsehn tsxali tvwGrstz r`d wGnyksa/au tvw tnykijd >aMi(dya5o nye adyu{r/b>n ww tsxa@e{dfk#n iF n Ugulnyija " bsbsxtzk#b#dfe{dfijalghaleet dlac b>nyxtultzawtvwdyk d ghghaciGbsgghn eac U#u a5ghrsa/tijalk#whtnyk6nyoghr#a b#ehn sxzaciM ijd tzks.ghr#a d uWghtnyk6gd yalk d gGghtija tkmghnjdfk5a bsxtultg}o Usk#u ghtnyk5nfo`grsaw"g{d ghanfo:ghrsau n ijb#dfk(d g ghtija t − 1 dfk#ghrsae{dfk#tsnyyi t sal~yacxzn bsijalkmg>a g" alack dfk# "@}rstvwa bsxztvu tzgo Usk#uWgtzn k tvwijd ajalb>alk#salkmgn k.d kUsi ace]nyob#dfe{dfija gale{w αt − rs1tvu{r@ at nybsghtiGet dlan ~yaceccyt~mtksT Uw`grsa bseny#xzaci o n ehb i Usxdfghtnyk . min f0 (x0 ) + α. w." g?".
(77).
(78). .
(79). − → − x 2 (ξ2 ), ξ2 )+ x 1 (ξ1 ), ξ1 ) + ξ2 |ξ1 [f2 (→ → − [f ( x (ξ ), ξ )] · · · ]],. ξ1 [f1 (. · · · + ξT |ξT −1 T T T T x0 = g(¯ x, y0 ), y 0 ∈ Y0 , xt (ξt ) = g(xt−1 (ξt−1 ), yt (ξt ), ξt ), yt (ξt ) ∈ Yt (xt−1 (ξt−1 ), ξt ), yt (ξt ) = γ(xt−1 (ξt−1 ), ξt , t, α),. ! ! !. # #. #. !. t = 1, . . . , T, t = 1, . . . , T, t = 1, . . . , T.. "fd " "fu " y " fa " o. #. ! !. # #. }r#aju{rdfe{dyuWgaletw ghtvulwnfo grstvwb#ehn sxzaciNtwe{d grsalet 2>acehack go enyi " `ghr#dfgbseny#xzaciNtw xvdfeya wuldyxza Ugxtzk#acdfersacehadyw " tw}xn stzijalkw tnyk#dyx Ugksn k xtksacdye?" k : 8 6i au nyijb#dyehaw ghnu{r#d w"gtujb#ehn ye{dfijijtzks gnIb#dfe{dfijalghaletedlac.b>nyxtultzawl >nfgr 6 Udfkmghtzgdfght~yalx dyk# 6WU#dyxztzgdfght~yacxz "}rsajijdytzk d ~ dfkmgdyyaGnfod w ghnu{r#d w"gtu xtzksadfe b#ehn ye{dfijijtzksdybsbsen dyu{rjtvwtzgwyeacdfg #a tstzxtzg" u n iGb#xza julnyk#w ghe{dftkmgwqi(d Gar#dfk#sxza tk/d(w"gedytz rmg o nye}dfe{( d sdfk#5ghr#ao dyuWggr#d g a@njksnygtzk Usalk#ulaghr#a@w n mx Ugtzn k5tzghr b#ehtnye dywcw Usijbgtzn k#w]nfo#r#dfg u{r#dyed uWgalemt dcacw`dynn@w n mx Usghtnyk " }rsa i(d "n e]n kw tva tvw:ghrsa ulnyij b Ug{d gtzn k#dfxUbsetula ghrsajk Usi >alenyo ~ dfetdysxacwyen wa b>nyk#alkmghtvdfxxz/tzghr.ghr#aGk Usi >ale nfo9gtzija w gdyyacw tzkghrsabsenysxaliM o nye{u tks U#w ghnGdfy ehac dfgha ghtija w"g{dfyawqo n ebseny#xzacijw tzghr d5rstyr Wk U#i >alenfo}w"g{dfya.w "(
(80) k grsajnfgrsaler#dyk#9`b#dyedyija ghacehet dla/bsenysxali(wyen n ksx.xtzk#acdfexItzk grsaQk Usi ace@nfo}gtzija w gdf acwc]dyk# bsen U#u a(bseny#xzacijw@nfoxzn Dtijalk whtnykMu nyijb#dyeha(gnjw ghnu{r#d w"gtuxtzk#acdfe}bseny edyijiGtks " }rsan kw tvanfo !:p bseny#xzacijw dywhac n k b#dfe{dfijalghaleet dlac b>nyxtvu tacw tw ghrd gGghrsac eacOw Usxzg tk6ksn ksxtzksadfejdfk# ksnyk u n k~ya b#ehn sxzaci(wr#tu{rMdfeardfe{(ghn(whnyx~y a "] k nye{ale gn >alksa sgo enyiDghr#a@dy~ dfkmg{dfyaw nyo:>nfghr ijalghrsnswc ulnyi stzk#aQgrsaMg"n dfb#bsehnmdyu{rsaw tkmghn d rmsetv ija grsn96ulnyk#w ghZe U#u ghtks d bdfe{dfija galemt dcac5bn xztvu rsaleaghr#a@bn xztvu Qo Usk#u ghtnyk/tvw#d w a5n k ghrsa whnyex Ugtzn k nyodjxdyeh a kUsi acenyow"gnu{rdyw ghtvuxztksadfeb#ehn ye{dfijijtzks(b#ehn sxzaci(.w " kI n Usegacw ggrsa rsetMb>nyxtvu ulxzadfexzj n Ughb>aleho nyei(wdGw"gnu{r#dyw ghtvuxtzk#acdfe bseny edyiGijtks@bseny#xzaci whrsn tzk# U#wghrd g ghrsa dyy ehac d gtzn kQnyoUw gdyyacwul=d Uw awdjstd wrstvu{rMksalmd gtz~ alxQd 2 acu gwgrsa whnyex Ughtnyk:#dfk#5gr#d g dGb#dye gnyo]grstwstd w}i(d aealijn ~ ac Uw tksGghrsarset5b>nyxtul " !.
(81) . !. #. . #. . !. #.
(82) . . . .
(83) X
(84) . }rsa/o n ehibUsxdfghtnyk,nfo@d i(d grsali(dfghtvuldfxbsehn ye{dfi tvw ehad gxz w tijbsxzt#a,m6ghrsaU#wha/nfo dfk dfxyacse{dftvu ijnalxtks/xdyksUdfya pI! 6" nfg@i(dfk pI!:w dfeaGr#n ac~yalewOUstzghac.o n e w"gnu{r#dyw ghtvubsenyye{dfijijtks@ " rsalkIghrsa@ nye Qn kMghrstvwghrsaw tvw}dyww gdye gac9#ghrsaceha a tvw"gac gnQn U#e ksn xac a }n ksx5g" nQwhw"gali(wo n ei(dfkdfytksQw ghnu{r#d w"gtu bsenyye{dfijijtks(ijn alxvw q!: h
(85) m dyxzx>dfk#(pId al e dfkQ
(86) @u eacdfghacjG Unyte dyk# U#whac n k#whtz xzt}dfk#acijb#w"gale "6 k dysstgtzn k`grsalea5a tw ghad w w ghaci uldfxxac
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