P. We make the following assumptions
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1 Inrnaonal Journal of Sn and Rsar IJSR ISSN Onln: Ind oprnus alu 5: 7896 Impa Faor 5: 639 Opmal Ddnd Prolm n ompound Posson Modl w orng Df a Run Yanan Wang Xong Song H nrs of nolog Sool of Sn anjn34 na Asra: In s papr w onsdr opmal ddnd prolm n ompound Posson modl w orng df a run w s rsrand o srags w rsrd dnss As w all known saroldrs sould rpa df a run rfor w wan o mamz paon of dffrn wn aumulad dsound ddnds unl run and dsound df a run and fnd ou opmal ddnd srag Furrwn lam amoun dsruon s ponnal w oan pl soluons of alu funon Kwords: alu Funon HJB quaon Opmal Barrr Srag Inroduon For onnuous-m rsk modl Grr969 frsl sudd opmal ddnd prolm nl 997 Asmussn and aksar rsard Brownan moon w drf modl mod of HJB quaon go opmal ddnd pol In lassal rsk modl opmal ddnd prolm go furr rsar For ampl Grr and Su prod a opmal srag s a rsold srag Kulnko and Smdl 8 dsussd opmal srag prolm n lassal rsk modl w apal njon go a mor gnral and mor omprns rsuls Dkson and Wars4 pond ou a saroldrs sould mor rsponsl o rpa ankrup of os dfs omplng w s w Grr Su and Sm 6 dd furr rsar for modfd modl Basd on ao or s papr s ddad o followng rsar Our goal s o mamz dffrn alu wn umula pd dsound ddnd and dsound df Furr w an dr opmal ddnd srag Wn lam sz o ponnal w alulad alu funon and fnd opmal ddnd arrr Opmal Ddnd Pamns W orng Df a Run Gn an nal surplus fr surplus nsuran ompan a m an wrn as X of N X Y In s papr lassal oll rsk modl s ompll drmnd prmum ra nns and lam-sz dsruon funon rgorous wa dfnng s flrd proal spa P W mak followng assumpons F on dsruon of szs and ourrns of lams: frs lam anno our a m zro wo lams anno our a sam m and numr of lams n an m nral s fn So 3 N and N s fn for an lam szs ar muuall ndpndn and ar also ndpndn of lam-arral ms 3 lam szs ar dnall dsrud 4 numr of lams n a m nral onl dpnds on lng of nral a s PN N k P N N for Assum k an s umula pd dsound ddnds unl m and s an adapd adlaglf onnuous w rg lms soas pross So surplus oms onrolld pross F PY W an dsr s modl n a opmal ddnd pol n an nral wou loss olum 6 Issu 5 Ma 7 wwwjsrn nsd ndr ra ommons Aruon BY Papr ID: AR734 X N Y W sa a a ddnd srag s admssl f s non drasng prdal w rsp o flraon F W dno s of all admssl srags Dfn orrspondng run m as : nf : X Baus w onsdr 66
2 Inrnaonal Journal of Sn and Rsar IJSR k ISSN Onln: Ind oprnus alu 5: 7896 Impa Faor 5: 639 of gnral wn X X sops So wn N Y For a srag n orrspondng alu funon s dfnd as E d X s formula sows a umula pd dsound ddnds sura df a s o sa n prof of saroldrs unl ankrup ms wr Our goal s o sk for opmal srag and mamz alu funon n alu funon s dfnd as sup 3 Bas Proprs of alu Funons In s son w sud proprs of alu funon undr onsran of ddnd pol mma 3 alu funon s monoonall nrasng on nral and for 3 Proof: For r s su a for all And w dfn a nw srag pol ddnd w an g mmdal And ak Baus of arrarnss of so Proof: In a wors as w nd o ompnsa for a lam sz Wn w onsdr no ddnd k lam sz os k n k EYk k k mma 33 For R alu funon as a ound and psz onnuous and sasfs lm 4 Proof Baus s monoonall nrasng so onsdr a pol d nf : u B mma 3 w an g X N Y Wn s unoundd And P X s onrg o zro E d X and s a srag w nal surplus Dfn s m from rsr o and r s no lam sz oos a nw srag ~ wr s s oous monoonall nrasng mma 3 For ound Papr ID: AR734 an ddnd pol as a olum 6 Issu 5 Ma 7 wwwjsrn nsd ndr ra ommons Aruon BY admssl srag w nal rsr Dno ~ or assum a frs lam sz ours asd on dns n 66
3 Inrnaonal Journal of Sn and Rsar IJSR ISSN Onln: Ind oprnus alu 5: 7896 Impa Faor 5: 639 E E P E d P E X E E d E E ~ d W ak suprmum from so B ounddnss of w an g psz onnuous: so Du o s psz ~ d X onnuous X X F X F 4 HJB Equaon and Opmal Srag In son w fnd ursall frs-ordr quaons w sasf alu funon of sal rra dfnd ao Now w us so-alld dnam programmng prnpl DPP n sor and sals HJB quaon a s o sa sup df n w df df df So HJB quaon oms sup df df 5 6 Mamz lf of formula 5 so oms Baus frs m of formula 7 s lnar so mamzd lf of formula 5 an wrn as : 7 If s ona on r s : nf : su a μ 8 Papr ID: AR734 olum 6 Issu 5 Ma 7 wwwjsrn nsd ndr ra ommons Aruon BY 66
4 Inrnaonal Journal of Sn and Rsar IJSR ISSN Onln: Ind oprnus alu 5: 7896 Impa Faor 5: 639 umula pd dsound ddnds sura df wr s nal rsr dfn : nf{ : } df u : u : : df 9 B dsussng w g quaon 9 and quaon ar qualn o quaon 5 Proposon 4 Assum a s ona on n s onnuousl dffrnal on Proof Baus of ona of formula opmal srag s arrr srag In fa du o surplus pross s on nral for orrspondng srag s admssl Wn pol s opmal Wn ddnd s pd dsound 8 s propr B usng HJB quaon and onnuous of w g df df So Smlarl w an proof Now w assum a s onnuous on dffrnal and As w all knownaloug for frs lam df ours surplus pross X ma nga - df W an proof a wn X orrspondng so w an g or If w an g or Bfor m of frs lamw an g df d df us w mus dp alu of df And pu no quaon 9 w an g so w g s pol s no opmal us wll our wn wo lams and dx d X X Y If Z Y n saroldrs sould rpa X -nf X Y Baus lar formula s mpossl so s onnuous Orws f w an g So alu funon sasfs X X Y And wn Papr ID: AR734 olum 6 Issu 5 Ma 7 wwwjsrn nsd ndr ra ommons Aruon BY 663
5 Inrnaonal Journal of Sn and Rsar IJSR ISSN Onln: Ind oprnus alu 5: 7896 Impa Faor 5: 639 X Y s unqu soluon of HJB quaon sasfs followng quaon: X X Y Proposon 4 Assum a quaon 5 s soluon of s nrasng and oundd and sasfs proposon n lm and opmal pol s gn quaon8 n n n Proof: Baus s ound and nrasng n s onrg so r s a srs of n n s n n B formula 8 and assum n l n n formula 5 n df n n So lm usng proposon 34 and HJB quaon w an g X s s ds X s a marngal Du o paon s zron s E X s ds X Du o s ound roug onrol onrgn orm w an g a wn X E and ors ar smoo so ang ordr wn lm and ngraln I s prod a wn surplus pross s undr arrr ddnd s zro wn surplus s or arrr X ddnd ra n w ddnd a mamum 5 Eponnall Dsrud lam Szs In ordr o oan an pl soluon o HJB quaon and an opmal ddnd pamn pol w assum a lam sz dsruon s gn F E o sol s HJB For arrar pol quaon 5 w an g l E X s s ds X so quaon w assum s a soluon on nral dfn a arrr E sds dx : nf : funon sasfs followng quaon: so Baus d d 3 d d 4 Papr ID: AR734 olum 6 Issu 5 Ma 7 wwwjsrn nsd ndr ra ommons Aruon BY 664
6 Inrnaonal Journal of Sn and Rsar IJSR ISSN Onln: Ind oprnus alu 5: 7896 Impa Faor 5: 639 olum 6 Issu 5 Ma 7 wwwjsrn nsd ndr ra ommons Aruon BY d d d d s d d So appl d d no formula 3 w an g 5 6 W larn from quaon 5 soluon o dffrnal quaon s: wr and sasf followng quaon: Baus s a parular soluon o quaon 6n w an g D 9 Wr s a nga soluon o quaon: B ound ondon and quaon 5 6 w an g and so w an drmn offn so Sol quaon s w an g: D D D wr D So an wrn as : W an know a s ona Fnall w nd o fnd arrr Du o n w an g pos soluon Papr ID: AR
7 Inrnaonal Journal of Sn and Rsar IJSR ISSN Onln: Ind oprnus alu 5: 7896 Impa Faor 5: 639 ln Propr 5: Assum a F and alu funon s ona on nral n al funon and opmal pol ar gn as followng: and opmal ddnd arrr an gn quaon Proof: Wn s soluon of du o HJB quaon 3 orm 3 w know a on nral and ar qualn And undr assumpons arrr an gn quaon Rfrns [] Asmussn Saksar M onrolld dffuson modls for opmal ddnd pa ou Insuran: Mamas and Eonoms 997:-5 [] Dkson D M Wars H R Som opmal ddnds prolmsasin Bulln 434:49-74 [3] Grr H Su E S W Sm N Mamzng Ddnds wou Bankrup Asn Bulln 636:5-3 [4] Grr H Su E S W On opmal ddnds srags n ompound posson modl Nor Amran Auuaral Journal 6a :76-93 [5] Kulnko N Smdl H Opmal ddnd srags n a ramr-undrg modl w apal njons Insuran: Mamas and Eonoms 843:7-78 [6] n X S Paloa K P ompound Posson rsk modl w a rsold srag[j] Insuran Ma Eonom 638:57-8 [7] u Zao-ang Xu W Sun D-a al On pd dsound pnal funons for wo lasss of rsk prosss undr a rsold ddnd srag [J]J omp Appl Ma93: [8] HAlrr and ausam Ddnds and m of run undr arrr srags w a apal-ang agrmn Prprnnrs of ausann4 olum 6 Issu 5 Ma 7 wwwjsrn nsd ndr ra ommons Aruon BY Papr ID: AR
Consider a system of 2 simultaneous first order linear equations
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