Optimal Stochastic control theory and its application to Landmark University Development ventures and investment model

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1 IOS Jonal of Mamais IOS-JM -ISSN: p-issn: X olm 3 Iss 5 I Sp - O 7 PP -3 wwwiosjonalsog Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns and insmn modl * NK Oladjo OJ Okoo MO Olwami HI kai Landmak Unisi Om-an Nigia agabaga ollg of daion amal Gana osponding o: NK Oladjo bsa: is pap dal wi opimal soasi onol o and is appliaion o Landmak Unisi Dlopmn ns and insmns modl H Soasi Diffnial qaions SD is onsidd as an odina diffnial qaions OD din b wi nois and w jsifid onnion bwn Io s ingal and wi nois in as of non-andom ingands inpd as os fnions Kwods: Insmn Opimal Soasi n Wi Nois Da of Sbmission: Da of apan: I Inodion Insmn onsis of making disions fo a manfaing ssm in wi diffn ps of ns s as opaions fails pni mainnan and aw maial sppl as wll as som dmand flaion o a sam o aios im o b ndsand and mo ffil dal wi andomnss fom aios sos Mamaial modls a an aaiz niq fa of majo n a ndd of wi Landmak Unisi managmn an b of an immns bnfiia Man ai sas and isoial figs a mad signifian onibion o onol o noabl among m inlds [] wo onsidd poblm of opimal insmn fo Nlo-ow modl nd a plnis-abl bdg b onsiding an opimal onol poblm wi wo sa aiabls fo dnamis of is modl and opind a opimal onol is a of insmn pndi a is qid o maimiz psn al of n sams o an infini im oizon sbj o a plnis-abl bdg [a] amind oninos-im onsmpion-pofolio poblm fo an indiidal wos inom is gnad b apial gains and insmns in asss wi pis assmd o saisf gomi ownion moion [b] nd s sls fo mo gnal ili fnion pi baio assmpions and fo inom gnad also fom non-apial gains sos [3] Sdid poblm of opimal ploaion and onsmpion of a naal so in soasi as wil [] also sdid lial onsmpion pans and aional addiion [5] onsidd opimal onsmpion aining woking im and lis o lif l [6] Insigad opimal onsmpion and insmn wi bankp In [7] soasi opimal onol poblm of onsmpion modl wi is sbj o I o diffnial qaion was disssd assmd a ili fnion is an inasing fnion of onsmpion as wil [8] Inodd maimm pinipl and bang-bang pinipl n insing ampl in on dimnsion aiss in Mon s modl of opimal pofolio b [9] and soasi appoa is sd o sd sabili and onol of mo and an modlld b [] w sd opimal onol appoa o sabiliz mo and an nsabl qalizd sas[] pplid soasi opimal onol o sd poblm of opimal onol of a soasi podion planning and insmn modl [] Poids an algoim fo opimal onol of nonlina soasi modls w nw sion of algoim OPON namd OPON was dlopd o obain appoima solion of onol opimm poblms w obji fnion is qadai and dnami mliaiabl ssm is nonlina II Soasi Opimal onol Pinipl W onsid diffnial qaion of fom: d f z f Gdz DOI: 979/ wwwiosjonalsog Pag

2 Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns n opimal onol poblm is spifid b giing a pfoman iion a gads possibl onol fnion in od of pfn b aaing a nmb J wi psn os of so a w an oos onol a minimizs i H d G W l os fnion b of fom: J W is infini minaion im and is no fid is al of on s ass H psn os diaion fom som dsid ajo of o s of oo m onol fos o ng G is os fail o a som spial ag s a minal Wil dminisi opimal onol poblm b of fom H z d G 3 Soling soasi opimal onol poblms dfind in qaions and 3 W l b n al fnion fo pd al of obji fnion of qaion 3 fom o : H z d G ma s opimal poli is followd fom o Gin X n b opimali pinipl z d d 5 ma H d ppling alo s pansion w g: d d d d d d d d 6 qaion an b fomall win s: X f d G dz fgdz d 7 d d f d Gdz d 8 mlipliaion ls of soasi alls: dz d dz d d 9 W sbsi fom qaion ino qaion appl qaions 3 and qaion 5 ilds: ma Hd d fdx G d d onding m on bo sids of qaion as d w di an Hamilon-Jaob- llman qaion of fom DOI: 979/ wwwiosjonalsog 5 Pag

3 Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns mah f G Fo n al fnion wi bonda ondiion S III ppliaion Of Soasi Opimal onol In Landmak Unisi Dlopmn nslmd nd Insmn Modl In is modl w onsidd a LMD n poding a singl omognos good wi a finisd goods waos and dfin paams and aiabls sd as follows: Y = inno ll a im = podion a a im = dmand a a im = lng of planning piod = fao inno goal ll = fao podion goal ll = iniial inno ll = inno olding os offiin = podion os offiin = salag al p ni of inno a im z = sandad Win poss = diffsion offiin n inno goal ll n podion goal ll is a saf sok ll a ompan wan o kp on and sids is mos ffiin ll a wi i is dsid o n fao ppling abo lisd paams and noaion w an n dsibd ondiion of modl b onsiding sok-flow qaion of modl as: d ; X H inno ll a im is inasd b podion and dasd b dmand Modifing qaion o gnaliz possibl soasi baio in modl w l inno ll Y o b onolld b Io soasi diffnial qaion s: X d dz X poss dz an b pssd fomall as w d w w is fs o as a wi nois poss qaion 3 an b inpd as sals ns o as inno spoilag a a andom in na podion a modl dos no sid o b nonngai i i do pmi disposabl and allowd inno ll o b ngai W adop modifid Hamilon-Jaob pinipl podd b Hamilon-Jaob-llman qaion saisfid b ain al fnion o sol poblm H w wis o minimiz pd oal os dmind b following fnional ingal: J Y d Y W assm a dnos minimm al of pd oal os fom o oizon wi X sing opimal poli fom o fnion is n gin as: 3 DOI: 979/ wwwiosjonalsog 6 Pag

4 Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns DOI: 979/ wwwiosjonalsog 7 Pag 5 min Y d Y Fo fnion al saisfing Hamilon-Jaobi-llman qaion; 6 ma Wi bonda ondiion 7 Minimizing pssion wi sp o b aking is diai wi sp o Sing pssion o zo w g: 8 n opimal podion a a minimizs oal os an b pssd as a fnion of n al fnion in following qaion: 9 Ping qaion 9 ino qaion 6 ilds: is is a nonlina paial diffnial qaion a ms b saisfid b n al fnion wi bonda ondiion 7 Fo podion a o b nonngai opimal podion a wold b ang ma W sk appliaion of Hamilon-Jaobi-llman qaion in soling soasi podion poblm aing diffn dmand a Soling nonlina paial diffnial qaion w assmd a solion aks fom: and 3 w do dnos diffniaion wi sp o im Sbsiing qaion 3 ino qaion and olling lik ms ilds: Fo an al of qaion old and following nonlina odina diffnial qaions ssm a did

5 Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns DOI: 979/ wwwiosjonalsog 8 Pag 5 Soling nonlina qaions fo diffn ass of dmand a wi bonda ondiions: ; ; D M and assm a im angd o W n a; ; Wi im oizon ; as W pand / in qaion 5 b paial faion n: I Gnal ass Fo Dmand as In LMD n W sall onsid fis as of dmand a as onsan ons and opimal podion a is gin as: 6 Wil fnions a as followd 7 8 d qaion 5 was sold af aking aag wi sp o sa aiabl b sbsiing fo opimal podion a fom qaion 6 o find pd inno ll 9 I was nod a wn dmand a is onsan and is qal o podion goal a n pd inno ll will b:

6 Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns DOI: 979/ wwwiosjonalsog 9 Pag 3 nd pd oal os is: 3 as wn dmand a is a im aing a i opimal podion a is n gin as: 3 b n fnion is 33 b pd al is n allad as: 3 In In nd 35 In pd inno ll is n allad as: 36 In In In Illsai ampls On Soasi n/inno Modl as: wn dmand a is onsan ons Gin a abl sowing pd opimal inno ll and n al pd opimal inno ll pd opimal n al a b a b

7 Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns Fig sows baio of onsan Dmand as s dmand kp onsan bo pd opimal inno ll and pd opimal n al onins dlin and la sabilizd o a paalll ll S II: Wn Dmand a is a im aing a i Gin a: abl blow sowing pd opimal n al a b Fig sows baio of aid Dmand as H as dmand kp inasing so also bo pd opimal inno ll and pd opimal n al inass I onlsion W a amind soasi opimal onol of wo soasi dnamial modls on Landmak Unisi diisional ns and insmn modl w bo inno ll and podion a sing lal o b soasi in na and dmand a is qall sing o b dminisi in na nmial illsai ampls sd fo soasi podion modl w o displas opimal pd inno ll agains im and pd n als agains im ass a DOI: 979/ wwwiosjonalsog 3 Pag

8 Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns wn dmand is onsan i ons and wn dmand a is a im aing a i a wo ass onsidd fns [] Si S P and L S Opimal adising fo Nlo-ow modl nd a plnisabl bdg Jonal of Opimal onol ppliaions and Modsol [] Mon ob Lifim Pofolio Slion nd Unain: oninos im as Jonal of onomis and Saisis ol [3] Mon Opimm onsmpion and pofolio ls in oninos im onol modl Jonal of onomi o ol [] Dzko N and Si S P Opimal ploaion and onsmpion of a naal so: soasi as Innaional Jonal of Poli nalsis ol [5] Dokn J and Fiingo G lial onsmpion pans and aional addiion Jonal of mian onomi iw ol [6] Fiing G and Noak Opimal onsmpion aining woking im and lis o lif l Jonal of Opimizaion o and ppliaions ol [7] Si S PDnami opimal onol modls in adising: s SIM iw ol No [8] Si SP and ompson GL Opimal onol o: ppliaions o Managmn Sin and onomis nd diion Klw admi Pbliss Dod [9] Ponagain Nssa ondiions fo opimali fo opimal onol poblm Jonal of onomis and Saisis ol [] Mon n inmpoal apial ass piing modl Jonal of onomia ol [] l-goa aos and opimal onol of an slf mission and mo ssm sad-sas soasi appoa Jonal of aos Solions and Faals ol [] l-goa adj L and l-ama Opimal onol of a soasi podion planning modl wi diffn dmand as Innaional Jonal of pplid Mamais ol [3] ikoia lsk-nikola lgoim fo opimal onol of nonlina soasi modl Jonal of Opimal onol ppliaions and Mods ol3; 7-98 NK Oladjo Opimal Soasi onol o and is appliaion o Landmak Unisi Dlopmn ns and insmn modl IOS Jonal of Mamais IOS-JM ol 3 no 5 7 pp 3 DOI: 979/ wwwiosjonalsog 3 Pag