News-generated dependence and optimal portfolios for n stocks in a market of Barndor -Nielsen and Shephard type.

Transcription

1 New-generaed dependence and opimal porfolio for n ock in a marke of Barndor -Nielen and Shephard ype. Carl Lindberg Deparmen of Mahemaical Saiic Chalmer Univeriy of Technology and Göeborg Univeriy Göeborg, Sweden. Sepember 8, 4 Abrac We conider Meron porfolio opimizaion problem in a Black and Schole marke wih non-gauian ochaic volailiy of Ornein-Uhlenbeck ype. The inveor can rade in n ock and a rik-free bond. We aume ha he dependence beween ock lie in ha hey parly hare he Ornein-Uhlenbeck procee of he volailiy. We refer o hee a new procee, and inerpre hi a ha dependence beween ock lie olely in heir reacion o he ame new. We how ha hi dependence generae covariance, and give aiical mehod for boh he ing and veri caion of he model o daa. Uing dynamic programming, we derive and verify explici rading raegie and Feynman-Kac repreenaion of he value funcion for power uiliy. Inroducion A claical problem in mahemaical nance i he queion of how o opimally allocae capial beween di eren ae. In a Black and Schole marke wih conan coe cien, hi wa olved by Meron in [5] and [6]. Recenly, [6] olved he ame problem for one ock and a bond in he more general marke model of [3]. In [3], Barndor -Nielen and Shephard propoe modeling he volailiy in ae price dynamic a a weighed um of non-gauian Ornein- Uhlenbeck (OU) procee of he form dy () = y () d + dz () ; The auhor would like o hank hi upervior Holger Roozén and Fred Epen Benh for valuable dicuion, a well a for carefully reading hrough preliminary verion of hi paper.

2 where z i a ubordinaor and >. Thi framework i a powerful modeling ool ha allow u o capure everal of he oberved feaure in nancial ime erie, uch a emi-heavy ail, volailiy cluering, and kewne. We exend he model by inroducing a new dependence rucure, in which he dependence beween ae lie in ha hey hare ome of he OU procee of he volailiy. We will refer o he OU procee a new procee, which implie he inerpreaion ha he dependence beween nancial ae i reacion o he ame new. We how ha hi dependence generae covariance, and give aiical mehod for boh he ing and veri caion of he model o daa. In hi exended model we conider an inveor who wan o maximize her uiliy from erminal wealh by inveing in n ock and a bond. Thi problem i an n-ock exenion of [6]. We allow for he inveor o have rericion on he fracion of wealh held in each ock, a well a borrowing and hor-elling rericion on he enire porfolio. For impliciy of noaion, we have formulaed and olved he problem for wo ock and a bond. However, he general cae i compleely analogou. The ochaic opimizaion problem i olved via dynamic programming and he aociaed Hamilon-Jakobi-Bellman (HJB) inegro-di erenial equaion. By ue of a veri caion heorem, we idenify he opimal expeced uiliy from erminal wealh a he oluion of a econd-order inegro-di erenial equaion. For power uiliy, we hen compue he oluion o hi equaion via a Feynman-Kac repreenaion, and obain explici opimal allocaion raegie. All reul are derived under exponenial inegrabiliy aumpion on he Lévy meaure of he ubordinaor. Recenly, porfolio opimizaion under ochaic volailiy ha been reaed in a number of aricle. In [9] and [], Meron problem i udied wih ochaic volailiy being modelled a a mean-revering proce. The paper [8] ue parial obervaion o olve a porfolio problem wih a ochaic volailiy proce driven by a Brownian moion correlaed o he dynamic of he riky ae. Going beyond he claical geomeric Brownian moion, [4], [5], and [8] rea di eren porfolio problem when he riky ae are driven by Lévy procee, and [] derive explici oluion for log-opimal porfolio in erm of he emimaringale characeriic of he price proce. For an inroducion o he marke model of Barndor -Nielen and Shephard we refer o [] and [3]. For opion pricing in hi conex, ee [7]. Thi paper ha even ecion. In Secion we give a rigorou formulaion of he marke and he porfolio opimizaion problem. We alo dicu he marke model and he implicaion of he dependence rucure. In Secion 3 we derive ome ueful reul on he ochaic volailiy model, and on momen of he wealh proce. We prove our veri caion heorem in Secion 4, and ue i in Secion 5 o verify he oluion we have obained. Secion 6 ae our reul, wihou proof, in he general eing. We dicu our reul and fuure reearch in Secion 7.

3 The opimizaion problem In hi ecion we de ne, and dicu, he marke model. We alo e up our opimizaion problem.. The marke model For T <, we aume a given a complee probabiliy pace (; F; P ) wih a lraion ff g T aifying he uual condiion. Inroduce m independen ubordinaor Z, and denoe heir Lévy meaure by l (dz); = ; :::; m: Remember ha a ubordinaor i de ned o be a Lévy proce aking value in [; ) ; which implie ha i ample pah are increaing. The Lévy meaure l of a ubordinaor ai e he condiion Z + min(; z)l(dz) < : We aume ha we ue he cádlág verion of Z : Le B i ; i = ; ; be wo Wiener procee independen of all he ubordinaor. We now inroduce our ochaic volailiy model. I i an exenion of he model propoed by Barndor -Nielen and Shephard in [3] o he cae of wo ock, under a pecial dependence rucure. To begin wih, our model i idenical o heir. We will dicu he di erence a hey occur. The nex exenion of he model, o n ock, i only a maer of noaion. Denoe by Y ; = ; :::; m, he OU ochaic procee whoe dynamic are governed by (.) dy () = Y ()d + dz ( ), where he rae of decay i denoed by > : The unuual iming of Z i choen o ha he marginal diribuion of Y will be unchanged regardle of he value of : To make he OU procee and he Wiener procee imulaneouly adaped, we ue he lraion f (B () ; B () ; Z ( ) ; :::; Z m ( m ))g T : From now on we view he procee Y ; = ; :::; m in our model a new procee aociaed o cerain even, and he ump ime of Z ; = ; :::; m a new or he releae of informaion on he marke. The aionary proce Y i repreenable a bu can alo be wrien a Y () = R exp (u) dz ( + u) ; ; (.) Y () = y e ( ) + R e ( u) dz ( u); ; where y := Y () ; and y ha he aionary marginal diribuion of he proce and i independen of Z () Z () ; : In paricular, if y = Y () ; 3

4 hen Y () ; ince Z i non-decreaing. We e Z () =, = ; :::; m; and e y := (y ; :::; y m ) : We aume he uual rik-free bond dynamic dr () = rr () d; wih inere rae r >. De ne he wo ock S ; S o have he dynamic (.3) ds i () = ( i + i i ()) S i () d + p i ()S i () db i (). Here i are he conan mean rae of reurn; and i are kewne parameer. We will call i + i i () he mean rae of reurn for ock i a ime : For noaional impliciy in our porfolio problem we denoe he volailiy procee by i inead of he more cuomary i : We de ne i a (.4) i () := ;y i () := P m! i;y () ; [; T ] ; where! i; are weigh umming o one for each i: The noaion ;y i denoe condiioning on Y () : Our model i here no he ame a u wo eparae model of Barndor -Nielen and Shephard ype. The di erence i ha he volailiy procee depend on he ame new procee. Thee volailiy dynamic give u he ock price procee (.5) S i () = S i () exp i + i i (u) du + p i (u)db i (u) : Thi ock price model doe no have aiically independen incremen and i i non-aionary. I alo allow for he incremen of he reurn R i () := log (S i () =S i ()) ; i = ; ; o have emi-heavy ail a well a boh volailiy cluering and kewne. The incremen of he reurn R i are aionary ince R i () R i () = log Si () S i () log where " = L " denoe equaliy in law. Si () S i (). Dicuion of he marke model = log Si () S i () = L R i ( ) ; Thi ecion aim o how ha he dependence rucure propoed in Secion. i no only imple from a aiical poin of view, bu alo ha very appealing economical inerpreaion. The paper [3] ugge a model wih n ock wih dynamic ds () = f + ()g S () d + () S () db(); where i a ime-varying ochaic volailiy marix, and are vecor, and B i a vecor of independen Wiener procee. Thi model include our a a pecial cae wih being a diagonal marix. However, in he claical Black 4

5 and Schole marke, dependence i modelled by covariance. In he cae of wo ock hi mean ha for ; S () = S () exp ( ) + p B () + p B () ; and S () = S () exp ( ) + p B () + p B () ; for a volailiy marix ; and B () = B () =. In our model, ock price develop independenly beide from reacing o he ame new. From an economic viewpoin, one can expec he model parameer o be more able han in he claical Black and Schole marke. For example, we do no require abiliy over expeced rae of reurn. Inead we ak ha every ime he marke i nervou o a cerain degree, i.e. for every peci c value of he volailiy i ; he mean rae of reurn i + i i will be he ame. We can inerpre hi a ha we only need abiliy in how he marke reac o new. A we will ee, for he purpoe of porfolio opimizaion we do no need o know he weigh! i; : More imporanly, he model generae a non-diagonal covariance marix for he incremen of he reurn over he ame ime period, which i he mo frequenly ued meaure of dependence in nance. Since he reurn have aionary incremen, i i u cien o how hi reul for R i ; i = ; : Noe ha we have Cov (R () R () ; R (u) R (v)) = Cov (R () ; R (u)) Cov (R () ; R (v)) Cov (R () ; R (u)) + Cov (R () ; R (v)) ; for ; ; u; v [; T ] : A will be hown below, for ; [; T ] ; we have ha (.6) Cov (R () ; R ()) = X m! ;! ; V ar (Y ()) e + e e + min (; ) ; which for = impli e o (.7) Cov (R () ; R ()) = m X! ;! ; V ar (Y ()) e + : Thi reul ay ha he model generae a covariance marix beween reurn, bu we do no immediaely know which correlaion ha can be obained. I 5

6 urn ou ha we can ge correlaion Corr (R () ; R ()) in he enire inerval ( ; ) : To derive Equaion (.6), by de niion of i we have ha E [R () R ()] = E + (u) du + Z Z + (u) du + X m Z = +! ; E Y (u) du X m +! ; E Y (u) du + m X i; p (u)db (u) p (u)db (u)! ;i! ; E Y i (u) du Z Y (u) du : Similarly, E [R ()] = + m X Z! ; E Y (u) du : Thi give ha Cov (R () ; R ()) = m X! ;! ; Cov Y (u) du; Z Y (u) du : By aionariy, we have ha E [Y ()] = Y ; for ome conan Y > ; for all R: If we aume ha u v; he independence of he incremen of Y give ha Cov (Y (u) ; Y (v)) = E Y (u) Y Y (v) Y Z v = E e (v u) Y (u) + Y (u) e (v ) dz ( ) u h = e (v u) E Y () i e (v u) Y Y = e (v u) V ar (Y ()) : The ame calculaion for v u how ha Cov (Y (u) ; Y (v)) = e v u V ar (Y ()) ; 6

7 and we ge (.8) Cov = Z Y (u) du; Z Y (u) du Cov (Y (u) ; Y (v)) dudv = V ar (Y ()) e + e e + min (; ) : By Iô iomery (ee []) we ge, imilar o above, V ar (R i ()) = for i = ; : Thi give and, for = ; Corr (R () ; R ()) = i! i; V ar (Y ()) e! ;! ; V ar (Y ()) e + e e + min (; ) v u P m! ; V ar (Y ()) e + v u P m! ; V ar (Y ()) e Corr (R () ; R ()) = v u P m! ; V ar (Y ()) e v u P m! ; V ar (Y ()) e! + +! i; Y ;! +!; Y + +!; Y! ;! ;! ; V ar (Y ()) e +! + +!; Y + +!; Y! : 7

8 There i alway a rade-o beween accuracy and applicabiliy when deigning model. An obviou advanage of our model i ha we do no have o eimae a ochaic volailiy marix, and hence we need le daa o obain good eimae of he model parameer. A drawback i ha, o obain high correlaion, we need he model o be very kew. Thi migh no oberved daa. I remain o ee if he rade-o wa a good one..3 Saiical mehodology In hi ecion we decribe a mehodology for ing he model o reurn daa. We will do hi for a Normalized Invere Gauian diribuion (NIG) ; which ha been hown o nancial daa well, ee e.g. [], [7] and []. Our choice play no formal role in he analyi. We aume ha we are oberving R i () ; R i () R i () ; :::; R i (k) R i ((k ) ) ; where i one day, and k + i he number of conecuive rading day in our period of obervaion. The NIG-diribuion ha parameer = p + ; ; ; and : I deniy funcion i f NIG (x; ; ; ; ) = exp p x q x K q e x ; where q (x) = p + x and K denoe he modi ed Beel funcion of he hird kind wih index : The domain of he parameer i R; > ; and : A andard reul i ha if we ake o have an Invere Gauian diribuion (IG) ; and draw a N (; )-diribued random variable "; hen x = ++ p " will be N IG-diribued. The IG-diribuion ha deniy funcion f IG (x; ; ) = p exp () x 3 exp x + x ; x > ; where and are he ame a in he NIG-diribuion. The exience and inegrabiliy of Lévy meaure l uch ha he volailiy procee i will have IG-diribued marginal i no obviou. See [] and [, Secion 7] for hi heory. The Lévy deniy l of he ubordinaor Z of an IG-diribued new proce Y i l (x) = () x + x e x ; where (; ) are he parameer of he IG-diribuion. The mehod decribed in [3], which we will furher exend, ue ha he marginal diribuion of he volailiy procee i are invarian o he rae of decay : Thee parameer are hen ued o he auocorrelaion funcion of he i o log-reurn daa. The auocorrelaion i de ned by (h) = Cov((h);()) V ar(()) ; h R: 8

9 For impliciy of expoiion we will aume ha we only need one o correcly model he auocorrelaion funcion of boh ock. However, for reaon o be explained laer, we will aume ha m = 3; and ha all = = 3 = : For our model calculaion how ha, for general m; i (h) =! i; exp ( h) + ::: +! i;m exp ( m h) ; where he! i; ; are he weigh from he volailiy procee ha um o one. Oberve ha ince we have aumed he rae of decay o be equal, we immediaely ge ha i (h) = exp ( h) : We proved hi more imple reul in Subecion.. The proof of he general cae i analogou. We aume ha we have ed N IG-diribuion o he empirical marginal diribuion of wo ock, and ha we have found a uch ha our model ha he righ auocorrelaion funcion. Thi can be done by empirically calculaing he auocorrelaion funcion i (h) for di eren value of h; and hen nd a o ha he heoreical and empirical auocorrelaion funcion mach. We denoe he IG-parameer of he volailiy procee i by ( i ; i ) ; i = ; : By Equaion (.7) we can now he covariance of he model o he empirical covariance from he reurn daa. Thi can be done by leing he wo ock hare he new proce Y 3, and each have one of he new procee Y i ; i = ; ; of heir own. In general, hi i done for each rae of decay. We formulae hi mahemaically a =! ; Y +! ;3 Y 3 IG ( ; ) =! ; Y +! ;3 Y 3 IG ( ; ) : We now ae wo properie of IG-diribued random variable ha we will need below. For X IG ( X ; X ), we have ha ax IG a X ; a X : Furhermore, if Y IG ( Y ; Y ) and i independen of X and we aume ha X = Y =: ; we have ha X + Y IG ( X + Y ; ) : Becaue of hi formula we can le! ; Y IG ( ; ; )! ;3 Y 3 IG ( ;3 ; ) ; where (.9) ; + ;3 = ; and where! ; Y IG ( ; ; )! ;3 Y 3 IG ( ;3 ; ) ; (.) ; + ;3 = ; 9

10 We ee, by he caling propery of he IG-diribuion, ha he wo expreion for he diribuion of Y 3, (.) Y 3 IG! ;3 ;3;! ;3 ; and (.) Y 3 IG! ;3 ;3;! ;3 ; mu be idenical. Wih he aid of Equaion (.8), in which we ue ha he variance V ar (Y ()) of a aionary invere Gauian proce Y i = 3, we ee ha Equaion (.7) become! ;3 ;3 e (.3)! ;3 3 + = C where C i he covariance ha we wan he reurn o have. I i now raighforward o check ha here are non-unique choice of! i; uch ha we can obain boh he righ auocorrelaion funcion of i and a peci c covariance for he reurn. The auocorrelaion funcion parameer i already correc by aumpion, and we conruced he new procee Y o ha heir marginal diribuion would no depend on i. Hence we only have o ake care of he covariance of he reurn R i. We do hi by uing Equaion (.9),..., (.3). Noe ha here i nohing crucial in our choice of covariance a meaure of dependence, nor doe i maer how many di eren rae of decay we ue. We now give a imple approach o deermine how well our model capure he rue covariance. We begin by ing a marginal diribuion o reurn daa, hereby obaining he parameer i and i ; i = ; : Since we have ha he reurn procee i ; i = ; ; are emimaringale, heir quadraic variaion, denoed by [] ; are R i (u) du; : Tha i, for a equence of random pariion ending o he ideniy, we have [log (S i =S i ())] () = i (u) du; where convergence i uniformly on compac in probabiliy. Thi i a andard reul in ochaic calculu. For each rading day we now empirically calculae he inegraed volailiy, ha i, we calculae he quadraic variaion of he oberved reurn over a rading day and, by he formula above, ue ha a a conan approximaion of he volailiy during ha day. If we do hi for a number of rading day, we ge approximaion of he volailiy procee i for ha period of ime. Uing he ed parameer i, i and generaed N (; )-diribued variable in Equaion (.5), we can now imulae alernaive reurn. We hen calculae he covariance-marix of boh he reurn daa e and he imulaed alernaive reurn and compare hem aiically.

11 .4 The conrol problem A main purpoe of hi paper i o nd rading raegie ha opimize he rader expeced uiliy from wealh in a deerminiic fuure poin in ime. The uiliy i meaured by a uiliy funcion U choen by he rader. Thi uiliy funcion U i a meaure of he rader averion oward rik, in ha i concreize how much he rader i willing o rik o obain a cerain level of wealh. Our approach o nding hee rading raegie, and he value funcion V, i dynamic programming and ochaic conrol. We will make ue of many of he reul found in [6], ince mo of heir idea are applicable in our eing. However, we need o adap heir reul o our cae. In hi ecion we e up he conrol problem under he ock price dynamic of Equaion (.3). Recall ha and ; are weighed um of he new procee, ee Equaion (.4). We begin by de ning a value funcion V a he maximum amoun of expeced uiliy ha we can obain from a rading raegy, given a cerain amoun of capial. We hen e up he aociaed Hamilon-Jakobi-Bellman equaion of he value funcion V: Thi equaion i a cenral par of our problem, a i i, in a ene, an opimaliy condiion. Mo of he laer ecion will be devoed o nding and verifying oluion o i. Denoe by i () he fracion of wealh inveed in ock i a ime, and e = ( ; ) : The fracion of wealh held in he rik-free ae i ( ). We allow no hor-elling of ock or bond, which implie he condiion i [; ] ; i = ; ; and + ; a.., for all T: However, hee rericion are parly for mahemaical convenience. We could equally well have choen conan a i ; b i ; c; d R; a i < b i ; c < d; uch ha he conrain would have aken he form i [a i ; b i ] ; i = ; ; and c + d; a.., for all T: The analyi i analogou in hi cae, bu more noaionally complex. Thi general eing allow u o conider, for example, law enforced rericion on he fracion of wealh held in a peci c ock, a well a hor-elling and borrowing of capial. We ae he main reul in hi eing, furher generalized o n ock, in Secion 6. The wealh proce W i de ned a W () = () W () S () S () + () W () S () S () + ( () ()) W () R () ; R () where i () W () =S i () i he number of hare of ock i which i held a ime : We alo aume ha he porfolio need o be elf- nancing in he ene ha no capial i enered or wihdrawn. Thi can be formulaed mahemaically a W () = W ()+ X i= Z i (u)w (u) ds i (u)+ S i (u) ( (u) (u)) W (u) dr (u) ; R(u) for all [; T ] : See [4] for a moivaing dicuion. The elf- nancing condi-

12 ion give he wealh dynamic for T a (.4) dw () = W () () ( + () r) d + W () () ( + () r) d + rw ()d + () p ()W ()db () + () p ()W ()db (); wih iniial wealh W () = w: Our de niion of he e of admiible conrol now eem naural. De niion. The e A of admiible conrol i given by A := f = ( ; ) : i i progreively meaurable, i () [; ] ; i = ; ; and + a.. for all T; and a unique oluion W of Equaion (.4) exig. An invemen raegy = f () : T g i aid o be admiible if A. Laer we will need ome exponenial inegrabiliy condiion on he Lévy meaure. We herefore aume ha he following hold: Condiion. For a conan c > o be peci ed below, Z + (e cz ) l (dz) < ; = ; :::; m: Recall ha he Lévy deniy l of he ubordinaor Z of an IG-diribued new proce Y i l (x) = () x + x e x ; where (; ) are he parameer of he IG-diribuion. Hence Condiion. i ai ed for c =: We know from he heory of ubordinaor ha we have h (.5) E e az()i Z = exp (e az ) l (dz) a long a a c wih c from Condiion. hold. Denoe (; ) by R + and [; ) by R + ; and aume ha y = (y ; :::; y m ) R m +. De ne he domain D by + D := f(w; y) R + R m +g: We will eek o maximize he funcional J(; w; y; ) = E ;w;y [U (W (T ))] ; where he noaion E ;w;y mean expecaion condiioned by W () = w; and Y () = y ; = ; :::; m: The funcion U i he inveor uiliy funcion. I i aumed o be concave, non-decreaing, bounded from below, and of ublinear growh in he ene ha here exi poiive conan k and (; ) o ha

13 U(w) k( + w ) for all w : Hence our ochaic conrol problem i o deermine he value funcion (.6) V (; w; y) = up A J(; w; y; ); (; w; y) [; T ] D; and an invemen raegy A, he opimal invemen raegy, uch ha V (; w; y) = J(; w; y; ): The HJB equaion aociaed o our ochaic conrol problem i (.7) = v + max i[;];i=;; + f( ( + r) + ( + r)) wv w + + X m w v ww + rwv w y v y + Z (v (; w; y + z e ) v (; w; y)) l (dz); for (; w; y) [; T ) D: We oberve ha we have he erminal condiion (.8) V (T; w; y) = U(w); for all (w; y) D; and he boundary condiion (.9) V (; ; y) = U(); for all (; y) [; T ] R m +: We now give a formal moivaion o hi equaion. The HJB equaion i obained by eing he upremum of he in nieimal generaor A of (W; Y ) applied o he value funcion V o zero. In oher word, if we aume ha V C ;, he HJB equaion i max i[;];i=;; + = max i[;];i=;; + (AV ) (w; y) E [V (; W () ; Y ())] V (; w; y) lim = ; # where we have ued he de niion of A in he r equaliy, and Iô formula (ee [9]) o evaluae E [V (; W () ; Y ())] : If we denoe he coninuou par of he quadraic covariaion by [; ] c, ue he noaion V () for V (; W () ; Y ()) ; 3

14 and e X () := X () X ( ) ; Iô formula give ha V () V () Z = V y ( ) dy () + + Z Z + + V ww ( ) d [W; W ] c () X < m : V () V ( ) X V y ( Z V w ( ) dw () + V ( + 9 = ) Y () ; : ) d Le N denoe he Poion random meaure in he Lévy-Khinchine repreenaion of Z : We have ued ha [Y ; Y ] c = ; = ; :::; m; by Theorem 6 in [9]. The Kunia-Waanabe inequaliy (ee [9, p. 69]) ell u ha d [X; Y ] c i a:e:(pah by pah) aboluely coninuou wih repec o d [Y ; Y ] c ; = ; :::; m; for a emimaringale X: Equaion (.) and (.4) now give Equaion (.7) once we have een ha, under quie general inegrabiliy condiion, 3 E 4 X fv (; W () ; Y ()) V ( ; W ( ) ; Y ( ))g5 = E 4 = Z Z Z + V (; W () ; Y ( ) + z e ) V (; W () ; Y ( )) N ( d; dz) 5 Z E [V (; W () ; Y () + z e ) + V (; W () ; Y ())] l (dz) d; where we have ued Fubini-Tonelli heorem and he fac ha, for Borel e ; N (; ) l () i a maringale, = ; :::; m. 3 Preliminary eimae Thi ecion aim a relaing he exience of exponenial momen of Y o exponenial inegrabiliy condiion on he Lévy meaure, a well a developing momen eimae for he wealh proce and howing ha he value funcion i well-de ned. Lemma 3. Aume Condiion. hold wih c = = for >. Then E exp( Y (u)du) exp y + Z + z exp l (dz)( ) 3 4

15 Proof. We ge from he dynamic (.) of Y ha Y (u)du = y + Z ( ) Z ( ) Y () y + Z ( ) Z ( ) = L y + Z ( ( )); ince Y () when y = Y () ; and " = L " denoe equaliy in law. Recall ha we have de ned Z () = : We hu have, uing Equaion (.5) in he la ep, ha E exp Y (u)du exp y = exp y + E exp Z + Z ( ( z exp )) l (dz)( ) : Lemma 3. Aume Condiion. hold for ome poiive conan c : Then Z E [exp(c Y ())] exp c y + fexp (c z) g l (dz)( ) + Proof. We ee from Equaion (.) ha c Y () c y + c Z ( ) c Z ( ) The reul follow from Equaion (.5). = L c y + c Z ( ( )) : Lemma 3.3 Aume Condiion. hold wih for ome > : Then where c = (+)!;+(+)!; ; = ; :::; m; up E ;w;y (W ()) A w ( + )! ; + ( + )! ; y + C()( ) A ; C() = ( r + r + r) + Z exp ( + )! ; + ( + )! ; z l (dz): + 5

16 Proof. We have by Equaion (.4) and Iô formula ha W () = w exp (u; (u) ; (u)) du where + Z (u) p (u)db (u) + Z (u) p (u)db (u) ; (u; ; ) = (u)( + r) + (u)( + r) + r ( (u)) ( (u)) : De ne X() = exp (u) p (u)db (u) + () ( (u)) (u) du (u) p (u)db (u) () ( (u)) (u) du : Since he procee Y () are righ-coninuou we have ha i (); i = ; ; are righ-coninuou. Due o he exponenial inegrabiliy condiion on Y we have ha "!# E exp Z T () + () d < : Thi implie ha R i(u) p i (u)db i (u); i = ; ; are well-de ned coninuou maringale. Then X() i a maringale by Novikov condiion (ee [9, p. 4]), and E [X()] = : Lemma 3. wih = (! ; +! ; ) ; = ; :::; m; give h E e E = R T () ( (u)) (u)du+ he R T my (u)du+ R T R i T () ( (u)) (u)du (u)dui E he (! R T ;+! ;) Y(u)dui < : Hence, by Hölder inequaliy and uing ha i [; ]; i = ; ; E (W ()) = w E exp + (u; (u); (u))du (u) p (u)db (u) + (u) p (u)db (u) 6

17 = w E exp (u; (u); (u))du + ( (u)) (u) + ( (u)) (u)du X() w E exp (u; (u); (u))du w e + ( (u)) (u) + ( (u)) (u)du E [X()] ( )( r+ r+r) E exp ( + ) (u)du + ( + ) = w ( )( r+ r+r) e my E exp (( + )! ; + ( + )! ; ) Applying Lemma 3. wih = ( + )! ; + ( + )! ; ; = ; :::; m; (u)du Y (u)du prove he reul. We now ue he reul above reul o how ha he value funcion of our conrol problem i well-de ned. Propoiion 3. Aume Condiion. hold wih c de ned a in Lemma 3.3. Then U() V (; w; y) + w +C () (T ))) ; where C() i de ned a in Lemma 3.3 and k >. ( + )! ; + ( + )! ; y Proof. We have ha U(w) U() ince U i non-decreaing. Thi give ha E [U (W (T ))] U(); for A ; which implie ha V (; w; y) U(): The upper bound follow from he ublinear growh condiion of U and Lemma 3.3: V (; w; y) = up E [U (W (T ))] k A + w +C () (T ))) : + up A E [U (W (T ))] ( + )! ; + ( + )! ; y 7

18 From now on we aume ha Condiion. hold wih c = (+)!;+(+)!; ; = ; :::; m: Thi enure ha he value funcion i well-de ned. 4 A veri caion heorem We ae and prove he following veri caion heorem for our ochaic conrol problem. Theorem 4. Aume ha v(; w; y) C ;; ([; T ) (; ) [; ) m ) \ C([; T ] D) i a oluion of he HJB equaion (.7) wih erminal condiion (.8) and boundary condiion (.9). For = ; :::; m; aume Z T up A and Z + E [v (; W () ; Y ( ) + z e ) v (; W () ; Y ( ))] l (dz)d < ; up A R T E h( i ()) i () (W ()) (v w (; W () ; Y ())) i d < ; i = ; : Then v(; w; y) V (; w; y); for all (; w; y) [; T ] D: If, in addiion, here exi meaurable funcion i (; w; y) [; ]; i = ; ; being he maximizer for he max-operaor in Equaion (.7), hen = (; ) de- ne an opimal invemen raegy in feedback form if Equaion (.4) admi a unique oluion W and h i V (; w; y) = v(; w; y) = E ;w;y U W (T ) ; for all (; w; y) [; T ] D: The noaion C ;; ([; T ) (; ) [; ) m ) mean wice coninuouly di ereniable in w on (; ) and once coninuouly di ereniable in ; y on [; T ) [; ) m wih coninuou exenion of he derivaive o = and y = ; = ; :::; m: Proof. Le (; w; y) [; T ) D and A ; and inroduce he operaor M v := ( ( + r) + ( + r)) wv w + + X m w v ww + rwv w y v y : 8

19 Iô formula give ha v (; W () ; Y ()) = v(; w; y) fv (u; W (u) ; Y (u )) + M v (u; W (u) ; Y (u))g du (u) p (u)w (u) v w (u; W (u) ; Y (u)) db (u) (u) p (u)w (u) v w (u; W (u) ; Y (u)) db (u) Z + (v (u; W (u) ; Y (u ) + z e ) v (u; W (u) ; Y (u ))) N ( du; dz); where N i he Poion random meaure coming from he Lévy-Khinchine repreenaion of he ubordinaor Z : We know from he aumpion ha he Iô inegral are maringale and ha he inegral wih repec o N are emimaringale. Thi give u ha E [v(; W (); Y ())] = v(; w; y) + E v(; w; y) + E 4 = v(; w; + (v + L v) (u; W (u); Y (u)) du max i[;];i=;; + 3 L va (u; W (u); Y (u)) du5 where Z L v := M v + (v(; w; y + z e ) + v(; w; y)) l (dz): We now ge ha v(; w; y) E [U (W (T ))] ; for all A, by puing = T and invoking he erminal condiion for v: The r concluion in he heorem now follow by oberving ha he reul hold for = T and w = : We prove he econd par by oberving ha ince for each i = ; ; i (; w; y) i aumed o be a meaurable funcion, we have ha i (; W (); Y ()) i F - meaurable for T: Thi, ogeher wih he aumpion ha i [; ] and he exience of a unique oluion W of Equaion (.4), implie ha (; W (); Y ()) i an admiible conrol. Moreover, ince i a maximizer, max i[;];i=;; + L v = L v 9

20 The above calculaion uing Iô formula go hrough wih equaliy by leing = : Hence, h i v (; w; y) = E U W (T ) V (; w; y) : Togeher wih he r par of he heorem, hi yield h i v (; w; y) = V (; w; y) = E U W (T ) ; for (; w; y) [; T ] D, ince he equaliy obviouly hold for = T and w = : 5 Explici oluion In hi ecion we conruc and verify an explici oluion o he conrol problem (.6), a well a an explici opimal conrol, when he uiliy funcion i of he form U(w) = w ; (; ): 5. Reducion of he HJB equaion In hi ubecion we reduce he HJB equaion (.7) o a r-order inegrodi erenial equaion by making a conecure ha he value funcion v ha a cerain form. We conecure ha he value funcion ha he form (5.) v(; w; y) = w h(; y); (; w; y) [; T ] D; for ome funcion h(; y): We de ne he funcion : [; ) [; )! R a ( ; ) = max i[;];i=; + f ( + r) + ( + r) + ( ) + r: If we iner he conecured value funcion ino he HJB equaion (.7) we ge a r-order inegro-di erenial equaion for h a (5.) h (; y) = ( ; )h (; y) + y h y (; y) Z (h (; y + z e ) h (; y)) l (dz) ; where (; y) [; T ) [; ) m : The erminal condiion become + h (T; y) = ; 8y [; ) m ; ince v(t; w; y) = U (w) = w. For our purpoe, we will need o be coninuouly di ereniable.

21 5. Coninuou di ereniabiliy of Here we prove ha i coninuouly di ereniable. We alo obain candidae for opimal fracion of wealh. A r-order condiion for an inerior opimum of ( ; ) i ( i + i i r) i i ( ) = ; i = ; : If we denoe he inerior opimum by i = i ( i ); i = ; ; hen we have i ( i ) = + i ; i = ; : i r i We ge from inpecion ha Equaion (5.) i coninuou and di ereniable whenever i (; ) ; i = ; ; + < : Elemenary calculu now give ha i = ; when i ; i = ; ; and ha for + ; he vecor of opimal fracion of wealh i of he form ( ; ) = (; ) ; [; ] : In he laer cae, Equaion (5.) aler o (5.3) ( ; ) = max f ( + r) + ( ) ( + r) [;] o + ( ) ( ) + r: Here he r-order condiion for an inerior opimum i ( + r) ( + r) ( ( ) ) ( ) = : In he name of conequence, we denoe he inerior opimum by = ( ; ) : Thi give ha ( ; ) = ( ) ( + r) ( + r) ( + ) + ( + ) ; and we eaily ee ha Equaion (5.3) i coninuou and di ereniable on (; ) : We will now prove ha lim!^i i (^ ; ^ ) = lim!i i (^ ; ^ ) ; i = ; ; for (^ ; ^ ) uch ha + = : We prove he reul for he derivaive aken in ; he reul for being analogou. The key o hi reul i o oberve ha when + = ; = = ( ) r + ( + r) ( + r) ( + ) + ( + ) :

22 For noaional impliciy, e i = ( i + i i r) ; i = ; : Calculaion how ha! ( ) ( + ) ( ) = ( ) ( + )! ( + ) ( ) ( ) + ( + ) ( + ) Bu we alo have ha = ( ) ( ) ( + ) ( ) ( + ) ( ) ( + ) ( ) = : = ( ) ( ) = : The proof ha i coninuouly di ereniable when i = ; i = ; ; i imilar o he reul above and we omi i. By he reul of hi ubecion we can now conclude ha for i = ; ; our candidae for opimal fracion of wealh are (5.4) i ( i ) = i whenever i (; ) and + < ; and (5.5) i = ; i r + i ; when i : When +, he opimal fracion of wealh are (5.6) ( + r) ( + r) ( ; ) = + ( ) ( + ) ( + ) ; and (5.7) = : Remark 5. Noe ha we can nd a conan > uch ha ( ; ) + + : 5.3 A Feynman-Kac formula In hi ubecion we de ne a Feynman-Kac formula ha we verify a a claical oluion o he relaed forward problem of Equaion (5.).

23 De ne he funcion g (; y) by R i g (; y) = E hexp y ( () ; ()) d ; (; y) [; T ] [; ) m ; where we denoe y i := ;y i = P m! i;y ; i = ; ; for y = Y () : Noe ha g (; y) = : We now how ha g i well-de ned under an exponenial growh hypohei in and : Lemma 5. Aume Condiion. hold wih c = (! ; +! ; ) for = ; :::; m: Then (! ; +! ; ) g (; y) + y A ; for ome poiive conan k: Proof. From Remark 5. we know ha ( ; ) + + for ome conan > : Therefore, Z g (; y) = E exp y ( () ; ()) d Z E exp y my e E y 4 + () + () d 3 e (!;+!;) R Y y ()d By independence of he Y ; = ; :::; m; we ge by Lemma 3. ha 3 my g (; y) e a E y 4 e (!;+!;) R Y y ()d5 e a m Y + Z (! ; +! ; ) exp + exp y (! ; +! ; ) Hence, here exi a poiive conan k uch ha z g (; y) e k+b P m (! ; +! ; ) y ; and we are done. We now aim o how ha g i coninuouly di ereniable in y. 5 : l (dz) 3

24 Lemma 5. Aume Condiion. hold wih c = (! ; +! ; ) ; = ; :::; m: Then g C ; (([; T ]) [; ) m ) ; ha i, g (; ) i coninuou for all [; T ] and g (; y) i once coninuouly di ereniable for all y [; ) m : Proof. We will ue he dominaed convergence heorem o prove ha we can inerchange expecaion and di ereniaion. Su cienly general condiion for u o do hi are conained in Theorem.7 in [], which eenially ay ha we need o bound he derivaive by an inegrable funcion independen of y: Le (; y) [; T ] R m + and e For each = ; :::; m; we (; = F (; y) = e R (y ();y ())d : Z ( y () ; y ()) d e R (y ();y ())d : Since i coninuouly di ereniable and i bounded ( () ; = ( y () ; y ())! e ce for ome ricly poiive conan c: Theorem.7(b) in [] now give ha (; y) = ( () ; y ()) e d e R (y ();y ())d : From he aumpion we have ha (; c e d e R ((y ();y ()))d c e T + R y ()d+ R y ()d ; where we once again can apply Lemma 3. o ge (; y) =@y i nie. Furhermore, wih he aid of i proof, we can wihdraw ha on a compac e wih y in i (; y) =@y i uniformly bounded in y by he random variable (! ; +! ; ) exp Z ( ) ; which i inegrable by Condiion.. Once again,.7(b) in [] can be applied o how ha g (; y) = E [F (; y)] i di ereniable in y: Bu di ereniabiliy i a local noion. Hence he reul i independen of he choice of compac e, and we = E i ; 8y R m + ; = ; :::; m: 4

25 We alo have ha y (; y) =@y i coninuou ince y 7! ( y () ; y ()) ; ( y () ; y ()) 7! (y () ; y ()) ; and (y () ; y ()) 7! ( y () ; y ()) are all coninuou mapping. By uing Theorem.7(a) in [] we now ge ha he mapping (; y) (; y) =@y i coninuou. Lemma 5.3 Aume Condiion. hold wih c = (! ; +! ; ) for = ; :::; m. Then " Z T Z # E g (u; Y (u) + z e ) g (u; Y (u)) l (dz) du < : + Proof. Di ereniabiliy of g and he mean value heorem give ha g (u; y + z e ) g (u; y) (u; y + x e ) z kz (! ; +! ; ) (y + Z ( u)) A ; where k i a poiive conan depending only on T and he parameer of he problem. Since we have (! ; +! ; ) (! ; +! ; ) Y y (u) + Z ( u) (Z ( u) + y + Z ( u)) = (! ; +! ; ) (y + Z ( u)) ; g (u; Y (u) + z e ) g (u; Y (u)) kz (! ; +! ; ) (y + Z ( u)) A : From he Tonelli heorem, he aumpion, and Equaion (.5) we have " Z T Z # E g (u; Y (u) + z e ) g (u; Y (u)) l (dz) du Z T + Z E kzl (dz) + (! ; +! ; ) 3 (y + Z ( u)) A5 du 5

26 = k < : Z + + u Z T zl (dz) Z + e (! ; +! ; ) z (! ; +! ; ) y l (dz) du We now how ha g (; y) i a claical oluion o he relaed forward problem of Equaion (5.). Propoiion 5. Aume here exi " > uch ha Condiion. i ai ed wih c = (! ; +! ; ) + " for = ; :::; m. Then g (; ) belong o he domain of he in nieimal generaor of Y and (; = ( ; ) g (; y) + (; Z (g (; y + z e ) g (; y)) l (dz) + for (; y) (; T ] [; ) m : (; y) =@ i coninuou, o ha g C ; ((; T ] [; ) m ) : Proof. We begin by oberving ha he condiion in Lemma 5., 5., and 5.3 are ful lled. The r wo erm on he righ-hand ide of Equaion (5.8) are coninuou ince i coninuou and g (; ) C by Lemma 5., for all [; T ] : The inegral operaor i alo coninuou in boh ime and pace. Thi can be deduced from he inegrabiliy condiion on he Lévy meaure l (dz) and Theorem.7 in [] ogeher wih argumen imilar o hoe of he proof of Lemma 5. and 5.3. Hence, if g olve Equaion (5.8) (; y) =@ i coninuou for (; y) (; T ) [; ) m ; and may be coninuouly exended o = T: Thu, g C ; ((; T ] [; ) m ) : Since y 7! g (; y) i coninuouly di ereniable by Lemma 5., we conclude from Iô lemma ha he mapping 7! g (; Y ()) i a local emimaringale wih dynamic g (; Y ()) = g (; y) + + Z + (; Y (u)) du (g (; Y (u ) + z e ) g (; Y (u ))) N ( du; dz) ; where N i he Poion random meaure in he Lévy-Khinchine repreenaion of Z : From Lemma 5.3 we have ha " Z T Z # E g (u; Y (u) + z e ) g (u; Y (u)) l (dz) du < ; + 6

27 and hu g (u; Y (u) + z e ) g (u; Y (u)) F ; = ; :::; m (ee [3, pp. 6-6], for hi noaion). Thi implie ha g (; Y ()) i a emimaringale, ince i belong o a ubcla of proper emimaringale ha [3], for impliciy, de ne o be emimaringale. Taking expecaion on boh ide and applying Fubini heorem give E [g (; Y ()) = + g (; y)] E Y (; Y (u)) Z + E [g (; Y (u ) + z e ) g (; Y (u ))] l (dz) du: Hence, if we noe ha Y i cádlág and y 7! g (; y) i coninuouly di ereniable, by leing # we ge ha g (; ) i in he domain of he in nieimal generaor of Y; which i denoed by G, Z Gg (; y) = y (; y) + (g (; y + z e ) g (; y)) l (dz) Since g (; Y ()) L (; P ) for all > in a neighborhood of zero, he Markov propery of Y ogeher wih oal expecaion yield Thu, E [g (; Y ()) E [g (; Y ())] = E E e h = E E = E = E + R Y y () (u); Y y () (u) du he R (y (u+);y (u+))du F ii he R + ( y (u))dui (u);y he R + ( y (u);y (u))du e R (y (u);y (u))dui : g (; y)] = E h e R + ( y (u);y (u))du e R (y (u);y (u))du e R (y (u);y (u))dui = h E e R + ( y (u);y (u))du e R (y (u);y (u))du e R + ( y (u))dui (u);y + n E he R + ( y (u))dui (u);y E he R (u))duio (y (u);y = E e R + ( y (u);y (u))du ne R o (y (u);y (u))du g ( + ; y) g (; y) + : 7

28 By he Fundamenal Theorem of Calculu we have ha e R + ( y (u);y (u))du ne R o (y (u);y (u))du! ( y ; y ) er (y (u);y (u))du ; # : We now need o how ha we may inerchange limi and inegraion. To do hi we de ne he funcion f () = e R (y (u);y (u))du : From he mean value heorem and he linear growh aumpion on we ge ha f () f up u[;t ] () f (u) + up u[;t ] e R T (+( y (u)+y (u)))du ( y (u) ; y (u)) e! + up ( y (u) + y (u)) : u[;t ] Since each Z i a non-decreaing proce, which implie up ( y (u) + y (u)) u[;t ] ( + ) + (! ; +! ; ) Z ( T ) ; e R + ( y (u);y (u))du ce R T (+ y (u)+y (u))du + d (! ; +! ; ) R (y (u);y (u))du ne R o (y (u);y (u))du e R T (+ y (u)+y (u))du Z ( T ) ; for ome poiive conan c; d: Bu by he independence of he Y and Equaion (.5) we have ha hce R T (+ y (u)+y (u))dui E c +! ; +! ; y A ; 8

29 for ome conan k: In addiion, we can ue ha here exi a poiive conan k " uch ha z k " e "z ; for all z ; which give ha d (! ; +! ; ) ^d E he R i T (+ y (u)+y (u))du Z ( T ) (! ; +! ; ) E " e (! ; +! ; ) +" Z ( T ) # = ^d (! ; +! ; ) exp T Z + e (! ; +! ; )+ " z l (dz) ; for ome conan ^d: The la um i nie by our inegrabiliy aumpion. Therefore, by dominaed convergence, we ee ha E e R + ( y (u);y (u))du ne R o (y (u);y (u))du = ( ; ) g (; y) : Analogouly, we may how exi. We now have ha which conclude he proof. De ne Gg (; y) = ( ; ) g (; y) (; y) h (; y) := g (T ; y) = E y h e R T ( y ();y ())di ; or equivalenly, by he ime-homogeneiy of Y; (5.9) h (; y) = E y h e R T (y ();y ())di : From our conecure of he form of he value funcion we now have our explici oluion candidae, namely (5.) v (; w; y) = w h (; y) : The candidae for he opimal feedback conrol i given in Subecion 5.. In he nex ecion we prove ha Equaion (5.) coincide wih he value funcion in Equaion (.6). 9

30 5.4 Explici oluion of he conrol problem We will apply he veri caion heorem o connec our explici oluion o he value funcion of he conrol problem. To hi end, we need wo inegrabiliy reul. Lemma 5.4 Aume Condiion. hold wih Then c = 8 (( + 4)! ; + ( + 4)! ; ) ; = ; :::; m: Z T for all A ; i = ; : h i E ( i ()) i () (W ()) h (; Y ()) d < ; Proof. We oberve ha he funcion h ha he ame growh a g: Therefore by Lemma 5. and i [; ] ; i = ; ; Z T h E (W ()) ( ()) () + ( ()) () i (W ()) ( ) h (; Y ()) d Z T h E (W ()) ( () + ())! ; +! ; + Z T h k " e kt E (W ())! ; +! ; + ") where k " i a poiive conan uch ha Hölder inequaliy now give Z T + k " e " P m! ; +! ; y : 3 Y () A5 d 3 Y () A5 d; h i E (W ()) ( ()) () + ( ()) () (W ()) ( ) h (; Y ()) k " e kt Z T my h (W ()) 4i d: ( + ") (! ; +! ; ) E exp Y () 3

31 I i obviou ha (+")(! ;+! ;) < c = ; :::; m; for " u cienly mall. Hence, he inegrabiliy condiion in Lemma 3. hold. We conclude by invoking Lemma 3.3. Lemma 5.5 For = ; :::; m; aume Condiion. hold wih Then Z T Z + c = 8 (( + 4)! ; + ( + 4)! ; ) : E [(W (u)) h (u; Y (u) + z e ) h (u; Y (u))] l (dz) du < : Proof. Following he argumen in he proof of Lemma 5.3, and hen applying Hölder inequaliy wih p = 4; q = 4=3, we ge Z T Z + Z T E E [(W (u)) h (u; Y (u) + z e ) " Z (W (u)) k zl (dz) e + Z = k e k+k zl (dz) + Z " T E (W (u)) e Z k e k+k zl (dz) + Pm Z " T h E (W (u)) 4i 4 E e Z k e k+k zl (dz) + Z T h E (W (u)) 4i 4 du Pm h (u; Y (u))] l (dz) du # (! ; +! ; ) (y +Z ( u)) du # (! ; +! ; ) Z ( u) du Pm " my E e for poiive conan k ; k ; k : We oberve ha and apply Lemma 3.3 o conclude ha Z T Z + # 3 8(! ; +! ; ) 4 3 Z ( u) du 8(! ; +! ; ) 3 Z ( T ) 8 (! ; +! ; ) 3 < c ; E [(W (u)) h (u; Y (u) + z e ) # 3 4 h (u; Y (u))] l (dz) du 3

32 Z k e k+k zl (dz) + 3 Z T 4 + Z T h E (W (u)) 4i 4 du < e 8(! ; +! ; ) 3 z! l (dz) A We um up he reul in hi ecion in he following heorem. Theorem 5. Aume Condiion. hold wih c = 8 (( + 4)! ; + ( + 4)! ; ) ; = ; :::; m: Then he value funcion of he conrol problem i V (; w; y) = w h (; y) ; where h i de ned in (5.9). Furhermore, he opimal invemen raegy i a given by Equaion (5.4),..., (5.7). Proof. We ee ha V i well-de ned under our aumpion. Furhermore, he inegrabiliy condiion implie by Lemma 5. ha h i well-de ned, ince g and h ha he ame growh. Oberve ha he opimal invemen raegy depend only on ( ; ) and no on he level of wealh w: Thi, ogeher wih he fac ha [; ] [; ] ; give ha here exi a unique oluion W o Equaion (.4). Hence, A : Se v (; w; y) = w h (; y) ; and noe ha he aumpion in Lemma 5.4 and 5.5 hold. We alo have ha he inegrabiliy condiion in Lemma 5. and Propoiion 5. hold, ince we may chooe " > a we like. Thi allow u o conclude ha v C [; T ] D, ince v i coninuou in w on [; ) ; and ha v i a claical oluion of he HJB equaion (.7). We now apply Theorem 4., and he proof i complee. 6 Generalizaion In hi ecion we ae, wihou proof, he mo imporan reul for he cae of n ock, i () [a i ; b i ] ; i = ; :::; n; and c nx i d: i= I can be een ha he addiional di culy in hi eing i merely noaional. 3

33 The HJB equaion aociaed o hi ochaic conrol problem i ( ) X n = v + max wv w i ( i + i i r) + X n i[a i;b i];i=;:::;n; c P w v ww i i n i= id i= i= X m Z + rwv w y v y + (v (; w; y + z e ) v (; w; y)) l (dz); for (; w; y) [; T ) D: We ill have he erminal condiion and he boundary condiion V (T; w; y) = U(w); for all (w; y) D; V (; ; y) = U(); for all (; y) [; T ] R m + : The oluion o hi equaion can be hown o be v (; w; y) = w h (; y) = w E y h e R T (y ();:::;y n ())di ; where i de ned a (6.) ( ; :::; n ) ( X n = max i ( i + i i r) i[a i;b i];i=;:::;n; c P n i= id i= ) nx i i + r: i= The opimal fracion of wealh are given by he parameer = ( ; :::; n) ha obain ( ; :::; n ) in Equaion (6.). 7 Fuure reearch We view hi paper a a aring poin for more reearch on our n-ae exenion of he Barndor -Nielen and Shephard model. Primarily, we would like o perform he aiical analyi propoed in Subecion., in order o clarify o wha exen our model capure he rue dependence beween nancial ae. Anoher queion of inere i porfolio opimizaion wih he incluion of uiliy of conumpion. Furher, i would be inriguing o conider he more general marke model ds () = ( + ()) S () d + p ()S () db () + S () Y () dz ( ) ; which i a modi caion of a model propoed by Barndor -Nielen and Shephard. I allow for he o-called leverage e ec o be di eren for he variou newprocee, bu alo implie a diincion beween good and bad new. A di culy 33

34 wih hi problem i ha he ock price are no longer coninuou. In boh hee cae i ough o be feaible o olve our n-ae exenion, once we have handled he one-ae problem. A nal iue ha we aim o conider i, given our ochaic volailiy marke, how much higher uiliy an inveor obain by rading according o our opimal porfolio model compared o omeone who follow he claical Meron policy. Reference [] Barndor -Nielen, O.E. (998): Procee of Normal Invere Gauian Type. Fin. Soch., [] Barndor -Nielen, O. E., Shephard, N. (a): Modelling by Lévy Procee for Financial Economeric. In Lévy Procee - Theory and Applicaion (ed O. E. Barndor -Nielen, T. Mikoch and S. Renick). Boon: Birkhäuer. [3] Barndor -Nielen, O. E., Shephard, N. (b): Non-Gauian Ornein- Uhlenbeck-baed model and ome of heir ue in nancial economic. Journal of he Royal Saiical Sociey: Serie B 63, 67-4 (wih dicuion). [4] Benh, F. E., Karlen, K. H., Reikvam K. (a): Opimal Porfolio Selecion wih Conumpion and Non-Linear Inegro-Di erenial Equaion wih Gradien Conrain: A Vicoiy Soluion Approach, Fin. Soch. 5, [5] Benh, F. E., Karlen, K. H., Reikvam K. (b): Opimal Porfolio Managemen Rule in a Non-Gauian Marke wih Durabiliy and Ineremporal Subiuion. Fin. Soch. 5, [6] Benh, F. E., Karlen, K. H., Reikvam K. (3): Meron porfolio opimizaion problem in a Black and Schole marke wih non-gauian ochaic volailiy of Ornein-Uhlenbeck ype. Mah. Fin. 3(), [7] Eberlein, E., and Keller, U. (995): Hyperbolic Diribuion in Finance, Bernoulli (3), [8] Emmer, S., Klüppelberg, C. (4): Opimal Porfolio when Sock Price Follow an Exponenial Lévy Proce. Fin. Soch. 8, [9] Fleming, W. H., Hernández-Hernández, D. (3): An Opimal Conumpion Model wih Sochaic Volailiy. Fin. Soch. 7, [] Folland, G. (999): Real Analyi, John Wiley & Son, Inc. [] Fouque, J.-P., Papanicolaou, G., and Sircar, K. R. (): Derivaive in Financial Marke wih Sochaic Volailiy. Cambridge: Cambridge Univeriy Pre. 34

35 [] Goll, T., Kallen, J. (3): A Complee Explici Soluion o he Log- Opimal Porfolio Problem. Ann. Appl. Prob., 3, [3] Ikeda, N., Waanabe, S. (989): Sochaic Di erenial Equaion and Diffuion Procee; nd ed. Kodanha: Norh-Holland. [4] Korn, R. (997): Opimal Porfolio, World Scieni c Publihing Co. Pe. Ld. [5] Meron, R. (969): Lifeime Porfolio Selecion under Uncerainy: The Coninuou Time Cae. Rev. Econ. Sa. 5, [6] Meron, R. (97): Opimum Conumpion and Porfolio Rule in a Coninuou Time Model. J. Econ. Theory 3, ; Erraum (973) 6, 3-4. [7] Nicolao, E., Venardo, E. (3): Opion Pricing in Sochaic Volailiy Model of he Ornein-Uhlenbeck Type. Mah. Fin. 3, [8] Pham, H., Quenez, M-C. (): Opimal Porfolio in Parially Oberved Sochaic Volailiy Model. Ann. Appl. Prob., -38. [9] Proer, P. (3): Sochaic Inegraion and Di erenial Equaion; nd ed. New York: Springer. [] Rydberg, T. H. (997): The Normal Invere Gauian Lévy Proce: Simulaion and Approximaion, Communicaion in Saiic: Sochaic Model 3(4), [] Sao, K. (999): Lévy Procee and In niely Diviible Diribuion. Cambridge: Cambridge Univeriy Pre. [] Økendal, B. (998): Sochaic Di erenial Equaion, 5h ed. Berlin: Springer. 35