Inroducion We sudy he problem of maximizing expeced isoelasic uiliy of erminal wealh in an incomplee coninuous ime marke wih coninuous price process.

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1 UILIY MAXIMIZAION AND DUALIY JOHANNES LEINER CENER OF FINANCE AND ECONOMERICS (COFE) UNIVERSIY OF KONSANZ Absrac. In an arbirage free incomplee marke we consider he problem of maximizing erminal isoelasic uiliy. he relaionship beween he opimal porfolio, he opimal maringale measure in he dual problem and he opimal value funcion of he problem is described by an BSDE. For a oally unhedgeable price for insananeous risk, isoelasic uiliy of erminal wealh can be maximized using a porfolio consising of he locally risk-free bond and a locally efficien fund only. In a markovian marke model we find a non-linear PDE for he logarihm of he value funcion. From he soluion we can consruc he opimal porfolio and he soluion of he dual problem. Keywords: Uiliy, Opimal Porfolios, Dualiy heory. AMS 91 Classificaions: 9A9, 9A1 JEL Classificaions: G11 I would like o hank Professor M. Kohlmann for his suggesions and suppor. I am also hankful o Professor M. Schweizer and Professor S. ang. 1

2 Inroducion We sudy he problem of maximizing expeced isoelasic uiliy of erminal wealh in an incomplee coninuous ime marke wih coninuous price process. he isoelasic uiliy of exponen p 6= 1 is defined as u (p) (x) :=sgn(1 p) jxjp and for p =by u () (x) :=ln(jxj). he wo p cases p < 1 and p > 1 are very differen in here economic inerpreaion, bu can be reaed o some exend by he same mahemaical mehods. Solving he opimizaion problem for p<1 is a plausible approach o find porfolios of opimal expeced growh. here are several papers on his opic: See, e.g. Meron (199), Pliska (1986), He and Pearson (1991), Karazas, Lehoczky, Shreve and Xu (1991), Karazas and Shreve (1999), Kramkov and Schachermayer (1999). For p = he problem is relaed o he mean-variance hedging problem, see Gourieroux, Lauren and Pham (1998), (GLP98), Pham, Rheinländer and Schweizer (1998) and Lauren and Pham (1999). he heory of sochasic dualiy, which goes back o Bismu (1973, 1975), is he cenral ool for solving hese problems. his heory allows o formulae an opimizaion problem over a se of maringale measures, being dual o he original opimizaion problem over a se of self-financing hedging-sraegies. Under quie general condiions, he soluion of one of he problems can be ransformed ino a soluion of he corresponding dual problem. Anoher imporan approach, is o ry o solve he opimizaion problem locally, i.e. by so-called myopic sraegies which maximize in some sense he expeced growh rae of he porfolio a every insan ofime. In some imporan cases hese sraegies urn ou o be globally opimal oo. See, e.g., Mossin (1968), Leland (197), Aase (1984, 1986, 1987, 1988), Foldes (1991), Goll and Kallsen (). his approach is relaed o he risk-sensiive sochasic conrol approach, see Bielecki and Pliska (1999, ). We consider an arbirage-free (in a sense o be specified laer) coninuous ime marke model wih unresriced rading. We use he modern equivalen maringale measure approach, see Harrison and Pliska (1981), Delbaen and Schachermayer (1994). Afer some echnical preparaions in Secion 1 and specificaion of he modelinsecion, we formulae he opimizaion problem and is corresponding dual Problem in Secion 3. (3.8)), relaing he erminal value V op We show a represenaion propery (formula of a porfolio o a maringale Research suppored by he Cener of Finance and Economerics, Projec Mahemaical Finance.

3 measure Z op op, o be sufficien for he opimaliy ofv for he uiliy maximizaion problem and he opimaliyofz op for he dual problem. he opimal values of he wo problem are relaed by a simple formula. In Secion 4 we inroduce he noion of a oally unhedgeable price for insananeous risk. In his siuaion we can explicily solve he uiliy opimizaion problem. he opimal porfolio is a locally efficien porfolio, a noion we inroduce in Secion 5. In Secion 6 we give an exisence resul for he soluions of he wo opimizaion problems. In Secion 7 we derive a backward sochasic differenial equaion, (BSDE), such ha from he soluion he opimal porfolio, he opimal value funcion and he soluion of he dual opimizaion problem can be consruced. See Yong and Zhou (1999) for an inroducion o BSDEs. In Secion 8 we consider a markovian marke model. We ransform he BSDE ino a non-linear PDE for he logarihm of he value funcion. From he parial derivaives of he soluion, we can consruc under addiional assumpions he opimal porfolio and he soluion of he dual opimizaion problem Self-financing Hedging Sraegies Le a filered probabiliy space Ω 1 := (Ω F (F s ) s P), saisfying he usual condiions be given. For simpliciywe assume F o be rivial up o ses of measure wih respec o P and F 1 = F 1 := F. For an adaped process X se X := X, X := lim h& X h for > if he limi exiss and define he processes X := (X )»<1 and X := X X if X exiss for all >. he componens of X are denoed as X i 1» i» d. For a process X and a map fi :Ω! μ R+, denoe he sopped process a ime fi by X fi. We will ofen resric a semimaringale X on Ω 1 o an inerval [ ]»» < 1, resp. o [ 1). herefore we inroduce he following filered probabiliy space (again saisfying he usual condiions), Ω [ ] := P jf Ω F F [ ] s for all»»»1, <1, where F s [ ] := F _s^ for» s<1. he process X s [ ] := X _s^ is hen a semimaringale on Ω [ ]. However, on [ ]we ofen wrie X insead of X [ ]. Se Ω := Ω [ ]. For q > 1 define L q (Ω [ ] ), respecively L q (Ω [ ] ), as he se of F - measurable random variables X, such ha E[X] < 1 a.s., respecively E [X] < 1 a.s., where E [ ] := E[ jf ] denoes he generalized condiional expecaion. Denoe he condiional variance by Var ( ). he sochasic exponenial of a semimaringale X is denoed as E(X) and we se E (X) :=E(1 [1)X). As a general references we cie Jacod and s

4 4 Shiryaev (1987), (J&S 87), and Jacod (1979). Denoe he se of predicable processes which are locally inegrable, resp. locally Riemann- Sieljes inegrable, wih respec o a local maringale M, resp. wih respec o a process A of finie variaion, by L 1 loc (M), resp. by L1 loc (A). If he semimaringale X admis a decomposiion X = X + A + M, where M is a local maringale and A is a process of finie variaion hen L 1 loc(x) :=L 1 loc(m) L 1 loc(a). We can now define he marke model: Le S = (S )»<1 be a R d - valued semimaringale. M := (Ω 1 S) = ((Ω F (F s ) s P)S) is a model for a marke, where S describes he price processes of d asses. We will ofen consider such a marke on an inerval [ ]» < < 1. his is equivalen o work wih he following marke model M [ ] defined by M [ ] := Ω [ ] S [ ]. Se M := M [ ]. We wan o model he economic aciviy of invesing money ino a porfolio of asses and changing he number of asses held over ime according o a cerain hedging sraegy. his is achieved wih he following definiion: Definiion 1.1. A hedging sraegy in he marke M is a H L 1 loc (S). he corresponding value process V H of H is defined as V H := HS. he gains process of H is defined as he semimaringale G H := H S. H is called self-financing if V H = V H + G H, i.e. H S = H S + R H sds s 8. Denoe he space of all self-financing hedging sraegies in M by SF(M). Noe ha for H SF(M), we have H [ ] SF(M [ ] ). he idea of a self-financing hedging sraegy is ha he changes over ime of he corresponding value process are solely caused by he changes of he value of he asses held in he porfolio and no by wihdrawing money from or adding money o he porfolio. Definiion 1.. A semimaringale B such hab and B are sricly posiive is called a numéraire for he marke M. he marke discouned wih respec o B is hen defined as M B := Ω 1 S B, where S B := S B.

5 For»» < 1, he marke resriced o he inerval [ ] is defined as M B [ ] := M B [ ] = Ω [ ] S B [ ]. 5 Noe ha for a numéraire B, B 1 is a numéraire oo and S B is a semimaringale. Usually here is in addiion o he marke M a numéraire B given and he marke μm := (Ω 1 μ S) μ S := (S B) is considered. Ofen B is he price process of a locally risk-free bond. If he numéraire is raded, i.e. he value process of a porfolio in M, one can ry o exend a hedging sraegy in M o a self-financing hedging sraegy in μm. Define he discouned marke μm B =(Ω 1 (S B 1)). he idea is o exend H o a self-financing hedging sraegy H μ =(H ^H) SF( M μ B ) by defining he process ^H := H S B + H SB HS B and hen o show ha H μ is a self-financing hedging sraegy in μm oo, see Geman, El Karui and Roche (1995) and Goll and Kallsen (). (Noe ha H S B HS B =(H S B ) + (H S B ) HS B =(H S B ) + H S B HS B =(H S B ) HS B is predicable, hence μ H as well.) Proposiion 1.3. Le B be a numéraire for he marke M. hen SF(M B )=SF(M) holds. Proof. Le H SF(M B ). Se V B = HS B. Firs, we have o show H L 1 loc (S). Since S = SB B = S + S B B + B S B +[S B B] his follows if we show ha H L 1 loc (SB B)L 1 loc (B S B )L 1 loc ([SB B]). Noe ha HS B = H(S B S B ) = V B (H S B ) = V B + H S B (H S B ) = V B +(H S B ) = V B, which is locally bounded. Since [S B B S B B] =(S B Ω S B ) [B B] andh(s B Ω S B )H =(V B ) is locally inegrable wih respec o [B B], we find H L 1 loc(s B B).

6 6 ha H L 1 loc (B S B ) L 1 loc ([SB B]) is easy o see. We calculae H S = H (S B B)=H (S B B + B S B +[S B B]) = (HS B ) B +(B H) S B +[H S B B] = V B B + B (H S B )+[V B B] = V B B V B B = HS B B H S B B = HS H S : his implies SF(M B ) SF(M). Now observe ha (M B ) B 1 = M, since B 1 is anuméraire. his implies he reverse inclusion. here is an alernaive way o consruc self-financing hedging sraegies: Lemma 1.4. Le H SF(M) be such ha V H 6= and V H 6= almos surely. Se H ~ := H. hen H ~ L 1 V loc(s), HS ~ H =1and (1.1) V H = V H + V H ( ~ H S) =V H E( ~ H S) holds. Conversely, le ~ H L 1 loc (S) wih ~ HS = 1 be given and se H := v E( ~ H S) ~ H for a F -measurable random variable v. hen H SF(M) and V H = v E( ~ H S). We call ~ H a generaor for he self-financing sraegy H and define V ( ~ H) := V H. Proof. Since V H 1 is locally bounded we have H ~ L 1 loc (S). We have V H V H = G H = H S =(V H ~ H) S = V H ( ~ H S). he second ideniy in (1.1) follows immediaely from he uniqueness of he soluion o he

7 Doléan-Dade SDE defining he sochasic exponenial, see J&S 87, I.4f. Conversely, we calculae 7 HS = v E( ~ H S) ~ HS = v E( ~ H S) ( ~ HS + ~ H S) = v E( ~ H S) (1 + ( ~ H S)) = v E( ~ H S) + E( ~ H S) ( ~ H S) = v E( ~ H S) + E( ~ H S) ( ~ H S) = v E( ~ H S) + E( ~ H S) 1 = v E( ~ H S) =v + v E( ~ H S) ( ~ H S) = v + v E( ~ H S) ~ H S = V H + G H :. Arbirage-free Markes So far we didnoworry abou arbirage. We consider in his secion he marke μm := (Ω 1 S), μ where S μ := (S B) isrd R-valued and B is a numéraire, wih B =1,which we assume o be uniformly bounded and uniformly bounded away from on finie inervals. For»»» 1 < 1, denoe he se of uniformly inegrable, resp. local maringales, living on Ω [ ] by L u (Ω [ ] ), resp. by L(Ω [ ] ). Define he following ses of local maringale measures: (.1) μd( M μ Φ [ ] ):= Z L(Ω [ ] )jz1 [] =1Z Ψ (S B ) [ ] Z L(Ω [ ] ) (.) D( μm Φ Ψ [ ] ):= Z L(Ω [ ] )jz1 [] =1Z > (S B ) [ ] Z L(Ω [ ] ) (.3) D abs ( M μ Φ [ ] ):= Z D( μ M μ Ψ [ ] )jz uniformly inegrable maringale

8 8 and (.4) D e ( M μ Φ [ ] ):= Z D( M μ Ψ [ ] )jz uniformly inegrable maringale : We will work wih he following No-Arbirage condiion: (.5) D e ( μm ) 6= 8 < 1: his condiion is known o be equivalen o he NFLVR-condiion, see Delbaen and Schachermayer (1994). I implies ha (.6) D e ( μm [ ] ) 6= 8»» < 1: We will ofen work wih he following ses of equivalen, resp. absoluely coninuous, local maringale measures, for q>1: (.7) (.8) (.9) (.1) D q ( μ M [ ] ):= Φ Z D( μm [ ] )jz L q (Ω [ ] ) Ψ ff D q ( M μ [ ] ):=ρ Z_ jz D q ( M μ [ ] ) Z μd q ( μ M [ ] ):= Φ Z D abs ( μ M [ ] )jz L q (Ω [ ] ) Ψ ff μd q ( μm [ ] ):=ρ Z_ jz μd q ( μm [ ] )Z > : Z Noe ha Z D q ( μm [ ] ) implies Z L p (Ω [ ] ). For q < 1 se D q ( μm [ ] ) := D( μm [ ] ) and μd q ( μm [ ] ) := μd q ( μm [ ] ) := μd( μm [ ] ). For Z D q ( M μ [ ] )and»»»,wehave Z[ ] Z D q ( M μ [ ]). Noe also ha D( q μm [ ] ) = D( μm [ ] ), since F was assumed o be rivial. p will always denoe a real number differen from 1. We define q := p, such ha for p 6= 1, p 1 p 1 + q 1 =1holds, bu for p =we have q =. V H Le B SF( μm [ ] ). We call a H B an B-arbirage, if V H =, and V H 6= almos surely. If here exiss no B-arbirage, hen B is called arbirage-free. In all probabilisic heories of financial markes allowing o rade a an infiniely large number of insances of ime one has o exclude cerain self-financing hedging sraegies, e.g. doubling sraegies, in order o avoid arbirage opporuniies. We will define several arbirage-free subses of SF( μ M [ ] ):

9 1. For p > 1 and D q ( μm [ ] ) 6=, (see Delbaen and Schachermayer (1996), (DS96)): SF p ( μm [ ] ) := (.11) resp. SF p ( μm [ ] ) := (.1) Noe ha SF p ( μ M [ ] ) = (.13) ρ H SF( μm [ ] )jv H L p (Ω [ ] ) V H B [ ] Z Lu (Ω [ ] ) 8 Z D q ( μm [ ] ) ρ H SF( μm [ ] )jv H L p (Ω [ ] ) V H B [ ] Z Lu (Ω [ ] ) 8 Z D q ( μm [ ] ) ρ H SF( μ M [ ] )jv H L p (Ω [ ] ) V H B [ ] Z Lu (Ω [ ] ) 8 Z μd q ( μm [ ] ) since for Z μd q ( μm [ ] ) we can find a Z μ μd q ( μm [ ] ) wih Z = μz _ μz and for Z D q ( μm [ ] ), we have Z ~ := Z+Z μ D q ( μm [ ] ), which implies ^Z := Z ~ _ ~Z D q ( μm [ ] ) and for H SF p ( μm [ ] ) ha V H B [ ] Z = (( B [ Z μ + Z) ^Z Z_ ) is a uniformly inegrable ] V H maringale.. For p<1 SF p ( μm [ ] ):=SF p ( μm Φ [ ] ):= H SF( μm [ ] )jv H Ψ (.14) : 3. For p>1 and S μ S p (Ω loc [ ]) G p ( μm Φ [ ] ):= H SF( μm [ ] )jv H S Ψ (.15) p (Ω [ ] ) where S p (Ω [ ] ) denoes he space of L p -inegrable semimaringales, see Delbaen, Mona, Schachermayer, Schweizer and Sricker (1997) (DMSSS97) for he case p = and Grandis and Krawczyk (1998), (GK98), for he general case p>1. Z ff ff ff : 9 Lemma.1. For p > 1 assume D q ( μm [ ] ) 6= and μ S o be coninuous. hen G p ( μm [ ] ) SF p ( μm [ ] ). In paricular G p ( μm [ ] ) is arbirage-free.

10 1 Proof. For H G p ( μm [ ] ) se fi n := inf on [fi n ) and since fi fi fi fi fifi V Hn B» n. [ ] ns fi fi fi fi fi V H s B s fi fifi n o, H n := H V fin H B fin Rd R on [fi n ]. hen H n SF p ( μm [ ] ), i Z jf s h V H n I follows E s» and all Z D q ( μm [ ] ). V Hn s and fi fi fi V H n B Z fi fifi» fi fififi sup»s» (V H B s ) B = V s Hn B s Z s for all» converges almos surely o V H s Z fi fififi L 1 (Ω [ ] ), since sup»s» V H s L p (Ω [ ] )by Doob's maximal inequaliy, hence we finde h V H B Z jf s i = V H s B s Z s for all» s». Define for F -measurable v (.16) and (.17) A Λp v ( μm [ ] ):= G p v( μm [ ] ):= ρ V H fi fih SF p ( μm [ ] ) V H B ρ V H B fi fih G p ( μm [ ] ) V H B B ff = v ff = v : For p > 1 and D q ( μm [ ] ) 6=, SF p ( μm [ ] ) has an imporan propery: A Λp 1 (M [ ] ) is known o be closed, if S B [ ] is locally in L p (Ω [ ] ) in he sense, ha here exiss a sequence U n n N of localizing sopping imes increasing o infiniy such ha for each n, he family fs fi [ ] jfi sopping imefi» U n g is bounded in L p (Ω [ ] ), see DS96. his condiion cerainly holds if μ S is coninuous. o work wih he spaces G p ( μ M [ ] ) is in some sense more naural, since is definiion involves only he objecive probabiliy measure P and no equivalen maringale measures. Furhermore G p ( μm [ ] ) is sable under sopping, a desirable propery from an economic poinofview. However, his space has in general weaker properies han SF p ( μ M [ ] ), see DMSSS97 and GK98. We will ofen work wih a coninuous price process S, resp. S. μ In his case L 1 (S) loc =L loc (S) holds. he price process admis a represenaion (.18) S = S + μ ff + M where μ =(μ i ) 1»i»d is predicable, ff is a predicable, increasing, coninuous, locally inegrable process such haμ is locally inegrable wih respec o ff. Furhermore, here exiss a symmeric non-negaive d dmarix-valued predicable process C = (C ij ) 1»ij»d, locally inegrable

11 wih respec o ff, such ha [S i S j ]=[M i M j ]=<M i M j >= C ij ff. ff can be chosen such ha B = E(r ff) for a predicable process r. In he coninuous case, D e ( μ M [ ] ) 6= implies μ = rs C, dffalmos surely for a predicable process L loc (M) and every Z D e ( μm [ ] ) is of he form Z = E ( M + N), where N is a no necessarily coninuous local maringale orhogonal o M wih [M N] =, see Ansel and Sricker (199) Opimal Porfolios Consider he problem of maximizing expeced uiliy from erminal wealh. We follow a sochasic dualiy approach, which goes back o Bismu (1973, 1975), see also Karazas, Lehoczky, Shreve and Xu (1991), (KLSX91), and Karazas and Shreve (1999), Kramkov and Schachermayer (1999) and Schachermayer () for general resuls. We have already defined he so-called isoelasic uiliy funcions u (p) p 6= 1, wih consan index of relaive risk-aversion, see Pra (1964) and Arrow (1976). For opimizaion muliplicaion of he uiliy funcion wih a consan facor or adding a consan has no effec. We choose o normalize he uiliy funcion such ha ju (p) (1)j =1for all p 6= 1 and define for p<1p6= (3.1) U (p) (x) = 1 for x<and (3.) U (p) (x) :=sgn(p)x p 8 x U () (x) :=ln(x) U () (x) = 1 for x». For p>1 se (3.3) We have for p<1p6= (3.4) and (3.5) 8 x> U (p) (x) := jxj p 8 x R du (p) dx (x) =jpjxp 1 du () dx (x) = 1 x 8 x> 8 x> (p) du and se () := 1 for p<1. dx We wan o solve he following opimizaion problem for fixed»» < 1 and p 6= 1: (3.6) V(p B) := ess sup HB H μ S =1 Λ E U (p) V H

12 1 where B fsf p ( μm [ ] ) fsf p ( μm [ ] ) G p ( μm [ ] )g for p > 1, resp. B = SF p ( μm [ ] ) for p<1, and he dual problem (3.7) W Λ (q C) := ess inf ZC E» U (q) B Z B where C fd q ( μm [ ] ) μd q ( μm [ ] )g. ( U (q) equals he convex dual o U (p) up o a consan facor, see Rockafellar (197)). See Karazas and Shreve (1999) for he definiion of ess sup and ess inf. If for H B wih V H = 1 and V(p B) =E U (p) V H Λ,henwesay V H solves Problem (3.6) for B. If for Z C, W Λ (q C) =E h U (q) B Z B i, hen we say Z solves he dual Problem (3.7) for C. For he momen we are ineresed in he Problem (3.6) for B = SF p ( μm [ ] ) and se V(p ) := V(p SF p ( μm [ ] )). For p > 1, we se W Λ (q ) := W Λ (q μ D q ( μ M [ ] )), respecively for p<1, we definew Λ (q ):= W Λ (q D q ( μ M [ ] )). (I will urn ou laer, ha W Λ (q ) = W Λ (q μd q (M [ ] )) for p>1andfor p<1 if V(p ) < 1.) he following proposiion shows he close relaion beween hese wo problems and gives he key idea how o handle he incompleeness of he marke. Proposiion 3.1. Assume ha here exiss an H SF p ( μm [ ] ) wih V H and a Z opλq μd ( μm [ ] ) such ha for some F -measurable random variable c> (3.8) Z opλq (p) du = cb sgn(1 p) dx V H and such ha V opp := V H V H V H B [ ] Z opλq is a uniformly inegrable maringale. hen solves Problem (3.6) for SF p ( μm [ ] ) and Z opλq solves for p > 1, resp. p < 1, he dual Problem (3.7) for μ D ( μ M [ ] ), resp. for D ( μm [ ] ) and μd ( μm [ ] ). here exiss a mos one such pair (V opp Z opλq ) wih a represenaion (3.8). For p 6= he

13 13 corresponding opimal values saisfy (3.9) jv(p )j p 1 jw Λ (q )j q 1 =1: Proof. Noe ha for p<1 (3.8) implies V H >. For H SFp ( μm [ ] ) wih V H = 1 and since U (p) is concave we have (3.1) U (p) V H» U (p) V opp (p) du + dx aking condiional expecaions we find E U (p) V H Λ» E "U (p) V opp» E h U (p) V opp V opp + Z opλq V H i (V H V opp ): # V opp sgn(1 p)cb since E h V H i B Z opλq» V H B = V opp B for p < 1, respecively since Z opλq μd q ( μm [ ] ) for p>1. Le Q μd q ( μm [ ] ) and calculae U (q) ψ dq dp B B 1 ψ Z opλq U (q) B B 1 = U (q) ψ! = U ZopΛq + Z opλq B B 1! du (q) dq dp B B 1 ψb Z opλq dx B B 1! sgn(1 p)k V opp Z opλq! dq dp B dq dp 1 A Z opλq B B 1 Z opλq for some F -measurable random variable k >. aking condiional expecaions we find " ψ (3.11) E U (q)!# " B Z opλq» E B ψ dq B U (q) dp B!#

14 14 since E h V opp B dq i dp» V opp B for p < 1, resp. since for p > 1 Z opλq μd q ( μm [ ] ). he uniqueness of he pair (V opp Z opλq ) follows from he sric concaviy of U (p). Since V opp B [ ] Z opλq is a uniformly inegrable maringale, i is deermined by V opp B Z opλq. Le H SF p ( μ M [ ] ) wih V H = 1 and Z D ( μ M [ ] ) such ha (3.1) Z = c (p) du B sgn(1 p) dx V H holds for a F -measurable random variable c >. hen V opp = V H, Z opλq = Z and V opp Z opλq = V H Z. B [ ] B [ ] Assume ha here exiss a» s» wih A := fv H s > V opp s g 6=. In his case we can change he self-financing hedging sraegy H on A [s ] o a H SF p ( μm [ ] ) such ha V H V opp and V H > V opp on A. From his we conclude he uniqueness of he pair (V opp Z opλq ). For p 6= we find U (p) (V opp ) = sgn(q) = sgn(q) = V opp V opp p Z opλp V opp cp sgn(1 p) B p 1 V opp hence (3.13) i V(p )=E hu (p) (V opp ) = 1 cp sgn(1 p)b :

15 15 For q 6= we find ψ! B U (q) Z opλp B = sgn(p)b q! q ψz opλp B = sgn(p)b q = sgn(p)b q ψz opλp B cjpj V opp! q 1 Z opλp B p 1 1 p 1 Z opλp B = sgn(p)b q (cjpj) 1 p 1 Z opλp V opp B hence (3.14) W Λ (q ) =E " U (q) ψ!# B Z opλp = sgn(p)(b cjpj) 1 p 1 : B (3.13) and (3.14) ogeher imply (3.9). We call (V opp Z opλq ) he opimal pair for he marke μm [ ] wih respec o opimizaion in SF p ( μm [ ] ). Wehave he following sabiliy propery for opimal pairs: Proposiion 3.. If he pair (V opp Z opλq ) admis a represenaion (3.8) wih V opp = V H for a H SF p ( μm [ ] ) wih V H = 1 and Z opλq D q ( μ M [ ] ), hen he opimal pair for he marke μm [ ] exiss and is given by V opp Z opλq (3.15) =! ψv opp _ Z opλq _ : V opp Z opλq Proof. Noe firs, ha by J&S87, Lemma III.3.6, wehave Z opλq >.» q ff For Z μd q ( μm [ ]) sea := ρe [Z q ] <E Z opλq. Define Z opλq ~Z := Z opλq on [ ) [ A c [ ], and Z ~ := Z opλq Z on A [ ].

16 16 We calculae h i E ~Z q = E he h ii ~Z q = E h 1 A Z opλq» E he q h E [Z q ]+1 A ce h q ii h = E Z opλq Z opλq Z opλq q i q ii hence ~ Z μd q ( μm [ ] ), ~ Z Z opλq = ZopΛq _ Z opλq = Z opλq and A = and we conclude. By assumpion we have V opp B Z opλq > and since V opp B [ ] Z opλq is a non-negaive supermaringale we have V opp >. Hence V opp V opp L p (Ω [ ]) and V opp _ Z is a uniformly V opp inegrable maringale for all Z D q ( μm [ ]). Since (3.16) Z opλq! (p) du ψv = c B sgn(1 opp p) dx V opp where (3.17) c := (p) du c V opp dx Z opλq we can apply Proposiion 3.1. Lemma 3.3. For p > 1, H SF ( μ M [ ] ) wih V H = 1, Z opλq μd ( μm [ ] ) and assume (V H Z opλq ) o admi a represenaion (3.8). If V H L p+ffl (Ω [ ] ), (or equivalenly Z opλq L q+ffl (Ω [ ] )), for some ffl>, hen H SF p ( μm [ ] ) and (V H Z opλq ) is he opimal pair for he marke μm [ ].

17 17 Proof. Observe ha V H B Z L 1+~ffl (Ω [ ] ) for some ~ffl > for all Z μd q ( μ M [ ] ). Hence V H B Z is a uniformly inegrable maringale. Now apply Proposiion 3.1. In he nex wo secions we look a an example and pospone an exisence resul for he opimal pair (V opp Z opλq ) unil Secion oally Unhedgeable Price for Insananeous Risk Assume S μ o be coninuous such ha we have a represenaion (.18). In his secion we seek a sufficien condiion ensuring cerain self-financing hedging sraegies o be opimal for problem 3.6. See Karazas and Shreve (1999), Example for a similar resul and he noion of oally unhedgeable coefficiens. his noion describes a marke model where he uncerainy in he coefficiens defining he model is in a cerain sense orhogonal o he uncerainy of he local maringale M driving he price process, such ha we can no hedge agains his risk. Se fi := p C L loc (ff). Definiion 4.1. For»» < 1, fi [ ] is called he insananeous price for risk process, or insananeous Sharpe-raio process, for he marke μm [ ]. Definiion 4.. For»» < 1, le an F -measurable random variable c > and a no necessarily coninuous local maringale N orhogonal o M, (or equivalenly wih [N M] = ), be given such ha 1. (4.1) E pr q fi ff = ce (N) :

18 18. For p<1, E (q M + N) is a uniformly inegrable maringale. We hen call he insananeous price for risk fi [ ] in he marke μm [ ] oally p-unhedgeable. resp. srongly oally p-unhedgeable if E (N) is a uniformly inegrable maringale. Remark 4.3. For p =we have q =and we find a unique represenaion (4.1) wih c = 1, N = for any»» < 1, hus fi [ ] is oally -unhedgeable in μm [ ]. For p =, he opimizaion problem (3.6) is also known as maximizing he Kelly-crierion, see Kelly (1956), Breiman (196) and Karazas and Shreve (1999). For general resuls see Aase (1986) and Goll and Kallsen (). Lemma 4.4. If fi [ ] is oally p-unhedgeable in μm [ ], hen fi [ ] is oally p-unhedgeable in μm [ ] for all»». Proof. For»» se c := ce pr q fi ff + N gives us a represenaion (4.1) and for p < 1, E (q M + N) uniformly inegrable maringale.. his is a Proposiion 4.5. Assume fi [ ] o be oally p-unhedgeable in μm [ ] wih a represenaion (4.1), hen he opimal pair for he marke μm [ ] for»» is given by (4.) V opp Z opλq =! ψv (Hp ) _ E V (Hp ) ( M + N)

19 where H p := p 1 S [ ] 1 p 1 generaes he value process B 19 (4.3) V (Hp ) := E r fi p 1 ff + p 1 M : Furhermore, for p 6= (4.4) V(p E )= sgn(q) h i E pr q fi ff E [E (N) ] resp. if fi [ ] is srongly oally p-unhedgeable in μm [ ], (4.5) V(p )= sgn(q)e» E pr q fi ff and (4.6) W Λ (p ) = sgn(p)(v(p )) 1 1 p : Proof. For p 6= calculae du (p) dx V (Hp ) = p sgn(q) V (Hp ) p 1 = p sgn(q)e r = p sgn(q)e fi p 1 (p 1)r q fi ff + = sgn(p 1)jpjcB B E ( M + N) p 1 Mp 1 ff + M

20 resp. for p = du () dx V (Hp ) = V (Hp ) 1 = E r + fi ff M 1 = E ( r ff + M) = B B E ( M) and find a represenaion (3.8), since E ( M + N) >. Se (4.7) Z opλq := E ( M + N) : V (Hp ) B [ ] ZopΛq = 1 E fi B p 1 ff + M p 1 E ( M + N) = 1 B E (q M + N) which is a uniformly inegrable maringale on [ ] for p < 1 by assumpion. For p>1 and ffl>1 observe V (Hp ) fflp = E r fi hence wefindv (Hp ) p 1 ff + Mfflp p 1 (p 1) = E fflpr fflqfi + fi ffl p fflp = E fflpr fflqfi p( ffl) 1 p 1 ff + fflq M ff + fflq M = E fflpr fflqfi (1 q(ffl 1)) ff + fflq M L p+~ffl (Ω [ ] ) for some ~ffl >, since (1 q(ffl 1)) > for ffl close o 1. By Lemma 3.3 we find (V (Hp ) ) E ( M + N) o

21 1 be he opimal pair. For p 6= we calculae U (p) V (Hp ) Since E (q M + N) (4.8) h E U V (p) (Hp ) i p = sgn(q) V (Hp ) = sgn(q)e r fi p 1 = sgn(q)e pr q fi = sgn(q)ce (q M + N) : ff + ff + q M p 1 Mp is a uniformly inegrable maringale we find = sgn(q)c = sgn(q) he las equaion follows from (3.9). E h E pr q fi E [E (N) ] ff In he nex secion we give aninerpreaion of he porfolios generaed by H p. i : 5. Locally Efficien Porfolios From Lemma 1.4 and μ = rs C, dff-a.s., we immediaely find V ( ^H) := E (H S) = E ((r HC ) ff + H M) H for a process ^H = 1 HS B L 1 loc (S). From Cauchy-Schwarz inequaliy i follows ha jhc j» p p p C HCH = fi HCH. We have [V ( ^H) V ( ^H) ]= V ( ^H) HCH ff [ ] : Weinerpre p HCH as a measure for he relaive insananeous risk of he porfolio generaed by ^H and ^Hμ = r HC as a measure for he p insananeously expeced relaive reurn rae. For fi 6= and HCH 6=, we find for he insananeous Sharpe-raio p ^Hμ r HCH = p HC HCH of insananeously expeced relaive excess reurn over he insananeously risk-free reurn rae and relaive insananeous risk, fi» HC p HCH» fi and HC p HCH = fi iff H k +Ker(C) for a predicable, sricly negaive, process k L HC loc (fi), resp. p HCH = fi iff H k +Ker(C) for a predicable, sricly posiive, process k L loc (fi). We call hese hedging sraegies locally efficien. We have seen in he las secion ha in

22 he case of a oally p-unhedgeable price for risk he opimal porfolios generaed by H p are locally efficien. See Markowiz (195, 1987) and Sharpe (1964, ). Define he following quaniies for p 6= 1: (5.1) (5.) and R (p ) := 1 d ln (V(p )) p d 1 R (p) := lim ln (V(p ))!1 p (5.3) R ( ) := dv() d (5.4) 1 R () := lim V( ):!1 Under some regulariy condiions, hese quaniies exis. By heorem 4.5 we find immediaely Proposiion 5.1. Under he assumpions, ff = and pr q fi consan for p 6= 1, resp. r + fi consan for p =, we have (5.5) V(p ) = exp pr q fi ( ) (5.6) R (p ) = R (p) = r + fi (1 p) resp. (5.7) V() = r + fi ( ) (5.8) R ( ) = R () = r + fi : See Bielecki and Pliska (1999, ) for an inerpreaion of he quaniies R (p) and he risk-sensiive sochasic conrol approach. R (p ) can be inerpreed as an implied forward growh rae of he expeced uiliy of wealh under he opimal self-financing hedging sraegy. As we will see in Secion 7, anoher relaed quaniy is he righ one o look a: We define (5.9) Y(p ):=ln(jv(p )j) :

23 6. Exisence of Opimal Porfolios Le» < 1 be fixed. In his secion we will assume μ S o be coninuous and D q ( μm [ ] ) 6= for p>1, resp. for p<1, D e ( μm [ ] ) 6= and V(p SF p ( μ M [ ] )) < 1. We assume for simpliciy in his secion ha B =1. he resuls can be generalized o he case of a B such ha B and B 1 are uniformly bounded on []. 3 heorem 6.1. Under he above assumpions, he opimal pair (V opp Z opλq ), saisfying (3.8) and Z opλq D( μ M [ ] ), exiss for he marke μm [ ] wih respec o opimizaion in SF p ( μm [ ] ). Proof. We firs prove he case p > 1. Since A Λp 1 ( μm [ ] ) is closed and convex and since L p (Ω [ ] ) is reflexive here exiss an elemen V opp wih minimal norm. As in GLP98, Lemma 4.1 and heorem 4.1, i is easily shown ha V opp. Since U (p) is concave we have for all Y A Λp ( μm [ ] ) (6.1) U (p) V opp + Y» U (p) V opp I follows from he opimaliy ofv opp + and since du (p) dx du (p) dx V opp V opp Y: L q (Ω [ ] )ha (6.) E»B du (p) dx V opp Y B = for all Y A Λp ( M μ [ ] ). From V opp Lq (Ω [ ]). Since V opp du (p) dx V opp L p (Ω [ ] ) i follows ha A Λp 1 ( μm [ ] ) we have

24 4 E» (6.3) V opp p 1 >. We herefore find Z opλp := E h du (p) dx h V opp du (p) dx V opp E fi fif i i μd q ( μm [ ] ): Opimaliy follows now from Proposiion 3.1. I was shown in GK98, Lemma 4.4, ha Z opλq D q ( μm [ ] ). For p<1 he resuls of KramkovandSchachermayer (1999), (KS99), can be applied. here, exisence and uniqueness of an opimal soluion V opp wih V opp > for problem (3.6) is proved. Furhermore, he exisence and uniqueness of a sricly posiive process Z op, such ha E Z op qλ = infzd( μ M [ ] ) E h Z B q i, and wih he following properies is shown: Z op = sgn(q)u (p) (V opp ), V opp Z op is a uniformly inegrable maringale and for an arbirary self-financing hedging sraegy wih non-negaive value process V, he process VZ op is a supermaringale. We will show in he nex lemma, ha for a coninuous price process Z op D( μm [ ] ). he worrying fac is of course ha Z op is in general only a supermaringale. However, he given example (Example 5.1' in KS99), showing ha Z op is in general no a local maringale, involves a non-coninuous price process. We define he following se of semimaringales living on Ω [ ] for < 1, see KS99: Y( μm [ ] ):= ρ Y jy =1 V H Y is a supermaringale B [ ] for all H SF ( μm [ ] ) ff :

25 Lemma 6.. Assume μ S o be coninuous and le Y Y( μm [ ] ) wih Y > be given. If here exiss a H SF ( μ M [ ] ) wih V H =1and 5 V H > and such ha V H Y is a uniformly inegrable maringale, hen Y D( μ M [ ] ). Proof. Since Y is a non-negaive supermaringale, we have by J&S87, Lemma III.3.6, ha Y > and Y > almos surely and hence Y = E(Z) for Z := 1 Y Y. Since Y is a supermaringale i is a special semimaringale and herefore Z oo. Z admis a represenaion Z = A+L, where A = A is a predicable process of finie variaion, L = L is a local maringale and A = L =. By J&S87, heorem III.4.11, we find a predicable process K L loc (M) and a local maringale N orhogonal o all componens of M, wih [M N] =, such ha L = K M + N and he represenaion Y = E(A + K M + N). Since V H is a local maringale wih respec o any equivalen maringale measure, V H > implies V H >. By Lemma 1.4 here exiss a ~ H L loc S such ha ~H 1 ~ H S B generaes V H. By assumpion and since M and ~ H C ff are coninuous, V H B Y = V ( ~ H ) B Y = E( ~ H C(K ) ff + A + (K + ~ H ) M + N) is a uniformly inegrable maringale. V H B Y ( H ~ C(K ) ff + A) = V H he Doléon-Dade SDE implies ha B V Y H Y B ((K + ~ H ) M + N) 1 is a predicable local maringale of finie variaion on

26 6 [], hence consan on [] almos surely, see J&S87, Corollary I We herefore find E( ~ H C(K ) ff + A) = 1. Now le H L loc S, se H μ := H 1 HS and consider he discouned value B process V Λ := V ( μ H) B = E( HC ff + H M) generaed by μ H. We have V Λ Y = E(HC(K ) ff+a+(k+h) M +N) = E((H ~ H )C(K ) ff+(k +H) M +N) is a supermaringale for all H L loc S by assumpion. For H := K + ~ H wefind(v Λ Y ) ((K )C(K ) ff) = V Λ Y (V Λ Y ) ((K + H) M + N) 1 o be a non-decreasing local supermaringale on []. herefore (K )C(K ) =, dff-a.s. and from 1 = E( ~ H C(K ) ff + A) = E(A) we conclude A = and Y = E( M + N) D( μm [ ] ). In DMSSS97, heorem A-C, (for p = ), and GK98, heorem 3.1 and heorem 4.1, (for p>1), necessary and sufficien condiions are given ensuring G p ( μm [ ] )obe closed. hese resuls imply Proposiion 6.3. If G p ( μ M [ ] ) is closed, hen (6.4) V(p SF p ( μm [ ] )) = V(p G p ( μm [ ] )) and V opp G p ( μm [ ] ). can be obained by a self-financing hedging sraegy in Furhermore, for»», he opimal pair for he marke (6.5) μm [ ] is given by ψv opp _ V opp ZopΛq _ Z opλq! G p ( μm [ ] ) D q ( μm [ ] ):

27 7. he BSDE Approach In his secion we will pu o use Proposiion 3.1 in a general seing. Assume he exisence of a coninuous local maringale N orhogonal o M such ha (M N) has he local maringale represenaion propery and [N N] = ~ C ff. Since he case p = is already solved (see Remark 4.3) we assume in his secion p 6= 1. Le»» be fixed. Consider he following formal calculaion for he opimal soluion for a maximizaion problem of erminal uiliy in he marke μm [ ] and an arbirary aainable Y B A Λp (M [ ] ): V opp 7 (7.1) U(V opp + Y )» U(V opp )+U (V opp )Y implies (7.) E»B U (V opp ) Y B = since ky is aainable for all F -measurable random variables k. Hence B U (V opp ) should define an absoluely coninuous maringale measure up o normalizaion. In general his argumen breaks down because of inegrabiliy problems. However, for isoelasic uiliy wih exponen p > 1 his approach works. None he less, we can ry he following ansaz: (7.3) c B U (V opp )=E ( M + ν N) respecively (7.4) ln c B U V opp 1 E ( M + ν N) ^H) V ( for ^H = H 1 HS B = where V opp [ ] = L ( μ loc S) and ν L (N). loc Ansaz (7.3) leads o a FBSDE. For he isoelasic uiliy funcions ansaz (7.4) will lead o a BSDE, where (H ν) form par of he soluion. For» s» define he adaped process (7.5) Y p s := ln ψ c B s du (p) dx! 1 V opp s E ( M + ν N) s :

28 8 Applying I^o's formula and by he definiion of he sochasic exponenial we find Y p = Y p + = Y p Z Z Z Z dy p s (p 1)H s C s H s ν s Cs ~ ν s s C s s dff s ((p 1)H s C s s pr s ) dff s ( s (p 1)H s ) dm s + Z ν s dn s : Because of Proposiion 3. and he formulas (3.16) and (3.17) we expec Y p o be independen of, hence we arrive a he following BSDE for»» : (7.6) Y (p ) = Z Z Z (p 1)H s C s H s ν s Cs ~ ν s s C s s dff s ((p 1)H s C s s pr s ) dff s ( s (p 1)H s ) dm s Z ν s dn s : Conversely, given an adaped soluion (Y (p ) Hν) o he BSDE (7.6) on [ ], we can define a self-financing hedging sraegy in μm [ ] by using (7.7) as a generaor for (7.8) We also have (7.9) ^H := H 1 HS [ ] B V ( ^H) := E ((r HC ) ff + H M) SF ( μm [ ] ): Z ν := E ( M + ν N) D( μm [ ] ): Lemma 7.1. V ( ^H) and Z ν (7.1) Z ν = exp Y (p ) B jpj saisfy (3.8): (p) du B sgn(1 p) dx V ( ^H)

29 9 Proof. Observe 1 = exp which implies exp exp ψz Z! (p 1)H s C s H s ν s Cs ~ ν s s C s s dff s (p 1)H s C s s pr s dff s Y (p ) + Z s (p 1)H s dm s + Z ν s dn s E ( M + ν N) = exp = = = exp exp Z (1 p)h s C s H s +(1 p)h s C s s + pr s dff s Y (p ) + Z Z (p 1)H s dm s r s H s C s s H sc s H s E (r ff) exp Y (p ) exp Y (p ) exp dff sp 1 Z H s dm sp 1 B E ((r HC ) ff + H M) p 1 B exp Y (p ) B jpj B (p) du sgn(1 p) dx V ( ^H) : Proposiion 7.. Assume (Y (p ) (H ν)) o be a soluion o he BSDE (7.6) on [ ]. Define ^H, resp. V ( ^H), Z ν by (7.7), resp. (7.8), (7.9). If for p<1, E ((H + ) M + ν N) is a uniformly inegrable marin- V ( ^H) Z ν gale, respecively if for p>1, V ( ^H) ^H SF p ( μm [ ] ), hen

30 3 is he opimal pair for he marke μm [ ] wih respec o opimizaion in SF p ( μ M [ ] ). Furhermore we have (7.11) V(p )= sgn(q) exp(y (p ) )= sgn(q) exp(y(p )): Proof. he firs asserion follows from Proposiion 3.1, (7.11) follows from (3.13). Conversely, he exisence of an opimal pair for he marke μm [ ] ogeher wih he local maringale represenaion propery of (M N), implies he exisence of a soluion (Y (p ) (H ν)) for he BSDE (7.6) on [ ] saisfying he assumpion of Proposiion Markovian Marke Model As an example, we will ransform in his secion he BSDE (7.6) for a (for simpliciy ime-homogeneous) markovian marke model ino a non-linear parial differenial equaion wih boundary condiion. Consider he following marke model: Assume he exisence of a (m + m )-dimensional Brownian moion W = (W 1 W ) on Ω 1 and assume F o be generaed by W. For simpliciy, le ^μ = (μ μ ) : R d+d! R d+d and ff : R d+d! R (d+d ) (m+m ) be smooh uniformly bounded funcions wih uniformly bounded derivaives of all orders, such ha for all x R d+d, ffff Λ (x) : R d+d! R d+d is inverible wih uniformly in x bounded inverse. Furhermore, assume ffff Λ (x) =C(x) C (x) :R d R d! R d R d. hen here exiss a R d+d -valued Markov process X =(S S ) solving he SDE for x R m+m (8.1) dx = ^μ(x )d + ff(x )dw X = x : Denoe by M he maringale par of S, andby N he maringale par of S. Noe ha M and N are orhogonal. Assume he ineres rae r o be a bounded funcion of X, and define B := expr r(x s)ds for all. Se := C 1 (μ rs) andfi := C. We now inerpre μs := (S B) asa price process and S as (non-raded) sae variables. Consider he following non-linear PDE for Y : R d R d [ 1)! R, (p + L 1Y + L Y = L 3 Y + L (p) Y + q fi pr

31 31 wih boundary condiion Y (s s ) = 8 s s, where L 1 := L := dx i=1 Xd i=1 μ + i + 1 i and for f C 11 (R d R d [ 1)), L 3 f := 1 L (p) f := Xd ij=1 1 ij j dx ij=1 dx ij=1 C ij j C j i C ij j : Assume Y (p) C 1 (R d R d [ 1)) o be a soluion of he PDE (8.), saisfying he boundary condiion Y (p) ( ) =. Applying I^o's formula o he process Y (p ) := Y (p) (S S ) we find (8.) Y (p ) H opp ν opλq := Y (p (p) (S S ) (S S p 1 (S S o be a soluion for he BSDE (7.6). We give (admiedly quie srong and no easy o check) condiions, ensuring Y (p ) (H ν) o be a useful soluion: heorem 8.1. If for p>1, E( M + ν opλq N) L q+ffl (Ω [ ] ) for an ffl>, resp. if for p<1p6=, E ( + H opp ) M + ν opλp N is a uniformly inegrable maringale, hen for all»» (8.3) V opp = E r H opp C ff + H opp M and (8.4) Z opλq = E ( M + ν opλq N) :

32 3 Proof. he asserion follows direcly from Lemma 3.3 and Proposiion 7.. Remark 8.. Because of heorem B in DMSSS97, resp. heorem 4.1 in GK98, and heorem.15 in DMSSS97, he condiion for p > 1 in he above heorem is more naural han i migh appear a he firs momen. Remark 8.3. For consan q fi pr one easily finds Y (p) (s s ) := q fi pr o be a soluion of he PDE (8.) saisfying he condiions of heorem 8.1. his also proves ha in his case he q- opimal maringale measure equals he minimal maringale measure, Z opλp = E ( M), see Föllmer and Schweizer (199). Remark 8.4. For p =, under he assumpion of heorem 8.1, we can consruc he hedging numéraire explicily. In Lauren and Pham (1999), Secion 6, his is achieved under sronger assumpions on fi r. See also Leiner () for an applicaion o he mean-variance efficiency problem and he calculaion of he ineremporal price for risk. 9. Conclusions One of he problems in he probabilisic heory of finance is, ha one has o make an ansaz for he probabiliy law of he fuure price processes. In a consanly changing world i is no a all clear how such

33 a law could ever be deermined. Wihou knowing he acual law (i is even difficul o argue ha such alaw exiss, since i is no clear how we can speak of probabiliies in experimens which can no be repeaed) i is difficul o define wha we mean by an opimal hedging sraegy. In a more pragmaic approach one could ry o parameerize a class of reasonable laws and look for parameers such ha he implied prices fi he observed prices bes. If observed and implied prices differ, or if he esimaed law differs from he law an invesor believes prices o follow, hen he invesor should buy underpriced and sell overpriced socks unil risk aversion, he rus in he used model and he confidence o be more clever han he marke, are in a balance. In our model his would involve solving parameerized (F)BSDEs and esimaing good parameers. In a discree ime model his can be achieved using a backward ieraive algorihm, bu would be compuaional very expensive, since observed prices have o be compared wih implied prices for a large number of parameer values. For a markovian model as in Secion 8, we would sill have o solve a (parameerized) non-linear PDE. Anoher problem in he probabilisic heory of finance is, ha whenever a new, i.e. non aainable, sock or securiy is inroduced o he marke, hen he marke daa C fi and he opimal pairs V opp, Z opλp will change in general. We have shown, ha he problem of maximizing isoelasic uiliy from erminal wealh is significanly faciliaed if he price for insananeous risk is assumed o be oally unhedgeable. In his siuaion opimal porfolios can be consruced from locally efficien porfolios and here holds a wo-fund-heorem for all invesors maximizing isoelasic uiliy of erminal wealh. he big advanage of locally efficien porfolios is ha hey can be deermined by esimaing he local characerisics of he price process. We only have o assume ha prices follow some probabiliy law (ha allows for successful esimaion of local characerisics), bu we don' have o decide for a specific law. 33 References Aase, K. K. (1984): Opimum Porfolio Diversificaion in a General Coninuous-ime Model. Sochasic Processes and heir Applicaions 18, Aase K. K. (1986): Ruin Problems and Myopic Porfolio Opimizaion in Coninuous rading. Sochasic Processes and heir Applicaions 1, Aase K. K. (1987): Sochasic Conrol of Geomeric Processes. J. Appl. Prob. 4,

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