Forward-backward systems for expected utility maximization

Transcription

1 Forward-backward sysems for expeced uiliy maximizaion Ulrich Hors, Ying Hu, Peer Imkeller, Anhony Reveillac, Jianing Zhang o cie his version: Ulrich Hors, Ying Hu, Peer Imkeller, Anhony Reveillac, Jianing Zhang. Forward-backward sysems for expeced uiliy maximizaion. Sochasic Processes and heir Applicaions, Elsevier, 214, 124 5, pp <1.116/j.spa >. <hal > HAL Id: hal hps://hal.archives-ouveres.fr/hal Submied on 13 Oc 211 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. he documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Forward-backward sysems for expeced uiliy maximizaion Ulrich Hors, Ying Hu, Peer Imkeller, Anhony Réveillac and Jianing Zhang Ocober 12, 211 Absrac In his paper we deal wih he uiliy maximizaion problem wih a general uiliy funcion. We derive a new approach in which we reduce he uiliy maximizaion problem wih general uiliy o he sudy of a fully-coupled Forward-Backward Sochasic Differenial Equaion FBSDE. AMS Subjec Classificaion: Primary 6H1, 93E2 JEL Classificaion: C61, D52, D53 1 Inroducion One of he mos commonly sudied opic in mahemaical finance and applied probably is he problem of maximizing expeced erminal uiliy from rading in a financial marke. In such a siuaion, he sochasic conrol problem is of he form V, x := sup EUX π + H 1.1 π A for a real-valued funcion U, where A denoes he se of admissible rading sraegies, < is he erminal ime, X π is he wealh of he agen when he follows he sraegy π A and his iniial capial a he iniial ime zero is x >, and H is a liabiliy ha he agen mus deliver a he erminal ime. One is ypically ineresed in esablishing exisence and uniqueness of opimal soluions and in characerizing opimal sraegies and he value funcion V, x which is defined as V, x := sup EUX, π + H F. π A Insiu für Mahemaik, Humbold-Universiä zu Berlin, Uner den Linden 6, 199 Berlin, Germany, hors@mahemaik.hu-berlin.de Universié de Rennes 1, campus Beaulieu, 3542 Rennes cedex, France, ying.hu@univ-rennes1.fr Insiu für Mahemaik, Humbold-Universiä zu Berlin, Uner den Linden 6, 199 Berlin, Germany, imkeller@mahemaik.hu-berlin.de CEREMADE UMR CNRS 7534, Universié Paris Dauphine, Place du Maréchal De Lare De assigny, PARIS CEDEX 16, France, anhony.reveillac@cremade.dauphine.fr Weiersrass Insiue for Applied Analysis and Sochasics, Mohrensr. 39, 1117 Berlin, Germany, jianing.zhang@wias-berlin.de 1

3 Here X, denoes he wealh of he agen when he invesmen period is, and where he filraion F, defines he flow of informaion. he quesion of exisence of an opimal sraegy π can essenially be addressed using convex dualiy. he convex dualiy approach is originally due o Bismu 2 wih is modern form daing back o Kramkov and Schachermayer 13. For insance, given some growh condiion on U or relaed quaniies such as he asympoic elasiciy condiion for uiliies defined on he half line exisence of an opimal sraegy is guaraneed under mild regulariy condiions on he liabiliy and convexiy assumpions on he se of admissible rading sraegies see e.g. 1 for deails. However, he dualiy mehod is no consrucive and does no allow for a characerizaion of opimal sraegies and value funcions. One approach o simulaneously characerize opimal rading sraegies and uiliies uses he heory of forward-backward sochasic differenial equaions FBSDE. When he filraion is generaed by a sandard Wiener process W and if eiher Ux := exp αx for some α > and H L 2, or Ux := xγ γ for γ, 1 or Ux = ln x and H =, i has been shown by Hu, Imkeller and Müller 9 ha he conrol problem 1.1 can essenially be reduced o solving a BSDE of he form Y = H Z s dw s fs, Z s ds,,, 1.2 where he driver f, z is a predicable process of quadraic growh in he z-variable. heir resuls have since been exended beyond he Brownian framework and o more general uiliy opimizaion problems wih complee and incomplee informaion in, e.g., 8, 19, 2, 21 and 17. he mehod used in 9 and essenially all oher papers relies on he maringale opimaliy principle and can essenially only be applied o he sandard cases menioned above exponenial wih general endowmen and power, respecively logarihmic, wih zero endowmen. his is due o a paricular separaion of variables propery enjoyed by he classical uiliy funcions: heir value funcion can be decomposed as V, x = gxv where g is a deerminisic funcion and V is an adaped process. As a resul, opimal fuure rading sraegies are independen of curren wealh levels. More generally, here has recenly been an increasing ineres in dynamic ranslaion invarian uiliy funcions. A uiliy funcion is called ranslaion invarian if a cash amoun added o a financial posiion increases he uiliy by ha amoun and hence opimal rading sraegies are wealh-independen 1. Alhough he propery of ranslaion invariance renders he uiliy opimizaion problem mahemaically racable, independence of he rading sraegies on wealh is raher unsaisfacory from an economic poin of view. In 18 he auhors derive a verificaion heorem for opimal rading sraegies for more general uiliy funcions when H =. More precisely, given a general uiliy funcion U and assuming ha here exiss an opimal sraegy regular enough such ha he value funcion enjoys some regulariy properies in, x, i is shown ha here exiss a predicable random field ϕ, x,x,, such ha he pair V, ϕ is soluion o he following backward 1 I has been shown by 6 ha essenially all such uiliy funcions can be represened in erms of a BSDE of he form

4 sochasic parial differenial equaion BSPDE of he form: V, x = Ux ϕs, xdw s ϕ x s, x 2 ds,, 1.3 V xx s, x where ϕ x denoes he parial derivaive of ϕ wih respec o x and V xx he second parial derivaive of V wih respec o he same variable. he opimal sraegy π can hen be obained from V, ϕ. Unforunaely, he BSPDE-heory is sill in is infancy and o he bes of our knowledge he non-lineariies arising in 1.3 canno be handled excep in he classical cases menioned above where once again one benefis of he separaion of variables see 11. Moreover, he uiliy funcion U only appears in he erminal condiion which is no very handy. In ha sense his is exacly he same siuaion as he Hamilon-Jacobi- Bellman equaion where U only appears as a erminal condiion bu no in he equaion iself. In his paper we propose a new approach o solving he opimizaion problem 1.1 for a larger class of uiliy funcion and characerize he opimal sraegy π in erms of a fully-coupled FBSDE-sysem. he opimal sraegy is hen a funcion of he curren wealh and of he soluion o he backward componen of he sysem. In addiion, he driver of he backward par is given in erms of he uiliy funcion and is derivaives. his adds enough srucure o he opimizaion problem o deal wih fairly general uiliies funcions, a leas when he marke is complee. We also derive he FBSDE sysem for he power case wih general non-hedgeable liabiliies; o he bes of our knowledge we are he firs o characerize opimal sraegies for power uiliies wih general liabiliies. Finally, we link our approach o he well esablished approaches using convex dual heory and sochasic maximum principles. he remainder of his paper is organized as follows. In Secion 2 we inroduce our financial marke model. In Secion 3 we firs derive a verificaion heorem in erms of a FBSDE for uiliies defined on he real line along wih a converse resul, ha is, we show ha a soluion o he FBSDE allows o consruc he opimal sraegy. Secion 4 is devoed o he same quesion bu for uiliies defined on he posiive half line. In Secion 5 we relae our approach o he sochasic maximum principle obained by Peng 22 and he sandard dualiy approach. We use he dualiy-bsde link o show ha he FBSDE associaed wih he problem of maximizing power uiliy wih general posiive endowmen has a soluion. 2 Preliminaries We consider a financial marke which consiss of one bond S wih ineres rae zero and of d 1 socks given by d S i := S i dw i + S i θ i d, i {1,..., d} where W is a sandard Brownian moion on R d defined on a filered probabiliy space Ω, F, F,, P, F, is he filraion generaed by W, and θ := θ 1,..., θ d is a predicable bounded process wih values in R d. Since we assume he process θ o be bounded, Girsanov s heorem implies ha he se of equivalen local maringale measures 3

5 i.e. probabiliy measures under which S is a local maringale is no empy, and hus according o he classical lieraure see e.g. 7, arbirage opporuniies are excluded in our model. For simpliciy hroughou we wrie ds i := d S i. S i We denoe by α β he inner produc in R d of vecors α and β and by he usual associaed L 2 -norm on R d. In all he paper C will denoe a generic consan which can differ from line o line. We also define he following spaces: { } S 2 R d := H 2 R d := β : Ω, R d, { β : Ω, R d, predicable, E sup β 2 <, predicable, E β 2 d, } <. Since he marke price of risk θ is assumed o be bounded, he sochasic process E θ W := exp θ s dw s 1 θ s 2 ds 2 has finie momens of order p for any p >. We assume d 1 + d 2 = d and ha he agen can inves in he asses S 1,..., S d 1 while he socks S d1+1,..., S d 2 canno be invesed ino. Denoe S H := S 1,..., S d 1,...,, W H := W 1,..., W d 1,...,, W O :=,...,, W d1+1,..., W d 2, and θ H := θ 1,..., θ d 1,..., he noaion H refers o hedgeable and O o orhogonal. We define he se Π x of admissible sraegies wih iniial capial x > as { } Π x := π : Ω, R d 1, E π 2 d <, π is self-financing 2.1 where for π in Π x he associaed wealh process X π is defined as X π := x + d 1 π r dsr H = x + i=1 Every π in Π x is exended o an R d -valued process by π := π 1,..., π d 1,,...,. π i rds i r,,. In he following, we will always wrie π in place of π, i.e. π is an R d -valued process where he las d 2 componens are zero. Moreover, we consider a uiliy funcion U : I R where I is an inerval of R such ha U is sricly increasing and sricly concave. We seek for a sraegy π in Π x saisfying EUX π + H < such ha π = argmax π Π x, E UX π +H < {EUX π + H} 2.2 where H is a random variable in L 2 Ω, F, P such ha he expression above makes sense. We concreize on sufficien condiions in he subsequen secions. 4

6 3 Uiliies defined on he real line In his secion we consider a uiliy funcion U : R R defined on he whole real line. We assume ha U is sricly increasing and sricly concave and ha he agen is endowed wih a claim H L 2 Ω, F, P. We inroduce he following condiions. H1 U : R R is hree imes differeniable H2 We say ha condiion H2 holds for an elemen π in Π x, if E U X π + H 2 < and if for every bounded predicable process h :, R, he family of random variables 1 h r dsr H U X π + H + εr h r dsr H dr is uniformly inegrable. ε,1 Before presening he firs main resul of his secion, we prove ha condiion H2 is saisfied for every sraegy π such ha E U X π +H < when one has an exponenial growh condiion on he marginal uiliy of he form: U x + y C 1 + U x 1 + expαy for some α R. Indeed, le G := h rdsr H and d >. We will show ha he quaniy 1 qd := sup E G U X π + H + εrgdr 1 G 1 ε,1 U X π +H+εrGdr >d vanishes when d goes o infiniy. For simpliciy we wrie δ ε,d := 1 1 G U X π +H+εrGdr >d. By he Cauchy-Schwarz inequaliy qd Since E U X π ha sup E ε,1 1 + U X π CE U X π + H 2 1/2 1 sup E G ε,1 1 + H G1 + expαεrgdr δ ε,d expαεrgdr 2 δ ε,d 1/2. + H 2 is assumed o be finie we deduce from he inequaliy expαζx 1 + expαx for all x R, < ζ < 1 1/2 qd C sup E G2 + expαg 2 δ ε,d. ε,1 Applying successively he Cauchy-Schwarz inequaliy and he Markov inequaliy, i holds ha qd CE G2 + expαg 4 1/4 sup Eδ ε,d 1/4 ε,1 5

7 CE G2 + expαg 4 1/4 d 1/4 sup ε,1 E G 1 U X π CE G2 + expαg 4 1/4 d 1/4 E G2 + expαg 2 1/8. Le p 2. Since h and θ are bounded i is clear ha E G 2p < and E G2 + expαg p E G 2p 1/2 E 2 + expαg 2p 1/2 expαg 2p 1/2 C = C C. 2 + E 2 + E exp 2pαh r dw H r exp 2pαh r 2 + 2pαh r θ r dr 2 Hence lim d qd = which proves he asserion. 1/2 2pαh r 2 dr 3.1 Characerizaion and verificaion: incomplee markes 1/4 + H + εrgdr We are now ready o sae and prove he firs main resul of his paper: a verificaion heorem for opimal rading sraegies. heorem 3.1. Assume ha H1 holds. Le π Π x be an opimal soluion o he problem 2.2 which saisfies assumpion H2. hen here exiss a predicable process Y wih Y = H such ha U X π + Y is a maringale in L 2 Ω, F, P and where Z := π i d Y,W d := = θ i U X π + Y U X π + Y Zi,,, i = 1,..., d 1 d Y,W i d,..., d Y,W d d. Proof. We firs prove he exisence of Y. Since E U X π +H 2 <, he sochasic process α defined as α := EU X π +H F, for in, is a square inegrable maringale. Define Y := U 1 α X π. hen Y is F, -predicable. Now Iô s formula yields Y + X π = Y + X π 1 U U 1 α s dα s By definiion, α is he unique soluion of he zero driver BSDE α = U X π U 3 U 1 α s U U 1 α s 3 d α, α s Y β s dw s,,, 3.2 6

8 where β is a square inegrable predicable process wih valued in R d. Plugging 3.2 ino 3.1 yields Y + X π =X π + H 1 U Xs π + Y s β sdw s U 3 Xs π + Y s U Xs π + Y s 3 β s 2 ds. Seing Z 1 := β, we have U X π +Y Y + X π =X π + H Z s dw s U 3 U Xs π + Y s Z s 2 ds. Now by puing Z i := Z i π i, i = 1,..., d, we have shown ha Y is a soluion o he BSDE Y = H Z s dw s fs, Xs π, Y s, Z s ds,,, 3.3 where f is given by fs, Xs π, Y s, Z s := 1 U 3 2 U Xs π + Y s πs + Z s 2 πs θ s. 3.4 Finally, by consrucion we have U X π + Y = α, hus i is a maringale. Now we deal wih he characerizaion of he opimal sraegy. o his end, le h :, R d 1 be a bounded predicable process. We exend h ino R d by seing h := h 1,..., h d 1,,..., and use he convenion ha h is again denoed by h. hus for every ε in, 1 he perurbed sraegy π + εh belongs o Π x. Since π is opimal i is clear ha for every such h i holds ha 1 lh := lim ε ε E Ux + Moreover we have 1 ε = Ux + h r ds H r πr + εh r dsr H + Y Ux + πrds r H + Y. 3.5 πr + εh r dsr H + Y Ux + 1 U X π + Y + θε h r dsr H πrds r H + Y dθ. Now using H2, Lebesgue s dominaed convergence heorem implies ha 3.5 can be rewrien as E U X π + Y h r dsr H 3.6 for every bounded predicable process h. Applying inegraion by pars o U Xs π +Y s s, and s h rdsr H, we ge s, U X π + Y h r dsr H 7

9 = U x + Y s s h r ds H r U X π s h r ds H r U X π s U X π s + Y s U 3 X π s + Y s h s ds H s + Y s h s π s + Z s ds. π s + Z s dw H s + π s θ s + fs, X π s + Y s π s + Z s 2 ds By definiion of he driver f, he previous expression reduces o U X π + Y = + s h r ds H r U Xs π + Y s θ s + U X π h r ds H r, Y s, Z s ds s + Y s πs + Z s h s ds U Xs π + Y s πs + Z s dws H + U Xs π + Y s h s dws H. 3.7 he nex sep would be o apply he condiional expecaions in 3.7, however he wo erms on he second line of he righ hand side are a priori only local maringales. We sar by showing ha he firs one is a uniformly inegrable maringale. Indeed, from he compuaions which have led o 3.3 we have ha U X π + Y π + Z = β, where we recall ha β is he square inegrable process appearing in 3.2. Using he BDG inequaliy we ge s r E sup h u dsu H U Xr π + Y r πr + Z r dw H r s, CE s h r dsr H 2 1/2 β s 2 ds s CE sup h r dsr H 2 1/2 1/2 β s 2 ds. s, Young s inequaliy furhermore yields s E sup h r dsr H 2 1/2 CE s, sup s, s h r ds H r 2 + CE 1/2 β s 2 ds β s 2 ds 8

10 C 1 + E sup s, s h r dw H r 2 where we have used ha h and θ are bounded. Applying once again he BDG inequaliy, we obain s E sup h r dwr H 2 4E h r 2 dr <. s, s, Puing ogeher he previous seps, we have ha s r E sup h u dsu H U Xr π + Y r πr + Z r dwr H <, hus we ge Noe ha E s h r ds H r U Xs π + Y s h s dws H U Xs π, + Y s π s + Z s dw H s U X π + Y = α is a square inegrable maringale and hus E U 2 Xs π + Y s h s ds <. =. is a square inegrable maringale. Similarly, E U X π + Y h r dsr H <. aking expecaion in 3.7 we obain for every n 1 ha E U X π + Y h r dsr H = E U Xs π + Y s θ s + U Xs π + Y s πs + Z s which in conjuncion wih 3.6 leads o E U Xs π + Y s θ s + U Xs π + Y s πs + Z s h s ds Indeed h s ds, 3.8 for every bounded predicable process h. Replacing h by h, we ge E U Xs π + Y s θ s + U Xs π + Y s πs + Z s h s ds =. 3.9 Now fix i in {1,..., d 1 }. Le A i s := U Xs π + Y s θ s + U Xs π + Y s πs i + Zs i and h s :=,...,, 1 A i s >,,..., where he non-vanishing componen is he i-h componen. From 3.9 we ge ha E 1 A i s >U X π s + Y s θ i s + U X π s 9 + Y s π i s + Zsds i =.

11 Hence, A i, dp d a.e.. deduce ha Similarly by choosing h s =,...,, 1 A i s <,,..., we U X π + Y θ i + U X π + Y π i + Z i =, dp d a.e. his concludes he proof since i {1,..., d 1 } is arbirary. he verificaion heorem above can also be expressed in erms of a fully-coupled Forward- Backward sysem. heorem 3.2. Under he assumpions of heorem 3.1, he opimal sraegy π for 2.2 is given by π i = θ i U X + Y U X + Y Zi,,, i = 1,..., d 1, where X, Y, Z R R R d is a riple of adaped processes which solves he FBSDE X = x U θ X s+y s s U X + Z s+y s s dws H U θ X s+y s s U X + Z s+y s s θs H ds Y = H Z s dw s 1 2 θh s 2 U 3 X s+y s U X s+y s 2 U X s+y s 3 + θ H s 2 U X s+y s U X s+y s + Z s θ H s 1 2 ZO s 2 U 3 U X s + Y s ds, 3.1 wih he noaion Z = Z 1,..., Z d 1 d, Z 1 +1,..., Z d. In addiion, he opimal wealh process }{{}}{{} =:Z H =:Z O X π is equal o X. Proof. From heorem 3.1 we know ha he opimal sraegy is given by π i = θ i U X π + Y U X π + Y Zi,,, i {1,..., d 1 } where Y, Z is a soluion o he BSDE 3.3 wih driver f like in 3.4. Now plugging he expression of π in relaion 3.4 yields X π = x U θ Xs π +Y s s dws H U θ Xs π +Y s s θs H ds Y = H U X π s Z s dw s +Y s + Z s 1 2 θh s 2 U 3 Xs π U X π s +Y s + Z s +Ys U Xs π +Ys 2 +Ys3 U X π s + θs H 2 U Xs π +Ys U Xs π +Ys + Z s θs H 1 2 ZO s 2 U 3 U X s π + Y s ds Recalling ha X π := x + π sdws H + θs H ds for any admissible sraegy π, we ge he forward par of he FBSDE. Remark 3.3. Using Iô s formula and he FBSDE 3.1, we have ha U X + Y = U x + Y + θ H s U X s + Y s dw H s + U X s + Y s Z O s dw O s. 1

12 Remark 3.4. Noe ha using he sysem 3.1, for α := U X π +Y, inegraion by pars yields for every in, U X π + Y X π X π = = + Xs π Xs π dα s + α s θ H s α s π s π sdw H s + U Xs π + Y s Zs H + πs π s πsds X π s X π s dα s + α s π s π sdw H s showing ha U X π + Y X π X π is a local maringale for every π in Π x. he converse implicaion of heorems 3.1 and 3.2 consiues he second main resul. heorem 3.5. Le H1 be saisfied for U. Le X, Y, Z be a riple of predicable processes which solves he FBSDE 3.1 saisfying: Z is in H 2 R d, E UX +H <, E U X + H 2 <, and U X + Y is a posiive maringale. Moreover, assume ha here exiss a consan κ > such ha U x U x κ for all x R. hen π i := U X + Y U X + Y θi Z i,,, i {1,..., d 1 }, is an opimal soluion of he opimizaion problem 2.2. Proof. Noe firs ha by definiion of π, X = X π. Since he risk olerance U x U x is bounded and since Z is in H 2 R d, we immediaely ge E π s 2 ds <, hus, π Π x. By assumpion, U X + Y is a posiive coninuous maringale, hence here exiss a coninuous local maringale L such ha U X + Y = EL. And we know from Remark 3.3 ha L = logu x + Y + Define he probabiliy measure Q P by θ H s dw H s + dq dp := U X + H EU X + H. U X s + Y s U X s + Y s ZO s dw O s. Girsanov s heorem implies ha W := W H + W O = W 1 +θ 1 d,..., W d 1 +θ d1 d, W d1+1 U X+Y U X+Y Zd 1+1 d,..., W d 2 U X+Y U X+Y Zd2 d is a sandard Brownian moion under Q. hus X π is a local maringale under Q for every π in Π x. Now fix π in Π x wih E UX π + H <. Le τ n n be a localizing sequence for he local maringale X π X π. Since U is a concave, we have UX π + H UX π + H U X π 11 + HX π X π. 3.12

13 aking expecaions in 3.12 we ge EUX π + H UXπ + H EU X + H E Q X π X π = E Q lim = lim n E Q τn n τn π s π sd W H s π s π sd W H s =, which evenually follows as a consequence of Lebesgue s dominaed convergence heorem. o his end we prove ha E Q sup π s πsd W s H <., Indeed he BDG inequaliy and he Cauchy-Schwarz inequaliy imply ha E Q sup π s πsd W H s, 1 CE Q π s πs 2 2 ds = CE U X + H EU X + H U X + H CE EU X + H π s πs 2 ds E π s πs ds <. We have proved in heorem 3.2 ha if 2.2 exhibis an opimal sraegy π Π x, hen here exiss an adaped soluion o he FBSDE 3.1. As a byproduc we showed he opimizaion procedure singles ou a pricing measure under which he asse prices and marginal uiliies are maringales. In ha sense, he process Y capures he impac of fuure rading gains on he agen s marginal uiliies. If we assume addiional condiions on he uiliy funcion U, we ge he following regulariy properies of he soluion X, Y, Z. Proposiion 3.6. Assume ha for H L Ω, F, P and ha he FBSDE 3.1 admis an adaped soluion X, Y, Z such ha Y is bounded. Le ϕ 1 x := U x U x, ϕ 2x := U 3 x U x 2 U x 3, ϕ 3 x := U 3 x U x, x R. Assume ha U is such ha ϕ i, i = 1, 2, 3 are bounded and Lipschiz coninuous funcions. hen X, Y, Z is he unique soluion of 3.1 in S 2 R S R H 2 R d. In addiion, Z W is a BMO-maringale. 12

14 Proof. Le X, Y, Z be a soluion o 3.1 such ha Y is bounded. hen, using he usual heory on quadraic growh BSDEs see for example 2, heorem 2.5 and Lemma 3.1 we have only from he backward par of he FBSDE ha Z is in H 2 R d and ha Z W is a BMO-maringale. In addiion here exiss a unique soluion o he backward componen in his space for a given process X. Now he previous regulariy properies of he processes Y, Z imply ha X is in S 2 R. We urn o he uniqueness of he X process. Assume ha here exiss anoher soluion X, Y, Z of 3.1. Hence, heorem 3.5 implies ha π := U X +Y U X +Y θi + Z i, i {1,..., d 1 } is an opimal soluion o our original problem 2.2 and X is he opimal wealh process. However, by sric concaviy of U and by convexiy of Π x he opimal sraegy has o be unique. So X and X are he wealh processes of he same opimal sraegy, hus, hey have o coincide for insance X = X, P a.s. which implies Y, Z = Y, Z. In he complee case we are able o consruc he soluion X, Y, Z. his is he subjec of he following Secion. 3.2 Characerizaion and verificaion: complee markes In his secion we consider he benchmark case of a complee marke. We assume d = 1 for simpliciy. H denoes a square inegrable random variable measurable wih respec o he Brownian moion W. In he complee case we can give sufficien condiions for he exisence of a soluion o he sysem 3.1. Our consrucion relies on he following remark. Remark 3.7. Using 3.1 he maringale U X π +Y becomes more explici, because Iô s formula applied o U X π + Y yields U X π + Y = U x + Y + = U x + Y U X π s + Y s π s + Z s dw s U X π s + Y s θ s dw s, where we have replaced π by is characerizaion in erms of X, Y, Z from heorem 3.1. Hence, U X π + Y = U x + Y E θ W,, his remark will allow us o prove exisence of a soluion o he sysem 3.1 under a condiion on he risk aversion coefficien U U of U. o his end, we give a sufficien condiion on U for he sysem 3.1 o exhibi a soluion. We have he following remark. Remark 3.8. If X, Y, Z is an adaped soluion o he sysem 3.1, hen P := X + Y is soluion of he forward SDE P = x + Y θ s U P s U P s dw s 1 2 θ s 2 U 3 P s U P s 2 U P s 3 ds,, In addiion, if X, Y, Z is in S 2 R S 2 R H 2 R d, hen P S 2 R. hus a necessary condiion for he FBSDE 3.1 o have a soluion is ha he SDE 3.14 admis a soluion. 13

15 We are now going o sae an exisence resul for he FBSDE sysem 3.1 ha characerizes opimal rading sraegies in erms of he funcions ϕ 1 x = U x U x and ϕ 2x = U 3 x U x 2 U x 3. Proposiion 3.9. Assume ha he funcions ϕ 1 and ϕ 2 are bounded and Lipschiz coninuous. hen he FBSDE X = x U θ X s+y s s U X + Z s+y s s dw s U θ X s+y s s U X + Z s+y s s θ s ds Y = H Z s dw s 1 2 θ s 2 U 3 X s+y s U X s+y s 2 U X s+y s 3 + θ s 2 U X s+y s U X s+y s + Z s θ s ds 3.15 admis a soluion X, Y, Z in S 2 R S 2 R H 2 R d such ha E UX + H < and E U X + H 2 <. Proof. Le m in R. Consider he following SDE Y m P m = x + m θ s ϕ 1 P m s dw s 1 2 θ s 2 ϕ 2 P m s ds,,. Since his SDE has Lipschiz coefficiens he exisence and uniqueness of a soluion in S 2 R is guaraneed see for example 23, V.3. Lemma 1. Nex, consider he BSDE = H Zs m dw s 12 θ s 2 ϕ 2 P ms + θ s 2 ϕ 1 P ms + Z ms θ s ds We denoe is driver by fs, p, z := 1 2 θ s 2 ϕ 2 p + θ s 2 ϕ 1 p + z θ s. Using he regulariy properies of ϕ 1 and ϕ 2 and he fac ha θ is bounded, here exiss a consan K > such ha fs, p, z K1 + z and he consan K depends only on α 1, α 2 and on θ, hus in paricular K does no depend on m. Since he driver f is Lipschiz in z, here exiss a unique pair of adaped processes Y m, Z m in S 2 R H 2 R d which solves In addiion, Y m K holds P- a.s. for all in,. We recall ha his consan K does no depend on m, hus Y m K. Using usual argumens we can show ha he map m Y m is coninuous. Even if his procedure is somehow sandard, we reprove his fac here o make he paper self-conained. Fix m, m in R wih m m. We se δy := Y m Y m, δz := Z m Z m. By 3.16 i follows ha δy, δz is soluion o he Lipschiz BSDE: δy = δz s dw s θ s δz s + hsds wih hs := 1 2 θ s 2 ϕ 2 Ps m ϕ 2 Ps m + θ s 2 ϕ 1 Ps m ϕ 1 Ps m. Using classical a priori esimaes for Lipschiz growh BSDEs see for example 16, Lemma 2.2 we ge ha: δy 2 E sup δy 2 CE hs 2 ds., 14

16 he boundedness of θ and he Lipschiz assumpion on ϕ 1 and on ϕ 2 immediaely imply ha E hs 2 ds CE Ps m Ps m 2 ds CE sup P m P m 2., Combining he inequaliies above wih classical esimaes on Lipschiz SDEs see for example 23, Esimae *** in he proof of heorem V.7.37 we finally ge ha δy 2 C m m 2 which concludes he proof by leing m ending o m. his conjuncion wih m Y m being bounded yields ha here exiss an elemen m R such ha Y m = m. Seing X m := P m Y m,,, i is sraighforward o check ha X m, Y m, Z m saisfies Moreover, we have X m S 2 R since Y m is bounded and since P m S 2 R. Nex, noe ha E U X + Y 2 < since U X + Y = U x + me θ W. Now using he concaviy of U, i holds ha Consequenly, Ux U x + U, Ux U xx U, x R. E UX + H E U X + H + U + E U X + HX + H + U <. 4 Uiliy funcions on he posiive half-line In his secion we sudy uiliy funcions U : R + R defined on he posiive half-line. Again, we assume ha U is sricly increasing and sricly concave. In he previous secion we have derived a FBSDE characerizaion of he opimal sraegy for he uiliy maximizaion problem 2.2. he key observaion was ha here exiss a sochasic process Y such ha U X π + Y is a maringale. However if U is only defined on he posiive half-line, i is no clear a priori ha he expression U X π + Y makes sense. We could generalize his approach by looking for a funcion Φ such ha ΦX π, Y is a maringale and such ha ΦX π, Y = U X π + H. When H =, i urns ou ha a good choice of funcion Φ is Φx, y := U x expy since he sysem we obain coincides up o a non-linear ransformaion wih he one obained by Peng in 22, Secion 4 using he maximum principle. Noe ha he sysem of Peng is no formulaed as a FBSDE bu raher as a sysem of equaions: one for he wealh process whose dynamics depend on he sraegy and one adjoin equaion, bu a reformulaion of his sysem of equaion allows o ge a FBSDE deails are given in Secion

17 In he previous secion, π denoed he oal amoun of money invesed ino he sock he number of shares being π/ S. Now we denoe by π i he proporion of wealh invesed in he i-h sock S i. Once again we denoe by Π x he se of admissible sraegies wih iniial capial x which is now defined by Π x := { } π : Ω, R d 1, π is predicable, E π s 2 ds <. 4.1 he associaed wealh process is given by X π := x + π s X π s ds H s,,. Again, we exend π o R d via π := π 1,..., π d 1,,..., and make he convenion ha we wrie π insead of π. hus, we have X π = xe π r dsr H,,. From now one we consider a posiive F -measurable random variable H. We furhermore need o impose he following assumpions on U. H3 U : R + R is hree imes differeniable, sricly increasing and concave H4 We say ha assumpion H4 holds for an elemen π in Π x, if i E X π U X π + H 2 < ; ii he sequence of random variables 1 +ερ ε Xπ X π is uniformly inegrable; 1 U X π + H + rx π +ερ X π dr ε,1 iii lim sup ε, 1 +ερ E ε Xπ X π ξ 2 =, where dξ = π ξ ds H + ρ X π ds H,,, and sup, E ξ 2 <. H5 here exiss a consan c > such ha U x xu x c for all x R+. 16

18 4.1 Characerizaion and verificaion: incomplee markes Noe ha in condiion H4, if U < or if H a > is saisfied, hen iii implies ii. heorem 4.1. Assume ha H3 holds and ha H is a posiive random variable belonging o L 2 Ω, F, P. Le π be an opimal soluion o 2.2 saisfying E UX π + H < and which saisfies assumpion H4. hen here exiss a predicable process Y wih Y = logu X π + H logu X π such ha Xπ U X π expy is a maringale and where Z := π i d Y,W 1 d s s = U X π Xs π U Xs π Zi s + θs, i s,, i = 1,..., d 1,.,..., d Y,W d d Proof. As in he proof of heorem 3.1, we prove he exisence of Y such ha X π U X π expy is a maringale wih Y = logu X π + H logu X π. Consequenly, U X π + H = U X π expy. By H4, he process α := EX π U X π + H F is a square inegrable maringale. In addiion i is he unique soluion o he BSDE α = X π U X π + H β s dw s,,, where β is a square inegrable predicable process wih values in R d. We se Y := logα logu X π logx π. As in he proof of heorem 3.1, Iô s formula implies ha Seing we ge ha βs U Xs π α s U Xs π Xπ s πs πs dw s 1 β s 2 U 2 α s 2 Xs π U Xs π Xπ s πs + πs θs H + Xπ s πs 2 U Xs π 2 2 U Xs π U 3 Xs π U Xs π Y = Y Z i = βi α π U X π Y = Y Z s dw s U X π U X π s s Xπ Xπ π s 2 ds. 2 U X π + U X π, i = 1,..., d, 4.2 U 3 X π U Xs π s Xπ s π s Z s 2 ds,,. U s πs 2 Zs H + θs H X π U Xs π s Xπ s π s + π s We now derive he characerizaion of π in erms of U and Y and Z. We employ an argumen pu forh in 22 and hen subsiue he Hamilonian by a BSDE. Fix π Π x. 17

19 Since he laer is a convex se, for ρ := π π, he π + ερ is an admissible sraegy for every ε, 1. We have 1 +ερ ε UXπ + H UX π + H = Since π is opimal we find 1 +ερ E ε Xπ X π Now le ξ be defined as 1 +ερ ε Xπ X π 1 U X π 1 U X π + H + rx π +ερ + H + rx π +ερ dξ = π ξ + ρ X π ds H,,. X π dr. X π dr, ε >. 4.3 By H4, we can apply Lebesgue s dominaed convergence heorem in inequaliy 4.3 which, possibly passing o a subsequence, yields Eξ U X π 1 +ερ + H = lim E ε ε Xπ X π Combined wih 4.3, i leads o 1 U X π + H + rx π +ερ X π dr. Eξ X π 1 U X π X π expy = Eξ U X π + H, π Π x. 4.4 We now resric consideraion o a paricular class of processes π, ha is, we choose ρ o be a bounded predicable process and we define π := ρ + π which is admissible sraegy since i is square inegrable. he inegraion by pars formula for coninuous semimaringales implies ha ξ X π 1 = ρ s dw H s + ρ s θ H s ρ s π sds,,. Anoher applicaion of inegraion by pars o α = U X π X π expy and ξx π 1 yields ξ U X π + Y = ξ X π 1 U X π X π expy = + ξ X π 1 dα + ρ expy X π α ρ dw H U X π Z H + θ H + U X π X π π d. 4.5 We now inend o ake he expecaion in he above relaion. o his end, we need he following momen esimaes. Using ha ρ is bounded, we have E sup ξ X π 1 2 = E sup ρ s dws H + ρ s θs H ρ s π 2 sds,, 18

20 CE sup ρ s dw H 2 s + E sup ρ s θs H ρ s π s ds,, C E ρ s 2 ds + E ρ s θ H 2 2 s ds + E ρ s π sds C 1 + E πs 2 ds <, 4.6 where we have used Doob s inequaliy. Consequenly, we ge E ξ X π 1 α E α 2 1/2 E ξ X π 1 2 1/2 <, which follows from he Cauchy-Schwarz inequaliy. Wih ρ being bounded, we ge for some generic consan C > E α s ρ s 2 ds CE α s 2 ds <. Hence α ρ dw H is a square inegrable maringale. Nex, le τ n n 1 be a localizing sequence for he local maringale ξ X π 1 dα. hen we have τn ξ X π 1 dα sup ξ X π 1 dα., o apply Lebesgue s dominaed convergence heorem and show ha E ξ X π 1 dα =, we need o prove E sup, ξ X π 1 dα < : E sup, ξ X π 1 dα CE CE <, ξ 2 X π 1 2 d α 1/2 1/2 sup ξ 2 X π 1 2 E α 1/2, where we have used he esimae 4.6. hus, by 4.5 i follows ha E ρ expy X π U X π Z H + θ H + U X π X π π d <, and from 4.4, i holds ha for every π in Π x such ha ρ is bounded, we ge E ρ expy X π U X π Z H + θ H + U X π X π π d. Subsiuing ρ wih ρ in he previous inequaliy, we obain for every ρ E ρ expy X π U X π Z H + θ H + U X π X π π d =

21 Now le A := U X π Z H + θ H + U X π X π π and le ρ ω := 1 Aω>. Recall ha we have dp d-a.s. expy X π >. Plugging ρ ino 4.7 yields Similarly choosing ρ ω := 1 Aω<, we find hus, we achieve A ω, dp d a.e. A ω =, dp d a.e. π i = U X π X π U X π Zi + θ, i,, i = 1,..., d 1. Le us now deal wih converse implicaion. heorem 4.2. Assume H3 and H5. Le X, Y, Z be an adaped soluion of he FBSDE X = x Y = log U X s U X s ZH s + θs H dws H U X s U X s ZH s + θ H s θ s ds, U X +H U X Zs H + θs H U 3 X su X s 2 Z s dw s U X s Z s 2 ds 4.8 such ha E UX π +H <, Z is an elemen of H2 R d and he posiive local maringale XU X expy is a rue maringale. π i := U X s X s U X s Zi s + θ i s, s,, i = 1,..., d 1 is an opimal soluion o he opimizaion problem 2.2. Proof. We firs noe ha π Π x since by he fac ha Z is in H 2 R d, here is a consan C > such ha E π 2 d C E Z H + θ H 2 d <. Now le π be an elemen of Π x. Le D := U X expy. Applying Iô s formula and plugging in he expression of π, we find ha hence, dd = D θ dw H + Z dw O, D = U x expy, D = U x expy E θ s dw H s + Z s dw O s,,, 4.9 which is a posiive local maringale. Now fix π in Π x. By definiion of X π and of D, he produc formula implies ha X π D saisfies DX π = xd Eπ θ W H + Z W O. 2

22 Hence, X π D is a supermaringale and so ED X π D x. By assumpion, X π D = XU X expy is a rue maringale so ED X π = D x. Finally combining he facs above, recalling ha D = U X π + H and using he concaviy of U, we obain EUX π + H UX π + H EU X π + HX π X π. 4.1 Remark 4.3. In he previous proof, if we apply inegraion by pars formula o D = U X expy and X π X π, we ge U X π expy X π X π = X π X π dd + D π X π π X π dw H, hus U X π expy X π X π is a local maringale for every admissible sraegy π. Remark 4.4. Noe ha using he regulariy assumpions of he FBSDE 4.8, we derived ha D := U X π expy is a rue maringale D = U x expy E θ W H + Z O W O. 4.2 Characerizaion and verificaion: complee markes We adop he seing and noaions of Secion 4 wih d 1 = d = 1 and H =. In he complee case we can give sufficien condiions for he exisence of a soluion o he sysem 4.8. o his end, noe he following remark. Remark 4.5. Similar o Remark 4.4, we can use 4.8 o characerize furher he maringale U X π expy : applying Iô s formula o U X π expy gives rise o hence, we have U X π expy = U x expy U X s expy s θ s dw s, U X π expy = U x expy E θ W,, his observaion allows o prove he exisence of 4.8 under a condiion on he risk aversion coefficien U U. Le ϕ 1 x := U x U x and ϕ 2x := 1 1 U 3 xu x 2. We will U x 2 now give sufficien condiion for he sysem 4.8 o exhibi a soluion. We begin wih he following remark. Remark 4.6. Noe ha if ϕ 2 is consan hen he sysem above decouples. An elemenary analysis shows ha his happens if and only is U is he exponenial, power, log or quadraic mean-variance hedging funcion. If Ux = exp α 1 x exp α 2 x hen ϕ 2 is bounded and Lipschiz and if Ux := xγ 1 γ 1 + xγ 2 γ 2 hen ϕ 2 is a bounded funcion. 21

23 heorem 4.7. Assume ha ϕ 2 is a coninuous bounded funcion. hen here exiss an adaped soluion X, Y, Z in S 2 R d 1 S 2 R H 2 R d o he FBSDE X = x Y = U X s U X Z s s + θ s dw s Z s dw s Moreover, E UX < and E U X 2 <. Proof. Fix m > and consider he BSDE = Y m U X s U X s Z s + θ s θ s ds Z s + θ s U 3 X su X s 2 U X s Z s 2 ds Zs m + θ s 2 ϕ 2 U 1 U x expme θ W exp Y m 1 2 Zm s 2 ds Z m s dw s. Since ϕ 2 is bounded, he driver of he BSDE above in Y m, Z m can be bounded uniformly in m, hence 12 yields a pair Y m, Z m S 2 R H 2 R soluion o his equaion wih Y m C where C does no depend on m and Z W is a BMO-maringale. In addiion once again using sandard argumens like in he proof of Proposiion 3.9 we have ha m Y m is coninuous. hus here exiss an elemen m > such ha Y m = m. Now applying Iô s formula o X m := U 1 U x expm E θ W exp Y m, we check ha X m, Y m, Z m saisfies I remains o show ha E UX <. From he concaviy of U we have ha E UX U E X + U + E U X X + U. Since X = xe U X XU X Z + θ W, U x xu x κ for x R and Z + θ W is a BMO-maringale, X is a rue maringale, and hus EX = x. Similarly we have ha X U X = X U X expy = xu x expy E U XU XZ + θ θ W and so XU X expy is a rue maringale. his hence proves E X U X <. 5 Links o oher approaches In his secion we link our approach o characerizing opimal invesmen sraegies o wo oher approaches based on he sochasic maximum principle and dualiy heory, respecively. 5.1 Sochasic maximum principle his secion links our approach in he complee marke seing o he approach using he sochasic maximum principle. As we are ineresed only in he link, we will only give a formal derivaion. In paricular, we suppose here ha U and U 1 are smooh enough 22

24 wih bounded derivaives. Le us consider he complee marke case wih d 1 = d = 1 for simpliciy and H = and recall ha in his seing, he wealh process is given by X π = x + π s dw s + π s θ s ds,,. We consider Jπ := EUX π and se X π := UX π. Iô s formula yields d X π = U U 1 X π π dw + U U 1 X π π θ U U 1 X π π 2 d and Jπ = E X π. Applying he maximum principle echnique described in 3 see also 22, Secion 4, we inroduce he adjoin equaion o ge d X π = U U 1 X π π dw + U U 1 X π π θ U U 1 X π π 2 d, Xπ = Ux, dp = U U U 1 X π θ π + 1 U 3 2 U U 1 X π π 2 U p + k U U 1 X π π d + k dw, p = We now inroduce he corresponding Hamilonian, defined as H, p, k, π := pu U 1 X π π θ U U 1 X π π 2 + ku U 1 X π π. A formal maximizaion gives π := U U U 1 X π k + θ. p Plugging his ino 5.1 yields d X π = U 2 U U 1 X π k p + θ dw 1 k 2 p θ d 2 dp = k p + θ p U 3 U 2 U 1 X π U 2, Xπ = Ux, d + k dw, p = We now relae his sysem wih 4.12 using a Cole-Hopf ype ransformaion. Firs we plug π ino 5.2 and obain dx π = U U X π k p + θ dw + θd, X π = x, 2 dp = k p + θ p U 3U 2 X U 2 π d + k dw, p = Nex consider he sysem dx π = U U X π dy = Z + θ Z + θ dw + θd, X π = x, U 3 X π U X π U 2 X π 1 2 Z 2 d + Z dw, Y =. 5.4 Seing p := expy, k := Z p and X := X, Iô s formula implies ha p, k is a soluion o

25 5.2 BSDE soluion via convex dualiy mehods Le us now urn o a very imporan link of our approach wih he convex dualiy heory. We have seen in Secions 3 and 4 ha our approach relies on choosing a process Y such ha he quaniies U X π + Y and X π U X π expy, respecively, are maringales. In fac, hese maringales are no any maringales. For insance in case of a uiliy funcion on he whole real line, U X π + Y is exacly U x + Y E θ W H + U U X π + Y Z O W O. So in he complee case i is exacly he maringale under which he price is iself a maringale. For uiliy funcions defined on he posiive half line his leads direcly o dualiy heory, since i is known from he original paper by Kramkov and Schachermayer 13 ha under some growh-ype condiion on U he opimal wealh process X π and he sochasic process Y soluion o he so-called dual-problem are such ha he sochasic process X π Y is a maringale. In addiion, wih our noaions, Kramkov and Schachermayer prove ha Y has he form Y = Y E θ W H + M where M is a maringale orhogonal o W H. Recall ha in our case X π U X π expy is a maringale and from 4.9, we have proved ha D := U X π expy is exacly of he form D E θ W H + Z O W O, in oher words Y = D and he Z O componen appearing in he soluion of our FBSDE exacly represens he orhogonal par in he dual opimizer of Kramkov and Schachermayer heory. Obviously, his needs o be derived more formally. his is he goal of his secion. he aim of his secion is o derive a soluion of he forward-backward equaion 4.12 by means of he resuls from he convex dualiy approach o 2.2. We denoe by Π 1 he se of admissible sraegies wih iniial capial one uni of currency. In he case of zero endowmen H =, he soluion o he concave opimizaion problem 2.2 is achieved by formulaing and solving he following dual problem: denoing he convex conjugae of he concave funcion U by { } V y := sup Ux xy, y >, x> where dx π via = X π π d S S, X π = x >, and defining a family of nonnegaive semimaringales Y := { Y : Y = 1, X π Y is a supermaringale for every π Π 1}, he primal problem 2.2 is solved by solving insead he dual convex opimizaion problem vy = inf Y Y E V yy, y >. 5.5 If his dual problem admis a unique soluion Y H = also yields a unique soluion Y, hen he primal problem 2.2 wih X π = x + = x + = IyY, 24 Xs π πs d S s S s α sds s

26 wih he corresponding opimal conrol π α S = X. Here we have I = U 1 and x = π v y 2. he case of bounded erminal endowmen H is deal wih in 5, where insead of 5.5 he following dual problem is considered vy = inf Y Y E V yy + yy H, y >. he case of general inegrable H has been sudied in 1, using he original dual problem 5.5 bu a sligh differen choice of he domain Y. A ubiquious propery of he convex dualiy mehod is ha once he primal and he dual opimizers are obained, heir produc X π Y is a nonnegaive rue maringale hence uniformly inegrable, see 13 for a economic inerpreaion. In he conex of uiliy maximizaion wih bounded random endowmens, his maringale propery of X π Y is poined ou in 5, Remark 4.6. his maringale propery of X π Y consiues he firs main ingredien for deriving a soluion for he forward-backward equaion A second main ingredien is consiued by he characerizaion of he dual domain Y. Noe in he coninuous process seing, Y is he family of all non-negaive supermaringales see e.g. 13, 1. According o a well known resul, every nonnegaive càdlàg supermaringale Y Y admis a unique muliplicaive decomposiion Y = AM where A is a predicable, non-increasing process such ha A = 1 and M is càdlàg local maringale. However, 15 characerize he elemens of Y Y by he muliplicaive decomposiion Y = AE θ H W H + K W O, 5.6 where A is a predicable non-increasing process such ha A = 1 and K H 2 loc Rd 2 see 15, Proposiion 3.2. Using ha he Fenchel-Legendre ransform V is sricly decreasing, 15, Corollary 3.3 shows ha he dual opimizer is a coninuous local maringale and admis he represenaion Y = E θ H W H + K W O 5.7 for a uniquely deermined K H 2 loc Rd 2. If vy = E V yy <, hen we can check ha he opimal K acually belongs o H 2 R d 2. his is done in he following lemma whose proof is in he same spiri as in 14, Lemma 3.2 Lemma 5.1. If for some y >, i holds ha vy = inf E V ye θ H W H + ν W O <, ν H 2 loc Rd 2 we have vy = inf E V ye θ H W H + ν W O, ν H 2 R d 2 i.e. he opimal K minimizing vy can be assumed o belong o H 2 R d 2. 2 his is equivalen o u x = y where ux = sup π E UX π + H. he differeniabiliy of boh vy and ux are shown in 5. 25

27 Proof. We inroduce he family of sopping imes τ n := inf { > : θ H s 2 + K s 2 ds n }, n N. Le y >, hen we have vy = E V ye θh W H + K W O = E E V ye θh W H + K W O F τ n E V ye τ n θh W H + K W O, where he las line follows by Jensen s inequaliy. Coninuing he las line and recalling ha V y is a sricly convex funcion, we have vy E V y exp τ n θ H s dws H + Ks dws O 1 τ n exp θ H 2 s 2 + Ks 2 ds V E y exp τ n θ H s dws H + Ks dws O 1 τ n exp θ H 2 s 2 + Ks 2 ds V y exp E 1 τ n θ H 2 s 2 + Ks 2 ds, where Jensen s inequaliy has been used wice. By coninuiy of V and of he exponenial funcion, i follows from he monoone convergence heorem ha vy lim V exp 1 τ n n 2 E θ H s 2 + Ks 2 ds = V exp 1 2 E θ H s 2 + Ks 2 ds. Since vy < and V exp = V = U =, i follows ha We deduce ha K H 2 R d 2. E θ H s 2 + Ks 2 ds <. Now using ha X π Y is a rue maringale and ha he dual opimizer Y is a local maringale saisfying 5.7, we ge he following resul. heorem 5.2. Le H be a non-negaive bounded random endowmen and assume ha he coefficien of relaive risk aversion xu x saisfies U x xu x lim sup <. 5.8 x U x hen here exiss x > such ha for all x > x he coupled FBSDE 4.8 has a soluion X, Y, Z such ha X = x. In addiion, X is he opimal wealh of he problem 2.2 and he dual opimizer Y associaed wih i is given by Y = U X expy so ha yy = U X + H. 26

28 Proof. he exisence of x > such ha for every x > x he quaniy ux = sup E U X π + H = E U X π + H π Π x is finie has been shown 5. We se X := X π. Also recall ha we have y = u x > for x > x and ha we have E yx Y = xy. Moreover, yy = U X + H. We define he rue maringale α := yx Y. We se Y := logα logx logu X. We have ha α Y = log X U X yy = log U X = logy + logy logu X. Recall ha by definiion of X and Y we have ha dy = Y θ H dw H + K dw O and Hence dx = X π dw H + π θ H d. We define: dy = θ H dw H + K dw O 1 2 θh 2 + K 2 d U X U X π X dw H so ha π X = Z H + θ H U X U X, and hen + π X θ H d 1 U 3 X U X U X 2 2 U X π 2 X 2 d. Z H := θ H U X U X π X, Z O := K. dy = Z H dw H + Z O dw O 1 2 θh 2 + K 2 d + θ H Z H + θ H 1 U 3 X U X U X 2 Z H + θ H 2 U X 2 2 U X 2 U X d 2 27

29 = Z H dw H + Z O dw O + Z H + θ H 2 Finally noe ha by consrucion Y = log soluion o 4.8 and U 3 X U X U X 12 2 ZH 2 d. U X +H U. Hence, X, Y, Z = X, Y, Z is a yy = U X expy. Le us recall ha he absolue risk aversion of Ux is defined as ARAx := U x U x. We say ha Ux has hyperbolic absolue risk aversion and he risk olerance as 1 ARAx 1 HARA if and only if is risk olerance ARAx shown ha a uiliy funcion Ux is HARA if and only if for given real numbers γ, a, b R. Ux = 1 γ ax γ, γ 1 γ + b ax 1 γ + b >, is linear in x. More precisely, i can be Corollary 5.3. Assume ha Ux is HARA. hen here exiss a consan κ R such ha he backward equaion from 4.8 can be wrien as U X Y = log + H U X = log U X + H U X Proof. Noice ha for he risk olerance i holds ha fx := Z s dw s Z s dw s 1 2 Z s 2 + κ Z H s + θ H s 2 ds 5.9 gs, Z s ds. 1 ARAx = U x U x f x = 1 + U xu 3 x U x 2. Since Ux being HARA implies ha f is linear in x, i follows ha here exis consans c, d R such ha f x = cx + d. Hence he BSDE from 4.8 can also be wrien as U X Y = log + H U X U X = log + H U X Z s dw s Z s dw s 1 2 Z s Z s 2 + κ Z H s f X s Z H s + θ H s 2 ds + θ H s 2 ds, for κ = c. 28

30 Obviously he driver of he BSDE 5.9, gs, z, saisfies he quadraic growh condiion gs, z α + γ 2 z 2 for suiably chosen real numbers α, γ >. In his seing 4, heorem 2 yields he following resul. U X Corollary 5.4. If ξ = log +H saisfies E e γ ξ <, hen he BSDE 5.9 admis U X a soluion Y, Z such ha Y is coninuous and Z H 2 loc Rd. 5.3 he power case wih general endowmen We finally deal wih an open quesion in mahemaical Finance namely he case of power uiliy wih general endowmen. We know from dualiy heory ha an opimal soluion exiss bu we would like o prove ha he sraegy is smooh i.e. square inegrable and o characerize i in erms of he soluion o an equaion for insance a FBSDE. We will use definiions and noaions of Secion 4. Le Ux := xγ γ wih γ a fixed parameer in, 1. Le H be a posiive bounded F -measurable random variable where we recall ha F, is he filraion generaed by W = W H, W O. We recall ha we denoe by Π x he se of admissible sraegies wih iniial capial x which is now defined by { } Π x := π : Ω, R d 1, π is predicable, E π s 2 ds < 5.1 where π i, i = 1,..., d 1 denoes he proporion of wealh invesed in he sock. he associaed wealh process is given by X π := x + π s X π s ds H s,,. Again, we exend π o R d via π := π 1,..., π d 1,,..., and make he convenion ha we wrie π insead of π. hus, we have X π = xe π r dsr H,,. Noe ha his seing covers he case of a purely orhogonal endowmen of he form H := φs O where φ is posiive. Now we can go in he analysis of he problem: X π sup E + H γ π Π x γ Indeed, wha is only known in ha case is ha an opimal sraegy exiss 1 bu in a much larger space ha Π x, in paricular i is no proved ha he opimal sraegy is square inegrable. Abou he characerizaion of his opimal sraegy one can wrie he Hamilon- Jacobi-Bellman PDE in he Markovian case bu no resuls allow us o solve i. We believe ha combining he dualiy heory, BSDEs echniques and our approach we could show firs ha he opimal sraegy belongs o he space Π x and ha we will give a characerizaion of i in erms of a FBSDE. Le us be more precise. 29

31 heorem 5.5. here exiss x > such ha for every x > x, he sysem X = x + X szs H+θH s 1 γ dws H + θh s X szs H +θs H 1 γ ds Y = γ 1 log 1 + H X Z s dw s γ 2γ 1 ZH s + θs H Z s 2 ds 5.12 admis an adaped soluion X, Y, Z. If in addiion Z H = Z 1,..., Z d 1 is in H 2 R d 1, hen π i := 1 1 γ Zi + θ i, i = 1,..., d is he opimal soluion o he maximizaion problem Proof. Firs noe ha he sysem 5.12 is exacly he sysem 4.8 wih Ux = xγ γ. Hence from heorem 5.2 here exiss x > such ha he sysem 5.12 admis a soluion X, Y, Z when x > x. We fix, x > x and consider he associaed soluion X, Y, Z ha is X = x. In addiion, we know from heorem 5.2 ha X = X. Hence π is given by I jus remains o prove ha π is in Π x, which is a direc consequence of he fac ha Z is in H 2 R d. Remark 5.6. Noe ha since we know ha he dual opimizer Y is given by Y = U X expy i is clear ha XU X expy is a rue maringale. Hence he square inegrabiliy of Z implies he condiion of heorem 4.2: EX + H γ <. Finally noice ha Z O is in H 2 R d 2 by Lemma 5.1. So he only elemen missing in he proof is indeed o show ha Z H is in H 2 R d 1 naurally, since he process π is inegrable wih respec o S H and so i is in H 2 R d 1. his quesion requires a deeper analysis of he sysem and is currenly invesigaed by he auhors. Acknowledgmens Hors acknowledges financial suppor hrough he SFB 649 Economic Risk. Imkeller and Réveillac are graeful o he DFG Research Cener MAHEON, Projec E2. Hu is parially suppored by he Marie Curie IN Projec Deerminisic and Sochasic Conrolled Sysems and Applicaions, call: F97-PEOPLE IN, n Zhang acknowledges suppor by DFG IRG 1339 SMCP. References 1 S. Biagini. Expeced uiliy maximizaion: he dual approach. Encyclopedia of Quaniaive Finance, J.-M. Bismu. héorie probabilise du conrôle des diffusions. Mem. Am. Mah. Soc., 167, J.-M. Bismu. An inroducory approach o dualiy in opimal sochasic conrol. SIAM Rev., 21:62 78,

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