Forward-backward systems for expected utility maximization

Transcription

1 Forward-backward sysems for expeced uiliy maximizaion Ulrich ors, Ying u, Peer Imkeller, Anhony Réveillac and Jianing Zhang December 23, 213 Absrac In his paper we deal wih he uiliy maximizaion problem wih general uiliy funcions including power uiliy wih liabiliy. We derive a new approach in which we reduce he resuling conrol problem o he sudy of a sysem of fully-coupled Forward- Backward Sochasic Differenial Equaion FBSDE ha promise o be accessible o numerical reamen. AMS Subjec Classificaion: Primary 61, 93E2 JEL Classificaion: C61, D52, D53 1 Inroducion One of he mos commonly sudied problems in mahemaical finance and applied probabiliy is relaed o maximizing expeced erminal uiliy from rading in a financial marke. In is version we consider here, he focus is more on an insurance issue: a small agen is ineresed in securiizing a random liabiliy arising in his usual business by invesing on a capial marke. e herefore has wo sources of income: his random liabiliy, and he wealh obained from rading on he capial marke up o a erminal ime wih appropriae invesmen sraegies. he agen s preferences are described by a uiliy funcion. So he sochasic conrol problem he faces resuls in he maximizaion of his erminal uiliy obained from boh sources of income wih respec o all admissible sraegies available o him. More formally, given his iniial wealh x >, he aims a aaining he value funcion V, x := sup EUX π π A Insiu für Mahemaik, umbold-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, umbold-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 Zenrum Mahemaik, U München, Bolzmannsr. 3, Garching, Germany, j.zhang@um.de 1

2 ere U is a general real-valued uiliy funcion wih properies o be specified, A denoes he se of admissible rading sraegies, < he rading horizon, X π is he agen s wealh obained upon following a sraegy π A, and he random variable describes a liabiliy he mus deliver a erminal ime. In he ypical focus of mahemaical ineres of his problem we find quesions relaed o exisence and uniqueness of opimal soluions, as well as he characerizaion of opimal sraegies and he value funcion V which is defined for imes and iniial wealh x > as V, x := sup EUX, π + F. π A ere X, denoes he wealh he agen is able o obain from rading in he capial marke in he invesmen period,, and he filraion F, describes he evoluion of informaion. he mos imporan echniques o ackle he exisence of opimal sraegies π are based on conceps of convex dualiy. his ool firs appears in Bismu 4, was furher developed on Brownian bases in he works by Pliska 35, Karazas and co-workers see for insance Karazas e al. 18, 19, 7, wih is modern and absrac form formulaed in he seing of general semimaringales due o Kramkov and Schachermayer 21. In his seing, growh condiions on U or relaed quaniies such as he asympoic elasiciy, if U is defined on he half line, can be formulaed. ogeher wih mild regulariy condiions on he liabiliy and convexiy assumpions on he se of admissible rading sraegies see e.g. 2 for deails hey guaranee he exisence of opimal invesmen sraegies. Dualiy echniques are general and far-reaching, ye no consrucive, especially from he perspecive of numerical approximaion: hey are so far no amenable o compuaion or simulaion of opimal sraegies and value funcions. hey reveal ye anoher shorcoming arising if one is ineresed in working wih non-convex consrains: convexiy breaks down for insance if he sraegies are supposed o ake ineger values. A direc sochasic approach o simulaneously characerize opimal rading sraegies and uiliies is provided by an inerpreaion of he maringale opimaliy principle by he ools of forward backward sochasic differenial equaions FBSDE. In he case of exponenial uiliy i was discussed in El Karoui e al. 12, and Sekine 37, sill wih elemens of convex analysis. Concepually no linked o convex dualiy mehods, in 14 i was seen o work in he seing of consrains ha are jus closed, no necessarily convex. If he filraion is generaed by a sandard Wiener process W, if eiher Ux := exp αx for some α > and L 2, or Ux := xγ γ for γ, 1 or Ux = ln x and =, and if he selecion of admissible sraegies is resriced o a closed se, i has been shown by u e al. 14 ha he conrol problem 1.1 can essenially be reduced o solving a BSDE of he form Y = 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. he mehod only works well in he cases of classical uiliy funcions, i.e. exponenial wih general endowmen, and power or logarihmic wih zero endowmen. In hese cases he forward par, he porfolio process, and he backward par given by 1.2, are decoupled. his is 2

3 due o a separaion of variables propery shared by he classical uiliy funcions: heir value funcion can be decomposed by 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. he sochasic approach has since been exended beyond he Brownian framework and o more general uiliy opimizaion problems wih complee and incomplee informaion in numerous papers, of which we only quoe 13, 29, 3, 31 and 27. 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 28 a verificaion heorem is derived for opimal rading sraegies for more general uiliy funcions in case =. More precisely, given a general uiliy funcion U and assuming ha here exiss an opimal sraegy regular enough such ha he value funcion possesses some regulariy in, x, i is shown ha here exiss a predicable random field ϕ, x,x,, such ha he pair V, ϕ solves a backward 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 heory of BSPDE is sill no well developed, 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 17. Moreover, he uiliy funcion U only appears in he erminal condiion which is no very handy. his corresponds o a general sochasic version of he amilon- Jacobi-Bellman equaion in he Markovian seing. In his paper we propose a new approach o solving he securiizaion problem wih liabiliy 1.1 for a larger class of uiliy funcions and characerize he opimal sraegy π in erms of a fully-coupled sysem of FBSDE. In conras o he classical Markovian framework corresponding o he analyical JB equaions for which hey are supposed o be funcions of he erminal value of he forward process, he general erminal liabiliies we work wih usually creae a sysem which is non-markovian. I is herefore essenially more general han he sysem of FBSDE oulined in Peng 34 in he conex of classical sochasic conrol problems. Coupled FBSDE have already been exensively sudied, bu essenially for Lipschiz coefficiens. he reamen has focused on mainly hree mehods: one using conracion mappings 1, 33, one based on PDE 24, 9, and he mehod of coninuaion 15, 38. We refer o 25 for he invesigaion of FBSDE wih Lipschiz coeffiens. 1 I has been shown by 1 ha essenially all such uiliy funcions can be represened in erms of a BSDE of he form

4 he derivaion of he FBSDE sysem appropriae for our purposes sars wih a verificaion ype observaion. In case of uiliy funcions defined on R if hey are defined on R +, a refinemen of he argumen will be applicable, given an opimal sraegy π of he forward porfolio process X π, o realize maringale opimaliy we posulae ha U X π + Y be a maringale, where Y, Z is he associaed backward process. As a consequence, Y, Z is given by a cerainy equivalen ype expression for marginal uiliy Y = U 1 EU X π + F X π. his idenificaion allows us o compue he driver of he BSDE relaed o Y, Z. I is given in erms of he derivaives of U, involves he opimal forward process X π, and provides he backward par of he FBSDE sysem. In a second sep, we consider possible soluion riples X, Y, Z of he FBSDE sysem obained in he firs sep, no assuming ha X corresponds o an opimal porfolio process. We hen use he variaional maximum principle in order o verify ha under mild condiions on U he riple X, Y, Z solves he original opimizaion problem. his in paricular means ha X coincides wih an opimal forward porfolio process X π. In summary, under mild regulariy condiions, soluions X, Y, Z of he FBSDE sysem provide soluions of he original securiizaion problem. Wih his we also exend he fully sochasic approach of 14 of he opimizaion problem wih erminal liabiliy by means of a direc ranslaion of maringale opimaliy ino sochasic equaions. Since i is no based on convex dualiy echniques, i should be able o cope wih closed, non-convex consrains as well. Alhough i appears ha we can wrie down sysems of FBSDE wih closed, non-convex consrains, we were no able o relae hem o he corresponding uiliy opimizaion problem ye. owever, i requires essenially more effor o solve a fully-coupled FBSDE sysem which in general will fail o possess soluions compared o he decoupled one of 14. In classical cases in which decoupling echniques apply our FBSDE sysem possesses soluions. In a more general seing, soluions are consruced using a compacness crierium due o Delbaen and Schachermayer 11 for which so far we canno avoid convexiy assumpions. Our approach provides in paricular an FBSDE sysem for he case of power uiliy wih general non-hedgeable liabiliies. o he bes of our knowledge ours is he firs reamen ha allows o characerize and calculae opimal sraegies in his case. 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 ranslaed ino a sysem of FBSDE for uiliies defined on he real line. Is converse shows ha a soluion o he FBSDE sysem obained allows o consruc he opimal sraegy. Secion 4 is devoed o he discussion of analogous quesions for uiliies defined on he posiive half line. In Secion 5 we relae our approach o he sochasic maximum principle obained by Peng 34 and he sandard dualiy approach. We use he dualiy-bsde link o consruc a soluion of he FBSDE associaed wih he problem of maximizing power uiliy wih general posiive endowmen. 4

5 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. Le us remark a his place ha a generalizaion of his model o one wih a volailiy marix σ for which σσ is uniformly ellipic as in u e al. 14 is sraighforward, and jus adds noaional complexiy o he reamen. Since we assume he process θ o be bounded, Girsanov s heorem implies ha he se of equivalen local maringale measures i.e. probabiliy measures under which S is a local maringale is no empy, and hus according o he classical lieraure see e.g. 11, arbirage opporuniies are excluded in our model. For simpliciy we wrie hroughou 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 := 2 R d := β : Ω, R d, { β : Ω, R d, coninuous and adaped, 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 2 θ s 2 ds 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 d 1+1,..., S d 2 are inaccessible o he agen. Denoe S := S 1,..., S d 1,...,, W := W 1,..., W d 1,...,, W O :=,...,, W d 1+1,..., W d 2, and θ := θ 1,..., θ d 1,..., he noaion refers o hedgeable and O o orhogonal. o any invesmen sraegy π in 2 R d 1, we define X π is associaed wealh process defined as: X π := X π + d 1 π r dsr = X π + i=1 π i rds i r,,, 5

6 and for every x > we denoe by Π x he se of rading sraegies wih iniial capial x, ha is: { } Π x := π 2 R d 1, X π = x. 2.1 Every π in Π x is exended o an R d -valued process by π := π 1,..., π d 1,,...,. 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 look for a sraegy π in Π x saisfying EUX π + < such ha π = argmax π Π x, E UX π + < {EUX π + } 2.2 where 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. 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. We inroduce he following condiions. 1 U : R R is hree imes differeniable 2 is an elemen of L 2 Ω, F, P 3 We say ha condiion 3 holds for an elemen π in Π x, if E U X π + 2 < and if for every bounded predicable process h :, Ω R, he family of random variables 1 h r dsr U X π + + εr h u dsu dr ε,1 is uniformly inegrable. Before presening he firs main resul of his secion, we prove ha condiion 3 is saisfied for every sraegy π such ha E U X π + < 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 rds r and d >. We will show ha he quaniy qd := sup E G ε,1 1 U X π + + εrgdr 1 G 1 U X π ++εrgdr >d 6

7 vanishes as d ends o infiniy. For simpliciy we wrie δ ε,d := 1 1 G U X π he Cauchy-Schwarz inequaliy qd sup E 1 + U 1 X π + G1 + expαεrgdr δ ε,d ε,1 CE U X π + 2 1/ /2 sup E G expαεrgdr δ ε,d. ε,1 Since E U X π + 2 is assumed o be finie we deduce from he inequaliy expαζx 1 + expαx for all x R, < ζ < 1 ++εrgdr >d. By ha 1/2 qd C sup E G2 + expαg 2 δ ε,d. ε,1 Applying successively he Cauchy-Schwarz inequaliy and he Markov inequaliy, we deduce qd CE G2 + expαg 4 1/4 sup Eδ ε,d 1/4 ε,1 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 r exp 2pαh r 2 + 2pαh r θ r dr 2 ence lim d qd = which proves he asserion. 1/2 2pαh r 2 dr 3.1 Characerizaion and verificaion: incomplee markes 1/4 + + εrgdr We are now ready o sae and prove he firs main resuls of his paper namely a characerizaion of he opimal sraegy cf. heorem 3.1 and a verificaion heorem presened in heorem

8 heorem 3.1. Assume ha 1 2 hold. Le π Π x be an opimal soluion o he problem 2.2 such ha 3 is saisfied. hen here exiss a coninuous adaped process Y wih Y = such ha U X π + Y is a square inegrable maringale and he opimal sraegy allows for he represenaion 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 + 2 <, he sochasic process α defined as α := EU X π + 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 where β is a square inegrable predicable process wih respec o d dp wih values in R d. Plugging 3.2 ino 3.1 yields Y + X π =X π + 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 + 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 = 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, which hus is a maringale. Now we deal wih he characerizaion of he opimal sraegy. o his end, le h :, Ω R d 1 be a bounded predicable process. We exend h ino R d by concaenaing zeros via h := h 1,..., h d 1,,..., and by abuse of noaion denoe h again by h. hus for every 8

9 ε in, 1 he perurbed sraegy π + εh belongs o Π x. Since π is opimal i is clear ha for every such h we have 1 lh := lim ε ε E Ux + πr + εh r dsr + Y Ux + πrds r + Y. 3.5 Moreover 1 Ux + ε = h r ds r πr + εh r dsr + Y Ux + 1 U X π + Y + θε h r dsr πrds r + Y dθ. Now by 3, Lebesgue s dominaed convergence heorem implies ha 3.5 can be rewrien as E U X π + Y h r dsr 3.6 for every bounded predicable process h. Applying inegraion by pars o U Xs π +Y s s, and s h rdsr, we ge s, U X π + Y h r dsr = U x + Y s s h r ds r U X π s h r ds r U X π s U X π s + Y s U 3 X π s + Y s h s ds s + Y s h s π s + Z s ds. π s + Z s dw 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 r U Xs π + Y s θ s + U X π h r ds r, Y s, Z s ds s + Y s πs + Z s h s ds U Xs π + Y s πs + Z s dws + U Xs π + Y s h s dws. 3.7 he nex sep would be o apply he condiional expecaions in 3.7. owever 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, 9

10 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 U Xr π + Y r πr + Z r dw r s, CE s h r dsr 2 1/2 β s 2 ds s CE sup h r dsr 2 1/2 1/2 β s 2 ds. s, Young s inequaliy furhermore yields s E sup h r dsr 2 1/2 CE C s, s, sup 1 + E s sup s, h r ds r s 2 + CE h r dw r 1/2 β s 2 ds 2, β s 2 ds where we have used ha h and θ are bounded. Applying once again he BDG inequaliy, we obain s E sup h r dwr 2 4E h r 2 dr <. s, s, Puing ogeher he previous seps, we have ha s r E sup h u dsu U Xr π + Y r πr + Z r dwr <, and hus we ge Noe ha E s h r ds r U Xs π + Y s h s dws U X π s, + Y s π s + Z s dw s U X π + Y = α is a square inegrable maringale and hus E U 2 Xs π + Y s h s ds <. Similarly, E U X π + Y h r dsr =. is a square inegrable maringale. <. Indeed 1

11 aking expecaion in 3.7 we obain for every n 1 ha E U X π + Y h r dsr = 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 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 θs i + 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 E 1 {A i s >}U Xs π + Y s θs i + U Xs π + Y s πs i + Zsds i =. ence, A i, dp d a.e.. Similarly by choosing h s =,...,, 1 {A i s <},,..., we deduce ha U X π + Y θ i + U X π his concludes he proof since i {1,..., d 1 } is arbirary. + Y π i + Z i =, dp d a.e. he characerizaion of he opimal sraegy above can also be given in erms of a fullycoupled 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 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 U θ X s+y s s U X + Z s+y s s θs ds Y = 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 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 =:Z O X π is equal o X. 11

12 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 of he BSDE 3.3 wih driver f as in 3.4. Now plugging he expression of π ino equaion 3.4 yields X π = x U θ Xs π +Ys s dws U θ Xs π +Ys s θs ds Y = U X π s Z s dw s +Ys + Z s 1 2 θ s 2 U 3 Xs π U X π s +Ys + Z s +Y s U X π s +Y s 2 U X π s +Y s 3 + θs 2 U Xs π +Y s U Xs π +Y + Z s θ s s 1 2 ZO s 2 U 3 U X s π + Y s ds Recalling ha X π := x + π sdws + θs 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 U X + Y = U x + Y + θ s U X s + Y s dw s + U X s + Y s Z O s dw O s. 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 s α s π s π sdw s + U Xs π + Y s Zs + πs π s πsds X π s X π s dα s + α s π s π sdw s. his shows 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 1 and 2 be saisfied. Le X, Y, Z be a riple of sochasic processes which solves he FBSDE 3.1 saisfying: Z is in 2 R d, E UX + <, E U X + 2 <, and U X + Y is a posiive maringale. Moreover, assume ha here exiss a consan κ > such ha for all x R. hen U x U x κ π i := U X + Y U X + Y θi Z i,,, i {1,..., d 1 }, is an opimal soluion of he opimizaion problem

13 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 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 s dw s + dq dp := U X + EU X +. U X s + Y s U X s + Y s ZO s dw O s. Girsanov s heorem implies ha W := W + W O = W 1 + θ1 d,..., W d 1 + θd 1 d, W d1+1 U X +Y U X +Y Zd 1+1 d,..., W d 2 U X +Y U X +Y Zd 2 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 + <. 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 + UX π aking expecaions in 3.12 we ge EUX π + UXπ + EU X + + U X π E Q X π X π = E Q lim = lim n E Q n τn + X π X π τn π s π sd W s π s π sd W 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 <., Indeed he BDG inequaliy and he Cauchy-Schwarz inequaliy imply ha E Q sup π s πsd W s, 1 CE Q π s πs 2 2 ds = CE U X + EU X + π s πs 2 ds

14 U X + CE EU X 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 ha he opimizaion procedure singles ou a pricing measure under which he asse prices and marginal uiliies are maringales. In his 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 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 2 R 2 R d. In addiion, Z W is a BMO-maringale. 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 3, heorem 2.5 and Lemma 3.1 we have only from he backward par of he FBSDE ha Z is in 2 R d and ha Z W is a BMO-maringale. In addiion here exiss a unique bounded soluion o he backward componen 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 process X. Assume ha here exiss anoher soluion X, Y, Z of 3.1. ence, heorem 3.5 implies ha π := U X +Y i {1,..., d 1 } is an opimal soluion o our original problem U X +Y θi + Z i, 2.2 and X is he opimal wealh process. owever, 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. 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. 14

15 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. ence, 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 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. 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 = 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 2 R d such ha E UX + < and E U X + 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 36, V.3. Lemma 1. Nex, consider he BSDE = Zs m dw s 12 θ s 2 ϕ 2 P ms + θ s 2 ϕ 1 P ms + Z ms θ s ds

16 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, we see ha 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 coninuous in z, here exiss a unique pair of adaped processes Y m, Z m in S 2 R 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 sandard argumens we can show ha he map m Y m is coninuous. Even if his procedure is sraighforward, we reprove his fac here o make he paper selfconained. 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 26, Lemma 2.2 we ge δy 2 E sup δy 2 CE hs 2 ds., 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 36, 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 end o m. his in 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, we see Ux U x + U, Ux U xx U, x R. Consequenly, we have E UX + E U X + + U + E U X + X + + U <, which concludes he proof. 16

17 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. owever 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 +. If =, i urns ou ha a good choice for Φ is given by Φx, y := U x expy, since he sysem we obain coincides up o a non-linear ransformaion wih he one obained by Peng in 34, 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 5.1. 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 2 R d 1, X π = x,. 4.1 where X π sands for he associaed wealh process given by X π := X π + π s X π s ds 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,,. We need o impose he following assumpions. 3 U : R + R is hree imes differeniable, sricly increasing and concave 4 is a posiive F -measurable random variable in L 2 Ω, F, P 5 We say ha assumpion 5 holds for an elemen π in Π x, if i E X π U X π + 2 < ; 17

18 ii he sequence of random variables 1 +ερ ε Xπ X π is uniformly inegrable; 1 U X π + + 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 + ρ X π ds,,, and sup, E ξ 2 <. 6 here exiss a consan c > such ha U x xu x c for all x R Characerizaion and verificaion: incomplee markes Noe ha in condiion 5, if U < or if a > is saisfied, hen iii implies ii. heorem 4.1. Assume ha 3 and 4 hold. Le π be an opimal soluion o 2.2 saisfying E UX π + < and assumpion 5. hen here exiss a predicable process Y wih Y = logu X π + 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 + logu X π. Consequenly, U X π + = U X π expy. By 5, he process α := EX π U X π + F is a square inegrable maringale. In addiion i is he unique soluion o he BSDE α = X π U X π + 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 βs Y = Y U X π α s U X π s s Xπ s π s π s dw s 18

19 Seing we ge Z i = βi α 1 β s 2 U 2 α s 2 Xs π U Xs π Xπ s πs + πs θs U Xs π 2 U Xs π U 3 Xs π U Xs π + Xπ s πs 2 2 π U X π Xπ Y = Y Z s dw s 1 2 U X π U X π s s 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 + θs 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 34 and hen subsiue he amilonian by a BSDE. Fix π Π x. Since he laer is a convex se, for ρ := π π, π + ερ remains an admissible sraegy for every ε, 1. We have 1 +ερ ε UXπ + UX π + = Since π is opimal we find 1 +ερ E ε Xπ X π Now le ξ be defined by 1 +ερ ε Xπ X π 1 U X π 1 U X π + + rx π +ερ + + rx π +ερ dξ = π ξ + ρ X π ds,,. X π dr. X π dr, ε >. 4.3 By 5, 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 + = lim E ε ε Xπ X π Combined wih 4.3, his leads o 1 U X π + + rx π +ερ X π dr. Eξ X π 1 U X π X π expy = Eξ U X π +, π Π x. 4.4 We now resric our aenion o a paricular class of processes π. We choose ρ o be a bounded predicable process and define π := ρ + π which is an admissible sraegy since 19

20 i is square inegrable. he inegraion by pars formula for coninuous semimaringales implies ha ξ X π 1 = ρ s dw s + ρ s θ 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 U X π Z + θ + 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 obain E sup ξ X π 1 2 = E sup ρ s dws + ρ s θs ρ s π 2 sds,, CE sup ρ s dw 2 s + E sup ρ s θs ρ s π 2 s ds,, C E ρ s 2 ds + E ρ s θ 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 <. ence α ρ dw 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, we need o prove E sup, ξ X π 1 dα <. In fac, E sup, ξ X π 1 dα CE 2 ξ 2 X π 1 2 d α 1/2 ξ X π 1 dα =

21 CE <, 1/2 sup ξ 2 X π 1 2 E α 1/2, where we have used he esimae 4.6. hus, 4.5 enails E ρ expy X π U X π Z + θ + 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 + θ + 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 + θ + U X π X π π d =. 4.7 Now le A := U X π Z + θ + 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 A ω, dp d a.e. Similarly choosing ρ ω := 1 {Aω<}, we find hus, we achieve π i A ω =, dp d a.e. = U X π X π U X π Zi + θ, i,, i = 1,..., d 1. Le us now deal wih he converse implicaion. heorem 4.2. Assume 3-4 and 6. Le X, Y, Z be an adaped soluion of he FBSDE X = x Y = log U X s U X s Z s + θs dws U X s U X s Z s + θ s θ s ds, U X + U X Zs + θs 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 + <, Z is an elemen of 2 R d and he posiive local maringale XU X expy is a rue maringale. hen π i := U X s X s U X s Zi s + θ i s, s,, i = 1,..., d 1 is a soluion o he opimizaion problem

22 Proof. We firs noe ha π Π x since by he fac ha Z is in 2 R d, here is a consan C > such ha E π 2 d C E Z + θ 2 d <. Now le π be an elemen of Π x. Le D := U X expy. Applying Iô s formula and insering he expression for π, we find ence dd = D θ dw + Z dw O, D = U x expy. D = U x expy E θ s dw 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 + Z W O. ence, 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 + and using he concaviy of U, we obain EUX π + UX π + EU X π + X π X π. 4.1 Remark 4.3. In he previous proof, if we apply he 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. 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 + 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 =. In he complee case we can give sufficien condiions for he exisence of a soluion o he sysem 4.8. o his end, ake noe of he following remark. Remark 4.5. Similarly 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 ence 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,,

23 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 now U x 2 give a 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, logarihmic 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. 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 S 2 R 2 R of 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. ence 2 yields a soluion pair Y m, Z m S 2 R 2 R of 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 as in he proof of Proposiion 3.9 we may sae 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 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 proves E X U X <. 23

24 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 his secion is solely of illusraive characer, we will only give a formal derivaion. In paricular, we assume here ha U and U 1 are sufficienly smooh funcions wih bounded and coninuous derivaives. Moreover, we confine he consideraion o he complee marke case wih d 1 = d = 1 and = 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 5 see also 34, 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 amilonian, defined as, 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 2 dp = k p + θ p U 3 U 2 U 1 X π U 2 p θ d, Xπ = Ux, d + k dw, p = We now relae his sysem wih 4.12 using a Cole-opf 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 24 d + k dw, p =

25 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, we see ha Iô s formula implies ha p, k is a soluion of FBSDE soluion via convex dualiy mehods Le us now urn o a very imporan link of our approach wih 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 + 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 18 and 21 ha under some growh-ype condiion on U he opimal wealh process X π and he sochasic process Y ha solve he dual problem are such ha he sochasic process X π Y is a maringale. In addiion, in our noaion, i is known from he dual approach ha Y has he form Y = Y E θ W + M where M is a maringale orhogonal o W. 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 + 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 of he dual opimizer in he language of he convex dual approach. Obviously, his needs o be derived more formally. Uiliy funcions defined on he real line he aim of his secion is o employ convex dualiy resuls o obain a soluion o he forward-backward sysem 3.1 ha has been derived for he case of uiliy funcions defined on he enire real line. o his end, we adop he convex dualiy resuls from 32 and summarize in he following heir framework. We also remark ha a more general seing is considered in 3. he uiliy funcion U : R R is assumed o saisfy he Inada condiions lim x U x = and lim x U x = as well as he reasonable asympoic elasiciy condiions xu x xu x lim inf > 1, lim sup < 1. x Ux x Ux he Fenchel-Legendre ransform of U is given by { } V y := sup Ux xy, y R. x R Raher han ackling he primal problem 2.2, he dual approach aacks he convex opimizaion problem v := inf E V µ + µ, 5.5 µ C 25

26 where C denoes he se of all measure densiies yν := y dq dp F where y, Q P is a probabiliy measure such ha S becomes a Q-local maringale and which has finie enropy E V dq dp <. wo condiions are essenial in 32: A1 he se C is non-empy; A2 here exiss consans C 1, C 2 R and ϕ 2 loc Rd 1 such ha ϕ sdss is bounded from below such ha he endowmen saisfies a.s. C 1 C 2 + ϕ s ds s, where for any ineger k, 2 loc Rk denoes he se of sochasic processes 2 X for which here exiss a sequence of sopping imes τ n n increasing o such ha for every n, X1,τn is an elemen of 2 R k. he key resuls heorem 1.1 and Proposiion 4.1 from 32 hen sae ha he dual problem 5.5 admis a unique soluion µ C ha saisfies where X := X π µ = U X +, = x+ π sds s is he soluion o he primal problem 2.2 and ha X µ is a rue maringale. he following lemma is an easy observaion on he srucure of he dual opimizer µ. Lemma 5.1. Under Assumpions A1 and A2, here exiss a process ν 2 loc Rd 1 such ha he dual opimizer µ yields he represenaion µ = µ E θ W + ν W O,. Proof. Le us assume w.l.o.g. ha µ is normalized, hence he densiy of a probabiliy measure ha is absoluely coninuous wih respec o P. hus, here exis κ, ν 2 loc Rd 1 such ha µ = E κ W + ν W O,. By Iô s formula, we obain dx µ = X µ κ + µ π dw + X µ ν dw O + µ π θ + κ d. Due o he maringale propery of X µ, he drif mus vanish leading o κ = θ. We are now in he posiion o consruc a soluion o he coupled FBSDE 3.1 by making use of he characerizaion of he dual opimizer from he previous lemma. heorem 5.2. Under he assumpions A1 and A2, he FBSDE 3.1 admis a soluion X, Y, Z which saisfies he ideniy µ = U X + Y. 2 noe ha here we idenify he se of ds -inegrable processes wih S 2 loc 26

27 Proof. Given he opimal wealh process X = x+ π sdss, le us define Y := U 1 µ X. Noe ha he Inada condiions imply ha U 1 is coninuous. his obviously induces he erminal condiion Y = and Iô s formula yields which by seing dy = Z θ 1 2 = θ µ U U 1 µ + π dw + U U 1 µ ν dw O U 3 U 1 µ µ 2 U U 1 µ 3 θ 2 + ν 2 + π θ d, µ U U 1 µ π, Z O = µ µ U U 1 µ ν, 5.6 and using he ideniy µ = U X + Y becomes dy = 1 2 θ 2 U 3 X + Y U X + Y 2 U X + Y 3 + θ 2 U X + Y U X + Y + Z θ 1 2 ZO 2 U 3 U X + Y d + Z dw + Z O dw O. his is he backward equaion from he sysem 3.1. owever we obain from 5.6 π = θ which gives rise o U X +Y U X s +Ys Z X = x θ U X + Y U X s + Y s + Z ence, puing X := X we finish he proof. ds s,. Using he framework of dualiy heory, we recall below as Proposiion 5.3 and Corollary 5.4 an alernaive verificaion heorem o heorem 3.5 proposed in Secion 3 under a differen se of assumpions. he difference beween hese assumpions lies in he fac ha he se of admissible sraegy consiss of predicable and inegrable wih respec o dsc S processes π he associaed wealh process of which is a supermaringale wih respec o any probabiliy measure whose densiy is in C wih y = 1. his se of sraegies will be denoed as P erm according o 32, Definiion 1.1. We provide a proof of Proposiion 5.3 in order o make his paper self-conained. Proposiion 5.3. Assume condiions A1-A2 are in force. If here exis a process π in P erm, and a riple µ, ν, X solving X = x + π sdw s + θ s ds µ = U + X + θ s µ sdw s µ sν s dw O s, such ha X µ is a rue maringale and Eµ <, hen π is an opimal sraegy for sup EUX π +. π P erm

28 Proof. Indeed, le π be an admissible sraegy in P erm. ence X π is a supermaringale under he probabiliy measure Q defined by µ µ. Using he convexiy of U, we have EUX π + EUX + EU X + X π X = µ E Q X π Eµ X. ere we have used he supermaringale propery of X π under Q and he maringale propery of µ X. Corollary 5.4. Assume Condiions A1-A2 are in force. If here exiss a process riple of adaped processes X, Y, Z soluion o he sysem 3.1 such ha XU X π + Y is a rue maringale and π := θ U X+Y U X+Y Z belongs o P erm, hen π is an opimal sraegy for sup EUX π +. π P erm Proof. If X, Y, Z is soluion o 3.1, hen by Remark 3.3, X, Y, Z is soluion o he sysem 5.7 wih ν := U U X + Y Z O. he conclusion follows from Proposiion 5.3. Uiliy funcions defined on he posiive half-line he aim of his secion is o employ convex dualiy resuls o obain a soluion o he forwardbackward sysem 4.12 ha has been derived for he case of uiliy funcions defined on he posiive half-line. We denoe by Π 1 he se of admissible sraegies wih iniial capial given by one uni of currency. In he case of zero endowmen =, he soluion o he concave opimizaion problem 2.2 is achieved by formulaing and solving he following dual problem. Denoe he convex conjugae of he concave funcion U by V y := sup x> { Ux xy }, y >, and consider wealh processes given by dx π nonnegaive semimaringales via = X π π d S S, X π = x >. Define a family of D := { D : D = 1, X π D is a supermaringale for every π Π 1}. hen he primal problem 2.2 is solved by finding a soluion o he convex dual opimizaion problem vy = inf E V yd, y >. 5.8 D D If his dual problem admis a unique soluion D = also yields a unique soluion D, hen he primal problem 2.2 wih X π = x + = x + 28 Xs π πs d S s S s α sds s

29 = IyY, wih he corresponding opimal conrol π α S = X. ere we have I = U 1 and x = π v y. 3 he case of bounded erminal endowmen is deal wih in 8, where insead of 5.8 a dual problem of he ype vy = inf E V yd + yd, y >, D D is considered wih a differen choice of he domain D. he case of general inegrable has been sudied in 16, using he original dual problem 5.8 bu a modificaion of he domain D for he case =. A ubiquious propery of he convex dualiy mehod is ha once he primal and he dual opimizers are obained, heir produc X π D is a nonnegaive rue maringale hence uniformly inegrable, see 21 for an economic inerpreaion. In he conex of uiliy maximizaion wih bounded random endowmens, his maringale propery of X π D is poined ou in 8, Remark 4.6. his maringale propery of X π D consiues he firs main ingredien for deriving a soluion o he forward-backward equaion A second main ingredien is consiued by he characerizaion of he dual domain D. Noe ha in he seing of coninuous processes, D is he family of all non-negaive supermaringales see e.g. 21, 16. According o a well known resul, every nonnegaive càdlàg supermaringale D D admis a unique muliplicaive decomposiion D = AM where A is a predicable, non-increasing process such ha A = 1 and M is càdlàg local maringale. owever, 23 characerizes he elemens of D D by he muliplicaive decomposiion D = AE θ W + K W O, 5.9 where A is a predicable non-increasing process such ha A = 1 and K 2 loc Rd 2 see 23, Proposiion 3.2. Using ha he Fenchel-Legendre ransform V is sricly decreasing, 23, Corollary 3.3 shows ha he dual opimizer is a coninuous local maringale and admis he represenaion D = E θ W + K W O 5.1 for a uniquely deermined K 2 loc Rd 2. If vy = E V yd <, hen we can check ha he opimal K acually belongs o 2 R d 2. his is done in he following lemma whose proof is in he same spiri as he one of 22, Lemma 3.2. Lemma 5.5. If for some y > we have vy = inf E V ye θ W + ν W O <, ν 2 loc Rd 2 3 his is equivalen o u x = y where ux = sup π E UX π +. he differeniabiliy of boh vy and ux are shown in 8. 29

30 we obain vy = inf E V ye θ W + ν W O, ν 2 R d 2 i.e. he opimal K minimizing vy can be assumed o belong o 2 R d 2. Proof. We inroduce he family of sopping imes τ n := inf { > : θ s 2 + K s 2 ds n }, n N. Le y >. hen we have vy = E V ye θ W + K W O = E E V ye θ W + K W O F τ n E V ye τ n θ W + K W O, where he las line follows by Jensen s inequaliy. Coninuing he las line and recalling ha V is a sricly convex funcion, we have vy E V y exp τ n θ s dws + Ks dw O 1 s exp 2 V E y exp τ n θ s dws + Ks dw O 1 s exp 2 V y exp E 1 τ n θ 2 s 2 + Ks 2 ds, τ n τ n θ s 2 + K s 2 ds θ s 2 + K s 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 = V exp 1 2 E θ s 2 + K s 2 ds θ s 2 + K s 2 ds. Since vy < and V exp = V = U =, i follows ha E θ s 2 + Ks 2 ds <. We deduce ha K 2 R d 2. Now using ha X π Y is a rue maringale and ha he dual opimizer Y is a local maringale saisfying 5.1, we ge he following resul. 3