Utility maximization in incomplete markets

Transcription

1 Uiliy maximizaion in incomplee markes Ying Hu IRMAR Campus de Beaulieu Universié de Rennes 1 F-3542 Rennes Cedex France Ying.Hu@univ-rennes1.fr Peer Imkeller Insiu für Mahemaik Humbold-Universiä zu Berlin Uner den Linden 6 D-199 Berlin Germany imkeller@mahemaik.hu-berlin.de Mahias Müller Insiu für Mahemaik Humbold-Universiä zu Berlin Uner den Linden 6 D-199 Berlin Germany muellerm@mahemaik.hu-berlin.de Ocober 15, 24 Absrac We consider he problem of uiliy maximizaion for small raders on incomplee financial markes. As opposed o mos of he papers dealing wih his subjec, he invesors rading sraegies we allow underly consrains described by closed, bu no necessarily convex, ses. he final wealhs obained by rading under hese consrains are idenified as sochasic processes which usually are supermaringales, and even maringales for paricular sraegies. hese sraegies are seen o be opimal, and he corresponding value funcions deermined simply by he iniial values of he supermaringales. We separaely rea he cases of exponenial, power and logarihmic uiliy. 2 AMS subjec classificaions: primary 6 H 1, 91 B 28; secondary 6 G 44, 91 B 7, 91 B 16, 6 H 2, 93 E 2. Key words and phrases: financial marke; incomplee marke; maximal uiliy; exponenial uiliy; power uiliy; logarihmic uiliy; supermaringale; sochasic differenial equaion; backwards sochasic differenial equaion. his work was parially suppored by he DFG research cener Mahemaics for key echnologies (FZ 86) in Berlin. 1

2 Inroducion In his paper we consider a small rader on an incomplee financial marke who can rade in a finie ime inerval [, ] by invesing in risky socks and a riskless bond. He aims a maximizing he uiliy he draws from his final wealh measured by some uiliy funcion. he rading sraegies he may choose o aain his wealh underly some resricion formalized by a consrain. For example, he may be forced no o have a negaive number of shares or ha his invesmen in risky socks is no allowed o exceed a cerain hreshold. We will be ineresed no only in describing he rader s opimal uiliy, bu also he sraegies which he may follow o reach his goal. As opposed o mos of he papers dealing so far wih he maximizaion of expeced uiliy under consrains we essenially relax he hypoheses o be fulfilled by hem. hey are formulaed as usual by he requiremen ha he sraegies ake heir values in some se, which is supposed o simply be closed insead of convex. We consider hree ypes of uiliy funcions. In he second secion we carry ou he calculaion of he value funcion and an opimal sraegy for exponenial uiliy. In his case, he invesor is allowed o have an addiional liabiliy, and maximizes he uiliy of is sum wih erminal wealh. In secion 3 we consider power uiliy, and in he final secion he simples one: logarihmic uiliy. he mehod ha we apply in order o obain value funcion and opimal sraegy is simple. We propose o consruc a sochasic process R ρ depending on he invesor s rading sraegy ρ, and such ha is erminal value equals he uiliy of he rader s erminal wealh. As menioned above, o model he consrain, rading sraegies are supposed o ake heir values in a closed se. In our marke, he absence of compleeness is no explicily described by a se of maringale measures equivalen o he hisorical probabiliy. Insead, we choose R ρ such ha ha for every rading sraegy ρ, R ρ is a supermaringale. Moreover, here exiss a leas one paricular rading sraegy ρ such ha R ρ is a maringale. Hereby, he iniial value is supposed no o depend on he sraegy. Evidenly, he sraegy ρ relaed o he maringale has o be he opimal one. hen he value funcion of he opimizaion problem is jus given by he iniial value of R ρ. Since we work on a Wiener filraion, he powerful ool of backward sochasic differenial equaions (BSDE) is available. I allows he consrucion of he sochasic conrol process ρ, and hus he descripion of he value funcion in erms of he soluion of a BSDE. In a relaed paper, El Karoui and Rouge [ER] compue he value funcion and he opimal sraegy for exponenial uiliy by means of BSDE, assuming more resricively ha he sraegies be confined o a convex cone. Sekine [Sek] relies on a dualiy resul obained by Cvianic and Karazas [CK], also describing consrains hrough convex cones. He sudies he maximizaion problem for he exponenial and power uiliy funcions, and uses an aainabiliy condiion which solves he primal and dual problems, finally wriing his condiion as a BSDE. In conras o hese papers, we do no use dualiy, and direcly characerize he soluion of he primal problem. his allows us o pass from convex o closed consrains. Uiliy maximizaion is one of he mos frequen problems in financial mahemaics 2

3 and has been considered by numerous auhors. Here are some of he milesones viewed from our perspecive of maximizaion under consrains using he ools of BSDEs. For a complee marke, uiliy maximizaion has been considered in [KLS]. Cvianic and Karazas [CK] prove exisence and uniqueness of he soluion for he uiliy maximizaion problem in a Brownian filraion consraining sraegies o convex ses. here are numerous papers considering general semimaringales as sock price processes. Delbaen e al. [DGR] give a dualiy resul beween he opimal sraegy for he maximizaion of he exponenial uiliy and he maringale measure minimizing he relaive enropy wih respec o he real world measure P. his dualiy can be used o characerize he uiliy indifference price for an opion. Also relying upon dualiy heory, Kramkov and Schachermayer [KS] and Cvianic e al. [CSW] give a fairly complee soluion of he uiliy opimizaion problem on incomplee markes for a class of general uiliy funcions no conaining he exponenial one. See also he review paper by Schachermayer [Sch] for a more complee accoun and furher references. he powerful ool of BSDE has been inroduced o sochasic conrol heory by Bismu [B]. Is mahemaical reamen in erms of sochasic analysis was iniiaed by Pardoux and Peng [PP], and is paricular significance for he field of uiliy maximizaion in financial sochasics clarified in El Karoui, Peng and Quenez [EPQ]. 1 Preliminaries and he marke model A probabiliy space (Ω, F, P ) carrying an m dimensional Brownian moion (W ) [, ] is given. he filraion F is he compleion of he filraion generaed by W. Le us briefly explain some special noaion ha will be used in he paper. sands for he Euclidean norm in R m. For q 1, L q denoes he se of F measurable random variables F such ha E[ F q ] <, for k N, H k (R d ) he se of all R d valued sochasic processes ϑ which are predicable wih respec o F and saisfy E[ ϑ k d] <. H (R d ) is he se of all F predicable R d valued processes ha are λ P a.e. bounded on [, ] Ω. Noe here ha we wrie λ for he Lebesgue measure on [, ] or R. Le M denoe a coninuous semimaringale. he sochasic exponenial E(M) is given by ( E(M) = exp M 1 ) 2 M, [, ], where he quadraic variaion is denoed by M. Le C denoe a closed subse of R m and a R m. he disance beween a and C is defined as dis C (a) = min a b. b C he se Π C (a) consiss of hose elemens of C a which he minimum is obained: Π C (a) = {b C : a b = dis C (a) }. his se is no empy and evidenly may conain more han one poin. 3

4 he financial marke consiss of one bond wih ineres rae zero and d m socks. In case d < m we face an incomplee marke. he price process of sock i evolves according o he equaion ds i S i = b i d + σ i dw, i = 1,..., d, (1) where b i (resp. σ i ) is an R valued (resp. R 1 m valued) predicable uniformly bounded sochasic process. he lines of he d m marix σ are given by he vecor σ i, i = 1,..., d. he volailiy marix σ = (σ i ) i=1,...,d has full rank and we assume ha σσ r is uniformly ellipic, i.e. KI d σσ r εi d, P a.s. for consans K > ε >. he predicable R m valued process θ = σ r (σ σ r ) 1 b, [, ], is hen also uniformly bounded. A d dimensional F predicable process π = (π ) is called rading sraegy if π ds is well defined, e.g. π S σ 2 d < P a.s. For 1 i d, he process π i describes he amoun of money invesed in sock i a ime. he number of shares is π i. he wealh process X π of a rading sraegy π wih iniial capial x saisfies he S i equaion X π = x + d i=1 π i,u S i,u ds i,u = x + π u σ u (dw u + θ u du), [, ]. In his noaion π has o be aken as a vecor in R 1 d. rading sraegies are self financing. he invesor uses his iniial capial and during he rading inerval [, ] here is no exra money flow ou of or ino his porfolio. Gains or losses are only obained by rading wih he sock. he opimal rading sraegy we will find in his paper happens o be in he class of maringales of bounded mean oscillaion, briefly called BMO maringales. Here we recall a few well known facs from his heory following he exposiion in [Kaz]. he saemens in [Kaz] are made for infinie ime horizon. In he ex hey will be applied o he simpler framework of finie ime horizon, replacing wih. Le G be a complee, righ coninuous filraion, P a probabiliy measure and M a coninuous local (P, G) maringale saisfying M =. Le 1 p <. hen M is in he normed linear space BMO p if M BMOp := sup E[ M M τ p G τ ] 1/p <. τ G sopping ime By Corollary 2.1 in [Kaz], M is a BMO p maringale if and only if i is a BMO q maringale for every q 1. herefore i is simply called BMO maringale. In paricular, M is a BMO maringale if and only if M BMO2 = sup E[ M M τ G τ ] 1/2 <. τ G sopping ime 4

5 his means, local maringales of he form M = ξ sdw s are BMO maringales if and only if M BMO2 = sup E[ ξ s 2 ds G τ ] 1/2 <. (2) τ G sopping ime τ Due o he finie ime horizon, his condiion is saisfied for bounded inegrands. According o heorem 2.3 in [Kaz], he sochasic exponenial E(M) of a BMO maringale M is a uniformly inegrable maringale. If Q is a probabiliy measure defined by dq = E(M) dp for a P BMO maringale M, hen he Girsanov ransform of a P BMO maringale is a BMO maringale under Q (heorem 3.6 in [Kaz]). Suppose our invesor has a liabiliy F a ime. his random variable F is assumed o be F measurable and bounded, bu no necessarily posiive. He ries o find a rading sraegy ha is opimal in presence of his liabiliy F, in a sense o be made precise in he beginning of he following secion. In order o compue he opimal rading sraegy we use quadraic Backward Sochasic Differenial Equaions (BSDE) and apply a resul of Kobylanski [Kob] o ge exisence of a soluion for our BSDE. his resul is proved for bounded erminal random variables. herefore we have o assume ha F is bounded. 2 Exponenial Uiliy In his secion, we specify he sense of opimaliy for rading sraegies by sipulaing ha he invesor wans o maximize his expeced uiliy wih respec o he exponenial uiliy from his oal wealh X p F. Le us recall ha for α > he exponenial uiliy funcion is defined as U(x) = exp( αx), x R. he definiion of admissible rading sraegies guaranees ha here is no arbirage. In addiion, we allow consrains on he rading sraegies. Formally, hey are supposed o ake heir values in a closed se, i.e. π (ω) C, wih C R 1 d. We emphasize ha C is no assumed o be convex. Definiion 1 (Admissible Sraegies wih consrains) Le C be a closed se in R 1 d. he se of admissible rading sraegies à consiss of all d dimensional predicable processes π = (π ) which saisfy E[ π σ 2 d] < and π C λ P a.s., as well as {exp( αx π τ ) : τ sopping ime wih values in [, ]} is a uniformly inegrable family. Remark 2 he condiion of square inegrabiliy in Definiion 1 guaranees ha here is no arbirage. In fac, he square inegrabiliy condiion on π and he boundedness of θ yields ha E[sup (X π ) 2 ] <. According o heorem 2.1 in Pardoux, Peng [PP], (X, π σ ) is he unique soluion of he BSDE X = X (π s σ s )dw s 5 (π s σ s )θ s ds,

6 wih E[ (Xπ s ) 2 ds] <, E[ (π sσ s ) 2 ds] <. So he iniial capial X π needed o aain X π is uniquely deermined. In paricular, heorem 2.2 in El Karoui, Peng, Quenez [EPQ] yields if X π = and X π P a.s., hen Xπ = P a.s. Remark 3 In accordance wih he classical lieraure (see Dellacherie, Meyer [DM]) he uniform inegrabiliy condiion in Definiion 1 coincides wih he noion of class D. Remark 4 If X π is square inegrable and π C λ P a.s., as well as X π is bounded from below on [, ], i is obvious ha π Ã. For [, ], ω Ω define he se C (ω) R m by C (ω) = Cσ (ω). (3) he enries of he marix valued process σ are uniformly bounded. herefore we ge min{ a : a C (ω) } k 1 for λ P a.e. (, ω) (4) wih a consan k 1. Furhermore, for every (ω, ), he se C (ω) is closed. his is crucial for our analysis. Remark 5 Wriing p = π σ, [, ], he se of admissible rading sraegies à is equivalen o a se A of R 1 m valued predicable sochasic processes p wih p A iff E[ p() 2 d] < and p (ω) C (ω) P a.s., as well as {exp( αx p τ ) : τ sopping ime wih values in [, ]} is a uniformly inegrable family. Such a process p A will also be named sraegy, and X (p) denoes is wealh process. So he invesor wans o solve he maximizaion problem [ ( ( ))] ds V (x) := sup E exp α x + π F, π à S where x is he iniial wealh. V is called value funcion. Losses, i.e. realizaions wih X π F <, are punished very srongly. Large gains or realizaions wih X π F > are weakly valued. Remark 6 We shall show below ha he sup is aken by a paricular sraegy p which is admissible in he sense of our definiion. Noe ha his process migh no lead o a wealh process which is bounded from below, and herefore no admissible in his sense. For furher deails see Schachermayer [Sch2] and Meron [M]. 6

7 he maximizaion problem is evidenly equivalen o [ ( ( V (x) = sup E p A exp α x + p (dw + θ d) F ))]. (5) In order o find he value funcion and an opimal sraegy we consruc a family of sochasic processes R (p) wih he following properies: R (p) = exp( α(xp F )) for all p A, R (p) = R is consan for all p A, R (p) is a supermaringale for all p A and here exiss a p A such ha R (p ) is a maringale. he process R (p) and is iniial value R depend of course on he iniial capial x. Given processes possessing hese properies we can compare he expeced uiliies of he sraegies p A and p A by E[ exp( α(x p F ))] R (x) = E[ exp( α(x p F ))] = V (x), (6) whence p is he desired opimal sraegy. o consruc his family, we se R (p) where (Y, Z) is a soluion of he BSDE := exp( α(x (p) Y )), [, ], p A, Y = F Z s dw s f(s, Z s )ds, [, ]. In hese erms we are bound o choose a funcion f for which R (p) is a supermaringale for all p A and here exiss a p A such ha R (p ) is a maringale. his funcion f also depends on he consrain se (C ) where (p ) akes is values (see (3)). We ge V (x) = R (p,x) = exp( α(x Y )), for all p A. In order o calculae f, we wrie R as he produc of a (local) maringale M (p) and a (no sricly) decreasing process Ã(p) ha is consan for some p A. For [, ] define M (p) ( = exp( α(x Y )) exp Comparing R (p) and M (p) à (p) yields α(p s Z s )dw s 1 2 ) α 2 (p s Z s ) 2 ds wih à (p) = exp( v(s, p s, Z s )ds), [, ], v(, p, z) = αpθ + αf(, z) α2 p z 2. 7

8 In order o obain a decreasing process Ã(p) evidenly f has o saisfy v(, p, Z ) for all p A and v(, p, Z ) = for some paricular p A. For [, ] we have Now se 1 α v(, p, Z ) = α 2 p 2 αp (Z + 1 α θ ) + α 2 Z 2 + f(, Z ) = α 2 p (Z + 1 α θ ) 2 α 2 Z + 1 α θ 2 + α 2 Z2 + f(, Z ) = α 2 p (Z + 1 α θ ) 2 Z θ 1 2α θ 2 + f(, Z ). f(, z) = α (z 2 dis2 + 1 ) α θ, C (ω) + zθ + 1 2α θ 2. For his choice we ge v(, p, z) and for ( p Π C (ω) Z + 1 ) α θ, [, ], we obain v(, p, Z) =. Here we see why he se C and hence C on which rading sraegies are resriced is assumed o be closed. In order o find he value funcion we have o minimize he disance beween a poin and a se. Furhermore here mus exis some elemen in C realizing he minimal disance. Boh requiremens are saisfied for closed ses. In a convex se he minimizer is unique. his would lead o a unique uiliy maximizing rading sraegy. However, we prove exisence of a possibly non unique rading sraegy solving he maximizaion problem for closed bu no necessarily convex consrains. heorem 7 he value funcion of he opimizaion problem (5) is given by V (x) = exp( α(x Y )), where Y is defined by he unique soluion (Y, Z) H (R) H 2 (R m ) of he BSDE wih Y = F Z s dw s f(s, Z s )ds, [, ], (7) f(, z) = α 2 dis2 ( z + 1 α θ, C ) + zθ + 1 2α θ 2. here exiss an opimal rading sraegy p A wih p Π C (ω)(z + 1 α θ ), [, ], P a.s. (8) 8

9 Proof In order o ge he exisence of soluions of he BSDE (7) we apply heorem 2.3 of [Kob]. According o Lemma 11 below, for fixed z R m, (f(, z)) [, ] defines a predicable process. A sufficien condiion for he exisence of a soluion is condiion (H1) in [Kob]: here are consans c, c 1 such ha f(, z) c + c 1 z 2 for all z R n P a.s. (9) By means of (4) we ge for z R m, [, ] dis 2 ( z + 1 α θ, C ) 2 z 2 + 2( 1 α θ + k 1 ) 2. So (9) follows from he boundedness of θ. heorem 2.3 in [Kob] saes ha he BSDE (7) possesses a leas one soluion (Y, Z) H (R) H 2 (R m ). o prove uniqueness, suppose ha soluions (Y 1, Z 1 ) H (R) H 2 (R m ), (Y 2, Z 2 ) H (R) H 2 (R m ) of he BSDE are given. hen we have Y 1 Y 2 = (Z 1 Z 2 )dw Now noe ha for s [, ], z 1, z 2 R m we may wrie (f(s, Z 1 s ) f(s, Z 2 s ))ds. f(s, z 1 ) f(s, z 2 ) = α 2 [dis2 (z α θ s, C s ) dis 2 (z α θ s, C s ) + (z 1 z 2 )θ s. Using he Lipschiz propery of he disance funcion from a closed se we obain he esimae Le us se f(s, z 1 ) f(s, z 2 ) c 1 z 1 z 2 + c 2 ( z 1 + z 2 )( z 1 z 2 ) β() = c 3 (1 + z 1 + z 2 ) z 1 z 2. { f(,z 1 ) f(,z 2 ) hen we obain from he preceding esimae, if Z Z 1 Z2 1 Z 2,, if Z 1 Z 2 =. β() c(1 + Z 1 + Z 2 ), [, ]. Moreover, from he boundedness of Y 1 and Y 2, he P BMO propery of Zi (s)dw s, i = 1, 2, follows, see Lemma 12. his in urn enails ha β(s)dw s is a P BMO maringale. Bu his allows us o give an alernaive descripion of he difference of soluions in Y 1 Y 2 = = (Z 1 s Z 2 s )dw s (Z 1 s Z 2 s )[dw s + β(s)ds]. 9 β(s) (Z 1 s Z 2 s )ds

10 his process is a maringale under he equivalen probabiliy measure Q which has densiy E( β()dw ) wih respec o P. Since Y 1 = F = Y 2 we herefore conclude Y 1 = Y 2 and Z 1 = Z 2, and uniqueness is esablished. o find he value funcion of our opimizaion problem, we proceed wih he unique soluion (Y, Z) H (R) H 2 (R m ) of (7). Le p denoe he predicable process consruced in Lemma 11 for a = Z+ 1 θ. hen Ã(p ) α (ω) = 1 for λ P almos all (, ω). By Lemma 12 below, (p s Z s )dw s is a P BMO maringale, whence R (p ) is uniformly inegrable (heorem 2.3 in [Kaz]). Since, moreover, Y is a bounded process, we obain he uniform inegrabiliy of he family {exp( αx (p ) τ ) : τ sopping ime in [, ]}. herefore p A. Hence R (p,x) is a maringale and [ ( ( ))] R (p ) = E exp α x + p s(dw s + θ s ds) F = exp( α(x Y )). I remains o show ha R (p) is a supermaringale for all p A. Since p A, he process M = M E( α (p s Z s )dw s ) is a local maringale. Hence here exiss a sequence of sopping imes (τ n ) n N saisfying lim n τ n = P a.s. such ha (M τn ) is a posiive maringale for each n N. he process Ã(p) is decreasing. hus R (p) τ n is a supermaringale, i.e. for s For any se A F s we have E[R (p) τ n F s ] R (p) s τ n. E[R (p) τ n 1 A ] E[R (p) s τ n 1 A ]. = M τn à (p) τ n Since {R (p) τ n } n and {R (p) s τ n } n are uniformly inegrable by he definiion of admissibiliy and he boundedness of Y, we may le n end o o obain E[R (p) 1 A ] E[R s (p) 1 A ]. his implies he claimed supermaringale propery of R (p). Remark 8 If he process p sdw s is a BMO maringale and E[exp( α(x (p) F))] <, a varian of an argumen of he above proof can be used o see ha p A. In fac, we see ha M (p) is a uniformly inegrable maringale, while A (p) is decreasing. Hence R (p) is a supermaringale. his jus saes ha for sopping imes τ Consequenly exp( α(x (p) τ Y τ )) E[ exp( α(x (p) F )) F τ ]. exp( αx τ (p) ) exp( αy τ ) E[exp( α(x (p) F )) F τ ]. his clearly implies uniform inegrabiliy of {exp( αx (p) τ ) : τ sopping ime in [, ]}. 1

11 We can show ha he sraegy p is opimal in a wider sense. In fac, an invesor who has chosen a ime he sraegy p will sick o his decision if he sars solving he opimizaion problem a some laer ime beween and. For his purpose, le us formulae he opimizaion problem more generally for a sopping ime τ and an F τ measurable random variable which describes he capial a ime τ, i.e. X τ = Xτ p for some p A. So we consider he maximizaion problem V (τ, X τ ) = ess sup p A E [ exp ( α ( X τ + p s (dw s + θ s ds) F τ ))]. (1) Proposiion 9 (Dynamic Principle) he value funcion x exp( α(x y)) saisfies he dynamic programming principle, i.e. V (τ, X τ ) = exp( α(x τ Y τ )) for all sopping imes τ where Y τ belongs o a soluion of he BSDE (7). An opimal sraegy ha aains he essenial supremum in (1) is given by p, he opimal sraegy consruced in heorem 7. Proof For [, ], se R = exp( α(x Y ))E ( ) α(p s Z s )dw s exp( v(s, p s, Z s )ds) and apply he opional sopping heorem o he sochasic exponenial. follows as in heorem 7. he claim Remark 1 If he consrain C on he sraegies is a convex cone, he value funcion V and he opimal sraegy p boh consruced in heorem 7 are equivalen o hose deermined in [Sek] and [ER]. Sekine considers he uiliy funcion x 1 exp( αx). He obains he value funcion α saring wih he BSDE V (x) = 1 exp( αx + Ȳ) α where Ȳ = αf z s dw s f(s, θ s, z s )ds, [, ], f(, θ, z) = θ Π C ( z + θ ) 1 2 z Π C ( z + θ ) 2. We evidenly have o show ha Ȳ = αy for [, ] or equivalenly αf(, θ, z α ) = f(, θ, z). Noe ha for a convex se C, he projecion Π C (a) is unique. If C is a convex cone and β >, hen βπ C (a) = Π C (βa). he equaliy for he funcions f and 11

12 f herefore follows. El Karoui and Rouge [ER] have obained he same BSDE and value funcion before Sekine. In he following Lemma we reurn o a echnical poin in he proof of heorem 7. We show ha i is possible o define a predicable process which saisfies (8). Insead of referring o a classical secion heorem, see Dellacherie and Meyer [DM], we prefer o give a direc and consrucive proof. Lemma 11 (measurable selecion) Le (a ) [, ], (σ ) [, ] be R 1 m resp. R d m valued predicable sochasic processes, C R d a closed se and C = Cσ, [, ]. (a) he process is predicable. d = (dis(a, Cσ )) [, ] (b) here exiss a predicable process a wih a Π C (a ) for all [, ]. Proof In order o prove (a), observe ha d is he composiion of coninuous mappings wih predicable processes. For k N le H k denoe he space of compac subses of R k equipped wih he Hausdorff meric and B(H k ) he Borel sigma algebra wih respec o his meric. he mapping dis : R m H m R is joinly coninuous hence B(R m ) B(H m ) B(R) measurable. Now consider j : R d m H d H m ha maps a compac subse C in R d by applying an d m marix σ o a compac subse K of R m. More formally, j maps C o he following se: K = {b R m c C : b = c σ}. he mapping j is also joinly coninuous and herefore B(R m d ) B(H d ) B(H m )- measurable. Hence (a) follows for compac C. If more generally C is closed bu no bounded, ake C n = C B n where B n is he closed ball wih radius n cenered a he origin. According o wha has already been shown, for n N, dis(a, C n σ ) defines a predicable process and dis(a, C n σ ) converges o dis(a, Cσ ), for n. his proves he firs claim. In order o prove he second claim, we firs concenrae on he case of compac C. We have o show ha for z R m and a compac se K R m here exiss a B(R m ) B(H m ) B(R m ) measurable mapping ξ(z, K) wih ξ(z, K) Π K(z). his is achieved by he definiion of a sequence of mappings ξ n (z, K) wih a subsequence of randomly chosen index ha converges o an elemen of Π K(z). he choice of he converging subsequence will depend in a measurable way on z and K. For n N, le G n = (x n i ) i N be a dyadic grid wih min x Gn dis( z, x) 1 for n all z R m. Le he elemens of he grid G n be numbered by G n = {gi n : i N}. Le K n be he elemens of he grid wih disance a mos 1 from G n n. Since we can describe he ses K n as he inersecions of he discree se G n wih he closed se of all poins in R m having disance a mos 1 from K, and his closed se depends n coninuously on K, Kn is measurable in K. For any z R m, le Π n (z, K) be he se of 12

13 all poins in K n wih minimal disance from z. Since K n is measurable in K, Π n (z, K) is obviously measurable in (z, K). o define ξ n (z, K), we have o choose one poin in Π n (z, K). Le i be he one wih minimal index in he enumeraion of G n. his choice preserves he measurabiliy in (z, K). Hence we obain ha ξ n (z, K) is B(R m ) B(H m ) B(R m ) measurable. Furhermore, lim inf n ξ n (z, K) < for all (z, K). his is one assumpion in Lemma 1.55 in [FS] ha we aim o apply. his lemma is saed for equivalence classes of random variables, where wo random variables are equivalen if hey are equal almos everywhere wih respec o a probabiliy measure. Considering carefully he proof we see ha we can apply his lemma also wihou reference o any measure, o obain a resul for every (z, K) R m H m. Lemma 1.55 in [FS] yields a sricly increasing sequence (τ n ) n N of ineger valued, B(R m ) B(H m ) B(R) measurable funcions and a mapping ξ : R m H m R m measurable wih respec o he corresponding produc σ algebra, saisfying lim ξ n τ n (z, K) (z, K) = ξ(z, K) z R m, K H m. Bu ξ is a selecion. Indeed, for every n N, dis(z, ξ τn (z, K)) dis(z, K) 1 τ n 1 n. Since ξ τn converges o ξ, we obain dis(ξ, K) =, hence ξ K and dis(z, ξ) = dis(z, K). hus by consrucion, ξ(z, K) Π K(z) for all (z, K) R m H m. We may hen choose a = ξ(a, Cσ) o saisfy he requiremens of he second par of he asserion in he compac case. Finally, if C is only closed, we may proceed similarly as in he proof for (a). Le a n = ξ(a, ( C B n )σ ), [, ]. his ime we apply Lemma 1.55 in [FS] o he sequence of predicable processes (a n ) n N and he measure P λ on Ω [, ]. We obain a sricly increasing sequence of random indices τ n (ω, ) measurable wih respec o he predicable σ algebra and a predicable process a such ha lim a τ n(ω,) (ω) = a (ω), for P λ a.e. (ω, ). n For he process a we have dis(a, Cσ ) = P λ a.e. Lemma 12 Le (Y, Z) H (R) H 2 (R m ) be a soluion of he BSDE (7), and le p be given by Lemma 11 for a = Z + 1 θ. hen he processes α are P BMO maringales. Z s dw s, p sdw s 13

14 Proof Le k denoe he upper bound of he uniformly bounded process Y. Applying Iô s formula o (Y k) 2, we obain for sopping imes τ [ ] E Zs 2 ds F τ = E[(F k) 2 F τ ] Y τ k 2 τ [ ] 2E (Y s k)f(s, Z s )ds F τ he definiion of f yields for all (, z) [, ] R m f(, z) zθ + 1 2α θ 2. herefore here exis posiive consans c 1, c 2 and c 1 such ha [ ] [ ] E Z s 2 ds F τ c 1 + c 2 E Z s + 1 ds F τ τ τ c [ ] 2 E Z s 2 ds F τ. Hence, Z sdw s is a BMO maringale. We nex deal wih he sochasic inegral process of p. implies p Z + 1 α θ + p (Z + 1 α θ). he definiion of p ogeher wih (4) yields for some consans k 1, k 2 p 2 Z + 2 α θ + k 1 2 Z + k 2, [, ], τ τ he riangle inequaliy and hus for every sopping ime τ [ ] [ ] E p 2 d F τ E 8 Z 2 d + 2 k2 2 F τ. τ his implies he P BMO propery of p sdw s. τ 3 Power uiliy In his secion we calculae he value funcion and characerize he opimal sraegy for he uiliy maximizaion problem wih respec o U γ (x) = 1 γ xγ, x, γ (, 1). his ime, our invesor maximizes he expeced uiliy of his wealh a ime wihou an addiional liabiliy. he rading sraegies are consrained o ake values in a closed 14

15 se C 2 R d. In his secion, we shall use a somewha differen noion of rading sraegy: ρ = ( ρ i ) i=1,...,d denoes he par of he wealh invesed in sock i. he number of shares of sock i is given by ρi X. A d dimensional F predicable process S i ρ = ( ρ ) is called rading sraegy (par of wealh) if he following wealh process is well defined: X ( ρ) = x + d i=1 X s ( ρ) ρ i,s ds i,s = x + S i,s X ( ρ) s ρ s σ s (dw s + θ s ds), (11) and he iniial capial x is posiive. he wealh process X ( ρ) can be wrien as: ( ) X ( ρ) = xe ρ s σ s (dw s + θ s ds), [, ]. As before, i is more convenien o inroduce ρ = ρ σ, [, ]. Accordingly, ρ is consrained o ake is values in C (ω) = Cσ (ω) [, ], ω Ω. he ses C saisfy (4). In order o formulae he opimizaion problem we firs define he se of admissible rading sraegies. Definiion 13 he se of admissible rading sraegies à consiss of all d dimensional predicable processes ρ = (ρ ) ha saisfy ρ C (ω) P λ a.s and ρ s 2 ds < P a.s. Define he probabiliy measure Q P by dq dp = E ( θ s dw s ). he se of admissible rading sraegies is free of arbirage because for every ρ Ã, he wealh process X ( ρ) is a local Q maringale bounded from below, hence a Q supermaringale. Since Q is equivalen o P, he se of rading sraegies à is free of arbirage. he invesor faces he maximizaion problem [ ( )] V (x) = sup E U X ( ρ). (12) ρ à In order o find he value funcion and an opimal sraegy we apply he same mehod as for he exponenial uiliy funcion. We herefore have o consruc a sochasic process R (ρ) wih erminal value R (ρ) = U ( ) ds s x + X s ρ s, S s 15

16 (ρ) and an iniial value R = R x ha does no depend on ρ, R(ρ) is a supermaringale for all ρ Ã and a maringale for a ρ Ã. hen ρ is he opimal sraegy and he value funcion given by V (x) = R. x Applying he uiliy funcion o he wealh process yields ( (X ρ,x ) γ = x γ exp γρ s dw s + γρ s θ s ds 1 2 his equaion suggess he following choice: ( R (ρ) = x γ exp γρ s dw s + γρ s θ s ds 1 2 where (Y, Z) is a soluion of he BSDE Y = Z s dw s ) γ ρ s 2 ds, [, ]. f(s, Z s )ds, [, ]. γ ρ s 2 ds + Y ), (13) In order o ge he supermaringale propery of R (ρ) we have o consruc f(, z) such ha for [, ] γρ θ 1 2 γ ρ 2 + f(, Z ) 1 2 γρ + Z 2 for all ρ Ã. (14) R (ρ ) will even be a maringale if equaliy holds for ρ Ã. his is equivalen o f(, Z ) 1 2 γ(1 γ) ρ 1 1 γ (Z + θ ) Hence he appropriae choice for f is 2 1 γ Z + θ γ 2 Z 2. f(, z) = ( ) γ(1 γ) 1 dis γ (z + θ ), C γ z + θ 2 2(1 γ) 1 2 z 2, and a candidae for he opimal sraegy mus saisfy ( ) 1 ρ Π C (ω) 1 γ (Z + θ ), [, ]. In he following heorem boh value funcion and opimal sraegy are described. heorem 14 he value funcion of he opimizaion problem is given by V (x) = x γ exp(y ) for x >, where Y is defined by he unique soluion (Y, Z) H (R) H 2 (R m ) of he BSDE Y = Z s dw s 16 f(s, Z s )ds, [, ], (15)

17 wih f(, z) = ( ) γ(1 γ) 1 dis γ (z + θ ), C γ z + θ 2 2(1 γ) 1 2 z 2. here exiss an opimal rading sraegy ρ Ã wih he propery ( ) 1 ρ Π C (ω) 1 γ (Z + θ ). (16) Proof According o Lemma 11, (f(, z)) [, ] is a predicable sochasic process which also depends on σ. Due o (4) and he boundedness of θ, Condiion (H1) for heorem 2.3 in [Kob] is fulfilled. We obain he exisence of a soluion (Y, Z) H (R) H 2 (R m ) for he BSDE (15). Uniqueness follows from he comparison argumens in he uniqueness par of he proof of heorem 7. Le ρ denoe he predicable process consruced wih Lemma 11 for a = 1 (Z+θ). 1 γ Lemma 17 below shows ha ρ Ã. By heorem 2.3 in [Kaz], he process R (ρ ) is a maringale wih erminal value R (ρ ) ( = x γ exp γρ sdw s + γρ sθ s ds 1 2 ) γ ρ s 2 ds. his is he power uiliy from erminal wealh of he rading sraegy ρ. herefore he expeced uiliy of ρ (ρ,x) is equal o R = x γ exp(y ). o show ha his provides he value funcion le ρ Ã. (14) yields ( ) ( ) R (ρ) = x γ exp(y )E (γρ s + Z s )dw s exp v s ds, [, ], for a process v wih v s λ P a.s. he sochasic exponenial is a local maringale. here exiss a sequence of sopping imes (τ n ) n N, lim n τ n = such ha E[ R (ρ) τ n F s ] R (ρ) s τ n, s for every n N. Furhermore, R(ρ) is bounded from below by. Passing o he limi and applying Faou s lemma yields ha R (ρ) is a supermaringale. he erminal value is he uiliy of he erminal wealh of he rading sraegy ρ. Consequenly R (ρ,x) E[U(X (ρ,x) )] R (x) = x γ exp(y ) for all ρ A. Again we can show ha an invesor saring o ac a some sopping ime in he rading inerval [, ] will perceive he sraegy ρ jus consruced as opimal. Le τ denoe a sopping ime and X τ an F τ measurable random variable which describes he capial a ime τ, i.e. X τ = Xτ ρ for a ρ Ã and an iniial capial x >. Consider he maximizaion problem V (τ, X τ ) = ess sup ρ Aτ E [ ( )] U X τ + X s ρ s (dw s + θ s ds). (17) τ 17

18 Proposiion 15 (Dynamic Principle) he value funcion x γ exp(y) saisfies he dynamic programming principle, i.e. V (τ, X τ ) = (X τ ) γ exp(y τ ) for all sopping imes τ, where Y τ is given by he unique soluion (Y, Z) of he BSDE (15). An opimal sraegy which aains he essenial supremum in (17) is given by ρ consruced in heorem 14. Proof See Proposiion 9. Remark 16 Suppose ha he consrain se C is a convex cone. hen he opimal sraegy ρ consruced in heorem 14 is he same as in [Sek]. Sekine uses he uiliy funcion x 1 γ xγ and obains he value funcion Ṽ (x) = 1 γ xγ exp((1 γ)ỹ), where Ỹ is defined by he unique soluion (Ỹ, Z) H (R) H 2 (R m ) of he BSDE Ỹ = Z s dw s g(s, Z s )ds, [, ]. Here g(, z) = θ ( 2 θ Π C z + θ ) 2 1 γ ( 1 γ 2 z Π C z + θ ) 2. 1 γ As for he exponenial uiliy funcion we have o show (1 γ)ỹ = Y or equivalenly z (1 γ)g(, ) = f(, z). In fac, we have 1 γ ( ) [ z θ 2 (1 γ)g, = (1 γ) 1 γ 2 1 ( ) ] z + 2 θ θ 2 Π C 1 γ ( ) (1 γ)2 z z γ Π θ 2 C 1 γ 1 = θ Π C (z + θ ) 2(1 γ) Π C (z + θ ) 2 o obain he las equaliy, we use 1 2 z 2 + zπ C (z + θ ) 1 2 Π C (z + θ ) 2 = (z + θ )Π C (z + θ ) 2 γ 2(1 γ) Π C (z + θ ) z 2 γ = 2(1 γ) Π C (z + θ ) z 2. (z + θ )Π C (z + θ ) = Π C (z + θ ) 2 18

19 (see (18) below). For he funcion f we obain f(, z) = γ(1 γ) γ (z + θ ) Π C γ (z + θ ) 2 2 (1 γ) 1 2 z 2 = γ 1 γ (z + θ )Π C (z + θ ) + γ = 2(1 γ) Π C (z + θ ) z 2. For [, ], z R m we herefore have (1 γ)g(, ( ) γ (z + θ ) γ 2(1 γ) Π C (z + θ ) z 2 z ) = f(, z). 1 γ I remains o prove ha for a convex cone C and a R m he following equaliy holds: Π C (a)(a Π C (a)) =. (18) If Π C (a) = hen he ideniy is saisfied. If no, consider he half line λπ C (a), λ. his half line is par of he cone C, so Π C (a) is also he projecion of a on he half line. Lemma 17 Le (Y, Z) H (R) H 2 (R m ) be a soluion of he BSDE (15), and le ρ be given by (16). hen he processes are P BMO maringales. Z s dw s, ρ sdw s Proof We can use he same line of reasoning as in he proof of Lemma 12. he argumen given here has o be slighly modified, however. We may ake a lower bound k for Y, and apply Iô s formula o Y k 2, o conclude in he same manner as before. 4 Log Uiliy o complee he specrum of imporan uiliy funcions, in his secion we shall consider logarihmic uiliy. As in he preceding secion, he agen has no liabiliy a ime. rading sraegies and wealh process have he same meaning as in secion 3 (see (11)). he rading sraegies ρ are consrained o ake values in a closed se C 2 R d. 19

20 For ρ = ρ σ he consrains are described by C = C 2 σ, [, ]. In order o compare he logarihmic uiliy of he erminal wealh of wo rading sraegies we have o impose a mild inegrabiliy condiion on ρ. Recall ha ρ i > 1 means ha he invesor has o borrow money in order o buy sock i and if ρ i < hen he invesor has a negaive number of sock i. An inegrabiliy condiion on ρ is no resricive. Definiion 18 he se of admissible rading sraegies A l consiss of all R d valued predicable processes ρ saisfying E[ ρ s 2 ds] < and ρ C P λ a.s. For he logarihmic uiliy funcion U(x) = log(x), x >, we obain a paricularly simple BSDE ha leads o he value funcion and he opimal sraegy. he opimizaion problem is given by V (x) = sup E[log(X (ρ) )] (19) ρ A l [ = log(x) + sup E ρ s dw s + (ρ s θ s 1 ] ρ A l 2 ρ s 2 )ds, (2) where he iniial capial x is posiive again. As in secion 2 we wan o deermine a process R (ρ) wih R (ρ) = log(x (ρ) ), and an iniial value ha does no depend on ρ. Furhermore, R (ρ) is a supermaringale for all ρ A l, and here exiss a ρ A l such ha R (ρ ) is a maringale. he sraegy ρ is he opimal sraegy and R ρ is he value funcion of he opimizaion problem (19). We can choose for [, ] ( R (ρ) = log x + Y + (ρ s + Z s )dw s ρ s θ s ) 2 θ2 s + f(s) ds, where f() = 1 2 dis2 (θ, C ) 1 2 θ 2, [, ], and (Y, Z ) is he unique soluion of he following BSDE: Y = Z s dw s f(s)ds, [, ]. Due o definiion 18, he boundedness of θ and (4), he sochasic inegral in R (ρ) is a maringale for all ρ A l. Hence R (ρ) is a supermaringale for all ρ A l. An opimal rading sraegy ρ which saisfies ρ Π C (θ ) can be consruced by means of Lemma 11. he iniial value Y saisfies [ ] Y = E f(s)ds. Hence [ ] V (x) = R ρ (x) = log(x) + E f(s)ds. 2

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