COVER S UNIVERSAL PORFOLIO, SOCHASIC PORFOLIO HEORY AND HE NUMÉRAIRE PORFOLIO CHRISA CUCHIERO, WALER SCHACHERMAYER AND ING-KAM LEONARD WONG Absrac. Cover s celebraed heorem saes ha he long run yield of a properly chosen universal porfolio is almos as good as ha of he bes rerospecively chosen consan rebalanced porfolio. he universaliy refers o he fac ha his resul is model-free, i.e., no dependen on an underlying sochasic process. We eend Cover s heorem o he seing of sochasic porfolio heory: he marke porfolio is aken as he numéraire, and he rebalancing rule need no be consan anymore bu may depend on he curren sae of he sock marke. By fiing a sochasic model of he sock marke his model-free resul is complemened by a comparison wih he numéraire porfolio. Roughly speaking, under appropriae assumpions he asympoic growh rae coincides for he hree approaches menioned in he ile of his paper. We presen resuls in boh discree and coninuous ime.. Inroducion In [9] he quesion was raised wheher here is a relaion beween. Cover s heory of universal porfolio (which appeared as he very firs paper of he presen journal, see [9] and sochasic porfolio heory (SP henceforh as iniiaed by R. Fernholz (see [7] and he references herein. Afer all, boh heories ask for general recipes for choosing in a preference-free way good (a leas in he long run porfolios among d asses, whose prices over ime are given by S = (S,..., S d. Here he ime varies in, where sands eiher for N = {0,,...} (discree ime or R + = [0, (coninuous ime. In many cases S is modeled by a sochasic process defined on some probabiliy space. We noe, however, ha one may also consider a model-free approach where S = (s,... s d is jus a deerminisic rajecory wih values in (0, d. Indeed, Cover and Ordenlich s discree ime resuls in [9, 0] are formulaed in his model-free sense. he siuaion is more suble in coninuous ime due o sochasic inegraion. In [25], F. Jamishidian eended Cover s universal porfolio o coninuous ime under a seing of Iô processes saisfying some asympoic sabiliy condiions. Dae: Augus 9, 208. 2000 Mahemaics Subjec Classificaion. 60G48, 9B70, 9G99. Key words and phrases. Sochasic porfolio heory, universal porfolio, log-opimal porfolio, long-only porfolios, funcionally generaed porfolios, diffusions on he uni simple, ergodic Markov process. he auhors wish o hank he anonymous referees and he associaed edior for heir useful commens. Pars of his paper were wrien while Chrisa Cuchiero was visiing EH Zürich; she is graeful o he Forschungsinsiu für Mahemaik. Waler Schachermayer graefully acknowledges he suppor by he Ausrian Science Fund (FWF under gran P2585 and P2886 and he Vienna Science and echnology Fund (WWF under gran MA4-008.
2 CUCHIERO, SCHACHERMAYER, WONG In sochasic porfolio heory one also seeks robus invesmen sraegies. More precisely, he sraegies should be consruced using only observable quaniies (such as marke weighs and heir quadraic variaions and should no depend on quaniies ha are non-observable or difficul o esimae. In paricular, no drif esimaion is involved which is usually required in epeced uiliy maimizaion. hese are eacly he principles behind he concep of funcionally generaed porfolios (see [7, Chaper 3]. While in mos of he lieraure an Iô process seing is assumed, much of SP can be developed in a model-free seing as done by S. Pal and L. Wong [30] in discree ime and by A. Schied e al. [3] in coninuous ime. he reason why i works in coninuous ime is ha he value processes of funcionally generaed porfolios can be defined wihou sochasic inegraion. In his paper we connec he wo heories and provide addiionally a comparison wih he numéraire porfolio, which corresponds o he classical log-opimal porfolio. Relaionships beween he wo heories were sudied in he recen papers by. Ichiba and M. Brod [23, 4] as well as L. Wong [33]. In paricular, Wong [33] eends Cover s approach o he family of funcionally generaed porfolios in discree ime and shows ha he disribuion of wealh in his family saisfies a pahwise large deviaion principle... Summary and discussion of he main resuls. In his aricle we work under he seing of SP. Namely, he marke porfolio is aken as he benchmark, or numéraire, so ha he primary asses are he marke weighs which ake values in he open d-simple defined by = { (0, d d i= i = }. Is closure is denoed by = { [0, ] d d i= i = }. his enables us o analyze sraegies which depend on he marke weighs, and he performance of relaive wealh wih respec o he marke porfolio.... Discree ime. We sar by summarizing our resuls in discree ime. We eend Cover s universal porfolio o a class of M-Lipschiz porfolio maps denoed by L M. Each elemen of L M maps he marke weighs o long-only porfolio weighs in (see Definiion 3.. Denoing by (V π =0 he relaive wealh process corresponding o a porfolio sraegy 2 (π =, we are ineresed in comparing he asympoic growh raes log(v π, for cerain opimal porfolio choices π. More precisely, under suiable condiions we esablish asympoic equaliy of he growh raes of he following porfolios: he bes rerospecively chosen porfolio a ime in he class L := M= L M (in his cone V,M will denoe he relaive wealh a ime achieved by invesing according o he bes sraegy in L M over he ime inerval [0, ]; he analogue of Cover s universal porfolio whose relaive wealh process (V (ν =0 is defined in (3.2 (here ν is a probabiliy measure on L wih full suppor on each L M ; he log-opimal porfolio among he class of long-only sraegies, whose relaive wealh process is denoed by ( V =0. Henceforh, we only use he erminlogy log-opimal porfolio. 2 Here, he porfolio weigh π is chosen a ime and is used over he ime inerval [, ].
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 3 he firs wo porfolios can be compared in a model-free way (see heorem 3.9. o compare hem wih he log-opimal porfolio we have o inroduce a probabilisic seing. Our main resul can hen be roughly saed as follows: heorem.. Le (µ =0 be a ime-homogenous ergodic Markov process in discree ime describing he dynamics of he marke weighs. hen,m log(v = M log(v (. (ν = log( V holds almos surely. Inuiively, his heorem says ha a suiable full suppor miure of sraegies (given by he universal porfolio is asympoically as good as he bes one chosen wih hindsigh, and he log-opimal porfolio consruced wih full knowledge of he underlying process...2. Coninuous ime. heorem., which involves Lipschiz porfolio maps, canno be eended direcly o coninuous ime because of sochasic inegrals. Insead, we consider funcionally generaed porfolios (see Secion 4 whose relaive wealh processes can be defined in a pahwise manner (see e.g. [3]. his choice no only allows model-free consideraions bu also perfecly connecs Cover s heory wih SP in coninuous ime. By replacing he se L M by cerain spaces of funcionally generaed porfolios and assuming ha he log-opimal porfolio is funcionally generaed, we ge essenially he same heorem as above. Apar from he work by F. Jamshidian [25], universal porfolio heory has only been sudied sparingly in coninuous ime; see for eample he paper [24] which sudied he performance of he universal porfolio under he Hybrid Alas model. o he bes of our knowledge generalizaions o nonparameric families of porfolio maps (in coninuous ime have no been considered so far. In his sense, our resuls significanly eend he coninuous ime lieraure. While our approach focuses on he mahemaical aspecs, universal porfolio sraegies have also been sudied eensively in an algorihmic framework. See [29] for a recen survey and in paricular [2]...3. Discussion of he resuls. Our model-free approach has clear advanages over classical ones which heavily rely on a paricular model choice. Even in he case when he model class (e.g. he Heson model or Lévy models is correcly specified, model parameers canno be esimaed precisely and always come wih a confidence inerval. So, in pracice he esimaed opimal porfolio is always differen from he rue opimal one. Our resuls suppor he idea ha a Bayesian average in he spiri of Cover s universal porfolio is, in he long run, beer han a subopimal esimae. As for he original heorems of Cover and Jamshidian, a valid criicism is of course ha we only esablish asympoic equaliy on a firs-order log-reurn basis. As such, a lo of imporan informaion is los in he i. However, one canno epec o obain any informaion on higher-order erms unless furher quaniaive assumpions are made on he considered models. Cover s aim and also he goal of he presen aricle is o be as model-free as possible. 3 Neverheless, i is of grea heoreical and pracical ineres o srenghen he asympoic resuls o quaniaive ones under suiable addiional condiions. We hope o address his imporan quesion in fuure research. 3 We are graeful o one of anonymous referees for poining his ou.
4 CUCHIERO, SCHACHERMAYER, WONG he remainder of he paper is organized as follows. In Secion 2 we provide a brief overview (in discree ime for convenience of he main opics of his paper, i.e., Cover s heorem, he seing of SP and he log-opimal porfolio. In Secion 3 we esablish heorem. in discree ime (see heorem 3.0 and Corollary 3., while Secion 4 is dedicaed o proving he corresponding saemens in coninuous ime in he seing of funcionally generaed porfolios and for he comparison wih he log-opimal porfolio under he assumpion ha he marke weighs follow an ergodic Iô diffusion (see heorem 4. and Corollary 4.3. Some auiliary and echnical proofs are gahered in he appendi. 2. Overview of he hree porfolios For eposiional simpliciy ime is discree in his secion. 2.. Cover s universal porfolio. Cover s insigh reveals ha he wisdom of hindsigh does no give significan advanages over a properly chosen universal porfolio consruced using only hisorical and curren prices of he asses. he relevan opimaliy crierion here is he asympoic growh rae of he porfolio. Le us skech his a firs glance surprising resul in a paricularly easy seing (compare [9, 0]: Fi N and hink of an invesor who a ime looks back which sock she should have bough a ime = 0 (by invesing her iniial endowmen and subsequenly holding he sock. here is an obvious soluion: pick i {,..., d} which maimizes he normalized logarihmic reurn (2. (log(si log(s0. i he problem wih his rading sraegy is, of course, ha we have o make our choice a ime = 0 insead of =. Here is he remedy (compare e.g., [3]: a ime = 0 simply divide he iniial endowmen, say AC, ino d porions of d AC, inves each porion in each of he socks and hen hold he resuling porfolio. A ime he normalized logarihmic reurn saisfies 4 log(v log d S j ( log(s i (2.2 d log(s0 i log d, j= S j 0 where again i denoes he sock which performed bes during he ime inerval [0, ]. Hence he difference beween (2. and (2.2 can be bounded by log(d which ends o zero as. Hence his buy-and-hold porfolio, which corresponds o a universal porfolio in he sense of Cover, has asympoically he same normalized logarihmic reurn as he only rerospecively known bes performing sock. Insead of hese pure invesmens Cover considered a more ambiious seing, namely all consan rebalanced porfolio sraegies: le b = (b,..., b d, i.e., b j 0 and d j= bj =. he value of he corresponding consan rebalanced porfolio (V (b =0 saring a V 0 (b = is defined by holding hroughou he proporion b j of he curren wealh in sock j, so ha V 0 (b = and (2.3 V + (b V (b (s = d j= b j sj + s j, 4 Laer we will use V o denoe insead he relaive wealh of he porfolio.
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 5 for each rajecory s = ((s j d j= =0 (0, d of he socks. Fi again and define he quaniy V by (2.4 V (s = ma b V (b(s, which is a funcion of he rajecory s = (s,..., s d =0. Again, he idea is ha, wih hindsigh, i.e., knowing (s,..., s d =0, one considers he bes weigh b which aains he maimum (2.4. Cover s goal is o consruc a porfolio which generaes wealh ha performs asympoically as well as he process (V =0 as, uniformly over all price pahs. For his reason he porfolio is said o be universal. In order o do so, le ν be a probabiliy measure on which replaces he previous uniform disribuion over he d socks. he universal porfolio is buil by invesing a ime 0 he porion dν(b of iniial capial in he consan rebalanced porfolio V (b and by subsequenly following he consan rebalanced porfolio process (V (b =0. he eplici formula for he wealh is (2.5 V (ν(s = V (b(sdν(b, where V (b is defined by (2.3. he porfolio weigh of he corresponding universal porfolio is given by he wealh-weighed average (2.6 b ν bv (s = d (b(sdν(b V (b(sdν(b. Le us now recall Cover s celebraed resul: heorem 2.. (Cover [9]: Le ν be a probabiliy measure on wih full suppor. hen (2.7 (log(v (ν(s log(v (s = 0, for all rajecories s = (s,..., s d =0 for which here are consans 0 < c C < such ha (2.8 c sj + s j he proof is given in he Appendi. C, for all j =,..., d and all N. Remark 2.2. As shown by. Cover and E. Ordenlich [0], he condiion (2.8 can be dropped a leas when ν is he uniform or Dirichle( 2,, 2 disribuion on (see also A. Blum and A. Kalai [3] for an elegan proof in case of he uniform disribuion. Remark 2.3. Le M ( be he se of probabiliy measures on. For each µ M (, consider he value V d (b(sdµ(b of he miure porfolio wih iniial measure µ. Noe ha he consan rebalanced porfolio V (b corresponds o he case where µ is he poin mass a b. I is easy o see ha V (b(sdµ(b = V (s, sup µ M (
6 CUCHIERO, SCHACHERMAYER, WONG where V (s is defined by (2.4. I follows ha he universal porfolio (2.5 (wih iniial measure ν is sill asympoically opimal in he larger class { ( } bv (bdµ(b (2.9 µ M ( V (b(sdµ(b. 2.2. Sochasic porfolio heory, porfolio maps and he corresponding universal porfolio. In SP we le (s,..., s d denoe he marke capializaions of he socks raher han heir prices. hen we define he vecor of marke weighs (µ,..., µ d by ( (µ,..., µ d = 0 s s + + s d,..., s d s + + s d. his amouns o aking he marke porfolio (whose value a ime is d j= sj as he numéraire (compare [] and [5]. he relaive wealh process (V π =0, epressed in unis of he marke porfolio and saring a V 0 =, is obained by he following recursive relaion 5 (2.0 V+ π V π = d π j µ j + + j= µ j In general, we allow all predicable, admissible rading sraegies (π =, where he porfolio weigh π is used over he ime inerval [, ]. In his paper all rading sraegies are fully invesed in he equiy marke, i.e., he porfolio weighs sum o for all. In paricular, he sraegies do no lend or borrow money. Henceforh all wealh processes are measured in unis of he marke porfolio. We will focus on rading sraegies defined by (deerminisic porfolio maps. hese are (Borel measurable funcions (2. π : which associae o he curren marke capializaion µ = (µ,..., µ d he weighs (π(µ = (π (µ,..., π d (µ according o which an agen disribues curren wealh among he d socks a ime. he consan rebalanced porfolio sraegies considered by Cover correspond o he consan funcions π :. In his paper we eend Cover s heory of consan rebalanced porfolios o cerain families of porfolio maps. Firs we noe ha Cover s and Jamshidian s definiion of a universal porfolio as in (2.6 and (2.5 can be easily eended o a general seing. Le G denoe some appropriae space of porfolio maps, B(G is Borel σ-algebra and ν some probabiliy measure on G. Definiion 2.4. Le ν be a probabiliy measure on (G, B(G. hen he corresponding universal porfolio a ime is given by he wealh-weighed average π ν G = πv π (2.2 dν(π dν(π. G V π From (2.0 i is easily seen ha he wealh generaed by π ν is given by (2.3 V (ν = V π dν(π. G 5 Here i is assumed implicily ha he socks do no pay dividends. his assumpion is common in universal and sochasic porfolio heory and allows us o focus on he main ideas..
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 7 2.3. he log-opimal porfolio. o define he log-opimal porfolio we consider a probabilisic seing. he sock price process S = (S,..., S d =0 and he corresponding relaive marke capializaions µ = (µ,..., µ d =0 are now assumed o be sochasic processes defined on a filered probabiliy space (Ω, F, (F =0, P. here is a large lieraure on he log-opimal porfolio (see e.g., [2], [26] and he references given here. For a fied horizon, his porfolio is by definiion he maimizer of he epeced logarihmic growh rae d E[log(V π µ j ] = E log + (2.4 =0 j= π j + over all predicable, admissible rading sraegies (π =. Under mild assumpions on he process a unique opimizer eiss; see e.g., [2, 28]. o connec he log-opimal porfolio wih universal porfolios in he sense of Definiion 2.4 we need appropriae assumpions. We will assume ha µ is a imehomogenous Markov process, and we will resric o long-only porfolios in he opimizaion of (2.4. hese imply ha he opimal porfolio in (2.4 (over he se of predicable processes aking values in has he form π = π(µ, where π : as in (2.. We denoe he corresponding opimizer by π. he Markovian assumpion can be moivaed by he sabiliy of capial disribuions of equiy markes (see [7, Chaper 5]. In SP, his led o sysems of ineracing Brownian paricles whose dynamics depend on heir relaive rankings. Under suiable condiions, hese sysems show behaviors observed in large equiy markes. See, for eample [, 24] for Alas -ype models and he references herein 6. We also refer o [27] which sudies he growh opimal porfolio in a Markovian seing wih uncerainies. 3. A comparison of he hree approaches - he discree ime case hroughou his secion we work in discree ime and assume ha he marke weighs are described by a d-dimensional pah µ = (µ =0 wih values in. We consider as far as possible a model-free approach, bu will inroduce a probabilisic seing when he log-opimal porfolio is involved. 3.. Definiions of he porfolios. We sar by defining rigorously, in he presen seing, he hree porfolios inroduced in Secion and Secion 2. 3... he bes rerospecively chosen porfolio. Consider Cover s heme of choosing rerospecively a ime a sraegy which is opimal wihin a cerain class of sraegies, in our case porfolio maps π :. A momen s reflecion reveals ha i does no make sense o allow o choose among all measurable funcions π :. Indeed, here is no resricion o choose π such ha π(µ = e j(, where j( {,..., d} maimizes µ j + /µj. his is asking for oo much clairvoyance and does no allow for meaningful resuls (compare [0] and [3, Secion 5]. However, i does make sense (economically as well as mahemaically o resric o more regular rading sraegies. In paricular, we work wih he following se of M-Lipschiz porfolio maps. For ɛ > 0 we le ɛ denoe he se of saisfying j ɛ d, for j =,..., d. Also we le be he usual -norm. µ j 6 A comparison beween he log-opimal porfolio and Cover s universal porfolio is sudied in [24].
8 CUCHIERO, SCHACHERMAYER, WONG Definiion 3.. For M > 0 we denoe by L M he se of all M-Lipschiz funcions M, i.e., π( π(y L y,, y. Remark 3.2. he se L M of M-Lipschiz funcions π : M is a compac meric space wih respec o he opology of uniform convergence induced by he norm π = sup{ π( : }. Remark 3.3. Insead of Lipschiz funcions we could jus as well consider oher compac funcion spaces, e.g., Hölder spaces equipped wih a proper norm. his is done in he cone of funcionally generaed porfolios in Secion 4. he rerospecively chosen bes performing porfolio among he above Lipschiz maps is defined as follows: Definiion 3.4. For a given rajecory (µ =0 ( + we define d V,M = sup V π = sup π j (µ µj + (3. π L M π L M µ j. By compacness (see Remark 3.2 and coninuiy of he map π V π here eiss an opimizer π,m L M (no necessarily unique such ha V,M = V π,m, hus he sup above can be replaced by ma. =0 j= 3..2. he universal porfolio. Our aim is o find a predicable process π M = (π M =, i.e., one which depends only on he hisory of he marke weighs, such ha he performance of (V πm =0 is asympoically as good as ha of (V,M =0. his can be achieved by he universal porfolio inroduced in Definiion 2.4, where he G is now L M as in Definiion 3.. As L M is a compac meric space, we may find a (Borel probabiliy measure ν on (L M, wih full suppor; his will be essenial for esablishing an analog o heorem 2.. he (relaive wealh of he universal porfolio is given, as in (2.3, by (3.2 V M (ν = V π dν(π. L M 3..3. he log-opimal porfolio. In order o relae he universal porfolio o he (long-only log-opimal porfolio, we assume ha µ = (µ =0 is a ime-homogeneous Markov process (see Secion 2.3. Here is a precise saemen. Assumpion 3.5. he process µ is a ime homogeneous, ergodic Markov process wih a unique invarian measure ϱ on he open simple. We denoe he ransiion kernel of he chain by by (ϱ(,, i.e., for all Borel ses A we have P[µ + A F ] = ϱ(µ, A. For furher noions concerning ergodic Markov processes we refer o [4]. he long-only log-opimal rading sraegy π, as noed above, is given in erms of a porfolio map. Given ha µ =, we know he condiional law ϱ(, of µ +. We herefore choose π( as he maimizer (3.3 ( π( = arg ma log( p, y ϱ(, dy p d
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 9 and assume ha π( can be chosen o be measurable (here, denoes he Euclidean do produc. For define he number L( as he value of he opimizaion problem (3.3, i.e., ( (3.4 L( = ma p log( p, y ϱ(, dy = log( π(, y ϱ(, dy. Considering π( = (which corresponds o he marke porfolio we clearly have L( 0 for each. We obain he a.s. relaion [ ( ] V+ L( = E log V µ =, where V = ( V =0 denoes he long-only log-opimal wealh process V π defined by he porfolio map π via (2.0. Assumpion 3.6. Using he above noaion we assume ha (3.5 L := L(dϱ( <. Applying Birkhoff s ergodic heorem for discree ime Markov processes (see [4, heorem 2.2, Secion 2..4] we have he following resul. heorem 3.7. Under Assumpions 3.5 and 3.6, we have ha, for ϱ-a.e. saring value µ 0, (3.6 log( V = L, he i holding rue a.s. as well as in L. More generally, le π : be any measurable porfolio map such ha ( ( (3.7 L π := log π(, y ϱ(, dy dϱ( >. d We hen have, for ϱ-a.s. saring value µ 0, ha (3.8 log(v π = L π a.s. as well as in L. In general here is lile reason why he funcion π should have beer regulariy properies han being jus measurable. On he oher hand, we may approimae π by more regular funcions, in paricular by funcions in L M. his will be crucial for comparing he asympoic growh raes. he following resul is inuiively obvious, bu he proof urns ou o be quie echnical and will be given in he appendi. Lemma 3.8. Under Assumpions 3.5 and 3.6, for any ɛ > 0 here eis M > 0 and an M-Lipschiz funcion π Lip L M such ha L π Lip > L ɛ, where L and L π are given in (3.5 and (3.7 respecively. In paricular, we have L = sup M sup π L M L π.
0 CUCHIERO, SCHACHERMAYER, WONG 3.2. Asympoically equivalen growh raes. We are now ready o compare he asympoic performance of he hree approaches. We firs esablish an analogue of heorem 2.. heorem 3.9. Fi M > 0 and a Borel probabiliy measure ν wih full suppor on L M. For every rajecory (µ =0 in we have (3.9,M (log(v log(v M (ν = 0. Proof. he inequaliy is obvious. For he reverse inequaliy we follow he argumen of [3]. As L M is compac and ν has full suppor, i is no difficul o see ha for any η > 0, here eiss δ > 0 such ha every η-neighbourhood of a poin π L M has ν-measure bigger han δ. Le a rajecory (µ =0 in be given. For a fied ime le π,m L M be an opimizer of (3.. Consider a porfolio map π M L M wih π M π,m < η, i.e., such ha, for every we have π M ( π,m ( = d j= πm ( j π,m ( j < η. Choose η > 0 small enough so ha α = ηmd < and define, for, (3.0 Rearranging, we have (3. π( = α πm ( α α π,m (. π M ( = ( απ,m ( + α π(. I is easy o see ha ha π maps ino. Using (3., we have he esimae (3.2 log V π M = log( π M (µ, µ + µ =0 log( ( απ,m (µ, µ + µ =0 =,M log(v + log( α. Fi ɛ > 0. Choosing η > 0 sufficienly small we can make α = ηmd small enough such ha he final erm is bigger han ɛ. Summing up, we have (3.3 [log(v,m log(v πm ] < ɛ whenever π M π,m < η. Denoe by B = B η (π,m he -ball wih radius η in L M which has ν- measure a leas δ > 0, where δ only depends on η. As each elemen π M of B saisfies (3.3 we have (3.4 log(v M (ν log(δ + log(v,m ɛ. Now (3.9 is proved by sending in (3.4 o infiniy and leing ɛ o zero.
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO Noe ha in heorem 3.9 we do no need he uniform boundedness condiion (2.8 (compare his resul wih [33, Lemma 3.3]. We now combine Lemma 3.8 (which is probabilisic wih heorem 3.9 (which is pahwise o obain under suiable assumpions equaliy of he asympoic performance among he hree porfolios. We firs consider he space L M for a fied M. In Corollary 3. below we hen formulae a resul for L = M L M. heorem 3.0. Le Ω = ( N be he canonical pah space equipped wih is naural filraion and a probabiliy measure P. Define µ = (µ =0 o be he canonical process, i.e., µ (ω = ω, which akes values in and saisfies Assumpions 3.5 and 3.6. Moreover, le M > 0 be a fied Lipschiz consan for he space L M. Consider he following objecs ha are defined for each rajecory (µ =0: 7 (i Define for each N he porfolio map π,m L M as well as he corresponding wealh V,M := V π,m as in (3.. (ii Fi a probabiliy measure ν on L M wih full suppor and consider he wealh process of he universal porfolio (V M (ν =0 as of (3.2. (iii Define he log-opimal porfolio among he porfolio maps π L M by [ π M = arg ma log( π(, y ] (3.5 ϱ(, dy dϱ( π L M d and he corresponding wealh process ( V M =0 = (V πm =0 via (2.0. hen, we have P-a.s. (3.6 inf,m log(v = inf log(v M (ν = log( V M = sup L π, π L M where L π is given in (3.7. In addiion, he firs equaliy holds for all rajecories (µ =0 in. Proof. We firs noe ha π M is well-defined; simply use he compacness of L M wih respec o (compare he proof of Lemma 3.8. Noe also ha by he ergodic heorem (heorem 3.7, we have for each π L M log V π = L π P-a.s., where L π is defined by (3.7. In paricular, as π M L M by definiion, we have (3.7 log V M = sup L π P-a.s. π L M ha he firs equaliy in (3.6 holds for all rajecories (µ =0 in was shown in heorem 3.9. For each fied N we obviously have log( V M,M (3.8 log(v P-a.s. Using (3.7, (3.8 and heorem 3.9 we hus have P-a.s. (3.9 sup L π = π L M log( V M inf 7 o simplify he noaions we will suppress ω.,m log(v = inf log(v M (ν.
2 CUCHIERO, SCHACHERMAYER, WONG On he oher hand, by he definiion of ( V M =0 as he log-opimizer wihin he class L M, we have (3.20 E[log(V M (ν] sup π L M E[log(V π ] = E[log( V M ]. o see his, noe ha he universal porfolio is given by (2.2. By he imehomogenous Markovianiy i is hus sufficien o dominae he lef hand side of (3.20 by aking he supremum over elemens in L M. Combining now (3.20, heorem 3.7 and (3.9 yields ha [ ] E inf log(v M (ν inf E[log(V M (ν] E[log( V M ] = log( V M,M inf log V = inf log(v M (ν, P-a.s. Here, he firs inequaliy follows from Faou s lemma (noe here ha log(v M (ν is bounded from below, see e.g., (3.4. From his we see ha he quaniy inf log(v M (ν is P-a.s. consan and equal o log( V M. his complees he proof of he heorem. Ne we will send M o infiniy in he following way. For M =, 2, 3,... choose a measure ν M on L M wih full suppor. Define ν = M= 2 M ν M and he wealh of he universal porfolio V (ν as in (3.2 by (3.2 V (ν = V π dν(π, N. L where L = M= L M. Recall ha ( V =0 is he wealh process of he (long-only log-opimal porfolio (3.3. Corollary 3.. Under he assumpions of heorem 3.0 we have P-a.s.,M log V = M log V (3.22 (ν = log V = L, where L is defined in (3.5. Proof. Leing M in (3.6, we have,m inf log V = M sup L π = L = M π L M log ˆV, where he las equaliy follows from heorem 3.7 and he second las follows from Lemma 3.8. By consrucion V (ν 2 M V M (νm for every M, so we have by heorem 3.9 for every M inf log V (ν inf ( M log 2 + log V M (ν M,M = inf log V, and hence also inf log V (ν inf M,M log V.
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 3 Using he same argumen as in he las par of he proof of heorem 3.0, we ge,m (3.23 inf log V M Now he corollary is proved if (3.24 sup = inf (log V (ν log V = sup log log V (ν = log V = L. ( V (ν = 0, V holds P-a.s. As by Lemma 3.2, ( V(ν V =0 is a non-negaive supermaringale, i converges P-a.s. o a finie i as. his in urn implies (3.24 and proves he asserion. Lemma 3.2. he process ( V(ν V =0 is a non-negaive supermaringale. Proof. Firs noe ha for any π :, ( V π V =0 is a non-negaive supermaringale. Indeed, by Lemma 3.3 below we have [ ] V π E + F = V π y π(µ, µ V + V π(µ d, y µ ϱ(µ, dy V π. V By Fubini s heorem we ge he supermaringale propery of ( V(ν V [ ] [ V + (ν E V+ π F = E dν(π ] F V + L V + [ ] V π = E + F dν(π V + L L V π V dν(π = V (ν V. =0, Here we esablish he supermaringale propery used in he previous proof. Lemma 3.3. Le π be given by (3.3. hen for every π : and every, y π(, π(, y dy. d ϱ(, Proof. We proceed as in he proof of [2, Proposiion 4.3]. Fi π and α (0, and define π α = απ + ( α π. hen by he (long only log-opimaliy of π we have for every 0 (log π(, y log πα (, y ϱ(, dy = y π(, πα (, y π α (, y ϱ(, dy = Hence, y π(, π α (, y dy d ϱ(, y π(, π α (, y dy ϱ(, ( π(, y π α (, y z dz ϱ(, dy y α( π( π(, π α (, y ϱ(, dy. y π(, ( α π(, y dy, ϱ(,
4 CUCHIERO, SCHACHERMAYER, WONG where he las equaliy follows from π α ( α π. By Faou s lemma we herefore have y π(, π(, y π(, y dy = y π(, d ϱ(, α 0 π α (, y dy ϱ(, α 0 π α (, y dy d ϱ(, y π(, α 0 α π(, y dy =. d ϱ(, 4. he coninuous ime case wih funcionally generaed porfolios his secion is dedicaed o a similar analysis in coninuous ime and wih funcionally generaed porfolio maps [7, Chaper 3]. Using he pahwise Iô calculus developed by H. Föllmer [20], we can define he corresponding wealh processes in a pahwise manner for any coninuous marke pah admiing a quadraic variaion process. his allows us o define he bes rerospecively chosen porfolio which is no well-defined in general (and in paricular for he Lipschiz porfolio maps. 4.. Funcionally generaed porfolios. We consider he following se of concave funcions. For some fied M > 0 and 0 α, we define { G M,α = G C 2,α (, concave such ha G C 2,α M and G }, M where C 2,α ( denoes he Hölder space of 2-imes coninuously differeniable funcions from R whose derivaives are α-hölder coninuous. ha is, where C 2,α ( = {G C 2 ( G C 2,α < }, G C 2,α = ma k 2 Dk G + ma sup k =2 y D k G( D k G(y y α wih k denoing a muli-inde in N 2. For α = 0 he second erm in his norm is lef away. Noe ha G is only defined on he simple. In order ha he parial derivaives are well defined, we assume ha each G is eended o an open neighborhood of such ha G( = G(, where is he orhogonal projecion of ono. he choice of he eension is irrelevan. Here is an analyical lemma whose proof is given in he appendi. Lemma 4.. For any M, α > 0 he se G M,α is compac wih respec o C 2,0. o he se of generaing funcions G M,α we associae now he se of funcionally generaed porfolios FG M,α in he spiri of [7] defined by (4. FG M,α = { π G :, (π G ( i = i Di G( G( + d j= D j G( G( j, i =,... d, G G M,α }. By he concaviy of G, π G akes values in, i.e., i is long-only (see e.g. [9, Remark.]. he corresponding wealh processes are denoed by V πg or V G.
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 5 For hese porfolios i is possible o obain a pahwise epression for V πg. We refer he reader o [3] for eensions of his pahwise approach o ime-dependen and pah-dependen generaing funcions. here his is achieved by applying he funcional Iô calculus developed by B. Dupire [3] and R. Con and D. Fournié [7, 8], which generalizes Föllmer s Iô calculus o pah-dependen funcionals. In his paper we only consider funcionally generaed porfolio maps as defined in (4.. We adop he noaion of [3] and fi a refining sequence of pariions ( n n= of [0,, i.e., n = { 0,,...} is such ha 0 = n 0 < n < and n k as k, and 2. Moreover, he mesh of n ends o zero on each compac inerval as n. Furhermore, we denoe he successor of n by. ha is, = min{u n, u > }. hroughou his secion he marke weighs are described by a d-dimensional coninuous pah µ = (µ 0 wih values in. Here and henceforh we le S + d be he se of d d posiive definie marices. Assumpion 4.2. he pah (µ 0 admis a coninuous S + d -valued quadraic variaion [µ] along ( n in he sense of [20], i.e., for any i, j d and all 0 he sequence (µ i s µi s(µ j s µj s s n,s converges o a finie i, as n, denoed [µ i, µ j ], such ha [µ i, µ j ] is coninuous. he dynamics of he relaive wealh process V πg buil by invesing according o π G FG M,α are given in his coninuous ime case by (4.2 dv πg V πg = d (π G (µ i dµi µ i i= = d i= D i G(µ G(µ dµi, V π 0 =, (compare (2.0 in he discree ime case, where he righ hand side has o be undersood as Föllmer s pahwise inegral (c.f. Equaion (2.5 in [3]. Noe ha he second equaliy holds by he definiion of π G and he fac ha d i= dµi = 0. Using (4.2 and Föllmer s Iô calculus, we have he following pahwise version of Fernholz s [7] maser equaion (also see [3, heorem 2.9]. Corollary 4.3. Le G C 2 ( and π G be defined as in (4.. Le (µ 0 be a coninuous pah saisfying Assumpion 4.2. hen V πg saisfies (4.3 V πg V G = G(µ G(µ 0 eg([0, ], 0 <, where g(d = 2G(µ i,j Dij G(µ d[µ i, µ j ]. 4.2. Definiions of he porfolios. We again consider (i he bes rerospecively chosen porfolio, (ii he universal porfolio and (iii he log-opimal porfolio. o define he log-opimal porfolio we will resric o a specific sochasic model inroduced in Secion 4.2.3. In Secion 4.2.4 we derive he asympoic growh rae for his model class under an addiional ergodiciy assumpion.
6 CUCHIERO, SCHACHERMAYER, WONG 4.2.. he bes rerospecively chosen porfolio. We consider he se of funcionally generaed porfolios FG M,α and a given coninuous pah (µ 0 saisfying Assumpion 4.2. For M, α > 0 fied, we define (4.4 V,M,α = sup V π G π G FG M,α = sup V G. G G M,α We firs prove ha an opimizer eiss by esablishing he following coninuiy propery whose proof can be found in he appendi. Lemma 4.4. Le, M, α > 0 be fied and (µ 0 be a coninuous pah saisfying Assumpion 4.2. Consider he funcion G V G where V G is given by (4.3. hen G V G is coninuous from (GM,α, C 2,0 o R. Proposiion 4.5. Le be fied and (µ 0 be a coninuous pah saisfying Assumpion 4.2. Le V,M,α be defined by (4.4. hen here eiss an opimizer G GM,α and in urn a porfolio π generaed by G such ha V,M,α = V π = V G. Proof. his is simply a consequence of coninuiy as proved in Lemma 4.4 and compacness of (G M,α, C 2,0 as shown in Lemma 4.. 4.2.2. Universal porfolio. o define he analogue of Cover s/jamshidian s porfolio in he presen seing, le m be a Borel probabiliy measure on (G M,α, C 2,0. Consider he map (4.5 F : G M,α FG M,α ; G F (G = π G, where π G is given by (4.. Define now on (FG M,α, a Borel probabiliy measure ν via he pushforward ν = F m. As in Definiion 2.4, we hen define he corresponding universal porfolio via (4.6 π FG G (µ π ν = M,α V πgdν(πg. V FG M,α πgdν(πg Analogous o (2.3, he value of he universal porfolio is given by V M,α (ν := V πν = V π G (4.7 dν(π G = V G dm(g. FG M,α G M,α Remark 4.6. More precisely, we need o verify ha he universal porfolio sill allows for pahwise inegraion and ha he value of he porfolio (as a pahwise inegral is given by he righ hand side of (4.7. hese claims can be easily checked using he definiions and resuls in [3], so we omi he deails. 4.2.3. Funcionally generaed log-opimal porfolios. By definiion, he log-opimal porfolios requires a sochasic model for he marke weighs. We suppose ha µ = (µ,..., µ d 0 follows a ime-homogeneous Markovian Iô diffusion, defined on (Ω, F, (F 0, P wih values in, given by (4.8 µ = µ 0 + c(µ s λ(µ s ds + 0 0 c(µs dw s, µ 0,
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 7 where denoes he mari square roo, W is a d-dimensional Brownian moion, λ is a Borel measurable funcion from R d and c is a Borel measurable funcion from S d +, saisfying (4.9 (4.0 0 λ (µ c(µ λ(µ d <, [0,, c( = 0, c ij (λ( j = 0,. i,j he requiremens in (4.0 are necessary o guaranee ha he process µ lies in. Noe ha (µ 0 given by (4.8 saisfies he so called srucure condiion (see [32] (because of (4.9 and he fac ha he drif par is of form 0 c(µ sλ(µ s ds. his srucural condiion characerizes he condiion of no unbounded profi wih bounded risk (NUPBR in he case of coninuous semimaringales (see e.g., [22]. In his seing he proporions of curren (relaive wealh invesed in each of he asses are described by processes π in he following se (4. Π = {π H d -valued, predicable, R-inegrable}, where he process R is defined componenwise by R i = dµ i s 0 µ. Here, H d denoes i s he hyperplane corresponding o porfolio weighs ha are no necessarily longonly, i.e., H d = { R d d j= j = }. Noe ha he se FG M,α is clearly a subse of long-only sraegies in Π. he relaive wealh process V π saisfies (4.2 dv π V π = d i= π i dµ i µ i, V0 π =. In conras o Secion 4., his is a usual sochasic inegral because we are dealing wih general inegrands π. Noe ha we can also wrie ( ( π V π = E((π R = ep dµ ( π (4.3 c(µ π d 0 µ 2 0 µ µ d = ep π i dµi 0 µ i π i π j 2 0 µ i c ij (µ d, i= where, for wo vecors p, q R d, p/q always denoes he componenwise quoien ( p q,..., pd. q d Ne we consider he log-opimal porfolio defined by (2.4 (bu in coninuous ime now. As in [6, Secion 3.], we derive he raio of wo wealh processes V π and V θ for π, θ Π. Using (4.2 (for he processes π and θ and Iô s lemma, his raio is given by d ( V π V θ = V π ( π V θ = V π V θ θ µ µ ( π θ µ µ i,j ( dµ c(µ θ µ j d µ ( c(µ dw + c(µ ( λ(µ θ µ d.
8 CUCHIERO, SCHACHERMAYER, WONG he finie variaion par of he epression vanishes for every π Π if we choose θ Π such ha ( θ (4.4 c(µ λ(µ = 0, P-a.s. for all 0. µ By passing from he scaled relaive weighs θ/µ o ordinary porfolio weighs via [6, Equaion (5], he generic soluion of (4.4, which we denoe by π 8, is given by d (4.5 π i = µ i λ i (µ + µ j λ j (µ. Le V be he associaed wealh process. From (4.4, he raio V π / V is, for any π Π, a non-negaive local maringale and herefore a supermaringale. Hence V yields he relaive wealh process corresponding o he log-opimal porfolio (see e.g., [26, 6]. Indeed, by he supermaringale propery and Jensen s inequaliy [ ] [ ( V E log(v π π log( V = E log V j= ] log ( E [ ] V π 0. V hus E[log(V π] E[log( V ] for all π Π. By (4.3, he epeced value of he log-opimal porfolio is given by [ ] sup E[log V π ] = π Π 2 E λ (µ c(µ λ(µ d. 0 So far we have opimized over all sraegies in Π. In he sequel we shall mainly consider suprema aken over smaller ses, in paricular over FG M,α. Noe ha in his case he opimizer will sill be a funcion of he marke weighs due o he Markov propery of (µ 0. In his cone le us also answer he quesion of when he log-opimal porfolio is funcionally generaed. his is needed o relae is asympoic growh rae o he one of he bes rerospecively chosen porfolio and he universal porfolio. Proposiion 4.7. Le (µ 0 be of he form (4.8. hen he log-opimal porfolio is generaed by a differeniable funcion G, i.e., π i = µ i Di G(µ d + µ j D j G(µ, i =,..., d, G(µ G(µ if he drif characerisic λ saisfies j= λ( = log G( = G( G(, d. Proof. he asserion follows from epression (4.5. 8 By a sligh abuse of noaion, we here wrie π alhough we do no resric o long-only porfolios as in Secion 2.3.
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 9 4.2.4. Asympoic growh raes for an ergodic marke weighs process. Assumpion 4.8. he process µ as given in (4.8 is an ergodic process wih saionary measure ϱ on. Wih his assumpion we derive an epression of he asympoic growh rae log V π. For he precise noion of ergodiciy in coninuous ime we refer o [4, Secion 2.2., heorem 2.4 and Secion 2.2.3]. Assumpion 4.8 is essenially saisfied under a mean reversion condiion. Eamples include polynomial models for he marke weighs saying in he inerior of he simple (see [5, heorem 5.] wih he subclass of volailiy sabilized models [8]. In he following heorem we consider porfolio maps which are no necessarily long-only, bu can ake values in he hyperplane H d. heorem 4.9. Under Assumpion 4.8 he following saemens hold rue: (i Le π : H d be any (ϱ-measurable porfolio map such ha ( π( c(λ( ϱ(d <, d ( ( π( π( (4.6 Q π := c( ϱ(d <. We hen have, for ϱ-a.e. saring value µ 0, ha ( π( log(v π = L π := c(λ(ϱ(d d ( ( π( π( c( ϱ(d, P-a.s. 2 d (ii Assume ha L := 2 λ (c(λ(ϱ(d <. hen, for ϱ-a.e. saring value µ 0, i holds ha log V = L, P-a.s. he proof of heorem 4.9 relies on he following lemma which is saed and proved in [7, Lemma.3.2]. Lemma 4.0. Le M be a coninuous local maringale such ha (4.7 2 M, M log log = 0, P-a.s. hen M = 0, P-a.s. Proof of heorem 4.9. Le us sar by proving saemen (i. By (4.2, log V π reads as (4.8 log V π = + 0 he local maringale par ( π(µ c(µ λ(µ d µ 2 ( π(µ c(µ dw. 0 µ M π := 0 0 ( π(µ ( π(µ c(µ dw µ µ c(µ π(µ µ d
20 CUCHIERO, SCHACHERMAYER, WONG saisfies Condiion (4.7 of Lemma 4.0 below. Indeed, by he ergodic heorem in coninuous ime (see e.g.,[4, heorem 2.4 and Secion 2.2.3] and (4.6 we have M π, M π = ( π(µ c(µ π(µ d Q π <, P-a.s. µ 0 µ Muliplying he lef hand side wih (log log /, herefore yields Condiion (4.7 and M π = ( π(µ c(µ dw 0, P-a.s. 0 µ Hence, evoking again he ergodic heorem yields ( log V π ( π(µ = c(µ λ(µ d ( π(µ c(µ π(µ d 0 µ 2 0 µ µ ( π( = c(λ(ϱ(d ( ( π( π( c( ϱ(d, 2 P-a.s. (and also in L (Ω, F, P and hus asserion (i. Concerning saemen (ii, noe from (4.4 ha he scaled relaive weighs corresponding o he log-opimal porfolio saisfy ( π( c( λ( = 0. hus, by (4.8 and (4.2, log V simplifies o log V = 2 In his case we have M π, M π = 0 λ (µ c(µ λ(µ d + which yields by he same argumen as above M π = 0 0 0 λ (µ c(µ dw. λ (µ c(µ λ(µ d 2L, P-a.s., λ (µ c(µ dw 0, P-a.s. and in urn log V = λ (µ c(µ λ(µ d = L, 2 0 P-a.s. 4.3. Asympoically equivalen growh raes. As in discree ime we will esablish asympoic equaliy of he growh raes of all hree porfolio ypes inroduced in Secion 4.2. Firs we compare he bes rerospecively chosen porfolio wih he universal one. For an analogous resul in he cone of opimal arbirage see heorem 4.5 of Kardaras and Roberson [27]. heorem 4.. Le M, α > 0 be fied and le (µ 0 be a coninuous pah saisfying Assumpion 4.2 such ha for all i {,..., d} (4.9 [µi, µ i ] <.
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 2 Consider a probabiliy measure m on G M,α wih full suppor and se ν = F m wih F defined in (4.5. hen,m,α (log V log V M,α (ν = 0, where V,M,α and V M,α (ν are defined in (4.4 and (4.7 respecively. Proof. he inequaliy is obvious. For he converse inequaliy we proceed similarly as in he previous secion (using only generaing funcions. As m has full suppor and G M,α is compac, we have ha, for η > 0 here eiss some δ > 0, such ha every η-neighborhood of a poin G G M,α has m-measure bigger han δ. Le and denoe by G he opimizer as of Proposiion 4.5. Consider now a generaing funcion G such G G C 2,0 η. hen i follows from (A.9 ha (4.20 ( log(v G log(v G =: K. ( ( M 2Mη 2 d2 η + M 3 2 d2 η ma[µ i, µ i ] i Fi ɛ > 0 and noe ha by assumpion (4.9 and coninuiy of [ui, u i ] on [,, sup [, [µi, µ i ] can be bounded by some consan. herefore we can choose η sufficienly small such ha K ɛ for all. Denoe by B = B η (G he C 2,0-ball wih radius η in GM,α which has m- measure a leas δ > 0, where δ only depends on η. We hen may esimae using Jensen s inequaliy and (4.20 ( V M,α ( (ν V G G = M,α m(dg ( B η(g V Gm(dG V G V G δ B η(g (V G m(dg (V G V G δ e K δ e ɛ. Leing for any given ɛ (which deermines η and in urn δ yields he asserion. o compare he asympoic performance wih ha of he log-opimal porfolio, we opimize over porfolio maps in FG M,α and suppose henceforh ha (µ 0 is of he form (4.8. Under Assumpion 4.8 and from heorem 4.9 define ( ( π π M,α G ( := arg ma c(λ(ϱ(d π G FG M,α (4.2 d ( π G ( ( π G ( c( ϱ(d 2 and he corresponding wealh process V M,α by V M,α = V πm,α, whenever π M,α is well defined. As ] sup E [log(v πg π G FG M,α yields π M,α as opimizer for all > 0, V M,α corresponds o he log-opimal porfolio among funcionally generaed porfolios wih generaing funcion in G M,α.
22 CUCHIERO, SCHACHERMAYER, WONG heorem 4.2. Le M, α > 0 be fied and le (µ 0 be a sochasic process of he form (4.8 saisfying Assumpion 4.8. Moreover, suppose ha (4.22 c ii (ϱ(d <, for all i {,..., d}, d (4.23 ma i {,...,d} (c(λ(i ϱ(d <. Consider a probabiliy measure m on G M,α wih full suppor and se ν = F m wih F defined in (4.5. hen (4.24 inf,m,α log V = inf M,α M,α log V (ν = log V, P-a.s. M,α where V denoes he log-opimal porfolio among FG M,α -maps defined via (4.2, V,M,α and V M,α (ν are defined pahwise in (4.4 and (4.7 respecively. Proof. We firs noe ha π M,α is well-defined. Indeed, he map ( π G ( G c(λ(ϱ(d ( π G ( ( π G ( c( 2 d ( d G( = c(λ(ϱ(d ( ( G( G( c( G( 2 G( G( ϱ(d ϱ(d is coninuous from (G M,α, 2,0 o R. his ogeher wih compacness of G M,α wih respec o 2,0 imply he well-definedness of π M,α. Noe also ha (4.22 and (4.23 as well as he condiions on G imply he assumpions of he ergodic heorem (heorem 4.9. Hence, we have for each π G FG M,α he P-a.s. i log V π G = L πg. In paricular, M,α log V = sup L π G (4.25 =: L M,α π G FG M,α holds P-a.s. Due o (4.22, we can now apply heorem 4. which implies he firs equaliy in (4.24. Moreover, we have by he definiion of V,M,α for each fied he inequaliy M,α log( V,M,α (4.26 log(v, P-a.s. Using (4.25, (4.26 and heorem 4., we hus have P-a.s., (4.27 L M,α = M,α log( V inf,m,α log(v = inf M,α log(v (ν. On he oher hand, by he definiion of ( V M,α 0 as log-opimizer wihin he class FG M,α (4.28 E[log(V M,α (ν] sup E[log(V π G π G FG M,α ] = E[log( V M,α ] holds. Concerning he firs inequaliy, noe ha he universal porfolio o build he wealh V M,α (ν is given by (4.6. By he ime-homogenous Markovianiy i is hus
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 23 sufficien o dominae he lef hand side of (4.28 by aking he supremum over elemens in FG M,α. Combining now (4.28, heorem 4.9 and (4.27 yields, E[ inf M,α log(v (ν] inf = inf M,α E[log(V (ν] E[log( V M,α ] log( V M,α M,α log(v (ν, P-a.s., where he firs inequaliy follows from Faou s lemma. From his we see ha inf is P-a.s. consan and equal o M,α log(v (ν M,α log( V. Hence he asserion is proved. As in he previous secion we can formulae a resul no depending eplicily on he consan M on α. Seing α = M we choose for M =, 2, 3,... a measure mm on G M, M wih full suppor. Define m = M= 2 M m M and he process V (ν by V (ν = V G m(dg. M= GM, M In order o compare he performance wih he one of he global log-opimal porfolio, whenever i is funcionally generaed, we combine he above resuls wih Proposiion 4.7. Corollary 4.3. Le (µ 0 be a sochasic process of form (4.8 saisfying Assumpion 4.8. Moreover, suppose ha λ and c saisfy (4.22 and (4.29 (4.30 λ( = Ĝ( Ĝ(, L = Ĝ( 2 Ĝ( c( Ĝ( Ĝ( ϱ(d < for some concave funcion Ĝ C2 (. hen we have P-a.s. (4.3 M log(v,m, M = log(v (ν = log( V = L. Proof. Noe firs ha L is well defined due o (4.30. Furhermore, noe ha for every ε > 0, here eiss some M > 0 and some funcion G G M, M such ha log(v G log( V + ε. Indeed his simply follows from coninuiy of G V G as assered in Lemma 4.4 and by choosing G G M, M close enough wih respec o he C 2,0 o he opimizing funcion Ĝ C2 ( whose generaed porfolio yields V due o (4.29 and
24 CUCHIERO, SCHACHERMAYER, WONG Proposiion 4.7. By heorem 4.2, we can herefore conclude (following he proof of Corollary 3. ha (4.32 inf M log(v,m, M = inf holds rue. As heorem 4. implies ha sup he asserion is proved if (4.33 sup log V (ν = log(v (ν = log( V = L sup M (log V (ν log V = sup log log V,M, M, ( V (ν = 0, P-a.s. V By he consideraions of Secion 4.2.3 (see also [2, Proposion 4.3], i follows ha ( V(ν V 0 is a non-negaive supermaringale. I converges P-a.s. o a finie i as. his in urn implies (4.33 and proves he saemen. Finally, a similar resul can be obained by resricing he log-opimal porfolio o he class of C 2 -funcionally generaed porfolios wihou imposing he drif condiion in Proposiion 4.7. We denoe by V fun he wealh process of he log-opimal porfolio among concave C 2 -funcionally generaed porfolios, i.e., π fun is defined as in (4.2, however by aking he arg ma over all concave C 2 -funcionally generaed porfolios. Corollary 4.4. Le (µ 0 be a sochasic process of form (4.8 saisfying Assumpion 4.8. Moreover, suppose ha (4.22 and (4.23 hold rue. hen (4.34 inf M log V,M, M = inf log V fun (ν = log V, P-a.s. Proof. he proof is he same as he firs par of Corollary 4.3 up o (4.32. Noe ha we canno ge rid of he inf because he supermaringale argumen from he proof of Corollary 4.3 does no hold. Appendi A. Proofs of cerain resuls and lemmas Proof of heorem 2.. Fi > 0 and he rajecory s = (s,..., s d =0 (R d +. For fied s he funcion b V (b(s is coninuous on. Hence here is b = b(s such ha (A. V (s = V ( b(s. In fac, condiion (2.8 implies ha he sequence of funcions (b log V (b = is Lipschiz on, uniformly in N and s saisfying (2.8 for some fied consans C > c > 0. Indeed, consider he disance on defined by b b = d j= bj b j. hen we may esimae log V (b log V ( b (log(c log(c b b.
UNIVERSAL PORFOLIOS, SP AND HE NUMÉRAIRE PORFOLIO 25 For ɛ > 0 we may herefore define δ := cɛ C > 0 such ha, for every δ-neighborhood U( b around any b we have log V (b log V ( b ɛ, for every b U( b. If he probabiliy measure ν has full suppor, we also may find η = η(ɛ, c, C > 0 such ha each such δ-neighborhood U( b, where b runs hrough, saisfies ν(u( b > η. Using (A. we herefore may conclude, similarly as in (A.8, ha (2.7 holds rue, uniformly in s = (s,..., s d =0 saisfying (2.8 for some fied consans C > c > 0. Proof of Lemma 3.8. Le π : be he opimizer of (3.3 and define, for 0 < ɛ <, ( π ɛ = ( ɛ π + ɛ d,...,. d Noe ha π ɛ akes values in ɛ (see Definiion 3., which is crucial for he subsequen argumens and he reason why we do no direcly work wih π. Also noe ha, for p ɛ, we have p, y d (A.2 = p j yj j ɛ d, j= for, y, as a leas one of he erms yj is greaer han or equal o one. j he average performance L πɛ defined via (3.7 for he porfolio map π ɛ is sill almos as good as he opimal average performance L L π : (A.3 [ L πɛ = log( π ɛ (, y ] ϱ(, dy dϱ( [ d log(( ɛ π(, y ] ϱ(, dy dϱ( d L + log( ɛ. o approimae π ɛ by a Lipschiz funcion π Lip aking is values in ɛ, we need some preparaion. By Assumpion 3.6 we can find δ > 0 such ha, for A, [ (A.4 (log( ɛ A log( π ɛ(, y ] ϱ(, dy dϱ( > ɛ, d provided ha ϱ[a] < δ. In paricular, we may find η > 0 such ha [ (A.5 (log( ɛ log( π ɛ(, y ] dϱ(, y dϱ( > ɛ. d \ η Now we find a Lipschiz funcion π Lip : ɛ such ha d (A.6 π Lip ( π ɛ ( = π Lip ( j π ɛ ( j < ηɛ 2, j= for all \A, where he ecepional se A saisfies ϱ[a] < δ. Indeed, he funcions from R d ɛ which are coninuously differeniable in a neighborhood of are dense wih respec o he L (R d, ϱ; R d -norm. Le M be a Lipschiz consan for π Lip such ha M ɛ.