Opmal Invesmen under Relave Performance Concerns Glles-Edouard ESPINOSA and Nzar TOUZI Cenre de Mahémaques Applquées Ecole Polyechnque Pars Absrac We consder he problem of opmal nvesmen when agens ake no accoun her relave performance by comparson o her peers. Gven N neracng agens, we consder he followng opmzaon problem for agen, N: sup EU λ XT π π A + λ X π T X,π T, where U s he uly funcon of agen, π hs porfolo, X π hs wealh, X,π he average wealh of hs peers and λ s he parameer of relave neres for agen. Togeher wh some mld echncal condons, we assume ha he porfolo of each agen s resrced n some subse A. We show exsence and unqueness of a Nash equlbrum n he followng suaons: - unconsraned agens, - consraned agens wh exponenal ules and Black-Scholes fnancal marke. We also nvesgae he lm when he number of agens N goes o nfny. Fnally, when he consrans ses are vecor spaces, we sudy he mpac of he λ s on he rsk of he marke. Inroducon The semnal papers of Meron [34, 35] generaed a huge leraure exendng he opmal nvesmen problem n varous drecons and usng dfferen echnques. We refer o Plska [36], Cox and Huang [7] or Karazas, Lehoczky and Shreve [24] for he complee marke suaon, o Cvanc and Karazas [8] or Zarphopoulou [39] for consraned porfolos, o Consanndes and Magll [6], Davs and Norman [], Shreve and Soner [37], Duffe Research suppored by he Char Fnancal Rsks of he Rsk Foundaon sponsored by Socéé Générale, he Char Dervaves of he Fuure sponsored by he Fédéraon Bancare Françase, and he Char Fnance and Susanable Developmen sponsored by EDF and Calyon. We are graeful o Jérôme Lebuchoux for nang hs sudy as well as for hs helpful commens and advces. We also hank Jean-Mchel Lasry for hs suppor and for many neresng dscussons and remarks, Chrsoph Fre and Gonçalo dos Res for many helpful dscussons as well as ponng ou some msakes, and fnally, Ramon van Handel and Ronne Srcar for some neresng dscussons.
and Sun [3] or Akan, Menald and Sulem [2] for ransacons coss, o Consanndes [5], Joun, Koehl and Touz [22, 23], Damon, Spa and Zhang [9] or Ben Tahar, Soner and Touz [3, 4] for axes, and o He and Pearson [9, 2], Karazas, Lehoczky, Shreve and Xu [25], Kramkov and Schachermayer [28, 29] or Kramkov and Srbu [3] for general ncomplee markes. However, n all of hese works, no neracon beween agens s aken no accoun. The mos naural framework o model such neracon would be a general equlbrum model where he behavour of he nvesors are coupled hrough he marke equlbrum condons. Bu hs ypcally leads o unracable calculaons. Insead, we shall model he neracons based on some smplfed conex of comparson of he performance o ha of he compeors or o some benchmark. A reurn of 5% durng a crss s no equvalen o he same reurn durng a fnancal bubble. Moreover, human bengs end o compare hemselves o her peers. In fac, economc and socologcal sudes have emphaszed he mporance of relave concerns n human behavors, see Veblen [38] for he socologcal par, and Abel [], Gal [7], Gomez, Presley and Zapaero [8] or DeMarzo, Kanel and Kremer [] for economc works, consderng smple models n dscree-me frameworks. In hs paper, we sudy he opmal nvesmen problem under relave performance concerns, n a connuous-me framework. More precsely, here are N parcular nvesors ha compare hemselves o each oher. Agens are heerogeneous dfferen uly funcons and dfferen consrans ses and nsead of consderng only hs absolue wealh, each agen akes no accoun a convex combnaon of hs wealh wh wegh λ, λ [, ] and he dfference beween hs wealh and he average wealh of he oher nvesors wh wegh λ. Ths creaes neracons beween agens and herefore leads o a dfferenal game wh N players. We also consder ha each agen s porfolo mus say n a se of consrans. In he conex of a complee marke suaon where all agens have access o he enre fnancal marke, we prove exsence and unqueness of a Nash equlbrum for general uly funcons. The opmal performances a equlbrum are explc, and herefore allow for many neresng qualave resuls. We nex urn o he case where he agens have dfferen access o he fnancal marke,.e. her porfolo consrans ses are dfferen. Our soluon approach requres o resrc he uly funcons o he exponenal framework, Then, assumng manly ha he agens posons are consraned o le n closed convex subses, and ha he drf and volaly of he log prces are deermnsc, we show he exsence and unqueness of a Nash equlbrum, usng he BSDE echnques nroduced by El Karou and Rouge [4] and furher developed by Hu, Imkeller and Muller [2]. The Nash equlbrum opmal posons are more explc n he case of consrans defned by lnear subspaces. In hs seng, we analyze he lm when he number of players N goes o nfny where he suaon consderably smplfes n he spr of mean feld games, see Lasry and Lons [3]. Noce ha our problem does no f n he framework of [3] for he wo followng reasons. Frs, n [3] he auhors consder smlar agens, whch s no he case n he presen paper, as he uly funcons, he parameers λ s and he ses of consran can be specfc. More mporanly, n [3], he sources of randomness of wo dfferen agens are ndependen. We fnally nvesgae he mpac of he neracon coeffcen λ. Under some addonal 2
assumpons, whch are sasfed n many examples, we show ha he local volaly of he wealh of each agen s nondecreasng wh respec o λ. In oher words, he more nvesors are concerned abou each oher λ large, he more rsky s he equlbrum porfolo of each nvesor. However n general, hs can fal o hold. Bu n he lm N goes o nfny, he same phenomenon holds for he average porfolo of he marke, whou any addonal assumpon. Roughly speakng, hs means ha he global rsk of he marke ncreases wh λ, alhough can fal for he porfolo of some specfc agen. Fnally, le us menon ha an earler verson of hs paper conaned n he PhD hess of he frs auhor [5] movaed a very neresng work by Dos Res and Fre [2]. In parcular, [2] hghlghs he dffculy n he exsence and unqueness of he quadrac muldmensonal backward SDE of he presen paper, and esablshed he exsence of a sequenally delayed Nash equlbrum n he general case. Ths paper s organzed as follows. Secon 2 nroduces he problem. In secon 3, we solve he complee marke suaon, for general uly funcons. In secon 4, we deal wh he general case wh exponenal uly funcons and porfolos ha are consraned o reman nsde closed convex ses. In secon 5, we resrc he ses of consrans o lnear spaces whch allows us n parcular o derve some neresng economc mplcaons. Noaons H 2 R m denoes he space of all predcable processes ϕ, wh values n R m, and sasfyng E ϕ 2 d <. The correspondng localzed space s denoed by H 2 loc Rm. When here s no rsk of confuson, we smply wre H 2 and H 2 loc. 2 Problem formulaon Le W be a d-dmensonal Brownan moon on he complee probably space Ω, F, P, and denoe by F = {F, } he correspondng compleed canoncal flraon. We assume ha F s generaed by W. Le T > be he nvesmen horzon, so ha [, T ]. Gven wo F-predcable processes θ akng values n R d and σ akng values n R d d, sasfyng: σ symmerc, defne posve, σ 2 d < + a.s, 2. and θ s bounded, d dp-a.e, 2.2 we consder a marke wh a non rsky asse wh neres rae r = and a d-dmensonal rsky asse S = S,..., S d gven by he followng dynamcs: ds = dags σ θ d + dw, 2.3 where for x R d, dagx s he dagonal marx wh -h dagonal erm equal o x. A porfolo s an F-predcable process {π, [, T ]} akng values n R d. Here π j s he amoun nvesed n he j-h rsky asse a me. Under he self-fnancng condon, he assocaed wealh process X π s defned by: X π = X + π r dags r ds r, [, T ]. 3
Gven an neger N 2, we consder N porfolo managers whose preferences are characerzed by a uly funcon U : R R, for each =,..., N. We assume ha U s C, ncreasng, srcly concave and sasfes Inada condons: U = +, U + =. 2.4 In addon, we assume ha each nvesor s concerned abou he average performance of hs peers. Gven he porfolo sraeges π, =,..., N, of he managers, we nroduce he average performance vewed by agen as: X,πj := Xπj. 2.5 The porfolo opmzaon problem of he h agen s hen defned by: V π j := V := sup E [U λ XT π + λ XT π X,πj π A = sup π A E [U X π T λ X,π j T T ], N, ] 2.6 where λ [, ] measures he sensvy of agen o he performance of hs peers, and he se of admssble porfolos A wll be defned laer. Roughly speakng, we mpose negrably condons as well as he consrans π akes values n A, a gven closed convex subse of R d. Our man neres s o fnd a Nash equlbrum n he conex where each agen s small n he sense ha hs acons do no mpac he marke prces S. Defnon 2. A Nash equlbrum for he N porfolo managers s an N uple ˆπ,..., ˆπ N A... A N such ha, for every =,..., N, gven ˆπ j, he porfolo sraegy ˆπ s a soluon of he porfolo opmzaon problem V ˆπ j. If n addon, for each =,..., N, ˆπ s a deermnsc and connuous funcon of [, T ], we say ha ˆπ,..., ˆπ N s a deermnsc Nash equlbrum. Our man resul s he followng: Man Theorem: Assume ha θ and σ are deermnsc and connuous funcons of [, T ], and ha for each =,..., N, U x = e x η for some consan η >, he porfolo consrans ses A are closed convex, and N = λ <. Then, here exss a unque deermnsc Nash equlbrum. In order o smplfy noaons, from now on, we wll wre X := X π and X := X,πj, [, T ]. In secon 3, we shall consder he complee marke suaon n whch he porfolos wll be free of consrans n oher words, A = R d for each. Ths wll be solved for general uly funcons. In he nex secons, we wll derve resuls for more general ypes of consrans, bu we wll focus on he case of exponenal uly funcons: U x = e x η. We wll frs consder he general case n secon 4, and hen n secon 5 we wll focus on he case of lnear consrans, where he A s are vecor subspaces of R d. 4
3 The complee marke suaon In hs secon, we consder he case where here are no consrans on he porfolos: A = R d, for all =,..., N. In he presen suaon, he densy of he unque equvalen marngale measure s: dq dp = T e θu dwu 2 θu 2du. 3. We shall denoe by E Q he expecaon under Q. In conras wh he general resuls n he subsequen secons, he complee marke suaon can be solved for general uly funcons. In hs case, he se of admssble sraeges A = A s he se of predcable processes π such ha: σπ H 2 loc Rd and X π s a Q-marngale. 3.2 To smplfy he noaons and presenaon n hs nroducory example, we also assume ha all agens have he same relave performance coeffcen λ: λ = λ [,, for all =,..., N, 3.3 see however Remark 3.. Neverheless, we allow he nvesors o have dfferen uly funcons U and dfferen nal endowmens x R. We denoe: x := xj, =,..., N. 3. Sngle agen opmzaon The frs sep s o fnd he opmal porfolo and wealh f hey exs of each agen, whle he sraeges of oher agens are gven. In oher words, we ry o fnd he bes response of agen o he sraeges of hs peers. As n he classcal case of opmal nvesmen n complee marke, we wll use he convex dual of U x. Snce U s srcly concave and C, we can defne I := U whch s a bjecon from R + ono R because of 2.4. The man resul of hs secon requres he followng negrably condons: for all y >, E U I y dq < and E Q I y dq <. 3.4 dp dp Lemma 3. For any =,..., N, le he sraeges π j A for j be gven. Then, under 3.4, here exss a unque opmal porfolo for he opmzaon problem 2.6 of agen wh opmal fnal wealh: X T = I y dq dp + λ X T, where y s defned by E Q I y dq dp = x λ x. 3.5 Proof. Usng he convex dual of U x, we have for any y > : U XT λ X T U I y dq dq + y XT λ dp dp X T + I y dq. dp The rgh-hand sde s negrable under P by he admssbly condons 3.2 and he negrably assumpons 3.4. The res of he proof s omed as follows he classcal marngale approach n he smple complee marke framework. 5
3.2 Paral Nash equlbrum The second sep s o search for a Nash equlbrum beween he N agens. Le X N := XT N be he vecor of ermnal wealh of he nvesors assocaed o π,..., π N. From Lemma 3., π,..., π N s a Nash equlbrum f and only f we have: λ A N X N = J N, where A N = M N R; J N = I y dq. λ dp N Under he condon λ n 3.3, follows ha A N s nverble and we can compue explcly ha: A N = λ + 2 λn+λ λ λn+λ λ λ λn+λ + 2 λn+λ hus provdng he exsence of a unque Nash equlbrum: Theorem 3. There exss a unque Nash equlbrum, and he equlbrum ermnal wealh for each =,..., N s gven by: ˆX T = λ + 2 λn+λ I y dq λ dp + λn+λ I j y j dq dp. Remark 3. In he case of specfc λ s, he prevous argumens can be adaped. In he expresson of A N, λ appears on he -h lne nsead of λ, A N s nverble f and only f N = λ < for more deals, see he proof of Lemma 4.3 below and hen s nverse s gven by:, A N = + λ N k k λ N k +λ N k λ N k +λn +λ N k, and A N j = k λ N +λ N j λ N k +λn +λ N k for j, where we denoed λ N := λ /N. The equlbrum performances are gven by ˆX T = N j= A N ji j y j dq dp, =,..., N. Remark 3.2 In he case λ =, urns ou ha here exs eher an nfny of Nash equlbra or no Nash equlbrum. Indeed, n hs case, A N s of rank N 2. Therefore f J N belongs o he mage of A N, hen here s an affne space of dmenson one of Nash equlbra, whle f J N s no n he mage of A N, hen here s no Nash equlbrum. In parcular, n he exponenal uly conex furher developed below, we drecly compue ha J N = A N x+ 2 η θ θd+dw, where x s he vecor of nal daa x and η s he vecor of rsk olerances η of each agen. Therefore J N belongs o he mage of A N f and only f η belongs o. 6
3.3 The exponenal uly case In order o push furher he analyss of he complee marke suaon, we now consder he exponenal uly case: U x = e x η, x R, 3.6 where η > s he rsk olerance parameer for agen,.e. he nverse of hs absolue rsk averson coeffcen. We denoe he average rsk olerance by: η N := N N j= η j. 3.7 In he presen conex, I y = η lnη y, so ha he equlbrum wealh process s: ˆX T = a η dq λ ln dp where a = x + η λ EQ ln dq dp. We denoe by ˆπ,N,λ he correspondng equlbrum porfolo sraegy of agen, where we emphasze s dependence on he parameers N and λ. In order o have explc formulas, we assume ha he rsk premum θ s a deermnsc connuous funcon of. Then, s well-known ha he classcal porfolo opmzaon problem wh no neracon beween managers leads o he opmal porfolos ˆπ, := η σ θ, [, T ]. Proposon 3. In he above seng, he equlbrum porfolo for agen s gven by: [ ] ˆπ,N,λ = k,n λ ˆπ, where k,n λ := λ λn N+λ + λn η N N+λ η. Remark 3.3 Assume furher ha η N η > as N. Then k,n λ + λ parcular, f all agens have he same rsk averson coeffcen η = η >, hen: ˆπ,N,λ = ˆπ λ := λ ˆπ. for all. η λ η. In Remark 3.4 In he case of smlar agens,.e. for any =,..., N, η = η and λ = λ, we can fnd he equlbrum porfolo very easly. Indeed, by symmery consderaons, all he X s mus be equal, X = X, and he opmzaon problem reduces o: sup π Ee λ η X T η λ Ths s he classcal case wh η replaced by, so ha he opmal porfolo s gven by ˆπ = ησ θ/ λ, n agreemen wh our resuls. In he general case of he followng secons, we wll no always be able o conclude anyhng on he behavor of every agen, herefore we nroduce he followng defnon: Defnon 3. The marke ndex and he correspondng marke porfolo are defned by: X := N N = X and π := N N = π, [, T ]. 7
We recall he defnon of he Sharpe rao SR and nroduce he varance rsk rao VRR: SR = expeced excess reurn, VRR := volaly expeced excess reurn. 3.8 varance For praccal purposes, he VRR s a beer creron for he wo followng reasons: VRR s robus o he nvesmen duraon, whle SR s no: for a me perod L and a scalar k >, we have SRkL = k SRL, whle VRRkL = VRRL. VRR accouns for he llqudy rsk relaed o he sze of he poson, whle SR does no: for a porfolo X and a scalar k >, we have SRkX = SRX and VRRkX = VRRX/k. We have he followng resuls for he mpac of λ: Proposon 3.2 For any lnear form ϕ, ϕˆπ,n,λ s ncreasng w.r. λ. The dynamcs of he marke ndex and he correspondng marke porfolo are gven by: d X = η N λ θ [θd + dw ] and π = η N λ σ θ. In parcular, for any lnear form ϕ, ϕ π s ncreasng w.r. λ. Proof. s mmedae, so we only prove. By Proposon 3., ˆπ,N,λ = k,n λ ˆπ,, and we decly compue ha: [ k,n λ λ = N + λ + N λ + N λ ] η N λ 2 N+λ 2 η. By defnon of η N n 3.7 and he fac ha η j for all j, we have η N η Therefore: N N. λ 2 N + λ 2 k λ λ NN + λ λn N 2 λ N λ + λn 2 N + >. In words, Proposon 3.2 saes ha he more nvesors are concerned abou each oher, he more rsk hey wll underake. In each nvesmen drecon, he global poson of agens, descrbed by ϕ π, wll ncrease wh λ and n he lm λ, we even have a lm of nfne posons ϕ π a.s. Furhermore, he drf and volaly of he marke ndex are boh ncreasng w.r. λ. The correspondng Sharpe rao s SR = θ, ndependen of λ, whle he varance rsk rao s VRR = λ η N, a decreasng funcon of λ. Ths s a perverse aspec of he presen fnancal markes whch may provde an explanaon of he emergence of fnancal bubbles, when managers use he Sharpe rao as a relable ndcaor. 8
3.4 General equlbrum In he prevous secons, he prce process S was gven exogeneously. We now analyze he effec of he relave performance coeffcen λ when he prce process S s deermned a he equlbrum. For each fxed prce process S, defned as n Secon 2, here exss a unque Nash equlbrum n he sense of Defnon 2.. Smlar o Karazas and Shreve [27], our objecve s o search for a marke equlbrum prce S whch s conssen wh marke equlbrum condons: N = π,j = K j S j for all j =,..., d and [, T ], 3.9 N = x = d j= Kj S j, 3. where K j s a consan such ha K j S j s he marke capalzaon of he j-h frm. Equaon 3.9 says ha he oal amoun nvesed n he socks of he j-h frm s equal o he marke capalzaon of hs frm. Equaon 3. says ha he nal endowmen of he nvesors equals he nal marke capalzaons. Wh :=,..., T R d, we observe ha 3.9 and 3. mply ha N = X = N = x + π dags ds = N = x + d j= Kj ds j = N = x + d j= Kj S j Sj = d j= Kj S j = N = π,.e. he oal amoun nvesed n he non-rsky asse s zero a any me [, T ]. Defnon 3.2 We say ha a process S s an equlbrum marke f here exss a Nash equlbrum ˆπ = ˆπ,..., ˆπ N assocaed o he prce dynamcs S, n he sense of Defnon 2., such ha S and ˆπ sasfy 3.9 and 3.. In order o smplfy noaons, we se K j = k j N, and k := k,..., k d. Proposon 3.3 Le θ be a deermnsc and connuous funcon of [, T ]. Then here exss an equlbrum marke whose rsk premum s θ. Moreover, n hs equlbrum marke, he marke ndex s gven by: X = x + η N λ θ θd + dw, [, T ]. Proof. By Proposon 3.2, follows ha S = η N λ dagkσ θ. Noce ha he prevous equaon does no defne σ unquely for d >. Conversely, le θ be some gven connuous funcon. Then we can choose a dagonal marx σ = σ, S, wh dagonal elemens σ, S = η N θ. λk S 9
Noce ha σ sasfes he condons for S o be a srong soluon of 2.3. Then, follows from Proposon 3. ha: d X = N N = dx = N N = ˆπ dags ds = η N λ θ θd + dw. We nex analyze he mpac of λ on he drf and he volaly of he marke ndex. Despe he mulplcy of marke equlbra, hey all lead o he smlar conclusons. Le us for example assume ha he rsk premum s ndependen of λ. Then he drf of he marke ndex s η θ 2 / λ and he volaly s η θ / λ, hus boh are ncreasng w.r. λ, and wh he same order. We may nerpre hs equlbrum as a fnancal bubble, where he reurn and he volaly are boh ncreased by he agens neracons. An alernave nerpreaon for a fund manager s ha, for he same gven reurn, he agens neracon coeffcen ncreases he volaly of he opmal porfolo. Noce ha n he presen seng, he varance rsk rao VRR = λ/η s decreasng n λ and ends o zero as λ. Ths ndcaes ha, accordng o hs creron, he agens neracons lead o marke neffcency. 4 General consrans wh exponenal uly In he res of hs paper, we consder a general case wh consraned porfolos. We assume: A s a closed convex se of R d, for all =,..., N. 4. We denoe by P he orhogonal projecon on σ A, whch s well-defned by 4.. For x R d, we denoe dsx, σ A := x P x he Eucldean dsance from x o he closed convex subse σ A. Remark 4. Recall ha for a closed convex se A n a Eucldean space, he orhogonal projecon on A, denoed P, s well-defned, s a conracon, and sasfes for any x, y R d : P x P y 2 x y P x P y x y 2. Moreover, P x s he only pon sasfyng x P x a P x for all a A. For echncal reasons, we resrc our analyss o exponenal uly funcons 3.6. Defnon 4. The se of admssble sraeges A s he collecon of all predcable processes π wh values n A, d dp a.e., such ha σπ H 2 loc Rd and such ha he famly { } e ±Xπ τ Xν π ; ν, τ soppng mes on [, T ] wh ν τ a.s 4.2 s unformly bounded n L p P for all p >. In comparson wh he admssbly condons of secon 3, he prevous defnon requres he unform boundedness condon of he above famly, whch s needed n order o prove a dynamc programmng prncple smlar o Lm and Quenez [32].
4. A formal argumen In hs secon, we provde a formal argumen whch helps o undersand he consrucon of Nash equlbrum of he subsequen secon. For fxed =,..., N, we rewre 2.6 as: V := sup E [U XT π ξ ], where ξ := λ X,π T =: λ x + ξ. 4.3 π A Then, followng El Karou and Rouge [4] or Hu, Imkeller and Müller [2], we expec ha he value funcon V and he correspondng opmal soluon be gven by: V = e x λ x Ỹ /η, σ ˆπ = P ζ + η θ for all [, T ], and Ỹ, ζ s he soluon of he quadrac BSDE: Ỹ = ξ + where he generaor f s gven by: ζ u θ u η 2 θ u 2 + f u ζ u + η θ u du ζ u dw u, T, 4.4 f z := 2η dsz, σ A 2, z R d. 4.5 Ths suggess ha one can search for a Nash equlbrum by solvng he BSDEs 4.4 for all =,..., N. However, hs rases he followng dffcules: - he fnal daa ξ does no have o be bounded as s defned n 4.3 hrough he performance of he oher porfolo managers; - he suaon s even worse because he fnal daa ξ nduces a couplng of he BSDEs 4.4 for =,..., N. To express hs couplng n a more ransparen way, we subsue he expressons of ξ and rewre 4.4 for = no: Ỹ = η ξ + f uζ udu ζu λ N Puζ j u j db u where B := W +. θ rdr s he Brownan moon under he equvalen marngale measure, λ N := λ N, ζ := ζ + η θ, [, T ], 4.6 and he fnal daa s expressed n erms of he unbounded r.v. ξ := θ u db u 2 Then Ỹ = Y, where Y, ζ s defned by he BSDE Y = η ξ + f uζ udu ζ u λ N θ u 2 du. 4.7 Puζ j u j db u. 4.8 In order o skech 4.8 no he BSDEs framework, we furher nroduce he mappng ϕ : R Nd R Nd defned by he componens: ϕ ζ,..., ζ N := ζ λ N P j ζj for all ζ,..., ζ N R d. 4.9
I urns ou ha he mappng ϕ s nverble under farly general condons. We shall prove hs resul n Lemma 4.3 for general convex consrans and n Lemma 5. n he case of lnear consrans. We denoe ψ := ϕ and ψz he -h block componen of sze d of z. Then one can rewre 4.8 as: ϕ Y = η ξ + where he generaor f s now gven by: f uz u du Z u db u, 4. f z := f ψ z for all z = z,..., z N R Nd. 4. A Nash equlbrum should hen sasfy for each : 4.2 Auxlary resuls ˆπ = σ P ψ Z, =,..., N. 4.2 Our frs objecve s o verfy ha he map ϕ nroduced n 4.9 s nverble. The crucal condon for he res of hs secon s: N λ <. 4.3 Recall he noaon λ N from 4.6. Lemma 4. Under 4. and 4.3, for any [, T ], he map I + λ N j P j on R d and s nverse s a conracon, for all j =,..., N. s a bjecon Proof. Le [, T ] be fxed, for ease of noaon, we om all subscrps. Snce σ A j s a closed convex se, from Remark 4., x y P j x P j y P j x P j y 2, for any x, y R d. Noce ha I + λ N j P j s a bjecon f and only f, for all y R d, he map f y x := y λ N j P j x 4.4 has a unque fxed pon. Snce P j s a conracon, we compue, for any x, x n R d : f y x f y x = λ N j P j x P j x λ N j x x = λ j x x. Case : If N 3 or λ j <, hen f y s a src conracon of R d. We prove now ha he nverse of I + λ N j P j s a conracon. Indeed f x y, we have: x y + λ N j P j x P j y 2 = x y 2 + λ N j 2 P j x P j y 2 + 2λ N j x y P j x P j y x y 2 >, 4.5 where we used he fac ha x y P j x P j y, see Remark 4.. Case 2: If N = 2 and λ j =, f y fals o be a src conracon. However, 4.5 sll holds, and mples ha I + P j s one-o-one. Usng Lemma 4.2 below, we ge he bjecon propery of I + P j and he conracon propery of he nverse funcon follows from 4.5. 2
Lemma 4.2 Le A be a closed convex se of R d. Then I + P A R d = R d. Proof. Le B := 2A = {y R d ; x A, y = 2x}, and le us prove ha P A y 2 P By = 2 P By for all y R d. 4.6 Ths mples ha y = I + P A y 2 P By I + P A R d for all y R d, whch gves he requred resul. To prove 4.6, defne x := 2 P By and z := y 2x. By Remark 4., P B y s he only pon n B sasfyng y P B y b P B y for all b B. In oher words, we have for any b B, z b 2x, or by defnon of B, for any a A, z 2a 2x. hence: x + z x a x for all a A, whch means ha x = P A x + z and herefore I + P A x + z = x + z + x = y. Recall he defnon of ϕ n 4.9. Lemma 4.3 Under 4. and 4.3, we have for [, T ]: ϕ s a bjecon of R Nd, and we wre ψ := ϕ. ψ s Lpschz connuous wh a consan dependng only on N and he λ s. Proof. For ease of noaon, we om all subscrps. For arbrary z = z,..., z N n R Nd, we wan o fnd a soluon ζ R Nd o he followng sysem: ϕ ζ = ζ λ N P j ζ j = z, N. 4.7 Subsracng λ j mes equaon o λ mes equaon j n 4.7, we see ha: λ I + λ N j P j ζ j = λ j I + λ N P ζ + λ z j λ j z,, j =,..., N. 4.8. From Lemma 4., we know ha I + λ N j P j s a bjecon, hus from 4.8, we compue: P j ζ j = λj λ P j I + λ N j P j I + λ N P ζ + λ z j λ j z, so ha, from 4.7: ζ = z + N P j I + λ N j P j λj I + λ N P ζ + λ z j λ j z =: g,z ζ. 4.9 2. We nex show ha, under Condon 4.3, g,z has a unque fxed pon. We have: I + λ N j P j x I + λ N j P j y 2 = x y 2 + 2λ N j x y P j x P j y + λ N 2 j P j x P j y 2 + 2λ N j + λ N 2 j P j x P j y 2 + λ N j 2 P j x P j y 2. 3
Therefore, P j I +λ N j P j s -Lpschz. Then, snce I +λ N +λ N P s +λ N -Lpschz: j g,z x g,z y λ j + λ N x y. Noce ha λ j +λ N j +λ N j + λ N maxλ, λ j, wh equaly f and only f λ = λ j. Therefore, Condon 4.3 mples ha K := λ j +λ N j + λ N <, where K depends only on N and he λ j s. Then, g,z s a src conracon and adms a unque fxed pon ha we wre {ψz}. I s hen mmedae ha ζ = ψz s he unque soluon of 4.7. 3. Fnally we prove ha ψ s Lpschz wh a consan dependng only on N and he λ j s. Le z, z 2 R Nd, from 4.9, we compue: ψz ψz 2 z z2 + K ψz ψz 2 + 2 sup z j zj 2. j N Snce K := sup j N K j <, we ge sup j N ψz j ψz 2 j 3 K sup j N z j zj 2, whch complees he proof snce K depends only on N and he λ j s. 4.3 The man resuls Smlar o he classcal leraure on porfolo opmzaon wh exponenal uly El Karou and Rouge [4], Hu, Imkeller and Muller [2], Mana and Schwezer [33], we frs esablsh a connecon beween Nash equlbra and a quadrac mul-dmensonal BSDE. Theorem 4. Under 4. and 4.3, le π,..., π N be a Nash equlbrum. Then: π = σ P ψ Z and V = e x η λ x Y, where Y, Z H 2 R N H 2 loc RNd s a soluon of he followng N-dmensonal BSDE: and ξ s defned by 4.7. Y = η ξ + I P 2η u ψuz u 2 du Zu db u, 4.2 Proof. See Secon 4.5. Unforunaely, he wellposedness of he BSDE 4.2 s an open problem n he presen leraure, hus prevenng Theorem 4. from provdng a characerzaon of Nash equlbra, see also [2]. Our second man resul focuses on he mul-dmensonal Black-Scholes fnancal marke, where we can guess an explc soluon o he BSDE 4.2. Alhough no unqueness resul s avalable for he BSDE 4.2 n hs conex, he followng complee characerzaon s obaned by means of a PDE verfcaon argumen. In vew of Lemma 4.3, under Condon 4.3, he maps ψ x := ψ η x,..., η N x for all x R d, =,..., N, [, T ], 4.2 are well-defned and Lpschz connuous on R d. 4
Theorem 4.2 Under 4. and 4.3, assume ha σ and θ are deermnsc connuous funcons. Then here exss a unque deermnsc Nash equlbrum: ˆπ = σ P ψ θ for all [, T ], 4.22 Moreover, he value funcon for agen a equlbrum s gven by: V = e x η λ x Y, Y = η θ 2 d + I P 2 2η ψ θ 2 d. Proof. See Secon 4.6. We conclude hs secon by wo smple examples. More neresng suaons wll be obaned laer under he addonal condon ha he consrans ses are lnear. Example 4. Common nvesmen Le σ = I d, λ = λ, η = η, and A = Bx, r for some x R d and r >, =,..., N. Here Bx, r s he closed ball cenered a x wh radus r > for he canoncal eucldean norm of R d. Usng Theorem 4.2, we compue he followng equlbrum porfolo: ηθ ηθ f ˆπ = P ηθ λ λ Bx, r λ = r x + ηθ ηθ λ x λ x oherwse. Noce n parcular ha, as one could expec, ˆπ x s colnear o ηθ λ x and ha ˆπ s n he boundary of Bx, r whenever ηθ λ Bx, r. One can prove ha ˆπ s nondecreasng w.r. λ and η. Noce also ha hs expresson s ndependen of N. Example 4.2 Specfc ndependen nvesmens Le σ = I d, λ = λ, η = η, and A = [a, b ]e, for some a b, =,..., N. Here e j, j d s he canoncal bass of R d. Usng Theorem 4.2, we compue he followng equlbrum porfolo for agen : ˆπ = P ηθ = a ηθ b. Ths s exacly he same expresson as n he classcal case wh no neracon beween managers. Hence, The equlbrum porfolo s no affeced by λ and N. Remark 4.2 Suppose ha he porfolo consrans ses A are no convex. Then, we have o face wo major problems. Frs, he projecon operaors A are no well-defned. Second, and more mporanly, he map ϕ may fal o be one-o-one or surjecve ono R Nd. The falure of he one-o-one propery means ha here could exs more han one Nash equlbrum. However he falure of he surjecvy ono R Nd, as llusraed by Examples 6. and 6.2 n he Appendx secon, would lead o a consraned N-dmensonal BSDE wh no addonal nondecreasng penalzaon process. Such BSDEs do no have soluons even n he case of Lpschz generaors, meanng ha here s no Nash equlbrum n hs conex. 5
4.4 Infne managers asympocs In he spr of he heory of mean-feld games, see Lasry and Lons [3], we examne he suaon when he number of mangers N ncreases o nfny wh he hope of geng some more explc qualave resuls wh behavoral mplcaons. In hs secon, we assume ha he number of asses d s no affeced by he ncrease of he number of managers, see however he examples of secon 5.3. We also specalze he dscusson o he case where he agens have smlar preferences and only dffer by her specfc access o marke. The followng resul s smlar o Proposon 5. n [6]. Therefore he proof s omed. Proposon 4. Le λ j = λ [, and η j = η > for all j. Assume N N = P U unformly on any compac subses, for all [, T ] resp. unformly on [, T ] K, for any compac subse K of R d. Then: ˆπ,N ˆπ, := σ P I U λi η θ + U for all [, T ] resp. unformly n [, T ]. 4.5 Proof of Theorem 4. Assume ha π,..., π N s a Nash equlbrum for our problem. Frs, by Hölder s nequaly, he admssbly condons for all =,..., N mply ha e X η T λ X T belongs o L p, for any p >. Le T be he se of all soppng mes wh values n [, T ], we defne he followng famly of random varables: [ J,π τ := E e η τ σuπu dbu λ X T x Fτ ], 4.23 V τ := ess sup π A J,π τ for all τ T so ha V = e η x λ x V. 4.24. By Lemma 4.4 below, he famly {V τ; τ T } sasfes a supermarngale propery. Indeed, le β,π := e η σuπu dbu for all π A, we have: β,π τ V τ Eβ,π θ V θ F τ for all soppng mes τ θ. Then, we can exrac a process V whch s càdlàg and conssen wh he famly defned prevously n he sense ha V τ = V τ a.s see Karazas and Shreve [26], Proposon I.3.4 p.6, for more deals. Moreover, hs process also sasfes he dynamc programmng prncple saed n Lemma 4.4, so ha for any π A, he process β,π V s a P-supermarngale. The defnon of a Nash equlbrum mples ha π s opmal for agen,.e. V = sup E e η X T π x λ X T x = E e X η T π x λ X T x, 4.25 π A whch mples ha he process β, π V s a square negrable marngale, as he condonal expecaon of a r.v. n L 2. 2. We now show ha he adaped and connuous process: γ := X π x + η ln β, π V, [, T ], 4.26 6
solves he requred BSDE. 2.a. Frs, by Jensen s nequaly, and he fac ha ln x x for any x >, we have: [ E XT π x λ η X ] T x X F ln β, π V E [ e η π T x λ X ] T x F. 4.27 By he admssbly condons, boh sdes of 4.27 belong o H 2, as condonal expecaons of random varables n L 2. Snce X π s also n H 2, we see ha γ s n H 2. Then, for all π A, we have ha: M,π := e η X π x γ = M e η X π X π, [, T ], where M = β, π V s a square negrable marngale. By Hölder s nequaly, follows ha e X η π X π L p for all p >. Then M,π s negrable. 2.b. In hs sep, we prove ha M,π s a supermarngale for all π A. Assume o he conrary ha here exss π A, s and A F s, wh PA > and such ha: E e X η π x γ F s > e X η s π x γ s on A, and le us work owards a conradcon. Defne: ˆπ u ω := π u ω {[s,t ] A} u, ω + π u ω {[s,t ] A c }u, ω. Snce A F s, usng Hölder s nequaly, we see ha ˆπ A and we have: [ { V E e X η ˆπ T x γt = E E e X η ˆπ T x γt }] F = E e X η ˆπ x γ by he fac ha ˆπ = π on [, T ]. Snce P[A] >, hs mples ha: [ { V E E e X η ˆπ x γ }] Fs > E e X η s ˆπ x γ s γ = e η = V, whch provdes he requred conradcon. 2.c. Snce M = β, π V s a marngale, follows from he marngale represenaon heorem ha M s an Iô process. Therefore 4.26 mples ha γ s also an Iô process defned by some coeffcens b and ζ : dγ = b d + ζ dw wh γ, ζ H 2 R H 2 loc Rd. 4.28 Moreover, by Jensen s nequaly, ln M, π s a supermarngale, and by 4.27 s bounded n L 2. Therefore adms a Doob-Meyer decomposon ln M, π = N + A, where N s a unformly negrable marngale and A a decreasng process. The marngale represenaon heorem hen mples ha here exss a process δ H 2 loc Rd such ha N = δ u dw u. Usng 4.27 and 4.28, we ge ζ = σ π + η δ. 2.d. We nex compue he drf of M,π. From he prevous supermarngale and marngale properes of M,π and M, respecvely, ogeher wh 4.28, we ge: b 2η σ π ζ + η θ 2 η 2 θ 2 ζ θ for all π A, and b = 2η σ π ζ + η θ 2 η 2 θ 2 ζ θ. 7
Ths mples ha: π = σ P ζ + η θ b = f, ζ = 2η dζ + η θ, σ A 2 η 2 θ 2 ζ θ, 4.29 and herefore γ, ζ H 2 R H 2 loc Rd s a soluon of he BSDE: dγ = ζ θ + η θ 2 I P ζ + η θ 2 d + ζ dw 2 2η γt = λ X T x = λ N π u j σ u dw u + θ u du. Recallng ha db = dw + θ d, we can wre : dγ = η θ 2 2 η I 2 P ζ + η θ 2 d + ζ db γt = λ X T x = λ N πj u σ u db u. 4.3 3. We fnally pu ogeher he N BSDEs obaned n Sep 2. Snce π,..., π N s a Nash equlbrum, equaon 4.3 holds for each =,..., N. Replacng he value of π j by 4.29 n he expresson of γ and wrng Γ := ζ + η θ, we see ha γ, Γ mus sasfy for each [, T ]: γ = λ N P j uγ j u db u η 2 θ u 2 du+ I P 2η uγ u 2 du so ha he adaped process Y := γ η 2 θ u 2 du+ η 2 θ u db u λ N [, T ], sasfes: Y = η ξ + I P 2η uγ u 2 du Γ u λ N wh Y, Γ H 2 R H 2 loc Rd. We fnally defne: Z := ϕ Γ = Γ λ N P j Γj. Γ u η θ u db u, P uγ j j u db u, PuΓ j j u db u. Under 4.3, usng Lemma 4.3, we know ha ϕ s nverble. As a consequence, Y, Z H 2 R N H 2 loc RNd s a soluon of he followng sysem of BSDEs: Y = η ξ + I P ψ 2η Z 2 d Z db. Moreover, for each, he equlbrum porfolo s gven by: σ π = P [ ψ Z ], [, T ]. The followng dynamc programmng prncple was used n Sep of he prevous proof. 8
Lemma 4.4 Dynamc Programmng For any soppng mes τ ν n T, we have: [ V τ = ess sup E e ν η τ σuπu,dbu V ν F τ ], =,..., N. π A Proof. Le τ ν T a.s. We frs oban by he ower propery ha: [ [ V τ = ess sup E E e η ν σuπu dbu λ X T x ] Fν e ν ] η τ σuπu dbu Fτ π A ess sup π A E [ e ν η τ σuπu dbu V ν F τ ]. To prove he converse nequaly, we fx π A and we observe ha J,π ν defned by 4.23 depends on π only hrough s values on ν, T. Therefore we have he deny: V ν = ess sup π A ν J,π ν, where A ν := {π A ; π = π on, ν, d dp-a.e}. We nex observe ha he famly {J,π ν, π A ν} s closed under parwse maxmzaon. Indeed, le π, π 2 n A ν, A := {ω Ω; J,π νω J,π 2 νω} and defne he process π := A π + Ω\A π 2. Snce π = π 2 = π on, ν, and A F ν, s mmedae ha π T. We compue Ee ± p η X π τ X π ν = Ee ± p η τ ϑ σπ db A + Ee ± p η τ ϑ σπ2 db Ω\A, so ha snce π, π 2 A ν, he famly {e ± η X π τ X π ϑ ; ϑ τ T } s unformly bounded n any L p, p >. Therefore, π A ν and s mmedae ha J,π ν = maxj,π ν, J,π 2 ν. Then follows from Theorem A.3, p.324 n Karazas and Shreve [27], ha here exss a sequence ˆπ n sasfyng: n, ˆπ n = π on, ν J,ˆπn ν s non-decreasng and converges o V ν. Then we have: J,ˆπn τ = E [J,ˆπn νe ν ] η τ σuπ u db u Fτ. Snce J,ˆπn ν s non-decreasng and converges o V ν, follows from he monoone convergence heorem ha: V τ E [e ν ] η τ σuπ u db u V ν F τ, and he requred nequaly follows from he arbrarness of π. 4.6 Proof of Theorem 4.2. We frs prove ha he porfolo 4.22 s ndeed a Nash equlbrum. The dea s o show ha we can make he formal compuaons of Secon 4. n he reverse sense..a. Le ξ := θu.db u 2 9 θu 2 du, [, T ], 4.3
Snce θ and σ are deermnsc and connuous funcons, he funcons P s are also deermnsc and connuous w.r.., z [, T ] R d. Le us prove ha he same holds for ψ, and herefore for ha ˆπ := σ P ψz s deermnsc and connuous w.r.. [, T ]. Indeed, s mmedae ha ϕ s deermnsc and connuous w.r.., ζ, so ha ψ s a deermnsc funcon of, z. Then, from Lemma 4.3, under Condon 4.3, ψ s Lpschz n z, unformly n, so ha here exss a consan K > such ha for all [, T ], and all z, z R d, ψ z ψ z z z. Le n, z R Nd, and ζ := ϕ z. We defne z n := ϕ n ζ for each n. Snce ϕ s connuous w.r., z n z, and we have, for all n, ψ n z n = ζ, so ha ψ n z ζ = ψ n z ψ n z n K z z n. Therefore ψ s connuous w.r... Then f z n z and n, we compue ψ n z n ψ z ψ n z n ψ n z + ψ n z ψ z, snce ψ s connuous w.r.. and Lpschz n z unformly n. As a consequence, we can defne he followng adaped and connuous processes: Z := η θ and Y := η ξ + 2η I P u ψ uz u 2 du, [, T ]. Then, we drecly verfy ha Y, Z sasfes he followng N-dmensonal BSDE: Y = η ξ + I P 2η u ψuz u 2 du Zu db u. Se: γ = Y + η 2 θu 2 du η θu db u + λ N ζ = ψ Z η θ = ψ η I θ. P j uψ j uz u db u By he same compuaons as n Secon 4., we see ha for all =,..., N, γ, ζ s a soluon of he -dmensonal BSDE: dγ = ζ θ + η θ 2 I P ζ + η θ 2 d + ζ dw 2 2η γt = λ N ˆπ u j σudw u + θudu. Then usng he defnon of ϕ and ψ we can rewre γ as: γ = η 2 = η 2 = η 2 θu 2 du + 2η θu 2 du + 2η θu 2 du + 2η I P u ψ uθu 2 du + λ N I P u ψ uθu 2 du + I P u ψ uθu 2 du + ζ u db u P j u[ψ u Z u ] j db u ψ u η I θu db u..b. Throughou hs sep, we fx an neger {,..., N}, and we defne: M π := e η X,π x γ for all π A. 2
By Iô s formula, follows ha M π s a local supermarngale for each π A, and M ˆπ s a local marngale. Then, here exs ncreasng sequences of soppng mes τ π n n T, such ha for each π, τ π n T a.s and for each n and any s : E[M π τ π n F s] M π s τ π n for all π A and E[M ˆπ τ ˆπ n F s] = M ˆπ s τ ˆπ n. 4.32 We nex nroduce he measure Q, equvalen o P, defned by s Radon-Nkodym densy: L = dq ψ η u I = e θu dw u 2 ψ η u I θu 2 du. 4.33 dp F We denoe by E he expecaon operaor under Q. Snce θ s a deermnsc and connuous funcon on [, T ], η 2 θu 2 du+ 2η I Pu ψ uθu 2 du s bounded. Then, for any π A : EM τ π n = [ L E e X η τn π x τn 2 θu 2 du+ 2η 2 τn I P u f u θu 2 du s ] e τn η ψ u I θu θudu+ 2 τn η ψ u I θu 2 du, 4.34 where we smply denoed τ n := τ π n. In 4.34, all he erms nsde he expecaon oher han e X η π τn are bounded. We shall prove n Sep.c below ha he famly {e X η π τ ; τ T } s unformly negrable under Q. Hence, he sequence of processes nsde he expecaon n 4.34 s unformly negrable under Q, and we may apply he domnaed convergence heorem o pass o he lm n, and we oban lm n EM τ π n = EM π. Togeher wh 4.32, hs mples ha: E e η X π x γ e η γ for all π A and E e η X ˆπ x γ = e η γ. Mulplyng by e x η λ x, we fnally ge V = e x η λ x Y, snce Y = γ, and ˆπ s opmal for agen. Hence ˆπ,..., ˆπ N s a Nash equlbrum..c. In hs sep, we prove ha he famly {Y τ := e X η τ,π : τ T } s Q unformly negrable for all π A. Fx some p >. Then by he admssbly condon, he famly {Y τ : τ T } s unformly bounded n L p P. Wh r := + p/2, follows ha he famly {Y r : τ T } s unformly negrable. Then for all c > and τ T, follows from Hölder s nequaly: E Q [Y τ Yτ c] = E [ L ] T Y τ Yτ c L T L q P Yτ r Yτ c L r P. where q s defned by /q + /r =. Snce {Y r : τ T } s unformly negrable, he las erm unformly goes o as c. 2. We now prove unqueness by usng a verfcaon argumen. 2.a. Le π,..., π N be a deermnsc Nash equlbrum, and defne for all =,..., N: u, x, y := e η x λ y 2 θu 2 du+ 2η 2 I Pu η θu+λ σu π N u 2 du 4.35 2
where π N u := πj u. Snce π j s a connuous funcon for all j =,..., N, he funcons u are C n he varable. Drec calculaon reveals ha u s a classcal soluon of he equaon: u sup p A L p u = and u T, x, y = e x λ y/η where for all p A, L p s he lnear second order dfferenal operaor: L p := σ π N θ y + σ π 2 N 2 yy 2 +σp.θ x + σp σ π N 2 xy + 2 σp 2 2 xx, and he supremum s aaned a a unque pon π := σ P η θ + λ σ π N. 4.36 2.b. In hs sep, we prove ha u, X, X = V. Frs, by Iô s formula we have for all π A : u, x, y = u τ n, Xτ π n, X τn τ n L π u r, Xr π, X τn rdr π π N r σrdwr, 4.37 where τ n := nf{r, X π r x n or X r x n}. Takng condonal expecaons n 4.37, and usng he fac ha L π u for any π A, we ge: u, x, y E,x,y u τ n, X π τ n, X τ n for all π A. 4.38 Snce he π j s, σ and θ are connuous deermnsc funcons and π A, follows from Hölder s nequaly ha {e η X τ λ X τ, τ T } s unformly bounded n any L p. By he defnon of U, hs propery s mmedaely nhered by he famly {u τ, X π τ, X τ, τ T }. Therefore, akng he lm n n 4.38, we ge u, x, y E,x,y e X η π T λ X T. By he arbrarness of π A, hs mples ha u, X, X V. We nex observe ha π A and he nequaly n 4.38 s urned no an equaly f π s subsued o π. By he domnaed convergence heorem, hs provdes: u, x, y = E,x,y e η X π T λ X T, whch, n vew of 4.38, shows ha u, X, X = V. 2.c. To see ha he connuous deermnsc Nash equlbrum s unque, consder anoher connuous deermnsc Nash equlbrum ˆπ,..., ˆπ N, and denoe by û he correspondng value funcons as n 4.35. I suffces o observe ha L π û < on any non-empy open subse B of [, T ] such ha π π on B, and he nequaly 4.38 s src. Therefore, any Nash equlbrum mus sasfy 4.36 for every =,..., N. 22
Se ˆγ := σˆπ, and le ˆγ be he marx whose -h lne s ˆγ. From he prevous argumen, ˆπ,..., ˆπ N s a Nash equlbrum f and only: Γ ˆγ := P η θ + λ N = ˆγ, =,..., N, [, T ], 4.39 ˆγ j.e. ˆγ s a fxed pon of Γ for all [, T ]. Usng Lemma 4.5 below, we have he unqueness of a Nash equlbrum. Fnally, he expresson for V a equlbrum follows from he las saemen of Lemma 4.5 ogeher wh 4.35. Recall he funcon ψ defned n 4.2. Lemma 4.5 Under 4.3, he funcon Γ defned n 4.39 has a unque fxed pon ˆγ for all [, T ], gven by: ˆγ = P ψ θ and sasfyng ψ θ = η θ + λ N ˆγj. Proof.. Snce P s a conracon, we compue: Γ x Γ x 2 := N = Γ x Γ x 2 N = xj xj 2 = x x 2, provng ha Γ s a conracon. 2. We nex show ha Γ 2 := Γ Γ s a src conracon. Indeed, under 4.3, we may assume whou loss of generaly ha λ <. Then: Γ Γ x Γ Γ x 2 λ N so ha: Γ 2 x Γ 2 x 2 N Γj x Γ j x 2 λ N λn j k j λn λn j xk xk 2 k j xk xk 2, = λ N 2 k= N 2 xk xk 2 + k xk xk 2 + N 2 x x 2 + +N 22 k 2 xk xk 2 λ N 2 + N 22 + x 2 2 x 2. Observe ha N 2 + N + N 2 2 = N 2. Then λ < mples ha Γ 2 s a src conracon. 3. Therefore Γ n s a src conracon as well for any n 2. As a consequence, Γ 2, Γ 3 and Γ 6 respecvely adm a unque fxed pon x 2, x 3 and x 6 resp. I s mmedae ha x 2 and x 3 are also fxed pons for Γ 6, herefore x 2 = x 3 = x 6, and fnally x 2 = Γ 3 x 2 = Γ Γ 2 x 2 = Γ x 2, so ha x 2 s a fxed pon of Γ. The unqueness s mmedae snce a fxed pon of Γ s also a fxed pon of Γ 2. 4. Le Θ R Nd be defned by Θ = η θ. By defnon of ψ n Lemma 4.3, ϕ ψ Θ = η θ for all =,..., N. Usng he defnon of ϕ n 4.9, hs mples ha: ψθ = η θ + λ N P j ψj Θ. 4.4 Applyng P and seng ˆγ = P ψθ, hs provdes ˆγ = Γ ˆγ, for each =,..., N. By he defnon of ψ ogeher wh he expresson of ψ, we have ψ θ = ψθ, so ha ˆγ = P ψ θ. Pluggng no 4.4 provdes he las saemen of he Lemma. 23
5 Lnear porfolo consrans We now focus on he case where he ses of consrans are such ha: A s a vecor subspace of R d, for all =,..., N. 5. Our man objecve n hs secon s o explo he lneary of he projecon operaors P n order o derve more explc resuls. 5. Nash equlbrum under lnear porfolo consrans In he presen conex, we show ha Condon 4.3 n Theorem 4.2 can be weakened o N = λ < or N = A = {}. 5.2 In vew of Lemma 4. whch s obvous n he presen lnear case, he map R := λ jp j I + λ N j P j I + λ N P 5.3 s well-defned. Moreover, for any j =,..., N, snce P j I + λ N j P j λ = I N j P j +λ N, so ha: j s a projecon, we compue ha R = λ N j +λ N j P j I + λ N P. The followng saemen s more precse han Lemma 4.3. Lemma 5. Le A N be vecor subspaces of R d. Then for all [, T ]: he lnear operaor ϕ s nverble f and only f 5.2 s sasfed, hs condon s equvalen o he nverbly of he lnear operaors I R, =,..., N, under 5.2, he -h componen of ψ = ϕ s gven by: ψ z = I R z + +λ N j P j λn zj λ N j z. The proof of hs lemma s repored n Secon 5.4. We now proceed o he characerzaon of Nash equlbra n he conex of he mulvarae Black-Scholes fnancal marke. From Lemma 5., f Condon 5.2 s sasfed, ψ defned by 4.2 s well-defned, s a lnear operaor and has he followng expresson: ψ = M := I λ N j +λ N j P j I + λ N P η I + λ N η j λ N j η P j +λ N j. 5.4 Theorem 5. Assume ha σ and θ are deermnsc, and ha 5.2 s sasfed. Then here exss a unque deermnsc Nash equlbrum gven by: ˆπ = σ P M θ for =,..., N, [, T ]. Moreover, he value funcon for agen a equlbrum s gven by: V = e η x λ x Y where Y = η 2 θ 2 d + 2η I P M θ 2 d. 24
Proof. Follow he lnes of he proof of Theorem 4.2, replacng Lemma 4.3 by Lemma 5. and Lemma 4.5 by he followng Lemma 5.2. Lemma 5.2 Le θ R d be arbrary and Γ : R Nd R Nd be defned for any γ R Nd by: Γ γ = P η θ + λ N γj. Then under 5.2, Γ adms a unque fxed pon ˆγ gven by ˆγ = P ψ θ. The proof of hs lemma s repored n Secon 5.4. We llusrae he prevous Nash equlbrum n he conex of symmerc managers wh dfferen access o he fnancal marke. Example 5. Smlar agens wh dfferen nvesmen consrans Assume ha σ and θ are deermnsc, and le λ j = λ [, and η j = η >, j =,..., N. Then here exss a unque deermnsc Nash equlbrum gven by: ˆπ = ησ P I λn +λ N P j I + λ N P θ, =,..., N. We conclude hs secon wh he followng qualave resu whch shows n parcular ha he managers neracons nduce an over-nvesmen on he rsky asses, and mply ha he marke porfolo π of Defnon 3. s nondecreasng n he neracon coeffcens λ, n agreemen wh Proposon 3.2. Ths resul requres a que resrcve condon whch however covers many examples, see also Remark 5. below. Proposon 5. Assume ha he projecon operaors P commue,.e. P P j = P j P for all, j =,..., N. Then, under he condons of Theorem 5., Agen s equlbrum porfolo s such ha σˆπ s nondecreasng w.r. λ j and η j, for all, j =,..., N and [, T ]. Proof. We fx an agen =,..., N, and we om all dependence. The assumpon ha he P s commue s equvalen o he exsence of an orhonormal bass {u k, k =,..., d} such ha, for all, u k s an egenvecor of P for all k. We wre P u k = ε,k u k, and we observe ha ε,k {, } by he fac ha P s a projecon. Then, by he explc expresson of ˆπ n Theorem 5., wrng θ = d k= θk u k, we drecly compue ha σˆπ 2 = d k= θk 2 l,k 2 where: l,k = ε,k m λ N mε m,k + λ N + λ N ε,k η + m m λ N η m λ N mη + λ N ε m,k. 5.5 m We now verfy ha l,k s nondecreasng w.r. λ j and η j, for all j =,..., N and k =,..., d, whch mples he requred resul by he orhogonaly of he bass {u k, k =,..., d}. - Tha l,k s nondecreasng n η j s obvous from 5.5. - Tha l,k s nondecreasng n λ s also obvous from 5.5. 25
- Fnally, for j, we drecly dfferenae 5.5, and see ha he sgn of l,k / λ N j s gven by he sgn of: ε,k ε j,k + λ N ε,k η + m = ε,k ε j,k λ N η + λ N + λ N m λ N η m λ N mη + λ N m η m + λ N ε m,k. m ε m,k η m λ N mε m,k + λ N + λ N ε,k m Remark 5. The saemen of Proposon 5. s no vald for general porfolo consrans, as llusraed by he followng example. Le N = d = 2, A = Re, A 2 = Re + e 2 and σ = I. Then he projecon operaors P and P 2 are defned by he followng marces n he bass e, e 2 : P = and P 2 = /2 /2 /2 /2 respecvely. By drec calculaon, ˆπ = 2 λ λ 2 2η + λ η 2 θ + λ η 2 θ 2, whch can be nceasng or decreasng n η and λ, =, 2 for approprae choces of he rsk premum θ. 5.2 Infne managers asympocs We now nvesgae he lmng behavor when he number of agens N goes o nfny wh fxed number of asses d. Recall ha. denoes he canoncal Eucldean norm on R d, and LR d s he space of lnear mappngs on R d endowed wh operaor norm U = sup x = Ux for all U LR d. Proposon 5.2 Le d be fxed and he sequence η N bounded n R. Assume ha N N = λ P U λ n LR d and N N = η P U η n LR d, 5.6 for all resp. unformly n [, T ]. Assume furher ha U λ <, [, ]. Then:, ˆπ,N ˆπ, := σ P I U λ η I U λ + λ U η θ for all resp. unformly n [, T ]. Proof. By Theorem 5., we have ˆπ,N = σ P A Bθ, where: A := I λ N j P j I + λn P and B := η I + +λ N j Snce P j, we have: λ j P j +λ N N N j= λ jp j j λ j P j +λ N λ jp j + j λ 2 j + N λ P + P j 2 +λ N j 26 λ N η j λ N j η +λ N j λ jp j N N j= λ jp j N λ jp j 3 N. P j.
Smlarly, by he boundedness of he sequence η : η j P j N N j= η jp j +λ N j 3 η N. Then as N, we have n LR d : I + λ N P I, λ j +λ N j P j U λ, η j +λ N j P j U η, and A I U λ, B η I + λ U η η U λ. Under he condon U λ <, he lm s fne. Moreover he convergence s unform n whenever he convergence 5.6 holds unformly n. Example 5.2 Symmerc agens wh dfferen access o he fnancal marke Le λ = λ [, and η = η >,. Then he lmng Nash equlbrum porfolo reduces o ˆπ, = ησ P I λu θ, [, T ],. Example 5.3 Symmerc agens wh fne marke access possbles In he conex of he prevous example, suppose furher ha {A, } = {A j, j =,..., p} for some neger p >. We denoe by kj N he number of agens wh porfolo consran A j, and we assume ha kj N/N κ j [, ] for all j =,..., p. Then, an mmedae applcaon of Proposon 5.2 provdes he lm Nash equlbrum porfolo: ˆπ, = ησ P I λ p θ. κ j P j Remark 5.2 We may also adop he followng probablsc pon of vew o reformulae Proposon 5.2. Assume ha here s a connuum of ndependen players modeled hrough a probably space, D, µ ndependen from he space Ω, F, P descrbng he fnancal marke uncerany. In such a seng, he marke neracons, he rsk olerance, and he projecon operaors are defned by he random varables λ, η and he process P = {P, [, T ]} akng values respecvely n [, ],, + and LR d. The lmng Nash equlbrum porfolo s hen gven by: ˆπ, := σ P I µλp η I µλp + λµ ηp θ, j= provded ha µλ P + µη P <, and µλp <. Our nex commen concerns he asympocs of he marke ndex X N and he marke porfolo π N of Defnon 3.. Remark 5.3 In he conex of Remark 5.2, we furher assume ha he random varables λ, η, and P are ndependen, and we denoe P := µp, λ := µλ, η := µη. Then, under he condon λ P <, he lm marke porfolo and marke ndex are gven by π = σ v, X = x + v dw + θd where v := ηū I λū θ. 27
In parcular, we have he followng observaons whch are conssen wh Proposon 3.2: - he drf of he marke ndex s non-negave, - he drf and he volaly of he marke ndex are nondecreasng n η and λ, - he VRR ndex of he marke porfolo s gven by VRR = θ P I λ P θ 2θ, θ PI λ P and s nonncreasng n η and λ. 5.3 Examples wh lnear consrans For smplcy, excep for Example 5.8, we assume ha he agens are symmerc λ = λ and η = η for =,..., N and only dffer by her access o he fnancal marke. Excep for he las Example 5.9, we shall consder a dagonal mul-dmensonal Black- Scholes model wh volaly marx σ = I d,.e. he rsky asses prce processes are ndependen. Under he condons of Theorem 5., he opmal Nash equlbrum s gven by: ˆπ = ηp I λ θ + I P j + λ λ P for =,..., N, 5.7 see Example 5.. Le e,..., e d be he canoncal bass of R d. Example 5.4 Le d = N and A = Re, =,..., N. Noce ha n = A = {}. Then Theorem 5. apples for all λ [, ]. The projecon marces P are all dagonal wh unque nonzero dagonal enry P, =. The calculaon of he Nash equlbrum s hen easy and provdes ˆπ = ησ θ e, =,..., N. Hence, n agreemen wh he economc nuon, he neracon has no mpac n hs example, and he opmal Nash equlbrum porfolo concdes wh he classcal case wh no neracons λ =. Example 5.5 Le d = 3, N = 2 and A = Re +Re 2, A 2 = Re 2 +Re 3. Snce A A 2 {}, Theorem 5. requres ha λ [,. In he presen conex, he projecon marces are dagonal wh P, = P 2,2 =, P 3,3 =, and P, 2 =, P 2,2 2 = P 3,3 2 =. An easy calculaon provdes he opmal Nash equlbrum: ˆπ = ηθ e + η λ θ2 e 2 and ˆπ 2 = η λ θ2 e 2 + ηθ 3 e 3. Noce ha he opmal nvesmen n he frs and he hrd sock for Agen and Agen 2, respecvely, s he same as n he classcal case λ =. However, he nvesmen n Sock 2, whch boh agens can rade, s dlaed by he facor λ [, +. 28