A financial market with interacting investors: does an equilibrium exist?

Transcription

1 A fnancal marke wh neracng nvesors: does an equlbrum exs? Chrsoph Fre Dep. of Mah. and Sa. Scences Unversy of Albera Edmonon AB 6G G Canada cfre@ualbera.ca Gonçalo dos Res echnsche Unversä Berln Insu für Mahemak Sr. des 7. Jun Berln Germany dosres@mah.u-berln.de hs verson: February 7, Mahemacs and Fnancal Economcs 4 ), 6 8) Absrac Whle radng on a fnancal marke, he agens we consder ake he performance of her peers no accoun. By maxmzng ndvdual uly subjec o nvesmen consrans, he agens may run each oher even unnenonally so ha no equlbrum can exs. However, when he agens are wllng o wave lle expeced uly, an approxmaed equlbrum can be esablshed. he sudy of he assocaed backward sochasc dfferenal equaon BSDE) reveals he mahemacal reason for he absence of an equlbrum. Presenng an llusrave counerexample, we explan why such muldmensonal quadrac BSDEs may no have soluons despe bounded ermnal condons and n conras o he one-dmensonal case. Runnng le: A fnancal marke wh neracng nvesors Key words: relave performance, neracng nvesors, approxmaed equlbrum, muldmensonal BSDE, quadrac generaor MSC subjec classfcaons: 9B5, 6H, 9G8 JEL classfcaon numbers: G, C6 correspondng auhor

2 Inroducon Assumng you have nvesed n a fund, are you sasfed wh he fund manager f she acheved a performance of 4 % n he las year? You may say ha he answer depends manly on wo facors: he rsk he manager has aken and he developmen of he markes n he las year. In mahemacal fnance, he frequenly used approach of maxmzng expeced uly from ermnal wealh ncorporaes smulaneously he performance and he rsk relaed o a radng sraegy. However, he relave performance compared o an ndex or oher nvesors s ypcally no aken no accoun, alhough benchmarkng may even be par of human naure and s mporan for a fund manager who needs o keep he fund compeve. he goal of hs paper s o sudy he mpacs of negrang relave-performance consderaons no he framework of uly maxmzaon. he model we consder consss of n agens who can rade n he same marke subjec o some ndvdual resrcons. Each agen measures her preferences by an exponenal uly funcon and chooses a radng sraegy ha maxmzes he expeced uly of a weghed sum conssng of hree componens: an ndvdual clam, he absolue performance and he relave performance compared o he oher agens. he queson s wheher here exss a Nash equlbrum n he sense ha here are ndvdual opmal sraeges smulaneously for all agens. We make he usual assumpon ha he fnancal marke s bg enough so ha he radng of our nvesors does no affec he prce of he asses. A model smlar o ours has been recenly suded n he PhD hess of Espnosa [7] bu n he absence of ndvdual clams and wh asses modeled as Iô processes wh deermnsc coeffcens. hese assumpons crucally smplfy he analyss and enable Espnosa [7] o show a nce exsence resul for a Nash equlbrum, whch wll also be presened n he forhcomng paper [8] by Espnosa and ouz. Addonally, hey sudy he form of a Nash equlbrum, whle our focus s on exsence quesons n a more general seng and nerpreaons as well as possble alernaves n he absence of a Nash equlbrum. We oban exsence and unqueness n a sochasc framework f all agens are faced wh he same radng resrcons. Under dfferen nvesmen consrans, however, an agen may run anoher one by solely maxmzng her ndvdual uly. Dfferen nvesmen possbles may allow an agen o follow a rsky and benefcal sraegy, and hereby negavely affec anoher agen who benchmarks her own sraegy agans he less resrced one. he bankrupcy of he agens can be avoded f agens wh more nvesmen possbles are showng soldary and wllngness o wave some expeced uly. hs leads o he exsence of an approxmaed

3 equlbrum, n he sense ha here exss an ɛ-equlbrum for every ɛ >. In an ɛ-equlbrum, every agen uses a sraegy whose oucome s a mos ɛ away from ha of he ndvdual bes response. Behnd hs well-known concep sands he dea ha agens may no care abou very small mprovemens. Our seng brngs up he addonal aspec of soldary: by accepng a small deducon from he opmum, an agen can help o save he ohers from falure. Applyng freely o our model Adam Smh s mos famous caon, we could say ha maxmzng ndvdual ules somemes leads o an equlbrum. Bu when one agen can domnae anoher because of less radng resrcons, he nvsble hand of he marke has o be accompaned wh soldary o guaranee an accepable oucome for every agen. hs fnancal nerpreaon goes along wh an neresng mahemacal bass, whch s due o he correspondence beween an equlbrum of he nvesmen problem and a soluon of a ceran backward sochasc dfferenal equaon BSDE). BSDEs provde a genune sochasc approach o conrol problems whch ypcally fnd her analyc analogues n he convex dualy heory and he Hamlon-Jacob-Bellman formalsm. A BSDE s of he form dy = f, Y, Z ) d + Z dw,, Y = ξ, where gven are a d-dmensonal Brownan moon W, an n-dmensonal random varable ξ and a generaor funcon f. A soluon Y, Z) consss of an n-dmensonal semmarngale Y and an n d)-dmensonal conrol process Z predcable wh respec o he flraon generaed by W. Exsence and unqueness resuls have frs been shown for BSDEs wh generaors f sasfyng a Lpschz condon; see for example Pardoux and Peng [5]. However, BSDEs relaed o mahemacal fnance, as n our suaon, ypcally nvolve generaors f whch are quadrac n he conrol varable. For such cases, Kobylansk [4] proved exsence, unqueness and comparson resuls when ξ s bounded and Y s one-dmensonal n = ). Her resuls were generalzed by Brand and Hu [] and Delbaen e al. [4] o BSDEs wh unbounded ermnal condons. Whle Kobylansk s proof canno be generalzed o n >, evzadze [] presens an alernave dervaon of Kobylansk s resuls va a fx pon argumen. hs yelds as a byproduc an exsence and unqueness resul also for n > f he generaor f s specfc purely quadrac) and ξ s suffcenly small he L -norm of ξ needs o be ny). he resul s n lne wh he manra ha paral dfferenal equaons PDEs) can ofen be solved for suffcenly small daa or on a suffcenly small me nerval, alhough he known exsence and unqueness By pursung hs own neres he frequenly promoes ha of he socey more effecually han when he really nends o promoe. Adam Smh n he Wealh of Naons 776). 3

4 resuls cover only some ypes of PDEs. hs dea s also refleced n he recen paper by Žkovć [], who shows exsence and unqueness of sochasc equlbra on a suffcenly small me nerval, where each agen maxmzes he expeced uly of her ermnal wealh n a class of ncomplee markes. For a muldmensonal quadrac BSDE.e., n > and f s quadrac n he conrol varable) lke ha relaed o our problem, no general exsence and unqueness resuls are known, even when ξ s bounded. On he oher hand, no explc counerexample s avalable so far o he bes of our knowledge. he paper s srucured as follows. he shor Secon presens an llusrave counerexample whch s easy o undersand and shows ha and why general muldmensonal quadrac BSDEs do no have soluons. hs gves a mahemacal flavor for he absence of an equlbrum n he fnancal model presened n Secon 3, because we esablsh here a relaon beween exsence of equlbra under regulary condons and soluons o such a BSDE. Secons 4 6 group he argumens and resuls explaned above on he non-)exsence of an equlbrum based on dfferen ypes of radng resrcons for he agens. Fnally, Secon 7 concludes, and he Appendx conans some proofs and auxlary resuls. An llusrave BSDE counerexample Afer some preparaon, we gve a counerexample o he exsence of soluons of muldmensonal quadrac BSDEs. hroughou he paper, we fx > and d, n N and work on a canoncal Wener space Ω, F, P) carryng a d-dmensonal Brownan moon W = W,..., W d ) resrced o he me nerval [, ]. We denoe by F = F ) s augmened naural flraon and assume F = F. For an equvalen probably measure Q, we defne: he space S of bounded predcable processes; he space H n,d Q) of n d)-dmensonal predcable processes Z ) normed by Z H n,d Q) := E Q [ racez Z ) d ] / ; he space BMOQ) of square-negrable marngales M wh M = and sasfyng M BMOQ) := sup E Q [ M M τ F τ ] L <, τ where he supremum s aken over all soppng mes τ valued n [, ]. In he case Q = P, we usually om he symbol P. A soluon of a BSDE dy = f, Y, Z ) d + Z dw,, Y = ξ,.) 4

5 wh gven n-dmensonal random varable ξ and generaor funcon f s a par Y, Z) sasfyng.) wh a semmarngale Y and Z H n,d. he counerexample, for whch we ake d = dmenson of W ), consss of he wo-dmensonal n = ) BSDE dy = Z dw,, Y = ξ,.) = Z + ) Z d + Z dw,, Y =,.3) dy where he ermnal condon ξ L s gven. here s an explc soluon for he frs componen, whch does no depend on he second. he generaor of he second componen depends quadracally on he conrol varables of boh he frs and he second dmenson of he BSDE. For some choces of he ermnal condon, he second componen explodes, leadng o nsolvably. heorem.. For some ξ L, he BSDE.),.3) has no soluon. Proof. From.), follows ha Y s explcly gven by Y = E[ξ F ] and Z s unquely defned va Iô s represenaon heorem hrough [ ] ξ = E[ξ] + Z dw, E Z d <. We now use Z n.3), whch mples [ E exp Z d = exp [ ) Y E E ) ] Z dw exp ) Y snce he sochasc exponenal E Z dw ) s a posve supermarngale. hs gves Y = f E [ exp Z d =, and he resul follows by seng ξ = ζ dw L for ζ gven n Lemma A. n he Appendx. he underlyng mahemacal reason presened n Lemma A. s ha here exss a bounded marngale whose quadrac varaon has an nfne exponenal momen. Snce he generaor n.3) depends quadracally on boh Z and Z, hs leads o exploson. Economcally speakng, f Y and Y n.) and.3) descrbe he wealh developmen of wo agens, hen he frs agen s wealh remans bounded, bu s flucuaon can desroy he second agen s wealh so ha he second agen collapses. hs rough dea wll be developed laer n Secon 5. Remarks. ) Our counerexample shows ha dmensons maer n sochascs. hs ssue of dmensonaly has already been poned ou by Emery [6]. 5

6 Whle he sochasc exponenal of any bounded connuous marngale s a rue marngale, he gave an example of a bounded connuous marxvalued marngale whose sochasc exponenal s no a rue marngale. Boh Emery [6] and our counerexample show ha negrably properes of sochasc processes may crucally depend on he dmenson, alhough Emery [6] and our counerexample are n compleely dfferen sengs. ) Because of he form of he ermnal condon, our BSDE counerexample s no Markovan, and hus has no analogue n erms of PDEs. However, s behavor s n some sense smlar o he well-known phenomenon of fneme graden blow-up n PDEs. Chang e al. [] also presened as heorem III.6.4 n Sruwe s book [9]) consder mappngs u from he closed un dsk n R no he un sphere n R 3 whch sasfy u = u + u u, u, x) = u x), u, ) D = u D..4) hey show ha for some smooh and bounded boundary condon u, he soluon of.4) blows up n fne me,.e., he maxmal exsence nerval [, ) has a fne. Whle boh he seng and he form of hs example are dfferen from ours, he underlyng spr s o some exen relaed: he dmensonaly and he appearance of u correspondng o Z + Z n.3) ) play crucal roles for he exploson. 3 Model seup and prelmnares Afer we have seen ha muldmensonal quadrac BSDEs need no have soluons, we sudy a fnancal problem, s lnk o exsence ssues for such BSDEs and how alerng he problem can lead o solvably. We sar n hs secon by nroducng he problem formulaon and hen group n Secons 4 6 he resuls based on dfferen ypes of radng resrcons for he agens. he fnancal marke we consder consss of a rsk-free bank accoun yeldng zero neres and m raded rsky asses S = S j ) j=,...,m wh dynamcs ds j = S j µ j d + d k= S j σ jk dw k,, S j >, j =,..., m; he drf vecor µ = µ j ) j=,...,m as well as he lnes of he volaly marx σ = σ jk ) j=,...,m, are predcable and unformly bounded. We assume ha σ k=,...,d 6

7 has full rank and ha here exss a consan C such ha C β β σσ β C β a.e. on Ω [, ] for all β R m. he marke prce of rsk θ := σ σσ ) µ s hen also unformly bounded and Ŵ := W + θ d s a Brownan moon under he probably measure ˆP gven by dˆp := E θ dw ). dp We consder n agens. Any agen can rade n S subjec o some personal resrcons and has o pay or s endowed wh) a clam F L a me. hs means ha agen uses some self-fnancng radng sraegy π = π,..., π m ) valued n A, where A s a closed and convex subse of R m. We denoe by P he projecon ono A σ,.e., P x) := argmn x z z A σ for x R d. If agen sars wh zero nal capal, her wealh a me relaed o a sraegy π s gven by X π := m j= π j s S j s ds j s = π sσ s dŵs. Any agen measures her preferences by an exponenal uly funcon U x) = exp η x), x R, for a fxed η >. Insead of maxmzng he classcal expeced uly E[U X π F, agen akes also he relave performance no consderaon and maxmzes over π he value V π := E [U λ )X π + λ X π ) X πj F j = E [U X π λ X πj F 3.) j for a fxed λ [, ] and gven he oher agens j use sraeges π j. he se A of admssble sraeges for agen s gven by A := { π R m -valued, predc. π A a.e. on Ω [, ], X π BMO ˆP)}. We se A := A A n. Because we assume ha each agen maxmzes her expeced uly whou cooperang wh he oher agens, we are neresed n Nash equlbra. Defnon 3.. In hs seng, a sraegy ˆπ A s a Nash equlbrum f for every, V ˆπ V π,ˆπ j for all π A. 7

8 he classcal problem of maxmzng E[U X π F has been suded by Hu e al. [] n he same seng, bu wh no necessarly convex A. hey gve n heorem 7 a BSDE characerzaon for he opmal sraegy and he maxmal expeced uly. Alhough her defnon of admssbly slghly dffers from ours class D)- nsead of BM O-condon), her heorem 7 sll holds under our defnon n he case λ = for all, whch can be seen from s proof and whch we laer use several mes. Our choce of admssbly allows for boh reganng he asseron of Hu e al. [] n he case λ = for all and dervng n Lemma 3. a BSDE characerzaon for general λ. By heorem 3.6 of Kazamak [3], he condon X π BMO ˆP) s equvalen o π σ dw BMOP) because θ s bounded. In conras o opmzng E[U X π F, we maxmze E[U X π F wh F := λ n j Xπj + F. Snce F s unbounded and depends on he oher agens sraeges, he sudy s more nvolved. hs problem of agens concernng he relave performance has also been consdered n he PhD hess of Espnosa [7] and wll be presened n Espnosa and ouz [8]. In a smpler seng where σ and θ are deermnsc and whou clams F, Espnosa [7] proved he exsence of a Nash equlbrum and gave a characerzaon of n hs heorem 4.4. Is proof conans a BSDE characerzaon smlar o Lemma 3. below. In our sochasc model, a counerexample n Secon 5 wll show ha here need no exs a Nash equlbrum and only a noon weaker han a Nash equlbrum mgh be sasfed. he followng resul, whch relaes a Nash equlbrum o a BSDE, s an analogue o heorem 7 of Hu e al. []. However, one has here no unqueness and exsence resul for he BSDE. In fac, he counerexample n Secon 5 shows ha exsence does no hold n general. Anoher dfference o Hu e al. [] s ha we have he BSDE characerzaon only for equlbra whn a ceran regulary class. hs regulary condon s needed o use he powerful ool of BMO-marngales. On he oher hand, s no essenal for he counerexample, as we wll see n he proof of heorem 5.. hs means ha we have a non-exsence resul for Nash equlbra whou mposng an addonal regulary condon. We recall he reverse Hölder nequaly R p Q). For p >, an equvalen probably measure Q and an adaped posve process M, we say M sasfes R p Q) C s.. ess sup E Q [M /M τ ) p F τ ] C. 3.) τ sop. me Lemma 3.. here s a one-o-one correspondence beween he followng: ) a Nash equlbrum ˆπ A such ha for any, here exss p > wh E [U X ˆπ λ ) ] X ˆπj F. sasfes R p P); 3.3) j 8

9 ) a soluon Y, Z) wh Z dw BMO of he muldmensonal BSDE dy θ = η Z + θ P Z + ) ) θ d + Z dŵ, η η η Y = λ P j Z j + ) θ dŵ + F, =,..., n. 3.4) η j j he relaon s gven by ˆπ σ = P Z + η θ ) and V ˆπ = expη Y ). Proof. Assume ) holds and fx. One can show by dynamc programmng smlarly o Lemma 4.5 of Espnosa [7] ha for any π A, M π gven by M π := e η X π ess sup E [U X κ X κ κ A λ j X ˆπj F ) F ] 3.5) has a connuous verson whch s a supermarngale and a marngale for π = ˆπ. hs uses ha for any π, π A and soppng me τ, we have π ],τ]] + π ]]τ, ]] A. A varan of Iô s represenaon heorem mples M ˆπ = M ˆπ E ) Z dw for Z wh Z d < a.s. and M ˆπ <. heorem 3.3 of Kazamak [3] yelds Z dw BMO because of 3.3) and he boundedness of F. We se Z := η Z + ˆπ σ, whch agan sasfes Z dw BMO because ˆπ A. For any π A, we oban M π = exp ) η X ˆπ η X π M ˆπ = M ˆπ N π B π, where ) N π := E η Z π σ) dw, η Z B π := exp + θ π σ Z + θ ˆπ σ ) ) d. η η he P-supermarngale propery of M π mples ha M π / N π = M ˆπ B π s a Q π -supermarngale where dqπ π π := N dp, usng ha N s a P-marngale by heorem.3 of Kazamak [3]. Because B π s a connuous Q π -submarngale and of fne varaon, s nondecreasng,.e., for any π A Z η θ π σ Z + η θ ˆπ σ a.e. Hence, we ge ˆπ σ = P Z + η θ ), usng ha a sraegy π sasfyng π σ = P Z + η θ ) can be chosen predcable by Lemma of Hu e al. []. We se Y := η log M ˆπ expη X ˆπ ) ) and oban for dy he expresson n 3.4) afer a sraghforward calculaon. 9

10 Moreover, 3.5) mples Y = λ n j ˆπj σ dŵ + F. Snce hs holds for any, we have ˆπ σ = P Z + η θ ) for all and 3.4) follows. Suppose ) holds, defne ˆπ by ˆπ j σ = P j Z j + η j θ ) for all j and fx. Lke n Lemma of Hu e al. [], we oban P Z + η θ ) dw BMO so ha ˆπ A. For π A, we se R,π := exp η X π Y ) ), whch sasfes ) R,π = expη Y ) E η Z π σ) dw η exp Z + θ π σ Z + θ P Z + ) ) θ d. η η η We deduce ha R,ˆπ s a marngale and V ˆπ R,π s a supermarngale and we have V ˆπ = expη Y ). For any π A, = R,π E [ ] R,π = V π,ˆπ j. In he specfc case where all F = and µ as well as σ are deermnsc, one can consruc a soluon o he) BSDE 3.4) by choosng a deermnsc Z wh Z = λ n j P j Z j + η j θ for all f Π n j=λ j <. hs s possble because for Π n j=λ j <, he mappng ϕ defned by z ϕ z) := z λ j P j z j ) 3.6) s nverble by Lemma 4.4 of Espnosa [7], who also shows ha ϕ s Lpschz-connuous unformly n. Snce Z dw s n BMO for hs deermnsc Z, he sraegy ˆπ sasfyng ˆπ σ = P Z + η θ ) s a Nash equlbrum by Lemma 3., and even fulflls 3.). Hence, we regan he form of a Nash equlbrum saed n heorem 4.4 of Espnosa [7], whose assumpons are F = and deermnsc µ and σ. In he followng, we gve a bref alernave dervaon whch does no use BSDEs. Remark. In hs remark, we fx and assume F = and ha µ and σ are deermnsc. Supposng π j A j for j are deermnsc, we oban from 3.) for any possbly sochasc) π A ha V π = EˆP = EˆP [ exp η λ ) π j σ η πσ + θ dŵ j [ η λ ) E π j σ η π σ + θ j exp η λ j ) dŵ e θ d π j σ η π σ + θ d e θ d

11 and hence V π EˆP [ η λ E η exp j λ ) ) ] π j σ η π σ + θ dŵ j π j σ + η θ ˆπ σ d ) θ d, where ˆπ σ = P λ n j πj σ + η θ ). hus we have η sup V π = exp λ π j σ + θ ˆπ ) π A η σ d θ d. hs shows he exsence of a Nash equlbrum ˆπ A gven by j ˆπ σ := P ϕ, θ,..., )) θ η η n = P λ = P λ P ϕ j,j θ,..., θ) ) + ) θ η η n η j j ˆπ j σ + ) θ, =,..., n, η where ϕ, denoes he -h componen of he nverse of ϕ gven n 3.6). Whle we do no need he BSDE formulaon n he presence of deermnsc parameers, s helpful n he general case. he muldmensonal BSDE 3.4) s coupled va s ermnal condon. By usng he mappng ϕ defned n 3.6), we can rewre 3.4) as dγ = η ϕ, Γ = F + η ζ ) P ϕ, ζ ) ) d + ζ dŵ,, θ s dŵs η where ζ := ϕ Z + η θ,..., Z n + η n θ ) and Γ := Y λ Ps j j Z js + η j θ s ) dŵs + η θ s ds, =,..., n, 3.7) θ s dŵs η θ s ds. 3.8) Because ϕ s Lpschz-connuous, 3.7) shows ha we are dealng wh a muldmensonal quadrac BSDE.

12 In he followng remark, we brefly menon wo arcles relaed o oher fnancal applcaons of muldmensonal quadrac BSDEs. Remark. El Karou and Hamadène [5] consder ceran games wh wo players. In a Markovan framework, hey gve a characerzaon for an equlbrum n erms of a soluon of a muldmensonal quadrac BSDE. For her seng, he couplng of ha BSDE s weak, namely s assumed ha he -h enry of he drver f s domnaed by C + z ) for some posve consan C. However, no exsence resul for such a BSDE s provded. Cherdo e al. [3] follow n he fooseps of Hors e al. [] o solve a problem of valung a dervave n an ncomplee marke by equlbrum consderaons. In Hors e al. [], he problem can be solved n a onedmensonal framework, snce he dervave s assumed o complee he marke. Cherdo e al. [3] do no mpose hs condon, whch makes he analyss much more nvolved. he auhors solve he problem n a dscree framework, bu close her work wh consderaons on he connuous case. he laer leads o a fully coupled muldmensonal quadrac BSDE, whose solvably s unknown. 4 Agens havng he same radng consrans In a suaon where all agens are faced wh he same radng consrans gven by a lnear subspace of R d, here exss a unque Nash equlbrum ˆπ and we can gve a BSDE characerzaon for ˆπ smlarly o Hu e al. []. Proposon 4.. Assume ha A = A are he same lnear subspace for all =,..., n and Π n =λ <, and se P = P. hen for =,..., n, he decoupled BSDEs dγ θ = η η ζ + θ P ζ + ) ) θ d + ζ dŵ, Γ = F 4.) η η have a unque soluon Γ, ζ) S H,d. here s a unque Nash equlbrum ) ˆπ A. I s gven by ψ ˆπ )σ = P ζ + η θ, where he lnear mappng ψ s defned by z ψ z) := z λ n j zj. Moreover, V ˆπ = expη Γ ). Proposon 4. shows ha he agens maxmal expeced uly s he same as n he case whou neracon. However, he opmal sraeges are dfferen. Snce all agens have he same consrans, an agen can compleely hedge agans he ohers agens behavor. hs mples ha he opmal sraegy accouns for he ohers agens behavor, whle he maxmal expeced uly s unalered compared o he suaon whou nerdependences.

13 Proof of Proposon 4.. Because A = A s a lnear space and ψ s nverble due o Π n =λ <, we have π A ψπ) A. hs mples sup π A V π,ˆπ j = sup p A E [ U X p F for all ˆπ j A j. Applyng heorem 7 of Hu e al. [] o he laer opmzaon problem yelds he resul. Remarks. ) he proof shows as well ha here exss no π A wh V π > V ˆπ for some and V π j V ˆπ j for all j, whch means ha ˆπ from Proposon 4. s also a Pareo opmum. ) he BSDEs 4.) correspond o 3.4). Indeed, defne Y, Z) by ζ + θ,..., ζ n + ) θ = ϕ Z + θ,..., Z n + ) θ, η η n η η n Y = Γ + λ P Z j + ) θ dŵ η, j j wh he nverble lnear mappng ϕ gven by 3.6). Because of A = A for all, n j P ) Z j + η j θ s n σ A, and we oban ζ + η θ P ζ + η θ ) = ζ + η θ + = λ P ζ + η θ + j λ P Z j + η j θ ) P Z j + ) ) θ η j j Z + η θ P Z + η θ ). herefore, he BSDEs 4.) are equvalen o 3.4). We now gve an easy counerexample for he case λ = where he BSDE 3.4) has no soluon. We ake n = number of agens), d = dmenson of W ), σ =, θ =, A = A = R no consrans), η = η = and λ = λ = only he relave performance maers). he BSDE 3.4) equals dy = d + Z dŵ, Y = dy = d + Z dŵ, Y = Z s + ) dŵs, Z s + ) dŵs. 3

14 By combnng hese equaons, we oban + hs mples Z dŵ = Z + ) dŵ Y = Ŵ Y + Z + ) dŵ Y. Ŵ = + Y + Y, whch s a conradcon, because he rgh-hand sde s sochasc whle he lef-hand sde s deermnsc. One can nerpre hs example as follows: Boh agens care only abou he relave wealh. Snce he marke prce of rsk θ s nonzero, here s some rsk nheren n he model and each agen wans o hedge agans hs rsk. For any gven sraegy π A of agen, he opmal sraegy of agen s ˆπ = π + θ = π + A. Analogously, ˆπ = π + A s he bes response of agen o any gven sraegy π A of agen. By ryng o hedge, he frs agen ransfers he rsk o he second agen, who hen ransfers back o he frs. Because of λ =, no agen reduces he rsk, bu nsead each agen eravely passes he buck o he oher. In he end, boh agens break down so ha here s no Nash equlbrum. hs counerexample can also be nerpreed n he conex of copyca hedge funds, whch ry o mae he sraegy of a successful hedge fund. If a hedge fund copes he sraegy of anoher fund whch self mmcs he former fund, hen no equlbrum can exs because he nerdependence muually amplfes he sraeges. 5 Agens wh ordered radng consrans hs secon deals wh a suaon where some agens have more radng possbles han ohers. hroughou hs secon, we assume Π n =λ < and ha A are lnear subspaces of R d sasfyng A A A n. We sar wh a counerexample for he case where wo agens have dfferen consrans. he frs agen copes wh a bounded clam F by choosng a suable hedgng sraegy. However, he second agen s affeced by he frs agen s hedgng sraegy, whch makes he second agen break down. heorem 5.. here exss a counerexample wh n =, lnear spaces A A and λ λ < where here s no Nash equlbrum. 4

15 Proof. We ake d = dmenson of W ), σ = )-deny marx, θ =, A = {x, x) x R}, A = {, )}, η = /π + ) hs choce wll laer smplfy compuaons; π denoes here he number and no a sraegy), η =, F =, F o be chosen laer and λ = λ = /. We oban for he correspondng BSDE 3.4) ha dy = π + ) Z, Z, d + Z dw, Y = F, 5.) dy = Z d + Z dw, Y = 4 Z, s + Z, s ) dw s + W s ). 5.) he frs componen 5.) does no depend on he second, and has for any bounded F a unque soluon Y, Z ) wh Z dw BMO. hs soluon s plugged n 5.) o solve for he second componen. Smlarly o he counerexample presened n Secon, we consruc an F such ha Y explodes. he dfference o he frs counerexample s ha 5.) has a quadrac generaor and 5.) depends on Z va a dw - and no a d-negral. We se F := π + ) log E ζ dw ) for ζ from Lemma A. wh log E ) [ π ζ dw S + and E exp ζ dw =. 5.3) 4 he BSDE 5.) has he explc soluon Y = π + ) log E ζ dw ), Z, = π + )ζ, Z, =. From 5.) and 5.3), follows ha e Y = E [exp Z, dw +W ) [ π + E exp 4 ζ dw = 4 5.4) by condonng on he σ-feld generaed by W and usng Jensen s nequaly. herefore, he coupled BSDE 5.), 5.) has no soluon and here s no Nash equlbrum sasfyng 3.3) by Lemma 3.. o see ha here exss no Nash equlbrum a all even whou 3.3) ), we noe ha a canddae Nash equlbrum ˆπ A mus sasfy ˆπ = radng consrans of agen ) and ˆπ = Z, +Z,, ) opmaly for agen, usng ˆπ = and heorem 7 of []). Bu hs gves V ˆπ = E [ U λ by 5.4). Z, + Z, ) dw + W ) = he radng consrans n he counerexample mgh look resrcve, bu s possble o generalze he counerexample o hgher-dmensonal W, 5

16 whle gvng he agens more radng possbles. For d >, one can deduce an analogous counerexample wh A = {x, x, y,..., y d 3 ) x, y R}, A = {,, y,..., y d 3 ) y R}; n ha case, Y sasfes dy = Z, + Z, ) d + Z dw, d e Y = E [exp Y Z, dw E [ exp E[Y G], =3 where G denoes he σ-feld generaed by W and W. heorem 5. shows ha havng A as ordered lnear spaces s no enough o guaranee he exsence of a Nash equlbrum. Even f he frs agen does no concern he relave performance, her choce of a hedgng sraegy for F may bankrup he oher agens. Whle he frs agen can hedge agans all oher sraeges, her sraegy may negavely nfluence he oher agens and run hem. Assumng ha he frs agen wans o avod he run of he oher agens, she mgh be wllng o reduce her wealh a lle b. Connung hs dea for he oher agens, we come o he followng relaxaon of a Nash equlbrum. Defnon 5.. We say ha here exss an addvely approxmaed equlbrum f for every ɛ >, here s ˆπ ɛ,,..., ˆπ ɛ,n ) A such ha for any, V ˆπɛ + ɛ V π,ˆπ ɛ,j for all π A. 5.5) A mulplcavely approxmaed equlbrum exss f for every ɛ >, here s ˆπ ɛ,,..., ˆπ ɛ,n ) A such ha for any, ɛ)v ˆπɛ V π,ˆπ ɛ,j for all π A. 5.6) Noe ha we use ɛ) and no +ɛ) n 5.6), because V s negave. In he leraure, here exss he noon of ɛ-equlbrum, whch corresponds o he suaon where 5.5) holds for a fxed ɛ >, nsead of all ɛ >. Gven he exsence of a Nash equlbrum, calculang such a fxed ɛ-approxmaon nsead of he rue Nash equlbrum can be more effcen and easer o mplemen; see for example Hémon e al. []. However, an ɛ-equlbrum need no be close o a Nash equlbrum; see Secon of Shoham and Leyon-Brown [8]. In our case, here may no even exs a Nash equlbrum as shown n heorem 5., whle here s always an approxmaed equlbrum by he nex resul. hs seemng weakness of he noon of ɛ-equlbrum s n fac an advanage n our suaon, because we am here o fnd approxmaed equlbra hemselves, raher han a convergence o Nash equlbra. here are 6

17 several reasons why agens can be sasfed wh a less-han-opmal sraegy. Radner [6], who has popularzed he noon of ɛ-equlbrum, menons ha a nearly opmal sraegy can be less cosly han a bes response. o see he reasons n our seng, le us brefly come back o he counerexample of heorem 5.. he wealh process correspondng o a bes sraegy s generally unbounded, whch may run anoher agen. If each agen resrcs her sraeges o hose wh bounded wealh processes, hs problem does no occur, and we wll see ha such bounded wealh processes can be used o consruc an approxmaed equlbrum. So movang he noon of an approxmaed Nash equlbrum can be done by jusfyng why agens may resrc hemselves o bounded wealh processes. Apar from ndvdual reasons and legal resrcons publc aemp o sablze he sysem by nroducng bounds), we can relae he noon of approxmaed equlbrum o he aspec of soldary. If he more powerful agens are wllng o devae lle from he expeced uly assocaed o he bes response, hen he oher agens do no break down and hey can even fnd hemselves nearly opmal sraeges n he sense of Defnon 5.. he move can be eher rue soldary or he hdden agenda o keep a weak agen n he compeon enablng an easy benchmarkng n fuure perods. Because Defnon 5. mposes on 5.5) and 5.6) o hold for every ɛ >, he agens do no need o agree on a parcular ɛ and each agen can choose ndvdually a sequence of) ɛ. heorem 5.3. here exss an addvely as well as mulplcavely approxmaed equlbrum. Because of s lengh, we presen he consrucve proof of heorem 5.3 n he Appendx, bu gve here a bref oulne. he man dea s ha for agen, only he sraeges of agens,..., really maer because she can hedge he oher sraeges. herefore, one sars o consder he frs agen s opmzaon problem when he sraeges of all oher agens are zero, and consrucs an auxlary sraegy whch leads o a devaon of a mos ɛ > from he opmum and whose wealh process s bounded. hen one bulds an auxlary sraegy for he second agen akng no accoun he frs agen s sraegy. o keep almos opmaly for he frs agen, her sraegy has o be updaed. One eravely connues wh he hrd unl he n-h agen. One could slghly adap he proof o show he exsence of an approxmaed equlbrum such ha addonally he sraegy for agen n s opmal,.e., 5.5) and 5.6) hold for = n also wh ɛ =. he underlyng reason s ha agen n canno negavely affec he oher agens because her sraegy s hedgeable by he ohers. he followng resul says more abou convergence of approxmaed equlbra n he case of wo agens. 7

18 Corollary 5.4. Assume n = and le ɛ k ) k N be a srcly posve sequence wh lm k ɛ k =, and le for each k, ˆπ ɛ k A be an approxmaed equlbrum consruced as n he Appendx proof of heorem 5.3 wh ɛ replaced by ɛ k ). Suppose ha here exss a Nash equlbrum ˆπ A wh X ˆπ, < BMO ˆP) 4η λ, where X ˆπ, BMO ˆP) := sup τ EˆP [ X ˆπ, Xτ ˆπ, ] L Fτ wh he supremum aken over all soppng mes τ valued n [, ]. hen we have lm k V ˆπɛ k = V ˆπ for =,. he proof of Corollary 5.4, whch s based on he convergence of he BSDEs relaed o V ˆπɛ k, s conaned n he Appendx. 6 A glmpse of general radng consrans We conclude by some resuls for general closed, convex ses A whou mposng any resrcons on he relaons of he A. For hs very general suaon, we gve n Secon 6. anoher relaxaon of a Nash equlbrum and dscuss n Secon 6. brefly he suaon where he rsky asses S are, n some sense, close o beng marngales. 6. Sequenally delayed equlbra We frs nroduce a furher relaxaon of a Nash equlbrum. Defnon 6.. We say ha here exss a sequenally delayed equlbrum f for any srcly posve sequence ɛ k ) k N wh lm k ɛ k =, here s ˆπ k ) k N A such ha for any k N and =,..., n, V ˆπk,,ˆπ k,j where we se ˆπ =. + ɛ k V π,ˆπ k,j for all π A, 6.) Roughly speakng, 6.) says ha ˆπ k, s almos opmal up o ɛ k ) for agen when he oher agens use he delayed sraeges ˆπ k,j. Defnon 5. would correspond o 6.) f ˆπ k,j were replaced by ˆπ k,j. In a way, he concep of sequenally delayed equlbra s opposed o ha of remblng-hand perfec equlbra. ha noon, whch has been nroduced by Selen [7], s a refnemen of a Nash equlbrum. Roughly speakng, a 8

19 remblng-hand perfec equlbrum s robus agans small devaons remblng hand ). In conras, a sequenally delayed equlbrum s a weaker noon han ha of a Nash equlbrum and gves a way of approachng a saus whch can be accepable for all agens. he dea behnd Defnon 6. s ha he delay makes he problem easer o handle and n he lm k, does no maer wheher one has ˆπ k,j or ˆπ k,j n 6.). Before makng hs saemen precse n Corollary 6.3, we gve an exsence resul. Proposon 6.. For any famly A ) =,...,n of closed ses, here exss a sequenally delayed equlbrum. Proof. Le ɛ k ) k N be a srcly posve sequence wh lm k ɛ k =. We consruc eravely a sequence ˆπ k ) k N A sasfyng 6.). Fx k N and {,..., n}, se ˆπ = and assume ha for any j {,..., n}, X ˆπk,j s bounded. By heorem 7 of Hu e al. [], here exss ˆp A such ha sup V π,ˆπk,j = V ˆp,ˆπk,j. π A We defne a sequence of soppng mes by τ l := nf { [, ] such ha } X ˆp l, l N and se p l) := ˆp ]],τl ]] A such ha X pl) = Xτ ˆp l. Usng ha he random varable F + λ n j X ˆπk,j s bounded, he a.s.-convergng sequence U X ˆp τj F λ n j X )) ˆπk,j s unformly negrable by he same argumen as above A.). herefore, we have j N pl),ˆπk,j lm V = lm E [U X pl) F λ X ˆπk,j = V ˆp,ˆπk,j. l l Choose L N such ha V pl),ˆπ k,j j V ˆp,ˆπk,j ɛ k, and se ˆπ k, := p L). By consrucon, 6.) s sasfed and X ˆπk, he proof follows from eravely usng he above procedure. s bounded. Corollary 6.3. Le ˆπ k ) k N A sasfy 6.). Fx and assume ha here exss ˆπ A wh ˆπk, dŵ ˆπ, dŵ a.s., and ha boh U X ˆπ k+, λ n j X ) ˆπk,j and U λ n j X ) ˆπk,j, k N, are unformly negrable. hen V ˆπ V π,ˆπ,j Proof. Fx π A wh bounded X π oban boh lm k V π,ˆπ k,j he asseron follows from 6.). for all π A wh bounded X π. = V π,ˆπ,j 9. Usng he unform negrably, we ˆπk+,ˆπ k,j and lm k V = V ˆπ.

20 6. Models close o he marngale case In he marngale case, where S s a P-marngale, θ s zero. hen he sraegy ˆπ = s a Nash equlbrum by Jensen s nequaly f F = for all. he dea behnd he followng resul s ha we sll can fnd a Nash equlbrum f we are no exacly n he marngale case, bu n some sense, close o. Proposon 6.4. Assume Π n =λ <, ha every A conans zero and ha for any here exss a consan c such ha F + η θ dŵ ) L θ d c ɛ 6.) η for some suffcenly small ɛ > dependng on η j ) j=,...,n, λ j ) j=,...,n and n. hen he BSDE 3.7) has a unque soluon Γ, ζ) wh suffcenly small sup Γ c L and ζ dŵ BMOˆP). Defnng ˆπ σ = P ϕ, ζ ) ) for ϕ gven n 3.6), ˆπ s a Nash equlbrum. Proof. We frs show exsence and unqueness of a soluon of 3.7) by applyng Proposon of evzadze []. o hs end, we verfy ha he generaor s purely quadrac,.e., here exss C such ha for all a, b R n d, ϕ, a) P ϕ, a) ) ϕ, b) P ϕ, b) ) ) C a + b a b. Seng ã = ϕ, a) and b = ϕ, b) and usng A, we have ha ã P ã) b b) P = ã P ã) + b b) ) P ã P ã) b b) P ã + ) b ã b + P ã) P ã + ) b ã b. b) 6.3) By Lemma 4.4 of Espnosa [7], ϕ s nverble and ϕ s Lpschz-connuous wh a consan L dependng on λ j ) j=,...,n and n. herefore, we oban ã b L a b as well as ã L a and b L b usng ϕ ) =. hs yelds 6.3) wh C := L. Proposon of evzadze [] now gves exsence and unqueness of a soluon Γ, ζ) of 3.7) under he assumpon 6.). Seng Z = ϕ, ζ) η θ and defnng Y va 3.8), he par Y, Z) solves he BSDE 3.4). Snce ζ dŵ BMOˆP) and θ s bounded, ζ dw s n BMOP) and so s Z dw because ϕ s Lpschz-connuous. Hence, he asseron follows from Lemma 3..

21 7 Concluson hs paper nroduces a model for a fnancal marke where agens maxmze expeced uly by consderng boh he absolue and he relave performance compared o her peers. In he case where all agens have he same radng resrcons or n a model wh deermnsc coeffcens, he exsence of an equlbrum s guaraneed. However, when some agens have more nvesmen possbles han ohers, her radng sraeges may negavely affec he weaker agens and hus only a relaxaon of a Nash equlbrum can be esablshed. hs reveals ha relave-performance consderaons n a fnancal marke may lead he sysem o collapse, whch can be avoded f he sronger agens show soldary. he resuls are based on he sudy of he relaed muldmensonal BSDE, makng he message wofold. In addon o he fnancal meanng, he BSDE counerexample shows boundares of BSDE heory n a muldmensonal framework. hs dual message also exemplfes he close relaonshp and nerplay beween mahemacs and fnancal economcs. Acknowledgmens We hank Nzar ouz, Ncole El Karou, Glles-Edouard Espnosa, Ulrch Hors and Marn Schwezer for smulang dscussons and helpful commens. he paper benefed from nsghful remarks by an anonymous referee. Chrsoph Fre graefully acknowledges fnancal suppor by he French Decson Scence Projec and he hospaly durng he me a École Polyechnque. hs research of Gonçalo dos Res was suppored by he Char Fnancal Rsks of he Rsk Foundaon durng hs say a École Polyechnque. Appendx A. Auxlary resuls Lemma A.. here exss ζ H, wh [ ζ dw S and E exp ζ d =. Proof. he followng consrucon s nspred by he proof of Lemma.7 of Kazamak [3]. Defne M := s dw s, [, ) A.)

22 so ha M ) u s a connuous marngale on [, u] for every u <. We π se τ := nf{ : M > } and ζ := [[,τ]]) so ha ζ dw s bounded by π. I remans o show ha E[ exp ζ d =. For hs, we defne an auxlary funcon h : [, ) [, ) by h) := e ), whch fulflls h) ds = log s h) =, [, ). We se B := M h), <, mplyng ha B ) < s an F h) ) < - Brownan moon. he random varable h τ) s he F h) ) < -soppng me when B frs leaves [, ]. From Lemma.3 of Kazamak [3], follows ha E [ exp α h τ) = for all α [, π/). herefore, we oban cosα) [ [ π E exp ζ τ d = E exp 8 d [ α = lm E exp α π/ h τ) [ π = E exp 8 h τ) = lm α π/ cosα) =. he resul s unchanged f one replaces n he defnon A.) of M he funcon s s by anoher connuous funcon g : [, ) R whch sasfes gs) ds = and gs) ds < for every [, ). For any gven ζ H,d wh ζ dw S, here exss a consan c such ha E [ exp c ζ d < by he John-Nrenberg nequaly heorem. of Kazamak [3]). Lemma A. shows conversely ha for any fxed consan c, here exss ζ H,d wh ζ dw S and E [ exp c ζ d =. Lemma A.. here exss ζ H, wh log E ) [ π ζ dw S + and E exp ζ dw =. 4 Proof. Smlarly o A.), we defne M := s dws for [, ) and se τ := nf { : M log > } and ζ := [[,τ]] ). Because we have M = ds = log for [, ), log E ζ dw ) s s bounded, and τ s he frs me ha EM) leaves [/e, e]. We recall he funcon h : [, ) [, ) gven by h) := e ), whch s he nverse of log. We se B := M h), <, so ha B ) < s an F h) ) < -Brownan moon. he random varable h τ) s he

23 F h) ) < -soppng me when he drfed F h) ) < -Brownan moon B /) < frs leaves [, ]. Lemma.3 of Kazamak [3] mples α E Q [exp h τ) = for all α [, π/), cosα) where dq := E B) = E M). For β dp h τ) τ π + )/8, we oban [ = E Q E P [exp β τ d E M) exp βh τ) τ [ = E Q EM) / τ e / E Q [exp β 8 exp βh τ) 8 M τ ) h τ) = and hence [ τ E P [expβm τ = E P EM) β τ exp β d τ e β E P [exp β d = so ha E P [ exp π + 4 ζ [ dw = EP exp π +M 4 τ =. A. Proofs of heorem 5.3 and Corollary 5.4 Proof of heorem 5.3. Fx ɛ > o show 5.5) and 5.6). We assume ɛ < whou loss of generaly.. Sep: Consrucon of an auxlary sraegy for agen. We sar by lookng a an auxlary problem for he frs agen. By heorem 7 of Hu e al. [], here exss ˆp A such ha sup E [ U X p F [ = E U X ˆp F. p A We defne a sequence of soppng mes by τ k := nf { [, ] such ha } X ˆp k, k N and se p k) := ˆp ]],τk ]] A such ha X pk) = Xτ ˆp k. Because F s bounded and )) U X ˆp can be wren as he produc of a marngale and a bounded process see he proof of heorem 7 of Hu e al. []), he process 3

24 )) U X ˆp F s of class D). Hence, he sequence U X ˆp τk F ))k N convergng almos surely s unformly negrable and hus, we have [ lm E U X pk) F = E [ U X ˆp F. A.) k Choose K N such ha { E [U X pk) F max E [ U X ˆp F ɛ, ɛ E[ U X ˆp F }. For noaonal convenence, we se π,) := p K), where π,j) sands for he auxlary sraegy of agen n he j-h eraon.. Sep: Consrucon of an auxlary sraegy for agen and adapaon of he frs agen s auxlary sraegy. We now consruc an auxlary sraegy π,) for agen n a smlar way; we smply replace η by η, U by U, and F by F + λ n Xπ,), whch s bounded by consrucon. Because here s nerdependence beween agens and, we need o adap he sraeges by seng π,) := λ λ /) π,), π,) := π,) + λ π,) o acheve ha π,) λ π,) = π,) λ π,), π,) λ π,) = π,). Snce A, A are lnear subspaces wh A A, we have π,) A and π,) A. 3. Sep: Consrucon of an auxlary sraegy for agen and adapaon of he auxlary sraegy of agens,...,. Lke above, we consruc an auxlary sraegy π 3,) for he hrd ) agen, replacng η by η 3, U by U 3, and F by F 3 + λ 3 X π,) + X π,). o accoun for he nerdependence, we se λ n, := π 3,3) := λ n, + λ n,)λ 3 /) π3,), n λ n + λ n ) λ λ n ) and defne π,3) := π,) + λ n,π 3,3), π,3) := π,) + λ n,π 3,3), achevng ha π 3,3) λ 3 π,3) + π,3)) = π 3,) λ 3 π,) + π,)), π,3) λ π,3) + π 3,3)) = π,) λ π,), π,3) λ π,3) + π 3,3)) = π,) λ π,). 4

25 Connung eravely lke hs, we fnally oban sraeges π,n),..., π n,n). he procedure works snce we can solve n each sep a sysem of lnear equaons wh non-zero deermnan because of he assumpon Π n =λ <.) 4. Sep: Defnon of ˆπ ɛ and verfcaon of 5.5) and 5.6). We se ˆπ ɛ,j := π j,n) A j for all j. For fxed, we have by consrucon ha V ˆπɛ = E [U X π,) λ j= where a := sup E [U X p λ p A [U = sup p A E X p λ { X πj, ) F max a ɛ, j= j X πj, ) F } ɛ a, X ˆπɛ,j F = sup V π,ˆπɛ,j. π A herefore, boh 5.5) and 5.6) are sasfed by hs ˆπ ɛ,,..., ˆπ ɛ,n ). Proof of Corollary 5.4. From A.), we oban lm k V ˆπɛ k = V ˆπ, where ˆπ ɛk, = ˆπ, ]],τ k ]] for some soppng me τ k. We sudy he BSDEs relaed o V ˆπɛ k. By consrucon and heorem 7 of Hu e al. [], we have exp η Y k) ) ɛk V ˆπɛ k exp η Y k) where Y k), Z k)) s he unque soluon n S, H,d ) of he BSDE dy k) θ = η Z k) + θ P Z k) + ) ) θ d + Z k) dŵ, η η η Y k) = λ X ˆπɛ k, + F = λ Xτ ˆπ, k + F. Because F and θ are bounded, here exs consans c and c no dependng on k) such ha for any soppng me ν, Y ν k) [ ] λ EˆP X ˆπ, Fν τ k + c and E η Z k) + θ ) dŵ ) ) E η Z k) + θ ) dŵ ) [ ]) exp θ dŵ+η λ X ˆπ, τ k EˆP X ˆπ, Fν τ +c k ν ν so ha Hölder s nequaly mples for any p, q > and some c 3 > dependng on p and q bu no on k or ν) [ E η Z k) + θ ) dŵ ) ] p EˆP E η Z k) + θ ) dŵ ) p F ν ν [ [ ]) ) ] /q. c 3 EˆP exp qpη λ X ˆπ, τ k EˆP X ˆπ, Fν τ k F ν ), 5

26 he assumpon X ˆπ, < BMO ˆP) 4η λ enables us o choose p, q > wh qp X ˆπ, < BMO ˆP) 4η λ. Usng X ˆπ, τ k. BMO ˆP) X ˆπ, BMO, we oban from he varan of he John-Nrenberg nequaly saed n heorem. ˆP) of Kazamak [3] ha [ [ ]) ) ] EˆP exp qpη λ X ˆπ, τ k EˆP X ˆπ, Fν τ k F ν, 4qpη λ X ˆπ, BMO ˆP) whch shows ha here exss p > such ha E η Z k) + θ) dŵ ) sasfes he reverse Hölder nequaly R p ˆP) unformly n k; compare 3.). hs mples by heorem 3.3 of Kazamak [3] ha he BMO ˆP) -norm of Z k) dŵ s bounded unformly n k. One can now show smlarly o he proof of heorem. of Fre [9] ha one has lm k Y k) = Y ), where Y ), Z )) s he soluon of he BSDE relaed o V ˆπ. herefore, we oban lm k V ˆπɛ k = lm k exp η Y k) ) = exp η Y ) ) = V ˆπ. References [] P. Brand and Y. Hu. Quadrac BSDEs wh convex generaors and unbounded ermnal condons. Probab. heory Rela. Felds, 43 4): , 8. [] K. Chang, W. Dng, and R. Ye. Fne-me blow-up of he hea flow of harmonc maps from surfaces. J. Dffer. Geom., 36:57 55, 99. [3] P. Cherdo, U. Hors, M. Kupper, and. Prvu. Equlbrum prcng n ncomplee markes under ranslaon nvaran preferences. Preprn, January. Avalable a hp:// kupper. [4] F. Delbaen, Y. Hu, and A. Rchou. On he unqueness of soluons o quadrac BSDEs wh convex generaors and unbounded ermnal condons. Preprn, June 9. Avalable a arxv:96.75v. [5] N. El Karou and S. Hamadène. BSDEs and rsk-sensve conrol, zerosum and nonzero-sum game problems of sochasc funconal dfferenal equaons. Sochasc Process. Appl., 7):45 69, 3. [6] M. Emery. Sur l exponenelle d une marngale BM O. Sémnare de Probablés XVIII. Lec. Noes Mah., 59:5, 98. [7] G.-E. Espnosa. Sochasc conrol mehods for opmal porfolo nvesmen. PhD hess, École Polyechnque Palaseau,. 6

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