Weak Kyle-Back equilibrium models for Max and ArgMax

Transcription

1 eak Kyle-Back equilibrium models for Max and ArgMax A. Kohasu-Higa and S. Oriz-Laorre July 2, 29 Absrac The goal of his aricle is o inroduce a new approach o model equilibrium in financial markes wih an insider. e prove he exisence and uniqueness in law of equilibrium for hese markes. Our seing is weaker han Back s one and i can be inerpreed as a firs heoreical sep owards developing saisical es procedures. Addiionally, i allows various forms of insider informaion o be considered under he same framework and compared. As major examples, we consider he cases of he maximum of he demand and he ime a which his maximum is aken, which have no been previously reaed in he lieraure of equilibrium in financial markes wih inside informaion. Simulaions indicae ha he expeced wealh for he maximum is greaer han he expeced wealh for is argumen. Key words: Large-insider rading, equilibrium heory, semimaringale decomposiion. JEL Classificaion: D53, D82, G, G2 Mahemaics Subjec Classificaion (2): 49J4, 6G48, 93E2 Inroducion In recen years, he sudy of mahemaical models for financial markes wih asymmery of informaion has been gaining an increasing aenion from mahemaical finance researchers. In a seminal paper and from he marke microsrucure poin of view, Kyle [6] inroduced a model in which an insider, who knows he value of he sock a some fuure ime, opimizes his wealh while he marke-maker makes prices raional, ha is, a raional expecaions equilibrium model. The main feaures of Kyle s model are ha i gives finie uiliies and ha i is a model of price formaion. Tha is, he insider conrols he price process hrough his demand of sock shares. Kyle s model has been exended by Back [2], Lasserre [7], Cho [5] and Campi and Çein [4], among ohers. e consider a coninuous ime marke composed of one risk-free asse and one risky asse. e assume, wihou loss of generaliy, ha he risk-free rae is zero. Trading in he risky asse is coninuous in ime and quaniy. Furhermore, he marke is order-driven, ha is, prices are deermined by he demand on he risky asse. There is o be a public release of informaion a ime : This informaion reveals he value of he risky asse, which we denoe by ξ : As he marke is order-driven, his enails ha ξ will be he price a which he asse will be raded jus afer he release of informaion and, herefore, he final profi obained hrough rading on his asse will depend on ξ : There are hree represenaive agens in he marke: he marke maker, he insider and he noise rader. The role of he marke maker is o organize he marke. Tha is, according o he asse s aggregae demand, he marke maker ses he price of he asse and clears he marke. The insider is assumed o Graduae School of Engineering Sciences, Osaka Universiy, Machikaneyama cho -3, Osaka , Japan. kohasu@sigmah.es.osaka-u.ac.jp Deparamen de Probabilia, Lògica i Esadísica, Universia de Barcelona, Gran Via 585, 87 Barcelona, Spain. soriz@ub.edu

2 know a he beginning of he rading period some srong informaion, say λ L(Y ), no necessarily equal o ξ ; which depends exclusively on he oal demand Y. This agen uses his informaion in order o maximize his/her expeced profi. The noise rader represens all he oher paricipans in he marke. Noise rader s orders are a consequence of liquidiy or hedging issues and are assumed o be independen of λ; bu no necessarily of ξ : Thanks o he demand of he noise rader, denoed by Z; he marke maker canno observe he demand of he insider. Our formulaion is weak in he sense ha he vecor (ξ ;λ;z) is no given beforehand, in conras wih he previous lieraure on his subjec. The iniial daa in our formulaion is (µ;l) where µ is he law of ξ and he oher ingrediens of an equilibrium are par of he problem. The mahemaical moivaion for using a weak se-up is due o he fac ha in a srong formulaion he relaionship beween λ;ξ and Z can no be simply saed in general. This relaionship is no unique if one only wans o give as iniial daa he law of he final price. Furhermore, in general, ξ is no independen of λ or Z. However, i is assumed ha ξ is made public a he end of he rading period. Hence, ξ is incorporaed in he funcional o be opimized in he equilibrium. From he economic modelling poin of view, he siuaion can be explained as follows. Suppose he exisence of a financial conroller, say a member of an exchange commission, which would like o es afer he ime inerval has been oally observed (say [; ]) he large rader/insider behavior in a secor of he marke. By a secor, we undersand a collecion of homogeneous companies sharing a similar aciviy, for which one can assign a law for ξ ; is value a. The financial conroller observes he daa for differen companies in he secor and afer some renormalizaion we can regard he daa as differen realizaions or sample poins in his universe. ih he daa, a law µ for ξ can be inferred and a funcional L of he oal demand is fixed for esing. The firs sep for he conroller is o know if i is possible ha here exiss insiders rading in he socks of his secor using he informaion L(Y ) and being in equilibrium. Our paper addresses his quesion. The nex sep would be o design a saisical es according o he probabilisic properies of he equilibrium, bu we do no pursue his goal in his paper. Now, we briefly discuss he concep of weak equilibrium used in his paper. The difference beween he classical noion of equilibrium used in Back [2], Secion, and he one proposed here is ha in Back he informaion is exogenously given while here is also par of he definiion of weak equilibrium. In paricular, condiion vii) in Definiion 7 saes ha if we fix he noise rade process and he informaion, he sraegy used by he insider maximizes his expeced final wealh wihin a suiable admissible space. This condiion can be also inerpreed as a local equilibrium condiion because he noise rade process and he informaion are fixed. This inerpreaion is linked o he noion of parial (or local) equilibrium. If he insider finds himself/herself a such parial equilibrium poin here is no paricular reason o move from such a poin. From he poin of view of a financial conroller, he procedure is carried ou afer all he daa is available. Tha is, he final price has already been announced and he conroller wans o es he exisence of some insiders in he marke. Once he ype of informaion is seleced, one can saisically check if he sraegy used by he insider(s) is locally opimal. I is imporan o poin ou ha his opimal sraegy has he same funcional form as he compensaor of he Brownian moion wih respec o is naural filraion enlarged wih he random variable L(). Besides he weak equilibrium feaure, here are various delicae mahemaical poins where our resuls and echniques differ from previously menioned research. Briefly summarizing, we menion: ) Due o he generaliy of he funcional L, we use variaional calculus (or dynamic principle) and we do no obain an HJB equaion formulaion. In paricular, opimal sraegies do no depend only on he insider s addiional informaion and he value process, he admissible sraegies do no form a linear space and he expeced profi depends on ξ which i is no measurable wih respec o he insider s filraion. These feaures inroduce some difficulies in order o obain he opimaliy resuls. 2) One of he condiions of admissibiliy require ha he opimal sraegy has o be adaped o he filraion generaed by noise raders process and he insider informaion. This resul which was 2

3 easy o obain in previous aricles (in fac, his was jus a propery of Brownian bridges) becomes exremely difficul in he generaliy presened here. In fac, we consider as examples he case where L(Y ) max Y corresponds o he maximum of he demand and o he argumen of his maximum L(Y ) argmax Y. This leads o he sudy of exisence and uniqueness of soluions of sochasic differenial equaions wih pah dependen coefficiens which degenerae a random imes. Finally, we compare he expeced wealh obained by he large rader/insider in he wo main examples considered. The simulaions indicae ha knowing τ, he ime a which he maximum of he oal demand is achieved, gives less expeced wealh han knowing M; he maximum of he oal demand. As a final remark, we wan o sae ha one of he main goals of he aricle is o raise/conribue o he discussion on he issue of he equilibrium concep for insider-large rader for general informaion as i is explained in he aricle. e do no preend ha his is he unique way o solve he problem. e hope ha oher researchers will also presen alernaive proposals and commens on his model. The paper is organized as follows. In Secion 2 we give some basic definiions and inroduce our weak formulaion of equilibrium. Secion 3 conains he discussion of he opimizaion problem for he insider and a opimaliy equaion is deduced. In Secion 4 we relae he properies of he soluions of he opimaliy equaion wih he raionaliy of prices. Secion 5 is devoed o sae he main resuls on he exisence and uniqueness in law of a weak equilibrium. In Secion 6 we deal wih wo basic examples, previously reaed in he lieraure, he Back s example and an example on binary informaion. Secion 7 aims o inroduce wo new examples in he lieraure of equilibrium for asymmeric markes. Firs, in he case ha he addiional informaion held by he insider is he maximum of he oal demand and second, in he case of he ime a which his maximum is aained. e sae and prove he exisence and uniqueness in law of a weak equilibrium in boh cases. Finally, we numerically compare he expeced wealh obained by he insider in hese wo examples. Secion 8 is dedicaed o he conclusions. Finally, Secion 9 conains an appendix devoed o prove some echnical resuls. Throughou he aricle C will denoe a consan ha may change from line o line. L (X ) denoes he law of he random elemen X. 2 eak formulaion of equilibrium In his secion we inroduce he concep of weak equilibrium. Firs we define he class of pricing rules and admissible sraegies. Definiion e say ha a funcion F : [;] R! R saisfies an exponenial growh condiion if here exis posiive consans A;B such ha jf (;y)j Ae Bjyj ; for all (;y) 2 [;] R. Definiion 2 A pricing rule is a funcion H 2 C ;3 ((;) R); such ha H y (;y) > ;8 2 [;] and H;H ;H y ;H yy saisfy an exponenial growh condiion. e denoe by H he se of funcions H saisfying hese properies. In he previous definiion we require he pricing rules o saisfy some regulariy and growh condiions for echnical reasons. From a modeling poin of view, he imporan assumpion is he requiremen ha H y (;y) > ;8 2 [;]: This implies ha he insider can inver he price process o obain he oal demand and, hence, he noise rader demand, see Remark 8 c) below. Definiion 3 Given a process Z and a random variable M, we define Θ sup (M;Z) as he class of F I F Z _ σ(m)-adaped càglàd processes in [;) which saisfy sup jθ s jds 2 L (Ω); () Z s H s; θ u du + Z s θ s ds 2 L (Ω); (2) 3

4 and for all H 2 H : sup Z s H y s; θ u du + Z s θ s ds 2 L+ε (Ω); for some ε > (3) exp C sup θ s ds 2 L (Ω); 8C > (4) Remark 4 e could replace he echnical condiions ();(2);(3) and (4) in he definiion of Θ sup (M;Z) by he sronger ones jθ s j +ε ds 2 L (Ω); for some ε > (5) and exp C sup θ s ds 2 L (Ω); 8C > : (6) The advanage is ha condiions (5) and (6) define a linear space. On he oher hand, condiion (5) is difficul o verify in specific examples. Definiion 5 Given a process Z and a random variable M, independen of Z; we say ha a process X is a (M;Z)-sraegy process if here exiss θ 2 Θ sup (M;Z) such ha X R θ sds; 2 [;]: Definiion 6 Given a final price ξ 2 L 2 (Ω), a sochasic process Z, a random variable M, independen of Z; a price semimaringale process P (P ) 2[;] wih respec o F Z _ σ (M) and a (M;Z)-sraegy process X (X ) 2[;], we denoe by V V (X;P;ξ ) he agen final wealh defined by V (X;P;ξ ) V + X s dp s + (ξ P )X ; whenever he above sochasic inegral is well defined. Here V is a consan. Definiion 7 (eak Equilibrium) Le L : C[;]! R k be a measurable funcional on he canonical iener space and µ be a probabiliy measure on R wih R R x2 µ (dx) <. e say ha here exiss a (L; µ)-weak equilibrium if here exiss some probabiliy space (Ω;F ;P) where here exiss hree processes Y, θ and Z ; a random variable ξ, a random vecor λ and a funcion H 2 H such ha i) Y X + Z ; where X R θ s ds for 2 [;]: ii) λ L(Y ) is independen of he process Z : iii) Z is a Brownian moion. iv) ξ has he law µ: v) θ 2 Θ sup (λ ;Z ): vi) Prices are raional. Tha is, P vii) For all θ 2 Θ sup (λ ;Z ); one has, H (;Y ) E[ξ jf Y ] for 2 [;]: E[V X;P;ξ ] E[V X ;P ;ξ ]; where X R θ sds; Y θ X + Z and P H (;Y θ ). Now we give a series of remarks relaed o his definiion. 4

5 Remark 8 a)i is clear ha Z is a F Z _σ(λ )-Brownian moion, as Z is adaped o his filraion and is independen of λ. b)the price of he asse will be equal o ξ ; jus afer he release of informaion a ime : This price has o have he pre-specified law µ. Furhermore he relaionship beween ξ and λ is specified hrough he raionaliy of prices (propery vi) above) and in general ξ is no independen from Z : c)the naural definiion of he insider filraion is F F Z _ σ(λ ): This is due o he monooniciy of he pricing rule and he fac ha he insider observes he prices, one has ha a ime he insider can infer Z : Noe ha in general F Y is no necessarily included in F Z : d)the se of (λ ;Z )-sraegies is usually non empy. Furhermore, in he opimizaion problem vii), one may hink ha is more naural o resric he opporuniy se of sraegies o he ones saisfying L(Y θ ) λ. This means ha he insider would realize he giving srong informaion on he oal demand. In he nex secion, we will see ha he opimum, wih or wihou his resricion, i is he same given condiion ii) in he Definiion 7. e)from now on we will always assume ha µ is a probabiliy measure on R saisfying R R x2 µ (dx) < wihou any furher menion. Noe ha in he above (parial) equilibrium se up he insider opimizes his expeced profi given he informaion λ and Z. In his aspec, he above equilibrium is a parial one. In oher words, if he agen uses he sraegy θ, here is no (local) reason o change sraegy. I can also be considered as an sable poin where he insider can acually realize all he condiions for a sable marke. The above se-up and he subsequen proofs o follow are no consrucive. 3 Opimizaion problem for he insider In his secion we give necessary condiions for a process o solve he opimizaion problem saed in propery vii). Given a iener process Z and a fixed random variable M; which is independen of Z; we define F I F Z _ σ (M): As poined ou in he inroducion, we use he classical approach of variaional calculus. From now on, we denoe by a super-index θ on Y he dependence of he oal demand on he sraegy of he insider. Then, Y θ R θ sds +. Before sudying he opimizaion problem we remark he following propery for he porfolio process θ. H ;Y θ is a F I -semimaringale and is de- Lemma 9 If θ 2 Θ sup (M;Z) hen he price process P θ composiion is given by P θ P θ + fh (s;ys θ ) + 2 H yy(s;ys θ ) + H y (s;ys θ )θ s gds + H y (s;ys θ )dz s : The proof is a sraighforward applicaion of Iô s formula, as H 2 C ;2 ((;) R) and Y θ is a semimaringale in he filraion F I F Z _ σ(m). The nex lemma is obained using he inegraion by pars formula. Lemma Le θ be any F I -adaped process such ha R jθ sjds < a.s. Then, we have ha V, V (X;P θ ;ξ ) V + (ξ H(;Y θ ))θ d: ihou loss of generaliy we assume from now on ha V : The opimizaion problem we consider in his secion is max J (θ); where θ2θ sup (M;Z) e also denoe by θ, arg J (θ), E max θ2θ sup (M;Z) (ξ H(;Y θ ))θ d ; θ 2 Θ sup (M;Z): (7) J (θ) when his process exiss. The difficulies o solve his problem are due o he nonlineariy of he funcional J and he fac ha Θ sup (M;Z) is no a linear space. 5

6 Remark Noe ha, for θ 2 Θ sup (M;Z); we have ha jj (θ)j E jξ j θ d + E H(;Y θ )θ d < ; due o ξ 2 L 2 (Ω) and ha θ saisfies condiions (4) and (2): Furhermore, if θ is F I -adaped and saisfies he inegrabiliy condiions ha define Θ sup (M;Z), bu is no necessarily càglàd, hen we also have ha jj (θ)j < : The firs sep in our sraegy o solve he problem is o sudy he properies of J (θ) in he following linear subse of Θ sup (M;Z) : Θ b (M;Z) fθ 2 Θ sup (M;Z) : here exiss K > ; s.. 8ω, jθ s (ω)j Kg: Lemma 2 If v;θ 2 Θ b (M;Z); hen D v J (θ), d dε J (θ + εv)j ε E v (ξ H(;Y θ ))d E v s ds H y (;Y θ )θ d : (8) Furhermore, he operaor D J (θ) : v! D v J (θ) is linear. Proof. Firs noe ha for any θ; v 2 Θ b (M;Z) and ε > one has ha θ + εv 2 Θ b (M;Z) and differeniaing under he inegral sign (see Lemma 46) and applying Fubini s heorem, one obains Z d dε J (θ + εv)j ε E v f(ξ H(;Y θ )) H y (s;ys θ )θ s dsgd E v (ξ H(;Y θ ))d E v s ds H y (;Y θ )θ d : Remark 3 If θ 2 Θ sup (M;Z) is such ha E (ξ H(;Y θ )) hen for any v 2 Θ b (M;Z) we have E v (ξ H(;Y θ ))d H y (s;ys θ )θ s ds jf I ; (9) E v s ds H y (;Y θ )θ d : From now on, we refer o equaion (9) as he opimaliy equaion. The nex sep is o prove he concaviy of J (θ) in Θ b (M;Z): This is done in he following proposiion, which makes use of a general resul on convex analysis, see Proposiion 45. Proposiion 4 If H 2 H saisfies hen J (θ) is concave in Θ b (M;Z). H y (;y) + 2 H yyy (;y) ; () 6

7 Proof. e will show ha for every θ;η 2 Θ b (M;Z); we have D η θ J (θ) J (η) J (θ); which hanks o Proposiion 45 is equivalen o J (θ) being concave. Given θ;η 2 Θ b (M;Z);α 2 [;]; define δ, η θ and Ψ α, θ + αδ: For α 2 [;]; define ϕ (α), D η θ J (Ψ α ) dα d J (Ψα ): e will show ha ϕ (α) d2 J (Ψ α ). Firs, by Lemma 46 we have ha dα 2 " Z # ϕ (α) E 2 δ s ds H yy(;y α )Ψ α d] E[2 δ s ds H y (;Y α )δ d Then we apply inegraion by pars in he second expecaion o obain ha " # 2 E 2 δ s ds H y (;Y Ψα )δ d E δ s ds H y (;Y Ψα ) " E 2 δ s ds fh y(;y Ψα ) + 2 H yyy(;y Ψα ) + H yy (;Y Ψα )Ψ α gd # : : Hence, ϕ (α) E " " + E 2 δ s ds H y (;Y Ψα ) # 2 δ s ds fh y(;y Ψα ) + 2 H yyy(;y Ψα )gd # : Using ha H y > and equaion () we conclude ha ϕ (α) and herefore ϕ is a decreasing funcion. On he oher hand, an applicaion of he mean value heorem gives ha J (η) J (θ) D η θ J(Ψ α ) ϕ (α ); for some α 2 [;]. Therefore J (η) J (θ) ϕ () D η θ J (θ): The following heorem gives a sufficien condiion o find he opimal process in Θ sup (M;Z): Theorem 5 Le H 2 H saisfy (): If θ 2 Θ sup (M;Z) is such ha E (ξ H(;Y θ )) H y (s;y θ s )θ s ds jf I ; hen J (θ) J (θ );θ 2 Θ sup (M;Z): Proof. According o Proposiion 48 and Proposiion 49, here exiss a sequence fθ ;n g n2n Θ b (M;Z) such ha lim n! J (θ ;n ) J (θ ) and lim n! D θ θ ;nj (θ ;n ) ;8θ 2 Θ b (M;Z): Using Proposiion 4, we obain ha J is concave in Θ b (M;Z) and herefore we have ha θ 2 Θ b (M;Z) J (θ) J (θ ;n ) + D θ θ ;nj (θ ;n ): Therefore, aking limis one ges ha J (θ) J (θ ) for all θ 2 Θ b (M;Z): Using Proposiion 48 again, we have ha for all θ 2 Θ sup (M;Z) here exiss a sequence fθ n g n2n Θ b (M;Z) such ha lim n! J (θ n ) J (θ) J (θ ): 4 Properies of he soluions o he opimaliy equaion The following proposiion is imporan o find a sraegy θ saisfying he opimaliy equaion and yielding a raional price. I ell us ha given an insider s sraegy saisfying he opimaliy equaion, hen he price process associaed o his sraegy is raional if and only if he marke maker sees he associaed oal demand as a Brownian moion. In oher words, he associaed price process is raional if and only if he marke maker sees he oal demand as if only he noise rader was buying or selling socks. Moreover his suggess he connecion beween he opimal insider s demand and he compensaor of a Brownian moion wih respec o a enlarged filraion, see Remark 9 below. 7

8 Proposiion 6 Assume here exiss a process θ 2 Θ sup (M;Z) saisfying he opimaliy equaion (9): Then H(;Y θ ) is a F Y θ -maringale if and only if Y θ is a F Y θ -Brownian moion. Proof. Assume ha θ and Y θ equivalen o R θ s ds + saisfy he opimaliy equaion. Noe ha his equaion is H(;Y θ ) H y (s;y θ s )θ s ds E[ξ jf I ] M () where M E[ R H y(s;y θ s )θ s dsjf I ]: Making in (), we obain H (;) + M E[ξ jf I ]: Applying Iô s formula o H(;Y θ ) in equaion (), we ge fh (s;y θ s ) + 2 H yy(s;y θ s )gds H y (s;y θ s )dz s + E[ξ jf I ] E[ξ jf] I (M M ); (2) for all 2 [;]: The r.h.s of equaion (2) is a coninuous F I - local maringale wih iniial value and he l.h.s. is a finie variaion process wih coninuous pahs. Therefore, boh processes mus be idenically zero. Therefore, we have ha Combining he above equaion wih Iô s formula, we have H (;Y θ ) + 2 H yy(;y θ ) ; 2 [;]: (3) H(;Y θ ) H (;) + H y (s;y θ s )dy θ s : (4) If Y θ is a F Y θ -Brownian moion hen he sochasic inegral R H y(s;y θ s )dy θ s is a maringale due o Lemma 44. Therefore H(;Y θ ) is a F Y θ -maringale. Conversely, noe ha as H y > ; we can wrie Y θ R dh(s;ys θ ) H y (s;ys θ ) : Hence, if we assume ha H(;Y θ ) is a F Y θ -maringale, hen Y θ is a F Y θ -local maringale. As Y θ has he same quadraic variaion as Z we obain ha Y θ is acually a Brownian moion wih respec o is own filraion. Corollary 7 If here exiss a process θ 2 Θ sup (M;Z) saisfying equaion (9) and H(;Y θ ) is an F Y θ - maringale, hen H and ξ mus saisfy H (;y) + 2 H yy (;y) (5) and H(;Y θ ) E[ξ jf I ]: (6) Proof. Equaion (3) and he fac ha Y θ is a Brownian moion in is own filraion leads o equaion (5): Making in he opimaliy equaion (9), one obains (6). 5 Exisence and uniqueness in law of weak equilibrium e sar his secion wih a resul giving sufficien condiions o obain a (L; µ)-weak equilibrium. The firs condiion in Theorem 8 essenially says ha he law µ of he asse value ξ mus be a smooh ransformaion of a sandard normal random variable. Acually, in he examples of he following secions we do no specify he law µ bu he pricing rule H; which gives his smooh ransformaion. e will consider he pricing rules sudied previously in he lieraure, which are H (;y) y and H (;y) e y+( )2, see [2]. Noice ha he exponenial pricing rule has much more economical inerpreaion as i implies 8

9 ha prices are lognormally disribued. The second condiion says ha here exiss a Brownian moion such ha is a semimaringale wih respec o enlarged filraion F _ σ (L()): According o Proposiion 6, if he insider wans o obain a raional price process hen he oal demand Y mus be a Brownian moion wih respec o is naural filraion. Therefore, i is naural o impose ha he addiional informaion of he insider, given by L(Y ); is such ha he oal demand remains a Brownian moion wih respec o he enlarged filraion F Y _ σ (L(Y )) and hen use he compensaor as he insider s sraegy. The echnical condiion in order o carry ou his argumen is F Y _σ (L(Y )) F I plus some inegrabiliy condiions, which is he hird condiion in he heorem. From he economic poin of view, i seems reasonable o expec ha he insider can no held "oo much informaion" for an equilibrium o hold. In our framework his is refleced in he semimaringale propery of : In fac, if L(Y ) gives oo much informaion o he insider, hen will no be a a semimaringale wih respec o he enlarged filraion and, herefore, prices will no be raional. Alhough here exiss a general crierion o ensure ha a given funcional saisfies his semimaringale condiion, known as Jacod s crierion, see for insance [9], his crierion does no apply o our main examples L() max or L() argmax. In oher models of insider rading, where he raionaliy of prices is no aken ino accoun, his condiion is no sufficien o provide realisic models wih finie expeced wealh for he insider opimizaion problem, see [2]. Usually he informaion held by he insider has o be perurbed by some noise, see [2] and [6]. Theorem 8 (Exisence) Given a measurable funcional L : C [;]! R k and µ a probabiliy measure on R saisfying R R x2 µ (dx) < : Assume: ) There exiss H 2 H such ha i saisfies (5) and µ (A) p 2π Z H(;) (A) e x2 2 dx; 8A 2 B (R): 2) There exiss a probabiliy space (Ω;F ;P) supporing a Brownian moion which is a semimaringale in he filraion F _σ (λ); λ, L(); wih semimaringale decomposiion R α (s;λ)ds+ λ ; where λ is a F _ σ (λ)-brownian moion. 3) α 2 Θ sup (λ; λ ): Then is a (L; µ)-weak equilibrium. (Y ;θ ;Z ;H ;ξ ;λ ) (;α(;λ); λ ;H;H (; );λ) Proof. Verificaion of properies i), iii), iv) and v) in he definiion of weak equilibrium is sraighforward. Propery ii) follows from he fac ha λ is an F _ σ (λ)-brownian moion and, hence, independen from F _ σ (λ) σ (λ): From hypohesis ) ogeher wih equaion (4); we have ha H (; ) is a F -maringale. As H (; ) E[H (; )jf ]; propery vi) follows. To check propery vii), we apply Iô s formula o H(; ) in he l.h.s. of he opimaliy equaion (9) wih θ α. Due o hypohesis ) we obain ha i is equal o E H y (s; s )ds λ jf λ _ σ (λ) : On he oher hand, hypohesis 3) implies ha α (;λ) is F λ _ σ (λ)-adaped, which enails ha is F λ _ σ (λ)-adaped and one can conclude ha F _ σ (λ) F λ _ σ (λ): Hence, due o Lemma 44, he above condiional expecaion equals o zero and he conclusion follows from Theorem 5. Remark 9 Of he hree hypohesis in he previous Theorem, hypohesis 3 is difficul o verify in general. Besides he inegrabiliy condiions in he definiion of Θ sup (λ; λ ), α (;λ) mus be F λ _ σ (λ)- adaped. This propery will follow if F _ σ (λ) F λ _ σ (λ). This problem seems o be difficul o solve in general. 9

10 e deal wih his problem in each of he examples o follow in he nex secions. The general sraegy is o show exisence and uniqueness for s.d.e. s of he form X α s;g;x [;s] ds +V ; where V is a Brownian moion, α is a (degenerae) funcional and G is a random variable independen of V: Therefore, X would be F V _ σ (G)-adaped. The following heorem gives a uniqueness resul for he (L; µ)-weak equilibrium found in he previous heorem. Condiion 6) in he following heorem deserves a commen. This assumpion roughly says ha wo weak equilibriums have he same law whenever are obained hrough a semimaringale decomposiion of a Brownian moion wih respec o a enlarged filraion. In oher words, if in Condiion 2) of Theorem 8 we use wo differen Brownian moions possibly defined in wo diferen probabiliy spaces, he wo differen weak equilibriums obained have he same law. From he economic poin of view, his assumpion saes ha if he marke maker knew he insider s adiional informaion hen he would have exacly he same informaion flow as he insider. Theorem 2 (Uniqueness in law) Assume he same hypoheses of Theorem 8 and denoe by (Y ;θ ; Z ;H ;ξ ;λ ) he (L; µ)-weak equilibrium. Suppose ha here exiss anoher probabiliy space supporing processes (Y;θ;Z) such ha ) Y R θ sds + ; 2) λ, L(Y ) is independen of Z; 3) Z is a Brownian moion in is own filraion; 4) θ 2 Θ sup (λ;z); 5) H (;Y ) E[H (;Y )jf Y ] for 2 [;]: 6) F Z _ σ (λ) F Y _ σ (λ): Then, we have ha L (Y ;X ;Z ;ξ ;λ ) L (Y;X;Z;ξ ;λ); where ξ, H (;Y ); and herefore E[V (X;P;ξ )] E [V X ;P ;ξ ]: Proof. Applying Iô s formula in he filraion F I F Z _ σ (λ); we have ha ξ H (;Y ) Hy (s;y s )θ s ds Hy (s;y s )dz s ; where in he las equaliy we have used ha H saisfies equaion (5): Afer aking condiional expecaion, his yields E ξ H (;Y ) Hy (s;y s )θ s ds j F I : Then, by Theorem 5 we have ha J (η) J (θ);8η 2 Θ sup (L(Y );Z). By hypohesis 5) and Proposiion 6 one ges ha Y is a Brownian moion in is own filraion. Therefore, L (Y;λ;ξ ) L Y ;λ ;ξ : As he process θ is adaped o F Y _σ (λ ), hen i can be wrien as θ Λ(;Y[;] [;] )); Pλ-a.s.. Then, defining ˆθ, Λ(;Y [;] ;L(Y [;] )) and using ha L (Y ) L (Y ); we have ha E Y Y s ˆθ u dujfs Y _ σ (λ) : s Thus, Y ˆθ s ds + M θ s ds + ;

11 where M is a F Y _ σ (λ)-maringale. Given he assumpion 6); he uniqueness of he semimaringale decomposiion of Y wih respec o F Y _ σ (λ) proves ha b θ θ;p λ a.s. The following resul is helpful when proving ha α 2 Θ sup (λ;z). Proposiion 2 Le Y be a Brownian moion and λ L(Y ): Assume ha Y has a semimaringale decomposiion wih respec o F Y _ σ (λ) given by Y R α sds + ; where Z is a F Y _ σ (λ)-brownian moion. Then, exp C sup α s ds 2 L p (Ω); p ; 8C > ; and sup F (s;y s )α s ds 2 Lp (Ω); p ; where F is any funcion saisfying an exponenial growh condiion. Proof. To prove he firs saemen, noice ha C exp p sup α s ds exp(pc sup jy j)exp(pc sup j j): By he Cauchy-Schwarz inequaliy, aking ino accoun ha Y and Z are Brownian moions, we obain ha p 2 E exp C sup α s ds E[ exp(2pc sup jy j)] < : To prove he second saemen, no ha p p F (s;y s )α s ds C (p) F (s;y s )dy s + Define M, F (s;y s )dy s and M 2, F (s;y s )dz s : F (s;y s )dz s p Here, M is a F Y -local maringale. By he BDG inequaliy (see Theorem 73, pag. 222 in [9]), aking ino accoun ha F saisfies an exponenial growh condiion and ha Y is a Brownian moion, we obain ha E sup M p C p E C p E C p A p E " " # p2 F (s;y s ) 2 ds) # p2 A 2 exp(2bjy s j)ds exp(pb sup jy j) Thus M is a F Y -maringale and sup M 2 L p (Ω); p : e can repea he same argumen for M 2, aking ino accoun ha M 2 is a F Y _ σ (λ)-local maringale. < : 6 Back s example and an example of binary informaion In his secion we commen on wo known examples where he general resul in Theorem 8 applies. Throughou his secion we will consider a Brownian moion defined on a complee probabiliy space (Ω;F ;P): From now on, we denoe by φ (x;) he densiy of a cenered Gaussian random variable wih variance, by Φ(x; ) is disribuion funcion and Φ(x; ) Φ(x; ).

12 In all he examples o follow in he nex secions, we assume ha µ is a probabiliy measure on R wih R R x2 µ (dx) < and ha here exiss H 2 H saisfying (5) and Z µ (A) φ (x;)dx; 8A 2 B (R): H(;) (A) Theorem 22 Le be a Brownian moion. Then is a semimaringale respec o he filraion F _ σ ( ) wih decomposiion where is a F _ σ ( )-Brownian moion, α (u; )du + ; 8 2 [;]; (7) for all 2 [;): α (; ) ; (8) The previous resul is well known and is proof can be found, for insance, in [], Théorème. In [], Corollaire., i is also discussed he connecion beween he Brownian bridge f g < and he Brownian moion f g < ; showing ha hese wo processes have he same naural filraion and ha is independen of f g < : This idea is laer used in order o consider equaion (7) as a linear equaion, where he unknown funcion is (ω) and (ω) and (ω) are given. The following resul is slighly more general han Corollaire., in [], in he sense ha if we assume ha we are given a Brownian moion B and a random variable G, independen from B and no necessarily Gaussian, we can consruc a process X wih erminal value G: In he paricular case ha L (G) N (;); he process X is a Brownian bridge wih X G: Theorem 23 Le B be a Brownian moion and G a random variable independen of B; boh defined in he same probabiliy space (Ω; F ; P): Then here exiss a unique srong soluion X adaped o he filraion F B _ σ (G) of he following sochasic differenial equaion X G X s s ds + B ; 2 [;]: (9) Furhermore, if we assume ha he law of G is N(;); hen X is a Brownian moion wih respec is own filraion. Proof. As G is independen of B; one has ha B is F B _ σ (G)-Brownian moion. Using as an inegraing facor ( ) ; we obain X G d ( ) 2 d + db : Therefore, one has ha X G + ( ha he process ) R db s s ; < : In lemma 6.9 of [3], pag. 358, i is proved B ( db s ) s ; < ; B ; is a coninuous, cenered Gaussian process wih covariance funcion s ^ s: Hence we have proved exisence and uniqueness for he soluions of he equaion (9). If we assume ha G N (;), we have ha G is a coninuous, cenered Gaussian process wih covariance funcion s. As he sum of wo independen Gaussian processes is sill a Gaussian process and B and G are independen, we obain ha X is a coninuous, cenered Gaussian process wih covariance funcion s ^ ; hus a sandard Brownian moion. 2

13 The following propery is imporan o deermine he finieness of opimal uiliies. For p > ; E[ R jα (;G)jp d] < if and only if p < 2; where α (;x) x X ; 8 2 [;]: Le s sae he weak equilibrium resul for his case. Theorem 24 Le L(Y ) Y : Then is a (L; µ)-weak equilibrium. Y ;θ ;Z ;H ;ξ ;λ (X;α (;G);B;H;H (;G);L(X)) In his paricular case he above weak equilibrium is in fac a srong ype equilibrium. For his, see Theorem in [2] or Proposiion 2 in [5]. Theorem 25 Assume ha we are given a Brownian moion Z and a srong informaion ξ : Assume ha H 2 H saisfies (5) and ξ H (;N (;)): Se θ, α(;(h ) (;ξ )): Then (H ;θ ) is an equilibrium. Tha is, H (;Y ) is a raional price, ha is H (;Y ) E[ξ jf Y ]. For all θ 2 Θ sup (ξ ;Z); one has where X () R θ () s E[V (X;P;ξ )] E[V (X ;P ;ξ )]; ds; Y () X () + Z and P () H (;Y () ): Now we consider he case in which he insider knows ha he oal demand a ime is greaer or equal o a fixed consan a: The nex wo resuls are quoed from [2], example 4.6. Theorem 26 Le be a Brownian moion. Then is a semimaringale respec o he filraion F _ σ [a; ) ( ) wih decomposiion α u; [a; ) ( ) du + a ; 8 2 [;]; where a is a F _ σ [a; ) ( ) -Brownian moion, for all 2 [;]: α ; [a; ) ( ) φ ( a; ) Φ( a; ) [a; ) ( ) + φ ( a; ) Φ(a ; ) [a; ) c ( ); Lemma 27 e have ha E[ R α ;[a; ) ( ) 2 d] < : Theorem 28 Le B be a Brownian moion and G a Bernoulli random variable independen of B; boh defined on he same probabiliy space (Ω;F ;P): Then here exiss a unique srong soluion X adaped o he filraion F B _ σ (G) of he following sochasic differenial equaion φ (X a; ) X Φ(X a; ) fg (G) + φ (X a; ) Φ(a X ; ) fg (G) ds + B ; < : (2) 3

14 Proof. Firs we will prove ha Ψ a (x;), φ (x a; )Φ(x a; ) is Lipschiz in he x variable for 2 [;); fixed. Noe ha we can ake a ; wihou loss of generaliy. Furhermore, Ψ (x;) Ψ (xp ;) p : e have ha x Ψ (x;) x φ (x; )Φ(x; ) (φ (x; ))2 (Φ(x; )) 2 x Ψ (x;) Ψ (x;) 2 x p Ψ (x p ;) + Ψ (x p 2 ;) : Fix < ; hen sup x Ψ (x;) 2[; ];x2r sup yψ (y;) + Ψ (y;) 2 : y2r Applying l Hospial s rule, i can be shown ha lim y! yψ (y;) + Ψ (y;) 2 ; lim y! yψ (y;) + Ψ (y;) 2 : which enails ha sup 2[; ];x2r x Ψ (x;) <. Therefore, Ψ a (x;) is Lipschiz in he x variable uniformly in 2 [; ]; < : To sudy he growh of Ψ a (x;) we ake a : Then, Ψ (x;) p sup Ψ (y;); y2r for 2 [; ]; < : I can be shown ha lim y! Ψ (y;) and lim y! Ψ (y;)y ; which implies ha sup y2r Ψ (y;) < : Hence, Ψ a (x;) saisfies a linear growh condiion, for 2 [; ]; < : Using he classical resuls on s.d.e. s, we have ha here exiss a unique srong soluion o he following equaion Y Ψ a Ys ;s ds + B ; < : e can use a similar reasoning for Ψ 2 a (x;), φ (x a; )Φ(a x; ) and ge he same conclusions. Finally, he F B _ σ (G)-adaped process X, Y fg (G) +Y 2 fg (G) solves our problem. Theorem 29 Le L(Y ) [a; ) (Y ): Then is a (L; µ)-weak equilibrium. Y ;θ ;Z ;H ;ξ ;λ (;α ; [a; ) ( ) ; a ;H;H (; );L()) Proof. e apply Theorem 8. The firs hypohesis of he heorem is assumed. The second hypohesis follows from Theorem 26. Finally ha α 2 Θ sup ( [a; ) ( ); a ) follows from Lemma 27, Proposiion 2 and Theorem 28 (see Remark 9). 7 The maximum and is argumen In his secion we deal wih wo examples ha are more complicaed, bu by far more ineresing. In paricular, he second example is new in he lieraure of insider rading wih iniial srong informaion. Throughou his secion we will consider a Brownian moion defined on a complee probabiliy space (Ω;F ;P): e consider he maximum process in he inerval [s;]; M s; ; s < defined by M s;, 4

15 max su u : To simplify noaion we use M, M ;, τ, argmax s s, M, M, τ, τ : and γ s;, M s; s. The densiy and disribuion funcion of γ s; are given by p 2 (x; s), 2φ (x; s) (; ) (x) and Π 2 (x; s), R x p 2 (z; s)dz. Similarly, he densiy of he random vecor (γ s; ; s ) is given by p (x;y; s), ( 2(2x y) p exp 2π( s) 3 (2x y) 2 2( s) ) (; )( ;x) (x;y): Le us recall a heorem by Lévy ha links he maximum process M wih he Brownian local ime L x (): Theorem 3 The pairs of processes f(m he same laws under P: ;M ); < g and f(j j;2l ()); < g have For more deails, see [3]; chaper 3, Theorem 6.7. Furhermore, i is easy o show ha, for a fixed ; M γ ; : Finally, we se ϕ (x;), p 2 (x;) Π 2 (x;) x 2 2 e R x e y2 2 dy (; ) (x): 7. L(Y ) max 2[;] Y In his subsecion we consider he case in which he insider knows he maximum of he oal demand. A more general version of he following resul is proved in Jeulin [] (see Proposiion 3.24, pag. 49). See also Mansuy and Yor [8] for an updae reference on enlargemen of filraions heory. Theorem 3 Le be a Brownian moion. Then is a semimaringale respec o he filraion F _ σ (M) wih decomposiion α (u;m)du + M ; 8 2 [;]; where M is a F _ σ (M)-Brownian moion, α (;M) M Noe ha fm <Mg [;τ) (): fm <Mg ϕ (M ; ) fm Mg: h R i Lemma 32 e have ha E R i jα (;M)jd < and Eh jα (;M)j2 d : Proof. To deduce he convergence of he firs expecaion, noice ha E α (;M)d E[ ] E[ M ] ; which implies M E fm <Mg d E fm Mgϕ (M ; )d : As he inegrands in he above expecaions are posiive, he problem is reduced o show M E fm <Mg d < : 5

16 Le s compue his expecaion E Condiioning wih respec F fm>m g Z Z y M d E E fm; >M g d γ ; d : M ; fγ; >M g and using Lemma 5, his expecaion is equal o r x 2 p 2 (x; ) p 2 (y;)dxdyd π < : To show he divergence of he second momen, noice ha E α (;M) 2 d " M 2 Z E fm>m g d# + E fmm g (ϕ (M ; )) 2 d : Therefore, i suffices o show he divergence of one of he above expecaions. The second expecaion above is equal o E fm >M ;g (ϕ (M ; )) 2 d Z M E (ϕ (M ; )) 2 p 2 (x; )dxd Z Z y Z Bu his inegral is infinie, because and his implies ha R (ϕ (y; )) 2 p 2 (x; ) p 2 (y;)dxdyd (p 2 (y; )) 2 Π 2 (y; ) p 2 (y;)dyd: lim y(p 2 (y; )) 2 y! + Π 2 (y; ) p 2 (y;) p 2 (; ) p 2 (;) 2 π p 6 ; ( ) (p 2 (y; )) 2 Π 2 (y; ) p 2 (y;)dy ;8 2 [ε; ε]; which is a se of posiive Lebesgue measure provided ε < 2. In order o verify ha α (;M) is F M _ σ (M)-adaped we prove ha is F M _ σ (M)-adaped, which follows from he following resul. Theorem 33 Le B be a Brownian moion and G a posiive random variable independen of B; boh defined in he same probabiliy space (Ω;F ;P): Then here exis a unique srong soluion X adaped o he filraion F B _ σ (G) of he following sochasic differenial equaion X where M X, max s X s. G Xs s fm X s <Gg ϕ (G X s ; s) fm X s Gg ds + B ; (2) Proof. Our approach o he soluion of (2) is o wrie X X [;ρ) () + X 2 [ρ;) (), where ρ, inf : X G ; X and X 2 are he soluions o he following s.d.e. s E : X G Xs s ds + B ; < ρ; 6

17 and E 2 : X 2 G ρ ϕ G X 2 s ; s ds + B B ρ ; ρ < ; which we denoe by E and E 2, respecively. The nex sep is o show he exisence and uniqueness of he soluions o E and E 2 : Noe, ha ρ is a F B _σ (G)-sopping ime. Exisence and uniqueness for he soluion of E : Follows as in he case of he Brownian bridge (see lemma 6.9 of [3], pag. 358). Exisence and uniqueness for he soluion of E 2 : Noe ha he drif has a singulariy a ρ: Tha is, lim x! + ϕ (x;) ; > : Insead of proving exisence and uniqueness for E 2 ; we will prove i for he following equivalen s.d.e. E2 : R ϕ (R s ; ρ s)ds + N ; < ρ: The s.d.e. E2 is obained from E 2 hrough he change of variables R G X+ρ 2 and N (B +ρ B ρ ): The exisence is proved in Proposiion 34. To prove he uniqueness, we may consider, R R 2 he difference of wo posiive soluions R and R 2 of E2 : Then, applying Iô s formula o ; we obain ha P-a.s. 2 ^( ρ) 2 ^( 2 ρ) ^( ρ) as ϕ (x;) ϕ (y;) if x y for all 2 (;). R s R 2 s ϕ R s ; ρ s ϕ R 2 s ; ρ s ds ; Proposiion 34 There exiss a posiive, coninuous, srong soluion wih respec o F N _ σ (ρ) o R ϕ (R s ; ρ s)ds + N ; < ρ; (22) where N is a iener process and ρ 2 (;) is a random variable independen of F N. Proof. Firs of all, noe ha xϕ (x;) ;8 > ;x 2 R: (23) e define ϕ n (x;), expf n ( x + )gϕ (x;); which saisfies (23) wih ϕn insead of ϕ: This sequence of funcions is monoone increasing in n, bounded and converges o ϕ (x;) for each x 2 R; > such ha x + > : Furhermore, x ϕ n (x;) expf n ( x + )g n x 2 ϕ (x;) + x ϕ (x;) x nx 2 ϕ (x;) ϕ n (x;): Using inequaliy (23); one obains ha sup x2[; );2[;] j x ϕ n (x;)j < ; which implies ha ϕ n (x;) is a Lipschiz funcion. Therefore, for a fixed n 2 N; we have he exisence and uniqueness of soluions for he following s.d.e. R n ϕ n (R n s ; ρ s)ds + N ; < ρ: By a comparison heorem, we have ha P R n+ R n ; < ρ ; which shows ha R, lim n! R n ; < ρ exiss almos surely in ( ; ] and i is a measurable process as i is a limi of measurable 7

18 processes. Now, we show ha for 2 [; ρ);r < ;P-a.s. and R saisfies equaion E2 : In order o prove he firs propery, we show he uniform inegrabiliy in n 2 N of R n ; < ρ: Applying Iô s formula, we obain (R n ) R n s ϕ n (R n s ; ρ s)ds + 2 R n s dn s ; < ρ: Nex, we bound he expecaion of he second erm above: e obain ^( ρ) E ^( ρ) R n s ϕ n (R n s ; ρ s)ds E ^( ρ) E fr n s >gr n s ϕ (R n s ; ρ s)ds h R ^( ρ) For he hird erm, one has E jr n s jϕ (Rn s ; ρ s)ds ^( ρ) E fr n s >gds : h(r n ^( ρ) )2 i 3: This implies he i R n udn u : Thus, sup n2n E uniform inegrabiliy of R^( n ρ) and herefore R ^( ρ) 2 L (Ω): Nex, we show ha R saisfies E2 : Firs noe ha R ^( ρ) lim To conclude he proof we show ha R n n! ^( ρ) ^( ρ) lim n! ϕ n (R n s ; ρ s)ds + N ^( ρ) : ^( ρ) ^( ρ) lim ϕ n (R n n! s ; ρ s)ds ϕ (R s ; ρ s)ds; < ; wih probabiliy : This will also give he coninuiy for he pahs of R: Fix ε > and define ρ ε, inff 2 (; ρ) : N εg; ρ ε l, inff 2 ρ ε l ; ρ : N N ρ ε R l ρ ε 2g; l : l By consrucion, he sequence fρ ε l g l2n is nondecreasing and herefore we can define σ ε, lim ρ ε l! l : For fixed ω 2 Ω; we apply he dominaed convergence heorem in each inerval [ρ ε l ;ρε l ]; l. One has ha, R R ρ ε l + ρ ε l ϕ R s ; ρ s ds + N N ρ ε l > R ρ ε l 2 ε 2 l ; for 2 [ρ ε l ;ρε l ] and l ; due o he posiiviy of he inegral. Then, using inequaliy (23), we have for s 2 [ρ ε l ;ρε l ] ha ϕ n (R n s ; ρ s) ϕ (R n s ; ρ s) ϕ R s ; ρ s R s 2l ε : Hence, by he dominaed convergence heorem This implies ha Z ρ ε Z l ρ ε lim ϕ n (R n l n! ρ ε s ; ρ s)ds ϕ (R s ; ρ s)ds: l ρ ε l R R ρ ε + ϕ (R s ; ρ ρ ε s)ds + N N ρ ε ; ρ ε < σ ε : 8

19 e prove now ha σ ε ρ: If ω 2 Ω is such ha here exiss l for which ρ ε l ρ; we have finished. By conradicion, assume ha he sequence ρ ε l is sricly increasing. Firs of all, by he l2n definiion of ρ ε l l2n and he fac ha he sequence is sricly increasing, one has ha N ρ ε N l ρ ε l R ρ ε Taking limis we obain ha R l 2: σ ε ; due o he coninuiy of Brownian pahs. Then R + R σ ε ϕ R s ; ρ s ds N N σ ε ; bu his conradics he law of ieraed logarihm when ends o σ ε, because he lef hand side is posiive almos surely for 2 [ρ ε ;σε ). Hence we can conclude ha he se of ω 2 Ω for which does no exis a finie l such ha ρ ε l ρ is a null se. Now, noice ha ρ ε # when ε # : Hence, N ρ ε! and by monoone convergence ε# Therefore, lim ϕ (R s ; ρ s)ds ϕ (R s ; ρ s)ds: ε# ρ ε R limr ρ ε ε# + ϕ (R s ; ρ s)ds + N ; < ρ: As R lim R n n! ; making in he above equaion we obain lim ε# R ρ ε : Furhermore, as jr j < ; P-a.s. we obain ha R ϕ (R s; ρ s)ds < ;P-a:s:; for < σ: Hence we have showed ha R saisfies equaion (22): Noe ha in paricular, we have also proved ha R > : Theorem 35 Le L(Y ) max Y. Then Y ;θ ;Z ;H ;ξ ;λ (;α (;M); M ;H;H (; );L()) saisfies all he requiremens o be a (L; µ)-weak equilibrium excep he càglàd propery in he condiion v). Proof. Properies i) hrough iv) in he definiion of weak equilibrium follow direcly. Propery v) wih he excepion of he càglàd propery follows from Lemma 32, Proposiion 2 and Theorem 33 (see Remark 9). From he assumpions on H and µ and equaion (4); we have ha H (; ) is a F -maringale. As H (; ) E[H (; )jf ]; propery vi) follows. Le s check propery vii). To simplify he noaion we se α, α (;M); : Noe ha α if τ and α if > τ: From his propery, i easily follows he following inequaliy Z τ jα jd α d which combined wih Proposiion 2 gives ha τ α d 3 sup α s ds ; (24) jα jd 2 L p (Ω); p : (25) For ε 2 (;); define τ ε;+ (τ + ε) ^ : Then he process α ε fα ε, α (τ;τ ε;+ ] c (); 2 [;]g converges Pλ; a.e. o α as ε # and i saisfies jα ε j jαj: Now we will prove ha α ε 2 Θ sup (M; M );8ε 2 (;): Firs, he càglàd propery of α ε follows from he fac ha his approximaion avoids he essenial disconinuiy of α in τ: The inegrabiliy propery () is rivial. Propery (4) follows from equaion (24): The proof of properies (2) and (3) are similar. e will prove propery (2): e have ha sup H(s;Y αε s )α ε s ds sup H(;Y αε ) jα jd; which belongs o L (Ω) by he Cauchy-Schwarz inequaliy, propery (25) and Lemma 44. According o Proposiion 48, lim n! J (α ε;n ) J (α ε ) for all ε 2 (;) where α ε;n is defined according o Definiion 47 wih θ α ε. As he funcional J is concave in Θ b (M; M ); we obain ha J (η) J (α ε;n ) + D η α ε;nj (α ε;n ) for η 2 Θ b (M; M ). 9

20 lim ε# J (α ε ) J (α): This is analogous o he proof of Proposiion 48. Noe ha using propery (24), we have ha R α α ε d 2 C sup R α 2 and H (;Y α )α H(;Y αε )α ε d H (;Y α )(α α ε )d + H (;Y α ) H(;Y αε ) α ε d C sup jh (;Y α )j jα jd +C sup H y (;Y αε +r(α α ε ) ) Z 2 dr jα jd : This gives sufficien inegrabiliy properies o apply he dominaed convergence heorem. Noe ha as in he proof of Lemma 44, sup H y (;Y αε +r(α α ε ) ) C exp 9B sup α s ds expfb sup j jg: (26) lim ε# lim n! D η α ε;nj (α ε;n ) : Repeaing he proof of Proposiion 49, we obain Dη α ε;nj (α ε;n ) B ε;n + B ε;n 2 ; where B ε;n, E η α ε;n H (;Y α ) H(;Y αε;n ) d and B ε;n 2, E (η s α ε;n s )ds H y (;Y α )α H y (;Y αε;n )α d ε;n : Le s show ha lim ε# lim n! B ε;n : This follows by dominaed convergence, once we have R shown ha sup ε;n η α ε;n H (;Y α ) H(;Y αε;n ) d 2 L (Ω); because lim n! α ε;n α ε ; P λ-a.s. and lim ε# α ε α P λ-a.s. Using inequaliies (24) and (26) we obain η α ε;n H (;Y α ) H(;Y αε;n ) d η α ε;n H y (;Y αε;n +r(α α ε;n ) )dr Y α Y αε;n d Z C + jα jd H y (;Y αε;n +r(α α ε;n ) )dr α α ε;n d; sup which is in L (Ω); because as in Lemma 44, sup j j and sup R α sds have exponenial momens. The proof of lim ε# lim n! B ε;n 2 can be obained similarly. Therefore, we have proved ha J (η) J (α);8η 2 Θ b (M; M ): The final resul follows from he applicaion of Proposiion 48, using an argumen as in he end of he proof of Theorem L(Y ) argmax 2[;] Y In his secion we consider he case in which he insider knows he ime a which he oal demand achieves is maximum. The firs par of his subsecion is devoed o obaining he compensaor of wih respec o he filraion F _σ (τ); which we will denoe by F τ ff τ ; g. This will be done dividing he problem ino wo pars: before he random ime τ and afer i. Bu firs, we give he condiional law of τ given F : 2

21 Proposiion 36 The condiional law of τ given F ; is ( P τ > ujf fmu γ u; + ug p 2 (M u ; ) if u < R u r (M ;v ; v)dv if u ; where r (x;s;) is given by r (x;s;), p 2 2π φ (x;s) (; ) (x) p 2π p 2 (x;s): Moreover, P τ > ujf is coninuous in u; P-a:s: Proof. If u < ; hen P τ > ujf P M u < M u; jf P Mu < M u; _ M ; jf P M u < M u; ;M u; > M ; jf + P Mu < M ; ;M u; M ; jf fmu <M u;g P M u; > γ ; + jf + P γ; > (M u _ M u; ) jf fmu <M u;g fmu <M u;g Z Mu; Z Mu; fmu γ u; + ug p 2 (z; )dz + p 2 (z; )dz + Z Mu p 2 (z; Z p 2 (z; (M u _M u;) Z (Mu _M u;) )dz: )dz p 2 (z; If u > ; he calculaions are more involved, he idea is o break he maximum processes ino pieces ha are independen of F and pieces ha are F -measurable. P τ > ujf P M u < M u; jf P M _ M ;u < M u; jf P M < M u; ;M M ;u jf + P M;u < M u; ;M < M ;u jf P M < γ u; + u ;M γ ;u jf +P α ;u < γ u; + u ;M < γ ;u jf : )dz Hence, P M < γ u; + u ;M γ ;u jf Z M Z x Z Z M Z x Z M Z M _y Z M Z M y p (x;y;u ) p 2 (z; u)dzdydx 2p (x;y;u )Φ(M y; u)dydx 2p (x;y;u )Φ(M y; u)dxdy 2(φ(jyj;u ) φ (2(M ) y;u ))Φ(M y; u)dy 2(φ (jm zj;u ) φ (M + z;u ))Φ(z; u)dz 2

22 On he oher hand, P γ ;u < γ u; + u ;M < γ ;u jf Z Z x M Z Z x M Z Z M Z x y p (x;y;u ) p 2 (z; u)dzdydx 2p (x;y;u )Φ(x y; u)dydx 2p (x;x z;u )Φ(z; u)dzdx 4φ (M + z;u )Φ(z; u)dz: Summing up, and aking ino accoun ha φ(jzj;u ) φ (z;u ); we obain P τ > ujf Z 2fφ (M z;u ) + φ (M + z;u )gφ(z; u)dz: Differeniaing under he inegral sign, we obain ha here exiss a densiy funcion r such ha P τ > ujf r (M ;v ; v)dv: u Furhermore, his densiy is smooh in all is variables due o he regulariy of φ and Φ: For he explici compuaion of his densiy we refer o [4]. To conclude he proof we only need o show ha P τ > ujf, as a funcion of u; is coninuous in u : e have ha lim P τ > ujf u! lim P M u < M u; jf u! P M M ; jf ; where we have used he dominaed convergence heorem for condiional expecaions and he P-a:s: coninuiy in of he pahs of M and M ; : Proposiion 37 If s ; we have ha M u u E fτ>g τ u du jf τ s fτ>g s Z s M u u τ u du : Proof. Le A 2 Fs and f a bounded Borel measurable funcion, hen aking ino accoun ha τ has a condiional densiy given F ; in he se fτ > g; we have ha E A f (τ) fτ>g ( s )] E[ A E f (τ) fτ>g jf ( s ) E A f (u)r (M ;u ; u)du( s ) : Applying Theorem 3 and Tanaka s formula, we obain ha he las expecaion is equal o E A f (u)r (j j;u ; u)du 2 dlv () d j v j s s E A f (u)r (j j;u ; u)du sgn( v )d v : 2 Noice ha r (j j;u ; u) p φ ( ;u ). Using Iô s formula, we can wrie 2π( u) r (j j;u ; u) Z 2 2 p φ (;u) + p x φ ( v ;u v)d v ; 2π ( u) 2π ( u) s 22

23 Then, he former expecaion is equal o " # 2 E A f (u) sgn( v ) p x φ ( v ;u v)dvdu s 2π ( u) E A f (u) E A f (u) E A E s j v j u v r (j vj;u v; u)dvdu s M v s v u v r (M v v ;u v; u)dvdu dv E fτ>g f (τ) M v v τ v jf v A f (τ) fτ>g s v τ v dv : As he σ-algebra Fs τ is generaed by elemens of he form A f (τ), where A 2 Fs and f is a bounded Borel funcion, we obain he resul using elemenary properies of he condiional expecaion. Noe also, ha (M ;M ) and (j j;2l ()) are no he same processes. e can inerchange hem because we are dealing wih expecaions, and herefore hey only depend on he law of he processes, which are equal by Theorem 3. Now, we are going o prove an analogous resul for he case afer he ime τ. In he proof we will use he decomposiion of wih respec o F _σ (M) (see Theorem 3). Proposiion 38 If s, we have ha E fτsg + ϕ (M u ; u)du jf τ s M v Z s fτsg s + ϕ (M u ; u)du Proof. Le A 2 Fs and f (τ) fτrg ; where r : e have ha E A f (τ) fτsg ( s ) E A fτrg fτsg ( s ) E A fτr^sg fτsg ( s ) E A fms^r Mg fτsg ( s ) E A f (τ) fτsg ϕ (M u ; u)du : s Noice ha fτr^sg fms^r Mg is Fs _σ (M)-measurable, and ha ϕ(m u ; u) is Fu τ -measurable because M M τ. The elemens of he form A f (τ);where A 2 Fs and f (τ) fτrg ; r ; generae he σ-algebra Fs τ. Therefore as in he proof of he previous proposiion we obain he resul using elemenary properies of condiional expecaions. The nex lemma gives us an inegrabiliy resul for he drif erm in he F τ -decomposiion of : h R i Lemma 39 e have ha E R i jα (;τ)jd < and Eh jα (;τ)j2 d : Proof. As in Lemma 32, we have ha E M [;τ) d τ E [τ;] ()ϕ (M ; )d ; where he inegrands are posiive. The second par of he saemen follows as in Lemma 32. The main resul of his secion is he following heorem which gives he semimaringale decomposiion of in he filraion F τ : Theorem 4 is a F τ -semimaringale wih he following decomposiion α (u;τ)du + τ ; (27) where α (u;τ) M u u τ u [;τ) (u) ϕ (M u ; u) [τ;] (u) and τ is a F τ -Brownian moion. 23