Backward Stochastic Differential Equations with Enlarged Filtration

Transcription

1 Backward Sochasic Differenial Equaions wih Enlarged Filraion Opion hedging of an insider rader in a financial marke wih Jumps Anne EYRAUD-LOISEL I.S.F.A. Universié Lyon 1 5 avenue Tony Garnier 697 Lyon FRANCE anne.eyraud@univ-lyon1.fr Absrac Insider rading consiss in having an addiional informaion, unknown from he common invesor, and using i on he financial marke. Mahemaical modeling can sudy such behaviors, by modeling his addiional informaion wihin he marke, and comparing he invesmen sraegies of an insider rader and a non informed invesor. Research on his subjec has already been carried ou by A. Grorud and M. Ponier since 1996 (see [8], [9], [1] e [12]), sudying he problem in a wealh opimizaion poin of view. This work focuses more on opion hedging problems. We have chosen o sudy wealh equaions as backward sochasic differenial equaions (BSDE), and we use Jeulin s mehod of enlargemen of filraion (see [6]) o model he informaion of our insider rader. We will ry o compare he sraegies of an insider rader and a non insider one. Differen models are sudied: a firs prices are driven only by a Brownian moion, and in a second par, we add jump processes (Poisson poin processes) o he model. Keywords: Enlargemen of filraion, BSDE, opion hedging, insider rading, asymmeric informaion, maringale represenaion. 1

2 1 Mahemaical Model Le W be a sandard d-dimensional Brownian moion, and (Ω, (F ) T, P ) a filered probabiliy space, wih Ω = C([, T ]; R d ) and (F ) [,T ] he naural filraion of Brownian moion W. We consider a financial marke wih k risky asses, whose prices are driven by: S i = S i + S i sb i sds + S i s(σ i s, dw s ), T, (1) and a bond (or riskless asse) evolving as: S = 1 + S s r s ds. Parameers b, σ, r are supposed o be bounded on [, T ], F-adaped, and ake values respecively on R d, R d k, R. Marix σ is inverible d dp -a.s. This is he usual condiions o have a complee marke. A financial agen has a posiive F -measurable iniial wealh X a ime = (X consan a.s. as F is rivial). His consumpion c is a nonnegaive Y-adaped process verifying c sds <,P -a.s. He ges θ i pars of i h asse. His wealh a ime is X = k i= θi S. i We consider he sandard self-financing hypohesis: dx = k θds i i c d (2) i= I means ha he consumpion is only financed wih he profis realized by he porfolio, and no by ouside benefis. Then, he wealh of he agen saisfies he following equaion: dx = θ S r d + k θs i b i i d + i=1 k θs i (σ i, i dw ) c d (3) Then, we denoe by π i = θs i i he amoun of wealh invesed in he i h asse for i = 1,..., k, and we noice ha θ S = X k 1 πi. We denoe also by π = (π, i i = 1,.., k) he porfolio (or sraegy), and so he oal wealh can be wrien as a soluion of a sochasic differenial equaion: dx = (X r c )d + (π, b r 1)d + (π, σ dw ), X L (F ) (4) where 1 is he vecor wih all coordinaes equal o 1. The previous line can also be rewrien by inegraing from o T : X T X = so: X = X T (X s r s c s )ds + i=1 (π s, b s r s 1)ds + (π s, σ s dw s ) a.s. [(X s r s c s ) + (π s, b s r s 1)] ds (σ }{{} sπ s, dw }{{} s ) a.s. (6) f(s,x s,z s) Z s 2 (5)

3 I is he form under which we will sudy he wealh process, as a soluion of a backward sochasic differenial equaion. We consider an opion-hedging problem, represened by a pay-off ξ, o be reached a mauriy T. As a ranscripion, we have a problem of porfolio duplicaion: we look for he iniial wealh X and he porfolio π such ha X T = ξ. The reason why BSDEs are ineresing in our case is ha hey allow us o model such a problem of opion hedging wih a unique equaion (see El Karoui, Peng and Quenez [7]). BSDEs are sochasic differenial equaions of he form: X = ξ + f(s, X s, Z s )ds (Z s, dw s ), T (7) ξ L 2 (Ω) is he final wealh, a goal o reach, f : Ω [, T ] R k R k d R k is a drif funcion, X is he oal wealh of he porfolio a ime Z represens he porfolio invesmens a ime One of he fundamenal resuls abou BSDEs is a heorem given by E.Pardoux (see [19]), which gives he exisence and uniqueness of he soluion of a BSDE under some Lipschiz hypoheses on he drif funcion. Theorem 1.1 (Pardoux I) Suppose f(., y, z) is F-prog. measurable, and 1. φ : R + R + increasing such ha f(, y, ) f(,, ) + φ( y ),, y a.s. 2. E P ( f(,, ) 2 ) < 3. f is globally Lipschiz w.r.. z and coninuous w.r.. y 4. y y, f(, y, z) f(, y, z) µ y y 2,, y, y, z, a.s. Then he BSDE has a unique soluion (X, Z) such ha E P Z 2 d < From now on, we suppose ha he financial agen is an insider rader: he has an addiional informaion compare o he sandard normally informed invesor. To model i, we use he mehod of enlargemen of filraion. We will suppose in all his paper ha r =, which means ha we don have ineres raes, because we will only consider small invesors, who do no influence ineres raes. In his model, we inroduce an insider, who has an informaion a ime denoed by L F T. L is F T -measurable, which means ha i will be public a ime T. To model he insider space, we 3

4 enlarge he iniial filraion wih L, in order o obain he filraion of he insider rader probabiliy space: Y = s>(f σ(l)) (8) Since he discouned asse prices are maringales in he iniial probabiliy space under a risk-neural probabiliy, i would be ineresing and naural ha hey sill have similar properies in he larger space. So he main problem is under which condiion do we have he following useful propery: Hypohesis 1 (H ) If (M ) T is a given (F, P )-maringale (or semimaringale), hen (M ) is a (Y, P )-semi-maringale. This problem has been developed by Jeulin [6] and Yor, and by Jacod [13], who shows ha his asserion is rue under he following hypohesis: Hypohesis 2 (H ) The condiional probabiliy law of L knowing F is absoluely coninuous wih respec o he probabiliy law of L, < T. Remark: if L is F T -measurable, and if is condiional probabiliy law given F T (δ L ) is absoluely coninuous wih respec o he disribuion of L, i implies ha σ(l) is aomic (see for a deeper sudy Meyer [17]). Bu his is no he case in his aricle, because we will suppose L F T and a erminal poin of view of our problem T < T. Under hypohesis (H ), Jacod gives he expeced decomposiion: one spli a (F, P )-maringale (he Brownian moion W in our example) ino a (Y, P )- maringale par and a finie variaion par as W = B + l sds where B is a (Y, P )-maringale (a (Y, P )-Brownian moion in case of W Brownian moion), and l is Y-adaped. This propery is also verified under a sronger hypohesis, for which we have sronger resuls, and which has been developed by Amendinger [1], Jeulin [6], Grorud and Ponier [1] : Hypohesis 3 (H 3 ) There exiss a probabiliy Q equivalen o P under which F and σ(l) are independen, < T. Among he remarkable consequences of his hypohesis, we can noice ha W is a (Y, Q)-Brownian moion. This aricle will successively sudy he exisence and uniqueness of he BSDE on he enlarged probabiliy space under (H 3 ) and under (H ). Remark: Before he sudy of hypohesis (H ), (H ) and (H 3 ), Bremaud and Yor [5] sudied hypohesis (H) under which (F, P )-(local) maringales are sill (Y, P )-(local) maringales. This hypohesis is no currenly used in insider models wih iniial enlargemen of filraions. In he case of iniial enlargemen, (H 3 ) implies (H). In fac, (H 3 ) implies he exisence of a probabiliy Q under which (H) is verified (see also Amendinger [2]). Conversely, i is easy o prove ha if (H) is rue under P, and if F is rivial, hen (H 3 ) 4

5 is rue. In a pracical and financial sense, i means ha i is no realisic o consider ha he "naural" probabiliy makes he informaion and he marke independen. Neverheless, hypohesis (H) appears o be relevan and useful in defaul risk models and progressive enlargemen of filraions. 2 BSDE under hypohesis (H 3 ) 2.1 Exisence and Uniqueness Theorem Le (H 3 ) be verified. We denoe by Q he new probabiliy. As (H 3 ) can no hold unil T exacly, bu only for < T, we chose L F T and we consider a problem of mauriy T < T. So we can enlarge our filraion unil T. We suppose also ha he BSDE wih parameer (ξ, f) has a unique soluion in he non insider space: we will suppose ha he hypoheses of Pardoux s exisence Theorem 1.1 are verified. To simplify he proof, we will even suppose ha f is globally Lipschiz wih respec o y and z. For he non insider invesor, he iniial BSDE is verified: { X = ξ + f(s, X s, Z s )ds (Z s, dw s ), T (Ω, (F ) T, P ), ξ L 2 (Ω, F T, P ) (9) As (W ) T is sill a Brownian moion under ((Y ) T, Q) hanks o hypohesis (H 3 ) (cf Jacod [13]), he equaion becomes in he insider space: { X = ξ + f(s, X s, Z s )ds ( Z s, dw s ), T (Ω, (Y ) T, Q), ξ L 2 (Ω, Y T, Q) where a soluion ( X, Z) is a couple of (Y)-adaped processes. We also suppose ha ξ L 2 (Ω, Y T, Q), such ha he problem is correcly given in he insider space. We have hen he following resul: Theorem 2.1 Under hypohesis of Theorem 1.1, and if E Q ( f(,, ) 2 d) < hen he backward equaion has a unique soluion in he insider space. Proof: The hypoheses of Pardoux s Theorem 1.1 can be checked. The filraion is no he naural filraion of he Brownian moion any more. We will have o cope wih his problem. f(., y, z) is F -progressively measurable y, z and F Y, so f(., y, z) is Y -progressively measurable. Moreover, as P Q he P -null ses are he same as he Q-null ses. So we sill have poin 1, 3 and 4 Q-a.s. For poin 2, under new probabiliy Q, he expeced value is no finie any more, so we have o suppose his poin rue in he hypoheses of Theorem 2.1. The las problem we have o cope wih is he new filraion which is no any more he naural Brownian filraion. This is annoying because he proof of Pardoux s Theorem 1.1 uses Iô maringale represenaion heorem, which supposes ha he filraion is he naural 5

6 Brownian filraion. Neverheless, as he new filraion Y is generaed by L and by he Brownian moion, we sill have a maringale represenaion resul in he case of a filraion generaed by he Brownian moion and H an iniial σ-algebra (see [14] Theorem III.4.33 p.189). And so Pardoux s proof can be adaped o our case. To simplify our proof, we suppose f globally Lipschiz in y. Le B 2 = (M 2 (, T )) k (M 2 (, T )) k d. We will define a funcion Φ : B 2 B 2 such ha (X, Z) B 2 is a soluion of he BSDE if i is a fixed poin of Φ. Le (U, V ) B 2, and (X, Z) = Φ(U, V ) wih: ] X = E Q [ξ + f(s, U s, V s )ds Y, T, X T = ξ. Then {Z, T } is obained by using Jacod and Shiryaev [14] generalized maringale represenaion heorem, applied o he maringale E Q ξ + [ f(s, U s, V s )ds Y ] [,T. ] So we obain: ) ξ + f(s, U s, V s )ds = E Q (ξ + f(s, U s, V s )ds σ(l) + (Z s, db s ) In his las equaliy, condiional expecaion is aken wih respec o Y and so T : X + Which implies X = ξ + and consequenly X = ξ + f(s, U s, V s )ds = X + f(s, U s, V s )ds f(s, U s, V s )ds (Z s, db s ) (Z s, db s ) (Z s, db s ) (1) This proves ha (X, Z) B 2 is soluion of he BSDE if i is a fixed poin of Φ. As f is globally Lipschiz wih respec o U, V and using Davis-Burkholder- Gundy s inequaliy, we deduce: ( ) E Q sup X T 2 < T consequenly { (X s, Z s db s ), T } is a maringale. Le (U, V ), (U, V ) B 2, (X, Z) = Φ(U, V ), (X, Z ) = Φ(U, V ), (Ū, V ) = (U U, V V ) and ( X, Z) = (X X, Z Z ). Then, from Iô formula, γ R, we have: e γ E Q X 2 + E Q 2KE Q 4K 2 E Q e γs (γ X s 2 + Z s 2 )ds e γs X s ( Ūs + V s )ds e γs X s 2 ds E Q 6 e γs ( Ūs 2 + V s 2 )ds (11)

7 We chose γ = 1 + 4K 2, and obain: E Q e γ ( X 2 + Z 2 )d 1 2 E Q e γ ( Ū 2 + V 2 )d (12) Then Φ is a sric conracion on B 2 wih norm T (X, Z) γ = (E Q eγ ( X 2 + Z 2 )d) 1 2. We deduce ha Φ has a unique fixed poin and we conclude ha he BSDE has a unique soluion. 2.2 Comparison of he soluions We firs look a an inuiive example. Suppose L = S T : he agen knows he final price (he deduces i for insance from an informaion on a former financial operaion, as a akeover). Suppose also ha he wans o hedge a digial opion 1 ST K. The insider will hen have wo possible invesmens: inves on he risky asse if S T K, or doing nohing oherwise. He has obviously a differen sraegy from he non insider agen. Moreover, in his special case, here is an arbirage opporuniy. In he general case, i is no so easy o deermine wheher he insider will have a differen invesmen sraegy from he non insider or no, especially when informaion is a ime T > T. So we have wo quesions: will he insider inves differenly from he non insider? Is here an arbirage in he insider space? Answering hese quesions can give us oher clues: is he informaion relevan? Is i useful? Moreover, when he insider has a very differen sraegy from he non insider, i will be possible o deec he former hrough saisical ess. This could be useful for marke fraud deecion agencies, as he French A.M.F. We can recall ha in a wealh opimizaion poin of view (see Grorud and Ponier [9]), he insider will immediaely have a compleely differen sraegy from he non insider. Is i he same in our hedging problem? We compare firs he sraegies of he wo agens (comparison of he soluions of he wo BSDE s), before sudying viabiliy and compleeness of he insider marke. Corollary 2.1 Suppose ha ξ L 2 (Ω, Y T, Q) L 2 (Ω, F T, P ), so ha he financial problem has a sense in he insider space as in he non insider space. We denoe by (X, Z) and (X, Z ) he soluions of he wo BSDE s. Then, if E Q Z 2 d <, he soluion of he insider s BSDE is he same as he non insider s one: (X, Z) = (X, Z ). Proof: according o Theorem 2.1, in he insider space (Ω, (Y ) T, Q) he BSDE has a unique soluion (X, Z ). Bu he non insider BSDE soluion (X, Z ) is (F ) T -progressively measurable, and so is i wih respec o 7

8 (Y ) T. As he BSDE is he same in boh spaces, we have X = ξ + f(s, X s, Z s )ds (Z s, db s ) So (X, Z ) is a soluion of he insider BSDE. As E Q Z 2 d <, we conclude ha i is he unique soluion of he insider BSDE. Remarks: Inuiively, as L (F T ), T < T and L ξ, from hypohesis (H 3 ), we can undersand ha if under Q he objecive is independen from he insider informaion, he will no have a differen sraegy, as soon as his sraegy is admissible in he insider space. In a cerain sense, he informaion is useless. In his case, here is no arbirage opporuniy, and he insider marke is viable. We have a hedging problem in a complee iniial marke, so here exiss a price for he opion, and a sraegy for hedging he risk. Wha is he use of he informaion? Eiher o creae an arbirage, which is impossible under (H 3 ) (see nex paragraph), or o propose a differen price for he opion in he marke. Bu hen wo problems appear: firs, who would buy such an opion? and second, proposing a differen price from he marke means exhibiing he fac ha we have an informaion... which is unineresing from he insider poin of view considering ha using he informaion is a fraud. 2.3 Viabiliy and compleeness of he insider marke We ry o ranslae our resuls in erm of viabiliy and compleeness of he marke. The main poin is o know if here is an arbirage opporuniy, and if he insider marke is complee. Theorem 2.2 Suppose ha he insider marke is viable, and le Q be a risk-neural probabiliy. If ξ L 2 (F, P ) L 2 (Y, Q), hen E P (ξ) = E Q (ξ). So he informaion does no creae any arbirage opporuniy: prices are he same in boh spaces. Proof: By a Girsanov ransformaion, risk-neural probabiliies allows us o remove drif in price processes, keeping volailiy. So in he insider space as in he non insider space, we obain ds = S (σ, dw ) where W is a (F, P ) and a (Y, Q)-Brownian moion. Then price processes under he wo risk neural probabiliies follow he same diffusion processes, and prices on boh markes are he same. In general, he insider marke is incomplee, bu has a paricular propery: Theorem 2.3 Le R 1 and R 2 be wo risk neural probabiliies in he insider space. Le Y L 1 R 1 (Q) L 1 R 2 (Q), hen prices are equal: E R1 (Y ) = E R2 (Y ). 8

9 proof: See Grorud [8]. The insider marke may have several risk neural probabiliies. I is no necessarily complee, neverheless i is always "pseudo-complee", in he sense ha all prices calculaed under differen risk neural probabiliies are he same. I could be inerpreed by he fac ha prices in he insider marke will only depend on informaion L and on he non insider marke: as he non insider has a unique risk neural probabiliy, here is only one price in he insider marke. Finally, following Amendinger [2] and Grorud and Ponier [1] we have he following resul: Theorem 2.4 Under (H 3 ), if he non insider marke is viable, hen he insider marke is also viable. Financially speaking he informaion L does no creae any arbirage opporuniy. On he oher hand, compleeness of he non insider marke does no necessarily imply compleeness of he insider marke. The enlarged space may have several risk neural probabiliies, bu which will have propery of pseudocompleeness of Theorem (2.3). 3 BSDE under hypohesis (H ) 3.1 Exisence and Uniqueness Theorem In his secion (H ) is supposed o hold: he condiional probabiliy law of L knowing F is absoluely coninuous wih respec o he law of L, T. We sill ake L F T, T < T. Le s recall he non insider BSDE: { X = ξ + f(s, X s, Z s )ds (Z s, dw s ), T (Ω, (F ) T, P ) (13) (H ) holds, say every (F, P )-maringale (M ) T is a (Y, P )-semi-maringale. So he Brownian moion W can be wrien: W = B + l sds where B is a (Y, P )-Brownian moion and l is a Y-adaped process. We deduce he new backward equaion in he insider space: { X = ξ + [f(s, X s, Z s ) (Z s, l s )] ds (Z s, db s ), T (Ω, (Y ) T, P ) (14) If we ake ξ L 1 (Ω, Y T, P ) in he insider space, we have a new BSDE wih a new drif, deduced from he previous drif according o he formula: g(ω,, y, z) = f(ω,, y, z) (z, l(ω, )). Le s consider Pardoux s exisence and uniqueness Theorem 1.1. The filraion is no generaed by he Brownian moion any more. So we don have 9

10 any maringale represenaion heorem. f(., y, z) and l are Y -progressively measurable, so he new drif g(., y, z) is Y -progressively measurable. As g(, y, ) = f(, y, ), he condiion on f sands also on g, so g(, y, ) g(,, ) + φ( y ), y, z P -a.s. Idenically, as g(,, ) = f(,, ) we sill have E P ( g(,, ) 2 d) <. On he oher hand, g is no globally Lipschiz, because: g(, y, z) g(, y, z ) = f(, y, z) f(, y, z )+l()(z z ) (K+l ) z z So if l is a.s. bounded, hen g is globally Lipschiz wih respec o z, bu if l is no bounded, his propery does no hold. Moreover, as g(y) = f(y) + consan, we sill have < y y, g(, y, z) g(, y, z) >=< y y, f(, y, z) f(, y, z) > µ y y 2. As f, g is also coninuous wih respec o y,, z a.s. Finally, all condiions are verified for he enlarged BSDE in he insider space, as soon as we suppose l bounded. Bu we need a maringale represenaion heorem. If l is almos surely bounded, hen E P (E( l.b)) = 1, < T. Then, according o proposiion 4.2 of Grorud and Ponier [12], hypohesis (H 3 ) is verified. We are in he previous case : under hypohesis (H 3 ), we have a maringale represenaion heorem, and we can conclude similarly o Theorem 2.1 (and wihou a change of probabiliy). We obained he following resul: Theorem 3.1 Under (H ) and hypoheses of Theorem 1.1, if l is a.s. bounded in he enlarged space (Ω, (Y ) T, P ), hen we deduce he exisence and uniqueness of he soluion of he enlarged BSDE. Remark: I will be useful o sudy wha happens on examples for which l is no bounded, and (H 3 ) does no hold. Bu a problem is ha we do no know any example of L in a coninuous model for which (H ) holds bu no (H 3 ). And if (H 3 ) holds, we have he resul of previous secion, and he problem is solved. This is he reason why i seems naural o inroduce jump processes ino our model, in order o have examples of L for which we have (H ) bu no (H 3 ). 4 Inroducion of Jump processes 4.1 Exended model We add jump processes in he price dynamics sudied in he previous secion. W is sill a m-dimensional sandard Brownian moion on (Ω W, F W, P W ) and (F W ) [,T ] is compleed naural filraion. We denoe by (Ω N, F N, P N ) anoher probabiliy space where N = (N 1,.., N n ) : Ω N R n is a n- dimensional mulivariae Poisson process, wih inensiy λ, [, T ]. We denoe by M = N λ sds he compensaed mulivariae Poisson process. N is denoed as a vecor (N k ) k=1,..,n of unidimensional mulivariae Poisson 1

11 processes wih inensiy (λ k ) k=1,..,n, F N-measurable. F N is generaed by F N and he jump imes of N. So he global probabiliy space is: (Ω, F, (F ) [,T ], P ) = (Ω W Ω N, F W F N, (F W F N ) [,T ], P W P N ). The marke model sill conains a bond and d = m + n risky asses whose prices (S i ) i=1,..,d follow a diffusion run by W and N: ds = S r, S = 1 ds i = S i b i d + S i (σ i, dw ) + S i (ρ i, dm ), S i = xi, i = 1,.., d. (15) We suppose he following, so ha he marke is viable and complee: b, r, σ, ρ are predicable and globally bounded processes, λ is a nonnegaive F -measurable process, which does no mee any neighborhood of, ρ i,k > 1, i, k,, Φ Φ is uniformly ellipic, where Φ = [σ ρ ] Le θ = Φ 1 (b r1), hen θ k < λ k, k = 1,.., n. We consider again an insider in his new marke wih jumps. The insider sill has informaion L L 1 (Ω, F T, P ) aking is values in R k, and he new filraion on he insider space is Y = s> (F s σ(l)), [, T ], T < T. We have he same hypohesis on wealh process and invesmen sraegy, and we sudy self-financing sraegies dx = d i= θi ds i c d, so he wealh process of he rader on his marke saisfies: X = X + θ sss r s ds c sds + [ d ( i=1 θ i s Ssb i i sds + θss i s(σ i s, i dw s ) + θss i s(ρ i i s, dm s ) )] As in he coninuous model, we obain he following BSDE for he wealh process: X = X T [(X s r s c s ) + (π s, b s r s 1)] ds }{{} 4.2 BSDE wih jumps (σsπ }{{} s, dw s ) Z s f(s,x s,z s,u s) T (ρ }{{} sπ s, dm s ) a.s. U s In his model wih jumps, and even in a more general model wih Poisson poin processes (see furher), Barles, Buckdahn and Pardoux [4] developed an exisence heorem for he soluion of BSDEs wih jumps. We denoe by B 2 = S 2 L 2 m(p ) L 2 n(p ) where: S 2 is he se of k-dimensional F -adaped càdlàg processes {Y } T such ha Y S 2 = sup T Y L 2 (Ω)< 11

12 L 2 m(p ) he se of all k m-dimensional F -progressively measurable ( ) 1 T processes {Z } T such ha Z L 2 m (P ) = E P Z 2 2 d < L 2 n(p ) he se of all k n-dimensional F -progressively measurable ( ) 1 T processes {U } T such ha U L 2 n (P ) = E P U 2 2 d < We have he following heorem (see Barles e al. [4]): Theorem 4.1 (Pardoux II) Le ξ L 2 (Ω, F T, P ) k and f : Ω [, T ] R k R k m R k n R k. If f is measurable, if E P f (,, ) 2 d < and if K such ha: f (y, z, u) f (y, z, u ) K( y y + z z + u u ), T, y, y, z, z, u, u hen here exiss a unique riple (X, Z, U) B 2 soluion of he BSDE: X = ξ + f s (X s, Z s, U s )ds (Z s, dw s ) (U s, dm s ), T Proof: The proof is he same as Pardoux s Theorem 1.1 proof: consrucing a sric conracion and using a maringale represenaion heorem. 4.3 Under hypohesis (H 3 ) Everyhing works globally as in he firs par of he paper. More precisely: Exisence and Uniqueness Theorem Thanks o Jacod and Shiryaev ([14] Theorem III.4.34 p.189), Grorud ([8] Theorem 3.1 p.648) shows a maringale represenaion heorem under (H 3 ) wih jumps. Wih his maringale represenaion heorem, we can adap he proof of Pardoux s Theorem (4.1), and as in he coninuous case in secion 2.1, we have he following resul: Theorem 4.2 Under hypohesis of Theorem 4.1 (so ( he iniial BSDE has a ) unique soluion), for ξ L 2 T (Ω, Y T, Q) and if E Q f (,, ) 2 d < hen he BSDE in he insider space has a unique soluion (X s, Z s, U s ) B 2. Comparison of soluions We have a similar resul as in secion 2.2: Proposiion 4.1 For ξ L 2 (Ω, Y T, Q) L 2 (Ω, F T, Q), we have: if E Q ( Z 2 L 2 m (Q) + U 2 L 2 n (Q) d ) <, hen he soluion of he enlarged BSDE is he same as he soluion of he iniial BSDE: (X, Z, U) = (X, Z, U ). Viabiliy and Compleeness of he marke As in he coninuous case, if he non insider marke is viable, hen he insider marke is also viable: here is no arbirage opporuniy (see Grorud [8]). 12

13 4.4 Under hypohesis (H ) In his case, he new model becomes ineresing, because now we have examples of L for which (H ) holds bu no (H 3 ). We summarize he resuls we have under his hypohesis before reaing an example. We use Jacod s resul on enlargemen of filraion under (H ) (see [13]), a bi differen from he resul in he coninuous model (see Grorud [8]). Proposiion 4.2 Under hypohesis (H ), we have: If Q is he condiional law of L knowing F, hen here exiss a measurable version of he condiional densiy dq : (ω,, x) p(ω,, x) which is a maringale and can be wrien, x R as: p(, x) = p(, x) + (α(s, x), dw s ) + (β(s, x), dm s ) where x, s α(s, x) and s β(s, x) are F-predicable processes. Moreover, s < T, p(s, L) > a.s. If Y is a maringale wrien as Y = Y + (u s, dw s ) + (v s, dm s ) hen d < Y, p(., x) > =< α(., x), u > d+ < β(., x), v > d a.s., and: Ȳ = Y (< α(., x), u > s + < Γ.β(., x), v > s ) x=l ds, p(s, L) T is a (Y, P )-local maringale where Γ is he diagonal marix of inensiies of N: d < M > s = Γ s ds We denoe by l s = α(s,l) p(s,l) and µ s = Γsβ(s,L) p(s,l). Then W = W l sds is a (Y, P )-Brownian moion and if 1 + β(,l) p(,l) hen M = M µ sds is a compensaed Poisson process wih inensiy λ (1 + β(,l) p(,l) ). Then he wealh process can be wrien in erm of a BSDE in he insider space: X = X T [(X s r s c s ) + (π s, b s r s 1) + σsπ s l s + ρ sπ s µ s ] ds }{{} (σsπ }{{} s, dw s ) Z s g(s,x s,z s,u s) (ρ }{{} sπ s, dm s ) a.s. U s wih a new drif g(s, X s, Z s, U s ) = f(s, X s, Z s, U s ) l s Z s µ s U s. 13

14 4.5 Sudy of an example of L For his example, le us ake L = N T : he insider rader knows he number of jumps a final ime T. In order o simplify he problem, we will consider a unidimensional process. The law of L is absoluely coninuous wih respec o he couning measure on N. We obain a measurable version of he condiional densiy: ( ) ( λ s ds) y N p(, y) = exp λ s ds (y N )! 1 [N; [(y). Then i is clear ha (H 3 ) does no hold (non equivalence of he laws), whereas (H ) is verified (law absoluely coninuous wih respec o he law of L). We give an explici expression of β in Proposiion 4.2: β(s, y) = k y s p(s, y) wih k y = y N λ s ds 1 and so µ = λ k L = λ In he insider space, M = M λ s ( N T N λ s ds 1 ( N T N s s λudu 1 ) ds is a Y -maringale. So N is a Y-Poisson process wih inensiy N T N T, T wih respec o Y. Indeed we should enlarge he iniial space unil T. Brownian moion does no change because he condiional densiy is represened only on he Poisson process, because of he independence beween Brownian moion and Poisson process. In his case, he enlarged BSDE is: ( )) T N T N T X T = ξ+ (f(s, X s, Z s, U s ) λ s s s λ udu 1 ds Z s dw s U s d M s The maringale represenaion heorem ha sands in (Ω, F, P ) allows us o find a soluion o he enlarged BSDE, bu we do no have any uniqueness resul in his case (µ is no bounded). 5 Inroducion of a Poisson measure Such a model is ineresing o develop because is incompleeness allows us o have hypohesis (H ) wihou (H 3 ). 5.1 The model In our las secion we inroduce jump processes where jumps are coninuous in ime and space, by using a Poisson measure. We consider a filered probabiliy space (Ω, F, (F ) T, P ) wih F he compleed filraion generaed by boh (W ) and (N ). (W ) is a sandard m-dimensional Brownian moion and (N ) a poin process wih random Poisson measure µ on R + E and compensaor ν(d, de) such ha { µ([, ] A) = (µ ν)([, ] A)} ) 14

15 is a maringale A E saisfying ν([, ] A) <. E = R l \ {} wih is Borel σ-algebra E. We can wrie as N = E µ(ds, de) he poin process, so dn = E µ(d, de). And we denoe by Ñ = N E ν(ds, de) he compensaed process. We use an addiional hypohesis on ν: ν(d, de) = dλ(de), λ supposed o be a σ-finie measure on (E, E) ha saisfies : E (1 e 2 )λ(de) < +. Le H be a finie-dimensional linear space, and le L 2 F ([, 1]; H) be he space of all (F )-adaped H-valued square inegrable processes L 2 F,P ([, 1]; H) be he space of (F )-predicable equivalen class versions. As previously we consider a financial marke wih one bond and k risky asses, in which asse prices are driven by he following sochasic differenial equaion ( [, T ], 1 i k) : S i = S i + Ssb i i sds + Ss(σ i s, i dw s ) + Ss i φ i s(e)µ(ds, de) (16) where b, σ and φ are predicable and globally Lipschiz processes. We rewrie he self-financing equaion as a BSDE, and he wealh-invesmen process is soluion of: X = X T E [(X s r s c s ) + (π s, b s r s 1)] }{{} f(s,x s,z s) E ds (σsπ s, dw s ) }{{} Z s (π s, φ(s, e)) µ(ds, de) a.s. (17) }{{} U s(e) As in he previous pars, an insider rader has an informaion L L 1 (Ω, F T, R k ) on he fuure. Y is sill he insider s naural filraion. In boh spaces, we sudy again exisence and uniqueness of he admissible wealh-porfolio processes in order o cover a pay-off represened by ξ = X T. 5.2 Exisence and uniqueness We use here wo main aricles: Barles, Buckdahn and Pardoux [4], and Tang and Li [23]. Le us firs define several process spaces. Le S 2 (F) be he se of all F -adaped cadlag k-dimensional processes squareinegrable {Y } T such ha Y S 2 (F)= sup T Y L 2 (Ω)<. Le L 2 (W ) be he se of all F -progressively measurable k d-dimensional ( 1/2 processes {Z } T such ha Z L 2 (W )= E P Z d) 2 <. Le L 2 ( µ) be he se of all mappings U : Ω [, T ] E R ha are P E- measurable (P being he σ-algebra of F -predicable subses of Ω [, T ]) 15

16 ( 1/2 such ha U L 2 ( µ)= E P E U (e) 2 ν(de, d)) <. Finally we define he funcional space B 2 (F) = S 2 (F) L 2 (W ) L 2 ( µ). Then we have he following resul: Theorem 5.1 (Barles e al. [4] Theorem 2.1, and Tang and Li [23] Lemma 2.4) Le ξ (L 2 (Ω, F T, P )) k and le f : Ω [, T ] R k R k d L 2 (E, E, ν; R k ) R k be a P B k B k d B(L 2 (E, E, ν; R k ))-measurable funcion saisfying: K >, E P f (,, ) 2 d < K (18) f (y, z, u) f (y, z, u ) K [ y y + z z + u u ] Then here exiss a unique riple (Y, Z, U) B 2 (F) soluion of he BSDE: Y = ξ+ f s (Y s, Z s, U s )ds Z s dw s U s (e) µ(ds, de), T E 5.3 BSDE under (H 3 ) : Adapaion of he Exisence and Uniqueness Theorem Under hypohesis (H 3 ), in his model wih coninuous random jumps, we can also adap he exisence heorem, as in he sandard model under he same hypohesis on he drif. An insider wih informaion L verifying (H 3 ) will have an admissible hedging sraegy for an opion wih pay-off ξ. We have he following heorem: Theorem 5.2 Le ξ (L 2 (Ω, Y T, Q)) k and le f be a drif funcion verifying hypohesis (18), and such ha E Q f (,, ) 2 d <. Then here exiss a unique riple (Y, Z, U) B 2 (Y) soluion of he BSDE: Y = ξ + f s (Y s, Z s, U s )ds Z s dw s U s (e) µ(ds, de) E We firs prove an imporan lemma for his proof: a maringale represenaion heorem in our conex, under (Y, Q) : Lemma 5.1 Le H be a finie-dimensional space and M an H-valued (Y )- adaped square inegrable maringale. Then here exiss Z i (.) L 2 (W ), i = 1,.., d and U(.,.) L 2 ( µ) such ha M = M + Z i sdw i (s) + E U(s, e) µ(ds, de) Proof of he Lemma: Ñ(ds, de) = N(ds, de) λ(de)ds is a local maringale. The couple (W, N) is a Brownian-Poisson process couple, and i is an independen incremen process (IIP) on space (F, P ). So (W, N) is he same 16

17 Brownian-Poisson process IIP in he enlarged space (Y, Q), from hypohesis (H 3 ). Then, Jacod and Shiryaev ( [14] h. III.4.34) gives us he expeced maringale represenaion heorem for independen incremen processes. We can now prove he heorem. Proof of he heorem: For all (Ȳ (.), Z(.), Ū(.,.)) B2 (Y), we know from he previous lemma ha here exiss Z i (.) L 2 (W ), i = 1,.., d and U(.,.) L 2 ( µ) such ha: [ ] Y T + f s (Ȳs, Z s, Ūs)ds = ξ + Z s dw s + U s (e) µ(ds, de) E Y Q This implies: ξ = Y T + f s (Ȳs, Z s, Ūs)ds We pu Y = E Y Q [ Y T + Z s dw s E E ] f s (Ȳs, Z s, Ūs)ds U s (e) µ(ds, de) We verify hen ha for each riple (Ȳ (.), Z(.), Ū(.,.)), he riple (Y (.), Z(.), U(.,.)) is characerized by he following equaion: Y = Y T + f s (Ȳs, Z s, Ūs)ds Z s dw s U s (e) µ(ds, de) which implies: Y = Y f s (Ȳs, Z s, Ūs)ds Z s dw s E E U s (e) µ(ds, de) The previous equaion defines a mapping Λ : (Ȳ (.), Z(.), Ū(.,.)) (Y (.), Z(.), U(.)). We inroduce, for k := (Y (.), Z(.), U(.,.)) B 2 (Y) he norm defined by: k := [ sup e b E Q Y 2 + sup e b E Q Z s 2 ds + T T E E Q U s (e) 2 ν(ds, de) wih b > a consan o be deermined laer. To complee he proof, i is sufficien o prove ha Λ maps B 2 (Y) ono iself, and is a sric conracion for he previous norm. Le (Ȳi(.), Z i (.), Ūi(.,.)) B 2 (Y) and (Y i (.), Z i (.), U i (.,.) := Λ(Ȳi(.), Z i (.), Ū(.,.)) for i = 1, 2. Then, using Iô s formula and equaion (18), we obain: E Q Y 1 () Y 2 () 2 + E Q d Z1(s) i Z2(s) i 2 ds i=1 +E Q U 1 (s, e) U 2 (s, e) 2 ν(ds, de) E [ T γk 2 E Q Y 1 (s) Y 2 (s) 2 ds + 1 γ E Q Ȳ1(s) Ȳ2(s) 2 ds +E Q d Z 1(s) i Z 2(s) i 2 ds + E Q i=1 17 E Ū1(s, e) Ū2(s, e) 2 ν(ds, de) ] ]

18 Which implies, from Gronwall inequaliy: E Q p 1 () p 2 () 2 + E Q 1 d q1(s) i q2(s) i 2 ds i=1 1 +E Q r 1 (s, e) r 2 (s, e) 2 ν(ds, de) E [ 1 γ 1 1 d E Q p 1 (s) p 2 (s) 2 ds + E Q q 1(s) i q 2(s) i 2 ds i=1 1 ] + E Q r 1 (s, e) r 2 (s, e) 2 ν(ds, de) +K E Q E e γk2 (s ) E [ 1 1 E Q p 1 (τ) p 2 (τ) 2 dτ + E Q ] r 1 (τ, e) r 2 (τ, e) 2 ν(dτ, de) where γ is a posiive real number. So we conclude: d q 1(τ) i q 2(τ) i 2 dτ (Y 1 Y 2, Z 1 Z 2, U 1 U 2 ) α (Ȳ1 Ȳ2, Z 1 Z 2, Ū1 Ū2) i=1 wih α = max{ 2 b γ, 4K 2 γb(b γk 2 ), 2K 2 b γk 2 } which complees he proof, wih an appropriae choice of γ and b such ha he consan α is sricly majored by 1. I means ha γ and b has o verify γ(1 + γ/2) < K 2 and b > 2/ γ. Thanks o his heorem, we have a similar resul as in he wo oher models: under (H 3 ) we have exisence and uniqueness of he soluion of he enlarged BSDE. Moreover, as before, if he problem is well defined in boh spaces, boh soluions are he same. Acknowledgemens I am deeply indebed o Axel Grorud for his paper. He guided my firs seps ino research, and I would like o express here all my hanks and graefulness, ogeher wih my regres for he loss of a mos respeced superviser. I am also very hankful o he anonymous referee for his careful reading and his help o improve his paper. Conclusion Successively in a coninuous process model, in a discree jump process model and finally in a coninuous jump process model, we have sudied and compared he sraegies of an insider rader and a non informed agen. Under cerain hypoheses we proved exisence and uniqueness of soluions for heir 18

19 hedging sraegies, and arbirage free model for he insider rader. In fac, wih correc hypoheses on he informaion on a complee iniial marke, he insider marke is viable, and even pseudo-complee. A limi o hese models can be raised: we have only considered small invesors. I is perhaps no relevan enough. A furher work would be o consider an opion hedging problem in a jump process model wih a large invesor. This would lead us o use Forward-Backward sochasic differenial equaions, insead of BSDEs. Wha is he pracical use of such resuls? I seems difficul o concreely apply hem a he momen. However such comparison resuls beween insider and non insider invesmen sraegies could be ineresing o esablish saisical ess for he deecion of insider raders. Applied o marke daas, i could help organisms like French A.M.F. deermining wheher an agen is informed or no. Unforunaely, heories are no ye enough performing o compue such ess, and A.M.F. s monioring agens do no use so specialized saisical ess. References [1] Amendinger, J., 1999, Iniial enlargemen of filraions and addiional informaion of financial markes, Docoral Thesis, T-U. Berlin [2] Amendinger, J., 2, Maringale Represenaion Theorems for Iniially Enlarged Filraions, Sochasic Processes and heir Applicaions [3] Bardhan, I., Chao, X., 1996, On maringale measures when asse reurns have unpredicable jumps, Soch. Proc. and heir Appl [4] Barles, G., Buckdahn, R., Pardoux, E., 1997, BSDE s and inegralparial differenial equaions, Sochasics and Sochasic Repors, Vol. 6, pp [5] Brémaud, P., Yor, M., Changes of filraions and of probabiliy measures. Z. Wahrsch. Verw. Gebiee 45 (1978), no. 4, [6] Chaleya-Maurel, M., Jeulin, T., 1985, Grossissemen gaussien de la filraion brownienne, Séminaire de Calcul sochasique, , Paris, Lecure Noes in Mahemaics 1118, 59 19, Springer-Verlag [7] El Karoui, N., Peng, S., Quenez, M.C., 1997, Backward Sochasic Differenial Equaions in Finance, Mahemaical Finance, Vol 7, 1,1 71. [8] Grorud, A., 2, Asymmeric informaion in a financial marke wih jumps, In. Journal of Theor. and Applied Fin., vol 3, 4, [9] Grorud, A., Ponier, M., 1997, Commen déecer le déli d iniié? C.R.Acad.Sci.Paris,.324, Serie I, p , Probabiliés. 19

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