On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: the critical case

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1 On he uniqueness of soluions o quadraic BSDEs wih convex generaors and unbounded erminal condiions: he criical case Freddy Delbaen Deparmen of Mahemaics ETH-Zenrum, HG G 54.3, CH-892 Zürich, Swizerland delbaen@mah.ehz.ch Ying Hu IRMAR, Universié Rennes 1 Campus de Beaulieu, F-3542 Rennes Cedex, France ying.hu@univ-rennes1.fr hal-8233, version 1-19 Mar 213 Adrien Richou Univ. Bordeaux, IMB, UMR 5251, F-334 Talence, France INRIA, Équipe ALEA, F-334 Talence, France adrien.richou@mah.u-bordeaux1.fr Absrac In 3, he auhors proved ha uniqueness holds among soluions whose exponenials are L p wih p bigger han a consan (p > ). In his paper, we consider he criical case: p =. We prove ha he uniqueness holds among soluions whose exponenials are L under he addiional assumpion ha he generaor is srongly convex. Key words and phrases. Backward sochasic differenial equaions, generaor of quadraic growh, unbounded erminal condiion, uniqueness resul. AMS subjec classificaions. 6H1. 1 Inroducion Since he seminar paper 5, backward sochasic differenial equaions (BSDEs in shor for he remaining of he paper) have found many applicaions in various domains. A lo of effors have been made in order o sudy he well posedness of hese equaions. Quadraic BSDEs is a kind of BSDE which has araced paricular aenion recenly and i is he subjec of he paper. In his aricle, we consider he following quadraic BSDE Y = ξ g(z s )ds+ Z s dw s, T, (1.1) where he generaor g is a coninuous real funcion ha is convex and has a quadraic growh wih respec o he variable z. Moreover ξ is an unbounded random variable (see e.g. 4 for he case of quadraic BSDEs wih bounded erminal condiions). Le us recall ha, in he previous equaion, we are looking for a pair of processes (Y,Z) which is required o be adaped wih respec o he filraion generaed by her d -valued Brownian moion W. In order o sae he main resul of his paper, le us suppose ha here exiss a consan > such ha ξ + L 1,exp( ξ) L 1 and g(z) 2 z 2. 1

2 1 INTRODUCTION 2 By a localizaion procedure similar o ha in 1, we prove easily ha he BSDE (1.1) has a leas a soluion (Y,Z) such hae Y andy belong o he class (D). Concerning he uniqueness issue, in 2, he auhors proved ha he uniqueness holds among soluions whose exponenials are in anyl p. In 3, he auhors proved ha he uniqueness holds among soluions whose exponenials are in L p for a givenp >, i.e. E sup e py T <. However, if we ake g(z) = 2 z 2, hen i is easy o see ha for he associaed BSDE, he uniqueness holds among soluions(y,z) such hae Y belongs o he class (D). I suffices o noe ha if(y,z) is a soluion such hae Y belongs o he class (D), hene Y is a uniformly inegrable maringale and Y = 1 lne e ξ F. hal-8233, version 1-19 Mar 213 So he aim of his paper is o sudy he uniqueness of soluion of BSDE (1.1) in he criical case: p =. We prove ha he BSDE (1.1) has a unique soluion (Y,Z) such ha e Y belongs o he class (D) under he addiional assumpion ha he generaor g is srongly convex. We do no know if his resul says rue wihou his addiional assumpion. The paper is organized as follows. Nex secion is devoed o an exisence resul, secion 3 conains a useful propery for soluions and he las secion is devoed o our main uniqueness resul. Le us close his inroducion by giving noaions ha we will use in all he aricle. For he remaining of he paper, le us fix a nonnegaive real number T >. Firs of all, (W ),T is a sandard Brownian moion wih values in R d defined on some complee probabiliy space (Ω,F,P). (F ) is he naural filraion of he Brownian moion W augmened by he P-null ses of F. As menioned before, we will deal only wih real valued BSDEs which are equaions of ype (1.1). The funciong is called he generaor andξ he erminal condiion. Le us recall ha a generaor is a funcionr 1 d R which is measurable wih respec ob(r 1 d ) and a erminal condiion is simply a realf T -measurable random variable. By a soluion o he BSDE (1.1) we mean a pair (Y,Z ),T of predicable processes wih values in R R 1 d such ha P-a.s., Y is coninuous, Z belongs o L 2 (,T), g(z ) belongs o L 1 (,T) and P-a.s. (Y, Z) verifies (1.1). For any realp 1,S p denoes he se of real-valued, adaped and càdlàg processes(y ),T such ha 1/p Y S p := E sup Y p < +. T M p denoes he se of (equivalen class of) predicable processes(z ),T wih values inr 1 d such ha ( ) p/2 Z M p := E Z s 2 ds 1/p < +. We also recall ha Y belongs o he class (D) as soon as he family {Y τ : τ T sopping ime} is uniformly inegrable. For any convex funcionf : R 1 d R, we denoef he Legendre-Fenchel ransform off given by f (q) = sup z R 1 d (zq f(z)), q R d. We also denoe f he subdifferenial of f. We recall ha he subdifferenial of f a z is he non-empy convex compac se of elemensu R d such ha f(z) f(z ) (z z )u, z R 1 d. Finally, for any predicable process (q ),T such ha q s 2 ds < + P-a.s., we denoe E(q) he Doléans-Dade exponenial ( ( exp q s dw s 1 )) q s 2 ds 2.,T

3 2 AN EXISTENCE RESULT 3 2 An exisence resul Le us begin by giving some assumpions used in his paper. Assumpion A. There exiss a consan > such ha 1. ξ + L 1 andexp( ξ) L 1, 2. g : R d R + is a convex funcion ha saisfies (a) g() =, (b) here exiss a consanc 1 such ha z R 1 d, g(z) C z 2. Assumpion B. There exis wo consansε > andc 2 such ha z,z R 1 d, s g(z ), Remark 2.1 g(z) g(z ) (z z )s ε 2 z z 2 C 2. hal-8233, version 1-19 Mar 213 If g is a C 2 funcion hen assumpion B is equivalen o he assumpion: here exis R and ε > such ha for all z R 1 d wih z > R, we haveg (z) εid. For a general convex generaorg wih quadraic growh i is easy o modify he erminal condiion and he probabiliy o obain a new generaor g : R 1 d R + such ha assumpion A.2. holds rue. The aim of his secion is o show he exisence of soluions under he assumpion A, using a localizaion mehod. Theorem 2.2 Le us assume ha assumpion A holds. Then he BSDE (1.1) has a leas a soluion (Y,Z) such ha: 1 lne e C1T e ξ F Y Eξ F. In paricular,e Y andy belong o he class (D). Proof of Theorem 2.2. To show his exisence resul we use he same classical localizaion argumen as Briand and Hu in 1. Le us fix n,p N and se ξ n,p = ξ + n ξ p. Then i is known from 4 ha he BSDE Y n,p = ξ n,p g(zs n,p )ds+ Zs n,p dw s, T, has a unique soluion(y n,p,z n,p ) S M 2. By applying Theorem2in 1, we have he esimae 1 lneφ ( ξ n,p ) F Y n,p where(φ (z)),t sands for he soluion o he inegral equaion φ (z) = e z + H(φ s (z))ds, T, wih H(p) = C 1 p1l 1,+ (p)+c 1 1l,1 (p). I is noiced in 1 ha φ (z) = e C1(T ) e z when z and z φ (z) is an increasing coninuous funcion. Thus, we have 1 lne e C1T e ξ F 1 lne e C1T e (ξn,p ) F 1 lneφ ( ξ n,p ) F Y n,p.

4 3 A UNIFORM INTEGRABILITY PROPERTY FOR SOLUTIONS 4 Moreover, g is a nonnegaive funcion, so We remark ha Y n,p = E ξ n,p g(zs n,p )ds,t, Y n,p+1 and we define Y p = sup n 1 Y n,p so ha Y p+1 heorem, we have F Eξ n,p F E ξ + F. Y n,p Y n+1,p, Y p and Y = inf p 1 Y p. By he dominaed convergence 1 lne e C1T e ξ F 1 lneφ ( ξ) F Y Eξ F, and in paricular, we remark ha lim + Y = ξ = Y T. Arguing as in 1 wih a localizaion argumen, we can show ha here exiss a process Z such ha (Y,Z) solves he BSDE (1.1). Finally, since processes E e C1T e ξ F and Eξ F belong o he class (D), we conclude ha e Y, Y + and so Y belong o he class (D). hal-8233, version 1-19 Mar A uniform inegrabiliy propery for soluions In his par we will show he following proposiion. Proposiion 3.1 We assume ha assumpion A holds rue. Le us consider (Y,Z) a soluion of he BSDE (1.1) such hay ande Y belong o he class (D). Then, for all predicable process(q s ) s,t wih values in R d and such haq s g(z s ) for alls,t,e(q) is a uniformly inegrable process and defines a probabiliyq P. Proof of Proposiion 3.1. Le us sar he proof by giving a simple lemma. Lemma 3.2 The family of random variables { e X X H } is uniformly inegrable if and only if here exiss a funcionk : R + R + such hak(x) + whenx +, and sup EK(X + ) < +, X H wih K(x) = x k()e d. Moreover, we can assume wihou resricion ha k C, k() = and k (x) > for allx R +. Proof of Lemma 3.2. We only prove he nonrivial implicaion. Firsly, le us remark ha { e X X H } { } is uniformly inegrable if and only if e X+ X H is also uniformly inegrable, so we can assume ha H is a family of posiive random variables. Now we apply he de la Vallée-Poussin heorem: here exiss a nondecreasing funciong : R + R + which is a consan funcion on each inervaln,n+1 forn N, ha saisfiesg(x) + whenx + and such ha sup EG(e X ) < +, X H wih G(x) = x g()d. Then, i is simple o consider a smooh approximaion g of g such ha g(1) = 1, 1 g (x) > for all x 1,+ and g +1 g(1) g g +C. This funcion g also saisfies g(x) + when x + and sup E G(e X ) < +, X H wih G(x) = x g()d. A simple calculus gives us 1 G(e x ) = g(e u )e u du and so we jus have o se k(x) = g(e x ) o conclude he proof.

5 3 A UNIFORM INTEGRABILITY PROPERTY FOR SOLUTIONS 5 Now, le us apply he previous lemma in our siuaion: since we consider a soluion (Y,Z) such ha e Y belongs o he class (D), hen here exiss a funcionk : R + R + given by Lemma 3.2 such ha wihk(x) = x k()e d. We define Ψ (x) = e x x 1 = sup τ T, sopping ime x EK(Yτ ) < +, (3.1) (e u 1)du and Ψ(x) = x k(u)(e u 1)du. ( ) ( x SinceΨ andψare convex funcions we can also consider heir dual funcions. Φ (x) = +1 x ln ) +1 x ) (x is he dual funcion ofψ sinceφ (x) = 1 ln +1 is he inverse funcion ofψ. Moreover, he dual funcion ofψis given byφ(x) = x Φ (u)du wihφ he inverse funcion ofψ. Now we consider a predicable process (q s ) s,t wih values in R d and such ha q s g(z s ) for all s,t. Firsly le us show has q s belongs ol 2 (,T)P-a.s.. Since assumpion A.2 holds rue forg, hen g saisfies g (q) C q 2 and g () =, (3.2) and hus, hal-8233, version 1-19 Mar 213 q s 2 ds C +C g (q s )ds = C +C Moreover, sinceq s g(z s ) we have and (Z s q s g(z s ))ds C +C Z s q s ds+c Z s 2 ds. Z s q s = (2Z s Z s )q s g(2z s ) g(z s ) Z s q s = ( Z s )q s g() g(z s ). So we finally obain q s 2 ds C +C Z s 2 ds < + P-a.s.. Now le us show hae(q) is a uniformly inegrable maringale. We sar by defining he sopping ime { ( ) } τ n = inf,t : sup q s 2 ds, Z s 2 ds n T, and he probabiliy dq n dp = M τ n, wih ( M = exp q s dw s 1 2 ) q s 2 ds. We will show ha (M τn ) n N is uniformly inegrable which is sufficien o conclude. Since (Y,Z) solves he BSDE (1.1), we have Y = Y τn g(z s )ds+ Z s dw s = Y τn + (Z s q s g(z s ))ds+ Z s (dw s q s ds) = E Y Qn τn + g (q s )ds. (3.3) Firsly, since Ψ and Φ are dual funcions, he Fenchel s inequaliy gives us Moreover, we have, hanks o (3.1), E Qn Y τn E Qn Y τ n E Ψ(Y τn ) EΦ(M τn ). E Ψ(Y τ n ) E K(Y τ n ) C,

6 3 A UNIFORM INTEGRABILITY PROPERTY FOR SOLUTIONS 6 wihc a consan ha does no depend onn. By puing hese inequaliies ino (3.3) we obain Y C EΦ(M τn )+E Qn g (q s )ds. (3.4) Thanks o he growh ofg given by (3.2) we have 1 E Qn g (q s )ds C +E Qn 2 Moreover, a simple calculus gives us E Qn 1 2 q s 2 ds. q s 2 ds = 1 EM τ n ln(m τn ). By puing hese wo resuls ino (3.4), and by seingλ = Φ Φ, we obain Le us remark ha Y C +EΛ(M τn ) EΦ (M τn )+ 1 EM τ n ln(m τn ). (3.5) hal-8233, version 1-19 Mar 213 EΦ (M τn ) 1 EM τ n ln(m τn ) ( = E ln 1+ ) ( ) ln +1 M τn ( = E ln 1+ ) ( ln +1 M τn EM τn +E ) +E An elemenary inequaliy gives us ( E ln 1+ ) E M τn and Thus, we have and inequaliy (3.5) becomes ( ) E ln +1 E ln M τn ( ln ) +1. ( 1. EΦ (M τn ) 1 EM τ n ln(m τn ) C Le us give a useful propery ofλha we will prove afer. 1, ) +1 Y C +EΛ(M τn ). (3.6) Proposiion 3.3 The funcion Λ saisfies Λ(x) lim x + x = +. Thanks o his proposiion and he inequaliy (3.6) we are allowed o apply he de la Vallée-Poussin Theorem: (M τn ) n N is uniformly inegrable and he proof is finished. Proof of Proposiion 3.3: I is sufficien o show haλ = Φ Φ is increasing andlim x + Λ (x) +. Firsly, le us show ha Ψ (Φ (x)) (x+), for all x : so we have Ψ (x) = k (x)(e x 1)+k(x)e x k(x)(e x 1)+k(x) Ψ (x)+ 2, Ψ (Φ (x)) Ψ (Φ (x))+ 2 = (x+). As a resul, we ge from he equaliy (Ψ (Φ (x))) = Ψ (Φ (x))φ (x) = 1 ha Φ 1 (x) obain Λ (x) = Φ (x) Φ 1 (x) (x+) 1 (x+) (x+). We finally

7 4 THE UNIQUENESS RESULT 7 and soλ is an increasing funcion. To conclude we will prove by conradicion haλ is an unbounded funcion: le us assume ha here exiss a consanasuch haλ A. Then we have ( ) ( ) x = Ψ (Φ (x)) = k(φ (x)) e Φ (x) 1 = k(φ (x)) e (Φ (x) Λ (x)) 1 ( ) (( ) ) k(φ (x)) e Φ x (x) e A 1 k(φ (x)) +1 e A 1, and so, we ge forxbig enough k(φ x (x)) ( C. x )e +1 A 1 Sincelim x + Φ (x) = +, previous inequaliy gives us hak is a bounded funcion, which is a conradicion. Remark 3.4 In 3, he auhors proved ha if for somep >, E sup e py T <, hal-8233, version 1-19 Mar 213 hene(q) has finie enropy, i.e., EE(q) T lne(q) T < +. However, in he criical case,e Y belongs o he class (D), his propery is no always rue. I suffices o ake againg(z) = 2 z 2, hen if Y, E(q) lne(q) = e Y e Y (Y +Y ). I follows ha ifg(z) = 2 z 2 andy,e(q) has finie enropy if and only ify e Y belongs o he class (D). 4 The uniqueness resul Remark 3.4 indicaes hae(q) does no always have finie enropy in he criical case. Hence we could no adop he verificaion argumen given in 3 o show he uniqueness. In his las secion, we show he uniqueness under he addiional assumpion B. Theorem 4.1 Le us assume ha assumpions A and B hold rue. Then he BSDE (1.1) has a unique soluion (Y,Z) such hay ande Y belong o he class (D). Proof of Theorem 4.1. The exisence resul is already given in Theorem 2.2. For he uniqueness, le us consider (Y,Z) and (Y,Z ) wo soluions of he BSDE (1.1) such ha Y, Y, e Y and e Y belong o he class (D). By a symmery argumen i is sufficien o show hay Y P-a.s. for all,t. For,T, le us denoe A := {Y < Y } and se he sopping ime τ = inf{s Y s Y s }. Then, for s,τ we have Y s Y s and Y τ = Y τ P-a.s. because Y is coninuousp-a.s.. Le us consider a predicable process(q s ) s,t wih values inr d and such haq s g(z s ) for alls,t. Thanks o Proposiion 3.1 we know ha E(q) defines a probabiliy ha we will denoeq. UnderQ, we ge Then, Iô formula gives us, for < α ε, d(y s Y s) = (g(z s ) g(z s) (Z s Z s)q s )ds (Z s Z s)dw Q s. (4.1) de α(ys τ Y s τ +C2(T s))1la = α1l A e α((ys τ Y s τ +C2(T s))1la (g(z s ) g(z s) (Z s Z s)q s +C 2 α 2 Z s Z s 2) 1l s τ ds α1l A e αc2(t s)1la 1l s>τ ds+1l s τ dm s, wih(m s ) s,τ a local maringale underq. From assumpion B we have ha g(z s) g(z s ) (Z s Z s )q s ε 2 Z s Z s 2 C 2.

8 REFERENCES 8 ( ) So, we obain ha e α(ys τ Y s τ +C2(T s))1la is a bounded supermaringale under Q and s T e α(ys τ Y s τ +C2(T s))1la E e α(yτ Y τ +C2(T T))1lA Fs = 1, s,t. I implies ha((y s τ Y s τ )1l A) s,t is a bounded process. Moreover,g is a convex funcion so g(z) g(z ) (z z )u, z,z R 1 d, u g(z). By using his inequaliy in (4.1), we obain ha((y s τ Y s τ )1l A) s,t is a bounded negaive supermaringale under Q such ha (Y τ Y τ)1l A =. We conclude ha (Y Y )1l A =, ha is o say, Y Y. Finally, i is raher sandard o show ha Z s Z s 2 ds = P-a.s.. References 1 P. Briand and Y. Hu. BSDE wih quadraic growh and unbounded erminal value. Probab. Theory Relaed Fields, 136(4):64 618, P. Briand and Y. Hu. Quadraic BSDEs wih convex generaors and unbounded erminal condiions. Probab. Theory Relaed Fields, 141(3 4): , 28. hal-8233, version 1-19 Mar F. Delbaen, Y. Hu and A. Richou. On he uniqueness of soluions o quadraic BSDEs wih convex generaors and unbounded erminal condiions. Ann. Ins. Henri Poincaré Probab. Sa., 47(2): , M. Kobylanski. Backward sochasic differenial equaions and parial differenial equaions wih quadraic growh. Ann. Probab., 28(2):558 62, 2. 5 E. Pardoux and S. Peng. Adaped soluion of a backward sochasic differenial equaion. Sysems Conrol Le., 14(1):55 61, 199.