Lecture notes on BSDEs Main existence and stability results

Transcription

1 Lecure noes on BSDEs Main exisence and sabiliy resuls Bruno Bouchard Universié Paris-Dauphine, CEREMADE, and CREST, Paris, France February 214 (revised May 215) Lecures given a he London School of Economics

2 Conens 1 Inroducion and moivaions Wha is a BSDE? Applicaion o he hedging of financial derivaives European opions Hedging wih consrains American opions Hedging according o a loss funcion Opimal conrol : he sochasic maximum principle Necessary condiion Sufficien condiion Examples Exponenial uiliy maximizaion wih consrains Risk measures represenaion Feynman-Kac represenaion of semi-linear parabolic equaions and numerical resoluion General exisence and comparison resuls The case of a Lipschiz driver The monoone case One dimensional case and non-lipschiz coefficiens The quadraic case Exisence for bounded erminal values Exisence for unbounded erminal values General esimaes and sabiliy for bounded erminal condiions using Malliavin calculus Comparison for concave drivers and general erminal condiions Addiional readings

3 3 Monoonic limis and non-linear Doob-Meyer decomposiion Monoonic limi Sabiliy of super-soluions of BSDEs g-supermaringale : decomposiion, monoonic sabiliy and down-crossings BSDE wih consrains Minimal supersoluion under general consrains Refleced BSDEs Exisence and minimaliy The case of a smooh barrier Link wih opimal sopping problems Furher readings Consrains on he gain process

4 Inroducion and general noaions Backward sochasic differenial equaions (BSDEs) are he non-markovian (sochasic) counerpar of semi-linear parabolic equaions. They have a wide range of applicaions in economics, and more generally in opimal conrol. In mahemaical finance, he sandard hedging heory can be wrien in erms of BSDEs (possibly refleced or wih consrains), bu hey are also naurally associaed o risk measures (g-expecaions), uiliy maximizaion under consrains, or recursive uiliies. These lecures are an inroducion o he heory of BSDEs and o heir applicaions. We will concenrae on various exisence and sabiliy resuls, saring from he classical Lipschiz coninuous case up o quadraic BSDEs, and BSDEs wih consrains. Our aim is o presen he echniques raher han he resuls by hemselves, so ha he reader can ener he subjec and furher sudy he references we provide. These noes should be read in he given order, some argumens ha are used repeaedly will only be explained he firs ime hey appear. We shall only consider BSDEs driven by a Brownian moion. Mos of he resuls presened here can be exended o BSDEs driven by a Brownian moion and a jump process, or even by a general rcll maringale. Very good complemenary readings are he lecures noes 1, 29 and he book 31. We collec here some general noaions ha will be used all over hese noes. We use he noaion x f o denoe he derivaive of a funcion f wih respec o is argumen x. For second order derivaives, we wrie xxf 2 and xyf. 2 The Euclydean norm of x R d is x, d is given by he conex. We will always work on a probabiliy space (Ω, F, P) ha suppors a d-dimensional Brownian moion W. We le F = (F ) T denoe he augmenaion of is raw filraion up o a fixed ime horizon T. In general, all ideniies are aken in he P a.s. or d dp-a.e. sense, his will be clear from he conex. We shall make use of he following spaces (he dimension of he random variables depends on he 3

5 conex): P : progressively measurable processes. L p (F ) : F -measurable random variables ξ such ha ξ L p := E ξ p 1 p <. We wrie L p if = T. S p : ζ in P wih coninuous pahs such ha ζ S p <. S p rcll : ζ in P wih rcll pahs such ha ζ S p := E sup,t ζ p 1 p <. A p : ζ in P wih non-decreasing rcll pahs and such ha ζ T L p and ζ =. 1 H p T : ζ in P such ha ζ H p := E ζ s p p ds H p : ζ in P such ha ζ p H := E T ζ s 2 ds p 1 p 2 <. <. T : se of sopping imes wih values in, T. We wrie T for T. 1 H 2 T BMO : ζ in P such ha ζ H 2 BMO := sup T E ζ s 2 2 ds F L <. Given wo processes X and X, we shall always use he noaion X for X X. We apply he same for wo funcions g and g : g = g g. In all his documen, C will denoe a generic consan which may change from line o line. Alhough i will no be said explicily, i will never depend on quaniies ha may change in he course of he argumen (like a parameer ha will be send o for insance). Proofs will be given for he one dimensional case alhough he resul is saed in a mulivariae framework. This is only o avoid heavy noaions. 4

6 Chaper 1 Inroducion and moivaions 1.1 Wha is a BSDE? Given an R d -valued random variable ξ in L 2 and g : Ω, T R d R d d, a soluion o he BSDE Y = ξ + g s (Y s, Z s )ds Z s dw s, T, P a.s., (1.1) is a pair of adaped processes (Y, Z), ypically in S 2 H 2, wih values in R d R d d such ha (1.1) holds. I means ha he process Y has he dynamics dy s = g s (Y s, Z s )ds + Z s dw s, bu, as opposed o forward SDEs, we prescribe is erminal condiion Y T = ξ raher han is iniial condiion Y. To fix ideas, le us consider he simple case g. Then, a couple (Y, Z) S 2 H 2 such ha (1.1) holds mus saisfy Y = E ξ F and he Z componen of he soluion is uniquely given by he maringale represenaion heorem ξ = E ξ + Z s dw s, i.e. E ξ F = E ξ + Z s dw s. From his paricular case, we see ha an adaped soluion o (1.1) can only be given by a pair: he componen Z is here o ensure ha he process Y is adaped. Unlike deerminisic ODEs, we can no simply rever ime as he filraion goes in one direcion. In he res of his Chaper, we provide various examples of applicaions. Oher examples can be found in he lecures noes 29 and in he book 31. 5

7 1.2 Applicaion o he hedging of financial derivaives Le us firs discuss some applicaions o he pricing and hedging of financial derivaives European opions We consider here a financial marke wih one risky asse S whose evoluion is given by ds = S µ d + S σ dw, in which µ and σ are some predicable and bounded processes. A rader can eiher inves in he risky asse S or borrow/lend money a an insananeous risk free ineres rae r, which is again bounded and predicable for sake of simpliciy. If π is he amoun of money invesed in S a, and Y is he oal wealh of he rader, hen Y π is he amoun of money ha is lend/borrowed, and he dynamics of he wealh process is dy = π S ds + r (Y π )d = {π (µ r ) + r Y } d + π σ dw. Le us now consider a European opion wih payoff a ime T given by a random variable ξ L 2. The aim of a rader who wans o sell his opion is o define he minima iniial amoun of capial Y such ha he can cover he payoff ξ. Obviously, if we can find a π such ha Y T = ξ, hen his minimal amoun is Y. Oherwise saed we look for a couple (Y, π) such ha Y = ξ {π s (µ s r s ) + r s Y s } ds π s σ s dw s. If here exiss a predicable process λ such ha (µ r) = σλ, which is called a risk premium in mahemaical finance, hen he above reads afer seing Z := πσ. Y = ξ {Z s λ s + r s Y s } ds Z s dw s, (1.2) Hence, he problem of hedging he opion is reduced o finding a soluion o a BSDE. In he above, he soluion is explicily given by Y = E Q e r sds ξ F in which Q is he equivalen probabiliy measure such ha W + λ sds is a Brownian moion, he so-called risk neural measure. However, he soluion is no more explici if he ineres raes for borrowing and lending are differen. Le us denoe hem by r b and r l respecively. Then, he dynamics of he wealh is given by Y = ξ { πs µ s + rs(y l s π s ) + rs(y b s π s ) } ds π s σ s dw s. 1 This secion can be compleed by he reading of El Karoui e al. 32, 3. 6

8 Assuming ha σ >, he corresponding BSDE is { } µ T s Y = ξ Z s + rl s (σ s Y s Z s ) + rb s (σ s Y s Z s ) ds Z s dw s. σ s σ s σ s Hedging wih consrains Le us now consider he case where he rader wans o confine himself o sraegies π saisfying cerain bounds: π m, M d dp-a.e. for given limis m, M >. Then, he needs o find (Y, Z) saisfying (1.2) and Z/σ m, M d dp-a.e. In general, his problem does no have a soluion and one needs o relax (1.2) ino dy {π (µ r ) + r Y } d + π σ dw wih Y T = ξ, (1.3) which we wrie as Y = ξ {Z s λ s + r s Y s } ds Z s dw s + A T A, (1.4) in which A is an adaped non-decreasing process. The A process can be viewed as a consumpion process. To ensure o keep π = Zσ wihin he prescribed bounds, one needs o sar wih a higher iniial wealh, which migh indeed no be used and can herefore be consumed. Hence, he soluion is now a riple (Y, Z, A). Obviously, uniqueness does no hold in general, as we can always sar wih an higher Y and compensae wih he A process. However, we are ineresed here in he characerizaion of he minimal soluion, in he sense ha Y is minimal among all possible soluions, since he rader wans o minimize he iniial capial required for he hedging. This problem has been widely sudied in he mahemaical finance lieraure, and we refer o 5, 14, 2, 22 for an analysis in he conex of BSDEs. The corresponding BSDE is usually referred as a BSDE wih consrain on he gain process American opions An American payoff can be viewed as an adaped process ζ: he amoun ζ is paid o he holder if he exercises his opion a before he mauriy T. Then, we wan o find (Y, Z) solving (1.3) such ha Y ζ on, T P a.s. Again, we can no expec o have an equaliy in (1.3) if we look for a minimal soluion, which in paricular should saisfy Y T = ζ T. Then, a soluion is again given by a riple (Y, Z, A), wih A adaped and non-decreasing, such ha (1.4) holds and Y ζ on, T. This is called a refleced BSDE, as he Y process should be refleced on he lower barrier ζ so has o say above i a any ime. This class of BSDEs has been firs inroduced in he conex of mahemaical finance by El Karoui e al. 3. 7

9 1.2.4 Hedging according o a loss funcion We now consider he problem of finding he minimal iniial wealh Y such ha here exiss π for which E l(y T ξ) m. In he above, m is a hreshold, and l is a loss funcion: a non-decreasing and concave funcion, which ypically srongly penalizes he loss (Y T ξ). This pricing crieria has be widely sudied by Föllmer and Leuker 33, 34. This leads o a class of BSDEs in which he erminal condiion Y T is no more fixed, bu has only o saisfy a cerain momen condiion. Their properies have been sudied by Bouchard, Elie and Réveillac Opimal conrol : he sochasic maximum principle Le us now urn o an applicaion in opimal conrol. We consider here he problem of maximizing an expeced gain of he form J(ν) := E g(xt ν ) + f (Xs ν, ν )d, in which X ν is he soluion of he one dimensional sde dx ν = b (X ν, ν )d + σ (X ν, ν )dw wih ν in he se U of predicable processes wih values in R. In he above, he random maps f, b and σ are such ha (, ω) (f (ω, x, u), b (ω, x, u), σ (ω, x, u)) is predicable for any (x, u) R 2 (we omi he ω argumen in he following). We also assume ha hey are d dp-a.e. bounded, C 1 in heir argumen (x, u), and ha hemselves as well as here firs derivaives are Lipschiz. The funcion g maps Ω R R, g() L, and g is a.s. C 1 wih bounded firs derivaive in x. In he following, we shall show how BSDEs permis o provide necessary and sufficien condiions for opimaliy. We refer o Peng 5, 51 for furher references Necessary condiion Le us sar wih a necessary condiion for a conrol ˆν o be opimal. The general idea is o used a spike variaion of he form ν ε, := ˆν1,) +ε,t +ν1,+ε) wih ε (, T ) and ν a F -measurable random variable, T. By opimaliy of ˆν, we mus have J(ˆν) J(ν ε, ), 8

10 and herefore, if ε J(ν ε, ) is smooh, ε J(ν ε, ) ε=. (1.5) The firs problem is herefore o show ha his map is smooh. From now on, we wrie ˆX for X ˆν and X ν,ε he general case. for X ν,ε, and we assume ha σ does no depend on ν for sake of simpliciy, see 51 for Under his addiional condiion, we can firs show ha X ν,ε is smooh wih respec o ε. Proposiion 1.1 Le us consider he process Ŷ,ν defined as he soluion of Y = 1 (b ( ˆX, ν) b ( ˆX ), ˆν ) + x b s ( ˆXs, ˆν s ) Y s ds + x σ s ( ˆXs ) Y s dw s. (1.6) Assume ha ˆν has P a.s. righ-coninuous pahs. Then, Ŷ ν, = ε Xν,ε ε= on, T P a.s. Moreover, ε J(νε, ) ε= = E + E x g( ˆX T )Ŷ ν, T + f ( ˆX, ν) f ( ˆX, ˆν ) x f s ( ˆX s, ˆν s )Ŷ s ν, ds. (1.7) The idea of he sochasic maximum principle is o inroduce a se of dual variables in order o exploi (1.7). Le us firs define he Hamilonian: H (x, u, p, q) := b (x, u)p + σ (x)q + f (x, u). Then, we assume ha here exiss a couple ( ˆP, ˆQ) of square inegrable adaped processes saisfying he BSDE ˆP = x g( ˆX T ) + x H s ( ˆX s, ˆν s, ˆP s, ˆQ s )ds ˆQ s dw s. (1.8) This equaion is called he adjoin equaion and ( ˆP, ˆQ) he adjoin process. The reason for inroducing his process becomes clear once we apply Iô s Lemma o ˆP Ŷ,ν. Indeed, assuming ha he local maringale par of ˆP Ŷ,ν is a rue maringale, we obain ha x g( ˆX T )Ŷ,ν T = ˆP T Ŷ,ν T is equal in expecaion o ˆP (b ( ˆX, ν) b ( ˆX, ˆν )) ( ) + x b s ˆXs, ˆν s Ŷ,ν s ˆP s ds + 9 Ŷs,ν x H s ( ˆX s, ˆν s, ˆP s, ˆQ s )ds x σ s ( ˆXs ) Ŷ,ν s ˆQ s ds,

11 which, by definiion of H, is equal o I follows ha ˆP (b ( ˆX, ν ) b ( ˆX, ˆν)) Y,ν s x f s ( ˆXs, ˆν s ) ds. ε J(ν ε, ) ε= = E H ( ˆX, ν, ˆP, ˆQ ) H ( ˆX, ˆν, ˆP, ˆQ ). By arbirariness of ν, his implies he necessary condiion for all T. H ( ˆX, ˆν, ˆP, ˆQ ) = max u R H ( ˆX, u, ˆP, ˆQ ) P a.s. (1.9) A similar analysis can be carried ou when σ does depend on he conrol ν bu i requires a second order expansion in he definiion of Y above. See Peng 5, Sufficien condiion We work wihin he same framework as above, excep ha we now allow σ o depend on he conrol process ν. We assume here ha he maps x g(x) and x Ĥ(x, ˆP, ˆQ ) := sup H (x, u, ˆP, ˆQ ) are P a.s. concave (1.1) u R for almos every, T, and ha x H ( ˆX, ˆν, ˆP, ˆQ ) = x Ĥ ( ˆX, ˆP, ˆQ ) (1.11) for all sopping imes. Noe ha he laer corresponds o he envelop principle along he pah of ( ˆX, ˆP, ˆQ). Under he above assumpions, he condiion H ( ˆX, ˆν, ˆP, ˆQ ) = max u R H ( ˆX, u, ˆP, ˆQ ), T (1.12) is acually a sufficien condiion for opimaliy. Indeed, we firs noe ha, by concaviy of g, E g( ˆX T ) g(xt ν ) E x g( ˆX T )( ˆX T XT ν ) = E ˆPT ( ˆX T XT ν ), 1

12 which, by Iô s Lemma and (1.11), implies E g( ˆX T ) g(xt ν ) E E + E ˆP s (b s ( ˆX s, ˆν s ) b s (Xs ν, ν s ))ds x Ĥ s ( ˆX s, ˆP s, ˆQ s )( ˆX s Xs ν )ds ( σ s ( ˆX s ) σ s (Xs )) ν ˆQs ds By definiion of H, Ĥ and (1.1)-(1.12), his leads o J(ˆν) J(ν) E (H s ( ˆX s, ˆν s, ˆP s, ˆQ s ) H s (Xs ν, ν s, ˆP s, ˆQ s ))ds E x Ĥ s ( ˆX s, ˆP s, ˆQ s )( ˆX s Xs ν )ds E Ĥ s ( ˆX s, ˆP s, ˆQ s ) Ĥs(Xs ν, ˆP s, ˆQ s )ds E x Ĥ s ( ˆX s, ˆP s, ˆQ s )( ˆX s Xs ν )ds. Remark 1.1 Le us now assume ha µ, σ and f are non-random and assume ha here exiss a smooh soluion ϕ o he Hamilon-Jacobi-Bellman equaion: = sup u R ( ϕ(, x) + b (x, u) x ϕ(, x) (σ (x, u)) 2 2 xxϕ(, x) + f (x, u) wih erminal condiion ϕ(t, ) = g. Assume ha he sup is aained by some û(, x). Se p := x ϕ and q := 2 xxϕσ. I follows from he envelop heorem, ha (p, q) formally solves (ake he derivaive wih respec o x in he above equaion) = Lû(,x) p(, x) + x Ĥ (x, p(, x), q(, x, û(, x))) wih he erminal condiion p(t, ) = x g. Le now ˆX be he conrolled process associaed o he Markov conrol ˆν = û(, ˆX ) (assuming ha i is well defined). Then, Iô s Lemma implies ha p(, ˆX ) = x g( ˆX T ) + q(s, ˆX s, ˆν s )dw s. x H s ( ˆX s, ˆν s, p(s, ˆX s ), q(s, ˆX s, ˆν s ))ds Under mild assumpions ensuring ha here is only one soluion o he above BSDE, his shows ha ˆP = p(, ˆX ) = x ϕ(, ˆX ) and ˆQ = q(, ˆX, ˆν ) = 2 xxϕ(, ˆX )σ ( ˆX, ˆν ). Oherwise saed, he adjoin process ˆP can be seen as he derivaive of he value funcion wih respec o he iniial condiion in space, while ˆQ is inimaely relaed o he second derivaive. 11. )

13 1.3.3 Examples Example 1.1 Le us firs consider he problem where X ν is defined as X ν = x + for some x > and where X ν s ν s ds s S s = x + max E ln(x ν T ) X ν s ν s µ s ds + S = S e (µs σ2 s /2)ds+ σsdws for some bounded predicable processes µ and σ > wih 1/σ bounded as well. X ν s ν s σ s dw s (1.13) This corresponds o he problem of maximizing he expeced logarihmic uiliy of he discouned erminal wealh in a one dimensional Black-Scholes ype model wih random coefficiens. Here, ν sands for he proporion of he wealh X ν which is invesed in he risky asse S. I is equivalen o maximizing E X ν T wih Xν now defined as The associaed Hamilonian is X ν = (ν s µ s ν 2 s σ 2 s/2)ds. H (x, u, p, q) = (uµ (u 2 σ 2 /2))p. Thus Ĥ(x, p, q) = 1 µ 2 p and he argmax is û(, x, p, q) := µ 2 σ 2 σ 2 adjoin process ( ˆP, ˆQ) is given by. I follows ha he dynamics of he ˆP = 1 ˆQ s dw s. This implies ha ˆP = 1 and ˆQ = d dp a.e. In paricular, for ˆX := X ˆν wih ˆν := µ/σ 2 he opimaliy condiions of he previous secion are saisfied. This implies ha ˆν is an opimal sraegy. Since he opimizaion problem is clearly sricly concave in ν, his is he only opimal sraegy. Observe ha he soluion is rivial since i only coincides wih aking he max inside he expecaion and he inegral in E XT ν = E T (ν sµ s νs 2 σs/2)ds 2. Example 1.2 We consider a similar problem as in he previous secion excep ha we now ake a general uiliy funcion U which is assumed o be C 1, sricly concave and increasing. We also assume ha i saisfies he so-called Inada condiions: x U( ) = and x U(+) =. 12

14 We wan o maximize E U(X ν T ) where Xν is given by (1.13). We wrie ˆX for X ˆν. In his case, he condiion (1.12) reads H ( ˆX, ˆν, ˆP, ˆQ ) = sup u R Bu, i is clear ha i can be saisfied only if (u µ ˆX ˆP + u σ ˆX ˆQ ). Thus, by (1.8), ˆP should have he dynamics ˆQ = λ ˆP wih λ = µ/σ. ˆP = x U( ˆX T ) + This implies ha we have o find a real ˆP > such ha ˆP = ˆP e 1 2 λ s ˆPs dw s. λ2 s ds λsdws and ˆP T = x U( ˆX T ). Hence, he opimal conrol, if i exiss, should saisfy ˆX T = ( x U) 1 ( ˆP e 1 2 λ2 s ds+ λsdws ). (1.14) Now, le Q P be defined by dq = ˆP T / ˆP so ha W Q = W + λ sds is a Q-Brownian moion, and ha X ν is a supermaingale under Q for all ν U. If ˆX is acually a rue Q-maringale, hen we mus have x = E Q ( x U) 1 ( ˆP e 1 2 λ2 s ds+ λsdws ). (1.15) Using he Inada condiions imposed above, i is clear ha we can find ˆP such ha he above ideniy holds. The represenaion heorem hen implies he exisence of an admissible conrol ˆν such ha (1.14) is saisfied. Since he sufficien condiions of Secion hold, his shows ha ˆν is opimal. We can also check his by using he concaviy of U which implies U(X ν T ) U( ˆX T ) + x U( ˆX T ) (X ν T X T ) = U( ˆX T ) + ˆP T ( X ν T ˆX T ) Since, by he above discussion, he las erm is non posiive in expecaion, his shows ha he opimal erminal wealh is acually given by (1.14) Exponenial uiliy maximizaion wih consrains We now consider a similar uiliy maximizaion problem, bu we add consrain on he financial sraegy. We resric o an exponenial uiliy funcion. Then, he following has been firs discussed 13

15 in El Karoui and Rouge 55. See also Hu e al. 38 for more deails and for he case of power uiliy funcions, or 37 for general uiliy funcions. Le U be predicable processes wih values in a compac (for simpliciy) subse A R. Given some predicable bounded processes ν and σ, we describe here he wealh associaed o a rading sraegy ν U by he dynamics We wan o compue dv ν = ν (µ d + σ dw s ), V ν =. u := sup E U(VT ν ) wih U(v) := e ηv, η >. ν U We use he following approach: find a process Y such ha L ν := U(V ν Y ) saisfies L ν is a super-maringale for all ν U. Lˆν is a -maringale for one ˆν U. L ν T = U(V T ν ) for all ν U. L ν does no depend on ν U, we call i L. If such a process exiss hen E U(VT ˆν ) = E Lˆν T = L E L ν T = E U(VT ν ), so ha ˆν is opimal and u = L. Le us ake Y of he form (1.1). Then, dl ν = ηl ν (ν µ + g (Y, Z ) η ) 2 (ν Z σ ) 2 d η(ν σ Z )L ν dw. Thus, we mus have ( η g (Y, Z ) = g(z ) = min a A 2 (aσ Z ) 2 η aµ) = min a A 2 ( aσ (Z + µ ) ησ 2 2Z µ ησ µ 2). ησ This provides BSDE wih a driver which is quadraic in Z. If a soluion exiss wih Z such ha Lˆν is a rue maringale for ˆν U defined by ( η ) g(z) = 2 (ˆνσ Z ) 2 ˆνµ, hen, ˆν is acually he opimal rading sraegy. We shall see laer, see Theorem 2.5 below, ha exisence holds and ha he corresponding Z belongs o H 2 BMO, which ensures ha Lˆν is indeed a rue maringale, see Kazamaki 39. Remark 1.2 For U(x) = x γ, we ake ν as he proporion of wealh and L ν := e γνsdws 1 2 γ νs 2 ds e γνs µs σs ds+y. We obain by he same argumens as above ha he value is x γ e Y. 14

16 1.5 Risk measures represenaion Backward sochasic differenial equaion can also be used o consruc risk measures. We briefly discuss his here and refer o Peng 53 for a complee reamen. Le us firs inroduce he noion of F-expecaion defined by Peng, which is inimaely relaed o he noion of risk measures. Definiion 1.1 A non-linear F-expecaion is an operaor E : L 2 R such ha X X implies EX EX wih equaliy if and only if X = X. Ec = c for c R. For each X L 2 and T, here exiss η X A F. We wrie E X for η X. L 2 (F ) such ha EX1 A = Eη X 1 A for all Remark 1.3 η X is uniquely defined. Indeed, if η saisfies he same, we can ake A = 1 η X > η and deduce from he firs iem in he above definiion ha Eη X 1 A > Eη 1 A if PA >. Noe ha i corresponds o he noion of condiional expecaion, in his non-linear framework. Le us now consider he soluion (Y, Z) of Y = ξ + g s (Y s, Z s )ds Z s dw s, T, and call he Y componen E g ξ. We omi when =. We se g µ (y, z) = µ z. The following remarkable resul shows ha no only BSDEs provides non-linear expecaions, bu ha a large class of hem (ypically he one used for cash invarian risk measures) are acually given by a BSDE. Theorem 1.1 Le g : Ω, T R R R be such ha g(x, y) H 2 for all (x, y), and g is uniformly Lipschiz in (y, z) d dp-a.e., hen E g is a non-linear expecaion. Conversely, le E be one non-linear F-expecaion such ha for all X, X L 2 EX + X EX + E gµ X and E X + X = E X + X if X L 2 (F ). Then, here exiss a random driver g which does no depend on y such ha g(z) µ z and E = E g. 15

17 1.6 Feynman-Kac represenaion of semi-linear parabolic equaions and numerical resoluion Le us conclude wih he link beween BSDEs and semi-linear parial differenial equaions. Consider he soluion X of he sde X = x + b s (X s )ds + σ s (X s )dw s, in which b and σ are deerminis maps ha are assumed o be Lipschiz in heir space variable. Assume ha here exiss a soluion v C 1,2 (, T ) R) C (, T R) o he PDE in which Then, he couple solves Indeed, by Iô s Lemma, G(X T ) = v(, X ) + = v(, X ) = Lϕ + g(, ϕ, x ϕσ) on, T ) R, wih v(t, ) = G Y = G(X T ) + Lϕ = ϕ + b x ϕ σ2 2 xxϕ. Y := v(, X), Z := x v(, X)σ(X) Lv(s, X s )ds + g s (X s, Y s, Z s )ds x v(s, X s )σ s (X s )dw s g s (X s, v(s, X s ), x v(s, X s )σ s (X s ))ds + Z s dw s. x v(s, X s )σ s (X s )dw s. In paricular, if he above BSDE has a mos one soluion, hen solving he BSDE or he PDE is equivalen. This provides an alernaive o he resoluion of PDEs by considering backward schemes of he form := E Y n + T n i+1 n g n(xn i, Y n n i, n Zn i+1 ) F n n, i i Y n n i Z n n i := n T E n i Y n n i+1 (W n i+1 W n i ) F n i in which YT n = g(xn T ) and Xn is he Euler scheme of X wih ime sep T/n, i n = it/n. When he coefficiens are 1/2-Hölder in ime and Lipschiz in he oher componens, his scheme converges a a speed n 1 2, see Bouchard and Touzi 8 and Zhang 58. Obviously his scheme is only heoreic as i requires he compuaion of condiional expecaions. Sill, one can use various Mone-Carlo ype approaches o urn i ino a real numerical scheme, see he references in he survey paper Bouchard and Warin 9 and in he book Gobe ,

18 Remark 1.4 a. A similar represenaion holds for ellipic equaions. In his case, we have o replace T in he BSDE by a random ime, ypically he firs exis ime of X from a domain, see e.g. he survey paper Pardoux 47. b. By considering BSDEs wih jumps, one can also provide a represenaion of sysems of parabolic equaions. The original idea is due o Pardoux, Pradeilles and Rao 49 and was furher discussed in Sow and Pardoux 56. The corresponding numerical scheme has been sudied by Bouchard and Elie 4. 17

19 Chaper 2 General exisence and comparison resuls The aim of his Chaper is o provide various exisence and sabiliy resuls for BSDEs of he form (1.1). From now on, a driver g will always be a random map Ω, T R d R d d R d such ha (g (y, z)) T P for all (y, z) R d R d d. 2.1 The case of a Lipschiz driver We firs consider he sandard case of a Lipschiz coninuous driver. Assumpion 2.1 g() H 2 and g is uniformly Lipschiz in (y, z). The following resuls are due o Pardoux and Peng 48. See also 32 for more properies such as differeniabiliy in he Malliavin sense and for applicaion in opimal conrol and in finance. We firs provide an easy esimae ha will be used laer on. Proposiion 2.1 Le Assumpion 2.1 hold. Fix ξ L 2. If (Y, Z) saisfies (1.1) (assuming i exiss) and (Y, Z) H 2 H 2, hen Y S 2. Proof. We use (1.1), he fac ha g has linear growh in (y, z) and he Burkholder-Davis-Gundy inequaliy o obain Y S 2 CE ζ 2 + Y s 2 + Z s 2 + g s 2 ()ds. Since he consrucion of a soluion will be based on a conracion argumen, we also need some a-priori esimaes on he sabiliy of soluions wih respec o heir drivers and erminal condiions. In paricular, he following ensures ha a BSDE can only have a mos one soluion. 18

20 Proposiion 2.2 (Sabiliy) Le Assumpion 2.1 for g and g holds. Fix ξ and ξ L 2. Le (Y, Z) and (Y, Z ) be associaed soluions (assuming hey exis) in S 2 H 2. Then, Y 2 S 2 + Z 2 H 2 C ( ζ 2 L 2 + g(y, Z) 2 H 2 ). Proof. We fix α R, and apply Iô s Lemma and he Lipschiz coninuiy of g o obain e α Y 2 + e αs Z s 2 ds = e αt ζ 2 + e ( αs 2 Y s (g s (Y, Z) g s(y, Z )) α Y s 2) ds 2 e αs Y s Z s dw s e αt ζ 2 + e αs C Y s Z s 2 + g s 2 (Y s, Z s ) α Y s 2 ds 2 Y s Z s dw s. The reason for inroducing he α is ha if we now choose α = C hen he Y erms cancel in he firs inegral on he righ-hand side: e α Y e αs Z s 2 ds e αt ζ 2 + C g s 2 (Y s, Z s )ds 2 e αs Y s Z s dw s. Noe ha eαs Y s Z s dw s is a uniformly inegrable maringale since by he Burkholder-Davis- Gundy inequaliy E sup,t e αs Y s Z s dw s CE ( Y s 2 Z s 2 ds) 1 2 CE 1 2 sup Y 2,T Taking expecaion in he previous inequaliy hen yields E sup E Y s 2 + Z 2 H C ( ζ 2 2 L + g(y, ) 2 Z) 2 H. 2 T 1 Z s 2 2 ds <. We now use he definiion of Y and he Burkholder-Davis-Gundy inequaliy o obain Y S 2 CE ζ 2 + Y s 2 + Z s 2 + g s 2 (Y s, Z s )ds, and he resul follows from he previous esimae. Remark 2.1 In he above, we did in fac no use he Lipschiz coninuiy of g. Exisence of a soluion associaed o g would be enough. 19

21 We are now in posiion o prove ha a soluion o (1.1) exiss. Theorem 2.1 (Exisence) Le Assumpion 2.1 holds. (1.1). Then, here exiss a unique soluion o Proof. In he case where g does no depend on (y, z) he resul follows from he maringale represenaion heorem. The general case is obained by a conracion argumen. Le H 2 α be he se of elemens ζ P such ha (e α 2 ζ ) T H 2, for α >. Given (U, V ) H α le us define (Y, Z) := Φ(U, V ) as he unique soluion of (1.1) for he driver (, ω) g (ω, U(ω), V (ω)). Define similarly (Y, Z ) from (U, V ). Then, e α Y 2 + e αs Z s 2 ds = 2 e αs 2 Y s (g s (U, V ) g s (U, V )) α Y s 2 ds e αs Y s Z s dw s. Since g is Lipschiz, Y s (g s (U, V ) g s (U, V )) C Y s ( U, V ) s ) α Y s 2 + C α ( U, V ) s 2, in which we used ha ab ηa 2 + η 1 b 2 for all a, b R and η >. Then, e α Y 2 + e αs Z s 2 ds C α e αs ( U, V ) s 2 ds 2 e αs Y s Z s dw s and herefore e α ( Y, Z) H 2 C α eα ( U, V ) H 2. For α large enough, he map Φ is conracing, and herefore we can find a fix poin (Y, Z) = Φ(Y, Z). This also prove uniqueness in H 2 α. We have (Y, Z) S 2 H 2 by Proposiion 2.1 and uniqueness in S 2 H 2 by Proposiion 2.2. We now sae a comparison resul. I is ineresing per-se, and i will be of imporan use for he consrucion of soluions wih more general divers. Also noe he echnique ha we use o prove i, i is a linearizaion procedure which is par of he sandard machinery. Proposiion 2.3 (Comparison) Le d = 1. Le Assumpion 2.1 holds for g and assume ha exisence holds for g. Assume ha ζ ζ and g(y, Z ) g (Z, Y ) d dp-a.e. Then, Y Y for all T. If moreover P ζ < ζ > or g(y, Z ) < g (Z, Y ) on a se of non-zero measure for d dp, hen Y < Y. 2

22 Proof. Since g is Lipschiz, he following processes are bounded: b := (Y Y ) 1 (g(y, Z) g(y, Z))1 Y Y and a := (Z Z ) 1 (g(y, Z) g(y, Z ))1 Z Z. Le hen Γ be he soluion of Since we obain Y = ζ + Γ = 1 + Γ sb s ds + Γ sa s dw s. b s Y s + a s Z s + g s (Y s, Z s)ds Z s dw s, Y = E Γ T ζ + Γ s g s (Y s, Z s)ds F. Remark 2.2 Noe ha he same argumens lead o Y = E Γ T ζ + Γ sg s ()ds F, wih b := Y 1 (g(y, Z) g(, Z))1 Y and a := Z 1 (g(, Z) g())1 Z. In paricular, if ξ + g() M for some real number M, hen Y is bounded. 2.2 The monoone case We now relax he Lipschiz coninuiy assumpion and replace i by a monooniciy condiion. The idea is originally due o Darling and Pardoux 23. Assumpion 2.2 (Monooniciy condiion) g() H 2, g is coninuous wih linear growh in (y, z), is Lipschiz in z and (g(y, ) g(y, )) (y y ) κ y y 2, for all y, y R d. Noe ha we can reduce o he case κ = in Assumpion 2.2 by considering (e κ Y, e κ Z ) in place of (Y, Z). Thus, he name monooniciy condiion. As in he Lipschiz coninuous case, we sar wih a-priori esimaes ha will hen be used o consruc a conracion. 21

23 Proposiion 2.4 Le g be such ha yg(y, z) y g() + κ ( y 2 + y z ) for all (y, z) R d R d d. Fix ξ L 2 and le (Y, Z) be a soluion of (1.1). Then, here exiss α > and C α (boh independen on T ) such ha E sup e α Y 2 +,T e αs Z s 2 ds ( C α E e αt ξ 2 + Moreover, here exiss γ >, independen on T, such ha Y 2 E e γ(t ) ξ 2 + e γ(t ) g s () 2 ds F. Proof. Apply Iô s Lemma o e α Y 2 o obain e α Y 2 + in which e αs Z s 2 ds = e αt ξ 2 + ) 2 e α 2 s g s () ds. e αs 2Y s g s (Y s, Z s ) α Y s 2 ds 2 e αs Y s Z s dw s (2.1) 2Y s g s (Y s, Z s ) α Y s 2 C Y s 2 + g s () 2 + C Y s Z s α Y s 2 Take α = C and use he above o deduce e α Y This provides he second asserion. e αs Z s 2 ds e αt ξ 2 + C C Y s 2 + g s () Z s 2 α Y s 2. The firs one follows from similar argumens, we firs wrie ha e αs g s () 2 ds 2 e αs Y s Z s dw s. 2Y s g s (Y s, Z s ) α Y s 2 C Y s Y s g s () + C Y s Z s α Y s 2 and ake α = C o deduce from (2.1) ha C Y s Y s g s () Z s 2 α Y s 2, Then, e α Y e αs Z s 2 ds = e αt ξ T 2 E e αs Z s 2 ds e αs 2 Y s g s () ds 2 e αs Y s Z s dw s. E e αt ξ 2 + e αs 2 Y s g s () ds 22

24 and E sup e α Y T 2 e αs Z s 2 ds E e αt ξ 2 + +E 2 sup T e αs 2 Y s g s () ds e αs Y s Z s dw s, in which, given ι >, 2E sup T e αs Y s Z s dw s 2E ( 2E ιe ιe e 2αs Y s 2 Z s 2 ds) 1 2 sup e αs Y s ( s Z s 2 ds) 1 2 sup e αs Y s 2 + C ι E s sup e αs Y s 2 + C ι E s Z s 2 ds e αt ξ 2 + e αs Y s g s () ds and, given η >, 2E e αs Y s g s () ds Combining he above leads o E sup e α Y T 2 e αs Z s 2 ds ηe sup e αs Y s 2 + C η E s ( g s () ds) 2. (1 + C ι )E e αt ξ 2 + (ι + C ι η + η)e +C η (1 + C ι )E ( g s () ds) 2. sup e αs Y s 2 s We conclude by choosing ι = 1/4 and η = 1/(4C ι + 4). Corollary 2.1 (Sabiliy) Le Assumpion 2.2 for g and g holds. Fix ξ and ξ L 2. Le (Y, Z) and (Y, Z ) be associaed soluions (assuming hey exis). Then, here exiss α > and C α (boh independen on T ) such ha Y 2 S 2 + Z 2 H 2 C αe e αt ζ 2 L 2 + ( ) 2 e α 2 s g s (Y s, Z s)ds Proof. ( Y, Z) solves he BSDE wih driver (y, z) ḡ(y, z) = g(y + Y, z + Z ) g (Y, Z ). I saisfies he requiremen of Proposiion 2.4. Theorem 2.2 (Exisence) Le Assumpion 2.2 holds. (1.1). Then, here exiss a unique soluion o 23

25 Proof. Uniqueness follows from Corollary 2.1. In he following, we separae he difficulies. Since g is Lipschiz in z, we can firs prove ha a conracion holds when a soluion exiss for he BSDE wih driver g(, V ) for any V H 2. This is a raher direc consequence of Corollary 2.1. Then, we will show ha a soluion acually exiss for g(, V ) by using he monooniciy condiion. Sep 1. Le us firs assume ha, for any V H 2, we can find a soluion o Y = ξ + g s (Y s, V s )ds Z s dw s, T. (2.2) Given (U, V ), (U, V ) le (Y, Z), (Y, Z ) be he corresponding soluions and le Φ be he corresponding mapping. Then, i follows from Corollary 2.1 ha ( ) 2 Y 2 S + 2 Z 2 H C αe 2 e α 2 s g s (Y s, V s ) g s (Y s, V s) ds C αt E e αs V s V s ds. Hence, Y 2 S 2 + Z 2 H 2 C αt e αt V 2 H 2. For T δ small, he map Φ is conracing. For larger values of T, we can glue ogeher he soluions backward: consruc a soluion on T, T δ, given Y T δ consruc a soluion on T δ, T 2δ wih erminal condiion Y T δ a T δ, and so on. This provides a soluion on, T. Sep 2. I remains o prove ha we can find a soluion o (2.2) for any V H 2. We now se h (y) = g (y, V ) and resric o he case where ξ + sup h () is bounded by a consan M. This will be imporan as we will modify he driver o reduce o he Lipschiz case and we need o ensure ha, in some sense, he modified driver shares he monooniciy propery of h (which will come from a bound on Y deduced from he bound on ξ and h()). By runcaing and mollifying he funcion h, we can find a sequence of (random) funcions h n which are Lipschiz in y, wih values if y n + 2, uniformly bounded on compac ses, and ha converges uniformly on compac ses o h. More precisely, we consider a smooh kernel ρ suppored by he uni ball, θ n C wih values in, 1 such ha θ n (y) = 1 if y n and θ n (y) = if y n + 1. We se h n (y) := n d ρ(nu)θ n (y u)h(y u)du. This allows us o come back o he case of a Lipschiz driver. Then, here exiss a soluion (Y n, Z n ) o he BSDE associaed o h n. We shall now show ha (Y n, Z n ) n is Cauchy, which will provide a soluion for he original driver g. Since ξ and h() are uniformly bounded, Proposiion 2.4 implies ha Y n A, where A is independen of n. This implies ha h n is monoone along he pah of Y. More precisely, if κ = (which we can always assume, see above), hen h n in monoone on he ball cener a of radius n 1: h n (y) = n d ρ(nu)θ n (y u)h(y u)du = n d ρ(nu)h(y u)du for y n 1, u 1 u 1 24

26 and herefore, for y, y n 1, (y y )(h n (y) h n (y )) = since κ =. u 1 In paricular, for n, m A + 1, we have n d ρ(nu)(y u (y u))(h(y u) h(y u))du, (Y m Y n )(h m (Y m ) h n (Y n )) = (Y m Y n )(h m (Y m ) h m (Y n )) + (Y m Y n )(h m (Y n ) h n (Y n )) + 2A sup h m (y) h n (y). y A By argumens already used in he proof of Proposiion 2.2, we deduce ha T Y m Y n S 2 + Z m Z n H 2 CE sup h m s (y) h n s (y) ds. y A Hence he sequence is Cauchy. Le (Y, Z) be he limi. Clearly, for all T, Y n Y and Zn s dw s Z sdw s in L 2. Moreover h n s (Y n s ) h s (Y s ) ds sup h n s (y) h s (y) ds + h s (Ys n ) h s (Y s ) ds. y A Since h is coninuous, convergence holds. Since equaliy beween opional processes on sopping imes implies indisinguishabiliy, see e.g. 24, Thm 86, Chap. IV, his shows ha (Y, Z) solves (2.2). Sep 3. We reduced above o bounded ξ and h() o ensure o be able o make profi of he monooniciy of h. The general case is obained by a sandard runcaion argumen. Fix p > and se ξ p := ξ1 ξ p, h p := h1 h() p. I saisfies he requiremens of he previous sep. Le Y p and Y q be soluions associaed o p q. By Corollary 2.1, Y 2 S 2 + Z 2 H 2 CE ξ 2 1 ξ >p + 1 { hs() >p}h 2 s(ys p )ds Since Y p Cp by Proposiion 2.4 and h(y) h() + C y, we have 1 { hs() >p}h 2 s(y p s ) C h () 2 1 { hs() >p}, and herefore Y 2 S 2 + Z 2 H 2 CE ξ 2 1 ξ >p + 1 { hs() >p}h 2 s()ds which goes o as q p. Hence here exiss a limi, and i is easy o check (by similar argumens as in he end of Sep 2) ha i provides a soluion. 25

27 2.3 One dimensional case and non-lipschiz coefficiens In he one dimensional case, we can even only assume ha g is coninuous. In general, we do no have uniqueness, bu only he exisence of a minimal soluion 1 (Y, Z). The following is due o Lepeleier and San Marin 41. Assumpion 2.3 g() H 2, g is coninuous wih linear growh in (y, z). Theorem 2.3 (Exisence) Le d = 1 and le Assumpion 2.3 holds. Then, here exiss a minimal soluion o (1.1). Proof. We proceed by inf-convoluion: g n (x) := inf x (g(x) + n x x ) o reduce o he Lipschiz coninuous case. Being n-lipschiz, he driver g n is associaed o a unique soluion (Y n, Z n ). By he comparison resul of Proposiion 2.3, he sequence (Y n ) n is non-decreasing and bounded from above by he soluion of he BSDE wih driver C(1 + y + z ). Hence, i converges a.s. and in H 2. Moreover, i follows from Iô s Lemma and he uniform linear growh propery of (g n ) n, which is an immediae consequence of he linear growh propery of g, ha Z n Z m 2 H 2 CE CE (Y n s (Y n s Ys m )(gs n gs m )(Ys n, Zs n )ds 1 Ys m ) 2 2 T 1 ds E (1 + Ys n + Zs n ) 2 2 ds Hence (Z n ) n is Cauchy and converges o some Z H 2. I remains o prove ha gm s (Ys m, Zs m )ds g s(y s, Z s )ds for all sopping imes T. Bu g m g uniformly on compac ses. In paricular g m (x m ) g(x) if x m x. Since, afer possibly passing o a subsequence (Y m, Z m ) (Y, Z) d dp-a.e., we acually obain ha g m (Y m, Z m ) g(y, Z) in H 2, by dominaed convergence (and up o a subsequence). We conclude wih he minimaliy of (Y, Z): if (Y, Z ) is anoher soluion, hen Y Y n by Proposiion 2.3, bu Y n Y. Remark 2.3 Noe ha comparison holds by Proposiion 2.3 for he minimal soluions consruced as above. Remark 2.4 Uniqueness can be obained under sronger condiions. I is for insance he case if here exiss a concave increasing funcion κ such ha κ() =, κ(x) > is x, + (x/κ(x))dx = and see 45. See also 44 for differen condiions. g(y, z) g(y, z ) 2 κ( y y 2 + x x 2 ), 1 i.e. if (Y, Z ) is anoher soluion, hen Y Y on, T. 26

28 2.4 The quadraic case We resric here o he one dimensional seing d = n = Exisence for bounded erminal values To prove exisence, we use he following idea. Assume ha g(y, z) = 1 2 z 2 and se Ȳ := ey. Then, dȳ = Ȳ dy + 1 2Ȳ Z2 d = Ȳ ZdW so ha (e Y, e Y Z) solves a linear BSDE, which soluion is even explici. The exension has been firs provided by Kobylanski 4. We sar wih he case where he driver is bounded in y and he erminal condiion is bounded. The exension o a linear growh condiion on y will be quie immediae, while working wih unbounded erminal condiion will require an addiional effor. Assumpion 2.4 ζ is bounded, g is coninuous, g(y, z) K g (1 + z 2 ) for all (y, z) R R. Theorem 2.4 (Exisence for bounded erminal values #1) Le Assumpion 2.4 holds. Then, here exiss a maximal soluion (Y, Z) o (1.1). Moreover, (Y, Z) S H 2 BMO. Proof. Saying ha (Ȳ, Z) solves (1.1) wih erminal condiion ζ is equivalen o saying ha (Y, Z) = (e 2KgȲ, 2K g Y Z) solves Y = ζ + g s (Y s, Z s )ds Z s dw s (2.3) in which ζ := e 2Kg ζ and which saisfies g(y, z) = 2K g y { g(ln y/2k g, z/(2k g y)) z 2 /(4K g y 2 ) } 2K 2 g y z 2 y g(y, z) 2K2 g y. In he following, we focus on finding a soluion o (2.3). The idea of he proof is he following. Firs, we runcae he driver in y and z so as o recover he case of a coninuous driver wih linear growh. Knowing ha ξ is bounded, we hen check ha he runcaion on y does no operae because 27

29 soluions are indeed (uniformly) bounded. By comparison, he sequence of Y -componens of he soluions will be non-decreasing and herefore convergen. I hen remains o prove he convergence of he Z-componens. Sep 1. Le θ p be a smooh, 1-valued funcion wih value 1 for z p and for z p + 1. Fix < β se ρ(y) = β 1 y β and ḡ p (x, y) := θ p (z) g(ρ(y), z) + 2K 2 g ρ(y)(1 θ p (z)). Since ḡ p is coninuous, exisence holds by Theorem 2.3. Le (Y p, Z p ) be a soluion. Then, we can choose (θ p ) p and (Y p ) p so ha he laer is non-increasing and bounded. Indeed, if (θ p ) p i is non-decreasing, hen ḡ p g(ρ, ), and we can appeal o Remark 2.3. Moreover, by Proposiion 2.3, Y p is sandwiched by he backward ODE wih divers 2K 2 g (y β 1 ) and 2K 2 g (y β), and erminal condiion given by A > and A 1 such ha A 1 e 2Kgζ A. For A large, his coincides wih he ode wih drivers 2K g y and 2K g y and he corresponding soluions a, a can be found such ha ρ(a) = a and ρ(a ) = a. Hence, we have Y p = ρ(y p ) so ha Z p does no depend on ρ as well. The corresponding lower bound is uniformly sricly posiive so ha 1/Y p is also bounded. Hence Y p converges a.s. and in H 2 o some Y as p, such ha Y and 1/Y are bounded. Sep 2. We now show ha (Z p ) p is uniformly bounded in H 2. To see his, le us se ψ(x) := e 3cx where c > is such ha 2c 2 Y p c Z p 2 g(y p, Z p ) 2c 2 Y p. (2.4) Then, ( ψ(ζ) = ψ(y p ) ψ (Ys p )ḡs(y p s p, Zs p ) 1 ) 2 ψ (Ys p ) Zs p 2 ds ψ (Ys p )Zs p dw s. By (2.4) and since ψ <, his implies ha, for any sopping ime, ( ψ(ζ) ψ(y p ) + ψ (Ys p )2c 2 Ys p + (cψ (Ys p ) + 1 ) 2 ψ (Ys p )) Zs p 2 ds ψ (Ys p )Zs p dw s. We now observe ha cψ ψ = 3c 2 ψ/2 and ha ψ(y p ) ι for some real ι > independen on p. Hence CE ψ(ζ) ψ(y p ) + ψ (Ys p ) 2c 2 Ys p F E Zs p 2 ds F. Noe ha he same is rue for Z p /(2K g Y p ). Sep 3. We can now prove ha (Z p ) p converges. As i is bounded in H 2, i converges weakly, up o a subsequence. I is no difficul o check ha ḡ p (Y p, Z p ) ḡ p (Y p, Z p ) λ(1 + Z q 2 ) λ ( 1 + Z q Z p 2 + Z p Z 2 + Z 2) 28

30 for some λ >. Se ψ(x) = e 4λx x/4 1. We can ake λ > 1/16 so ha ψ is sricly increasing and ψ() =. Again, we apply Iô s lemma o obain E ψ(y p Y q ) + E ψ (Y p s { 1 2 ψ λψ }(Y p s Ys q ) Zs p Zs q 2 ds. Y q s )λ ( 1 + Z p s Z s 2 + Z s 2) ds Noe ha 1 2 ψ λψ = 4λ 2 e 4λx + λ 4 >. Then, being bounded, { 1 2 ψ λψ } 1 2 (Y p Y q ) converges srongly in H 2 as q o { 1 2 ψ λψ } 1 2 (Y p Y ), so ha { 1 2 ψ λψ } 1 2 (Ys p Ys q ) Z p Z q converges weakly as q. From his and he fac ha Y q Y and ψ is non-decreasing, we ge E { 1 2 ψ λψ }(Y p s Y s ) Z p s Z s 2 ds lim inf E { 1 q 2 ψ λψ }(Ys p Ys q ) Zs p Zs q 2 ds E ψ (Ys p Y s )λ ( 1 + Zs p Z s 2 + Z s 2) ds E ψ(y p Y ). Hence E { 1 2 ψ 2λψ }(Ys p Y s ) Zs p Z s 2 ds lim inf q E E { 1 2 ψ λψ }(Ys p Y q ψ (Y p s E ψ(y p Y ). Y s )λ ( 1 + Z s 2) ds s ) Zs p Zs q 2 ds Since { 1 2 ψ 2λψ } = λ/2 >, i holds ha Z p Z in H 2. Sep 4. I remains o prove ha (Y, Z) is a soluion o (2.3). This is done by similar argumens as already used. Remark 2.5 (comparison) The fac ha he sequence (Y p ) p is non-increasing implies ha comparison holds for he maximal soluions. One could similarly provide a minimal soluion. We now consider a more general seing Assumpion 2.5 ζ is bounded, g is coninuous, g(y, z) K g (1 + y + z 2 ). Theorem 2.5 (Exisence for bounded erminal value #2) Le Assumpion 2.5 holds. Then, here exiss a unique maximal soluion (Y, Z) o (1.1). Moreover, (Y, Z) S H 2 BMO. Proof. We can runcae he y erm in he driver. Then, since ζ is bounded, we can again sandwich he soluion is a way ha he runcaion does no operae. 29

31 2.4.2 Exisence for unbounded erminal values We nex urn o he case where ξ is no bounded. We follow Briand and Hu 12. Assumpion 2.6 g is coninuous, here exiss β, γ > α β/γ such ha There exiss λ > γe βt such ha E e λ ξ <. g(y, z) α + β y + γ 2 z 2. Theorem 2.6 (Exisence for unbounded erminal value) Le Assumpion 2.6 holds. Then, here exiss a soluion (Y, Z) o (1.1). Moreover, in which φ is given by (2.5) below. Proof. 1 γ ln E φ ( ξ) F Y 1 γ ln E φ (ξ) F, The proof is based on Theorem 2.5 as we shall firs runcae he erminal condiion. However, we need global esimaes on he Y -componen of he corresponding soluions, which do no depend on he runcaion. To do his, we exend he deerminisic bounds obained in Sep 1 of he proof of Theorem 2.4 ino (2.6) below. The exisence of a soluion in he general case will hen be obained hanks o his global conrol and a localizaion argumen, see Seps 2 and 3. In Sep 4, we prove ha he Z-componen is in H 2. Sep 1. Se P = e γy BSDE wih erminal condiion e γξ and driver and Q = γe γy Z. If (Y, Z) S H 2 solves (1.1) hen (P, Q) solves he F (p, q) := 1 p> (γpf(ln p/γ, q γp ) 1 2 q 2 p ). We se H(p) := 1 p 1 p(αγ + β ln p ) + αγ1 p<1 F (p, q) and le φ(x) be he soluion of φ (x) = e γx + H(φ s )ds. (2.5) We firs prove ha any (possible) soluion (Y, Z) S H 2 of (1.1) wih ξ bounded saisfies 1 γ ln E φ ( ξ) F Y 1 γ ln E φ (ξ) F. (2.6) Se Φ := E φ (ξ) F = E e γξ + H(φ s (ξ))ds F s. 3

32 Then, by applying he represenaion heorem o we can find χ such ha e γξ + E H(φ s (ξ)) F s ds On he oher hand Since H is convex, we obain P Φ Φ = e γξ + P = e γξ + E H(φ s (ξ)) F s ds F (P s, Q s )ds (F (P s, Q s ) H(Φ s )) ds χ s dw s. Q s dw s. (Q s χ s )dw s. Since H is locally Lipschiz and P, Φ are uniformly conrolled in (, ), we can apply comparison based on F H. This provides he upper-bound of (2.6). We hen consider ( Y, Z) and work as above o obain he lower bound. Sep 2. We assume here ha ξ and se ξ n := ξ n. By Theorem 2.5 and Sep 1, we can find a soluion (Y n, Z n ) such ha 1 γ ln E φ ( ξ n ) F Y n 1 γ ln E φ (ξ n ) F. (2.7) By considering maximal soluions, we mus have Y n Y n+1, see Remark 2.5. We define Y := sup n Y n and observe from (2.7) ha i well defined in L 1 and saisfies (2.6). Se k := inf{ : 1 γ ln E φ (ξ) F k} T. Since H, (φ ) is non-increasing so ha ln E φ (ξ) F k implies ln E φ (ξ n ) F k. In view of (2.7) his implies ha Y n k is bounded from above. We can hen define Y k := sup n Y n k and show, by similar argumens as in he proof of Theorem 2.4, ha here exiss Z k which is he limi in H 2 of Z n k such ha Since ( k ) k is non-decreasing, we have Y k k k Y k = sup YT n k + g s (Ys k, Zs k )ds Zs k dw s. n coninuous process wih limi ξ a T. Se Z = Z k ha Y k = Y k + k = Y k. By k T and (2.7), we deduce ha Y is a k g s (Y s, Z s )ds 31 for k. Since Z k+1 = Z k on, k, we obain k k Z s dw s.

33 I remains o send k. Sep 3. We now consider he general case. Se ξ p := ( p) ξ. Then, define (Y p, Z p ). We do he same as a above by aking inf over p. Sep 4. I remains o prove ha Z H 2. Le n := inf{ : e2γ Ys Z s 2 ds n}. Se ψ(x) := (e γx 1 γx)/γ 2. Then, since x ψ( x ) is C 2 and ψ on R +, n ψ( Y ) = ψ( Y n ) + (ψ ( Y s )sign(y s )g s (Y s, Z s ) 12 ) ψ ( Y s ) Z s 2 ds n ψ ( Y s )sign(y s )Z s dw s ψ( Y n ) + n n ψ ( Y s )(α + β Y s )ds ψ ( Y s )sign(y s )Z s dw s. Bu γψ (Y s ) ψ ( Y s ) = 1, so ha ψ( Y ) + 1 n 2 E Z s 2 ds n Z s 2 (γψ (Y s ) ψ ( Y s )) ds n E ψ( Y n ) + ψ ( Y s )(α + β Y s )ds. I remains o appeal o (2.6) and our inegrabiliy assumpions on ξ General esimaes and sabiliy for bounded erminal condiions using Malliavin calculus The conen of his secion is due o Briand and Elie 11. The firs idea of his paper is o rely on he fac ha he Y -componen of a quadraic BSDE wih bounded erminal value should be bounded, while he Z-componen should be BMO, i.e. belong o H 2 BMO. We make his asserion precise in he nex proposiion. Assumpion 2.7 g is deerminisic, g() and ζ are bounded by K, g(y, z) g(y, z ) L y y y + K z (1 + z + z ) z z. Proposiion 2.5 (Equivalence of he classes of definiion) Le Assumpion 2.7 hold. (a.) Assume ha (Y, Z) S H 2 solves (1.1), hen Z H 2 BMO and here exiss κ > such ha E Z s 2 ds F κ(1 + Y H )e κ Y H. (b.) Assume ha (Y, Z) S 2 H 2 BMO solves (1.1), hen Y S e LyT (1 + T )K. 32

34 (c.) Any soluion in S H 2 or in S 2 H 2 BMO is in S H 2 BMO. Proof. a. Noe ha Le ψ be defined by g(y, z) K + K z 2 + L y y + 3K z 2 z 2. ψ(x) = e3kz x 1 3K z x 3K z 2, and apply Iô s Lemma o obain ( ) 3Kz ψ(y ) ψ(ξ) + 2 ψ (Y s ) ψ (Y s ) T Zs 2 ds + C ψ (Y s ) (1 + Y s ) ds ψ (Y s )Z s dw s. 2 Since ψ 3K z ψ = 1 and ψ, his implies 1 T 2 E Z s 2 ds F E ψ(ξ) + ( ψ (Y s ) K + K ) z 2 + L y Y s ds F. b. We can 2 use he linearizaion procedure of Proposiion 2.3 because Z H 2 BMO. Now ha we know ha Z should be BMO, we can use he linearizaion procedure of Proposiion 2.3 o obain comparison of soluions wih differen erminal condiions. Proposiion 2.6 (Sabiliy in he erminal condiion) Le Assumpion 2.7 hold for (g, ξ) and (g, ξ ). Le (Y, Z) and (Y, Z ) be associaed soluions in S H 2 BMO. Then, here exiss an equivalen probabiliy measure P and a bounded process b such ha Y Y = E P e bsds (ξ ξ ) F for all sopping ime T. Moreover, here exiss p > 1 such ha for all p p we can find C p which depends only on p, L y, K z and K such ha Y Y S 2p + Z Z H p Cp ξ ξ L 2p. Proof. Again, we can use he linearizaion argumen of Proposiion 2.3 o obain ha here exiss an adaped process b, bounded by L y, and a H 2 BMO such ha Y = ζ + b s Y s + a s Z s ds Z s dw s. The bound on a H 2 BMO depends only on K z. Since a H 2 BMO, we can again define an equivalen measure P and a P-Brownian moion W such ha Y = ζ + b s Y s ds Z s d W s. 2 For any ζ H 2 BMO, he Doléans-Dade Exponenial E( ζ sdw s ) is a maringale, see Kazamaki

35 This proves he firs ideniy. Le E a be he Doleans-Dade exponenial of a sdw s. Since he laer is BMO, his is a maringale and, since b is bounded by L y, Y e LyT (E a ) 1 E E a T ζ F e LyT (E a ) 1 E (E a T ) q F 1 q E ζ p F 1 p in which 1/p + 1/q = 1. Bu, by he reverse Hölder inequaliy, E (E a T )q F 1 q C E a for 1 < q < q in which q depends a H 2 BMO, see Kazamaki 39, Theorem 3.1. For he Z erm, we again apply Iô s Lemma o Y 2 o obain in which Z s 2 ds = ζ Y s (g s (Y s, Z s ) g s (Y s, Z s))ds 2 Y s Z s dw s Y s (g s (Y s, Z s ) g s (Y s, Z s)) (sup Y 2 )(L y + 2K 2 z (1 + Z s 2 + Z s 2 ) Z 2. We conclude by using Burkholder-Davis-Gundy inequaliy, he energy inequaliy for BMO maringales 3 which implies ha Z 2p + Z 2p p!( Z 2p + Z 2p H 2p H 2p H 2 BMO HBMO), 2 ogeher wih he bound on Y of Proposiion 2.5. Remark 2.6 Exension of he comparison resul o differen drivers is sraighforward. We can now look for anoher proof of exisence. The general idea is he following. When g, hen he Clark-Ocone formula implies ha Z = E D ξ F whenever he Malliavin derivaive process Dξ = (D ξ) is well-defined, see Nualar 46. If Dξ is bounded, hen Z is bounded. The same essenially holds for BSDEs. Thus, if Dξ is bounded, hen he Z-componen of he soluion is bounded and everyhing works as if he driver was uniformly Lipschiz in z. Thanks o Proposiion 2.6, he general case can be obained by approximaing any bounded erminal condiion by a sequence ha is smooh in he Malliavin sense. Theorem 2.7 (Shor exisence proof) Le Assumpion 2.7 hold. Then, here exiss a unique soluion (Y, Z) S H 2 BMO o (1.1). Proof. Sep 1. If g is C 1 and ξ has a bounded Mallivian derivaive, hen i follows from El Karoui e al. 32 ha for s D s Y = D s ξ + 3 See Kazamaki 39, Secion 2.1 ( y g(y u, Z u )D s Y u + z g(y u, Z u )D s Z u ) ds D s Z u dw u, 34