Locally Lipschitz BSDE driven by a continuous martingale path-derivative approach Monash CQFIS working paper

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1 Locally Lipschiz BSDE driven by a coninuous maringale pah-derivaive approach Monash CQFIS working paper 17 1 Absrac Using a new noion of pah-derivaive, we sudy exisence and uniqueness of soluion for backward sochasic differenial equaion BSDE driven by a coninuous maringale M wih [M, M] = msm sdr[m, M] s: Kihun Nam School of Mahemaical Sciences, Monash Universiy and Cenre for Quaniaive Finance and Invesmen Sraegies kihun.nam@monash.edu Y = ξm [,T ] + fs, M [,s], Y s, Z sm sdr[m, M] s Z sdm s N T +N Here, for [, T ], M [,] is he pah of M from o, and ξγ [,T ] and f, γ [,], y, z are deerminisic funcions of, γ, y, z [, T ] D R d R d n. In paricular, we are ineresed in he case when fs, γ, y, z is locally Lipschiz in y, z. The pah-derivaive is defined as a direcional derivaive wih respec o he pah-perurbaion of M in a similar way o he verical funcional derivaive inroduced by Dupire 9, and Con and Fournie 13. We firs prove he exisence, uniqueness, and pahdiffereniabiliy of soluion in he case where f, γ [,], y, z is Lipschiz in y and z. Afer proving Z is a pah-derivaive of Y when f is Lipschiz and M has maringale represenaion propery, we exend he exisence and uniqueness resuls o locally Lipschiz f. When he BSDE is onedimensional, we could show he exisence and uniqueness of soluion. On he conrary, when he BSDE is mulidimensional, we show exisence and uniqueness only when [M, M] T is small enough: oherwise, we provide a counerexample ha has blowing-up soluion. Lasly, we invesigae he applicaions o uiliy maximizaion problems under power and exponenial uiliy funcion. Cenre for Quaniaive Finance and Invesmen Sraegies

2 Locally Lipschiz BSDE driven by a coninuous maringale pah-derivaive approach Kihun Nam School of Mahemaical Science Monash Universiy Clayon, VIC 38, Ausralia November 13, 17 Absrac Using a new noion of pah-derivaive, we sudy exisence and uniqueness of soluion for backward sochasic differenial equaion BSDE driven by a coninuous maringale M wih [M, M] = m sm sdr[m, M] s : Y = ξm [,T ] + fs, M [,s], Y s, Z s m s dr[m, M] s Z s dm s N T + N Here, for [, T ], M [,] is he pah of M from o, and ξγ [,T ] and f, γ [,], y, z are deerminisic funcions of, γ, y, z [, T ] D R d R d n. In paricular, we are ineresed in he case when fs, γ, y, z is locally Lipschiz in y, z. The pah-derivaive is defined as a direcional derivaive wih respec o he pah-perurbaion of M in a similar way o he verical funcional derivaive inroduced by Dupire 9, [18], and Con and Fournie 13, [14]. We firs prove he exisence, uniqueness, and pah-differeniabiliy of soluion in he case where f, γ [,], y, z is Lipschiz in y and z. Afer proving Z is a pah-derivaive of Y when f is Lipschiz and M has maringale represenaion propery, we exend he exisence and uniqueness resuls o locally Lipschiz f. When he BSDE is one-dimensional, we could show he exisence and uniqueness of soluion. On he conrary, when he BSDE is mulidimensional, we show exisence and uniqueness only when [M, M] T is small enough: oherwise, we provide a counerexample ha has blowing-up soluion. Lasly, we invesigae he applicaions o uiliy maximizaion problems under power and exponenial uiliy funcion. MSC 1: 6H1, 6H7, 93E Key words: Backward sochasic differenial equaion, pah differeniabiliy, funcional derivaive, coefficiens of superlinear growh, uiliy maximizaion 1 Inroducion Le M be a square inegrable coninuous n-dimensional local maringale wih quadraic covariaion marix [M, M] = m sm sdr[m, M] s for a R n n -valued process m. We le D be he se of càdlàg R n -valued funcions on [, T ]. Consider he following backward sochasic differenial equaion BSDE driven by M where he erminal condiion is ξ : D R d and he driver is f : [, T ] D R d R d n R d : Y = ξm [,T ] + fs, M [,s], Y s, Z s m s dr[m, M] s Z s dm s N T + N 1.1 1

3 Here, we denoe M [,s] o be he pah of M sopped a s. The soluion of above BSDE is a riple Y, Z, N of adaped processes saisfying [N, M] =. We sudy he exisence and uniqueness of soluion when ξγ and fs, γ, y, z are Lipschiz in γ and locally Lipschiz in y, z. In order o do so, we sudy he differeniabiliy of soluions under he perurbaion of he pah of M. Then, we apply our resul o various uiliy maximizaion problems. BSDE was firs inroduced by Bismu 1973, [] as a dual problem of sochasic opimizaion under he assumpions d = 1, Brownian moion M, and a linear funcion f. Then, Pardoux and Peng 199, [39] exended he well-posedness resul o d 1 and Lipschiz funcion f. One can find classical resuls and applicaions in he survey paper wrien by El Karoui e al. 1997, []. Since Pardoux and Peng s seminal paper, researchers exended he well-posedness resul in various direcions. One direcion of exension is o incorporae he case where f grows superlinearly in z. The wellposedness resuls for such BSDEs have numerous applicaions including uiliy maximizaion in incomplee marke Hu e al. 5, [3], dynamic coheren risk measure Gianin, 6, [], equilibrium pricing in incomplee marke Cheridio e al., 16, [7], and more recenly, sochasic Radner equilibrium in incomplee marke Kardaras e al., 15, [3]. When d = 1 and ξ is bounded, Kobylanski, [33] proved he exisence and uniqueness of soluion when fs, γ, y, z grows quadraically in z. Briand and Hu 6, [4], 8, [5], and Delbaen e al. 11, [17] furher exended he resuls o unbounded erminal condiion ξ. Superquadraic BSDE driven by a Brownian moion also araced he ineres among mahemaician. Delbaen e al. 1, [16] showed ha such BSDE is ill-posed if here is no regulariy assumpion on he erminal condiion and he driver. Richou 1, [4] sudied he exisence and uniqueness of soluion for superquadraic Markovian BSDE. Cheridio and Nam 14, [8] showed he exisence and uniqueness of soluion for he non-markovian case using Malliavin calculus and is connecion o semilinear parabolic PDEs under he Markovian assumpion. On he conrary, when d > 1, Frei and dos Reis 1, [1] showed ha a mulidimensional BSDE wih a quadraic driver migh no be well-posed. By choosing a erminal condiion which is irregular wih respec o he underlying Brownian moion, hey were able o consruc an example such ha he soluion Y blows up. When one does no assume regulariy condiions on ξ and f, only a few posiive resuls are known when he erminal condiion is small, or he driver saisfies cerain resricive srucural condiions: see Tevzadze 8, [43], Cheridio and Nam 14, [9], Hu and Tang 14, [5], Jamneshan e al, 14, [7], Kupper e al. 15, [34], and Xing and Zikovic 16, [44]. When ξ and f are assumed o be regular, Nam14,[36], Kupper e al. 15, [34], and Cheridio and Nam 17, [1] Researchers also ried o generalize Brownian moion o a general maringale M. When M is a coninuous maringale, El Karoui and Huang 1997, [19] provided he exisence and uniqueness of soluion in he case where f, γ [,], y, z is Lipschiz wih respec o y, z when d 1. When d = 1, Morlais 9, [35] invesigaed he exisence and uniqueness of soluion when f, γ [,], y, z has quadraic growh in z. Researchers generalized even o he case where M is a general maringale wih jumps. To name a few, Possamai e al. 15, [3] sudied he case where d = 1, f has quadraic growh in z, and M has jumps. On he oher hand, Papapanoleon e al. 16, [38] rea he case whered 1 and f is sochasically Lipschiz in y, z. However, he following quesions have no been answered when M is a general maringale: If d = 1, does one have well-posedness when fs, γ, y, z grows superquadraically in z? If d > 1, does one have well-posedness when fs, γ, y, z grows superlinearly in z? In his aricle, we answer hese quesions when M is a coninuous maringale; ξγ is Lipschiz wih respec o γ; and fs, γ, y, z is Lipschiz in γ and locally Lipschiz in y, z. To be more specific, we were able o esablish exisence and uniqueness of soluion when d = 1 and find a uniform almos sure bound

4 of he soluion Z. In he case where d > 1, we have he exisence and uniqueness of soluion as well as he bound of Z only if [M, M] T is small enough: oherwise, we provide a counerexample such ha Z blows up. We apply he 1D resul o various kinds of conrol problems for SDE driven by M using he maringale mehod inroduced by Hu e al. 5, [3]. In he case where M has jumps, our mehod does no work anymore, 1 and we leave his quesion for fuure papers. The argumen is based on he analysis of he sabiliy under perurbaion of M. We call his sabiliy pah-differeniabiliy of he soluion. In oher words, when we model sochasic opimizaion problem as a BSDE, he pah-derivaive of Y implies he sabiliy of value process wih respec o he perurbaion of underlying noise. An imporan propery, which is called dela-hedging formula, is ha Z is he pahderivaive of Y under appropriae noion if M possesses maringale represenaion propery. If one can find a uniform bound of Z by esimaing he derivaive of Y and using dela-hedging formula, we can use he localizaion argumen o prove he well-posedness of BSDEs wih locally Lipschiz drivers. Using Malliavin calculus on BSDE as in El Karoui e al. 1997, [] and Hu e al. 1, [4], his sraegy was used in Briand and Elie 13, [3], Cheridio and Nam 14, [8], and Kupper e al. 15, [34] when M is a Brownian moion. However, his mehod canno be rivially exended o a coninuous maringale M because M is no Malliavin differeniable in general. For example, consider he case where M is a Brownian moion sopped a a hiing ime. Even when ξ is smooh, ξw τ is no Malliavin differeniable in general. Therefore, classical Malliavin calculus mehod used in he papers menioned above canno be used o sudy he pah-differeniabiliy of soluion for his ype of BSDE. 3 One may define anoher pah-derivaive noion for BSDE by assuming Markovian srucure, ha is one assumes ξγ = ξx T and f, γ [,T ], y, z = f, X, y, z where dx = b, X da + σ, X dm for some deerminisic funcion b and σ. In many cases, here is a deerminisic measurable funcion u such ha Y = u, X, M. Then one can define pah-differeniabiliy as a classical differeniabiliy of he funcion u. This approach was used in Imkeller e al. 1, [6] o sudy exisence, uniqueness, and pah-differeniabiliy of 1.1 wih d = 1 and f grows quadraically in z. However, his mehod canno be exended BSDE wih fully pah-dependen coefficiens. One of he recen definiions of pah-derivaive is he funcional Iô derivaive developed by Dupire 9, [18], and Con and Fournie 13,[14]. The perurbaion in funcional Iô derivaive is given by eiher horizonal or verical displacemen of he pah a he las ime. The funcional Iô calculus is general in a sense ha i assume neiher Markovian srucure nor Gaussian propery of M. Using verical funcional Iô derivaive, Con 16, [13] was able o ge a dela-hedging formula for 1.1 when M is a coninuous semimaringale deermined by forward SDE driven by Brownian moion. Even hough funcional Iô calculus has is own srengh, i is no suiable for obaining a uniform bound of Z. The reason is ha we do no know he equaion he funcional Iô derivaive of Y saisfies. In order o find a uniform esimae of Z, we modify he verical funcional Iô derivaive o imeparamerized version similar o Malliavin derivaive and use such noion o obain BSDE for pah-derivaives of Y and Z. Then, by he classical mehod in BSDE, we ge a uniform bound of Z. However, we should noe ha he pah-funcional represenaion of random variables and sochasic processes are no unique and our pah-derivaive definiion crucially depends on he represenaion. Therefore, i is imporan o selec logically consisen represenaions of he coefficiens ξ, f and our soluion Y, Z, N. This is done by Theorem 3.7 and i is he main reason why we canno exend our resul o he case where M has jumps. 1 see Remark 3.8 Cheridio and Nam 14, [8] If τ is a sopping ime such ha W τ is Malliavin differeniable, hen τ mus be a consan. Indeed, for W τ = 1 {s<τ} dw s D 1,, one obains from Proposiion 5.3 of El Karoui e al. 1997, [] ha 1 {s<τ} D 1, for almos all s, and herefore, by Proposiion 1..6 of Nualar 6, [37], P[s < τ] = or 1. 3 This ype of BSDE is also known as BSDE wih random erminal ime and sudied by numerous researchers including Darling and Pardoux 1997, [15] and Jeanblanc e al. 15, [8]. 3

5 The aricle is organized as follows. In Secion, we give he definiions, noaions, and assumpions we use hroughou his aricle. In Secion 3, we review he basic properies of BSDE wih Lipschiz driver and driven by a coninuous maringale. Then we sudy he differeniabiliy of BSDE in Secion 4. Using resuls from Secion 3 and 4, we sudy he exisence and uniqueness of soluion for BSDEs wih locally Lipschiz drivers in Secion 5. In paricular, we show he exisence and uniqueness of soluion when [M, M] T is small enough or d = 1. Oherwise, he soluion may blow-up, and i is shown by an example in subsecion 5.3. We sudy uiliy maximizaion of conrolled SDE in Secion 6. In Secion 7, for power and exponenial uiliy funcion, he scheme is applied o opimal porfolio selecion under hree differen ypes of resricion: 1 when he invesmen sraegy is resriced o a closed se; when he diversificaion of porfolio gives he invesor exra benefi; and 3 when here is informaion processing cos for invesmen. Preliminaries Real space We denoe R he se of real number and R + he se of nonnegaive real numbers. For any naural numbers l and m, R l m is he se of real l-by-m marices. R m is he se of m-dimensional real vecors and we idenify wih R m 1 unless oherwise saed. For any marix X, we le X o be is ranspose and we define X o be he Euclidean norm, ha is X := rxx. We always endow Borel σ-algebra on R l m wih respec o he norm and denoe i by BR l m. For X R l n, we denoe i, j-enry of X as X ij. We denoe I o be he ideniy marix of appropriae size. Probabiliy space and he driving maringale Le Ω, F, F, P be a filered probabiliy space. We assume he filraion F := F [,T ] is complee, quasi-lef coninuous, and righ coninuous. Le M be a square inegrable coninuous n-dimensional maringale wih a coninuous predicable quadraic covariaion marix [M, M] and M =. We assume ha here exiss a R n n -valued predicable process m such ha where [M, M] = A := r[m, M] = m s m sda s n [M i, M i ]. Moreover, we always assume ha A T is bounded by K. Then, we have wo consequences: m s = 1 ds dp-a.e. A = n n i=1 k=1 mik s da s = i=1 [M, M] T A T A T / n [M, M] T = m s da s T m sm sda s m s da s = A T In addiion, we assume here exiss a Poisson random measure ν on [, T ] R n wih mean Leb µ where µ is he uniform probabiliy measure on a uni ball cenered a. We le ˆM := xνds, dx [,] R n and R n -valued càdlàg { maringale M } := M + ˆM. We assume ha M and ν are independen and moreover, for any given γ ˆMω : ω Ω and ω Ω, here is ω Ω such ha ˆMω = γ and Mω = Mω. 4

6 We also assume ha F M, he augmenaion of σm s : s, is quasi-lef coninuous and righ coninuous. This condiion is rue when M is a Hun process: see Proposiion.7.7 of Karazas and Shreve 1991, [9] and Secion 3.1 of Chung and Walsh 5, [11]. Therefore, any Feller process M saisfies his propery. This implies, F M, he augmenaion of σm s : s, is also quasi-lef coninuous and righ coninuous. I is noeworhy o observe F M F M because M is a coninuous process while ˆM is a pure jump process. Noe ha since F is coninuous, for any R d -valued F, P-maringale is of he form ZdM + N where Z is a F-predicable R d n -valued process and N is a R d -valued F, P-maringale wih [N, M] =. This saemen also holds wih F M or F M insead of F. As always, we undersand equaliies and inequaliies in P-almos sure sense. The space of càdlàg pahs We le D be he se of all càdlàg R n -valued funcions on [, T ]. For γ D, we denoe γ o be he value a ime and γ [,] o be he funcion γ [,] s := γ s. For γ, γ D, we define γ + γ := γ + γ. On D, we endow a sup norm, γ := sup [,T ] γ and le D be is Borel σ-algebra. Then, we have he following lemma whose proof is given a he appendix. Lemma.1. A R k -valued sochasic process X is adaped o F M if and only if here exiss a pah funcional X : [, T ] D R k such ha X = X, M [,] holds almos surely for each [, T ] and X, is D-measurable. Le x γ = γ and define a filraion H := σ{x s : s }. We le P be he predicable σ-algebra on [, T ] D associaed wih filraion {H }. Then, i is easy o check ha if a funcion f : [, T ] D R d is P-measurable, hen f, M [,] is a predicable processe since M [, ] : [, T ] Ω D is a predicable processe. Banach space We se he following Banach spaces: L : all d-dimensional random vecors X saisfying X := E X < S : all R d -valued càdlàg adaped processes Y T saisfying Y S := sup T Y < H : all R d -valued càdlàg adaped processes Y T saisfying Y H := E Y da < H m: all R d -valued predicable processes Z T saisfying Z H m := E Z m da < M : all càdlàg maringale N T saisfying N M := Er[N, N] T <, [N, M] =, and N =. BSDE and is soluion Assume ha ξ : D R d is D-measurable and f : [, T ] D R d R d n R d is P BR d BR d n - measurable. The soluion of BSDEξ, f is a riple of adaped processes Y, Z, N H H m M which saisfies Y = ξm [,T ] + fs, M [,s], Y s, Z s m s da s Z s dm s N T + N. BSDEξ, f Wih a sligh abuse of noaion, someimes we denoe he above BSDE as BSDEξM [,T ], f. 5

7 3 Properies of BSDE wih Lipschiz Driver In his secion we presen exisence, uniqueness, sabiliy, comparison, and pah-represenaion resuls regarding BSDEξ, f when fs, γ, y, z is Lipschiz wih respec o y, z. Excep for pah-represenaion of soluions provided in Theorem 3.6 and Theorem 3.7, mos of resuls are well-known: see e.g. El Karoui and Huang 1997, [19]. However, we provide he proof for he readers convenience. Se: STD The erminal condiion ξ is in L. Le P be he progressively measurable σ-algebra on [, T ] Ω. The driver f : [, T ] Ω R d R d n R d is P BR d BR d n -measurable funcion such ha E fs,, da s <. Moreover, we assume ha here are C y, C z R + such ha fs, y, z fs, y, z C y y y + C z z z. Proposiion 3.1. Assume STD. Then here exiss a unique soluion Y, Z, N H H m M of and moreover, Y S. Y = ξ + fs, Y s, Z s m s da s Z s dm s N T + N 3.1 Remark 3.. Noe ha fs, Y s, Z s may no be predicable. Therefore, he inegral wih respec o A should be inerpreed as Lebesgue-Sieljes inegral. Proof. Le us define he following Banach spaces: H a: all R d -valued càdlàg adaped processes Y T saisfying Y H a := E eaa Y da <. H m,a: all R d -valued predicable processes Z T saisfying Z H m,a := E eaa Z m da <. M a: all càdlàg maringale N T saisfying N M a := E eaas dr[n, N] < and N =. For a = C y C z +, we will use conracion mapping heorem for φ : y, z, n H a H m,a M a Y, Z, N H a H m,a M a. where Y, Z, N is given by he soluion of BSDE or equivalenly, Y = ξ + fs, y s, z s m s da s Z s dm s N T + N, Y + ] Z s dm s + N = E [ξ + fs, y s, z s m s da s Then, since e aas is beween 1 and e ak, he space H a H m,a M a is equivalen o H H m M and he fixed poin we ge by conracion mapping heorem is he unique soluion in H H m M. Firs, le us show ha φy, z, n = Y, Z, N is in H H m M and herefore in H a H m,a M a. From Theorem 7 Corollary 3 of II.6 of Proer 4, [41], E E [ ξ + ] fs, y s, z s m s da s E ξ + fs, y s, z s m s da s < 6

8 for all [, T ] implies On he oher hand, [ Z H + N m M = Er Z s dm s + N, Y ξ + fs, y s, z s m s da s + sup [,T ] From Burkholder-Davis-Gundy inequaliy, for some consan C, E sup [,T ] Z s dm s + N T N E ] Z s dm s + N T Z s dm s + N T + E sup [,T ] <. Z s dm s + N T N. Z s dm s + N C Z H + N m M Therefore, Y S and his implies Y H. Nex, le us show he conracion. Le Y, Z, N := φy, z, n and Y, Z, N := φy, z, n. Le us denoe δy s := Y s Y s, δz s := Z s Z s, δn s := N s N s, and δf s := fs, y s, z s m s fs, y s, z sm s. Then, By Iô formula, δy = δf s da s δz s dm s δn T + δn. δy = e aas δys δf s δz s m s a δy s da s e As δy s δz s dm s e As δy s dδn s e aas dr [δn, δn] s e aas a 1 δy s + 1 a 1 δf s δz s m s a δy s da s e As δy s δz s dm s e As δy s dδn s. If we ake expecaion on boh side and rearrange i, by Lemma A.1, we ge e aas dr [δn, δn] s E e aas δy s + Z s m s da s + E e aas dr [δn, δn] s 1 a 1 E e aas δf s da s a a 1 E e aas y s y s + z s m s z sm s da s and φ is a conracion on H a H m,a M a. Therefore, here exiss a unique fixed poin, which is our soluion, in H a H m,a M a. Therefore, here is a unique soluion in Y, Z, N H H m M and Y S from he argumen a he beginning of he proof. Proposiion 3.3. Assume STD. Moreover, assume ha here exis C ξ, C f R + such ha ξ C ξ and T fs,, da s Cf. Then, for soluion Y, Z, N of 3.1, we have Y Cξ + C f e 1 KCy+C z +1. 7

9 Proof. Since STD are saisfied, here exiss a unique soluion Y, Z, N H H m M. By Iô formula, when a = C y + Cz + 1, we have e aa Y = e aa T ξ + e aas Ys fs, Y s, Z s m s Z s m s a Y s da s e aas dr [N, N] s e aas Ys Z s dm s e aas Ys dn s because e aa T ξ + e aas fs,, da s e aas Y s Z s dm s e aas Y s dn s. Y s fs, Y s, Z s m s fs,, + C y + C z + 1 Y s + Z s m s. If we ake E F on boh side, by Lemma A.1, we ge Y E [ e aa T ξ ] + E [e aa T ] fs,, da s Cξ + C f ekcy+c z +1 Proposiion 3.4. Sabiliy Assume ha ξ, f saisfies STD wih Lipschiz coefficiens C y and C z. Also assume ha ξ, f saisfies STD possibly wih differen Lipschiz coefficiens. Le Y, Z, N and Ȳ, Z, N are soluions of Y = ξ + fs, Y s, Z s m s da s Z s dm s N T + N Ȳ = ξ + fs, Ȳs, Z s m s da s Z s dm s N T + N, Then, we have he following esimae: Y Ȳ + Y Ȳ + Z Z + N N H H m M e KCy+C z + ξ ξ + f, Ȳ, Z m f, Ȳ, Z m H Proof. Denoe δy := Y Ȳ, δz := Z Z, δn := N N, δξ := ξ ξ, and Then, we have gs, y, z := fs, Ȳs + y, Z s m s + z fs, Ȳs, Z s m s. δy = δξ + gs, δy s, δz s m s da s δz s dm s δn T + δn. where δξ, g saisfies STD. By applying Iô formula on e aa δy where a = C y + Cz +, we have e aa δy = e aa T δξ + e aas δys gs, δy s, δz s m s a δy s δz s m s da s e aas dr[δn, δn] s e aas δys Z s dm s e aas δys dn s. 8

10 and his implies, by Lemma A.1, Ee aa δy + E e aas δy s da s + 1 E since Therefore, Ee aa T δξ + E e aas δz s m s da s + E e aas gs,, da s e KCy+C z + e aas dr[δn, δn] s E δξ + E gs,, da s δy s gs, δy s, δz s m s gs,, + C y + C z + 1 δy s + 1 δz sm s. sup E δy + E δy s da s + E δz s m s da s + Er[δN, δn] T [,T ] sup Ee aa δy + E e aas δy s da s + 1 [,T ] E e KCy+C z + E δξ + E gs,, da s. T e aas δz s m s da s + E e aas dr[δn, δn] s Now le us prove he comparison heorem when d = 1. This resul will be used in Secion 4.. We will denoe EX = exp X 1 [X, X]. Theorem 3.5. Comparison Theorem Le d = 1 and assume ha m is inverible for all [, T ]. Assume STD for ξ, f and ξ, f. Le Y, Z, N and Ȳ, Z, N be he soluions of Y = ξ + Ȳ = ξ + fs, Y s, Z s m s da s fs, Ȳs, Z s m s da s Z s dm s N T + N Z s dm s N T + N. If ξ ξ almos surely and f, y, z f, y, z d dp dy dz-almos everywhere, hen Y Ȳ a.s. for all [, T ] Proof. Le us denoe and δξ := ξ ξ δf s := fs, Ȳs, Z s m s fs, Ȳs, Z s m s δy s := Y s Ȳs, δz s := Z s Z s, δn s := N s N s Zm i := Zm 1, Zm,, Zm i, Zm i+1, Zm i+,, Zm n F s := fs, Y s, Z s m s fs, Ȳs, Z s m s, G i s := fs, Ȳs, Zm i 1 s fs, Ȳs, Zm i s δy s δzm i s dγ s := Γ s Fs da s + G sm s 1 dm s ; Γ = 1. 9

11 Noe ha F and G i are uniformly bounded by C y and C z, respecively, and Γ for all. Moreover, Γ = E F s da s + G sm s 1 dm s e CyK E G sm s 1 dm s where E G sm s 1 dm s is a maringale because of Novikov condiion. Noe ha, by Doob s maximal inequaliy, E sup E G sm s 1 dm s 4E E G sm s 1 dm s s s 4Ee Gs da s E G sm s 1 dm s 4e nc z K < and herefore, Γ S. On he oher hand, if we subrac boh equaions, we ge δy = δξ + If we apply Iô formula o Γ s δy s, we ge Γ δy = δy δf s + F s δy s + δz s m s G s da s Γ s δf s da s + This implies ΓδY + Γ s δf s da s is a local maringale. Noe ha E sup s E sup s s Γ s δy s 1 Γ S + 1 δy S Γ u δf u da u E sup Γ s s δz s dm s δn T + δn s. δy s Γ s G sm s 1 + Γ s δz s dm s + Γ s dδn s 3. δf u da u 1 Γ S + 1 E δf u da u < Therefore, ΓδY + Γ s δf s da s and δy s Γ s G sm s 1 + Γ s δz s dm s + Γ s dn s are maringales. If we ake E on boh side of he backward version of 3., we ge δy = 1 Γ E [Γ T δξ + Γ s δf s da s ]. Now le us give he exisence and uniqueness resul when he erminal condiion and he driver depends on he pah of M or M. Consider he following assumpions: S For any γ D wih γ 1, ξm [,T ] +γ, ξm [,T ] L and E Lip There are nonnegaive consans C y and C z such ha fs, M [,s] +γ,, da s, E fs, M [,s],, da s <. f, γ [,], y, z f, γ [,], y, z C y y y + C z z z for all γ D, [, T ], y, y R d, z, z R d n. 1

12 Theorem 3.6. Assume S and Lip. The following BSDE Y = ξm [,T ] + fs, M [,s], Y s, Z s m s da s Z s dm s N T + N, 3.3 has a unique soluion Y, Z, N H H m M. Moreover, Y S and here are pah funcionals Y : [, T ] D R d, Z : [, T ] D R d n, and N : [, T ] D R d such ha Y,, Z,, N, are D-measurable and for each [, T ]. Y = Y, M [,], Z = Z, M [,], and N = N, M [,] Proof. Since ξm [,T ] and fs, M [,s], y, z saisfies STD, he exisence and uniqueness of soluion follows from Proposiion 3.1. Consider a BSDE wih he same erminal condiion and driver under he filraion F M and noe ha M, m, A are adaped o F M. By he same logic above, here exiss a unique soluion which is adaped o F M. Moreover, since F M F, his soluion and he soluion Y, Z, N under original filraion should be he same. Therefore, Y, Z, N should be F M -adaped. The exisence of pah funcionals Y, Z, and N is a resul of Lemma.1. Theorem 3.7. Assume S and Lip. Le h [, 1/] and ξ h γ [,T ] := ξγ [,T ] + he i 1 [u,t ] f h s, γ [,s], y, z := fs, γ + he i 1 [u,t ] [,s], y, z. Then, BSDEξ h, f h has a unique soluion Y h, Z h, N h H H m M. Moreover, Y h S. In addiion, for he same pah funcionals Y, Z, N defined in Theorem 3.6, we have Y h = Y, M [,] + he i 1 [u,t ], Z h = Z, M [,] + he i 1 [u,t ], and N h = N, M [,] + he i 1 [u,t ] for each [, T ]. In paricular, his holds when h =. Proof. The exisence of unique soluion Y h, Z h, N h and Y h being in S comes from Proposiion 3.1. Le us denoe M h := M + he i 1 [u,t ]. Le Y, Z, N be he unique soluion of 3.3 and Y, Z, N be he corresponding pah funcionals. Le us define Ω Ω so ha PΩ = 1 and Y ω = Y, M [,] ω, Z ω = Z, M [,] ω, and N ω = N, M [,] ω for all ω Ω. Noe ha, by our assumpion on ˆM and M, for all ω Ω, here exiss ω Ω such ha Mω = Mω and ˆMω = he i 1 [u,t ]. For such ω, and Y ω = Y, M h [,] ω, Y ω = ξm h [,T ] ω+ Z ω = Z, M h [,] ω, and N ω = N, M h [,] ω. fs, M[,s] h ω, Y s ω, Z sωm s ωda s ω Z sdm s ω N T ω+n ω. Since his holds for all ω realizing all possible pahs of M h and M, he riple is he unique soluion of BSDEξ h, f h. Y h, Z h, N h := Y, M[,] h h h, Z, M[,], N, M[,] Remark 3.8. One may ask wheher we can consider M wih jumps. If M is a maringale wih jumps, we know ha he soluion Y, Z, N is adaped o he filraion generaed by boh M and M. However, M is no adaped o F M and i is no obvious ha wheher he soluion Y, Z, N is acually adaped o he filraion generaed only by M. 11

13 4 Pah-differeniabiliy of BSDE wih Lipschiz Driver Assuming ha every maringale can be represened by sochasic inegral wih respec o M, Z is ofen a pah-derivaive of Y in some sense: see, for example, El Karoui 1997, [] for Malliavin calculus sense or Con 16, [1] for funcional Iô calculus sense. This propery is also called dela-hedging formula due o is relaionship wih finance. We will prove his propery and use i o sudy locally Lipschiz BSDEs. More precisely, we will find he almos sure uniform bound of he pah-derivaive of Y o conclude Z is uniformly bounded. Then, our locally Lipschiz BSDE becomes essenially Lipschiz BSDE and exisence, uniqueness, and sabiliy auomaically follows from he resuls of Secion 3. However, boh Malliavin calculus and funcional Iô calculus are no suiable for our problem as we described in he inroducion. Therefore, we will firs define a new sense of pah-derivaive. Then, we will prove he dela-hedging formula for BSDE and show ha he pah-derivaive of Y, Z, N is he soluion of he differeniaed BSDE. This resul, combined wih Proposiion 3.3, will be used o find he bound of Z in Secion 5. Definiion 4.1. For a random variable V = VM [,T ] and a vecor e R 1 n, we say V is e -differeniable a u if VM [,T ] + he 1 [u,t ] VM [,T ] lim h,l h exiss and we denoe i by e uv. For a sochasic process X = X, M [, ] and a vecor e R 1 n, we say X is e -, e,m -, and e,n -differeniable a u and define he e -, e,m -, and e,n -derivaive a u by e ux := e,m u X := lim h,h m X, M [,T ] + he 1 [u,t ] [, ] X, M [, ] lim h,h h X, M [,T ] + he 1 [u,t ] [, ] X, M [, ] e,n u X := lim h,m X, M [,T ] + he 1 [u,t ] [, ] X, M [, ] h h if X H if X H m if X M if he corresponding limi exiss. In general, we denoe u V = n n n e i u Ve i, m u X = e i,m u X e i, and N u X = e i,n u X e i. i=1 i=1 i=1 where {e i } i=1,,,n is he sandard basis of R 1 n. If a random variable or a sochasic process e -, e,m -, and e,n -differeniable for every e R 1 n and a almos every u [, T ], hen we say he random variable or sochasic process is differeniable wih respec o M, or -, m, N -differeniable. Remark 4.. This is a modified version of verical funcional Iô derivaive: see Dupire 9, [18], Con and Fournie 13, [14]. The key differences are ha i is ime-paramerized and he convergence is in L -sense wih respec o an appropriae measure. Noe ha he above definiion crucially depends on he represenaion pah-funcional V or X. For example, le c : γ D cγ C[, T ] : R n be a funcion ha removes any jump par of γ. Then, he random variable V = VM [,T ] can also be wrien as V cm [,T ]. Noe ha V c is always -differeniable wih derivaive. 1

14 Therefore, in order o incorporae above definiions o esablish meaningful resul in BSDE, we need o choose pah represenaion carefully. Since sochasic inegral are defined as a limi of ime-pariioned sum, we have already resriced our represenaion of ξ and f by wriing a BSDE. Therefore, one should choose he pah-funcional represenaion of ξ and f as limis of pah-dependen funcionals which depends on finie number of ime secions of he pah of M. Oherwise, he pah-derivaive of lef hand side of BSDE may be differen from pah-derivaive of righ hand side which is wrien by sochasic inegrals. Le C k be he se of coninuously differeniable funcions from R n k o R d. Le { } S := H : D R d : k N, g C k, = k k 1 k k = T s.. Hγ = gγ k γ 1 k,, γ k γ k k k 1 For ξ k S, we selec g C k such ha ξ k γ = gγ k 1 γ k,, γ k k γ k k 1 and hen, we have e uξ k M [,T ] = i gm k 1 M k,, M k k M k k 1 e where i saisfies k i 1 < u k i. For he driver f which depends on finiely many values γ k i i=1,...,k, we choose is pah funcional and define he derivaive similarly: for each s, y, z [, T ] R d R d n, we rea fs,, y, z as in S. For soluion Y, Z, N of BSDEξ, f, we will always refer o he pah funcional Y, Z, N defined in Theorem 3.7. Noe ha he choice of such Y, Z, N may no be unique bu heir derivaives are unique sochasic processes which ogeher forms a soluion of differeniaed BSDE as we will see soon. This is consisen wih he resul of Con and Fournie 13, [14]. We would like o emphasize ha hese definiions are only needed in order o esimae he bound of Z process by dela-hedging formula Z = Y which is he nex secion s main resul Theorem 5.1. The key idea is he following: i Proposiion 4.3: Consider BSDEξ k, f k where ξ k, f k converges o ξ, f. Choose a represenaion of Y k, Z k, N k of BSDEξ k, f k by Theorem 3.7 and esablish BSDE saisfied by Y k, m Z k, N N k. ii Theorem 4.5 and Corollary 4.6: Prove Z k = Y k when M has maringale represenaion propery. iii Theorem 5.1: Using Proposiion 3.3, Proposiion 4.3, and he fac ha Z k converges o Z, find he bound of Z. Proposiion 4.3. Assume ξ and f saisfy S, Lip Diff For each s, y, z [, T ] R d R d n, ξ S and fs,, y, z S. In addiion, for all e R 1 n and almos every u [, T ], e uξm [,T ] L and e uf, M [, ], y, z y,z =Y,Zm H, D For all [, T ], γ D, f, γ [,], y, z is coninuously differeniable wih respec o y and z. and le Y, Z, N o be he soluion of BSDEξ, f. Le Y, Z, N be he corresponding pah funcional as in Theorem 3.7. Then, he soluion Y = Y, M [, ], Z = Z, M [, ], and N = N, M [, ] are -, m -, and N -differeniable, respecively. Moreover, for each i = 1,,, n and almos every u [, T ], e i u Y, e i,m u Z, e i,m u N = {,, if u > U, V, W if u d dp-almos everywhere where U, V, N H H m M is he unique soluion of he 3.1 wih he erminal condiion e i u ξm [,T ] and he driver g, y, z = ζ + η y + θ z 13

15 Here, we defined ζ := e i u f, M [,], y, z y,z =Y,Z m η := y f, M [,], Y, Z m θ := z f, M [,], Y, Z m and θ z := i,j z ijf, M [,], Y, Z m z ij Proof. Noe ha Y = Y, M [,], Z = Z, M [,], and N = N, M [,], and herefore, if u >, hen e i u Y, e i,m u Z, e i,m u N =,, because Y, Z, N is unaffeced by he perurbaion of M a u. Le us denoe M h := M + he i 1 [u,t ] and ξ h γ [,T ] := ξγ [,T ] + he i 1 [u,t ] f h s, γ [,s], y, z := fs, γ + he i 1 [u,t ] [,s], y, z. Noe ha Y, M[,] h, Z, M h Le us define, for u, [,], N, M h [,] is he unique soluion of BSDEξh, f h by Theorem 3.7. Then, for u, we have Ξ h,u,i = ξm h [,T ] ξm [,T ] h U h,u,i V h,u,i W h,u,i = Y, M h [,] Y, M [,] h = Z, M h [,] Z, M [,T ] h := N, M h [,] N, M [,] h U h,u,i = Ξ h,u,i + Here, we defined δ h,u,i fs, M [,s], U h,u,i s, V s h,u,i m s da s Vs h,u,i dm s W h,u,i T + W h,u,i δ h,u,i f, M [,], y, z = 1 [ ] f h, M h [,], Y, M [, ] + hy, Z, M [,] m + hz f, M [,], Y, M [, ], Z, M [,] m = ζ h,u,i + η h,u,i y + θ h,u,i z where ζ h,u,i := 1 h [ ] f h, M [,], Y, M [, ], Z, M [,] m f, M [,], Y, M [, ], Z, M [,] m η h,u,i y + θ h,u,i z := 1 h [f h, M [,], Y, M [, ] + hy, Z, M [,] m + hz 14 f h, M [,], Y, M [, ], Z, M [,] m ].

16 This BSDE has a unique soluion U h,u,i, V h,u,i, W h,u,i H H m M because Ξ h,u,i and δ h,u,i f saisfies STD. In paricular, η h,u,i C y and θ h,u,i C z d dp-a.s. uniformly for all h and u. Also noe ha lim h,l Ξ h,u,i = e i u ξm [,T ], lim h,h ζ h,u,i = ζ, lim h η h,u,i = η, and lim h θ h,u,i = θ wih η C y and θ C z. Then, by Proposiion 3.4, U h,u,i U U + h,u,i U V + h,u,i V + W h,u,i W H H m M Ξ e KCy+C z + h,u,i Ξ ζ + h,u,i ζ + η h,u,i ηu + θ h,u,i θ V m. H By dominaed convergence heorem, as h, we have η h,u,i θ ηu and h,u,i θ V m. H H Therefore, U h,u,i L U = e i u Y for all [, T ] U h,u,i H U = e i u Y V h,u,i H m W h,u,i M V = e i,m Z u W = e i,n N This implies Y, Z, and N are -, m -, and N -differeniable, respecively, and u e i u Y, e i,m u Z, e i,n u N = U, V, W I is widely known ha he densiy process Z can be hough as a derivaive of Y wih respec o he driving maringale. Under he assumpion ha M possesses maringale represenaion propery, we can prove his is indeed he case wih our definiion of pah-derivaive. To prove i, we need he following lemma. Lemma 4.4. Consider Z := Z, M H m for Z : [, T ] D R d n such ha Z is e,m u Then, Z s dm s is e u-differeniable a u [, T ], and e u Proof. Le us denoe M h,u Z s dm s = := M + he 1 [u,t ]. Then Zs, M h,u h,u [,s]dms = lim Π Z u e + u e,m u Z s dm s for u, T ] e,m u Z s dm s for u [, ] N i= Z i, M h,u h,u [, i ]M i+1 M h,u i -differeniable. 15

17 where Π is a pariion { = 1 N = T } including a poin u [, T ] and Π is he larges inerval of Π. Since he limi is convergence in probabiliy, we can ake an appropriae subsequence of Π so ha he convergence is almos sure sense. Likewise Zs, M [,s] dm s = lim Π N Z i, M [,i ]M i+1 M i i= using he same sequence of pariion as above, by aking anoher subsequence if necessary. Then we have Zs, M h,u h,u [,s]dms = lim Π = lim Π = u N i= N i= Therefore, Zs, M [,s] dm s [ ] Z i, M h,u h,u [, i ]M i+1 M h,u i Z i, M [,i ]M i+1 M i [ ] Z i, M h,u [, i ] Z i, M [,i ] M h,u i+1 M h,u i + Z i, M [,i ] M h,u i+1 M h,u i M i+1 + M i Zs, M h,u [,s] Zs, M [,s] dm s + hz u e 1 u,t ]. e u Z s dm s = Z u e 1 {u,t ]} + lim h,l = Z u e 1 {u,t ]} + u = Z u e 1 {u,t ]} + u Zs, M h,u [,s] Z, M [,s] dm s u h Z, M h,u [, ] lim Z, M [, ] dm s h h,h m e,m u Z s dm s s Theorem 4.5. Assume ha ξ and f saisfy S, Lip, Diff, and D and le Y, Z, N o be he soluion of BSDEξ, f. Le Y, Z, N be he corresponding pah funcional as in Theorem 3.7. Then, u Y u = Z u + u N u, du dp-almos everywhere. Proof. Since M has maringale represenaion propery and Y, Z, N is F M -adaped by Theorem 3.7, we know N and Y has coninuous pah. Therefore, we have he following forward SDE. Le u, T ] and Y = Y fs, M [,s], Y s, Z s m s da s + M h s := M s + he i 1 [u,t ] s Z s dm s + N fs h := fs, M[,s] h, Ys, M [,s ] h h, Zs, M[,s] m s 16

18 where Y and Z are defined as in he proof of Proposiion 4.3. Le us define ζ, η, θ, ζ h,u,i, η h,u,i, θ h,u,i, and U h,u,i, V h,u,i, W h,u,i as in he proof of Proposiion 4.3. Noe ha ζ h,u,i converges o ζ in H ; U h,u,i, V h,u,i, W h,u,i converge o e i u Y, e i,m u Z, e i,n u W in H H m M ; η h,u,i, θ h,u,i converge o η, θ in d dp-a.e. sense; and η h,u,i, θ h,u,i, η, θ are bounded d dp-a.e. sense. Therefore, E f h s f s 1 h [u,t ] s [ζ s + η s e i u Y s + θ s e i,m u Z s m s ] da s KE fs h f s 1 h [u,t ] s [ζ s + η s e i u Y s + θ s e i,m u Z s m s ] da s ζ K h,u,i η ζ + h,u,i U h,u,i η e i θ H u Y + h,u,i H s Vs h,u,i m s θ s e i,m H u Z s m s K ζ h,u,i η ζ + h,u,i U h,u,i e i η H u Y + h,u,i η e i H u Y H + θs h,u,i Vs h,u,i m s e i,m θ u Z s m s + h,u,i H s θ s e i,m H h u Z s m s by dominaed convergence heorem. As a resul, fs h f s lim da s = ζ s + η s e i h,l u Y s + θ s e i,m u Z s m s da s. h u Then, by our previous lemma, for u, ] e i u Y = Z u e i u ζ s + η s e i u Y s + θ s e i,m u Z s m s da s + u e i,m u Since u Y and u N are righ coninuous, we prove he claim by leing u. Z s dm s + e i When M has maringale represenaion propery, hen N and above heorem implies he following corollary which we will use in secion 5 and 6. Corollary 4.6. Assume he condiions in Theorem 4.5. In addiion, assume ha M M has maringale represenaion propery; ha is, any F M, P maringale X such ha Er[X, X] T < can be expressed as X = X + Z sdm s for some Z H m. Le Y, Z, N o be he soluion of BSDEξ, f. Le Y, Z, N be he corresponding pah funcional as in Theorem 3.7. Then, u Y u = Z u, du dp-almos everywhere. 5 BSDE wih Locally Lipschiz Driver In his secion, we always assume M. This implies Y has coninuous pah, herefore, Y s = Y s for all s and N. Using Corollary 4.6, his maringale represenaion propery enables us o bound Z process by bounding -derivaive of Y. u N 17

19 5.1 A Priori Esimae of Z Le k be an posiive ineger and = k k 1 k k = T be a pariion of [, T ]. For γ D, we define P k γ := γ k γ 1 k,, γ k γ k k k 1 L k a 1,..., a k := k a i 1 [ k i,t ]. i=1 Le us denoe x R kn and le ϕ Cc R kn ; R be he mollifier { λ exp 1 if x < 1 ϕx := 1 x oherwise, where he consan λ R + is chosen so ha R ϕxdx = 1. Se ϕ k x := k kn ϕkx, and define kn [ ξ k := ξ L k ϕ k] P k f k s, γ, y, z := fs, L k P k γ x, y, zϕ k x dx. R kn Theorem 5.1. Assume ha ξ and f saisfy M, Lip. In addiion, assume he following condiion: Diff ξ < and fs,,, da s <. M[,T ] L k P k M [,T ] L k. There are D ξ, D f R + such ha, for all γ, γ D, s, y, z [, T ] R d R d n, ξγ ξγ D ξ γ γ and fs, γ, y, z fs, γ, y, z D f γ γ. Then, BSDEξ, f has a unique soluion Y, Z, N H H m M. Moreover, Y has coninuous pah, N and Z Dξ + D f Ke 1 KCy+C z +1 d dp-a.e. Remark 5.. I is easy o see ha he second condiion of Diff holds when M is a Brownian moion and = it/k. More generally, le W be Brownian moion and assume ha M has a maringale represenaion k i M = η s dw s where here exiss a consan C such ha η s C almos surely. Then E M [,T ] L k P k M [,T ] = E sup sup i=,...,k 1 i < i+1 [ k 1 E i= sup i < i+1 i i η s dw s η s dw s [ k 1 ] 1/ CC T/k i= 4 ] 1/ E [ k 1 sup i= i < i+1 i i+1 [ k 1 i= CE CC T 1 k k. i η s dw s 4 ] 1/ ] 1/ η s ds 18

20 The inequaliies are based on sup a i i=,...,k 1 k 1 a i, Jensen inequaliy, and Burkholder-Davis-Gundy inequaliy. Here, we used C for he conan of Burkholder- Davis-Gundy inequaliy. Therefore, such M saisfies he second condiion of Diff. Before we proceed o he proof, le us observe he following facs. Lemma 5.3. Under he assumpion of Theorem 5.1, we have he following resuls. i ξ, f saisfies S and ξ k, f k saisfies S, Diff, Lip ii u ξ k M [,T ] D ξ and u f k, M [,], y, z D f for all u [, T ], y, z R d R d n in d dp-a.e. iii For soluion Y, Z, N H H m M of BSDEξ, f, Y has a coninuous pah, N, and ξ k M [,T ] i= L ξm [,T ] and f k H, M [, ], Y, Zm f, M [, ], Y, Zm. k k Proof. I is easy o see ξ k, f k saisfies Lip. Noe ha ξ M[,T ] + γ ξ + D M[,T ξ ] + γ < ξm [,T ] ξ + D ξ M [,T ] + D ξ ˆM[,T ] < since M [,T ] L by Burkholder-Davis-Gundy inequaliy and ˆM [,T ] is bounded by he number of jumps of M which is a Poisson random variable. Therefore, ξ saisfies S. We can use he same arguemen o show ha f saisfies S as well. Noe ha ξ k γ ξ k γ ξl k P k γ x ξl k P k γ x ϕ k x dx R kn D ξ L k P k γ x L k P k γ x ϕ k x dx Likewise, R kn D ξ γ γ. f k s, γ, y, z f k s, γ, y, z D f γ γ. Using he same argumen for ξ, f, his implies ξ k, f k saisfies S. Moreover, i also implies Diff and ii because of Lipschizness and he convoluion wih he mollifier ϕ k. Lasly, since M has maringale represenaion propery and Y, Z, N is F M -adaped by Theorem 3.7, we know N and Y has coninuous pah. Also, noe ha ξm [,T ] ξ k M [,T ] ξm [,T ] ξ L k P k M [,T ] + ξ L k P k γ ξ L k P k γ x ϕ k x dx R kn M[,T D ξ ] L k P k M [,T ] + D ξ max x R i ϕ k x dx k kn We can argue similarly for f k o conclude iii holds. i 19

21 Proof of Theorem 5.1. Le us denoe x = y, z R d R d n and le β Cc { λ exp 1 if x < 1 βx := 1 x oherwise, R d R d n ; R be he mollifier where he consan λ R + is chosen so ha R βxdx = 1. Se β m x := m d+dn βmx, m N \ {}, dn+d and define f k,m, γ [,], x := f k, γ [,], x x β m x dx. R dn+d Then, i is easy o check ha S, Lip, Diff for f k implies ha ξ k, f k,m saisfies S, Lip, Diff, and D. Therefore, here exiss soluion Y k,m, Z k,m of he BSDE Y k,m = ξ k M [,T ] + f k,m s, M [,s], Y k,m s From our Proposiion 3.3 and Proposiion 4.3, we know u Y k,m, Z k,m D ξ + D f Ke 1 KCy+C z +1 for all u [, T ]. Then, from Corollary 4.6, for all k, m N, Z k,m D ξ + D f Ke 1 KCy+C z +1 s m s da s Z s k,m dm s d dp -almos everywhere. By Proposiion 3.4, we have Y Y k,k Z + Z k,k H H m ξm[,t e KCy+C z +1 ] ξ k M [,T ] f, + M[, ], Y, Z m f k,k, M [, ], Y, Z m. H Since f k is Lipschiz, f k, M [,], Y, Z m f k,k, M [,], Y, Z m f k, M [,] Y, Z m f k, M [,], Y, Z m x β k x dx R dn+d C y + C z x β k x dx m. R dn+d Combined wih he previous lemma, his implies ha f, M [, ], Y, Z m f k,k, M [, ], Y, Z m H k. Therefore, Y k,k Y in H and Z k,k Z in H m wih Z k,k Dξ + D f Ke 1 KCy+C z +1. Therefore, Z D ξ + D f Ke 1 KCy+C z +1.

22 5. Exisence and uniqueness when d 1 and [M, M] T is small. Theorem 5.4. Assume he following condiions: M, Diff, and Loc There exiss a nondecreasing funcion ρ : R + R + such ha f, γ [,], y, z f, γ [,], y, z ρ z z y y + z z for all [, T ], y, y R d and z, z R d n. Assume ha K is small enough so ha here is R R + saisfying D ξ + D f Ke 1 KρR+1 R Then, he BSDE Y = ξm [,T ] + fs, M [,s], Y s, Z s m s da s Z s dm s has a unique soluion Y, Z H H m such ha Z is bounded. Moreover, Z R in d dp-almos everywhere sense. Proof. For R in he assumpion, consider he BSDE wih he erminal condiion ξm [,T ] and he driver Rz g, γ [,], y, z := f, γ [,], y,. z R Then, g saisfies Diff and Lip wih Lipschiz coefficien of he driver ρr = C y = C z. Therefore, here exiss a unique soluion U, V H H m for he following BSDE and V is bounded by U = ξm [,T ] + gs, M [,s], U s, V s m s da s V s dm s V D ξ + D f Ke 1 KρR+1. Therefore, since m = 1 for all see Secion, V m V m Dξ + D f Ke 1 KρR+1 R. This implies ha U, V is also a soluion of Y = ξm [,T ] + fs, M [,s], Y s, Z s m s da s Z s dm s. Now le us show he uniqueness. Assume ha Y, Z is anoher soluion such ha Z is bounded by Q. Wihou loss of generaliy, we can assume Q R. Then, if we consider Qz h, γ [,], y, z := f, γ [,], y,, z Q hen BSDEξ, h has a unique soluion in H H m. Since Y, Z and Y, Z are boh soluion o such BSDE, we have Y, Z = Y, Z. 1

23 5.3 Explosion of soluion when d > 1 and [M, M] T is large. If d > 1, he resul on he previous subsecion canno exend o arbirary large K in general. This can be shown by he following counerexample which is inspired by Chang e al. 199, [6]. For δ >, le M := W τ W τ T δ 1 [T δ, where W is a wo dimensional Brownian moion and τ := inf { [T δ, : W W T δ 1/ } T. Noe ha M saisfies M because of Lemma.1 of Peng 1991, [4]. Define he erminal condiion ξ as cos θ T sin g R T ξm T := sin θ T sin g R T cos g R T where g : R + R is a smooh funcion wih bounded derivaives of all order and R s, θ s is he polar coordinae of M s. Noe ha ξ is a smooh funcion wih bounded derivaive. We le, for ε, 1, λ r 1+ε φr := arccos λ + r 1+ε where λ is big enough so ha cos φr 1 + ε 1 for r [, 1]. We choose a smooh funcion g so ha 1 r g r arccos 1 + r + φr for r [, 1] and g =. Noe ha g has bounded derivaives all orders on [, 1]. Le us show he following BSDE driven by M have a soluion Y, Z such ha Z is bounded only when [M, M] T is small enough: Y = ξm T + 1 Z sm s Y T s Y s 1 da s Z s dm s. 5.4 Since we have [M, M] := s τ T δ I, we have m s = 1 I and A s = 4s τ T δ. Noe ha ess sup ω Ω A T = 4δ. I is easy o check ha Loc and Diff if we le ρx = x + 1 x. Then, by Theorem 5.4, 5.4 has a unique soluion Y, Z H H m such ha Z is bounded if δ is small enough so ha here exiss R such ha D ξ e δr+r /+1 R where D ξ is he bound on he derivaive of ξ wih respec o M T. In order o prove he nonexisence of soluion for large δ, we need he following proposiion. Proposiion 5.5. Consider he PDE of g : [, [, 1] R: g = rr g + 1 r rg sin g cos g r ; g, r = g r, g, =, and g, 1 = π. This PDE admis a unique classical soluion on [, T for some T R + and lim T T r g, =. Proof. See he proof par i in Chang e al. 199, [6].

24 Now, for T in Proposiion 5.5, assume ha δ > T and 5.4 has a soluion Y, Z H H m such ha Z is bounded. For g, r in Proposiion 5.5, if we le u : [, T R R 3 be x 1 x sin g, x x u, x = x sin g, x, cos g, x we can easily deduce ha Y = ut, M and Z := ut, M T T, T ] by using Iô formula and he uniqueness of soluion for BSDE. Noe ha ut, as T T by Proposiion 5.5 and u, x is coninuous in, x. Therefore, for any large L, here exiss ε > such ha ut, x L for all, x T T, T T + ε ε, ε. Since M is a scaled Brownian moion saring a T δ and sopped a τ, P Z L for all T T, T T + ε PM ε, ε for all T T, T T + ε >. This implies BSDE 5.4 canno have a soluion such ha Z is bounded when δ T. Remark 5.6. Our counerexample above shows ha here is no soluion Y, Z such ha Z is bounded and herefore, Theorem 5.4 is sharp in his sense. We do no exclude he possibiliy ha here is a soluion Y, Z H H m such ha Z is no bounded. However, if δ = T in above example, lim T T Z = almos surely. Remark 5.7. Anoher counerexample for he exisence of soluion for mulidimensional quadraic BSDE is given by Frei and dos Reis 1, [1]. They proved he Y par of soluion explodes when he erminal condiion is singular wih respec o he perurbaion of underlying maringale, i.e., Brownian moion. In our example, Y is uniformly bounded and he erminal condiion is smooh wih bounded derivaive. However, in our case, Z explodes wih posiive probabiliy. 5.4 Exisence and uniqueness when d = 1. When d = 1 and m is inverible for all [, T ], we can remove he smallness condiion on K if we assume Lipschizness of f, γ [,], y, z wih respec o y. Theorem 5.8. Assume ha M and Diff hold, d = 1, and m is inverible for all [, T ]. In addiion, assume ha Loc here exis C y R + and a nondecreasing funcion ρ : R + R + such ha f, γ [,], y, z f, γ [,], y, z C y y y + ρ z z z z for all [, T ], y, y R d and z, z R d n. Then, he BSDE Y = ξm [,T ] + fs, M [,s], Y s, Z s m s da s Z s dm s has a unique soluion Y, Z H H m such ha Z is bounded. Moreover, Z n [ D ξ + D f C y e CyK D f C y ], d dp -almos everywhere. In he case where C y =, he bound changes o nd ξ + D f K. If we assume D and Diff as well, Y and Z are - and m -differeniable, respecively, and Z u = u Y u. 3

25 Proof. We already know ha ξ, f saisfies S by Lemma 5.3. Noe ha if we can prove he heorem under assumpion D, and Diff, we can generalize i o Diff using he same argumen in Theorem 5.1. Therefore, wihou loss of generaliy, we will assume D and Diff, and moreover, D ξ, C y >. Le R := [ n D ξ + D f e CyK D ] f C y C y Firs le Y, Z be he soluion of Y = ξm,t + gs, M [,s], Y s, Z s m s da s Z s dm s where g is a smooh exension of f such ha { f, γ[,], y, z if z R g, γ [,], y, z := f, γ [,], y, R + 1z/ z if z R + 1 and z g ρr + 1. By our Proposiion 4.3 and 5.1, for u, and Z = Y, where Le us compare 5.5 wih e i u Y = Ξ + ζ s + η s e i u Y s + θ s e i,m u Z s m s da s U = D ξ + Ū = D ξ + Ξ := e i u ξm [,T ] ζ := e i u g, M [, ], y, z y,z=y,z m η := y g, M [, ], Y, Z m θ := z g, M [, ], Y, Z m D f + C y U s + ρr + 1 V s m s da s Df C y Ūs ρr + 1 V s m s da s e i,m u Z s dm s 5.5 V s dm s 5.6 V s dm s The BSDEs 5.6 have unique soluions in H H m such ha U, Ū S. Le us define { v hv := v if v oherwise dγ Γ = C y hu s da s + ρr + 1hV s m s m 1 s dm s ; Γ = 1. Here, we use h as defined on eiher R or R 1 n depending on he conex. If we apply Iô formula o Γ U, dγ U = D f Γ da + Γ U ρr + 1hV m m 1 + Γ V dm. 4

26 Since hu, hv m are bounded by 1, by he same logic as in he proof of Theorem 3.5, Γ S and his implies ha Γ s ρr + 1hV s m s m 1 s dm s and ΓU + D f Γ s da s are rue maringales. Therefore, Γ = E C y hu s da s + ρr + 1 hv s m s m 1 s dm s U = 1 Γ E [ Γ T D ξ + D f Γ s da s F ]. ] = E [Γ T C y Γ s hu s da s F > Since U >, we have hu s = 1 and he firs par implies ha, since Γ := e CyA EρR+1 hv sm s m 1 s dm s, ] ] Therefore, we ge [ E Γ s Γ da s F U 1 [ ΓT E C y Γ F D ξ + D f e CyK A D f. C y C y 1 C y 1 C y e CyK A 1 C y. We can ge he same upper bound for Ū using he same argumen. By he comparison heorem 3.5, we know e i u Y D ξ + D f e CyK A D f R C y C y n which implies Z m R. Therefore, Y, Z is a soluion of he original BSDE Y = ξm [,T ] + fs, M [,s], Y s, Z s m s da s Z s dm s. Uniqueness can be easily checked by he same argumen in he proof of Theorem Uiliy Maximizaion of Conrolled SDE Driven by M In his secion, we apply he previous resuls o he uiliy maximizaion problem for conrolled SDEs driven by M. Our conrol is and we require { } A := X H : ess sup X ω <,ω [,T ] Ω For a given conrol, consider one of he wo SDEs driven by M where M saisfies M: X = x + X = x + X s bs, M [,s], s da s + bs, M [,s], s da s + X s σs, M [,s], s dm s 6.7 σs, M [,s], s dm s 6.8 Here, b : [, T ] D R 1 n R and σ : [, T ] D R 1 n R 1 n are joinly measurable funcions such ha b, M [, ], and σ, M [, ], are predicable for any A. 5

27 6.1 Power uiliy Our objecive in his subsecion is o find a conrol A ha maximize [ 1 E X ] T e ξm κ [,T ] κ where κ,, 1] and X is given by 6.7. Theorem 6.1. Assume ha here exis an increasing coninuous funcion ρ : R + R and a B[, T ] D BR n -measurable funcion k : [, T ] D R 1 n R 1 n ha saisfy he following condiions: bs, M [,s], π + σs, M [,s], π ρ π for all π R 1 n. ks, γ, z ρ z for all s, γ, z [, T ] D R 1 n. For all s, γ, π, z [, T ] D A R d n, he following inequaliy holds: Gs, γ, π, z := 1 κ σs, γ [,s], πm s z bs, γ [,s], π + 1 σs, γ [,s], πm s Gs, γ [,s], ks, γ [,s], z, z := fs, γ [,s], z The BSDEξ, f has a soluion Y, Z H H m such ha Z is bounded. Then, s := ks, M [o,s], Z s m s A is he opimal conrol and he opimal value is xκ e κy κ. Proof. Firs of all, since M [, ] : [, T ] Ω D is an adaped coninuous funcion, noe ha is a predicable process because i is a deerminisic measurable funcion of predicable processes. Moreover, since Z is bounded, is bounded by our assumpion. Therefore, A. As in Hu e al. 5, [3], we use he maringale echnique o prove he heorem. Noe ha, since bs, M [,s], s and σs, M [,s], s are bounded for A, 6.7 has a unique srong soluion X = x exp bs, M [,s], s 1 σs, M [,s], s m s da s + σs, M [,s], s dm s. For noaional convenience, le us use b s := bs, M [,s], s and σs := σs, M [,s], s. Le us define a family of sochasic process { U } given by A U = 1 κ X e Y κ = xκ κ exp = xκ e κy E κ = xκ e κy E κ κy + κ κ κ b s 1 σ s m s + fs, M [,s], Z s m s da s + κ σ s Z s dms σ s Z s dms exp κ exp κ σ s Z s dms b s 1 σ s m s + κ σ s m s Z s m s + fs, M [,s], Z s m s fs, M[,s], Z s m s Gs, M [,s], s, Z s m s da s da s 6