Classical and Variational Differentiability of BSDEs with quadratic growth

We consider Backward Stochastic Differential Equations (BSDE) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect to the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDE to be differentiable in the variational sense (Malliavin differentiable).


Introduction
Problems of stochastic control treated by the crucial tool of backward stochastic differential equations (BSDEs) have been encountered in many areas of application of mathematics in recent years. A particularly important area is focused around optimal hedging problems for contingent claims in models of financial markets. Recently, a special class of hedging problems in incomplete financial markets has been considered in the area where finance and insurance concepts meet. At this interface problems of securitization arise, i.e. insurance risk is transferred to capital markets. One particularly interesting risk source is given by climate or environmental hazards affecting insurance companies or big branches of the economy that depend on weather such as agriculture and fishing, transportation and tourism. The public awareness of climate hazards such as floods or hurricanes is continually increasing with the intensity of the discussion about irreversible changes due to human impact.
BSDEs typically appear in the following setting. On a financial market some small investors are subject to an external risk source described for instance by weather or climate influences. There may also be big investors such as re-insurance companies that depend in a possibly different way on the same risk source. In this situation market incompleteness stems from the external risk not hedgeable by the market assets. One may complete the market either by making the external risk tradable through the introduction of an insurance asset traded among small agents, or by introducing a risk bond issued by a big agent. In this setting, treating the utility maximization problem for the agents under an equilibrium condition describing basically market clearing for the additional assets, leads to the determination of the market price of external risk through a BSDE which in case of exponential utility turns out to be quadratic in the control variable (see [HM06], [CHIM05] and [CIM04]). Alternatively, instead of maximizing utility with respect to exponential utility functions we might minimize risk measured by the entropic risk measure. In this setting we again encounter a BSDE with quadratic nonlinearity, of the type where W is a finite-dimensional Wiener process of the same dimension as the control process Z, with a generator f that depends at most quadratically on Z, and a bounded terminal condition ξ. In the meantime, the big number of papers published on general BSDEs is rivalled by the number of papers on BSDEs of this type of nonlinearity. For a more complete list of references see [CSTV05] or [Kob00]. In particular, there are papers in which the boundedness condition on ξ is relaxed to an exponential integrability assumption, or where the stochastic integral process of Z is supposed to be a BMO martingale.
In a particularly interesting case the terminal variable ξ is given by a function g(X x T ) at terminal time T of the solution process X of a forward SDE with initial vector x ∈ R. Similarly, the driver f may depend on the diffusion dynamics of X x . Via the famous link given by the generalized Feynman-Kac formula, systems as the above of forward-backward stochastic differential equations are seen to yield a stochastic access to solve nonlinear PDE in the viscosity sense, see [Kob00].
In this context, questions related to the regularity of the solutions (X x , Y x , Z x ) of the stochastic forward-backward system in the classical sense with respect to the initial vector x or in the sense of the stochastic calculus of variations (Malliavin calculus) are frequently encountered. Equally, from a more analytic point of view also questions of smoothness of the viscosity solutions of the PDE associated via the Feynman-Kac link are seen to be very relevant.
For instance, Horst and Müller (see [HM06]) ask for existence, uniqueness and regularity of a global classical solution of our PDE from the analytic point of view. Not attempting a systematic approach of the problem, they use the natural access of the problem by asking for smoothness of the solutions of the stochastic system in terms of the stochastic calculus of variations. But subsequently they work under the restrictive condition that the solutions of the BSDE have bounded variational derivatives, which is guaranteed only under very restrictive assumptions on the coefficients.
The question of smoothness of the stochastic solutions in the parameter x arises for instance in an approach of cross hedging of environmental risks in [AIP05]. Here the setting is roughly the one of an incomplete market generated by a number of big and small agents subject to an external (e.g. climate related) risk source, and able to invest in a given capital market. The risk exposure of different types of agents may be negatively correlated, so that typically one type profits from the risky event, while at the same time the other type suffers. Therefore the concept of hedging one type's risk by transferring it to the agents of the other type in a cross hedging context makes sense. Mathematically, in the same way as described above, it leads to a BSDE of the quadratic type, the solution (Y x , Z x ) of which depends on the initial vector x of a forward equation with solution X x . Under certain assumptions, the cross-hedging strategy can be explicitly given in a formula depending crucially on x, and in which the sensitivity with respect to x describes interesting quality properties of the strategy.
In this paper, we tackle regularity properties of the solutions (Y x , Z x ) of BSDEs of the quadratic type such as the two previously sketched in a systematic and thorough way. Firstly, the particular dependence on the starting vector x of the forward component of a forwardbackward system will be generalized to the setting of a terminal condition ξ(x) depending in a smooth way to be specified on some vector x in a certain Euclidean state space. We both consider the smoothness with respect to x in the classical sense, as well as the smoothness in the sense of Malliavin's calculus.
The common pattern of reasoning in order to tackle smoothness properties of any kind starts with a priori estimates for difference and differential quotients, or for infinite dimensional gradients in the sense of variational calculus. In the estimates, these quantities are related to corresponding difference and differential quotients or Malliavin gradients of the terminal variable and the driver. To obtain the a priori estimates, we make use to changes of probability of the Girsanov type, by which essentially nonlinear parts of the driver are eliminated. Since terminal conditions in our treatment are usually bounded, the exponential densities in these measure changes are related to BM O martingales. Known results about the inverse Hölder inequality allow to show that as a consequence the exponential densities are r-integrable for some r > 1 related to the BM O norm. This way we are able to reduce integrability properties for the quantities to be estimated to a natural level. In a second step, the a priori inequalities are used to derive the desired smoothness properties from corresponding properties of driver and terminal condition. To the best of our knowledge, only Malliavin differentiability results of this type have been obtained so far, with strong conditions on the coefficients restricting generality considerably (see [HM06]).
The paper is organized as follows. In section 1 we fix the notation and recall some process properties needed in the proofs of the main body of the paper. Section 2 contains the main results on classical differentiability. In sections 3, 4 and 5 we give a priori bounds for classes of non-linear BSDEs. Section 6 contains the proofs of the theorems stated in Section 2. Section 7 is devoted to the application of the proven results to the forward-backward SDE setting. In Section 8 we state and prove the Malliavin differentiability results.

Preliminaries
Throughout this paper let (Ω, F, P ) be a complete probability space and W = (W t ) t≥0 a d−dimensional Brownian motion. Let {F t } t≥0 denote the natural filtration generated by W , augmented by the P −null sets of F.
Let T > 0, ξ be an F T -measurable random variable and f : Ω × [0, T ] × R × R d → R. We will consider Backward Stochastic Differential Equations (BSDEs) of the form (1) As usual we will call ξ the terminal condition and the function f the generator of the BSDE (1). A solution consists of a pair (Y, Z) of adapted processes such that (1) is satisfied. To be correct we should write since W and Z are d−dimensional vectors; but for simplicity we use this notation as it is without ambiguity. It is important to know which process spaces the solution of a BSDE belongs to. We therefore introduce the following notation for the spaces we will frequently use. Let p ∈ [1, ∞]. Then, for m ∈ N * is the space of bounded measurable processes.
We will omit reference to the space or the measure when there is no ambiguity.
Furthermore, we use the notation Suppose that the generator satisfies, for a ≥ 0 and b, c > 0 Kobylanski has shown in [Kob00] that if ξ is bounded and the generator f satisfies (2), then there exists a solution (Y, Z) ∈ R ∞ × L 2 . Moreover, it follows from the results in [Mor07], that in this case the process Z is such that the stochastic integral process relative to the Brownian motion · 0 ZdW is a so-called Bounded Mean Oscillation (BMO) martingale. Since the BMO property is crucial for the proofs we present in this paper we recall its definition and some of its basic properties. For an overview on BMO martingales see [Kaz94].
If we want to stress the measure P we are referring to we will write BMO(P ). It can be shown that for any p, q ∈ [1, ∞] we have BMO p = BMO q (see [Kaz94]). Therefore we will often omit the index and simply write BMO for the set of BMO martingales.
In the following Lemma we state the properties of BMO martingales we will frequently use.  2) Let M be a BMO martingale relative to the measure P . Then the processM = M − M is a BMO martingale relative to the measure Q (see Theorem 3.3 in [Kaz94]).

Differentiability of quadratic BSDEs in the classical sense
Suppose that the terminal condition and the generator of a quadratic BSDE depend on the Euclidean parameter set R n for some n ∈ N * . We will show that the smoothness of the terminal condition and the generator is transferred to the solution of the BSDE where terminal condition and generator are subject to the following conditions (C1) f : Ω×[0, T ]×R n ×R×R d → R is an adapted measurable function such that f (ω, t, x, y, z) = l(ω, t, x, y, z) + α|z| 2 , where l(ω, t, x, y, z) is globally Lipschitz in (y, z) and continuously differentiable in (x, y, z); for all r ≥ 1 and (t, y, z) the mapping (C2) the random variables ξ(x) are F T −adapted and for every compact set K ⊂ R n there exists a constant c ∈ R such that sup x∈K ξ(x) ∞ ≤ c; for all p ≥ 1 the mapping R n → L p , x → ξ(x) is differentiable with derivative ∇ξ.
If (C1) and (C2) are satisfied, then there exists a unique solution (Y x , Z x ) of Equation (3). This follows from Theorems 2.3 and 2.6 in [Kob00]. We will establish two differentiability results for the pair (Y x , Z x ) in the variable x. We first consider differentiability of the vector valued map with respect to the Banach space topology defined on R p (R 1 ) × L p (R d ). This will be stated in Theorem 2.1. A slightly more stringent result will be obtained in the subsequent Theorem 2.2.
Here, we consider pathwise differentiability of the maps in the usual sense, for almost all pairs (ω, t). In both cases, the derivatives will be identified with (∇Y x , ∇Z x ) solving the BSDE We emphasize at this place that it is not immediate that this BSDE possesses a solution. In fact, without considering it as a component of a system of BSDEs also containing the original quadratic one, it can only be seen as a linear BSDE with global, but random (and not bounded) Lipschitz constants.
Theorem 2.1. Assume (C1) and (C2). Then for all p ≥ 1, the function , is differentiable, and the derivative is a solution of the BSDE (4).
Under slightly stronger conditions one can show the existence of a modification of Y x which is P -a.s. differentiable as a mapping from R n to R. Let e i = (0, . . . , 1, . . . , 0) be the unit vector in R n where the ith component is 1 and all the other components 0. For x ∈ R n and h = 0 let For the existence of differentiable modifications we will assume that (C3) for all p ≥ 1 there exists a constant C > 0 such that for all i ∈ {1, . . . , n}, x, x ′ ∈ R n and Theorem 2.2. Suppose, in addition to the assumptions of Theorem 2.1, that (C3) is satisfied and that l(t, x, y, z) and its derivatives are globally Lipschitz continuous in (x, y, z). Then there such that for almost all ω, Y x t is continuous in t and continuously differentiable in x, and for all x, (Y x t , Z x t ) is a solution of (3).

Moment estimates for linear BSDEs with stochastic Lipschitz generators
By formally deriving a quadratic BSDE with generator satisfying (C1) and (C2) we obtain a linear BSDE with a stochastic Lipschitz continuous generator. The Lipschitz constant depends on the second component of the solution of the original BSDE. In order to show differentiability, we start deriving a priori estimates for this type of linear BSDE with stochastic Lipschitz continuous generator. For this purpose, we first need to show that the moments of the solution can be effectively controlled. Therefore this section is devoted to moment estimates of solutions of BSDEs of the form We will make the following assumptions concerning the drivers: , the process l(ω, t, u, v) is (F t )predictable and there exists a constant M > 0 such that for all (ω, t, u, v), Moreover, we assume that (U, V ) is a solution of (5) satisfying Under the assumptions (A1), (A2), (A3), (A4) and (A5) one obtains the following estimates.
Theorem 3.1 (Moment estimates). Assume that (A1)-(A5) are satisfied. Let p > 1 and r > 1 such that E( HdW ) T ∈ L r (P ). Then there exists a constant C > 0, depending only on p, T , M and the BMO-norm of HdW ), such that with the conjugate exponent q of r we have Moreover we have The proof is divided into several steps. First let β > 0 and observe that by applying Itô's formula to e βt U 2 t we obtain By (A2), the auxiliary measure defined by Q = E(H · W ) T · P is in fact a probability measure.
We therefore first prove moment estimates under the measure Q.
Lemma 3.2. For all p > 1 there exists a constant C, depending only on p, T and M , such that Moreover we have Proof. Throughout this proof let C 1 , C 2 , . . ., be constants depending only on p, T and M . Inequality (8) implies and (A5) together with the existence of the rth moment for E( Integrating both sides and using Young's inequality, we obtain and hence Inequality (12), (A5) and Doob's L p inequality imply for p > 1 In order to complete the proof, note that (8) implies By Young's inequality, 2 T t e βs M |U s ||V s |ds ≤ 1 2 T t e βs |V s | 2 ds + 8M 2 T t e βs U 2 s ds, and hence which, combined with (13) leads to the desired Inequality (10). Equation (15), Young's inequality, Doob's L p -inequality and the Burkholder-Davis-Gundy inequality imply Thus, with Inequality (14), the proof is complete.
Proof of Theorem 3.1. Notice that by the second statement of Lemma 1.2, the process HdŴ = HdW − · 0 H 2 s ds belongs to BMO(Q), and hence − HdŴ also. Moreover, E( HdW ) −1 = E(− HdŴ ). Consequently, by the third statement of Lemma 1.2, there exists an r > 1 such that E(H · W ) T ∈ L r (P ) and Hölder's inequality and Lemma 3.2 imply that for the conjugate exponent q of r we have where C 1 , C 2 represent constants depending on p, M, T and the BM O norm of HdW . Sim- q 2 , and hence (6). By applying the same arguments to (10) we finally get (7).

A priori estimates for linear BSDEs with stochastic Lipschitz constants
In this section we shall derive a priori estimates for the variation of the linear BSDEs that play the role of good candidates for the derivatives of our original BSDE. These will be used to prove continuous differentiability of the smoothly parametrized solution in subsequent sections. Let (ζ, H, l 1 , A) and (ζ ′ , H ′ , l 2 , A ′ ) be parameters satisfying the properties (A1), (A2), (A3) and (A4) of Section 3 and suppose that l 1 and l 2 are globally Lipschitz continuous and differentiable in resp.
Theorem 4.1 (A priori estimates). Suppose we have for all β ≥ 1, T 0 δU 2 s |δV s | 2 ds ∈ L β (P ) and T 0 |δU s δA s |ds ∈ L β (P ). Let p ≥ 1 and r > 1 such that E( H ′ dW ) T ∈ L r (P ). Then there exists a constant C > 0, depending only on p, T , M and the BMO-norm of H ′ dW , such that with the conjugate exponent q of r we have We proceed in the same spirit as in the preceding section. Before proving Theorem 4.1 we will show a priori estimates with respect to the auxiliary probability measure Q defined by There exists a constant C > 0, depending only on p, T and M , such that Proof. The difference δU satisfies Let β > 0. Applying Itô's formula to e βt δU 2 t , t ≥ 0, yields the equation where Using the Lipschitz property of l 1 we obtain We will now derive the desired estimates from Equation (20). First observe that by taking conditional expectations, we get Let p > 1. Then for some constants C 1 , C 2 , . . ., depending on p, T and M , we obtain and by Doob's L p inequality we get By using Young's and Hölder's inequalities we have Therefore, we may further estimate Equation (22), Doob's L p -inequality and the Burkholder-Davis-Gundy inequality imply Consequently, Young's inequality allows to deduce Finally, (17) and (21) imply

A priori estimates for quadratic BSDEs
Consider the two quadratic BSDEs and where ξ and ξ ′ are two bounded F T -measurable random variables, and l 1 and l 2 are globally Lipschitz and differentiable in (y, z).
The a priori estimates we shall prove next will serve for establishing (moment) smoothness of the solution of the quadratic BSDE with respect to a parameter on which the terminal variable depends smoothly. Note first that by boundedness of ξ and ξ ′ we have that both ZdW and Z ′ dW are BM O martingales, so that we may again invoke the key Lemma 1.2.
. Then there exists a constant C > 0, depending only on p, T , M and the BMO-norm of (α (Z s +Z ′ s )dW ), such that with the conjugate exponent q of r we have Moreover we have We give only a sketch of the proof since the arguments are very similar to the ones used in the proofs in Sections 3 and 4.
First observe that By applying Itô's formula to e βt |δY t | 2 we obtain We start with a priori estimates under the auxiliary probability measure Q defined by Q = Note that the process t 0 e βs δY s δZ s dW s is a strict martingale because δY s is bounded and (δZ ·W ) is BMO relative to Q.
Notice that Equation (26) is of similar but simpler form than Equation (20). This is because the (H s − H ′ s ) term in (20) has been completely absorbed by the Girsanov measure change. As a consequence, following the proof of Lemma 4.2, we obtain the following estimates: Lemma 5.2. For all p > 1 there exists a constant C > 0, depending only on p, M and T , such that

Moreover we have
Proof of Theorem 5.1. The arguments are similar to those of the proof of Theorem 3.1. Just make use of Lemma 1.2.

Proof of the differentiability
We now approach the problem of differentiability of the solutions of a quadratic BSDEs with respect to a vector parameter on which the terminal condition depends differentiably. We start with the proof of the weaker property of Theorem 2.1. Our line of reasoning will be somewhat different from the one used for instance by Kunita [Kun90] in the proof of the diffeomorphism property of smooth flows of solutions of stochastic differential equations. He starts with formally differentiating the stochastic differential equation, and showing that the resulting equation possesses a solution. The latter is then used explicitly in moment estimates for its deviation from difference quotients of the original equation. The estimates are then used to prove pathwise convergence of the difference quotients to the solution of the differentiated SDE. We emphasize that in our proofs, we will have to derive moment estimates for differences of difference quotients instead. They will allow us to show the existence of a derivative process in a Cauchy sequence type argument using the completeness of underlying vector spaces, which of course will be the solution process of the formally differentiated BSDE. So our procedure contains the statement of the existence of a solution of the latter as a by-product of the proof of the Theorem 2.1. It is not already available as a good candidate for the derivative process, since, as we stated earlier, the formally differentiated BSDE is a globally Lipschitz one with random Lipschitz constants for which the classical existence theorems do not immediately apply. Throughout assume that f (t, x, y, z) = l(t, x, y, z) + α|z| 2 and ξ(x) satisfy (C1) and (C2) respectively.
For all x ∈ R n let (Y x t , Z x t ) be a solution of the BSDE (3). It is known that the solution is unique and that (Y x , Z x ) ∈ R ∞ (R 1 ) × L 2 (R d ) (see [Kob00]).
It follows from Lemma 1 in [Mor07] that there exists a constant D > 0 such that for all x ∈ R n we have (Z x · W ) T BM O 2 ≤ D. Now let r > 1 be such that Ψ(r) > 2αD (see property 3) of Lemma 1.2), and denote as before by q the conjugate exponent of r.
Proof of Theorem 2.1. To simplify notation we assume that M > 0 is a constant such that ξ(x), x ∈ R n , and the derivatives of l in (y, z) are all bounded by M . We first show that all the partial derivatives of Y and Z exist. Let x ∈ R n and e i = (0, . . . , 1, . . . 0) be the unit vector in R n the ith component of which is 1 and all the others 0. For all h = 0, let To simplify the last term we use a line integral transformation. For all (ω, t) Observe that these functions satisfy (A3) and that they are Lipschitz continuous and differentiable in (u, v). In these terms, and thus we obtain an equation as modelled by (5). Notice that for all h, h ′ = 0 the pairs for all β ≥ 1, and therefore, with Theorem 5.1, the third summand on the right hand side of (28) converges to zero as h, h ′ → 0. In order to prove convergence of the second summand let P ⊗ λ be the product measure of P and the Lebesgue measure λ on [0, T ]. It follows from Theorem 5.1 that Z x+he i converges to Z x in measure relative to P ⊗ λ. Moreover, for all t ∈ [0, T ], Y x+he i t converges to Y x t in probability. Since the partial derivatives l y and l z are continuous and bounded, dominated convergence Now let (h n ) be a sequence in R \ {0} converging to zero. Then, since R 2p (R 1 ) and L 2p (R d ) are Banach spaces, the sequence U hn converges to a process ∂ ∂x i Y x t , and V hn to a process ∂ ∂x i Z x t with respect to the corresponding norms. By convergence term by term for the difference quotient version of the quadratic BSDE and its formal derivative, which follows from our a priori estimates, we see that the pair Similarly to the first part one can show that lim is partially differentiable. The a priori estimates of Theorem 4.1 imply that the mapping x → (∇Y x t , ∇Z x t ) is continuous and hence, (Y x t , Z x t ) is totally differentiable. Since differentiability with respect to 2pth moments implies differentiability with respect to all inferior moments above 1, we have established the result.
As a byproduct of the previous proof we obtain that for every x ∈ R n there exists a solution (∇Y x t , ∇Z x t ) of the BSDE (4). We now proceed with the proof of Theorem 2.2, in which we claim pathwise continuous differentiability. To be consistent with the previous proof, we will again compare difference quotients varying in h. To this end we need the following estimates.
Lemma 6.1. Suppose (C3) is satisfied and that l and the derivatives of l are all Lipschitz continuous in (x, y, z). Then for all p > 1 there exists a constant C > 0, dependent only on p, T , M and D, such that for all x, x ′ ∈ R n , h, h ′ ∈ R and i ∈ {1, . . . , n}, Proof. This follows from Theorem 5.1, where we put l 1 (s, y, z) = l(s, x + he i , y, z), l 2 (s, y, z) = l(s, x ′ + h ′ e i , y, z).
The preceding Lemma immediately implies a first pathwise smoothness result in x for the process Y x . In fact, Kolmogorov's continuity criterion applies and yields a modification of Y x which is continuous in x. More precisely: Corollary 6.2. There exists a processŶ x such that for all (t, ω) ∈ [0, T ] × Ω, the function x →Ŷ x t (ω) is continuous, and for all (t, x) we haveŶ x t = Y x t almost surely.
Let e i be a unit vector in R n . For all x ∈ R n and h = 0, let U x,h . The proof of Theorem 2.2 will be based on the following result on the usual difference of difference quotients. Knowing a "good candidate" for the derivative from Theorem 2.1 we allow h = 0 this time, by replacing the difference quotient with this candidate.
Since O is bounded, (C3) implies that for every r > 1 there exists a constant C 1 such that for all (x, h) ∈ O we have E(sup t∈[0,T ] |ζ x,h t | 2r ) < C 1 . Now let s x,h , m x,h , A x,h , G x,h , I x,h and U x,h be defined as in the proof of Theorem 2.2, and denote A x,0 = ∂l ∂x (x, Y x , Z x ), G x,0 = ∂l ∂y (x, Y x , Z x ), etc. Then the estimate (29) will be deduced from the inequality E sup which follows from Theorem 4.1. We first analyze the order of the convergence of To this end notice that and, by applying Hölder's inequality we obtain with Lemma 6.1 Similarly, , and so we con- By using similar arguments we get Theorem 5.1 and the Lipschitz continuity of l imply Proof of Theorem 2.2. To simplify notation we may assume that (29) is satisfied for O = R n+1 . Assume that Y x t is continuous in x (see Corollary 6.2). Lemma 6.3 and Kolmogorov's continuity criterion imply that U x,h t has a modificationÛ x,h and note that we obtain thus a continuous version of the solution of the BSDE (4). For all (x, h) ∈ Q n+1 let N (x, h) be a null set such that for all ω / N (x, h) is a null set such that for all ω / ∈ N the following implication holds: If q k ∈ Q n and r k ∈ Q \ {0} are sequences with lim k→∞ q k = x ∈ R n and lim k→∞ r k = 0, then lim As a consequence of this and the subsequent Lemma 6.4, Y x t (ω) is continuously partially differentiable relative to x i if ω / ∈ N . Since we can choose such a null set for any i ∈ {1, . . . , n}, total differentiability follows and the proof is complete.
Lemma 6.4. Let f : R n → R be a continuous function and g : R n → R n a continuous vector field. Suppose that for all sequences q k ∈ Q n with q k → x ∈ R n and r k ∈ Q \ {0} with r k → 0 we have lim where 1 ≤ i ≤ n. Then f is differentiable and ∇f = g.

Proof.
To simplify notation assume that n = 1. Let x k ∈ R with x k → x ∈ R and h k ∈ R \ {0} with h k → 0. Since f is continuous we may choose q k ∈ Q and r k ∈ Q \ {0} such that and hence f is partially differentiable. Since the partial derivatives g i are continuous, f is also totally differentiable.

Differentiability of quadratic Forward-Backward SDEs
In this section we will specify the results obtained in the preceding sections to BSDEs where the terminal conditions are determined by a forward SDE driven by the same Brownian motion as the BSDE. When considering BSDEs with terminal condition determined by a forward SDE we will need regularity of the forward equation. This will be guaranteed if the coefficients are functions belonging to the following space. Throughout this section let again n be a positive integer and W a d-dimensional Brownian motion.
Definition 7.1. Let k, m ≥ 1. We denote by B k×m the set of all functions h : With any pair h ∈ B n×1 and σ ∈ B n×d we associate the second order differential operator We will consider Forward-Backward SDEs (FBSDEs) of the form where the coefficients satisfy the following assumptions: x, y, z) = l(ω, t, x, y, z) + α|z| 2 , where l(ω, t, x, y, z) is globally Lipschitz and continuously differentiable in (x, y, z), It follows from standard results on SDEs and from Theorem 2.3 in [Kob00] that there exists a solution (X x , Y x , Z x ) of Equation (31). As we will show, the results of Section 2 imply that (X x , Y x , Z x ) is differentiable in x and that the derivatives (∇X x , ∇Y x , ∇Z x ) solve the FBSDE (32) Our first result parallels Theorem 2.1 in which differentiability with respect to vector space topologies is treated.
Theorem 7.2. Let (D1) and (D2) be satisfied and assume that g : R n → R is bounded and differentiable. Moreover, suppose that ∂l ∂x (t, x, y, z) is Lipschitz continuous in x. Then for all , is differentiable, and the derivative is a solution of the BSDE (32).
Proof. By standard results, the mapping R n → R p (R 1 ), x → X x has a continuous version (which we assume being identical to the given one), and for all p > 1 there exists a constant C ∈ R + such that for x, See for example Lemma 4.5.4 and Lemma 4.5.6 in [Kun90]. In order to be able to apply Theorem 2.1, we need to verify Condition (C1). For this purpose, note that This proves (C1). Moreover, notice that ξ(x) = g(X x T ) satisfies Condition (C2). Thus the statement follows from Theorem 2.1.
If in addition Condition (D3) is satisfied, we again obtain a sharper result stating pathwise continuous differentiability of an appropriate modification of the solution process.
Let M > 0 be a constant such that g, the derivatives of g, b and σ, and the partial derivatives of l in (x, y, z) are all bounded by M . For all x ∈ R let (X x t , Y x t , Z x t ) be the solution of the FBSDE (31). To correspond formally to Theorem 2.2, in the setting of our FBSDE we have to work withl (ω, t, x, y, z) = l(ω, t, X x t (ω), y, z). But this functional fails to be globally Lipschitz in x. This is why we have to modify slightly the proof of Theorem 2.2, and cannot just quote it. We start by showing that ξ(x) = g(X x T ) satisfies Condition (C3).
. Then for every p > 1 there exists a C > 0, dependent only on p and M , such that for all x, x ′ ∈ R n and h, h ′ = 0, Moreover, for all t ∈ [0, T ], Proof. Note that by Ito's formula g(X ∂x i ∂x j . By (D3) we haveσ ∈ B 1×d andb ∈ B 1×1 . Therefore, by using standard results on stochastic flows (see Lemma 4.6.3 in [Kun90]), we obtain the result.
Proof of Theorem 7.3. First note that it is well-known that X x may be chosen to be continuous in t and continuously differentiable in x (see for example Theorem 39, Ch. V, [Pro04]). In order to prove that Y x has such a modification as well, note that Lemma 7.4 implies that ξ(x) = g(X x T ) satisfies Condition (C3). Now let again U x,h As in Lemma 6.3 we will derive this estimate from Inequality (30). Notice that the assumptions of Theorem 7.3 guarantee that all the terms appearing in (30), satisfy the same properties and thus provide the same estimates. There is one essential difference which is due to the appearance of X x instead of x in the first component of the line described by the integral In fact, with ∆ The first summand satisfies From this we can easily deduce Similarly,

Malliavin differentiability of quadratic BSDEs
In this section we shall ask for a different type of smoothness for solutions of quadratic BSDEs, namely differentiability in the variational sense or in the sense of Malliavin's calculus. Of course, this will imply smoothness of the terminal condition in the same sense. If the terminal condition is given by a smooth function of the terminal value of a forward equation, it will also involve variational smoothness of the forward equation.
Let us first review some basic facts about Malliavin calculus. We refer the reader to [Nua95] for a thorough treatment of the theory and to [KPQ97] for results related to BSDEs. To begin with, let C ∞ b (R n×d ) denote the set of functions with partial derivatives of all orders defined on R n×d whose partial derivatives are bounded.
Let S denote the space of random variables ξ of the form where F ∈ C ∞ b (R n×d ), h 1 , · · · , h n ∈ L 2 ([0, T ]; R d ). To simplify the notation assume that all h j are written as row vectors.
If ξ ∈ S of the above form, we define the d-dimensional operator D = (D 1 , · · · , D d ) : For ξ ∈ S and p > 1, we define the norm It can be shown (see for example [Nua95]) that the operator D has a closed extension to the space D 1,p , the closure of S with respect to the norm · 1,p . Observe that if ξ is F t −measurable then D θ ξ = 0 for θ ∈ (t, T ]. We shall also consider n-dimensional processes depending on a time variable. We define the space L a 1,p (R n ) to be the set of R n −valued progressively measurable processes u(t, ω) t∈[0,T ],ω∈Ω such that i) For a.a. t ∈ [0, T ], u(t, ·) ∈ (D 1,p ) n ; ii) (t, ω) → D θ u(t, ω) ∈ (L 2 ([0, T ])) d×n admits a progressively measurable version; iii) u a 1,p = E[ Here, for y ∈ R d×n we use the norm |y| 2 = i,j (y i,j ) 2 . We also consider the space D 1,∞ = ∩ p>1 D 1,p .
We cite for completeness a result from [Nua95] that we will use in the next section.
Lemma 8.1 (Lemma 1.2.3 in [Nua95]). Let {F n , n ≥ 1} be a sequence of random variables in D 1,2 that converges to F in L 2 (Ω) and such that Then F belongs to D 1,2 , and the sequence of derivatives {DF n , n ≥ 1} converges to DF in the weak topology of L 2 (Ω × [0, T ]).
Let us now consider the BSDE Our assumptions on driver and terminal condition this time amount to (E1) f : Ω × [0, T ] × R × R d → R is an adapted measurable function such that f (ω, t, y, z) = l(ω, t, y, z) + α|z| 2 , where l(ω, t, y, z) is globally Lipschitz and continuously differentiable in (y, z); for all p > 1 we have E P [( T 0 |l(ω, t, 0, 0)| 2 ds) 2p ] < ∞; (E2) for all (t, y, z), the mapping Ω → R, ω → l(ω, t, y, z) is Malliavin differentiable and belongs to L a 1,p (R) for all p > 1. For any (ω, t, y, z) and θ ∈ [0, T ], the (a.e. valid) inequality holds true |D θ l(ω, t, y, z))| ≤K θ (ω, t) + K θ (ω, t)(|y| + |z|) where K θ andK θ are positive adapted processes satisfying for all p ≥ 1 We first consider the case where the terminal variable has no further structural properties, such as depending on the terminal value of a forward equation. For notational simplicity we shall treat the case of one dimensional z and Wiener process and so may omit the superscript i in D i etc. We will this time use the typical Sobolev space approach, hidden in Lemma 8.1, to describe Malliavin derivatives, which are in fact derivatives in the distributional sense. In this approach we shall employ an approximation of the driver of our BSDE by a sequence of globally Lipschitz continuous ones, for which the properties we want to derive are known.
Let us therefore introduce a family of truncated functions starting with describing their derivatives by g ′ n (z) =    −2n , z < −n 2z , |z| ≤ n 2n , z > n.
Then we have g n (z) = z 2 for |z| ≤ n, g n (z) = 2n|z| − n 2 for |z| > n, and thus |g n (z)| ≤ z 2 and g n (z) → z 2 locally uniformly on R for n → ∞. A similar statement holds for the derivative of g n (z): |g ′ n (z)| ≤ 2|z| and g ′ n (z) → 2z locally uniformly on R for n → ∞. With these truncation functions we obtain the following family of BSDEs: [l(s, Y n s , Z n s ) + αg n (Z n s )]ds, n ∈ N * .
From Proposition 2.4 of [Kob00] we obtain that there exists (Y s , Z s ) ∈ R ∞ (R) × L 2 (R) such that Y n s → Y s uniformly in [0, T ] and Z n s → Z s in L 2 (R). Since the truncated equations have Lipschitz continuous drivers, Proposition 5.3 of [KPQ97] guarantees that (Y n t , Z n t ) ∈ D 1,2 × D 1,2 with the following Malliavin derivative D θ Y n t = 0 and D θ Z n t = 0, if t ∈ [0, θ), D θ Y n t = D θ ξ + T t ∂ y l(Y n s , Z n s )D θ Y n s + ∂ z l(Y n s , Z n s )D θ Z n s +D θ l(s, Y n s , Z n s ) + αg ′ n (Z n s )D θ Z n s ds − Now we aim at showing that the sequences DY n and DZ n are bounded in D 1,2 , in order to use Lemma 8.1. This will be done by deriving a priori estimates in the style of the preceding sections, this time uniform in n. We therefore first show boundedness relative to the auxiliary measures Q n := E α g ′ n (Z n )dW · P , in the form of the following a priori inequality.
Lemma 8.2. Let p > 1. If the driver and terminal condition satisfy hypotheses (E1), (E2) and (E3), then the following inequality holds for the BSDE (36): In a second step, we have to estimate the last term of the preceding equation. From Condition (E2) we obtain with another universal constant Our main result can now be proved.
This means Z ∈ L ∞ . This way we recover the Malliavin differentiability results of [HM06] from our main result. Remarks: 1. The methods of proof of this Section, building upon a truncated sequence of Lipschitz BSDEs, could also be used in the treatment of the differentiability problem in Section 6. This sequence would allow the use of the results in [KPQ97], which, combined with the a priori estimates of sections 3 and 4 would imply differentiability.
2. Our main results allow less restrictive hypotheses. For example in Section 3, we assume for our a priori estimations that ζ ∈ L p for all p ≥ 1. An analysis of the proof clearly reveals that to obtain estimates in R p or L p we only need that ζ ∈ L p for all p ∈ (2, 2pq 2 ]. We chose to write ζ ∈ L p for all p ≥ 1 not to produce an overload of technicalities in a technically already rather complex text.