Christel Geiss
email: christel.geiss@uibk.ac.at

Stefan Geiss
email: stefan.geiss@uibk.ac.at

Emmanuel Gobet
email: emmanuel.gobet@polytechnique.edu

Generalized fractional smoothness and L p -variation of BSDEs with non-Lipschitz terminal condition

Keywords:

We relate the Lp-variation, 2 ≤ p < ∞, of a solution of a backward stochastic differential equation with a path-dependent terminal condition to a generalized notion of fractional smoothness. This concept of fractional smoothness takes into account the quantitative propagation of singularities in time.

Introduction

During the last years the concept of fractional smoothness in the sense of function spaces has been used in the theory of stochastic processes to analyze approximation and variational properties. It turned out that phenomena known for special examples can be explained in terms of fractional smoothness. For example, approximation properties of certain stochastic integrals can be explained by the fractional smoothness of the integral itself, see [START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF][START_REF] Geiss | Interpolation and approximation in L 2 (γ)[END_REF]. Similarly, variational properties of backward stochastic differential equations (BSDEs) can be upper bounded in case that the fractional smoothness of the terminal condition is known. To explain the latter aspect consider the BSDE

Y t = ξ + T t f (s, X s , Y s , Z s )ds - T t Z s dW s
with a Lipschitz generator f , where X = (X t ) t∈[0,T ] is a forward diffusion, and define the L p -variation var p (ξ, f, τ ) := sup i=1,...,n

sup ti-1<s≤ti Y s -Y ti-1 p + n i=1 ti ti-1 Z t -Z ti-1 2 p dt 1 2 where τ = (t i ) n i=0 is a deterministic time-net 0 = t 0 < • • • < t n = T , Z ti-1 := 1 t i -t i-1 ❊ ti ti-1 Z s ds|F ti-1 ,
and where 2 ≤ p < ∞, which we will assume throughout this paper. Note that by interchanging the L p -and L 2 -norms (where we use p ≥ 2) and using the Burkholder-Davis-Gundy inequality, the L p -distance between the stochastic integral T 0 Z s dW s and its discrete counterpart n i=1 Z ti-1 (W ti -W ti-1 ) is upper bounded by a multiple of n i=1 ti ti-1 Z t -Z ti-1 2 p dt 1 2 . Hence the quantity var p (ξ, f, τ ) is stronger compared to what is needed to quantify the discretization of the stochastic integral term of our BSDE. Besides the fact that this variation gives a strong insight into the quantitative behavior of the BSDE, in particular var 2 (ξ, f, τ ) was used to describe the error in adapted backward Euler schemes for ξ = g(X T ) with g being a Lipschitz function; see [START_REF] Bouchard | Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations[END_REF][START_REF] Zhang | A numerical scheme for BSDEs[END_REF] for implicit schemes and [START_REF] Gobet | A regression-based Monte Carlo method to solve backward stochastic differential equations[END_REF][START_REF] Gobet | Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations[END_REF] for explicit schemes possibly with jump processes. In [14, Theorems 3.1 and 3.2] upper bounds for

n i=1 ti ti-1 Z t -Z ti-1 2 2 dt
were obtained for ξ = g(X T ) satisfying ❊|g(X T ) -❊(g(X T )|F t )| 2 ≤ c 2 (Tt) θ for some 0 < θ ≤ 1, where g is not assumed to be a Lipschitz function. On the other hand, path-dependent settings without taking into account fractional smoothness were considered, for example, in [START_REF] Hu | Nonlinear Feynman-Kac formula and discrete-functionaltype BSDEs with continuous coefficients[END_REF][START_REF] Ma | Representation theorems for backward stochastic differential equations[END_REF][START_REF] Zhang | Discretizing a backward stochastic differential equation[END_REF] In this paper, results are generalized and extended into the following directions:

• We consider a path-dependent setting by terminal conditions of the form ξ = g(X r1 , ..., X r L ) with 0 = r 0 < • • • < r L = T , where g is not necessarily a Lipschitz function and introduce a corresponding path-dependent fractional smoothness in the Malliavin sense. This concept of smoothness extends the classical concepts, based on real interpolation, to a time-dependent one taking care about the propagation of smoothness in time. In the classical case one would assign to a random variable ξ some 0 < θ ≤ 1 which describes the fractional smoothness of ξ while here we assign to the parameters (ξ, f ) of our BSDE a vector Θ = (θ 1 , ..., θ L ), where θ l stands for the local smoothness of the BSDE at time r l . It turns out that this vector is completely characterized by the L p -variation of Y and Z. In case our terminal condition depends on X T only our generalized smoothness coincides with earlier approaches from, for example, [START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF] and [START_REF] Gobet | L 2 -time regularity of BSDEs with irregular terminal functions[END_REF].

• Instead of the L 2 -variation we consider the stronger L p -variation with 2 ≤ p < ∞. In addition, the integrated Z-variation

n i=1 ti ti-1 Z t -Z ti-1 2
2 dt is replaced by the variation Z s -Z t p with s and t being fixed, and the L p -variation of the process Y is included as well. To our knowledge the weaker criterion for 0 < p < 2 in the context of this paper has not been considered yet and might require different arguments as some of our proofs rely on the condition that p ≥ 2.

• We provide equivalences showing that the results are sharp.

• In Corollary 2.4 we show, given the terminal condition ξ = g(X r1 , ..., X r L ) has a certain fractional smoothness, how to obtain time-nets τ n of cardinality Ln + 1 such that sup n √ n var p (ξ, f, τ n ) < ∞.

These time-nets compensate the possible singularities of the Z-process when approaching a time-point r l from the left.

Organization of the paper. After introducing the setting in Section 1, we formulate in Section 1.2 our concept of functional fractional smoothness of a BSDE and characterize this smoothness in various ways. Here we partly transfer the results from [START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF] and [START_REF] Gobet | L 2 -time regularity of BSDEs with irregular terminal functions[END_REF] from the case ξ = g(X T ) to the path-dependent one. In Section 2.2 we present two sufficient conditions for our fractional smoothness. The point of these two conditions (Corollary 2.6 and Theorem 2.10) is that they only involve the terminal condition ξ and do not use the solution Y nor the generator f of our BSDE. The proofs of the main results are contained in Section 3. Finally, B(η 1 , η 2 ) := 1 0 x η1-1 (1x) η2-1 dx where η 1 , η 2 > 0, will denote the Beta-function.

Setting and basic concepts 1.Forward-backward stochastic differential equations

We fix a complete probability space (Ω, F, P), T > 0, d ≥ 1 and a d-dimensional standard Brownian motion W = (W t ) t∈[0,T ] with W 0 ≡ 0. Furthermore, we assume that (F t ) t∈[0,T ] is the augmentation of the natural filtration of W .

The forward equation. Let (A b,σ ) We have b, σ ∈ C 0,2 b ([0, T ] × | d ), where the derivatives up to order two are taken with respect to the space-variables and, for some γ ∈ (0, 1], are assumed to be γ-Hölder continuous (w.r.t. the parabolic metric) on all compact subsets of [0, T ] × | d . Moreover, there is a δ > 0 such that Ax, x ≥ δ|x| 2 for x ∈ | d and b and σ are 1 2 -Hölder continuous in time, uniformly in space.

We work with the usual stochastic flow (X t,x s ) s,t∈[0,T ],x∈| d that solves for (t, x) ∈ [0, T ] × | d the SDE X s = x on [0, t] and dX t,x s = σ(s, X t,x s )dW t s + b(s, X t,x s )ds on [t, T ], where W t s := W s -W t and the augmented natural filtration (

F t s ) s∈[t,T ] of (W t s ) s∈[t,T ] is used (i.p. X = X 0,x0
). With our assumptions we can assume that (X t,x s ) s,t∈[0,T ],x∈| d is a continuous process in (s, t, x).

If g : | d → | is a polynomially bounded Borel function, 0 < R ≤ T , and
F (t, x) := ❊g(X t,x R ) for 0 ≤ t ≤ R, (1)
then

F ∈ C 1,2 ([0, R) × | d ) and ∂ ∂t F (t, x) + 1 2 A(t, x), D 2 F (t, x) + b(t, x), ∇ x F (t, x) = 0
by Proposition B.1 below where

D 2 := ∂ 2 ∂x i ∂x j d i,j=1
∇ x F (r, X t,x r ) = ❊ g(X t,x R )N r,1,(t,x) R |F t r , D 2 F (r, X t,x r ) = ❊ g(X t,x R )N r,2,(t,x) R |F t r for the Malliavin weights N r,i,(t,x) R
that satisfy, for any given 0 < q < ∞, that

❊ N r,i,(t,x) R q |F t r 1 q ≤ κ q (R -r) i 2 a.s. and ❊ N r,i,(t,x) R |F t r = 0 a.s.
for i = 1, 2 and all 0 ≤ t ≤ r < R ≤ T with a constant κ q > 0 independent from (t, r, R, x) (see [START_REF] Gobet | Sensitivity analysis using Itô-Malliavin calculus and martingales, and application to stochastic optimal control[END_REF], [14, Proof of Lemma 1.1] and Remark B.2 below). A typical application of these estimates are the crucial inequalities

∇ x F (r, X t,x r ) p ≤ κ p ′ g(X t,x R ) -❊(g(X t,x R )|F t r ) p √ R -r , (2)
D 2 F (r, X t,x r ) p ≤ κ p ′ g(X t,x R ) -❊(g(X t,x R )|F t r ) p R -r , (3)
for 1 < p, p ′ < ∞ with 1 = (1/p) + (1/p ′ ).
The backward equation. We are interested in the backward equation

Y t = ξ + T t f (s, X s , Y s , Z s )ds - T t Z s dW s for t ∈ [0, T ] a.s.
and assume the following conditions:

(A f ) The function f : [0, T ] × | d × | × | d → | is continuous in (t, x, y, z)
and continuously differentiable in x, y and z with uniformly bounded derivatives. In particular, there are

K f > 0 and L f > 0 such that |f (s, x 1 , y 1 , z 1 ) -f (s, x 2 , y 2 , z 2 )| ≤ L f [|x 1 -x 2 | + |y 1 -y 2 | + |z 1 -z 2 |], |f (s, x, y, z)| ≤ K f + L f (|x| + |y| + |z|).
(A g ) There are R = {r 0 , ..., r L } with 0 = r 0 < r 1 < • • • < r L = T and a measurable function of at most polynomial growth g :

(| d ) L → | such that ξ := g(X r1 , ..., X r L ).
In this setting, the solution (Y, Z) to the above BSDE is uniquely defined in any L p -space for 1 < p < ∞; see [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF]Theorem 4.2]. Additionally, we assume in the paper that the solution (Y, Z) is realized such that, on [r l-1 , r l ),

Y t = u l (X l-1 ; t, X t ) and Z t = v l (X l-1 ; t, X t )σ(t, X t ),
where we set X l-1 := (X r1 , . . . , X r l-1 ). The above functions u l and v l are well defined due to the next proposition, which is an extension of [24, Theorem 3.2] and follows from Lemma A.2, see also [START_REF] Hu | Nonlinear Feynman-Kac formula and discrete-functionaltype BSDEs with continuous coefficients[END_REF].

Proposition 1.1. Assume that (A b,σ ), (A f ) and (A g ) are satisfied. Then, for l = 1, ..., L there exist measurable

u l : (| d ) l-1 × [r l-1 , r l ) × | d → | and v l : (| d ) l-1 × [r l-1 , r l ) × | d → | 1×d and Borel sets D l ⊆ | d(l-1) , l = 2, ..., L,
such that D c l is of Lebesgue measure zero, and such that

(i) u l (x l-1 ; •, •) : [r l-1 , r l ) × | d → | is continuous and continuously differen- tiable w.r.t. the space variable with ∇ x u l (x l-1 ; t, x) = v l (x l-1 ; t, x), where x l-1 = (x 1 , . . . , x l-1 ), (ii) there are α l , q l,1 , ..., q l,l ∈ [1, ∞) such that sup t∈[r l-1 ,r l ) |u l (x l-1 ; t, x)| + sup t∈[r l-1 ,r l ) √ r l -t|v l (x l-1 ; t, x)| ≤ α l (1 + |x 1 | q l,1 + • • • + |x l-1 | q l,l-1 + |x| q l,l ),
(iii) for all l = 1, ..., L, x 1 , ..., x l-1 , x ∈ | d and r l-1 ≤ s < r l the triplet

X s,x t , u l (x l-1 ; t, X s,x t ), v l (x l-1 ; t, X s,x t )σ(t, X s,x t ) t∈[s,r l )
solves the BSDE with generator f and terminal condition u l (x l-1 ; r l , X s,x r l ) where

u l (x l-1 ; r l , x) := u l+1 (x l-1 , x; r l , x)χ D l (x l-1 ) : 2 ≤ l < L, g(x l-1 , x)χ D l (x l-1 ) : l = L.
and u 1 (r 1 , x) := u 2 (x; r 1 , x).

In the above proposition we used the convention that h(x 0 ; •) := h(•). It should be noted that by Proposition 1.1 we modify at each level l = 2, ..., L the functional for the Y -process on a nullset. However, because of

P(X r1 ∈ D 2 , ..., (X r1 , ..., X r L-1 ) ∈ D L ) = 1, (4)
this does not affect the L p -solution of our BSDE so that Proposition 1.1 is sufficient for our purpose.

Piece-wise linearization of the backward equation. We let F l (x l-1 ; •, •) :

[r l-1 , r l ] × | d → | be given by F l (x 1 , ..., x l-1 ; t, x) = F l (x l-1 ; t, x) := ❊u l (x 1 , ..., x r l-1 ; r l , X t,x r l ).

The function F l solves the backward PDE

∂F l ∂t (x l-1 ; t, x) + 1 2 A(t, x), D 2 F l (x l-1 ; t, x) + b, ∇ x F l (x l-1 ; t, x) = 0 on the interval [r l-1 , r l ) for fixed x 1 , ..., x l-1 ∈ | d .
Two facts that are frequently used in the paper. Firstly, for a filtered prob-

ability space (M, Σ, ◗, (G t ) t∈[r,R] ), 1 ≤ q ≤ ∞, r ≤ t ≤ R and ξ ∈ L q , one has that ξ -❊(ξ|G t ) q ≤ sup t≤s≤R ξ -❊(ξ|G s ) q ≤ 2 ξ -❊(ξ|G t ) q (5)
as a consequence that ❊(•|F s ) is a contraction on L q . Secondly, given the assumptions on our forward diffusion, a polynomially bounded Borel function

g : | d → |, r ≤ t ≤ R ≤ T and 1 ≤ q < ∞, we have that g(X r,x R ) -❊(g(X r,x R )|F r t ) q ≤ | d | d | d
|g(ξ)g(η)| q Γ(r, x; t, y)Γ(t, y; R, ξ)Γ(t, y; R, η)dydξdη

1 q ≤ 2 g(X r,x R ) -❊(g(X r,x R )|F r t ) q . (6)

Functional fractional smoothness

The usage of fractional smoothness in the investigation of variational properties of BSDEs is the central idea of this paper. Fractional smoothness can be defined in various ways. One way is the so-called K-method, a method where functions are decomposed into differentiable parts and parts that are not differentiable.

A quantitative analysis of these decompositions leads to fractional smoothness.

To be more precise, assume two Banach spaces X 0 and X 1 , where (say) X 1 is continuously embedded into X 0 , 0 < t < ∞ and x ∈ X 0 , and recall that the K-functional is given by

K(x, t; X 0 , X 1 ) := inf{ x 0 X0 + t x 1 X1 : x = x 0 + x 1 }.
For 0 < θ < 1 and 1 ≤ q ≤ ∞ this leads to the real interpolation spaces

x (X0,X1) θ,q := t -θ K(x, t; X 0 , X 1 ) Lq((0,∞), dt t ) with X 1 ⊆ (X 0 , X 1 ) θ1,q ′ 1 ⊆ (X 0 , X 1 ) θ1,q1 ⊆ (X 0 , X 1 ) θ0,q0 ⊆ X 0
where 0 < θ 0 < θ 1 < 1 and 1 ≤ q 0 , q 1 , q ′ 1 ≤ ∞ with q ′ 1 ≤ q 1 (see [START_REF] Bennet | Interpolation of operators[END_REF][START_REF] Bergh | Interpolation spaces. An introduction[END_REF]). Applying this concept to the Malliavin Sobolev space D 1,p , we obtain the Malliavin Besov (or fractional Sobolev) spaces

B θ p,q := (L p , D 1,p ) θ,q (7)
where 0 < θ < 1 is the main parameter of the smoothness and 1 < q ≤ ∞ the fine-tuning parameter. In a context close to this paper these spaces and related ones have been exploited for example in [START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF][START_REF] Geiss | Interpolation and approximation in L 2 (γ)[END_REF][START_REF] Gobet | L 2 -time regularity of BSDEs with irregular terminal functions[END_REF][START_REF] Toivola | On fractional smoothness and approximations of stochastic integrals[END_REF]. The classical setting of the Wiener space is changed in [START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF][START_REF] Gobet | L 2 -time regularity of BSDEs with irregular terminal functions[END_REF] into a setting where the standard Gaussian measure is replaced by the distribution of the forward diffusion. Here we go one step ahead and replace 0 < θ < 1 by a vector Θ = (θ 1 , ...., θ L ), where θ l describes the smoothness at time r l :

Definition 1.2. Let Θ = (θ 1 , ..., θ L ) ∈ (0, 1] L , 2 ≤ p < ∞ and ξ ∈ L p .
If Y is the solution of the BSDE with generator f and terminal condition ξ, then we let (ξ, f ) ∈ B Θ p,∞ (X) provided that there is some c > 0 such that

Y r l -❊(Y r l |F s ) p ≤ c(r l -s) θ l 2
for all l = 1, ..., L and r l-1 ≤ s < r l . The infimum over all possible c > 0 is denoted by

c B Θ p,∞ = c B Θ p,∞ (ξ, f
). In the case that f = 0 we will simply write ξ ∈ B Θ p,∞ (X).

Specializing to p = 2 and to the linear one-step Gaussian case (X = W , T = L = 1 and f = 0) it holds (see [START_REF] Geiss | Interpolation and approximation in L 2 (γ)[END_REF]Corollary 2.3]) that

g(W 1 ) ∈ B (θ) 2,∞ (W ) if and only if g ∈ B θ 2,∞ (| d , γ d ),
where the Wiener space over the standard Gaussian measure γ d on | d is considered. In particular, for d = 1 and for the orthonormal basis consisting of Hermite polynomials

(h k ) ∞ k=0 ⊆ L 2 (|, γ 1 ) we obtain that g = ∞ k=0 α k h k ∈ B θ 2,∞ (|, γ 1
) if and only if there is some c > 0 such that for all 0 ≤ t < 1 one has that

∞ k=1 kt k-1 α 2 k ≤ c 2 (1 -t) 1-θ ,
see [START_REF] Geiss | Interpolation and approximation in L 2 (γ)[END_REF]Theorem 2.2]. These connections explain the notation (p, ∞) in Definition 1.2. For a more general connection between the speed of convergence of the conditional expectations used in Definition 1.2 and the real interpolation method the reader is referred to [START_REF] Geiss | Interpolation and approximation in L 2 (γ)[END_REF]. Our definition of fractional smoothness by an upper bound of Y r l -❊(Y r l |F s ) p has the advantage that (2) and (3) give the upper bounds

∇ x F l (X l-1 ; s, X s ) p ≤ κ p ′ Y r l -❊(Y r l |F s ) p √ r l -s ≤ κ p ′ c B Θ p,∞ (r l -s) θ l -1 2 and D 2 F l (X l-1 ; s, X s ) p ≤ κ p ′ Y r l -❊(Y r l |F s ) p r l -s ≤ κ p ′ c B Θ p,∞ (r l -s) θ l -2 2
for r l-1 ≤ s < r l and 1 = (1/p) + (1/p ′ ), so that we can control the gradient and the Hessian of F l . For our paper the fine-tuning parameter q = ∞ in (the generalization of) ( 7) turns out to be the right one.

Finally, we want to mention the coincidence, that most of the relevant examples are naturally linked to this fine-tuning parameter q = ∞ in (7).

Time-nets, splines and entropy numbers

In our BSDE system the Z-process gets possibly singular at any of the particular time points r l when r l is approached from the left. The degree of this singularity is determined by the parameter θ l describing the fractional smoothness in r l . To keep the variation var p (g(X r1 , ..., X r L ), f, τ ) small, we have to choose time-nets which refine on the left of r l with an order given by the fractional smoothness θ l while each of the intervals [r l-1 , r l ] is divided into n sub-intervals.

Definition 1.3. For Θ ∈ (0, 1] L we let τ n,Θ = (t n,Θ k ) nL k=0 be given by t n,Θ 0 := 0 and

t n,Θ k := r l-1 + (r l -r l-1 ) 1 -1 - k -(l -1)n n 1 θ l for (l -1)n < k ≤ ln.
Estimates on the L p -variation Y t -Y s p are close to estimates how good the process Y can be approximated in L p by linear adapted splines, i.e. we simply compute adapted approximations of Y at the time-points t 0 , ..., t n and interpolate them linearly. So the notion adapted spline refers to the fact that the knots are adapted, however the spline itself is not an adapted process. The adapted splines are typically used in complexity theory for stochastic processes to find efficient approximation schemes for stochastic processes where the whole path needs to be approximated but the adaptedness of the approximation is not fully needed, see [START_REF] Creutzig | Free-knot spline approximation of stochastic processes[END_REF]. Here we use the following notation:

Definition 1.4. Given a time-net τ = (t k ) n k=0 with r = t 0 < • • • < t n =
R ≤ T we say that the process S = (S t ) t∈[r,R] is an adapted spline based on τ provided that S t k is F t k -measurable for all k = 0, ..., n and

S t := t k -t t k -t k-1 S t k-1 + t -t t k-1 t k -t k-1 S t k for t k-1 ≤ t ≤ t k .
Finally, we recall the notion of entropy numbers to measure and compare compactness properties of Y = (Y s ) s∈[t,r l ] as t ↑ r l where the process gets singular.

Definition 1.5. Given a normed space E and A ⊆ E we define e n (A|E) := inf ε, where the infimum is taken over all ε > 0 such that there are x 1 , ..., x n ∈ E with

A ⊆ n i=1 {x i + εB E } with B E := {x ∈ E : x ≤ 1}.
2 Functional fractional smoothness and BSDEs

A general equivalence

The basic result of this paper is Theorem 2.1. Assume that (A b,σ ), (A f ) and (A g ) are satisfied. For 2 ≤ p < ∞ and fixed l ∈ {1, ..., L} and θ l ∈ (0, 1] consider the following conditions:

(C1 l ) There is some c 1 > 0 such that, for r l-1 ≤ s < t < r l , Z t -Z s p ≤ c 1 t s (r l -r) θ l -2 dr 1 2
.

(C2 l ) There is some c 2 > 0 with Z t p ≤ c 2 (r l -t) θ l -1 2 for r l-1 ≤ t < r l .
(C3 l ) There is some c 3 > 0 such that, for r l-1 ≤ s < t ≤ r l ,

Y t -Y s p ≤ c 3 t s (r l -r) θ l -1 dr 1 2
.

(C4 l ) There is some c 4 > 0 such that, for r l-1 ≤ s < r l ,

Y r l -❊(Y r l |F s ) p ≤ c 4 (r l -s) θ l 2 .
(C5 l ) There is some c 5 > 0 such that, for r l-1 ≤ t < r l ,

t r l-1 |(D 2 F l )(X l-1 ; s, X s )| 2 ds 1 2 p ≤ c 5 (r l -t) θ l -1 2
.

(C6 l ) There is some c 6 > 0 such that for all n = 1, 2, ... there is an adapted spline S n = (S n t ) t∈[r l-1 ,r l ] based on

r l-1 + (r l -r l-1 ) 1 -1 - k n 1 θ l n k=0 such that √ n sup t∈[r l-1 ,r l ] Y t -S n t p ≤ c 6 .
The spline can be arranged such that S n r l-1 = Y r l-1 and S n r l = Y r l .

(C7 l ) There is some c 7 > 0 such that for r l-1 ≤ t < r l one has that

sup n≥1 √ ne n (Y s ) s∈[t,r l ] |L p ≤ c 7 (r l -t) θ l 2 .
Then one has that

(C1 l ) θ l ∈(0,1) =⇒ (C2 l ) ⇐⇒ (C3 l ) ⇐⇒ (C4 l ) ⇐⇒ (C5 l ) ⇐⇒ (C6 l ) ⇐⇒ (C7 l ) =⇒ (C1 l ).
Remark 2.2. The implication (C1 l ) =⇒ (C2 l ) does not hold in general. To see this we consider d = T = L = l = 1, f = 0, θ 1 = 1 and p = 2, and let

g = ∞ n=0 α n h n with ∞ n=0 α 2 n < ∞ where (h n ) ∞ n=0 ⊆ L 2 (|, γ 1 )
is the orthonormal basis of Hermite polynomials. Then, as in [START_REF] Geiss | Interpolation and approximation in L 2 (γ)[END_REF]Lemma 3.9], we get that

∂ 2 F 1 ∂x 2 (t, W t ) 2 2 = ∞ n=0 α 2 n+2 (n + 2)(n + 1)t n and Z t -Z s 2 2 = t s ∞ n=0 α 2 n+2 (n + 2)(n + 1)r n dr.
Choosing α n := (n(n -1)) -1/2 for n ≥ 2 and α 0 = α 1 = 0 gives (C1 l ) but sup 0≤t<1 Z t 2 = ∞.

From Theorem 2.1 the multi-step case directly follows. For its formulation we introduce for Θ = (θ 1 , ..., θ L ) ∈ (0, 1] L and 0 ≤ t < T the function

ϕ(t) := L l=1 χ [r l-1 ,r l ) (t)(r l -t) θ l -1 2 . Theorem 2.3. Assume that (A b,σ ), (A f ) and (A g ) are satisfied. For 2 ≤ p < ∞ and Θ ∈ (0, 1] L consider

the following conditions:

(C1) There is some

c 1 > 0 such that, for r l-1 ≤ s < t < r l , Z t -Z s p ≤ c 1 t s ϕ(r) 2 r l -r dr 1 2
.

(C2) There is some c 2 > 0 with Z t p ≤ c 2 ϕ(t) for 0 ≤ t < T .

(C3) There is some

c 3 > 0 such that, for r l-1 ≤ s < t ≤ r l , Y t -Y s p ≤ c 3 t s ϕ(r) 2 dr 1 2 . (C4) (ξ, f ) ∈ B Θ p,∞ (X).
(C6) There is some c 6 > 0 such that for all n = 1, 2, ... there is an adapted spline

S n = (S n t ) t∈[0,T ] based on τ n,Θ such that √ n sup t∈[0,T ] Y t -S n t p ≤ c 6 .
Then one has that

(C1) Θ∈(0,1) L =⇒ (C2) ⇐⇒ (C3) ⇐⇒ (C4) ⇐⇒ (C6) =⇒ (C1).
The remaining properties (C5 l ) and C7 l ) could be included as well. By using the properties (C3) and (C1) we deduce by a simple computation

Corollary 2.4. For 0 < θ ′ l < θ l < 1, l = 1, ..., L and (ξ, f ) ∈ B Θ p,∞ (X) one has that sup n √ n var p (ξ, f, τ n,Θ ′ ) < ∞.
Examples will be considered in Example 2.9 and Theorem 2.10. The proof of Theorem 2.1 is postponed to Section 3.1.

Sufficient conditions for fractional smoothness

In this section we describe sufficient conditions on ξ for the condition (ξ, f ) ∈ B Θ p,∞ (X) which are independent from the generator f . Note that in the case L = 1 it follows by definition that (ξ, 0)

∈ B Θ p,∞ (X) implies that (ξ, f ) ∈ B Θ p,∞ (
X). To our knowledge it is open whether it still holds for L > 1.

The first sufficient condition

The first sufficient condition is based on the concept to measure the fractional smoothness of a random variable on the Wiener space by mixing the underlying Gaussian structure with an independent copy and to look how sensitive the given random variable is with respect to this operation (see, for example, [START_REF] Hirsch | Lipschitz functions and fractional Sobolev spaces[END_REF]). In our setting this would correspond to comparing, for example, g(X 1 ) with g(X η 1 ) where X η 1 is defined via a Brownian motion W η t := 1η 2 W t + ηB t with B being a Brownian motion independent from W and 0 ≤ η ≤ 1. Because we have a time-dependent structure we extend this concept by allowing more general operations with W and its independent copy B.

W η t := t 0 1 -η(s) 2 dW s + t 0 η(s)dB s
and denote by (F η t ) t∈[0,T ] the augmentation of its natural filtration. We also define X η to be the strong (F η t ) t∈[0,T ] -measurable solution of

X η t = x 0 + t 0 b(s, X η s )ds + t 0 σ(s, X η s )dW η s .
For a given

F η T -measurable terminal condition ξ η ∈ L p with 2 ≤ p < ∞ we let (Y η , Z η ) be the L p -solution in the filtration (F η t ) t∈[0,T ] of Y η t = ξ η + T t f (s, X η s , Y η s , Z η s )ds - T t Z η s dW η s .
In the case η ≡ 0 we simply write

W = W 0 , ξ = ξ 0 , (X, Y, Z) = (X 0 , Y 0 , Z 0 ), and
F t = F 0 t .
Our aim is to bound the distance between (X η , Y η , Z η ) and (X, Y, Z) by the following stability result: Theorem 2.5. Assume that (A b,σ ) and (A f ) are satisfied. Then for 2 ≤ p < ∞ and ξ, ξ η ∈ L p we have that

sup 0≤t≤T |X η t -X t | p + sup 0≤t≤T |Y η t -Y t | p + T 0 |Z η t -Z t | 2 dt 1/2 p ≤ c ξ η -ξ p + [1 + ξ p ] T 0 η(t) 2 dt
where c > 0 depends at most on (p, T, b, σ, K f , L f ) and is non-decreasing with respect to K f and L f .

The proof can be found in Section 3.2. The motivation for the result is Corollary 2.6 below. To formulate it, given 0 ≤ t < r ≤ T we let

η t,r (s) := χ (t,r] (s),
i.e. we replace the Brownian paths on (t, r] by an independent copy.

Corollary 2.6. Assume 2 ≤ p < ∞, (A b,σ ), (A f ) and ξ = g(X r1 , ..., X r L ) ∈ L p for some Borel measurable function g : | L → |. Let ξ t,r := g(X ηt,r r1 , ..., X ηt,r r L ) for 0 ≤ t < r ≤ T and let Θ = (θ 1 , ..., θ L ) ∈ (0, 1] L . If there is a constant c > 0 such that one has that ξ -ξ t,r l p ≤ c(r l -t) θ l 2 (8)
for all l = 1, ..., L and r l-

1 ≤ t < r l , then (ξ, f ) ∈ B Θ p,∞ (X). Proof. For r l-1 ≤ t < r l we get by (6) that Y r l -❊(Y r l |F t ) p ≤ Y r l -Y ηt,r l r l p ≤ c (2.5)   ξ -ξ t,r l p + [1 + ξ p ] T 0 η t,r l (r) 2 dr   ≤ c (2.5) c(r l -t) θ l 2 + [1 + ξ p ] √ r l -t .
Using a truncation argument, we obtain a modified version of Theorem 2.3, without assuming that g is polynomially bounded nor that f is continuously differentiable in (x, y, z).

Corollary 2.7. Assume (A b,σ ) and that the generator f :

[0, T ]×| d ×|×| d →
| is continuous in (t, x, y, z) and that there is some

L f > 0 such that |f (s, x 1 , y 1 , z 1 ) -f (s, x 2 , y 2 , z 2 )| ≤ L f [|x 1 -x 2 | + |y 1 -y 2 | + |z 1 -z 2 |]. Let 2 ≤ p < ∞, ξ = g(X r1 , ..., X r L ) ∈ L p for some Borel measurable function g : | L → |, Θ ∈ (0, 1]
L and let (Y, Z) be the L p -solution of our BSDE. Assume that condition (8) is satisfied. Then there are sets N l ⊆ [r l-1 , r l ) of Lebesgue measure zero such that the following is satisfied:

(C1') There is some c 1 > 0 such that for s, t ∈ [r l-1 , r l ) \ N l with r l-1 ≤ s < t < r l one has Z t -Z s p ≤ c 1 t s ϕ(t) 2 r l -r dr 1 2 . (C2') There is some c 2 > 0 with Z t p ≤ c 2 ϕ(t) for t ∈ L l=1 ([r l-1 , r l ) \ N l ). (C3') There is some c 3 > 0 such that, for r l-1 ≤ s < t ≤ r l , one has Y t -Y s p ≤ c 3 t s ϕ(r) 2 dr 1 2 . Proof. (a) Let (f N ) N ≥1 be a sequence of generators satisfying assumption (A f ) such that (i) lim N T 0 |f N (s, X s , Y s , Z s ) -f (s, X s , Y s , Z s )|ds p = 0, (ii) K f N ≤ 2K f and L f N ≤ L f . (b) Letting y N = -N ∨ y ∧ N for y ∈ | and N ≥ 1, ξ N satisfies (A g ) and ξ N -ξ p → 0 as N → ∞.
In addition, for all l = 1, ..., L and r l-1 ≤ t < r l we have

ξ N -(ξ N ) t,r l p = ξ N -(ξ t,r l ) N p ≤ ξ -ξ t,r l p ≤ c (8) (r l -t) θ l 2 .
(c) To (ξ N , f N ) we associate (Y N , Z N ) as BSDE solution in L p . In view of the inequality above and according to Corollary 2.6, (ξ

N , f N ) ∈ B Θ p,∞ (X). Because K f N , L f N and ξ N p are bounded independently of N , we have sup N ≥1 c B Θ p,∞ (ξ N , f N ) < ∞,
which follows by the proof of Corollary 2.6. Theorem 2.3 applies to (Y N , Z N ) for each N and there are c N > 0 such that

Z N,t -Z N,s p ≤ c N t s ϕ(t) 2 r l -r dr 1 2
for r l-1 ≤ s < t < r l . Looking at the constants in the proof of (C4 l ) ⇒ (C1 l ) we realize that we can take sup N c N =: c < ∞. By Lemma A.1 applied to ξ (0) = ξ, f 0 (ω, s, y, z) := f (s, X s (ω), y, z), (Y (0) , Z (0) ) = (Y, Z), and ξ (1) = ξ N , f 1 (ω, s, y, z) := f N (s, X s (ω), y, z), (Y (1) , Z (1) ) = (Y N , Z N ), there is a subsequence (N k ) ∞ k=1 such that Z N k ,t converges to Z t a.s. for t ∈ [r l-1 , r l ) \ N l for some N l of Lebesgue measure zero. Fatou's lemma gives

Z t -Z s p ≤ c t s ϕ(t) 2 r l -r dr 1 2 for r l-1 ≤ s < t < r l with s, t ∈ [r l-1 , r l ) \ N l . As in the proof of (C1 l ) =⇒ (C2 l ) =⇒ (C3 l ) below we can deduce (C2 ′ ) and C3 ′ ) where in the case r l-1 ∈ N l in (C1 l ) =⇒ (C2 l ) we have to replace Z r l-1 p by lim inf n Z ρn p with ρ n ∈ [r l-1 , r l ) \ N l and ρ n ↓ r l-1 Definition 2.8. A measurable function g : | → | is of bounded variation, in short g ∈ BV, provided that g BV := sup N sup -∞<x0<•••<x N <∞ N k=1 |g(x k ) -g(x k-1 )| < ∞.
The following Example 2.9 is more general than needed in this paper, however this generality does not require any extra effort and constitutes the natural setting.

Example 2.9. Assume 0 < θ < 1 p ≤ α ≤ 1, g j ∈ BV with ∞ j=1 g j α BV < ∞, and linear and continuous functionals µ 1 , µ 2 , ... ∈ (C[0, T ]) * with µ j ≤ 1 such that the laws of X, µ 1 , X, µ 2 , X, µ 3 , ... have densities bounded uniformly by a constant β > 0. Define

ξ := Φ(g 1 ( X, µ 1 ), g 2 ( X, µ 2 ), ...),
where Φ is a measurable function such that

|Φ(x 1 , x 2 , ...) -Φ(y 1 , y 2 , ...)| ≤ κ ∞ j=1 |x j -y j | α
for some κ > 0. Then there is a constant c > 0 such that for all measurable

η : [0, T ] → [-1, 1] we have that ξ -ξ η p ≤ c T 0 η(r) 2 dr θ 2 . Consequently, given Θ ∈ (0, 1/p) L there is a constant c ′ > 0 such that ξ -ξ t,r l p ≤ c ′ (r l -t) θ l 2 for r l-1 ≤ t < r l . Proof. Using [1, Theorem 2.4] for 1 ≤ q < ∞ we get that ξ -ξ η p ≤ κ ∞ j=1 |g j ( X, µ j ) -g j ( X η , µ j )| α p ≤ κ ∞ j=1 g j ( X, µ j ) -g j ( X η , µ j ) α αp ≤ κ3 α+ 1 p β q q+1 1 p ∞ j=1 g j α BV X, µ j -X η , µ j q q+1 1 p q ≤ κ3 α+ 1 p β q q+1 1 p ∞ j=1 g j α BV sup j X, µ j -X η , µ j q q+1 1 p q ≤ κ3 α+ 1 p β q q+1 1 p   ∞ j=1 g j α BV   sup 0≤t≤T |X t -X η t | q q+1 1 p q ≤ κ3 α+ 1 p β q q+1 1 p   ∞ j=1 g j α BV     c (21) T 0 η(r) 2 dr 1 2   q q+1 1 p
, where inequality (21) below is used. Taking 1 ≤ q < ∞ large enough the assertion follows.

The second sufficient condition

The second sufficient condition relies on a simple iteration procedure: Theorem 2.10. Assume that (A b,σ ) and (A f ) are satisfied and that ξ := g(X r1 , ..., X r L ),

where |g(x 1 , ..., x L ) -g(x ′ 1 , ..., x ′ L )| ≤ L l=1 [|g l (x l ) -g l (x ′ l )| + ψ l (x 1 , ..., x l ; x ′ 1 , ..., x ′ l )|x l -x ′ l |]
with polynomially bounded Borel functions g, g l and ψ l such that

g l (X r l ) -❊(g l (X r l )|F t ) p ≤ c(r l -t) θ l 2 (9)
for l = 1, ..., L, 0 < θ l ≤ 1, and r l-1 ≤ t < r l . Then,

(ξ, f ) ∈ B Θ p,∞ (X). The proof of Theorem 2.10 is given in Section 3.3.
Example 2.11. Let Φ : | L → | be Lipschitz and g 1 , ..., g L be as in Theorem 2.10, and define g(x 1 , ..., x L ) := Φ(g 1 (x 1 ), ..., g L (x L )).

To verify [START_REF] Friedman | Partial Differential Equations of Parabolic Type[END_REF] for concrete functions g l , it is sufficient to check the inequality for the Brownian motion and for an appropriately rescaled function:

Proposition 2.12. Let c (B.1) > 0 be the constant from Proposition B.1 so that

Γ(t, x; s, ξ) ≤ c (B.1) γ d s-t x -ξ c (B.1)
and let h l (x) := g l x 0 + c (B. [START_REF] Avikainen | On irregular functionals of SDEs and the Euler scheme[END_REF] x and assume that

h l (W r l ) -❊(h l (W r l )|F t ) p ≤ c l (r l -t) θ l 2 for 0 ≤ t < r l , (10)
then ( 9) holds true for some c > 0.

The proof of this proposition can be found in the appendix. One can rescale the argument of the function h l in [START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF] as well to assume that r l = 1. Examples for [START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF] with d = 1 and r l = 1 are the following: A precise investigation about the relation of ( 10) to B θ p,q (| d , γ d ) can be found in [START_REF] Geiss | BMO and L p -approximation of stochastic integrals[END_REF].

(a) If h l (x) = χ [K,∞) (x) for some K ∈ |, then θ = 1/p according to [12,
3 Proofs of the main results 3.1 Proof of Theorem 2.1

(C1 l ) =⇒ (C2 l ) for 0 < θ l < 1 is obvious as Z t p ≤ Z r l-1 p + c 1 t r l-1 (r l -r) θ l -2 dr 1 2 = Z r l-1 p + c 1 1 1 -θ l [(r l -t) θ l -1 -(r l -r l-1 ) θ l -1 ] 1 2 ≤ Z r l-1 p + c 1 (1 -θ l ) -1 2 (r l -t) θ l -1 2
.

(C2 l ) =⇒ (C3 l ) We observe that Y t -Y s p = t s f (r, X r , Y r , Z r )dr - t s Z r dW r p ≤ t s f (r, X r , Y r , Z r ) p dr + a p t s Z r 2 p dr 1 2 ≤ K f (t -s) + L f t s |X r | + |Y r | + |Z r | p dr + a p t s Z r 2 p dr 1 2 ≤ (t -s) K f + L f sup r∈[0,T ] X r p + L f sup r∈[0,T ] Y r p +c 2 (L f √ T + a p ) t s (r l -r) θ l -1 dr 1 2
where we used that 2 ≤ p < ∞ and where a p > 0 is the constant from the Burkholder-Davis-Gundy inequality.

(C3 l ) =⇒ (C4 l ) Here we get that

Y r l -❊(Y r l |F s ) p ≤ Y r l -Y s p + Y s -❊(Y r l |F s ) p ≤ 2 Y r l -Y s p ≤ 2c 3 r l s (r l -r) θ l -1 dr 1 2 = 2c 3 1 θ l (r l -s) θ l 2 . (C4 l ) =⇒ (C5 l ) We consider t r l-1 |(D 2 F l )(X l-1 ; s, X s )| 2 ds 1 2 p 18 = d k=1 t r l-1 |(∇ x (∂ x k F l ))(X l-1 ; s, X s )| 2 ds 1 2 p ≤ 1 η d k=1 t r l-1 |(∇ x (∂ x k F l )σ)(X l-1 ; s, X s )| 2 ds 1 2 p ≤ d k=1 1 η t r l-1 |(∇ x (∂ x k F l )σ)(X l-1 ; s, X s )| 2 ds 1 2 p ≤ d k=1 b p η t r l-1 (∇ x (∂ x k F l )σ)(X l-1 ; s, X s )dW s p
where b p > 0 is the constant from the Burkholder-Davis-Gundy inequality and the ellipticity condition on σ implies that there exists an η > 0 such that

η|y| | d ≤ |y * σ(t, x)| | d for all x, y ∈ | d .
To upper-bound the terms of the last sum we use Itô's formula and our PDE (which reduces the number of terms) to obtain

∂ x k F l (X l-1 ; t, X t ) -∂ x k F l (X l-1 ; r l-1 , X r l-1 ) = - t r l-1 ∂ x k b, ∇ x F l + 1 2 ∂ x k A, D 2 F l (X l-1 ; s, X s )ds (11)
+ t r l-1 (∇ x (∂ x k F l )σ) (X l-1 ; s, X s )dW s which implies that t r l-1 (∇ x (∂ x k F l )σ)(X l-1 ; s, X s )dW s p ≤ ∇ x F l (X l-1 ; t, X t ) p + ∇ x F l (X l-1 ; r l-1 , X r l-1 ) p + t r l-1 ∂ x k b, ∇ x F l + 1 2 ∂ x k A, D 2 F l (X l-1 ; s, X s )ds p ≤ κ p ′ R t √ r l -t + κ p ′ R r l-1 √ r l -r l-1 + κ p ′ ∂ x k b ∞ r l r l-1 R s √ r l -s ds +κ p ′ ∂ x k A ∞ 2 r l r l-1 R s r l -s ds with R s := Y r l -❊(Y r l |F s )
p and r l-1 ≤ s < r l where we used (A b,σ ) and inequalities (2) and (3). Consequently,

t r l-1 |(D 2 F l )(X l-1 ; s, X s )| 2 ds 1 2 p ≤ c 4 db p η κ p ′ (r l -t) θ l -1 2 + (r l -r l-1 ) θ l -1 2 + sup 1≤k≤d ∂ x k b ∞ r l r l-1 (r l -s) θ l -1 2 ds + sup 1≤k≤d ∂ x k A ∞ 2 r l r l-1 (r l -s) θ l
2 -1 ds .

(C5 l ) =⇒ (C2 l ) Here we start with Lemma 3.1. Assume that (A b,σ ), (A f ) and (A g ) are satisfied. There exists a constant c > 0, depending at most on σ, b, T, d and 2 ≤ p < ∞, such that, for all r l-1 ≤ s < t < r l ,

∇ x F l (X l-1 ; t, X t ) -∇ x F l (X l-1 ; s, X s ) p ≤ c(t -s) ∇ x F l (X l-1 ; r l-1 , X r l-1 ) p +c(t -s) s r l-1 |D 2 F l (X l-1 ; v, X v )| 2 dv 1 2 p +c t s |D 2 F l (X l-1 ; v, X v )| 2 dv 1 2 p .
Proof. For simplicity we will omit X l-1 in the computation. Using [START_REF] Geiss | Interpolation and approximation in L 2 (γ)[END_REF] with r l-1 replaced by s we get that

∇ x F l (t, X t ) -∇ x F l (s, X s ) p ≤ d k=1 ∂ x k F l (t, X t ) -∂ x k F l (s, X s ) p ≤ d k=1 ∂ x k b ∞ t s |∇ x F l (v, X v )|dv p + d k=1 ∂ x k A ∞ 2 t s |D 2 F l (v, X v )|dv p +a p d k=1 t s |(∇ x (∂ x k F l )σ)(v, X v )| 2 dv 1 2 p ≤ d k=1 ∂ x k b ∞ t s |∇ x F l (v, X v )|dv p + d k=1 ∂ x k A ∞ (t -s) 1 2 2 + da p σ ∞ t s |D 2 F l (v, X v )| 2 dv 1 2 p 20
where a p > 0 is the constant from the Burkholder-Davis-Gundy inequality, so that

∇ x F l (t, X t ) -∇ x F l (s, X s ) p ≤ c 1 t s |∇ x F l (v, X v )|dv p + c 2 t s |D 2 F l (v, X v )| 2 dv 1 2 p ( 12
)
with

c 1 := d k=1 ∂ x k b ∞ and c 2 := √ T 2 d k=1 ∂ x k A ∞ + da p σ ∞ .
Using this relation for s = r l-1 and applying Gronwall's lemma implies

∇ x F l (t, X t ) p ≤ e c1T   ∇ x F l (r l-1 , X r l-1 ) p + c 2 t r l-1 |D 2 F l (r, X r )| 2 dr 1 2 p   .
Now we return to [START_REF] Geiss | Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces[END_REF] and get that

∇ x F l (t, X t ) -∇ x F l (s, X s ) p ≤ c 1 t s ∇ x F l (r, X r ) p dr + c 2 t s |D 2 F l (r, X r )| 2 dr 1 2 p ≤ c 1 e c1T t s   ∇ x F l (r l-1 , X r l-1 ) p + c 2 r r l-1 |D 2 F l (v, X v )| 2 dv 1 2 p   dr +c 2 t s |D 2 F l (r, X r )| 2 dr 1 2 p ≤ c 1 e c1T (t -s) ∇ x F l (r l-1 , X r l-1 ) p +c 1 c 2 e c1T t s s r l-1 |D 2 F l (v, X v )| 2 dv 1 2 p dr +c 1 c 2 e c1T t s r s |D 2 F l (v, X v )| 2 dv 1 2 p dr +c 2 t s |D 2 F l (r, X r )| 2 dr 1 2 p ≤ c 1 e c1T (t -s) ∇ x F l (r l-1 , X r l-1 ) p +(t -s)c 1 c 2 e c1T s r l-1 |D 2 F l (v, X v )| 2 dv 1 2 p +[c 1 c 2 e c1T (t -s) + c 2 ] t s |D 2 F l (v, X v )| 2 dv 1 2 p .
For r ∈ [r l-1 , r l ) we consider

δv l (x l-1 ; r, x) := v l (x l-1 ; r, x) -∇ x F l (x l-1 ; r, x) (13)
and get that, a.s.,

u l (x l-1 ; r l-1 , x l-1 ) -F l (x l-1 ; r l-1 , x l-1 ) = r l r l-1 f (x l-1 ; r, X r l-1 ,x l-1 r )dr - r l r l-1 δv l (x l-1 ; r, X r l-1 ,x l-1 r )σ(r, X r l-1 ,x l-1 r )dW r l-1 r with f (x l-1 ; r, x) := f (r, x, u l (x l-1 ; r, x), v l (x l-1 ; r, x)σ(r, x)). Letting λ r (x l-1 ; s, x) := | d f (x l-1 ; r, ξ)∇ x Γ(s, x; r, ξ)dξ
and applying a stochastic Fubini argument, it follows that

δv l (x l-1 ; s, X r l-1 ,x l-1 s )σ(s, X r l-1 ,x l-1 s ) = r l s λ r (x l-1 ; s, X r l-1 ,x l-1 s )dr σ(s, X r l-1 ,x l-1 s ) a.s.
for s ∈ [r l-1 , r l )\N l (x l-1 ), where N l (x l-1 ) is a Borel set of measure zero. Hence for s ∈ [r l-1 , r l ) \ N l (x l-1 ) we get by (2) and Proposition 1.1 that

δv l (x l-1 ; s, X r l-1 ,x l-1 s )σ(s, X r l-1 ,x l-1 s ) p ≤ r l s λ r (x l-1 ; s, X r l-1 ,x l-1 s )σ(s, X r l-1 ,x l-1 s ) p dr ≤ σ ∞ κ p ′ r l s f (x l-1 ; r, X r l-1 ,x l-1 r ) p √ r -s dr ≤ σ ∞ κ p ′ r l s K f + L f X r l-1 ,x l-1 r p + u l (x l-1 ; r, X r l-1 ,x l-1 r ) p √ r -s + L f v l (x l-1 ; r, X r l-1 ,x l-1 r )σ(r, X r l-1 ,x l-1 r ) p √ r -s dr ≤ σ ∞ κ p ′ r l s 1 √ r -s K f + L f X r l-1 ,x l-1 r p + α l 1 + σ ∞ √ r l -r 1 + |x 1 | q l,1 + • • • + |x l-1 | q l,l-1 + |X r l-1 ,x l-1 r | q l,l p dr.
By continuity of both sides in s one can estimate the first term by the last term in the above display for all s ∈ [r l-1 , r l ). Using the stochastic flow we obtain the inequality

Z s -∇ x F l (X l-1 ; s, X s )σ(s, X s ) p ≤ σ ∞ κ p ′ r l s 1 √ r -s K f + L f X r p + α l 1 + σ ∞ √ r l -r 1 + |X r1 | q l,1 + • • • + |X r l-1 | q l,l-1 + |X r | q l,l p dr ≤ c 0 < ∞
where c 0 > 0 does not depend on s. The assertion (C2 l ) follows from this and Lemma 3.1 applied to s = r l-1 because

Z r p ≤ Z r -∇ x F l (X l-1 ; r, X r )σ(r, X r ) p + σ ∞ ∇ x F l (X l-1 ; r, X r ) p ≤ c 0 + σ ∞ (1 + c (3.1) T ) ∇ x F l (X l-1 ; r l-1 , X r l-1 ) p + σ ∞ c (3.1) t r l-1 |D 2 F l (X l-1 ; v, X v )| 2 dv 1 2 p .
(C4 l ) =⇒ (C1 l ) To make our assumption (C4 l ) more transparent, the constant c 4 > 0 of this condition is denoted by c B Θ p,∞ in the following. Using (13) and letting r l-1 ≤ r < r l , by condition (A b,σ ) we get that

Z r l-1 ,x l-1 t -Z r l-1 ,x l-1 s p ≤ Z r l-1 ,x l-1 t σ(t, X r l-1 ,x l-1 t ) -1 -Z r l-1 ,x l-1 s σ(s, X r l-1 ,x l-1 s ) -1 p σ ∞ + Z r l-1 ,x l-1 s σ(s, X r l-1 ,x l-1 s ) -1 (σ(t, X r l-1 ,x l-1 t ) -σ(s, X r l-1 ,x l-1 s )) p ≤ ∇ x F l (x l-1 ; t, X r l-1 ,x l-1 t ) -∇ x F l (x l-1 ; s, X r l-1 ,x l-1 s ) p σ ∞ + δv l (x l-1 ; t, X r l-1 ,x l-1 t ) -δv l (x l-1 ; s, X r l-1 ,x l-1 s ) p σ ∞ +L σ σ -1 ∞ Z r l-1 ,x l-1 s p × ( ❊( X r l-1 ,x l-1 t -X r l-1 ,x l-1 s p |F r l-1 s ) 1 p ∞ + |t -s| 1 2 ) ≤ c σ,b,p,T [D 1 (x l-1 ) + D 2 (x l-1 ) + D 3 (x l-1 )] with D 1 (x l-1 ) := ∇ x F l (x l-1 ; t, X r l-1 ,x l-1 t ) -∇ x F l (x l-1 ; s, X r l-1 ,x l-1 s ) p , D 2 (x l-1 ) := δv l (x l-1 ; t, X r l-1 ,x l-1 t ) -δv l (x l-1 ; s, X r l-1 ,x l-1 s ) p , D 3 (x l-1 ) := (t -s) 1 2 Z r l-1 ,x l-1 s p .
Now we show that each D i (X l-1 ) p , i = 1, 2, 3, is bounded by a constant times t s (r lr) θ l -2 dr 1 2 which implies (C1 l ).

The term D 1 (X l-1 ): Here we use Lemma 3.1 to get

D 1 (X l-1 ) p = ∇ x F l (X l-1 ; t, X t ) -∇ x F l (X l-1 ; s, X s ) p ≤ c (3.1) (t -s) ∇ x F l (X l-1 ; r l-1 , X r l-1 ) p +c (3.1) (t -s) s r l-1 |D 2 F l (X l-1 ; v, X v )| 2 dv 1 2 p +c (3.1) t s |D 2 F l (X l-1 ; v, X v )| 2 dv 1 2 p ≤ c (3.1) (t -s) ∇ x F l (X l-1 ; r l-1 , X r l-1 ) p +c (3.1) (t -s) s r l-1 D 2 F l (X l-1 ; v, X v ) 2 p dv 1 2 +c (3.1) t s D 2 F l (X l-1 ; v, X v ) 2 p dv 1 2 ≤ c (3.1) (t -s)κ p ′ c B Θ p,∞ (r l -r l-1 ) θ l -1 2 +c (3.1) (t -s) s r l-1 κ 2 p ′ c 2 B Θ p,∞ (r l -v) θ l -2 dv 1 2 +c (3.1) t s κ 2 p ′ c 2 B Θ p,∞ (r l -v) θ l -2 dv 1 2
where we have used (3). Finally we apply

(t -s) s r l-1 (r l -v) θ l -2 dv 1 2 ≤ (t -s) s -r l-1 (r l -s) θ l -2 2 ≤ √ t -s s -r l-1 t s (r l -v) θ l -2 dv 1 2
.

The term D 2 (x l-1 ) and a linearization: First we follow the approach of [START_REF] Gobet | L 2 -time regularity of BSDEs with irregular terminal functions[END_REF] done for the one-step scheme, that shows that the difference process ((v l -∇ x F l )(X l-1 ; r, X r )) r∈[r l-1 ,r l ) solves the linear BSDE with the generator f lin defined below. We fix x 1 , ..., x l-1 ∈ | d and define

f lin : [r l-1 , r l ) × | d × | 1×d × | d×d → | 1×d by f lin (x l-1 ; r, x, U, V ) := A 0 l (x l-1 ; r, x) + U B 0 l (x l-1 ; r, x) + d j=1 V j C j,0 l (x l-1 ; r, x),
where V j is the j-th row of V , with A 0 l (x l-1 ; r, x) := ∇ x f r, x, u l (x l-1 ; r, x), v l (x l-1 ; r, x)σ(r, x) + ∂f ∂y r, x, u l (x l-1 ; r, x), v l (x l-1 ; r, x)σ(r, x) ∇ x F l (x l-1 ; r, x)

+ d j=1 ∂f ∂z j r, x, u l (x l-1 ; r, x), v l (x l-1 ; r, x)σ(r, x) × × ∇ x d k=1 ∂F l ∂x k (x l-1 ; r, x)σ kj (r, x) , B 0 l (x l-1 ; r, x) := ∂f ∂y (r, x, u l (x l-1 ; r, x), v l (x l-1 ; r, x)σ(r, x))I | d + ∇ x b(r, x) + d j=1 ∂f ∂z j (r, x, u l (x l-1 ; r, x), v l (x l-1 ; r, x)σ(r, x))∇ x σ j (r, x)
and

C j,0 l (x l-1 ; r, x) := ∂f ∂z j (r, x, u l (x l-1 ; r, x), v l (x l-1 ; r, x)σ(r, x))I | d + ∇ x σ j (r, x),
with σ j = (σ kj ) d k=1 ∈ | d , δv l defined as in [START_REF] Geiss | BMO and L p -approximation of stochastic integrals[END_REF], and δu l (x l-1 ; r, x) := u l (x l-1 ; r, x) -F l (x l-1 ; r, x).

This implies

|f lin (x l-1 ; r, x, u, v)| ≤ |A 0 l (x l-1 ; r, x)| + c (14) [|u| + |v|]. (14)
To associate a BSDE to the driver f lin , we first check that

r l r l-1 A 0 l (x l-1 ; r, X r l-1 ,x l-1 r ) p dr < ∞. (15)
For this purpose we let

ψ l (x l-1 ; r) := 1 + ∇ x F l (x l-1 ; r, X r l-1 ,x l-1 r ) p + D 2 F l (x l-1 ; r, X r l-1 ,x l-1 r ) p , which implies that A 0 l (x l-1 ; r, X r l-1 ,x l-1 r ) p ≤ c (16) ψ l (x l-1 ; r). (16)
In view of ( 2) and (3) we have that

ψ l (x l-1 ; r) ≤ 1 + (1 + √ r l -r) κ p ′ r l -r × F l (x l-1 ; r l , X r l-1 ,x l-1 r l ) -F l (x l-1 ; r, X r l-1 ,x l-1 r ) p . ( 17
)
To obtain the integrability of the upper bound on ψ l (x l-1 ; r) (and thus that of A 0 l (x l-1 ; r, X r l-1 ,x l-1 r

) p ), we show that the assumption on global fractional smoothness implies a local fractional smoothness. Indeed, our global assumption reads as

F l (X l-1 ; r l , X r l ) -F l (X l-1 ; s, X s ) p ≤ c B Θ p,∞ (r l -s) θ l 2 (18)
for r l-1 ≤ s < r l . For any 0 < δ < 1 this implies that

r l r l-1 (r l -s) -pθ l 2 -δ F l (X l-1 ; r l , X r l ) -F l (X l-1 ; s, X s ) p p ds < ∞.
Using the transition density of X and Fubini's theorem implies the existence of a Borel set E l ⊆ (| d ) l-1 such that E c l has Lebesgue measure zero and

r l r l-1 (r l -s) -pθ l 2 -δ F l (x l-1 ; r l , X r l-1 ,x l-1 r l ) -F l (x l-1 ; s, X r l-1 ,x l-1 s ) p p ds < ∞
for all (x 1 , ..., x l-1 ) ∈ E l . For those (x 1 , ..., x l-1 ) ∈ E l we may deduce (using (5)) for s ∈ ((r l-1 + r l )/2, r l ) and a l := s -(r ls) that

F l (x l-1 ; r l , X r l-1 ,x l-1 r l ) -F l (x l-1 ; s, X r l-1 ,x l-1 s ) p p ≤ 2 p (s -a l ) -1 (r l -a l ) δ+ pθ l 2 s a l (r l -r) -pθ l 2 -δ F l (x l-1 ; r l , X r l-1 ,x l-1 r l ) -F l (x l-1 ; r, X r l-1 ,x l-1 r ) p p dr ≤ 2 p+δ+ pθ l 2 (r l -s) δ+ pθ l 2 -1 r l r l-1 (r l -r) -pθ l 2 -δ F l (x l-1 ; r l , X r l-1 ,x l-1 r l ) -F l (x l-1 ; r, X r l-1 ,x l-1 r ) p p dr.
Taking 0 < δ < 1 such that δ+ pθ l 2 -1 > 0 we obtain a local fractional smoothness for all (x 1 , ..., x l-1 ) ∈ E l . Then for x l-1 ∈ E l the inequality (15) is satisfied. Thus, because of [14, Theorem 2.1] the process (δv l (x l-1 ; s, X

r l-1 ,x l-1 s )) s∈[r l-1 ,r l ) solves the U -component of the BSDE U r l-1 ,x l-1 s = r l s f lin (x l-1 ; r, X r l-1 ,x l-1 r , U r l-1 ,x l-1 r , V r l-1 ,x l-1 r )dr - r l s (V r l-1 ,x l-1 r ) * dW r l-1
r * for all x l-1 ∈ E l (according to [START_REF] Gobet | L 2 -time regularity of BSDEs with irregular terminal functions[END_REF], [START_REF] Gobet | A regression-based Monte Carlo method to solve backward stochastic differential equations[END_REF] and [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF]Theorem 4.2] this BSDE has a unique L p -solution). Upper bound for D 2 (X l-1 ) p : Applying Lemma A.3 to h = f lin (the function κ from Lemma A.3(iii) is obtained by Proposition B.1 and ( 15) is used) it follows that

s p ≤ c (A.3) r l s |A 0 l (x l-1 ; r, X r l-1 ,x l-1 r )|dr p ≤ c (A.3) c (16) r l s ψ l (x l-1 ; r)dr ≤ c (A.3) c (16) [r l -s] + κ p ′ r l s (1 + √ r l -r) × F l (x l-1 ; r l , X r l-1 ,x l-1 r l ) -F l (x l-1 ; r, X r l-1 ,x l-1 r ) p r l -r dr =: ϕ l (x l-1 ; s), that means U r l-1 ,x l-1 s p ≤ ϕ l (x l-1 ; s) (19)
with

ϕ l (X l-1 ; s) p ≤ c (A.3) c (16) [r l -s] + κ p ′ (1 + √ T )c B Θ p,∞ r l s (r l -r) θ l 2 -1 dr or ϕ l (X l-1 ; s) p ≤ c (20) [r l -s] + c B Θ p,∞ r l s (r l -r) θ l 2 -1 dr . (20)
Exploiting again Lemma A.3 also gives that

V r l-1 ,xr-1 s p ≤ c (A.3) r l s A 0 l (x l-1 ; r, X r l-1 ,x l-1 r ) p √ r -s dr ≤ c (A.3) c (16) r l s ψ l (x l-1 ; r) √ r -s dr for s ∈ [r l-1 , r l ) \ N l (x l-1 )
, where N l (x l-1 ) has Lebesgue measure zero. Hence

U r l-1 ,x l-1 s -U r l-1 ,x l-1 t p = t s f lin (x l-1 ; r, X r l-1 ,x l-1 r , U r l-1 ,x l-1 r , V r l-1 ,x l-1 r )dr - t s V r l-1 ,x l-1 r dW r l-1 r p ≤ t s |A 0 l (x l-1 ; r, X r l-1 ,x l-1 r )|dr p +c (14) t s [|U r l-1 ,x l-1 r | + |V r l-1 ,x l-1 r |]dr p +a p t s |V r l-1 ,x l-1 r | 2 dr 1 2 p 27 ≤ t s |A 0 l (x l-1 ; r, X r l-1 ,x l-1 r )|dr p + c (14) t s U r l-1 ,x l-1 r p dr +[c (14) √ t -s + a p ] t s V r l-1 ,x l-1 r 2 p dr 1 2 ≤ t s |A 0 l (x l-1 ; r, X r l-1 ,x l-1 r )|dr p + c (14) t s ϕ l (x l-1 ; r)dr +[c (14) √ t -s + a p ]c (A.3) c (16) t s r l r ψ l (x l-1 ; w) √ w -r dw 2 dr 1 2
.

Because P((X r1 , ..., X r l-1 ) ∈ E l ) = 1 we can use the stochastic flow property and can bound D 2 (X l-1 ) p from above by the L p -norms of the following three expressions: Taking the L p -norm of the last term gives

t s r l r ψ l (X l-1 ; w) √ w -r dw 2 dr 1 2 p ≤ t s r l r 1 + ∇ x F l (X l-1 ; w, X w ) p + D 2 F l (X l-1 ; w, X w ) p √ w -r dw 2 dr 1 2 ≤ t s r l r dw √ w -r 2 dr 1 2 +κ p ′ c B Θ p,∞   t s r l r (r l -w) θ l -1 2
+ (r l -w) θ l -2 2 √ w -r dw 2 dr   1 2 ≤ t s r l r dw √ w -r 2 dr 1 2 +κ p ′ c B Θ p,∞ (1 + √ T )   t s r l r (r l -w) θ l -2 2 √ w -r dw 2 dr   1 2 ≤ 2 √ T √ t -s + κ p ′ c B Θ p,∞ (1 + √ T )γ l t s (r l -r) θ l -1 dr 1 2
with γ l :=

1 0 (1-t) θ l 2 -1 √ t dt.
For the next to the last term we obtain

t s ϕ l (X l-1 ; r)dr p ≤ c (20) t s (r l -r) + c B Θ p,∞ r l r (r l -w) θ l 2 -1 dw dr ≤ c (20) T + c B Θ p,∞ 2 θ l T θ l 2 (t -s).
Finally, we get by ( 17) and (18) that

t s |A 0 l (X l-1 ; r, X r )|dr p ≤ c (16) t s ψ l (X l-1 ; r) p dr ≤ c (16) (t -s) + √ T (1 + √ T )κ p ′ c B Θ p,∞ t s (r l -r) θ l -2 dr 1 2
.

The term D 3 (X l-1 ): Let r l-1 ≤ s < t < r l and recall

Z r l-1 ,x l-1 t = v l (x l-1 ; t, X r l-1 ,x l-1 t )σ(t, X r l-1 ,x l-1 t
).

From inequality [START_REF] Hu | Nonlinear Feynman-Kac formula and discrete-functionaltype BSDEs with continuous coefficients[END_REF] we obtain (ts)

1 2 Z r l-1 ,X l-1 s p ≤ (t -s) 1 2 σ ∞ v l (X l-1 ; s, X r l-1 ,Xr l-1 s ) p ≤ (t -s) 1 2 σ ∞ ∇ x F l (X l-1 ; s, X r l-1 ,Xr l-1 s ) p + U r l-1 ,X l-1 s p ≤ (t -s) 1 2 σ ∞ (κ p ′ c B Θ p,∞ (r l -s) θ l -1 2 + ϕ l (X l-1 , s) p ) ≤ (t -s) 1 2 σ ∞ κ p ′ c B Θ p,∞ (r l -s) θ l -1 2 + c (20) [r l -s] + c B Θ p,∞ r l s (r l -r) θ l 2 -1 dr ≤ c(t -s) 1 2 [1 + (r l -s) θ l -1 2 ] ≤ c (t -s) 1 2 + t s (r l -r) θ l -1 dr 1 2
.

(C3 l ) =⇒ (C6 l ) Let t n,θ l k := r l-1 + (r l -r l-1 ) 1 -1 - k n 1 θ l for k = 0, ..., n and S n t n,θ l k := Y t n,θ l k . One obtains for t ∈ (t n,θ l k-1 , t n,θ l k ) ⊆ [r l-1 , r l
] and an appropriate η ∈ (0, 1), that

S n t -Y t p = (1 -η)Y t n,θ l k-1 + ηY t n,θ l k -Y t p ≤ (1 -η) Y t n,θ l k-1 -Y t p + η Y t n,θ l k -Y t p ≤ (1 -η)c 3 t t n,θ l k-1 (r l -r) θ l -1 dr 1 2 + ηc 3 t n,θ l k t (r l -r) θ l -1 dr 1 2 29 ≤ c 3 1 θ l [(r l -t n,θ l k-1 ) θ l -(r l -t n,θ l k ) θ l ] 1 2 = c 3 (r l -r l-1 ) θ l 2 √ θ l 1 √ n . (C6 l ) =⇒ (C4 l ) We consider Y r l +t n,θ l n-1 2 -S n r l +t n,θ l n-1 2 p = Y r l +t n,θ l n-1 2 - 1 2 S n r l + S n t n,θ l n-1 p ≥ Y r l +t n,θ l n-1 2 - 1 2 Y r l + S n t n,θ l n-1 p - 1 2 Y r l -S n r l p so that Y r l -2Y r l +t n,θ l n-1 2 + S n t n,θ l n-1 p ≤ 3c 6 √ n .
But this means that

Y r l -❊ Y r l |F r l +t n,θ l n-1 2 p ≤ 6c 6 √ n . Because r l - r l + t n,θ l n-1 2 = 1 2 (r l -r l-1 )n -1 θ l
we get that

Y r l -❊ (Y r l |F t ) p ≤ 6c 6 r l -r l-1 2 - θ l 2 (r l -t) θ l 2 for t = r l + t n,θ l n-1 2 .
Using (5) proves our assertion for r l-1 + r l -r l-1 2 ≤ t < r l . For the remaining

r l-1 ≤ t < r l-1 + r l -r l-1 2 we can simply use Y r l -Y r l-1 p < ∞. (C7 l ) =⇒ (C4 l ) Let t ∈ [r l-1 , r l ).
We use (C7 l ) for n = 1 so that Y t and Y r l can be covered by one ball with any radius bigger than c 7 (r lt) θ l 2 . Taking the infimum of these radii we get that Y r l -Y t p ≤ 2c 7 (r lt)

θ l 2 which implies that Y r l -❊(Y r l |F t ) p ≤ 4c 7 (r l -t) θ l 2 . (C3 l ) =⇒ (C7 l ) Fix t ∈ [r l-1 , r l ) and n ≥ 1. Let N ≥ 1 and choose k ∈ {1, ..., N } such that t ∈ [t N,θ l k-1 , t N,θ l k ) ⊆ [r l-1 , r l ).
For those time-nets we computed in (C3) =⇒ (C6) that

Y u -Y v p ≤ c 3 (r l -r l-1 ) θ l 2 √ θ l 1 √ N for u, v ∈ [t N,θ l k-1 , t N,θ l k ] ⊆ [r l-1 , r l ]. Now we choose N ≥ 1 such that the cardi- nality of t N,θ l k : k = 0, ..., N ∩ [t N,θ l k , r l ] is equal to n, i.e. n = 1 + N r l -t N,θ l k r l -r l-1 θ l .
For n ≥ 2 this implies that

n 2 ≤ n -1 = N (r l -r l-1 ) θ l (r l -t N,θ l k ) θ l ≤ N (r l -r l-1 ) θ l (r l -t) θ l and e n ((Y s ) s∈[t,r l ] |L p ) ≤ c 3 (r l -r l-1 ) θ l 2 √ θ l 1 √ N ≤ c 3 √ θ l 2(r l -t) θ l √ n . The case n = 1 implies that t N,θ l k-1 ≤ t < t N,θ l k = r l . As in (C3 l ) =⇒ (C4 l ) we have Y r l -Y s p ≤ c 3 1 θ l (r l -s) θ l 2 ≤ c 3 1 θ l (r l -t) θ l 2
for all s ∈ [t, r l ] so that

e 1 ((Y s ) s∈[t,r l ] |L p ) ≤ c 3 1 θ l (r l -t) θ l 2 . ✷ 3.2 Proof of Theorem 2.5 (a)
We get, a.s., that

X η s -X s = s 0 [b(r, X η r ) -b(r, X r )]dr + s 0 [σ(r, X η r ) -σ(r, X r )] 1 -η(r) 2 dW r + s 0 σ(r, X η r )η(r)dB r - s 0 σ(r, X r )(1 -1 -η(r) 2 )dW r .
Using the Burkholder-Davies-Gundy inequalities we estimate

+a p p σ p ∞ s 0 η(r) 2 dr p 2 + a p p σ p ∞ s 0 (1 -1 -η(r) 2 ) 2 dr p 2 ,
where L b and L σ are the Lipschitz constants (with respect to x) of b and σ, and a p the constant from the Burkholder-Davis-Gundy inequality. Note that 1 -

1 -η(r) 2 = η(r) 2 1+ √ 1-η(r) 2 ≤ |η(r)| using |η(r)| ≤ 1. Thus, applying Gronwall's lemma implies sup 0≤r≤s |X η r -X r | p ≤ c (21) s 0 η(r) 2 dr 1 2 (21)
where c ϕ(η)

1 -η 2 + 1 -ϕ(η) |η| = 2 (22)
using the convention 0 0 = 0. Thus, we can define the parameterized driver

f η (t, ω, y, z) := f t, X η t (ω), y, z W ϕ(η(t)) 1 -η(t) 2 + z B 1 -ϕ(η(t)) η(t)
where z = (z W , z B ) is 2d-dimensional. In view of ( 22), the driver f η is Lipschitz with respect to y and z. Thus, for any F W,B T -measurable terminal condition ξ ∈ L p , there is an unique solution in L p in the filtration F W,B to the BSDE

Y t = ξ + T t f η (s, Y s , Z s )ds - T t Z W s dW s - T t Z B s dB s
(Y η , [Z η,W , Z η,B ]) with Z η,W s = Z η s 1 -η(s) 2 and Z η,B s = Z η s η(s) solves our BSDE because Y η t = ξ η + T t f (s, X η s , Y η s , Z η s ϕ(η(s)) + Z η s (1 -ϕ(η(s))))ds - T t Z η s 1 -η(s) 2 dW s - T t Z η s η(s)dB s = ξ η + T t f η (s, Y η s , [Z η,W s , Z η,B s ])ds - T t Z η,W s dW s - T t Z η,B s dB s .
(e) To sum up, we have proved that (Y, [Z, 0]) and (Y η , [Z η .

1η(.) 2 , Z η . η(.)]) solve the BSDEs with data (ξ, f 0 ) and (ξ η , f η ) in the filtration (F W,B t ) t∈[0,T ] . Then, we are in a position to apply Lemma A.1 (with d replaced by 2d) and get

sup 0≤t≤T |Y η t -Y t | p + T 0 (Z η t 1 -η(t) 2 -Z t , Z η t η(t)) 2 dt 1/2 p ≤ c (A.1)   ξ η -ξ p + T 0 |f η (t, Y t , [Z t , 0]) -f 0 (t, Y t , [Z t , 0])|dt p   ≤ c (A.1) ξ η -ξ p + L f T 0 |X η t -X t |dt p +L f T 0 |Z t | ϕ(η(t)) 1 -η(t) 2 -1 dt p
where c (A.1) (here and thereafter) is not identical with the constant c in Lemma A.1 but only refers to the fact that the inequality of Lemma A.1 is used. Now,

since ϕ(η) √ 1-η 2 -1 ≤ c ϕ |η| for some constant c ϕ > 0, we have T 0 |Z t | ϕ(η(t)) 1 -η(t) 2 -1 dt ≤ c ϕ T 0 |Z t | 2 dt 1/2 T 0 η(t) 2 dt 1/2
.

With the previous estimate on X η -X from (21) this leads to

sup 0≤t≤T |Y η t -Y t | p + T 0 (Z η t 1 -η(t) 2 -Z t , Z η t η(t)) 2 dt 1 2 p ≤ c (A.1) ξ η -ξ p +L f   T c (21) + c ϕ T 0 |Z t | 2 dt 1 2 p   T 0 η(t) 2 dt 1 2
.

Applying Lemma A.1 to ξ (0) = 0, f 0 = 0, Y (0) s ≡ 0, Z (0) s
≡ 0, ξ (1) = ξ, f 1 (ω; s, y, z) := f (s, X s (ω), y, z) and our solution (Y, Z) we obtain

α s (ω) = |f (s, X s (ω), 0, 0)| ≤ K f + L f sup 0≤t≤T |X t (ω)| and T 0 |Z t | 2 dt 1 2 p ≤ c (A.1) [K f + L f + ξ p ].
To complete the proof, it remains to use the inequality

|(Z η t 1 -η(t) 2 -Z t , Z η t η(t))| 2 = |Z η t | 2 + |Z t | 2 -2 1 -η(t) 2 Z η t , Z t ≥ 1 2 |Z η t -Z t | 2 . ✷ 3.3 Proof of Theorem 2.10
(a) In this step we assume that all (x 1 , ..., x L ) (and similarly (x ′ 1 , ..., x ′ L )) that appear have the property that x 1 ∈ D 2 , (x 1 , x 2 ) ∈ D 3 , ..., (x 1 , ..., x L-1 ) ∈ D L where the sets D 2 , ..., D L are taken from Proposition 1.1. By backward induction we prove the following estimate regarding the terminal condition function Φ l (x 1 , ..., x l ) := u l (x 1 , ..., x l-1 ; r l , x l ) of the BSDE at time r l :

|Φ l (x l ) -Φ l (x ′ l )| ≤ c l l i=1 [|g i (x i ) -g i (x ′ i )| + ψ i (x i ; x ′ i )|x i -x ′ i |] . (23)
This is true for l = L by our assumption. Assume now that (23) holds for some 2 ≤ l ≤ L and let us prove the inequality for l -1. We have

Φ l-1 (x 1 , ..., x l-1 ) -Φ l-1 (x ′ 1 , ..., x ′ l-1 ) ≤ u l (x 1 , ..., x l-1 ; r l-1 , x l-1 ) -u l x ′ 1 , ..., x ′ l-1 ; r l-1 , x l-1 + u l x ′ 1 , ..., x ′ l-1 ; r l-1 , x l-1 -u l x ′ 1 , ..., x ′ l-1 ; r l-1 , x ′ l-1 ≤ u l (x 1 , ..., x l-1 ; r l-1 , x l-1 ) -u l x ′ 1 , ..., x ′ l-1 ; r l-1 , x l-1 + α l √ r l -r l-1 (1 + |x ′ 1 | q l,1 + • • • + |x ′ l-1 | q l,l-1 + |x l-1 | q l,l + |x ′ l-1 | q l,l )|x l-1 -x ′ l-1 |
A Some lemmas about BSDEs

We fix a complete probability space (M, Σ, ◗), 0 ≤ r < R ≤ T (the upper bound T is used to bound some constants independently from R), d ≥ 1 and a d-dimensional standard Brownian motion B = (B t ) t∈[r,R] with B r ≡ 0. Furthermore, we assume that (G t ) t∈[r,R] is the augmentation of the natural filtration of B. The diffusion (X s ) s∈[r,R] is considered with respect to the same σ and b as used before, restricted to the corresponding time interval. Regarding the flow (X t,x s ) s,t∈[r,R],x∈| d and the filtrations (G t s ) s∈[t,R] we use the same convention as in Section 1.1.

Lemma A.1 (L p -stability of solutions of BSDEs). Let 2 ≤ p < ∞, f i : M × [r, R] × | k × | k×d → | k be measurable with respect to P rog(M × [r, R]) × B(| k ) × B(| k×d ) with P rog(M × [r, R]
) being the σ-algebra of progressively measurable subsets, and assume that, a.s.,

Y (i) t = ξ (i) + R t f i (s, Y (i) s , Z (i) s )ds - R t Z (i) s dB s for i = 0, 1 and r ≤ t ≤ R with R r |f i (s, Y (i) s , Z (i) s )|ds + sup r≤t≤R |Y (i) t | + R r |Z (i) s | 2 ds 1 2 ∈ L p . Let α s (ω) := |f 1 (ω; s, Y (0) s (ω), Z (0) s (ω)) -f 0 (ω; s, Y (0) s (ω), Z (0) s (ω))| and suppose that there is a L f1 > 0 such that |f 1 (ω; s, u 1 , v 1 ) -f 1 (ω; s, u 2 , v 2 )| ≤ L f1 [|u 1 -u 2 | + |v 1 -v 2 |].
Then there exists a c p > 0, depending on p only, such that for a ≥ L f1 + L 2 f1 one has

❊   sup t∈[r,R] e ap(t-r) |∆Y t | p + R r e 2a(s-r) |∆Z s | 2 ds p 2   ≤ c p p ❊ e ap(R-r) |∆ξ| p + R r e a(s-r) α s ds p . Proof. The result is a direct consequence of [6, Proposition 3.2]. For ∆Y t := Y 1 t -Y 0 t , ∆Z t := Z 1 t -Z 0 t and ∆ξ := ξ (1) -ξ (0) we get that ∆Y t = ∆ξ + R t f (s, ∆Y s , ∆Z s )ds - R t ∆Z s dB s with f (s, ∆y, ∆z) := f 1 (s, ∆y + Y (0) s , ∆z + Z (0) s ) -f 0 (s, Y (0)
s , Z

s ) and

| f (ω; s, ∆y, ∆z)| = |f 1 (ω; s, ∆y + Y (0) s (ω), ∆z + Z (0) s (ω)) -f 0 (ω; s, Y (0) s (ω), Z (0) s (ω))| ≤ |f 1 (ω; s, Y (0) s (ω), Z (0) s (ω)) -f 0 (ω; s, Y (0) s (ω), Z (0) s (ω))| +|f 1 (ω; s, ∆y + Y (0) s (ω), ∆z + Z (0) s (ω)) -f 1 (ω; s, Y (0) s (ω), Z (0) s (ω))| ≤ α s (ω) + L f1 [|∆y| + |∆z|].
Applying [6, Proposition 3.2] implies the assertion.

The following lemma shows that [24, Theorem 3.2] transfers to our path dependent setting as expected. The proof is presumably only included in this preprint version for the convenience of the reader as it is a copy of that one in [START_REF] Zhang | Representation of solutions to BSDE's associated with a degenerate FSDE[END_REF] (see also [START_REF] Hu | Nonlinear Feynman-Kac formula and discrete-functionaltype BSDEs with continuous coefficients[END_REF]Section 5]).

Lemma A.2 (Representation of a BSDE parameterized by a parameter y ∈| K ).

Assume that (A b,σ ) and (A f ) are satisfied, that K, d ≥ 1 and that H : ) s∈[t,R] is the Y -component of the BSDE with respect to the forward diffusion (X t,x s ) s∈[t,R] , the terminal condition G(y; X t,x R ) with terminal time R ∈ (0, T ], and the generator f , the following assertions are satisfied:

| K × | d → | is Borel-measurable with |H(y; x)| ≤ α(1 + |y| γ + |x| β ) =: ψ(y, x) for some α, β, γ ∈ [1, ∞). Then there exists a Borel set F ⊆ | K such that F c
(i) For fixed y ∈ | K we have that U (y; •, •) ∈ C 0,1 ([r, R) × | d ). (ii) The functions U : | K × [r, R] × | d → | and ∇ x U : | K × [r, R) × | d → | 1×d are measurable.
(iii) There exists a constant c > 0 depending at most on (b, σ, T, α, γ, β, K f , L f ) such that

(a) |U (y; t, x)| ≤ cψ(y; x) for (y, t, x) ∈ | K × [r, R] × | d , (b) |∇ x U (y; t, x)| ≤ c ψ(y;x) √ R-t for (y, t, x) ∈ | K × [r, R) × | d . (iv)
For any y ∈ | K , the solution of the BSDE with the terminal condition G(y; X r,x R ), generator f , and forward diffusion (X r,x s ) s∈[r,R] can be represented as (a) Y y;r,x t = U (y; t, X r,x t ) on [r, R], (b) Z y;r,x t = ∇ x U (y; t, X r,x t )σ(t, X r,x t ) on [r, R).

Proof. We find solves our BSDE on the interval [t, R], that U n (y;

H n ∈ C ∞ 0 (| K × | d ), n ≥ 1,
•, •) ∈ C 0,1 ([r, R] × | d ) and that ∇ x U n (y; t, x) = ❊ G n (y; X t,x R )N t,1,(t,x) R + R t f (s, X t,x s , Y y,n;t,x s , Z y,n;t,x s )N t, 1,(t,x) s ds .
Properties of the function U n (a) To estimate U n (y; t, x) we let U n 0 (y; t, x) be the corresponding solution with the zero generator and denote by (Y y,n;t,x s,0 , Z y,n;t,x s,0

) s∈[t,R] the corresponding solution to our BSDE. By Lemma A.

)|ds 2 ≤ c (A.1) K f R + c (A.1) L f R t [|X t,x s | + |Y y,n;t,x s,0 | + |Z y,n;t,x s,0 |]ds 2 ≤ c (A.1) K f R + c (A.1) L f × R t [|X t,x s | + |❊(G n (y; X t,x R )|G t s )| + σ ∞ |❊(G n (y; X t,x R )N s,1,(t,x) R |G t s )]ds 2 ≤ c (A.1) K f R + c (A.1) L f × R t [|X t,x s | + 2|❊(ψ(y; X t,x R ) 2 |G t s )| 1 2 (1 + σ ∞ |❊((N s,1,(t,x) R ) 2 |G t s )| 1
2 )]ds

K f R + c (A.1) L f × R t [|X t,x s | + 2|❊(ψ(y; X t,x R ) 2 |G t s )| 1 2 (1 + σ ∞ κ 2 (R -s) -1/2 )]ds 2 ≤ 2❊ψ(y; X t,x R ) + c (A.1) K f R + c (A.1) L f × R t X t,x s 2 ds + 2 R t [ ψ(y; X t,x R ) 2 (1 + σ ∞ κ 2 (R -s) -1/2 )ds ≤ ψ(y; X t,x R ) 2 2 + 2c (A.1) L f [R + 2 σ ∞ κ 2 R 1/2 ] +c (A.1) L f R t X t,x s 2 ds + c (A.1) K f R ≤ β 2 ψ(y; x) 2 + 2c (A.1) L f [R + 2 σ ∞ κ 2 R 1/2 ] +c (A.1) L f R t α 2 [1 + |x|]ds + c (A.1) K f R
) 2 √ s -t ds ≤ c (24) κ 2 ψ(y; X t,x ρ ) 2 √ ρ -t + κ 2 × ρ t K f + L f [ X t,x s 2 + c (24) ψ(y; X t,x s ) 2 + ∇ x U n (y; s, X t,x s )σ(s, X t,x s ) 2 ] √ s -t ds ≤ c (24) κ 2 ψ(y; X t,x ρ ) 2 √ ρ -t + κ 2 × ρ t K f + L f [ X t,x s 2 + c (24) ψ(y; X t,x s ) 2 + σ ∞ ∇ x U n (y; s, X t,x s ) 2 ] √ s -t ds ≤ c (24) κ 2 ψ(y; X t,x ρ ) 2 √ ρ -t + κ 2 ρ t K f + L f X t,x s 2 + c (24) ψ(y; X t,x s ) 2 + σ ∞ A ρ s ψ(y;X t,x s ) √ ρ-s 2 √ s -t ds ≤ c (24) κ 2 ψ(y; X t,x ρ ) 2 √ ρ -t +κ 2 2 √ ρ -r[K f + L f + L f c (24) ] sup s∈[t,ρ] [1 + X t,x s 2 + ψ(y; X t,x s ) 2 ] +κ 2 σ ∞ B ρ t sup s∈[t,ρ] ψ(y; X t,x s ) 2 B 1 2 , 1 2
≤ c (24) κ 2 β 2 ψ(y; x) √ ρ -t + κ 2 2 √ ρ -r[K f + L f + L f c (24) ][1 + α 2 [1 + |x|] +β 2 ψ(y; x)] + κ 2 σ ∞ B ρ t β 2 ψ(y; x)B 1 2 , 1 2
≤ A ψ ψ(y; x) 1 √ ρ -t + B ρ t
where A ψ > 0 depends at most on (b, σ, T, α, β, γ, K f , L f ). Consequently,

A ρ t ≤ A ψ 1 + √ ρ -tB ρ t and B ρ t ≤ A ψ 1 + √ ρ -tB ρ t . In case of |ρ -t| ≤ (2A ψ ) -2 this gives B ρ t ≤ 2A ψ and |∇ x U n (y; t, x)| ≤ 2A ψ ψ(y; x) √ ρ -t .
Moreover, in case of 1 4 (2A ψ ) -2 ≤ |ρ -t| ≤ (2A ψ ) -2 we also get that

|∇ x U n (y; t, x)| ≤ 2A ψ ψ(y; t, x) √ ρ -t ≤ 8A 2 ψ ψ(y; x).
The latter inequality means that

|∇ x U n (y; t, x)| ≤ 8A 2 ψ ψ(y; x)
whenever r ≤ t ≤ R and |R -t| ≥ 1 4 (2A ψ ) -2 . On the other side,

|∇ x U n (y; t, x)| ≤ 2A ψ ψ(y; x) √ R -t for r ≤ t ≤ R and |R -t| ≤ (2A ψ ) -2 .
Combining both estimates yields to

|∇ x U n (y; t, x)| ≤ c (25) ψ(y; x) √ R -t (25)
for all t ∈ [r, R].

(c) We show that U n (y; t, x) is measurable as function on

| K × [r, R] × | d . Let
y, y ′ ∈ F . Then it follows by Lemma A.1 that

|U n (y; t, x) -U n (y ′ ; t, x)| ≤ c (A.1) G n (y; X (t,x) R ) -G n (y ′ ; X (t,x) R ) 2 ≤ c (A.1) Lip(H n )|y -y ′ | and |U n (y; t, x) -U n (y ′ ; t ′ , x ′ )| ≤ |U n (y; t, x) -U n (y ′ ; t, x)| + |U n (y ′ ; t, x) -U n (y ′ ; t ′ , x ′ )| ≤ c (A.1) Lip(H n )|y -y ′ | + |U n (y ′ ; t, x) -U n (y ′ ; t ′ , x ′ )|. Hence (U n ) -1 (B) ∩ (F × [r, R] × | d ) ∈ B(| K × [r, R] × | d ) for all open sets B ⊆ |. On F c we have (U n ) -1 (B) ∩ (F c × [r, R] × | d ) = F c × U -1 (B) ∈ B(| K × [r, R] × | d )
where U (t, x) is the functional for the Y process with zero terminal condition. Consequently, U n is measurable. This implies that for all fixed parameters y ∈ | K there is a uniform convergence on D of U n towards U , so that U (y; •, •) is continuous on [r, R) × | d . Moreover, as limit of measurable functions U n , the function

|U (y; t, x) -U n (y; t, x)| 2 ≤ c 2 (A.1) G(y; X t,x R ) -G n (y; X t,x R ) 2 2 = c 2 (A.1) | d Γ(t, x; R, ξ)|G(y; ξ) -G n (y; ξ)| 2 dξ ≤ c 2 (A.1) | d c (B.1) γ d R-t x -ξ c (B.
U : | K × [r, R) × | d → | is measurable as well. Because U (y; R, x) = G(y; x) the function U : | K × [r, R] × | d → | is measurable.
(e) Now we get )N t,1,(t,x) s ds .

Y y;r,x t -U (y; t, X r,x t ) 2 ≤ Y y;r,x t -U n (y; t, X r,x t ) 2 + U n (y; t, X r,x t ) -U (y; t, X r,x t ) 2 ≤ c (A.1) G(y; X r,x R ) -G n (y; X r,x R ) 2 + U n (y; t, X r,x t ) -U (y; t, X
By dominated convergence we have that

lim n ∇ x U n (y; t, x) = lim n ❊ G n (y; X t,x R )N t,1,(t,x) R + R t f s, X t,x s , U n (y; s, X t,x s ), ∇ x U n (y; s, X t,x s )σ(s, X t,x s ) N t,1,(t,x) s ds = ❊ G(y; X t,x R )N t,1,(t,x) R + R t f s, X t,x s , Y y;t,x s , Z y;t,x s N t,1,(t,x) s ds = V (y; t, x) for all (t, x) ∈ [r, R) × | d , which also implies |V (y; t, x)| ≤ c (25) ψ(y; x) √ R -t by (25). Consequently, lim n R-δ r ∇ x U n (y; t, X r,x t )σ(t, X r,x t ) -V (y; t, X r,x t )σ(t, X r,x t ) 2 2 dt = 0
for all δ ∈ (0, Rr) and Z y;r,x t = V (y; t, X r,x t )σ(t, X r,x t ) a.s. for almost every t ∈ [r, R). So we can re-define

Z y;r,x t := V (y; t, X r,x t )σ(t, X r,x t ). (g) Next we show that V (y; t, x) = ❊ G(y; X t,x R )N t,1,(t,x) R + R t f (s, X t,x s , Y y;t,x s , Z y;t,x s )N t,
χ (t,R) (s) √ R -s √ s -t ❊ [ √ R -sf (s, X t,x s , Y y;t,x s , Z y;t,x s )][ √ s -tN t,1,(t,x) s ] ds = (r,R) ϕ t (s)ψ t,x (s)ds with ϕ t (s) := χ (t,R) (s) √ R -s √ s -t , ψ t,x (s) := χ (t,R) (s)❊ [ √ R -sf (s, X t,x s , Y y;t,x s , Z y;t,x s )][ √ s -tN t,1,(t,x) s ] .
The family (ϕ t ) t∈[r,b] is uniformly integrable for b ∈ [r, R). The boundedness of (ψ t,x ) (t,x)∈D follows from

|ψ t,x (s)| ≤ √ R -s f (s, X t,x s , Y y;t,x s , Z y;t,x s ) 2 √ s -t N t,1,(t,x) s 2 ≤ √ R -s K f + L f ( X t,x s 2 + U (y; s, X t,x s ) 2 + σ ∞ V (y; s, X t,x s ) 2 ) κ 2
and the previous estimates on U and V obtained by ( 24) and [START_REF] Zhang | Discretizing a backward stochastic differential equation[END_REF]. Moreover, lim n ϕ tn (s) = ϕ t (s) for all s ∈ (r, R)\{t}.

As for ❊[G(y;

X t,x R )N t,1,(t,x) R
], we show that lim n ψ tn,xn (s) = ψ t,x (s) for all s ∈ (r, R) \ {t}.

(h) Finally, we show that ∇ x U = V . For x 0 , x 1 ∈ | d we have that

U n (y; t, x 0 ) -U n (y; t, x 1 ) = 1 0 ∇ x U n (y; t, x 0 + λ(x 1 -x 0 )), x 1 -x 0 dλ for r ≤ t < R. By dominated convergence we have that U (y; t, x 0 ) -U (y; t, x 1 ) = 1 0 V (y; t, x 0 + λ(x 1 -x 0 )), x 1 -x 0 dλ
so that we are done.

Lemma A.3 (L p -bound for the Z-process for a singular generator). Assume condition (A b,σ ), 0 ≤ r < R ≤ T , 2 ≤ p < ∞ and assume that X = (X s ) s∈[r,R] is the diffusion with parameters (b, σ) started in some x r ∈ | d . 1 Consider the BSDE

U t = R t h(s, X s , U s , V s )ds - R t V s dB s (26) with a generator h : [r, R) × | d × | d × | d×d → | d which is measurable with respect to B([r, R)) × B(| d ) × B(| d ) × B(| d×d
) and assume the following:

(i) h(s, •, u, v) is continuous in x for fixed s, u, v. (ii) |h(s, x, u 1 , v 1 ) -h(s, x, u 2 , v 2 )| ≤ L(|u 1 -u 2 | + |v 1 -v 2 |) for some L > 0. (iii) |h(s, x, u, v)| ≤ α(s, x) + λ|u| + µ|v| where α : [r, R) × | d → | is non- negative and B([r, R)) × B(| d )-measurable, α(s, •
) is continuous for fixed s and satisfies α(s, x) ≤ κ(s)[1 + |x| q ] for some q ≥ 0, where the function κ(.) ≥ 0 is bounded on compact subintervals of [r, R) and R r α(s, X s ) p ds < ∞.

Then there exists an unique solution (U, V ) such that

sup r≤t≤R |U t | + R r |V t | 2 dt 1 2 ∈ L p and a constant c = c(p, σ, b, T, L, λ, µ) > 0 such that (1) U t p ≤ c R t |α(s, X s )|ds p for t ∈ [r, R), (
(t) (s, x, u, v) := h(s, x, u, v) if s ∈ [t,
v : | d → [0, ∞) ∈ C ∞ 0 with v(x) = 0 for |x| ≥ 1 and | d v(x)dx = 1. For N ≥ 1, ε > 0, x ∈ | d and ξ ∈ | define v ε (x) := 1 ε d v x ε , h ε,N (s, x, u, v) := (v x ε * h N )(s, x, u, v)
where h N := (h

N/ √ d 1 , ..., h N/ √ d d
) with ξ N = (ξ ∧ N ) ∨ (-N ) for ξ ∈ | (so that |h N | ≤ N ) and the notation v x ε indicates that the convolution is taken with respect to x. Assumption (ii) implies that

|h ε,N (s, x, u, v)| ≤ (v x ε * α N )(s, x) + λ|u| + µ|v|. The function h ε,N is uniformly Lipschitz in (x, u, v) as |h ε,N (s, x 1 , u 1 , v 1 ) -h ε,N (s, x 2 , u 2 , v 2 )| ≤ L(|u 1 -u 2 | + |v 1 -v 2 |) + sup s ′ ,x ′ ,u ′ ,v ′ |∇ x ′ h ε,N (s ′ , x ′ , u ′ , v ′ )| |x 1 -x 2 |,
where we note that ∇ x ′ h ε,N is a matrix, and where the matrix λ s t is obtained via the PDE approach, so that we get, a.s., ). Hence, rewriting the dependence with respect to N and ǫ in our estimates, we have proved Applying Fatou's lemma on the left-hand side of ( 27) and dominated convergence on the right-hand side (note that |v x ε * α N | ≤ N and that α is supposed to be continuous in x), we derive Hence there is some N 0 ⊆ [r, R] of Lebesgue measure zero such that

|∇ x ′ h ε,N (s ′ , x ′ , u ′ , v ′ )| = ε -d-1 |ξ-x ′ |≤ε (∇v) x ′ -ξ ε h N (s, ξ, u ′ , v ′ )dξ ≤ ε -1 vol(B 1 (| d ))N ∇v ∞ . (
U 0 r + R r V 0 t dB t = R r h 0 (s, X s , U 0 s , V 0
V N,ε t p ≤ d 2 R t (v x ε * α N )(s, X s ) p √ s -t ds (
V N t p ≤ d 2 R t α N (s, X s ) p √ s -t ds ≤ d 2
V N k s → k V s a.s. for s ∈ N 0 .
Again applying Fatou's lemma gives that are essential so that we briefly recall their construction. For notational simplicity we let t = 0 and omit the superscripts (t, x). For i = 1 one has N r,1 R := 1 R-r R r (σ(s, X s ) -1 ∇X s ∇X -1 r ) * dW s * where ∇X t = ∇ x b(t, X t )∇X t dt + ∇ x σ(t, X t )∇X t dW t with ∇X 0 = I | d , the identity matrix (see, for example, [START_REF] Ma | Representation theorems for backward stochastic differential equations[END_REF][START_REF] Gobet | Sensitivity analysis using Itô-Malliavin calculus and martingales, and application to stochastic optimal control[END_REF]). To consider i = 2 we follow [START_REF] Gobet | Sensitivity analysis using Itô-Malliavin calculus and martingales, and application to stochastic optimal control[END_REF] and let 0 ≤ r < R ≤ T , ρ := (r + R)/2, g : | d → | be a Borel measurable polynomially bounded function and F like in [START_REF] Avikainen | On irregular functionals of SDEs and the Euler scheme[END_REF]. For k = 1, ..., d we have that (∂F/∂x k )(r, X r ) = ❊(F(ρ, X ρ )N r,1 ρ (k)|F r ) a.s. Applying the ∇-operator, which can be justified by standard methods, we derive that, a.s. Therefore we can take N r,2 R (k) := [N ρ,1 R ∇X ρ N r,1 ρ (k) + ∇N r,1 ρ (k)](∇X r ) -1 to obtain ∇ x (∂F/∂x k )(r, X r ) = ❊(g(X R )N r,2 R (k)|F r ) a.s.

V t p ≤ d 2

  Some notation. Given a vector x ∈ | d we denote by |x| its Euclidean norm, for a linear operator D ∈ L(| n , | m ) the symbol |D| stands for the Hilbert-Schmidt norm, where | n and | m are equipped with the standard Euclidean structure. Given D(t, x) ∈ L(| n , | m ) with (t, x) ∈ [0, T ] × | d and 0 < T < ∞, we use D ∞ := sup x∈| d ,t∈[0,T ] |D(t, x)|.

X t = x 0 + t 0 b

 0 (s, X s )ds + t 0 σ(s, X s )dW s with x 0 ∈ | d , where b : [0, T ] × | d → | d and σ : [0, T ] × | d → L(| d , | d ) satisfy the following conditions:

.

  The standard tail estimates for the transition density Γ are re-called in Proposition B.1. They ensure that ∂ ∂t ∇ x F , ∇ x ∂ ∂t F and D m x F with |m| ≤ 3 exist and are continuous on [0, R) × | d . For 0 ≤ t ≤ r < R ≤ T one has that, a.s.,

  Let us consider two independent d-dimensional Brownian motions W and B on the same complete probability space (Ω, F, P) starting in zero, and let us denote by (F W t ) t∈[0,T ] (resp. (F B t ) t∈[0,T ] and (F W,B t ) t∈[0,T ] ) the P-augmentation of the natural filtrations of W (resp. B and (W, B)). For a measurable function η : [0, T ] → [-1, 1] we define the standard d-dimensional F W,B -Brownian motion

Example 4 . 7 ,

 47 Proposition 4.5]. (b) If h l (x) = x α for x ≥ 0 and h(x) = 0 otherwise, and 0 < α < 1 -(1/p), then θ = α + (1/p) according to [22, Example 5.2, Lemma 4.7] and [12, Proposition 4.5].

( 21 )

 21 > 0 depends at most on (p, T, b, σ). (b) We consider Y η -Y and Z η -Z and relate (Y, Z) and (Y η , Z η ) to two BSDEs driven by the same Brownian motion (W, B). This is the purpose of the construction below. Let ϕ = χ [-1/2,1/2] so that sup η∈[-1,1]

because of [ 6 ,

 6 Theorem 4.2]. (c) For the driver f 0 (i.e. η ≡ 0) and terminal condition ξ we have that (Y, [Z, 0]) solves our BSDE. (d) For the driver f η and the terminal condition ξ η we have that

  is of Lebesgue measure zero and such that for G(y; x) := χ F (y)H(y; x) and U (y; t, x) := Y y;t,x t a.s. : r ≤ t < R G(y; x) : t = R , where (Y y;t,x s

  Properties of the function U (d) Let D be the product of [r, b] ⊆ [r, R) where b ∈ (r, R) and a compact subset of | d . For (t, x) ∈ D Lemma A.1 and Proposition B.1 yield

1 ) 2 ( 2 (

 122 |G(y; ξ) -G n (y; ξ)| 2 dξ ≤ c r) |G(y; ξ) -G n (y; ξ)| 2 dξ ≤ c A.1) | d c (B.1) (2π(Rb)) ξ) -G n (y; ξ)| 2 dξ.

  R) and h (t) := 0 otherwise, and the accordingly modified α. So we turn to the statement (2).

  (a) Fix a bump-function

1 ss

 1 b) Fix N ≥ 1 and ε > 0, leth 0 (s, x, u, v) := h ε,N (s, x, u, v)χ [r,R) (s) and α 0 (s, x) := (v x ε * α N )(s, x)χ [r,R) (s). Let (U 0 , V 0 ) be the solution of our BSDE (26) with h replaced by h 0 according to[START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF] Theorem 2.6], where U 0 s := A 0 (s, X s ) for a continuous and bounded functionA 0 : [r, R] × | d → | d . It is also shown that λ × ◗({(t, ω) ∈ [r, R] × M : |V 0 s | > c})= 0 for some c > 0. By considering a Picard iteration U s, X s , A 0 (s, X s ), V 0,k-can show by induction that V 0,k s can be realized as a measurable functional of s and X s and obtains finally that there is a measurablefunction B 0 : [r, R] × | d → | d×d with B 0 ∞ ≤ c,such that one can realize (using uniqueness from [8, Theorem 2.6]) V 0 as V 0 s = B 0 (s, X s ). Now h 0 (s, X s , U 0 s , V 0 s ) = ❊h 0 (s, X s , U 0 dB t a.s.,

  27) for t ∈ M = [r, R]\N N,ε with d 2 := d 1 (1 + 2T ). (c) Let N := N,n N N,1/n and let ((U N t , V N t )) t∈[r,R] be the solution of (26) with the generator h N . Becauselim ε↓0 R r h N,ε (s, X s , U N s , V N s )h N (s, X s , U N s , V N s ) 2 ds = 0by dominated convergence (here we use the continuity of h in x) and|h N,ε (r, x, u 1 , v 1 )h N,ε (r, x, u 2 , v 2 )| ≤ L[|u 1u 2 | + |v 1v 2 |], all N = 1,2, ... Hence there are sub-sequences (n N l ) 0 ◗ × λ a.e. and a Borel set N N ⊆ [0, T ] of Lebesgue measure zero such that V N,1/n N l s → l V N s a.s. for s ∈ N N .

2 2

 2 , X s ) p √ st ds for all t ∈ [r, R]\( ∞ N ′ =1 N N ′ ∪ N ). In the same way, Lemma A.1, R r h N (s, X s , U s , V s )h(s, X s , U s , V s ) 2 ds → N 0 and |h N (s, x, u 1 , v 1 )h N (s, x, u 2 , v 2 )| ≤ L[|u 1u 2 | + |v 1v 2 |] give lim ds = 0 and the existence of a subsequence (N k ) ∞ k=1 such that lim k→∞ |V N k s -V s | = 0 ◗ × λ a.e.

1 (

 1 , X s ) p √ st ds for all N = 1, 2, ... and t ∈ [r, R]\(∞ N ′ =0 N N ′ ∪ N ).B AppendixProposition B.1([9, pp. 260, 72, 74, 44]). For b, σ satisfying (A b,σ ), there exists a continuous transition densityΓ : {(t, x, s, ξ) : 0 ≤ t < s ≤ T and x, ξ ∈ | d } → (0, ∞) such that P(X t,x s ∈ B) = B Γ(t, x; s, ξ)dξ for 0 ≤ t < s ≤ T and B ∈ B(| d ), where X t,x s = x + , X t,x r )dW r ,such that the following is satisfied:(i)For all multi-indices m and k with |m|+2k ≤ 3 one has that the derivatives D k t D m x Γ(t, x; s, ξ) exists and are continuous on [0, s) × | d and that the differentiation can be done in any order. (ii) For 0 ≤ t < s ≤ T and (x, ξ) ∈ | d × | d one has 2 Γ + b, ∇ x Γ = 0 where A = σσ * and D 2 = For all multi-indices m with |m| ≤ 3 there exists a constant c = c m > 0 such that for 0 ≤ t < s ≤ T and (x, ξ) ∈ | d × | d one has that |D m x Γ(t, x; s, ξ)| ≤ c (st) -|m| 2 2πt) d/2 e -|η| 2 2t . Remark B.2. The weights N r,i,(t,x) R

∇ 1 R

 1 x (∂F/∂x k )(r, X r )∇X r = ❊(∇ x F (ρ, X ρ )∇X ρ N r,1 ρ (k) + F (ρ, X ρ )∇N r,1 ρ (k)|F r ) = ❊(❊(g(X R )N ρ,|F ρ )∇X ρ N r,1 ρ (k) + ❊(g(X R )|F ρ )∇N r,1 ρ (k)|F r ).

  K+d -a.e. and |H n (y; x)| ≤ 2ψ(y, x).

			such that
	lim n H n = H λ Hence there is a Borel set F ⊆ | K such that F c is of Lebesgue measure zero and such that
	for all y ∈ | K with	lim n	G n (y; •) = G(y; •) λ d -a.e.
	G n (y; x) := χ F (y)H n (y; x) and G(y; x) := χ F (y)H(y; x).
	Let U n be defined as U with G replaced by G n . Applying [20, Theorems 3.1
	and 4.2] gives that
	(U n (y; s, X t,x s ), ∇ x U n (y; s, X t,x s )σ(s, X t,x s )) s∈[t,R]

  According to [20, Theorem 3.1, Corollary 3.2] the gradient ∇ x U n (y; t, x) exists and is bounded by a constant that might depend on n and y. For r ≤ t ≤ ρ ≤ R and y ∈ | K we define Although A ρ and B ρ might depend on n and y, we do not indicate this for the purpose of notational simplicity. Using [20, Theorem 4.2] yields

	so that, for some c (24) ≥ 1,
			|U n (y; t, x)| ≤ c (24) ψ(y; x).	(24)
	(b) A ρ t	:=	√	ρ -t sup x∈| d	|∇ x U n (y; t, x)| ψ(y; x)	,
		B ρ t	:= sup s∈[t,ρ]	A ρ s .
	|∇ x U n (y; t, x)| = ❊ U n (y; ρ, X t,x ρ )N t,1,(t,x) ρ	+	t	ρ	f (s, X t,x s , Y y,n;t,x s	, Z y,n;t,x s	)N t,1,(t,x) s	ds
	≤ c (24) κ 2	ψ(y; X t,x ρ ) 2 √ ρ -t	+ κ 2		t	ρ	f (s, X t,x s , Y y,n;t,x s	, Z y,n;t,x s

  2) and there exists a Borel set N ⊆ [r, R) of Lebesgue measure zero such that The existence of the unique L p -solution (U, V ) follows from [6, Theorems 4.1 and 4.2] and the statement (1) follows from [6, Proposition 3.2] where we consider the BSDE with the generator h

	V t p ≤ c	t	R	α(s, X s ) p √ s -t	ds
	for all t ∈ [r, R)\N .
	Proof. The local boundedness of κ ensures	R t	α(s,Xs) p √ s-t

ds < ∞ for t ∈ [r, R).

  Applying the same inequality to s ∈ M gives by iteration for t ∈ M that

		V 0 t p
	≤ κ p ′ = κ p ′	t t	R R	ψ(s) + µκ p ′ ψ(s) √ ds + µκ 2 R s √ s -t ψ(w)+µ V 0 w p √ w-s p ′ B 1 2 , 1 2 s -t	dw R t +(µκ p ′ ) 2 B ds ψ(s)ds +		1 2	,	1 2	t	R	V 0 s p ds
	≤	κ p ′ +	√	T µκ 2 p ′ B	1 2	,	1 2	t	R	ψ(s) √ s -t +(µκ p ′ ) 2 B ds	1 2	,	1 2	t	R	V 0 s p ds.
	It follows from the boundedness properties of V 0 s that this reason we can apply Gronwall's lemma to derive		R r	V 0 s p ds < ∞. For
	s )ds s , V 0 ❊h 0 (s, X s , U 0 s )ds + by a stochastic Fubini argument and = R r V 0 t = V 0 t p ≤ R t λ s t p ds V 0 t p ≤ (κ p ′ + √ T µκ 2 p R s α 0 (w, X w ) p dw R r R t λ s t dsdB t √ ds s -t ≤ d 1 (1 + 2T ) R t α 0 (s, X s ) p √ ds s -t with d 1 := κ p ′ + √ T µκ 2 p ′ B 1 2 , 1 2 e (µκ p ′ ) 2 B( 1 2 , 1 2 )T (1 + λc (A.3)(1)
						≤ κ p ′ ≤ κ p ′ = κ p ′	t t t	R R R	h 0 (s, X s , U 0 s , V 0 s ) p √ s -t a 0 (s, X s ) p + λ U 0 s p + µ V 0 ds s p √ s -t ψ(s) + µ V 0 s p √ s -t ds	ds

R t λ s t ds a.s. for a.e. t ∈ [r, R]. If the set of those t is denoted by M, then for t ∈ M, with ψ(s)

:= α 0 (s, X s ) p + λ U 0 s p . ′ B(1/2, 1/2))e (µκ p ′ ) 2 B(1/2,1/2)(R-t) R t ψ(s) √ st ds for t ∈ M. Next we estimate ψ(s) by ψ(s) ≤ α 0 (s, X s ) p + λc (A.3)(1) R s α 0 (w, X w ) p dw,

where we use Lemma A.3(1) (with the same (L, λ, µ)), and get

V 0 t p ≤ d 1 R t α 0 (s, X s ) p +

We would need to write X r,xr s but use simply Xs to shorten the notation.

The author is supported by the Project 133914 of the Academy of Finland. † This research is part of the Chair Financial Risks of the Risk Foundation, the Chair Derivatives of the Future and the Chair Finance and Sustainable Development.

‡ A significant part of this work was done when the third author was affiliated to Grenoble Institute of Technology.

where we used Proposition 1.1. To estimate the remaining first term we use Lemma A.1 and get that u l (x 1 , ..., x l-1 ; r l-1 , x l-1 )u l x ′ 1 , ..., x ′ l-1 ; r l-1 , x l-1 ≤ c (A.1) u l x 1 , ..., x l-1 ; r l , X r l-1 ,x l-1 r l u l x ′ 1 , ..., x ′ l-1 ; r l , X r l-1 ,x l-1

(b) In the second step we verify the fractional smoothness, where we use (4) and therefore the inequalities from step (a). For r l-1 ≤ s < r l , we have

In particular, this expression depends on x 0 , b, σ, r 1 , ..., r l , s and Φ l but not on the specific realization of the diffusion X. Hence we can assume the extended setting from Section 2.2.1. Using inequalities ( 6) and the estimate [START_REF] Zhang | A numerical scheme for BSDEs[END_REF] implies that

✷ 4 Perspectives

As natural steps, which could follow this paper, we see the investigation of more sufficient conditions for the fractional smoothness of a BSDE and the investigation of the limiting case as the number of points r 1 , ..., r L tends to infinity. In this connection the question, to what extend the generator might be pathdependent, is of interest as well. Moreover, the investigation of the above results in the context of other types of BSDEs (for example including reflection) and the development of numerical algorithms based on the discretizations proposed in this paper would be important.

Proof of Proposition 2.12. Assume that we have the diffusions

tively, satisfying our assumptions with the corresponding transition densities Γ 1 and Γ 2 , and assume that they satisfy Γ 1 (t, x; s, ξ) ≤ M Γ 2 (µt, νx; µs, νξ)

for some M, µ, ν > 0 and all x, ξ ∈ | d and 0 ≤ t ≤ s ≤ T 1 and with T 2 = µT 1 .

Let g : | d → | be a polynomially bounded Borel function. Then, for

In fact, we have that

Γ 2 (0, x 2 ; µt, x)Γ 2 (µt, x; T 2 , ξ)Γ 2 (µt, x; T 2 , η)dxdξdη

where we used relation [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF]. This implies our assertion by taking (Γ 1 , x 1 , T 1 ) = (Γ, x 0 , r l ) and T 2 = T 1 , ν = 1/c (B.1) and X 2 t = νx 0 + W t . ✷