Donsker-Type Theorem for BSDEs: Rate of Convergence

(1)

HAL Id: hal-02165750

https://hal.archives-ouvertes.fr/hal-02165750

Preprint submitted on 26 Jun 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Donsker-Type Theorem for BSDEs: Rate of Convergence

Philippe Briand, Christel Geiss, Stefan Geiss, Celine Labart

To cite this version:

Philippe Briand, Christel Geiss, Stefan Geiss, Celine Labart. Donsker-Type Theorem for BSDEs:

Rate of Convergence. 2019. �hal-02165750�

(2)

Donsker-Type Theorem for BSDEs:

Rate of Convergence

Philippe Briand ^∗† Christel Geiss Stefan Geiss Céline Labart^† This version: 2019-06-24

Abstract

In this paper, we study in the Markovian case the rate of convergence in the Wasserstein distance of an approximation of the solution to a BSDE given by a BSDE which is driven by a scaled random walk as introduced in Briand, Delyon and Mémin (Electron. Comm. Pro- bab. 6(2001), 1–14).

1. Introduction.

In this paper, we are concerned with the discretization of solutions to BSDEs of the form Y_t=G(B) +

Z _T

t

f(B_s, Y_s, Z_s)ds− Z _T

t

Z_sdB_s, 0≤t≤T,

where B is a standard Brownian motion. These equations have been introduced by Jean- Michel Bismut for linear generators in [2] and by Étienne Pardoux and Shige Peng for Lipschitz generators in [14].

In one of the first studies on this topic, in the case where the generator f may depend on z as well, Philippe Briand, Bernard Delyon and Jean Mémin [5] proposed an approximation based on Donsker’s theorem. They showed that the solution (Y, Z) to the previous BSDE can be approximated by the solution (Yⁿ, Zⁿ) to the BSDE

Y_tⁿ=G(Bⁿ) + Z

]t,T]

f(Bⁿ_s−, Y_s−ⁿ, Z_sⁿ)dhBⁿi_s+ Z

]t,T]

Z_sⁿdBⁿ_s, 0≤t≤T, where Bⁿ is the scaled random walk

Bⁿ_t = q

T /n

[nt/T]

X

k=1

ξ_k, 0≤t≤T,

and (ξ_k)_k≥1 is an i.i.d. sequence of symmetric Bernoulli random variables. They proved, in full generality, meaning that G(B) is only required to be a square integrable random variable, that (Yⁿ, Zⁿ) converges to (Y, Z). However, the question of the rate of convergence was left open. Right now it seems to be hopeless to get a result in this direction for such a general path- dependent terminal condition G(B).But in the Markovian case, meaning that G(B) = g(B_T),

∗In memory of Jean Mémin from whom I learned lots of mathematics.

†Many thanks to Pierre Baras for very fruitful discussions about the heat equation.

(3)

this problem seems to be tractable, in particular due to the PDE structure behind. Indeed, (Y, Z) is related to the semilinear heat equation

∂tu(t, x) +1

2∆u(t, x) +f(t, x, u(t, x),∇u(t, x)) = 0, (t, x)∈[0, T[×R, u(T,·) =g, (1) where, under certain regularity conditions, we can choose

Ys=u(s, Bs) and Zs=∇u(s, B_s).

In the case where Bⁿ is the discretized Brownian motion, the link to PDEs was exploited in [18,3] to get the rate of convergence, in the Markovian case, of the classical scheme for BSDEs.

The convergence of this scheme was already proved in [6, Proposition 13] for a general terminal condition and a generator that is Lipschitz in its spatial coordinates but without any rate of convergence.

Even though the link with PDEs was pointed out in [5], the rate of convergence of the approximation of BSDEs given by scaled random walks was completely open. In two recent papers, Christel Geiss, Céline Labart and Antti Luoto [10,9] give a first answer to this question.

They showed that the error between (Yⁿ, Zⁿ) and (Y, Z) is of order n^−ε/4 when g is assumed to be ε-Hölder continuous and f(t,·) Lipschitz continuous. One of the main arguments in these papers consists in constructing the random walk from the Brownian motion B using the Skorohod embedding (see [17]) together with generalizations of the pioneering work of Jin Ma and Jianfeng Zhang [13] on representation theorems for BSDEs. This approach allows to work with convergence in the L²-sense even if the problem naturally arises in the weak sense. The drawback is that the rate of convergence n^−ε/4 obtained in these papers is not optimal as one can expect n^−ε/2.

The objective of our study is to improve the previous rate of convergence by using a weak limit approach, the error being measured in the Wasserstein distance. Our starting point is a result of Emmanuel Rio [15] who proved that, whenT = 1, for allr≥1, there exists a constant Cr such that, for alln≥1,Wr(Bⁿ₁, G)≤Crn^−1/2, whereWr is the L^r-Wasserstein distance and Ga standard normal random variable (see Section 3). Firstly, we generalize this result to cover the case wheref ≡0 which corresponds to the heat equation. Then, using the associated PDE, in particular representation formulas in the spirit of [13], we are able to prove that

Wr(Y_tⁿ, Yt)≤Crn^−ε/2, and Wr(Z_tⁿ, Zt)≤ √Cr

T −tn^−ε/2 for all t∈[0, T[,

whengandf(t,·, y, z) areε-Hölder continuous, which gives the correct rate of convergence when the error is measured in the Wasserstein-distance. We refer to Theorem 10 in Section5 for the precise statement.

2. Notation.

In all the sequel, T > 0 is a fixed positive real number. We work on a complete probability space (Ω,F,P) carrying a standard real Brownian motion{B_t}0≤t≤T,and{F_t}0≤t≤T stands for the augmented filtration of B which is right continuous and complete.

We consider the following BSDE Yt=g(B_T) +

Z T t

f(s, Bs, Ys, Zs)ds− Z T

t

ZsdBs, 0≤t≤T. (2) Throughout this article, we will assume for the function g defining the terminal condition and the generatorf the following:

(4)

Assumption (A1). There exist 0< ε≤1 and 0< α≤1 such that it holds:

(i) The functiong:R−→R isε-Hölder continuous: for all (x, x⁰)∈R² one has

g(x)−g x⁰≤ kgk_ε x−x⁰

ε.

(ii) The function f : [0, T]×R×R×R −→ R is α-Hölder continuous in time, ε-Hölder continuous in space and Lipschitz continuous with respect to (y, z): for all (t, x, y, z) and (t⁰, x⁰, y⁰, z⁰) in [0, T]×R×R×R one has

f(t, x, y, z)−f t⁰, x⁰, y⁰, z⁰

≤ kf_tk_α t−t⁰

α+kf_xk_ε x−x⁰

ε+kf_yk_Lip y−y⁰+kf_zk_Lip z−z⁰. (3) Most of the time, we do not need to distinguish between kf_yk_Lip and kf_zk_Lip and we let kfk_Lip:= max (kf_yk_Lip,kf_zk_Lip).

Convention: Later the phrase that a constant C >0 depends on (T, ε, f, g) stands for the fact that C can be expressed in terms of (T, ε,kf_xk_ε,kf_yk_Lip,kf_zk_Lip, K_f,kgk_ε, g(0)) where

K_f := sup

t∈[0,T]

|f(t,0,0,0)|.

Similarly, a dependence on (T, α, ε, f, g) means an additional dependence on (α,kf_tk_α).

From [4, Theorem 4.2] it is known that under (A1), the BSDE (2) has a unique L^p-solution (Y, Z) for anyp∈]1,∞[.So for (t, x)∈[0, T[×Rwe let Y_s^t,x, Z_s^t,x_s∈[t,T]be the square integrable solution to the BSDE

Y_s^t,x =gB^t,x_T + Z T

s

fr, B_r^t,x, Y_r^t,x, Z_r^t,xdr− Z T

s

Z_r^t,xdBr, t≤s≤T, (4) whereB_r^t,x:=x+B_r−B_t, and set, as usual, forx∈R,u(T, x) :=g(x), and, for (t, x)∈[0, T[×R,

u(t, x) :=Y_t^t,x =E

"

g(B_T^t,x) + Z T

t

fr, B_r^t,x, Y_r^t,x, Z_r^t,xdr

# .

It is well known that the function u is continuous on [0, T]×R (see also Lemma 6 below) and under Lipschitz assumptions in (x, y, z) and for α ≥ ¹₂ it is the viscosity solution to (1), see [20, Theorem 5.5.8]. Moreover, in this Markovian setting, for (t, x) ∈ [0, T]×R, we have Y_s^t,x =u(s, B_s^t,x) a.s. for alls∈[t, T]. In [19, Theorem 3.2], for a generator which is Lipschitz continuous in all space variables and a measurable g with polynomial growth, J. Zhang proved that u belongs to C^0,1([0, T[×R) and that Z_s^t,x = ∇u(s, B_s^t,x) a.e. on [t, T[×Ω. Moreover, the following representation holds

∇u(t, x) =E

"

g(B_T^t,x)B_T −B_t T −t +

Z T t

fr, B_r^t,x, Y_r^t,x, Z_r^t,xB_r−B_t r−t dr

#

, (t, x)∈[0, T[×R.

If F is the function given by

F(s, x) :=f(s, x, u(s, x),∇u(s, x)) for (s, x)∈[0, T[×R, (5) we thus have

u(t, x) =E

"

g(B_T^t,x) + Z T

t

F(r, B^t,x_r )dr

#

, (t, x)∈[0, T[×R, (6)

(5)

together with

∇u(t, x) =E

"

g(B_T^t,x)B_T −B_t T−t +

Z T t

F(r, B_r^t,x)B_r−B_t r−t dr

#

, (t, x)∈[0, T[×R. (7) These formulas play an important role in the sequel.

In Section 4 and in the appendix, we extend these results to the case where f(t,·, y, z) is ε-Hölder continuous and make the regularity of uand ∇u precise.

As mentioned before, we are concerned with the approximation of the solution Y^t,x, Z^t,x to (4) by a solution to the BSDE driven by a scaled random walk. To do this, let us consider, on some probability space, not necessarily (Ω,F,P), an i.i.d. sequence (ξk)_k≥1 of symmetric Bernoulli random variables. For n ∈ N^∗ := {1,2,3, ...} we set h := T /n and we consider the scaled random walk

B_tⁿ:=

√ h

[t/h]

X

k=1

ξ_k, 0≤t≤T,

where [x] := max{r∈Z:r ≤x} for any real numberx.As we did for the Brownian motion, for x∈Rand 0≤t≤s≤T we put

B_s^n,t,x:=x+B_sⁿ−B_tⁿ.

Let us introduce some further notation. We denote the ceiling function bydxe := min{r ∈ Z:r≥x}for x∈R.Moreover, we set

nt:= [t/h], t:=h[t/h] =hnt and t:=hdt/he, t∈[0, T].

For n∈N^∗ let us consider the following BSDE driven byBⁿ: Y_tⁿ=g(B_Tⁿ) +

Z

]t,T]

f(s, B_sⁿ−, Y_sⁿ−, Z_sⁿ)dhBⁿi_s− Z

]t,T]

Z_sⁿdB_sⁿ, t∈[0, T].

It was shown in [5] that, as soon as hkfk_Lip < 1, this BSDE has a unique square integrable solution (Yⁿ, Zⁿ), Yⁿ being adapted and Zⁿ being predictable with respect to the filtration generated by Bⁿ. By construction, Yⁿ is a piecewise constant càdlàg process with Y_tⁿ =Y_tⁿ. The process Zⁿ is defined as an element of L²(Ω×[0, T], dP⊗dhBⁿi), where we start with a Zⁿ defined only on the points {kh : k = 1, . . . , n} and extend it to ]0, T] as a càglàd process (Z_tⁿ)_t∈]0,T] by setting Z_tⁿ = Z_tⁿ. The previous BSDE is actually a discrete BSDE that can be solved by hand since, for k= 0,· · · , n−1, we have

Y_khⁿ =Y_(k+1)hⁿ +h f(k+ 1)h, B_khⁿ , Y_khⁿ, Z_(k+1)hⁿ −√

h Z_(k+1)hⁿ ξ_k+1, Y_nhⁿ =g(B_Tⁿ).

Thus, ifY_(k+1)hⁿ is given, Z_(k+1)hⁿ =h^−1/2E

Y_(k+1)hⁿ ξ_k+1| F_khⁿ , (8)

Y_khⁿ =Y_(k+1)hⁿ +h f(k+ 1)h, B_khⁿ , Y_khⁿ, Z_(k+1)hⁿ −√

h Z_(k+1)hⁿ ξk+1

=E

Y_(k+1)hⁿ | F_khⁿ +h f(k+ 1)h, B_khⁿ , Y_khⁿ, Z_(k+1)hⁿ , (9) where the last equality follows by taking the conditional equation w.r.t. F_khⁿ of the second line.

(6)

Since we are in a Markovian setting, there is also an analog of the Feynman-Kac formula. If u is a given function we set

Dⁿ₊u(x) := 1 2

u(x+√

h) +u(x−√

h), Dⁿ₋u(x) := 1 2

u(x+√

h)−u(x−√ h), and

∇ⁿu(x) :=h^−1/2Dⁿ₋u(x). (10)

Remark 1. From the definition ofDⁿ₊ and Dⁿ₋, we get that ifu isε-Hölder,Dⁿ₊u and D₋ⁿu are also ε-Hölder with constantkuk_ε.

LetUⁿbe the solution to the finite difference equation, where forx∈Randk= 0, . . . , n−1 we require

( Uⁿ(kh, x) =Dⁿ₊Uⁿ((k+ 1)h, x) +hf((k+ 1)h, x, Uⁿ(kh, x), h^−1/2D₋ⁿUⁿ((k+ 1)h, x)), Uⁿ(nh, x) =g(x).

(11) Then, we obtain from (8) and (9) (cf [5, Proposition 5.1]) that

Y_khⁿ =Uⁿ kh,

√ h

k

X

i=1

ξ_i

!

, k= 0, . . . , n, Z_khⁿ =∇ⁿUⁿ kh,

√ h

k−1

X

i=1

ξi

!

, k= 1, . . . , n.

These formulas rewrite in continuous time to

Y_tⁿ=Y_tⁿ=Uⁿ(t, B_tⁿ), t∈[0, T], and Z_tⁿ=Z_tⁿ=∇ⁿUⁿ t, B_t−ⁿ , t∈]0, T].

If we set, for 0≤t≤T and x∈R,Uⁿ(t, x) :=Uⁿ(t, x), we haveY_tⁿ=Uⁿ(t, B_tⁿ).

More generally, for 0≤t < T, we define (Y^n,t,x, Z^n,t,x) as the solutionY^n,t,x= (Ys^n,t,x)_s∈[t,T]

and Z^n,t,x= (Zs^n,t,x)_s∈]t,T] to the BSDE Y_s^n,t,x=g(B_T^n,t,x) +

Z

]s,T]

f(r, B_r^n,t,x− , Y_r^n,t,x− , Z_r^n,t,x)dhBⁿi_r− Z

]s,T]

Z_r^n,t,xdBⁿ_r, s∈[t, T]. (12) We set Y_T^{n,T ,x} =g(x). Then,

Y_s^n,t,x=Y_s^n,t,x=Uⁿ(s, B_s^n,t,x), 0≤t≤s≤T. (13) Let us observe that Z^n,t,x is first defined at the pointst=kh, k=nt+ 1, . . . , n. As before we let Zs^n,t,x:=Z_s^n,t,x fors∈]t, T]. We have

Z_s^n,t,x=Z_s^n,t,x=∇ⁿUⁿ(s, B^n,t,x_s− ) fors∈]t, T].

In particular,

Z_s^n,t,x=∇ⁿUⁿ(s+h, B_s^n,t,x) whenever s∈]t, T]\{kh:k=nt+ 1, . . . , n}. (14)

(7)

Of course, we have

Uⁿ(t, x) =Y_t^n,t,x=Y_t^n,t,x fort∈[0, T].

Similarly, we define, for (t, x)∈[0, T[×R,

∆ⁿ(t, x) :=∇ⁿUⁿ(t+h, x) =Z_t+h^n,t,x. (15) With this notation, (14) rewrites as

Z_s^n,t,x= ∆ⁿ(s, B_s^n,t,x) whenever s∈]t, T]\{kh:k=nt+ 1, . . . , n}. (16) It follows that

Uⁿ(t, x) =E



gB_T^n,t,x+h

n

X

k=nt+1

fkh, B_(k−1)h^n,t,x , Y_(k−1)h^n,t,x , Z_kh^n,t,x





=E

"

gB_T^n,t,x+ Z T

t

fs, B_s^n,t,x, Y_s^n,t,x, Z_s^n,t,xds

# ,

which rewrites, taking into account (13) and (16), to Uⁿ(t, x) =E

"

gB_T^n,t,x+ Z T

t

fs, B_s^n,t,x, Uⁿ(s, B_s^n,t,x),∆ⁿ(s, B_s^n,t,x)ds

#

=E

"

gB_T^n,t,x+ Z T

t

fs,Θ^n,t,x_s ds

#

, (17)

where

Θ^n,t,x_s :=B_s^n,t,x, Uⁿ(s, B_s^n,t,x),∆ⁿ(s, B_s^n,t,x). (18) We will prove in Section 5 that (Uⁿ,∆ⁿ) converges to (u,∇u).

From now on we assume thatn≥n0(T,kfk_Lip) wheren0(T,kfk_Lip)∈N^∗ is the integer given in Lemma12 in the appendix and which automatically implies also existence and uniqueness of solutions because n₀ > Tkfk_Lip.

3. Scaled random walk and Wasserstein distance.

One starting point of our paper is the following result of Emmanuel Rio [15] (Theorem 2.1); see also [16]. This result covers, up to a generalization, the case where the generator vanishes, i.e.

f ≡0.

Letψbe the convex function defined by ψ(x) =e^|x|−1. The Orlicz norm associated to this functionψ of any real random variable X is given by

kXk_ψ := inf{a >0 :E[ψ(X/a)]≤1}, inf∅:= +∞.

Let us recall that, for anyr ≥1, sup

x>0

n x^r ψ(x)

o

<+∞, kXk_L^r ≤

sup

x>0

n x^r ψ(x)

o1/r

kXk_ψ. (19) LetX andY be two random variables end let us denote byµthe law ofX and by ν the law of Y. With the usual abuse of notation, the Wasserstein distance associated to ψ is defined by

Wψ(µ, ν) =Wψ(X, Y) := inf{kX−Yk_ψ : law(X) =µ, law(Y) =ν}.

(8)

Let (X_k)_k≥1 be an i.i.d. sequence of random variables withE[X] = 0, EX²= 1,and such that, for some σ > 0, E

he^σ|X^|ⁱ <+∞. Let G be a standard normal random variable. In [15, Theorem 2.1], Emmanuel Rio proved that there exists a constant C >0 such that, forn≥1,

Wψ

n^−1/2Sn, G≤C n^−1/2, where Sn=X1+. . .+Xn. As a byproduct, for anyr ≥1, there exists a constant c_r>0 such that

W_rn^−1/2S_n, G≤c_rn^−1/2, where Wr stands for the L^r-Wasserstein distance

Wr(µ, ν) =Wr(X, Y) := infⁿE[|X−Y|^r]^1/r: law(X) =µ, law(Y) =ν^o. (20) We have also the result of Kantorovich-Rubinstein, i.e.

W₁(µ, ν) =W₁(X, Y) = sup{E[f(X)]−E[f(Y)] :kfk_Lip≤1}. (21) Remark 2. We could also consider the case where 0 < r < 1 by using the fact that, in this case, E(|X −Y|^r) is a distance (see the arguments in [1, Section 7.1]). In general, we have W_p(µ, ν) =W_q(µ, ν) for 0< p < q <∞.

Let us start with a straightforward generalization of Rio’s result.

Proposition 3. There exists a C >0 such that, for all x∈R and all0≤t≤s≤T, W_ψB_s^n,t,x, B^t,x_s ≤C

T n

1/2

.

As a byproduct, taking into account (19), for anyr≥1, there exists a cr>0 such that, for all x∈R and all 0≤t≤s≤T,

Wr

B_s^n,t,x, B_s^t,x≤cr

T n

1/2

. (22)

Proof of Proposition 3. We have, for anyx∈Rand all 0≤t≤s≤T, W_ψ(x+Bⁿ_s −B_tⁿ, x+Bs−Bt) =W_ψ(Bⁿ_s −B_tⁿ, Bs−Bt). If s=t, thenB_tⁿ−B_sⁿ= 0,and we have

Wψ(B_sⁿ−B_tⁿ, Bs−Bt) =kB_s−Btk_ψ =√

s−tkGk_ψ ≤√

hkGk_ψ. Let us assume thatt < s and let us write

W_ψ(Bⁿ_s −B_tⁿ, B_s−B_t) =W_ψB_sⁿ−Bⁿ_t, B_s−B_t

≤W_ψB_sⁿ−B_tⁿ, B_s−B_t+W_ψ B_s−B_t, B_s−B_t. (23) Let us treat each term separately. For the first one, Rio’s result gives

W_ψ





√ 1

n_s−n_t

ns

X

k=nt+1

ξ_k, G



≤C (n_s−n_t)^−1/2,

(9)

and multiplying by√

ns−nt

√

h, we get, since^ph(ns−nt)Gis equal toBs−B_tin distribution, W_ψ B_sⁿ−B_tⁿ, B_s−B_t≤C√

h.

Let us deal with the second term of (23). Let β(s, t) := min(s−t, s−t). Then W_ψ Bs−Bt, Bs−Bt

=W_ψ(N(0, s−t),N(0, s−t))

=Wψ(N(0, β(s, t)),N(0, β(s, t))∗ N(0,|s−t−(s−t)|))

≤W_ψ(0,N(0,|s−t−(s−t)|))

= q

|s−t−(s−t)| kGk_ψ. But |s−t−(s−t)| ≤h,and this concludes the proof.

Let us finish with a simple consequence of this result that we will use in the sequel.

Corollary 4. Let 0 < ε ≤ 1 and let g : R −→ R be an ε-Hölder continuous function. Then there exists a C >0 depending on T such that, for all x∈R and all0≤t≤s≤T,

E

hgB_s^n,t,xⁱ−E

hgB_s^t,xⁱ≤Ckgk_εn^−ε/2,

and, setting δ(t, s) := max (s−t, s−t),

E

hgB_s^n,t,x B_s^n,t,x−xⁱ−E

hgB_s^t,x B_s^t,x−xⁱ≤Ckgk_εδ(t, s)^1/2n^−ε/2.

Proof. Letx∈Rand 0≤t≤s≤T. For any coupling (X, Y) ofB_s^n,t,xand B_s^t,x, using Hölder’s inequality when 0< ε <1,

W₁gB_s^n,t,x, gB_s^t,x≤ kgk_εW₁B_s^n,t,x, B_s^t,x^ε≤ kgk_εc^ε₁ ^T_n^ε/2. Choosing f(x) =x in (21), this implies the first result.

Let us prove the second assertion. We start by observing that, sinceBs−Bt and B_sⁿ−B_tⁿ are centered random variables, we have, setting h(y) := (g(x+y)−g(x))y,

E

hgB_s^t,x B_s^t,x−xⁱ=E[(g(x+Bs−Bt)−g(x)) (Bs−Bt)] =E[h(Bs−Bt)], E

h

gB_s^n,t,x B^n,t,x_s −xⁱ=E[(g(x+Bⁿ_s −B_tⁿ)−g(x)) (B_sⁿ−B_tⁿ)] =E[h(B_sⁿ−B_tⁿ)]. Let us remark that, for any real numbers y and z,|h(y)| ≤ kgk_ε|y|^1+ε, and using the fact that

|y−z|^1−ε≤ |y|^1−ε+|z|^1−ε,

|h(y)−h(z)| ≤ |g(x+y)−g(x)| |y−z|+|(g(x+y)−g(x))−(g(x+z)−g(x))| |z|

≤ kgk_ε(|y−z| |y|^ε+|y−z|^ε|z|)

≤ kgk_ε|y−z|^ε|y−z|^1−ε|y|^ε+|y−z|^ε|z|

≤ kgk_ε|y−z|^ε|y|+|y|^ε|z|^1−ε+|z|.

(10)

Young’s inequality,|y|^ε|z|^1−ε≤ε|y|+ (1−ε)|z| ≤ |y|+|z|, leads to

|h(y)−h(z)| ≤2kgk_ε|y−z|^ε(|y|+|z|). (24) In the case where s=t we have

W1(h(B_sⁿ−Bⁿ_t), h(Bs−Bt))≤E[|h(B_s−Bt)|]≤ kgk_εE

h|B_t−Bs|^(1+ε)ⁱ

≤ kgk_ε(s−t)^(1+ε)/2E

h|G|^(1+ε)ⁱ,

where law(G) =N(0,1). Sinces=timplies (s−t)^1/2=δ(t, s)^1/2 and (s−t)^ε/2 ≤h^ε/2,we have W1(h(B_sⁿ−B_tⁿ), h(Bs−Bt))≤ kgk_εδ(t, s)^1/2(^T_n)^ε/2

using the fact that E

h|G|^(1+ε)ⁱ≤1.

Let us turn to the case t < s. For any coupling (X, Y) of B_sⁿ−B_tⁿ andB_s−B_t, using (20) and (24),

W1(h(B_sⁿ−B_tⁿ), h(Bs−Bt))≤E[|h(X)−h(Y)|]

≤2kgk_εE[|X−Y|^ε(|X|+|Y|)], and, by Hölder’s inequality with p= 2/εand q = 2/(2−ε),

W1(h(B_sⁿ−B_tⁿ), h(Bs−Bt))≤2kgk_εE

h|X−Y|²ⁱ^ε/2 E[|X|^2/(2−ε)]^1−ε/2+E[|Y|^2/(2−ε)]^1−ε/2

≤2kgk_εE

h|X−Y|²ⁱ^ε/2(s−t)^1/2+ (s−t)^1/2. From (20) it follows that

W₁(h(B_sⁿ−B_tⁿ), h(B_s−B_t))≤2kgk_εW₂(B_sⁿ−B_tⁿ, B_s−B_t)^ε (s−t)^1/2+ (s−t)^1/2

≤4 c^ε₂kgk_ε T

n ε/2

δ(t, s)^1/2,

where we have used (22) forr = 2.

Thus, for 0≤t≤s≤T,

W1(h(B_sⁿ−Bⁿ_t), h(Bs−Bt))≤Ckgk_εn^−ε/2δ(t, s)^1/2, and the result follows as before by choosing f(x) =x in (21).

4. Regularity results on u, Uⁿ, ∇u and ∆ⁿ.

Let us start by known regularity properties of the function u that follow from classical a priori estimates for BSDEs.

Lemma 5. Under Assumption(A1) there exists a constantC >0depending on (T, ε, f, g)such that, for all (t, x)∈[0, T]×R,

|u(t, x)| ≤C(1 +|x|)^ε, ku(t,·)k_ε ≤C, ku(·, x)k_ε/2 ≤C(1 +|x|)^ε.

(11)

Proof. The first two results follow directly from classical a priori estimates for BSDEs, see e.g.

[8, Proposition 2.1]. The last one ensues from the following upper bound: for any realx and for 0≤r ≤t≤T,

|u(r, x)−u(t, x)|=|E[Y_r^r,x]−E[Y_t^t,x]|

≤ |E[Y_r^r,x−Y_t^r,x]|+|E[Y_t^r,x−Y_t^t,x]|

≤ Z t

r E[|f(s, B_s^r,x, Y_s^r,x, Z_s^r,x)|]ds+E[|Y_t^r,x−Y_t^t,x|]

≤C Z t

r E[1 +|B^r,x_s |^ε+|Y_s^r,x|+|Z_s^r,x|]ds+E[|Y_t^r,x−Y_t^t,x|].

Since the norm inS²× H² of (Y^r,x, Z^r,x) is of order (1 +|x|)^ε, we use Cauchy-Schwarz inequality to bound the first term and a priori estimates enable (similarly as in the proof of [8, Proposition 4.1]) to bound the second term.

Next we extend [19, Theorem 3.2] to the case wheref(t,·, y, z) is Hölder continuous.

Lemma 6. Recall the notation (5) and let Assumption (A1) hold.

(a) The function u belongs to C^0,1([0, T[×R) and, for all (t, x)∈[0, T[×R, we have,

Z_s^t,x =∇u(s, B_s^t,x) for a.e.(s, ω)∈[t, T[×Ω, (25) as well as (7) i.e.

∇u(t, x) =E

g(B_T^t,x)BT −Bt

T−t

+E Z T

t

F(s, B_s^t,x)Bs−Bt

s−t ds

! .

(b) Moreover, there exists a constant C >0 depending on (T, ε, f, g) such that, (i) k∇u(t,·)k_ε≤ ^√^C

T−t for all t∈[0, T[, (ii) |∇u(t, x)| ≤ ^C

(T−t)^(1−ε)/2 for all (t, x)∈[0, T[×R. Consequently, forEr :=E[· |F_r],

∇u(r, B_r^t,x) =Er

g(B_T^t,x)B_T −B_r T −r

+Er

Z T r

F(s, B_s^t,x)B_s−B_r s−r ds

!

a.s. for r∈[t, T[.

(26) Proof of Lemma 6. The proof is divided into two steps.

Step 1. We assume in addition thatf is Lipschitz continuous w.r.t. x. Then according to [19], we have only the second point to prove and we know that, for some constant C,

|∇u(t, x)| ≤ C(1 +|x|)

√T−t on [0, T[×R. (27)

(bi) The representation (26) yields to k∇u(r, B_r^t,x)− ∇u(r, B^t,y_r )k_L2 ≤

Er

g(B_T^t,x)−g(B_T^t,y)B_T −B_r T −r

_L₂ +

Z T r

Er

F(s, B_s^t,x)−F(s, B_s^t,y)B_s−B_r s−r

_L2

ds. (28)

(12)

Since g isε-Hölder continuous we get

Er

(g(B_T^t,x)−g(B_T^t,y))BT −Br

T −r

≤ kgk_ε|x−y|^ε 1

√T−r. Similarly, we obtain by the conditional Cauchy-Schwarz inequality the estimate

Er

(F(s, B_s^t,x)−F(s, B_s^t,y))Bs−Br

s−r

_L₂

≤F(s, B_s^t,x)−F(s, B_s^t,y)

L²

√ 1 s−r. Using (3) for f, we have

|F(s, B_s^t,x)−F(s, B_s^t,y)|

|F(s, B_s^t,x)−F(s, B_s^t,y)|

≤(kf_xk_ε+Ckf_yk_Lip)|x−y|^ε+kf_zk_Lip|∇u(s, B_s^t,x)− ∇u(s, B_s^t,y)|. (29) By combining the above estimates we conclude from (28) that

k∇u(r, B_r^t,x)− ∇u(r, B^t,y_r )k_L2 ≤ kgk_ε|x−y|^ε

√T−r + 2 (kf_xk_ε+Ckf_yk_Lip)|x−y|^ε√ T−r

+ Z T

r

∇u(s, B_s^t,x)− ∇u(s, B_s^t,y)

L²

kf_zk_Lip

√s−rds

≤C₁|x−y|^ε

√T −r + Z T

r

∇u(s, B_s^t,x)− ∇u(s, B^t,y_s )

L²

kf_zk_Lip

√s−rds forC1 :=kgk_ε+ 2T(kf_xk_ε+Ckf_yk_Lip).Because of (27) we have

∇u(s, B_s^t,x)− ∇u(s, B_s^t,y)

L² ≤C2

1 +|x|+|y|

(T−s)^1/2

with C₂ =C₂(C, T)>0. Hence we may apply Gronwall’s lemma (Lemma14) and get k∇u(r, B_r^t,x)− ∇u(r, B^t,y_r )k_L2 ≤C₁c₀|x−y|^ε

√T−r, for somec₀ =c₀(T,kf_zk_Lip)>0.Especially, for r=tthis implies

|∇u(t, x)− ∇u(t, y)| ≤C|x−y|^ε

√T −t for someC =C(T, ε, f, g)>0.

(bii) We first notice that for any ε-Hölder continuous function k and for all 0≤t < s ≤T we have

E

k(B_s^t,x)Bs−Bt

s−t

= E

(k(B^t,x_s )−k(x))Bs−Bt

s−t

≤ kkk_ε

(s−t)^(1−ε)/2. (30) Therefore, we obtain from (7) that

|∇u(t, x)| ≤ E

g(B_T^t,x)B_T −B_t T−t

+ E

Z _T

t

F(s, B_s^t,x)B_s−B_t s−t ds

!

≤ kgk_ε

(T−t)^(1−ε)/2 + Z T

t

kF(s,·)k_ε (s−t)^(1−ε)/2ds.