1Introduction FedericaMasiero AdrienRichou AnoteontheexistenceofsolutionstoMarkoviansuperquadraticBSDEswithanunboundedterminalcondition

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.18(2013), no. 50, 1–15.

ISSN:1083-6489 DOI:10.1214/EJP.v18-2124

A note on the existence of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition

Federica Masiero

Adrien Richou

Abstract

In [17], the author proved the existence and the uniqueness of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition when the generator and the terminal condition are locally Lipschitz. In this paper, we prove that the existence result remains true for these BSDEs when the regularity assumptions on the terminal condition is weakened.

Keywords: Backward stochastic differential equation ; Generator of superquadratic growth ; Unbounded terminal condition ; Existence result.

AMS MSC 2010:60H10.

Submitted to EJP on June 29, 2012, final version accepted on April 12, 2013.

1 Introduction

Since the early nineties and the work of Pardoux and Peng [15], there has been an increasing interest for backward stochastic differential equations (BSDEs for short) because of the wide range of applications. A particular class of BSDE is studied since few years: BSDEs with generators of quadratic growth with respect to the variablez (quadratic BSDEs for short). See e.g. [12, 2, 6] for existence and uniqueness results and [19, 11, 13] for applications.

Naturally, we could also wonder what happens when the generator has a superqua- dratic growth with respect to the variablez. Up to our knowledge the case of super- quadratic BSDEs was firstly investigated in the recent paper [5]. In this article, the authors consider superquadratic BSDEs when the terminal condition is bounded and the generator is convex in z. Firstly, they show that in a general way the problem is ill-posed: given a superquadratic generator, there exists a bounded terminal condition such that the associated BSDE does not admit any bounded solution and, on the other hand, if the BSDE admits a bounded solution, there exist infinitely many bounded solu- tions for this BSDE. In the same paper, the authors also show that the problem becomes

∗Dipartmento di Matematica e Applicazioni, Università di Milano Bicocca, Italy.

E-mail:[email protected]

†Université Bordeaux 1, IMB, UMR 5251, France. INRIA, Équipe ALEA, France.

E-mail:[email protected]

well-posed in a Markovian framework: when the terminal condition and the generator are deterministic functions of a forward SDE, we have an existence result. More pre- cisely, let us consider(X, Y, Z)the solution to the (decoupled) forward backward system

X_t = x+ Z t

0

b(s, X_s)ds+ Z t

0

σ(s)dW_s,

Yt = g(XT) + Z T

t

f(s, Xs, Ys, Zs)ds− Z T

t

ZsdWs, with growth assumptions

|f(t, x, y, z)| 6 C(1 +|x|^pf +|y|+|z|^l+1), l >1,

|g(x)| 6 C(1 +|x|^pg).

In [5], the authors obtain an existence result by assuming that pg = pf = 0, f is a convex function that depends only on z and g is a lower (or upper) semi-continuous function. As in the quadratic case it is possible to show that the boundedness of the terminal condition is a too strong assumption: in [17], the author shows an existence and uniqueness result by assuming thatp_g 61 + 1/l,p_f 61 + 1/l, f andg are locally Lipschitz functions with respect toxandz. When we consider this result, two questions arise:

• Could we have an existence result whenp_gorp_f is greater than1 + 1/l?

• Could we have an existence result whenf orgis less smooth with respect toxor z, that is to say, is it possible to have assumptions on the growth ofgandfbut not on the growth of their derivatives with respect toxandz?

For the first question, the answer is clearly “no” in the quadratic case: see e.g. [6]. In the superquadratic case, the authors of [10] have obtained the same limitation on the growth of the initial condition for the so-called generalized deterministic KPZ equation ut=uxx+λ|ux|^qand they show that this boundary is sharp for power-type initial condi- tions. So, it seems that the answer of the first question is also “no” in the superquadratic case.

For the second question, the answer is clearly “yes” in the quadratic case. Indeed, a smoothness assumption onf is required for uniqueness results (see e.g. [3, 6]) but not for existence results (see e.g. [3, 1]). In the superquadratic case, the authors of [5] show an existence result wheng is only lower (or upper) semi-continuous but also bounded. Neverthelessf(z)is assumed to be convex, that implies that it is a locally Lipschitz function. The aim of this note is to mix results of articles [5, 17] to obtain an existence result when the terminal condition is only lower (or upper) semi-continuous and unbounded. Let us remark that we answer only partially to the second question because we do not relax smoothness assumptions onf.

For completeness, in the paper [4], Cheridito and Stadje show an existence and uniqueness result for superquadratic BSDEs in a Lipschitz or bounded “path-dependent”

framework: the terminal condition and the generator are Lipschitz or bounded func- tions of Brownian motion paths. To the best of our knowledge, [5, 17, 4] are the only papers that deal with superquadratic BSDEs.

The paper is organized as follows. In section 2 we obtain some general a priori es- timates onY andZfor Markovian superquadratic BSDEs whereas section 3 is devoted to the existence result described before.

Notations Throughout this paper,(Wt)t>0 will denote ad-dimensional Brownian mo- tion, defined on a probability space(Ω,F,P). For t > 0, let Ft denote the σ-algebra

σ(W_s; 06s6t), augmented with theP-null sets ofF. The Euclidean norm onR^{d will} be denoted by|.|. The operator norm induced by|.|on the space of linear operators is also denoted by|.|. The notationE^tstands for the conditional expectation givenFt. For p>2,m∈N, we denote further

• S^p the space of real-valued, adapted and càdlàg processes (Yt)_t∈[0,T] normed by kYk_Sp=E[(sup_t∈[0,T]|Yt|)^p]^1/p;

• M^p(R^m), orM^p, the space of all progressively measurable processes (Z_t)_t∈[0,T]

with values inR^{mnormed by}kZk_Mp=E[(RT

0 |Zs|²ds)^p/2]^1/p.

In the following, we keep the same notationC for all finite, nonnegative constants that appear in our computations.

In this paper we considerX the solution to the SDE Xt=x+

Z t 0

b(s, Xs)ds+ Z t

0

σ(s)dWs, (1.1)

and(Y, Z)∈ S²× M²the solution to the Markovian BSDE Y_t=g(X_T) +

Z T t

f(s, X_s, Y_s, Z_s)ds− Z T

t

Z_sdW_s. (1.2)

By a solution to the BSDE (1.2) we mean a pair(Yt, Zt)_t∈[0,T] of predictable processes with values in R×R^{1×d such that} P^-a.s., t 7→ Yt is continuous, t 7→ Zt belongs to L²([0, T]), t 7→ f(t, X_t, Y_t, Z_t) belongs to L¹([0, T]) and P−a.s. the equation (1.2) is verified.

2 Some a priori estimates on Y and Z

For the SDE (1.1) we use standard assumption.

Assumption (F.1). Let b : [0, T]×R^d → R^{d and} σ : [0, T] → R^d×d be continuous functions and let us assume that there existsKb>0such that:

(a) ∀t∈[0, T],|b(t,0)|6C,

(b) ∀t∈[0, T],∀(x, x⁰)∈R^d×R^d,|b(t, x)−b(t, x⁰)|6Kb|x−x⁰|.

Let us now consider the following assumptions on the generator and on the terminal condition of the BSDE (1.2).

Assumption (B.1). Letf : [0, T]×R^d×R×R^1×d →Rbe a continuous function and let us assume that there exist five constants,l >1,06rf < ¹_l,β >0, γ>0andδ>0 such that:

(a) for each(t, x, y, y⁰, z)∈[0, T]×R^d×R×R×R^1×d,

|f(t, x, y, z)−f(t, x, y⁰, z)|6δ|y−y⁰|; (b) for each(t, x, y, z, z⁰)∈[0, T]×R^d×R×R^1×d×R^1×d,

|f(t, x, y, z)−f(t, x, y, z⁰)|6 C+γ

2(|z|^l+|z⁰|^l)

|z−z⁰|;

(c) for each(t, x, x⁰, y, z)∈[0, T]×R^d×R^d×R×R^1×d,

|f(t, x, y, z)−f(t, x⁰, y, z)|6

C+β

2(|x|^rf +|x⁰|^rf)

|x−x⁰|.

Assumption (TC.1). Letg:R^d→Rbe a continuous function and let us assume that there exist0 6r_g < ¹_l and α>0 such that: for each(t, x, x⁰, y, z)∈ [0, T]×R^d×R^d× R×R^1×d,

|g(x)−g(x⁰)|6 C+α

2(|x|^rg+|x⁰|^rg)

|x−x⁰|.

We also use more general growth assumptions that are more natural for existence results.

Assumptions (B.2). Letf : [0, T]×R^d×R×R^1×d →Rbe a continuous function and let us assume that there exist constants,l >1,06rf < ¹_l,β¯>0,¯γ>0,δ¯>0,06η < l+ 1, ε >0such that: one of these inequalities holds, for all(t, x, y, z)∈[0, T]×R^d×R×R^1×d, (a) |f(t, x, y, z)|6C+ ¯β|x|^rf+1+ ¯δ|y|+ ¯γ|z|^l+1,

(b) −C−β¯|x|^rf+1−δ¯|y| −γ¯|z|^η 6f(t, x, y, z)6C+ ¯β|x|^rf+1+ ¯δ|y|+ ¯γ|z|^l+1, (c) −C−β¯|x|^rf+1−δ¯|y|+ε|z|^l+16f(t, x, y, z)6C+ ¯β|x|^rf+1+ ¯δ|y|+ ¯γ|z|^l+1.

Assumption (TC.2). Letg : R^d →Rbe a lower semi-continuous function and let us assume that there exist06pg<1 + 1/landα¯>0such that: for eachx∈R^d,

|g(x)|6C+ ¯α|x|^pg. Remark 2.1. The following relations hold true:

• (B.2)(c)⇒(B.2)(b)⇒(B.2)(a).

• (B.1)⇒(B.2)(a).

• (TC.1)⇒(TC.2) withpg=rg+ 1.

• We only consider superquadratic BSDEs, so l > 1. l = 1 corresponds to the quadratic case.

Firstly, let us recall the existence and uniqueness result shown in [17].

Proposition 2.2. We assume that (F.1), (B.1) and (TC.1) hold. There exists a solution (Y, Z)of the Markovian BSDE (1.2) inS²× M²such that,

|Z_t|6A+B(|X_t|^rg+ (T−t)|X_t|^rf), ∀t∈[0, T]. (2.1) Moreover, this solution is unique amongst solutions(Y, Z)such that

• Y ∈ S²,

• there existsη >0such that

E

e^(12+η)γ

2 4

RT 0|Zs|^2lds

<+∞.

Remark 2.3. To be precise, in the Proposition 2.2 of the article [17] the author shows the estimate

|Z_t|6A+B|X_t|^rg∨rf, ∀t∈[0, T],

but it is rather easy to do the proof again to show the estimate (2.1) given in Proposition 2.2.

Such a result allows us to obtain a comparison result.

Proposition 2.4. We assume that (F.1) holds. Letf₁,f₂two generators andg₁,g₂two terminal conditions such that (B.1) and (TC.1) hold. Let (Y¹, Z¹)and (Y², Z²) be the associated solutions given by Proposition 2.2. We assume that g1 6 g2 and f1 6 f2. Then we have thatY¹6Y²almost surely.

Proof of Proposition 2.4. The proof is the same than the classical one that can be found in [7] for example. Let us setδY :=Y¹−Y²andδZ :=Z¹−Z². The usual linearization trick gives us

δYt=g1(XT)−g2(XT)+

Z T t

f1(s, Xs, Y_s¹, Z_s¹)−f2(s, Xs, Y_s¹, Z_s¹)+δYsUs+δZsVsds−

Z T t

δZsdWs,

with|Us|6δand

|Vs|6C+γ 2

Z_s¹

l+ Z_s²

l

6C(1 +|Xs|^(rg∨rf)l).

Since(rg∨rf)l < 1, Novikov’s condition is fulfilled and we are allowed to apply Gir- sanov’s transformation:

δY_t = E^Qt

"

e^RtTUudu(g₁(X_T)−g₂(X_T)) + Z T

t

e^RtsUudu(f₁(s, X_s, Y_s¹, Z_s¹)−f₂(s, X_s, Y_s¹, Z_s¹))ds

#

6 0, with

dQ dP = exp

Z T 0

V_sdW_s−1 2

Z T 0

|V_s|²ds

! .

Now we are ready to prove estimates onY andZ.

Proposition 2.5. Let us assume that (F.1), (B.1), (B.2), (TC.1) and (TC.2) hold. Let (Y, Z)be the solution of the BSDE (1.2) given by Proposition 2.2. Then we have, for all t∈[0, T],

|Yt|6C(1 +|Xt|^pg + (T −t)|Xt|^rf+1)

with a constantCthat depends on constants that appear in assumptions (F.1), (B.2) and (TC.2) but not in assumptions (B.1) and (TC.1).

Proof of Proposition 2.5. Let us consider the terminal condition

¯

g(x) =C+ ¯α(|x|+ 1)^pg, and the generator

f¯(t, x, y, z) =C+ ¯β|x|^rf+1+ ¯δ|y|+ ¯γ|z|^l+1,

withCsuch thatg6¯gandf 6f¯. (B.1) holds forf¯and (TC.1) holds for¯g, so, according to Proposition 2.2, there exists a unique solution( ¯Y ,Z¯)to the BSDE

Y¯t= ¯g(XT) + Z T

t

f¯(s, Xs,Y¯s,Z¯s)ds− Z T

t

Z¯sdWs.

Thanks to Proposition 2.4, we know that

Y 6Y ,¯ and Y¯ >0.

Moreover, since Z¯s

6C(1 +|Xs|^{(pg−1)∨rf}),(pg−1)l <1andrfl <1, we have Y¯t 6 E^t

"

e^¯δ(T−t)(C+ ¯α(|Xt|+ 1)^pg) + Z T

t

e^¯δ(s−t)(C+ ¯β|Xs|^rf+1+ ¯γ Z¯s

l+1)ds

#

6 C

1 +Et

sup

t6s6T

|X_s|^pg

+ (T−t)Et

sup

t6s6T

|X_s|^rf+1

.

Let us remark that the constantCin the a priori estimate forZ¯ depends on constants that appear in assumptions (F.1), (B.2) and (TC.2) but not in assumptions (B.1) and (TC.1). Thanks to classical estimates on SDEs we have, for allp>1,

Et

sup

t6s6T

|X_s|^p

6C(1 +|X_t|^p),

so we obtain

Yt6Y¯t6C(1 +|Xt|^pg+ (T−t)|Xt|^rf+1).

By the same type of argument we easily show that

−C(1 +|Xt|^pg+ (T−t)|Xt|^rf+1)6Yt, and this concludes the proof.

Proposition 2.6. Let us assume that (F.1), (B.1), (B.2)(c), (TC.1) and (TC.2) hold. Let (Y, Z)be the solution of the BSDE (1.2) given by Proposition 2.2. Then, for allt∈[0, T], we have

E^t

"

Z T t

|Zs|^l+1ds

#

6C(1 +|Xt|^pg+ (T−t)|Xt|^rf+1),

with a constantC that depends on constants that appear in assumptions (F.1), (B.2)(c) and (TC.2) but not in assumptions (B.1) and (TC.1).

Proof of Proposition 2.6. To show the proposition we just have to write E^t

"

Z T t

|Zs|^l+1ds

# 6 1

ε E^t

"

Z T t

f(s, Xs, Ys, Zs)ds+ Z T

t

C+ ¯β|Xs|^rf+1+ ¯δ|Ys| ds

#!

6 1 ε E^t

"

Yt−g(XT) + Z T

t

C+ ¯β|Xs|^rf+1+ ¯δ|Ys|ds

#!

6 C(1 + (T−t)|Xt|^rf+1+|Xt|^pg) thanks to Proposition 2.5.

Remark 2.7. Proposition 2.6 stays true if we replace assumption (B.2)(c) by

−C−β¯|x|^rf+1−δ¯|y| −γ¯|z|^l+16f(t, x, y, z)6C+ ¯β|x|^rf+1+ ¯δ|y| −ε|z|^l+1. Remark 2.8. In Propositions 2.5 and 2.6 we insist on the fact thatC does not depend on constants that appear in assumptions (B.1) and (TC.1) when the local Lipschitzianity of the coefficients is stated. Thanks to this property, we can use these a priori estimates on Y and Z in the following section where we obtain an existence result when the terminal condition is not locally Lipschitz.

3 An existence result

Let us now introduce new assumptions.

Assumption (F.2). b is differentiable with respect to xand σ is differentiable with respect tot. There existsλ∈R^{+such that}∀η∈R^d

tησ(s)[^tσ(s)^t∇b(s, x)−^tσ⁰(s)]η 6λ

^tησ(s)

2, ∀(s, x)∈[0, T]×R^d.

Remark 3.1. It is shown in part 5.5.1 of [18] that if σdoes not depend on time, as- sumption (F.2) is equivalent to this kind of commutativity assumption:

• there existA: [0, T]×R^d→R^d×dandB: [0, T]→R^{d×dsuch that}Ais differentiable with respect tox,∇xAis bounded and∀x∈R^d,∀s ∈[0, T],b(s, x)σ=σA(s, x) + B(s).

It is also noticed in [18] that this assumption allows us to reduce assumption on the regularity ofbby a standard smooth approximation ofA.

Assumption (B.3). f is differentiable with respect tozand for all(t, x, y, z)∈[0, T]× R^d×R×R^1×d,

f(t, x, y, z)− h∇_zf(t, x, y, z), zi6C−ε|z|^l+1.

Remark 3.2. Let us give some substantial examples of functions such that (B.3) holds.

If we assume thatf(t, x, y, z) :=f1(t, x, y, z)+f2(t, x, y, z)withf1a differentiable function with respect tozsuch that,∃p∈[0, l[,∀(t, x, y, z)∈[0, T]×R^d×R×R^1×d,

|∇zf1(t, x, y, z)|6(1 +|z|^p),

andf2is a twice differentiable function with respect tozsuch that,∀(t, x, y, z)∈[0, T]× R^d×R×R^1×d,∀u∈R^d,

tu∇²_zzf₂(t, x, y, z)u>(−C+ε|z|^l−1)|u|², then we easily see that

f1(t, x, y, z)− h∇zf1(t, x, y, z), zi6C+C|z|^p+1, and a direct application of Taylor expansion with integral form gives us

f₂(t, x, y, z)− h∇zf₂(t, x, y, z), zi6C−C⁰|z|^l+1,

so (B.3) holds. For example, (B.3) holds for the functionz7→C|z|^l+1+h(|z|^l+1−η)with C >0,0< η6l+ 1andha differentiable function with a bounded derivative.

Proposition 3.3. Let us assume that (F.1), (F.2), (B.1), (B.3), (TC.1) and (TC.2) hold.

Let(Y, Z)be the solution of the BSDE (1.2) given by Proposition 2.2. If we assume that 06pgl <1, then we have, for allt∈[0, T[,

|Zt|6 C(1 +|Xt|^pg/(l+1))

(T−t)^1/(l+1) +C|Xt|^rf

+1 l+1 .

The constantCdepends on constants that appear in assumptions (F.1), (F.2), (B.1), (B.3) and (TC.2) but not in assumption (TC.1).

Proof of Proposition 3.3. Firstly we approximate our Markovian BSDE by another one.

Let(Y^M, Z^M)the solution of the BSDE Y_t^M =gM(XT) +

Z T t

fM(s, Xs, Y_s^M, Z_s^M)ds− Z T

t

Z_s^MdWs, (3.1)

withg_M = g◦ρ_M and f_M = f(., ρ_M(.), ., .)where ρ_M is a smooth modification of the projection on the centered Euclidean ball of radiusM such that|ρ_M| 6M, |∇ρ_M| 61 andρM(x) =xwhen|x| 6M −1. It is now easy to see thatgM andfM are Lipschitz functions with respect tox. Proposition 2.3 in [17] gives us thatZ^M is bounded by a constantC0 that depends on M. So,fM is a Lipschitz function with respect to z and BSDE (3.1) is a classical Lipschitz BSDE. Now we use the following Lemma that will be shown afterwards.

Lemma 3.4. Let us assume that (F.1), (F.2), (B.1), (B.3), (TC.1) and (TC.2) hold. We also assume that06pgl <1. Then we have, for allt∈[0, T[,

Z_t^M

6 An+Bn|Xt|^pg/(l+1)

(T−t)^1/(l+1) +Dn|Xt|^rf

+1 l+1 ,

with(A_n, B_n, D_n)_n∈Ndefined by recursion:B₀= 0,D₀= 0,A₀=C₀T^1/(l+1), An+1=C(1 +A^al_n +B_n^alp+D^al¯_n^p), Bn+1=C, Dn+1=C,

wherea:= (p_g∨(r_f+ 1))/(l+ 1),p >1,p >¯ 1andCis a constant that does not depend onM and constants in assumption (TC.1).

Sinceal <1, the recursion function that define the sequence(An)n>0 is a contrac- tor function, so An → A∞ whenn → +∞, withA∞ that does not depend on M and constants in assumption (TC.1). Finally, we have, for allt∈[0, T[,

Z_t^M

6C(1 +|Xt|^pg/(l+1))

(T−t)^1/(l+1) +C|Xt|^rf

+1 l+1 .

The constantCdepends on constants that appear in assumptions (F.1), (F.2), (B.1), (B.3) and (TC.2) but not in assumption (TC.1). MoreoverC does not depends on M. Now, we want to come back to the initial BSDE (1.2). It is already shown in the proof of Proposition 2.2 of the article [17] that(Yⁿ, Zⁿ)→ (Y, Z)inS²× M². So our estimate onZ^M stays true for a version ofZ.

Proof of Lemma 3.4. Let us prove the result by recursion. Forn = 0we have already shown the result. Let us assume that the result is true for somen∈Nand let us show that it stays true forn+ 1. In a first time we suppose thatf andgare differentiable with respect toxandy. Then(Y^M, Z^M)is differentiable with respect toxand(∇Y^M,∇Z^M) is the solution of the BSDE

∇Y_t^M = ∇gM(XT)∇XT − Z T

t

∇Z_s^MdWs+ Z T

t

∇xfM(s, Xs, Y_s^M, Z_s^M)∇Xsds

+ Z T

t

∇yfM(s, Xs, Y_s^M, Z_s^M)∇Y_s^M +∇zfM(s, Xs, Y_s^M, Z_s^M)∇Z_s^Mds,

and a version ofZ^M is given by(∇Y_t^M(∇X_t)⁻¹σ(t))_t∈[0,T]. Let us introduce some nota- tions: we set

dW˜_t := dW_t− ∇zf_M(t, X_t, Y_t^M, Z_t^M)dt, α_t :=

Z t 0

e^{R0s∇yfM(u,Xu,YuM,ZuM)du}∇_xf_M(s, X_s, Y_s^M, Z_s^M)∇X_sds(∇X_t)⁻¹σ(t), Z˜_t^M := e^{R0t∇yfM(s,Xs,YsM,ZsM)ds}Z_t^M +αt.

By applying Girsanov’s theorem we know that there exists a probabilityQ^M under which W˜ is a Brownian motion with

dQ^M dP = exp

Z T 0

∇zf_M(t, X_t, Y_t^M, Z_t^M)dW_t−1 2

Z T 0

∇zf_M(t, X_t, Y_t^M, Z_t^M)

2dt

! .

Then, exactly as in the proof of Theorem 3.3 in [16], we can show the following lemma.

Lemma 3.5.

e^λtZ˜_t^M

2

is aQ^M-submartingale.

For the reader’s convenience, we recall this proof in the appendix. It results that

e^λtZ˜_t^M

l+1

is also aQ^M-submartingale and we have:

E^Qt^M

"

Z T t

e^2λs

Z˜_s^M

l+1

ds

#

> e^2λt

Z˜_t^M

l+1

(T−t)

> e^2λt

e^{R0t∇yfM(s,Xs,YsM,ZsM)ds}Z_t^M+αt

l+1

(T−t), which implies

Z_t^M

l+1(T −t)

6 C

e^2λt

e^{R0t∇yfM(s,Xs,YsM,ZsM)ds}Z_t^M +αt

l+1

+|αt|^l+1

(T−t)

6 C E^Qt^M

"

Z T t

e^2λs

Z˜_s^M

l+1

ds

#

+ (T−t)

1 +|X_t|^(l+1)rf

!

6 C 1 +E^Qt^M

"

Z T t

Z_s^M

l+1ds

# +E^Qt^M

"

Z T t

|Xs|^(l+1)rfds

#

+ (T−t)|Xt|^(l+1)rf

! . (3.2) Let us recall that(Y^M, Z^M)is solution of BSDE

Y_t^M =gM(XT) + Z T

t

f˜M(s, Xs, Y_s^M, Z_s^M)ds− Z T

t

Z_s^MdW˜s,

with

f˜_M(s, x, y, z) :=f_M(s, x, y, z)− hz,∇zf_M(s, x, y, z)i.

Since assumption (B.3) holds forf, assumption (B.2)(c) holds for −f˜M with constants that do not depend on M. Then we can mimic the proof of Proposition 2.6 (see also Remark 2.7) to show that

E^Qt^M

"

Z T t

Z_s^M

l+1ds

#

6C 1 +E^Qt^M[|XT|^pg] + Z T

t

E^Qt^M[|Xs|^pg] +E^Qt^M

h|Xs|^rf+1i ds

! , (3.3) with a constantC that does not depend onM and constants that appear in assumption (TC.1). Then, by putting (3.3) in (3.2), we see that we just have to obtain an a priori estimate forE^Qt^M[|Xs|^c]withc∈R^{+∗. We have}

|Xs| =

Xt+ Z s

t

b(u, Xu)du+ Z s

t

σ(u)dW˜u+ Z s

t

σ(u)∇zfM(u, Xu, Y_u^M, Z_u^M)du 6 |Xt|+C+C

Z s t

|Xu|du+

Z s t

σ(u)dW˜u

+C Z s

t

Z_u^M

ldu,

withCthat does not depend onM. Now we use the recursion assumption to obtain Z s

t

Z_u^M

ldu 6 C Z s

t

A^l_n

(T−u)^l/(l+1)+ B_n^l

(T−u)^l/(l+1)|Xu|^lpg/(l+1)+D_n^l |Xu|^{(rf+1)l/(l+1)}

du.

Obviously we haveRT t

A^l_n

(T−u)^l/(l+1)du6CA^l_n. For the other terms we use Young inequal- ity: Sincelpg/(l+ 1)<1and(rf+ 1)l/(l+ 1)<1, we have

Z s t

Z_u^M

ldu 6 CA^l_n+C Z s

t

B_n^lp

(T−u)^lp/(l+1)+D^l¯_n^p+|X_u|

du,

withp= 1/(1−lp_g/(l+ 1))andp >¯ 1. Since we assume thatlp_g <1, thenlp/(l+ 1)<1 andRs

t

B_n^lp

(T−u)^lp/(l+1)du6CB_n^lp. Finally, we obtain Z s

t

Z_u^M

ldu6CA^l_n+CB_n^lp+CD^l¯_n^p+C Z s

t

|Xu|du, and

|X_s|6|X_t|+C+C Z s

t

|X_u|du+ sup

t6r6T

Z r t

σ(u)dW˜_u

+CA^l_n+CB_n^lp+CD_n^l¯p.

Gronwall’s lemma gives us

|X_s| 6 C

1 + sup

t6r6T

Z r t

σ(u)dW˜_u

+A^l_n+B_n^lp+D_n^l¯p+|X_t|

that implies

E^Qt^M[|Xs|^c] 6 C 1 +A^cl_n +B_n^clp+D_n^cl¯p+|Xt|^c

. (3.4)

By putting (3.4) in (3.3) and (3.2), we obtain Z_t^M

l+1(T−t) 6 C 1 +E^Qt^M[|XT|^pg] + Z T

t E^Qt^M

h|Xs|^pg∨(rf+1)i

ds+ (T−t)|Xt|^(l+1)rf

!

6 C

1 +A^(l+1)al_n +B_n^(l+1)alp+D_n^(l+1)al¯p+|Xt|^pg + (T−t)|Xt|^rf+1 , witha = (pg∨(rf+ 1))/(l+ 1) and C that does not depend onM and constants that appear in assumption (TC.1). So, we easily see that we can take

An+1=C(1 +A^al_n +B_n^alp+D^al¯_n^p), Bn+1=C, Dn+1=C, and then the result is proved.

Whenf andg are not differentiable we can prove the result by a standard approxi- mation and stability results for BSDEs with linear growth.

Since the estimate on Z given by Proposition 3.3 does not depend on constants that appear in assumption (TC.1), we can use it to show an existence result for su- perquadratic BSDEs with a quite general terminal condition.

Theorem 3.6(Existence result for superquadratic BSDEs). Let assume that (F.1), (F.2), (B.1), (B.2)(b), (B.3) and (TC.2) hold. We also assume that06pgl <1, then there exists a solution(Y, Z)to the BSDE (1.2) such that(Y, Z)∈ S²× M². Moreover, we have for allt∈[0, T[,

|Zt|6C(1 +|Xt|^pg/(l+1))

(T−t)^1/(l+1) +C|Xt|^rf

+1

l+1 , (3.5)

and, if we assume that (B.2)(c) holds,

E

"

Z T 0

|Zs|^l+1ds

#

<+∞.

Proof of Theorem 3.6. The proof is based on the proof of Proposition 4.3 in [5]. For each integern>0, we construct the sup-convolution ofgdefined by

gn(x) := sup

u∈R^d

{g(u)−n|x−u|}.

Let us recall some well-known facts about sup-convolution:

Lemma 3.7. Forn>n0withn0big enough, we have,

• gn is well defined,

• (TC.1) holds forg_n withr_g= 0,

• (TC.2) holds forgn with same constantsCandα¯than forg(they do not depend on n),

• (gn)nis decreasing,

• (gn)nconverges pointwise tog.

Since (TC.1) holds, we can consider (Yⁿ, Zⁿ)the solution given by Proposition 2.2.

It follows from Propositions 2.4 and 2.5 that, for alln>n0,

−C(1 +|X_t|^pg+ (T−t)|X_t|^rf+1)6Y_tⁿ⁺¹6Y_tⁿ6Y_tⁿ⁰ 6C(1 +|X_t|^pg+ (T−t)|X_t|^rf+1), (3.6) withCthat does not depend onn: indeed, the constant in Proposition 2.5 just depends on the growth of the terminal condition and here the growth ofgn can be chosen in- dependently ofn(see previous lemma). So(Y_n)_n converges almost surely and we can define

Y = lim

n→+∞Yⁿ.

Passing to the limit into (3.6), we obtain that the estimate of Proposition 2.5 stays true forY. Now the aim is to show that(Zn)nconverges in the good space. For anyT⁰ ∈]0, T[, (Yⁿ, Zⁿ)satisfies

Y_tⁿ=Y_Tⁿ0+ Z T⁰

t

f(s, Xs, Y_sⁿ, Z_sⁿ)ds− Z T⁰

t

Z_sⁿdWs, 06t6T⁰. (3.7) Let us denoteδY^n,m := Yⁿ−Y^m and δZ^n,m := Zⁿ−Z^m. The classical linearization method gives us that(δY^n,m, δZ^n,m)is the solution of BSDE

δY_t^n,m=δY_T^n,m0 + Z T⁰

t

U_s^n,mδY_s^n,m+V_s^n,mδZ_s^n,mds− Z T⁰

t

δZ_s^n,mdWs, where|U^n,m|6Cand, by using estimates of Proposition 3.3,

|V^n,m|6C(1 +|Zⁿ|^l+|Z^m|^l)6C(1 +|X|^p), (3.8) with p < 1 and C that depends on T⁰ but does not depend onn and m. Since p <

1, Novikov’s condition is fulfilled and we can apply Girsanov’s theorem: there exists a probability Q^{n,m such that} dW˜t := dWt−V_t^n,mdt is a Brownian motion under this probability. By classical transformations, we have that(δY^n,m, δZ^n,m)is the solution of the BSDE

δY_t^n,m=δY_T^n,m0 e^RT

0 t U_s^n,mds

− Z T⁰

t

e^RtsUun,mduδZ_s^n,mdW˜s.

SinceU^n,mis bounded, classical estimates on BSDEs give us (see e.g. [7])

E^Qn,m



 Z T⁰

0

|δZ_s^n,m|²ds

!2

6CE^Qn,mh

|δY_T^n,m0 |⁴i

. (3.9)

Now, we would like to have the same type of estimate than (3.9), but with the classical expectation instead ofE^Qn,m. To do so, we define the exponential martingale

E_T^n,m0 := exp Z T⁰

0

V_s^n,mdWs−1 2

Z T⁰ 0

|V_s^n,m|²ds

! .

Then, for allp∈R^,

E[(E_T^n,m0 )^p]< Cp, (3.10) with C_p that does not depend on n and m: indeed, by applying (3.8) and Gronwall’s lemma we have

Eh e^pRT

0

0 V_s^n,mdWs−^p₂RT0

0 |V_s^n,m|²dsi

= E

e¹²

RT0

0 2pV_s^n,mdWs−¹₂RT0

0 |2pV_s^n,m|²ds

+(^p2−p2)^R₀^T0|Vs^n,m|²ds

6 Eh e^RT

0

0 2pV_s^n,mdWs−¹₂RT0

0 |2pV_s^n,m|²dsi^1/2 Eh

e(^2p2−p)^R₀^T0|V_s^n,m|²dsi^1/2 6 Eh

e^C|^2p2−p|(^1+sup06s6T|Xs|^2p)i^1/2

< +∞,

because2p <2. By applying Cauchy Schwarz inequality and by using (3.10) and (3.9), we obtain

E

"

Z T⁰ 0

|δZ_s^n,m|²ds

#

= E

"

(E_T^n,m0 )^−1/2(E_T^n,m0 )^1/2 Z T⁰

0

|δZ_s^n,m|²ds

#

6 E

(E_T^n,m0 )^−11/2 E^Qn,m



 Z T⁰

0

|δZ_s^n,m|²ds

!²



1/2

6 CE^Qn,mh

|δY_T^n,m0 |⁴i1/2

6 CE

(E_T^n,m0 )^21/2 Eh

|δY_T^n,m0 |⁸i1/4

6 CEh

|δY_T^n,m0 |⁸i1/4 n,m→0

−−−−−→0.

SinceM²is a Banach space, we can define Z= lim

n→+∞Zⁿ, dP×dt-a.e..

If we apply Proposition 2.6, we have thatkZⁿk_M2 < Cwith a constantC that does not depend onn. So, Fatou’s lemma gives us thatZ ∈ M². Moreover, the estimate onZⁿ given by Proposition 3.3 stays true forZ and, if we assume that (B.2)(c) holds, then Proposition 2.6 gives us that

E

"

Z T 0

|Z_sⁿ|^l+1ds

#

< C

with a constantCthat does not depend onnand so E

"

Z T 0

|Z_s|^l+1ds

#

< C.

Finally, by passing to the limit whenn →+∞in (3.7) and by using the dominated convergence theorem, we obtain that for any fixedT⁰ ∈[0, T[,(Y, Z)satisfies

Yt=YT⁰+ Z T⁰

t

f(s, Xs, Ys, Zs)ds− Z T⁰

t

ZsdWs, 06t6T⁰. (3.11) To conclude, we just have to prove that we can pass to the limit whenT⁰→T in (3.11).

Let us show thatYT⁰ T⁰→T

−−−−→g(XT)a.s.. Firstly, we have

lim_s→TYs6lim_s→TY_sⁿ=gn(XT)a.s. for anyn>n0,

which implieslim_s→TY_s 6g(X_T), a.s.. On the other hand, we use assumption (B.2)(b) and we apply Propositions 2.5 and 3.3 to deduce that, a.s.,

Y_tⁿ = gn(XT) + Z T

t

f(s, Xs, Y_sⁿ, Z_sⁿ)ds− Z T

t

Z_sⁿdWs

> g_n(X_T)−C Z T

t

1 +|Xs|^rf+1+|Y_sⁿ|+|Z_sⁿ|^ηds− Z T

t

Z_sⁿdW_s

> E^t

"

gn(XT)−C Z T

t

1 +|Xs|^(rf+1)∨pg+1 +|Xs|^ηpg/(l+1) (T−s)^η/(l+1) ds

#

> Et[g_n(X_T)]−C(T−t)(1 +|X_t|^(rf+1)∨pg)−C(T−t)^1−η/(l+1)(1 +|X_t|^ηpg/(l+1)), and

Yt= lim

n→+∞Y_tⁿ>E^t[g(XT)]−C(T−t)(1+|Xt|^(rf+1)∨pg)−C(T−t)^1−η/(l+1)(1+|Xt|^ηpg/(l+1)), which implies

lim_t→TY_t>lim_t→TEt[g(X_T)] =g(X_T).

Hence,lim_t→TYt=g(XT)a.s. .

Now, let us come back to BSDE (3.11). Since we have Z T

t

|f(s, Xs, Ys, Zs)|ds6 Z T

t

C(1 +|Xs|^rf+1+|Ys|+|Zs|^l+1)ds <+∞a.s., then

Z T⁰ t

f(s, X_s, Y_s, Z_s)ds ^T

0→T

−−−−→

Z T t

f(s, X_s, Y_s, Z_s)ds <+∞a.s..

Finally, passing to the limit whenT⁰ →T in (3.11), we conclude that(Y, Z)is a solution to BSDE (1.2).

Remark 3.8. The functionz7→C|z|^l+1+h(|z|^l+1−η)withC >0,0< η6l+ 1andha differentiable function with a bounded derivative is an example of generator such that (B.1), (B.2)(b) and (B.3) hold.

Remark 3.9. The estimate

|Z_t|6 C(1 +|Xt|^pg)

√T−t +C|X_t|^rf+1

is already known in the Lipschitz framework as a consequence of the Bismut-Elworthy formula (see e.g. [8]). For the superquadratic case, the same estimate was obtained whenpg = 0and f does not depend onxandy in [5] (see also [16] for the quadratic case). In [5], Remark 4.4. gives the same type of estimate than (3.5) for the example f(z) = |z|^l. This result was already obtained by Gilding et al. in [9] using Bernstein’s technique whenf(z) =|z|^l,b= 0andσis the identity.

Remark 3.10. In this article, estimate (3.5) for the processZ allows us to obtain an existence result. But this type of deterministic bound is also interesting for numeri- cal approximation of BSDEs (see e.g. [16]) or for studying stochastic optimal control problems in infinite dimension (see e.g. [14]).

A Appendix

Proof of Lemma 3.5. Let us set F_t^M :=e^{R0t∇yfM(s,Xs,YsM,ZMs )ds}∇Y_t^M+

Z t 0

e^{R0s∇yfM(u,Xu,YuM,ZuM)du}∇xf_M(s, X_s, Y_s^M, Z_s^M)∇Xsds,

and

F˜_t^M :=e^λtF_t^M(∇Xt)⁻¹.

Since d∇Xt = ∇b(t, Xt)∇Xtdt, thend(∇Xt)⁻¹ = −(∇Xt)⁻¹∇b(t, Xt)dt and thanks to ItÃt’’s formula,

dZ˜_t^M =dF_t^M(∇Xt)⁻¹σ(t)−F_t^M(∇Xt)⁻¹∇b(t, Xt)σ(t)dt+F_t^M(∇Xt)⁻¹σ⁰(t)dt, and

d(e^λtZ˜_t^M) = ˜F_t^M(λId− ∇b(t, Xt))σ(t)dt+ ˜F_t^Mσ⁰(t)dt+e^λtdF_t^M(∇Xt)⁻¹σ(t).

Finally, d

e^λtZ˜_t^M

2

=dhNit+ 2

λ

F˜_t^Mσ(t)

2

−F˜_t^Mσ(t)[^tσ(t)^t∇b(t, Xt)−^tσ⁰(t)]^tF˜_t^M

dt+dN_t^∗,

withNt:=Rt

0e^λsdF_s^M(∇Xs)⁻¹σ(s)andN_t^∗aQ^M-martingale. Thanks to the assumption (F.2) we are able to conclude that

e^λtZ˜_t^M

2

is aQ^M-submartingale.

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