Shengjun Fan

Ying Hu
email: ying.hu@univ-rennes1.fr

Shanjian Tang
email: sjtang@fudan.edu.cn

L 1 solution to scalar BSDEs with logarithmic sub-linear growth generators ✩

Keywords: Backward stochastic differential equation, L 1 solution, Existence and uniqueness, Logarithmic sub-linear growth generator, Comparison theorem

By developing the test function method and combining the localization technique, we prove existence of an L 1 solution to a one-dimensional backward stochastic differential equation (BSDE for short) with L 1 terminal condition when the generator g has a one-sided linear growth in the first unknown variable y and a logarithmic sub-linear growth in the second unknown variable z, which improves some existing results. A new idea to study existence of an adapted solution to a BSDE is given. When the generator g additionally satisfies a one-sided Osgood condition in y and a logarithmic uniform continuity condition in z, we further establish a comparison theorem for the L 1 solutions to the above BSDEs, which yields immediately the uniqueness of the solution.

Introduction

Fix a positive integer d and a real number T > 0. For a, b ∈ R, define a ∧ b := min{a, b}, a + := max{a, 0} and a -:= -min{a, 0}, and sgn(x) := 1 x>0 -1 x≤0 with 1 A being the indicator function of set A. Let (Ω, F, P) be a complete probability space with augmented filtration (F t ) t∈[0,T ] generated by a d-dimensional standard Brownian motion (B t ) t∈[0,T ] . For each p > 0, denote by L p the collection of all F T -measurable real-valued random variables ξ satisfying E[|ξ| p ] < +∞, L p the collection of all (F t )-adapted real-valued processes (X t ) t∈[0,T ] satisfying

∥X∥ L p := E T 0 |X t |dt p 1 p ∧1
< +∞, S p the collection of (F t )-adapted continuous real-valued processes (Y t ) t∈[0,T ] satisfying

∥Y ∥ S p := E[ sup t∈[0,T ] |Y t | p ] 1 p ∧1
< +∞, and M p the collection of all (F t )-adapted R 1×d -valued processes (Z t ) t∈[0,T ] satisfying

∥Z∥ M p :=    E   T 0 |Z t | 2 dt p/2      1 p ∧1 < +∞.
Recall that an (F t )-adapted real-valued process (X t ) t∈[0,T ] belongs to class (D) if the {X τ : τ ∈ Σ T } is a family of uniformly integrable random variables, where Σ T represents the collection of all (F t )-stopping times τ taking values in [0, T ]. The equality and inequality between random elements are stood in P-a.s..

We study the following scalar backward stochastic differential equation (BSDE for short):

Y t = ξ + T t g(s, Y s , Z s )ds - T t Z s dB s , t ∈ [0, T ], (1.1)
where ξ is an F T -measurable real-valued random variable called the terminal condition, the function (called the generator)

g(ω, t, y, z) : Ω × [0, T ] × R × R 1×d → R
is (F t )-adapted for each (y, z), and the pair of processes (Y t , Z t ) t∈[0,T ] taking values in R × R 1×d is called the adapted solution of (1.1), which is (F t )-adapted such that P -a.s., t → Y t is continuous,

t → |g(t, Y t , Z t )| + |Z t | 2
is integrable, and verifies (1.1).

Throughout the paper, we always suppose that β ≥ 0 and γ, c > 0 are three given nonnegative constants, and (f t ) t∈[0,T ] is a given (F t )-adapted nonnegative process. By BSDE(ξ, g), we mean the BSDE with terminal condition ξ and generator g. For narrative convenience, let α > 0 and the generator g satisfy the following growth condition:

|g(ω, t, y, z)| ≤ f t (ω)| + β|y| + γ|z| α , (ω, t, y, z) ∈ Ω × [0, T ] × R × R 1×d . (1.2)
In the case of α ∈ (0, 1), α = 1, α ∈ (1, 2), α = 2 and α > 2, it is usually called that the generator g has a linear growth in y, and a sub-linear, linear, sub-quadratic, quadratic and super-quadratic growth in z respectively. The following existence results are well known and classical. For the case of α ∈ (0, 1), it follows from [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF][START_REF] Fan | Bounded solutions, L p (p > 1) solutions and L 1 solutions for one-dimensional BSDEs under general assumptions[END_REF] [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF][START_REF] El Karoui | Backward stochastic differential equations in finance[END_REF][START_REF] Lepeltier | Backward stochastic differential equations with continuous coefficient[END_REF][START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF][START_REF] Fan | L p (p > 1) solutions for one-dimensional BSDEs with linear-growth generators[END_REF] for more details. For the case of α = 2, by [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF] we know that if (ξ,

that if (ξ, f • ) ∈ L 1 × L 1 , then BSDE(ξ, g) admits a solution (Y t , Z t ) t∈[0,T ] such that Y • belongs to class (D) and (Y • , Z • ) ∈ S p × M p for each p ∈ (0, 1). For the case of α = 1, if (ξ, f • ) ∈ L p × L p for some p > 1, then BSDE(ξ, g) admits a solution (Y • , Z • ) ∈ S p × M p , see e.g.
f • ) is bounded, then BSDE(ξ, g) admits a solution (Y t , Z t ) t∈[0,T ] such that Y • is a bounded process and Z • ∈ M 2 .
For the case of α > 2, it follows from [START_REF] Delbaen | Backward SDEs with superquadratic growth[END_REF] that even if (ξ, f • ) is bounded, BSDE(ξ, g) may not admit a solution.

When the generator g satisfies (1.2) with α = 2, [START_REF] Briand | BSDE with quadratic growth and unbounded terminal value[END_REF] gave a general existence result on an adapted solution to BSDE(ξ, g), which weakens the boundedness condition on (ξ, f • ) in [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF] to a certain exponential moment condition. To the best of our knowledge, this is the first work investigating the weakest possible integrability condition on (ξ, f • ) for existence of an adapted solution to BSDE(ξ, g) when the generator g has a certain growth condition on both unknown variables. In this spirit, [START_REF] Hu | Existence of solution to scalar BSDEs with L exp 2 λ log(1 + L)-integrable terminal values[END_REF] and [START_REF] Fan | Existence and uniqueness of solution to scalar BSDEs with L exp µ 2 log(1 + L) -integrable terminal values: the critical case[END_REF] studied the same problem in the case that the generator g satisfies (1.2) with α = 1 by applying the dual representation of solution to BSDE with convex generator used in [START_REF] El Karoui | Backward stochastic differential equations in finance[END_REF][START_REF] Tang | Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations[END_REF] and utilizing the test function method respectively.

Very recently, the case that the generator g satisfies (1.2) with α ∈ (1, 2) was tracked in [START_REF] Hu | Well-posedness of scalar BSDEs with sub-quadratic generators and related PDEs[END_REF], and the case that the generator g satisfies (1.2) with |z|| ln |z|| α instead of |z| α was dealt with in [START_REF] Hu | Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs[END_REF], where g has a super-linear growth in z. We would like to especially mention that in some of known results mentioned above, the generator g can be allowed to have a one-sided linear growth in y.

Generally speaking, more assumptions are required in order to obtain the uniqueness and the comparison theorem of solutions of scalar BSDEs. In particular, when the generator g is of a linear growth in z, it is additionally supposed that g satisfies a one-sided Osgood condition in y and a uniform continuity condition in z to obtain the desired results (see [START_REF] El Karoui | Backward stochastic differential equations in finance[END_REF][START_REF] Fan | Bounded solutions, L p (p > 1) solutions and L 1 solutions for one-dimensional BSDEs under general assumptions[END_REF][START_REF] Buckdahn | Uniqueness of solution to scalar BSDEs with L exp µ 2 log(1 + L) -integrable terminal values[END_REF][START_REF] Hu | Existence of solution to scalar BSDEs with L exp 2 λ log(1 + L)-integrable terminal values[END_REF][START_REF] Kim | Uniqueness of solution to scalar BSDEs with L exp µ 0 2 log(1 + L) -integrable terminal values: an L 1 -solution approach[END_REF] for details, where Gronwall's inequality, Bihari's inequality and Girsanov's transform are usually used as important tools). And, when the generator g has a super-linear, sub-quadratic or quadratic growth in z, some extended convexity (concavity) assumptions of the generator g in (y, z) are added to get the uniqueness or the comparison theorem (see [START_REF] Briand | Quadratic BSDEs with convex generators and unbounded terminal conditions[END_REF][START_REF] Delbaen | On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions[END_REF][START_REF] Fan | Bounded solutions, L p (p > 1) solutions and L 1 solutions for one-dimensional BSDEs under general assumptions[END_REF][START_REF] Hu | On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions[END_REF][START_REF] Hu | Well-posedness of scalar BSDEs with sub-quadratic generators and related PDEs[END_REF][START_REF] Hu | Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs[END_REF] for more details, where the θ difference method and the a priori estimate technique usually play the key roles in the proof). In addition, we would like to mention that under the condition that the generator g satisfies (1.2) with α ∈ (0, 1), if g further satisfies a one-sided Osgood condition in y and a uniform continuity condition with a sub-linear growth condition in z, then the uniqueness and the comparison theorem of solutions of scalar BSDEs hold, see [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF] and [START_REF] Fan | Bounded solutions, L p (p > 1) solutions and L 1 solutions for one-dimensional BSDEs under general assumptions[END_REF] for details.

In this paper, we put forward a new idea to study existence of an adapted solution to BSDE(ξ, g).

More specially, the following question is the starting point of this paper. That is, if only (ξ, f

• ) ∈ L 1 × L 1
is satisfied, then what growth conditions (as weakest as possible) of the generator g in (y, z) can ensure existence of the adapted solution to BSDE(ξ, g)? As mentioned before, [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF] Furthermore, by using the same a priori estimate technique as above and combining Theorem 2.1 in [START_REF] Fan | Bounded solutions, L p (p > 1) solutions and L 1 solutions for one-dimensional BSDEs under general assumptions[END_REF], we establish a comparison theorem for the adapted L 1 solutions to the previous BSDEs when the generator g additionally satisfies a one-sided Osgood condition in y and a logarithmic uniform continuity condition in z, which naturally yields the uniqueness of the L 1 solution, see Proposition 2.5 in Section 2.

The rest of this paper is organized as follows. In next section we state the main result, and in section 3 we prove it.

Statement of main result

Let us first introduce the following assumptions on the generator g.

(H1) dP × dt -a.e., g(ω, t, •, •) is continuous.

(H2) g has a one-sided linear growth in y and a logarithmic sub-linear growth in z, i.e., there exists a

constant λ ∈ (1/2, 1] that dP × dt -a.e., for each (y, z) ∈ R × R 1×d , sgn(y)g(ω, t, y, z) ≤ f t (ω) + β|y| + γ|z| (ln(e + |z|)) λ .
(H3) g has a general growth in (y, z), i.e., there exists a continuous nondecreasing function h(•) :

[0, +∞) → [0, +∞) with h(0) = 0 such that dP × dt -a.e., for each (y, z) ∈ R × R 1×d , |g(ω, t, y, z)| ≤ f t (ω) + h(|y|) + c|z| 2 .
(H4) g satisfies a one-sided Osgood condition in y, i.e., there exists a continuous, nondecreasing and

concave function ρ(•) : [0, +∞) → [0, +∞) with ρ(0) = 0, ρ(u) > 0 for u > 0, and 0 + du ρ(u) = +∞ such that dP × dt -a.e., for each (y 1 , y 2 , z) ∈ R × R × R 1×d , sgn(y 1 -y 2 )(g(ω, t, y 1 , z) -g(ω, t, y 1 , z)) ≤ ρ(|y 1 -y 2 |).
(H5) g satisfies a logarithmic uniformly continuity condition in z, i.e., there exists a nondecreasing continuous function κ(•) : [0, +∞) → [0, +∞) with linear growth and κ(0) = 0 such that dP × dta.e., for each (y,

z 1 , z 2 ) ∈ R × R 1×d × R 1×d , |g(ω, t, y, z 1 ) -g(ω, t, y, z 2 )| ≤ κ |z 1 -z 2 | (ln(e + |z 1 -z 2 |)) λ ,
where the constant λ ∈ (1/2, 1] is the same as in (H2).

The following existence and uniqueness theorem is the main result of this paper. Then BSDE(ξ, g) admits a solution

(Y t , Z t ) t∈[0,T ] such that Y • belongs to class (D) and (Y • , Z • ) ∈ S p ×M p
for each p ∈ (0, 1). And, there exists a constant C > 0 depending only on (β, γ, λ, T ) such that

|Y t | ≤ |Y t | + t 0 f s ds ≤ CE |ξ| + T 0 f t dt F t + C, t ∈ [0, T ]. (2.3)
Moreover, if g also satisfies assumptions (H4) and (H5), then the solution

(Y • , Z • ) such that Y • belongs to class (D) is unique. Remark 2.2.
With respect to Theorem 2.1, we make the following several remarks.

(i) It is not hard to verify that for each α ∈ (0, 1), there exists a constant k α > 0 depending only on α such that for each z ∈ R 1×d ,

|z| α ≤ k α + |z| ln(e + |z|)
.

This means that for the one-dimensional case, (H2) is weaker than the sub-linear growth assumption (H6) used in [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF], and then Theorem 2.1 improves the corresponding existence result of [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF].

(ii) Note that for each λ > 1 and z ∈ R 1×d ,

|z| (ln(e + |z|)) λ ≤ |z| ln(e + |z|)
.

Therefore, the constant λ appearing in assumption (H2) can take values in (1/2, +∞). However,

it is still open whether Theorem 2.1 holds still when α ∈ (0, 1/2].

(iii) It is not hard to verify that the constant e appearing in (H2) and (H5) can be replaced with any constant bigger than one. And, it is clear that if g satisfies assumption (H5), then it is uniformly continuous in z. In addition, it can be proved that assumptions (H3)-(H5) imply (H2).

In the sequel, we give two examples to which Theorem 2.1 applies, but no existing results could.

Example 2.3. For each (ω, t, y, z) ∈ Ω × [0, T ] × R × R 1×d , define g(ω, t, y, z) := e Bt(ω) + e -y cos 2 |z| + |z| sin |z| (ln(e + |z|)) 2/3 -|z| 2 sgn(y).
It is easy to check that this g satisfies (H1)-(H3) with

f • = e B• + 1, β = 0, γ = 1, λ = 2/3, c = 1 and h(u) = e u .
It then follows from Theorem 2.1 that for each ξ ∈ L 1 , BSDE(ξ, g) admits an adapted solution

(Y t , Z t ) t∈[0,T ] such that Y • belongs to class (D) and (Y • , Z • ) ∈ S p × M p for each p ∈ (0, 1). Example 2.4. For each (ω, t, y, z) ∈ Ω × [0, T ] × R × R 1×d , define ḡ(ω, t, y, z) := B t (ω) + y 2 1 y≤0 + l(|y|) + l |z| ln(e + |z|) + |z|,
where

l(u) := u| ln u|1 0≤u≤ε + l ′ -(ε)(u -l(ε)), u ∈ (0, +∞)
with ε > 0 being sufficiently small. Note that if F (•) : [0, +∞) → [0, +∞) is a nondecreasing concave function with F (0) = 0, then for each 0 < x < y, by letting k := x/y ∈ (0, 1) we have

F (y) -F (x) = F (y) -F (ky + (1 -k) • 0) ≤ F (y) -(kF (y) + (1 -k)F (0)) = (1 -k)F (y) + kF (0) ≤ F ((1 -k)y + k • 0) = F (y -x),
and then for each x 1 , x 2 ≥ 0,

|F (x 1 ) -F (x 2 )| ≤ F (|x 1 -x 2 |).
With the above observation, it is not very hard to verify that for each x 1 , x 2 ≥ 0,

|l(x 1 ) -l(x 2 )| ≤ l(|x 1 -x 2 |) and 0 + du l(u) = +∞, x 1 ln(e + x 1 ) - x 2 ln(e + x 2 ) ≤ |x 1 -x 2 | ln(e + |x 1 -x 2 |) and | √ x 1 - √ x 2 | ≤ |x 1 -x 2 | ≤ 2 |x 1 -x 2 | ln(e + |x 1 -x 2 |) + |x 1 -x 2 | ln(e + |x 1 -x 2 |)
.

Thus, it can be checked that this ḡ satisfies assumptions (H1)-(H5) with

f • = B • + 2, β = 1, γ = 2, λ = 1, c = 2, h(u) = u 2 + u, ρ(u) = l(u) and κ(u) = l(u) + u + 2 √ u.
It then follows from Theorem 2.1 that for each ξ ∈ L 1 , BSDE(ξ, g) admits a unique adapted solution

(Y t , Z t ) t∈[0,T ] such that Y • belongs to class (D) and (Y • , Z • ) ∈ S p × M p for each p ∈ (0, 1).
The following proposition establishes a comparison result for the L 1 solutions of two BSDEs, which naturally yields the uniqueness part of Theorem 2.1.

Proposition 2.5. Let ξ and ξ ′ be two terminal conditions, g and g ′ be two generators, and

(Y t , Z t ) t∈[0,T ]
and (Y ′ t , Z ′ t ) t∈[0,T ] be respectively an adapted solution to BSDE(ξ, g) and BSDE(ξ ′ , g ′ ). Assume that g (resp. g ′ ) satisfies assumptions (H4)-(H5), and

(Y • -Y ′ • ) + belongs to class (D). If ξ ≤ ξ ′ and 1 Yt>Y ′ t (g(t, Y ′ t , Z ′ t ) -g ′ (t, Y ′ t , Z ′ t )) ≤ 0 (resp. 1 Yt>Y ′ t (g(t, Y t , Z t ) -g ′ (t, Y t , Z t )) ≤ 0 ), (2.4
)

then for each t ∈ [0, T ], we have Y t ≤ Y ′ t .

Proof of main result

We first introduce the following technical lemma. Proof. Let f (x) := (e + x) ln(e + x) -2x, x ∈ [0, +∞). A simple calculate gives

f ′ (x) = ln(e + x) -1 ≥ 0, x ≥ 0.
Then, (3.5) follows since f (x) ≥ f (0) = e > 0 for each x ≥ 0.

In order to prove (3.6), we let h(x) := (e + x) 2 -2x ln(e + x) -e, x ∈ [0, +∞). A simple calculate gives that for each x ≥ 0,

h ′ (x) = 2(e + x) -2 ln(e + x) - 2x e + x and h ′′ (x) = 2 - 2 e + x - 2e (e + x) 2 ≥ 2 - 4 e > 0.
Hence, for each x ≥ 0 we have

h ′ (x) ≥ h ′ (0) = 2e -2 > 0 and then h(x) ≥ h(0) = e 2 -e > 0,
which yields the desired inequality (3.6).

By virtue of the above lemma we can prove the following key inequality. (3.10)

The previous inequality (3.10) means that f is a strictly convex function on [0, x]. Furthermore, noticing by (3.9) that f ′ (0) = -2x < 0 and

f ′ (x) = 2x - 2x (ln(e + x)) λ + 2λx 2 (e + x)(ln(e + x)) λ+1 > 0,
we deduce that there exists a unique y 0 ∈ (0, x) such that f ′ (y 0 ) = 0 and

f (x, y) = f (y) ≥ f (y 0 ) = f (x, y 0 ), y ∈ [0, x].
(3.11)

In the sequel, let

y 1 = x 2(ln(e + y 1 )) λ .
(3.12)

Then, y 1 ∈ (0, x/2) and it follows from (3.9), (3.12) and (3. with respect to the second variable. Observe from (3.7) that

-ϕ x (s, x) γ|z| (ln(e + |z|)) λ + 1 2 ϕ xx (s, x)|z| 2 = ϕ xx (s, x) - γϕ x (s, x) ϕ xx (s, x) |z| (ln(e + |z|)) λ + 1 2 |z| 2 ≥ -2γ 2 ϕ 2 x (s, x) ϕ xx (s, x) ln e + γϕ x (s, x) ϕ xx (s, x) -2λ
.

Hence, it is sufficient for the function ϕ(•, •) to satisfy that for each (s, x)

∈ [0, T ] × [0, +∞), -βϕ x (s, x)x -2γ 2 ϕ 2 x (s, x) ϕ xx (s, x) ln e + γϕ x (s, x) ϕ xx (s, x) -2λ + ϕ s (s, x) ≥ 0. (3.16)
In the sequel, we choose the following test function

ϕ(s, x) := (k + x) 1 - 1 (ln(k + x)) δ µ s , (s, x) ∈ [0, T ] × [0, +∞)
to explicitly solve the inequality (3.16), where δ is a positive constant to be assigned, µ s : [0, T ] → (0, +∞) is a nondecreasing and continuous differentiable function to be assigned and

k := exp 2 δ + 2(δ + 1) + 2δ (δ + 1)γ . (3.17)
First of all, a simple computation gives that for each (s, x) ∈ [0, T ] × [0, +∞),

ϕ x (s, x) = 1 - 1 (ln(k + x)) δ 1 - δ ln(k + x) µ s > 0, ϕ xx (s, x) = δ (k + x)(ln(k + x)) δ+1 1 - δ + 1 ln(k + x) µ s > 0,
and

ϕ s (s, x) = (k + x) 1 - 1 (ln(k + x)) δ µ ′ s > 0.
Combining (3.17) and the last three inequalities yields that for (s, x)

∈ [0, T ] × [0, +∞), 1 2 µ s ≤ ϕ x (s, x) ≤ µ s , (3.18)
δµ s 2(k + x)(ln(k + x)) δ+1 ≤ ϕ xx (s, x) ≤ δµ s (k + x)(ln(k + x)) δ+1 (3.19)
and

ϕ s (s, x) ≥ 1 2 (k + x)µ ′ s . (3.20) It follows from (3.18), (3.19) and (3.17
-βϕ x (s, x)x -2γ 2 ϕ 2 x (s, x) ϕ xx (s, x) ln e + γϕ x (s, x) ϕ xx (s, x) -2λ + ϕ s (s, x) ≥ -β(k + x)µ s - 2γ 2 µ 2 s δµs 2(k+x)(ln(k+x)) δ+1 (ln(e + k + x)) 2λ + 1 2 (k + x)µ ′ s ≥ (k + x) -β + 4γ 2 δ (ln(k + x)) δ-(2λ-1) µ s + 1 2 µ ′ s , (s, x) ∈ [0, T ] × [0, +∞).
Thus, if we pick δ = 2λ -1 and

µ s := exp 2 β + 4γ 2 δ s = exp 2 β + 4γ 2 2λ -1 s , s ∈ [0, T ],
then (3.16) and then (3.15) holds.

In conclusion, we have the following proposition on the test function.

Proposition 3.3. Let λ ∈ (1/2, 1] and

k 0 := exp 2 2λ -1 + 4λ + 2λ -1 λγ . (3.22) For (s, x) ∈ [0, T ] × [0, +∞), define φ(s, x) := (k 0 + x) 1 - 1 (ln(k 0 + x)) 2λ-1 exp 2 β + 4γ 2 2λ -1 s . (3.23)
Then, the test function φ(•, •) satisfies that for each (s, x, z)

∈ [0, T ] × [0, +∞) × R 1×d , -βφ x (s, x)x -φ x (s, x) γ|z| (ln(e + |z|)) λ + 1 2 φ xx (s, x)|z| 2 + φ s (s, x) ≥ 0.
(3.24)

The following Proposition 3.4 establishes an important a priori estimate for the solution to a BSDE.

Proposition 3.4. Assume that ξ is a terminal condition, g is a generator satisfying assumption (H2),

and

(Y t , Z t ) t∈[0,T ] is a solution of BSDE(ξ, g). If the process |Y • | + • 0 f s ds belongs to class (D)
, then there exists a constant C > 0 depending only on (β, γ, λ, T ) such that for each t ∈ [0, T ],

|Y t | ≤ |Y t | + t 0 f s ds ≤ CE |ξ| + T 0 f t dt F t + C. ( 3
+ x) ≤ φ(s, x) ≤ k 1 (k 0 + x), (s, x) ∈ [0, T ] × [0, +∞), (3.28)
where k 0 is defined in (3.22) and

k 1 := exp 2 β + 4γ 2 2λ -1 T . (3.29)
Combining (3.27) and (3.28) yields that for each n ≥ 1 and t ∈ [0, T ],

1 2 (k 0 + Ȳt ) ≤ φ(t, Ȳt ) ≤ E φ(τ t n , Ȳτ t n ) F t ≤ k 1 E (k 0 + Ȳτ t n ) F t .
Since Ȳ• belongs to class (D) and τ t n → T as n → ∞ for each t ∈ [0, T ], by sending n to infinity in the last inequality we deduce that

Ȳt ≤ 2k 1 k 0 + 2k 1 E[ ȲT |F t ] -k 0 , t ∈ [0, T ],
which is the desired assertion.

Before proving Theorem 2.1, we introduce the following proposition, which is a direct corollary of Proposition 3.8 in [START_REF] Hu | Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs[END_REF]. Proposition 3.5. Assume that the generator g satisfies assumption (H2) and

(Y t , Z t ) t∈[0,T ] is a solution of BSDE(ξ, g). If |Y • | + • 0 f s ds ∈ S p for p > 0, then Z • ∈ M q for each q ∈ (0, p). Proof of Theorem 2.1. For positive integers n, p ≥ 1, let ξ n,p := ξ + ∧ n -ξ -∧ p and g n,p (ω, t, y, z) := g + (ω, t, y, z) ∧ n -g -(ω, t, y, z) ∧ p.
As the terminal condition ξ n,p is bounded and g n,p is a bounded generator, it follows from [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF] that BSDE(ξ n,p , g n,p ) admits a maximal bounded solution (Y

[f s ∧ (n ∨ p)]dt ≤ CE |ξ n,p | + T 0 [f t ∧ (n ∨ p)]dt F t + C ≤ CE |ξ| + T 0 f t dt F t + C. (3.30) It follows from Theorem 2.3 in [8] that Y n,p
• is nondecreasing in n and non-increasing in p. Then, in view of (3.30) and assumptions (H1) and (H3), by using the localization technique put forward initially in [START_REF] Briand | BSDE with quadratic growth and unbounded terminal value[END_REF] we deduce that there exists an (F t )-adapted process (Z t ) t∈[0,T ] such that (Y Let us further show that (Y • , Z • ) ∈ S p × M p for each p ∈ (0, 1). Indeed, in view of (2.3), it follows from Lemma 6.1 in [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF] that for each p ∈ (0, 1),

• := inf p sup n Y n,p • , Z • ) is a solution to BSDE(ξ, g).
E sup t∈[0,T ] |Y t | + t 0 f s ds p ≤ (2C) p 1 1 -p E |ξ| + T 0 f s ds p + 1 < +∞,
which means that Y • ∈ S p for each p ∈ (0, 1), and then by Proposition 3.5, Z • ∈ M p for each p ∈ (0, 1).

Thus, the existence part of Theorem 2.1 is proved.

The uniqueness part is a direct corollary of Proposition 2.5, which will be proved below. The proof of Theorem 2.1 is then complete.

Proof of Proposition 2.5. We only prove the case that the generator g satisfies assumptions (H4)-(H5), and dP × dt -a.e., With the last inequality in hand, noticing that Ỹ• belongs to class (D) and using a same argument as that in the proof of Proposition 3.4 we deduce that for each t ∈ [0, T ],

1 Yt>Y ′ t (g(t, Y ′ t , Z ′ t ) -g ′ (t, Y ′ t , Z ′ t )) ≤ 0. ( 3
Ỹt ≤ 2k 1 k 0 + 2k 1 E[ ỸT |F t ] -k 0 ,
where the constants k 0 and k 1 are defined respectively in (3.22) and (3.29) with A instead of β and γ.

Then, we have for each t ∈ [0, T ],

(Y t -Y ′ t ) + + 2At ≤ 2k 1 k 0 + 2k 1 E[(ξ -ξ ′ ) + |F t ] + 2AT -k 0 ≤ 2k 1 k 0 + 4Ak 1 T, which means that (Y • -Y ′ • ) + is a bounded process.
Finally, in view of assumptions (H4)-(H5) of g, ξ ≤ ξ ′ and (3.31) together with the fact that (Y • -Y ′ • ) + is a bounded process, we can apply Theorem 2.1 in [START_REF] Fan | Bounded solutions, L p (p > 1) solutions and L 1 solutions for one-dimensional BSDEs under general assumptions[END_REF] to obtain the desired assertion.

Theorem 2 . 1 .

 21 Let the terminal condition ξ ∈ L 1 and the generator g satisfy (H1)-(H3) with f • ∈ L 1 .

Lemma 3 . 1 .

 31 For each x ≥ 0, we have (e + x) ln(e + x) > 2x (3.5) and (e + x) 2 -2x ln(e + x) > e. (3.6)

Proposition 3 . 2 . 2 + 4x 2 (

 3222 For each x, y ≥ 0 and λ ∈ [0, 1], it holds that xy (ln(e + y)) λ ≤ 2x 2 (ln(e + x)) The conclusion is obvious in the case of λ = 0. Let us prove the case of λ ∈ (0, 1]. For (x, y) ∈ [0, +∞) × [0, +∞), define the function f (x, y) := y 2 -2xy (ln(e + y)) λ + 4x 2 (ln(e + x)) 2λ = y -x (ln(e + y)) λ ln(e + x)) 2λ -x 2 (ln(e + y)) 2λ . (3.8)Clearly, in order to prove (3.7), it suffices to prove that f (x, y) ≥ 0 for each x, y ≥ 0. By (3.8) it is obvious when y ≥ x, hence we only need to prove that f (x, y) ≥ 0 for each x ∈ (0, +∞) and y ∈ [0, x]. Now, fix arbitrarily x ∈ (0, +∞) and let f (y) := f (x, y), y ∈ [0, x]. Then we have f ′ (y) = 2y -2x (ln(e + y)) λ + 2λxy (e + y)(ln(e + y)) λ+1 , y ∈ [0, x](3.9)and in view of (3.5), f ′′ (y) = 2 + 2λx (e + y)(ln(e + y)) λ+1 + 2λx (e ln(e + y) -(λ + 1)y) (e + y) 2 (ln(e + y)) λ+2 = 2 + 2λx [(e + y) ln(e + y) + e ln(e + y) -(λ + 1)y] (e + y) 2 (ln(e + y)) λ+2 > 2 + 2λx [(e + y) ln(e + y) -2y] (e + y) 2 (ln(e + y)) λ+2 > 0, y ∈ [0, x].

2 + 4x 2 ( 1 λ 1 λ

 2211 5) that f ′ (y 1 ) = x (ln(e + y 1 )) λ -2x (ln(e + y 1 )) λ + 2λxy 1 (e + y 1 )(ln(e + y 1 )) λ+1 = x [2λy 1 -(e + y 1 ) ln(e + y 1 )] (e + y 1 ) ln(e + y 1 )) λ+1 ≤ x [2y 1 -(e + y 1 ) ln(e + y 1 )] (e + y 1 )(ln(e + y 1 )) λ+1 < 0.Therefore, y 1 < y 0 and then it follows from (3.11) and (3.8) thatf (x, y) ≥ f (x, y 0 ) = y 0 -x (ln(e + y 0 )) λ ln(e + x)) 2λ -x 2 (ln(e + y 0 )) 2λ ≥ 4x 2 (ln(e + x)) 2λ -x 2 (ln(e + y 1 )) 2λ , y ∈ [0, x].(3.13) Finally, by (3.6) we can deduce that (e + y 1 ) 2 -2y 1 (ln(e + y 1 )) λ ≥ (e + y 1 ) 2 -2y 1 ln(e + y 1 ) > e. It follows from the last inequality and (3.12) that 2 ln(e + y 1 ) ≥ ln(e + 2y 1 (ln(e + y 1 )) λ ) = ln(e + x), which means that 4(ln(e + y 1 )) 2λ ≥ (ln(e + x)) 2λ . (3.14) Combining (3.13) and (3.14) yields that f (x, y) ≥ 0 for each x ∈ (0, +∞) and y ∈ [0, x], which is the desired assertion. Now, we fix arbitrarily λ ∈ (1/2, 1] and search for a positive, continuous, strictly increasing and strictly convex function ϕ(s, x) : [0, T ] × [0, +∞) → (0, +∞) satisfying -βϕ x (s, x)x -ϕ x (s, x) γ|z| (ln(e + |z|)) λ + 1 2 ϕ xx (s, x)|z| 2 + ϕ s (s, x) ≥ 0, (s, x, z) ∈ [0, T ] × [0, +∞) × R 1×d , (3.15) where and hereafter, ϕ s (•, •) denotes the first-order partial derivative of ϕ(•, •) with respect to the first variable, and ϕ x (•, •) and ϕ xx (•, •) respectively the first-order and second order partial derivative of ϕ(•, •)

  ) that γϕ x (s, x) ϕ xx (s, x) ≥ γ 2δ (k + x)(ln(k + x)) δ+1 ≥ k + x. (3.21) Substituting (3.18)-(3.21) into the left hand side of (3.16) we can deduce that

  Furthermore, by letting n and p to infinity in (3.30) yields the desired inequality (2.3), and then the process Y • belongs to class (D).

t 1 t≤ A Ŷ + s + A 1

 11 .31)The other case can be proved in the same way. Without loss of generality, we always assume that the function ρ(•) in assumption (H4) and the function κ(•) in assumption (H5) verify that for each x ≥ 0,ρ(x) ≤ Ax + A and κ(x) ≤ Ax + A,(3.32)where A > 0 is a universal constant.Define Ŷ• := Y • -Y ′ • and Ẑ• := Z • -Z ′ • . Then, ( Ŷ• , Ẑ• ) verifies Ŷt = ξ -ξ ′ + T t (g(s, Y s , Z s ) -g ′ (s, Y ′ s , Z ′ s )) ds -T Ẑs dB s , t ∈ [0, T ]. (3.33) Let Ỹt := Ŷ + t + 2At and Zt := 1 Ŷt>0 Ẑt , t ∈ [0, T ].Itô-Tanaka's formula yieldsỸt = ỸT + T t Ŷs>0 (g(s, Y s , Z s ) -g(s, Y ′ s , Z ′ s )) -2A ds -T Zs dB s -T t dL s , t ∈ [0, T ],where L • is the local time of Ŷ• at 0.Furthermore, in view of assumptions (H4) and (H5) of the generator g together with (3.31) and (3.32),we have dP × ds -a.e.,1 Ŷs>0 (g(s, Y s , Z s ) -g ′ (s, Y ′ s , Z ′ s )) -2A ≤ ρ( Ŷ + s ) + 1 Ŷs>0 κ | Ẑs | (ln(e + | Ẑs |)) λ -2A Ŷs>0 | Ẑs | (ln(e + | Ẑs |)) λ ≤ A Ỹs + A| Zs | (ln(e + | Zs |)) λ . (3.34) Let the function φ(•, •) be defined in Proposition 3.3 with A instead of β and γ in (3.22) and (3.23). In view of (3.34), applying Itô-Tanaka's formula to the process φ(s, Ỹs ) yields that dφ(s, Ỹs ) = -φ x (s, Ỹs ) 1 Ỹs>0 (g(s, Y s , Z s ) -g(s, Y ′ s , Z ′ s )) -2A ds + φ x (s, Ỹs ) Zs dB s +φ x (s, Ỹs )dL s + 1 2 φ xx (s, Ỹs )| Zs | 2 ds + φ s (s, Ỹs )ds ≥ -φ x (s, Ỹs ) A Ỹs + A| Zs | (ln(e + | Zs |)) λ + 1 2 φ xx (s, Ỹs )| Zs | 2 + φ s (s, Ỹs ) ds +φ x (s, Ỹs ) Zs dB s , s ∈ [0, T ]. Then, by (3.24) in Proposition 3.3 with A instead of β and γ we have dφ(s, Ỹs ) ≥ φ x (s, Ỹs ) Zs dB s , s ∈ [0, T ].

  .25) • is the local time of Y • at 0. Now, let the function φ(•, •) be defined in (3.23) and apply Itô-Tanaka's formula to the process φ(s, Ȳs ) to deduce, in view of (H2), dφ(s, Ȳs ) = φ x (s, Ȳs ) (-sgn(Y s )g(s, Y s , Z s ) + f s ) ds + φ x (s, Ȳs ) Zs dB s Ȳs )|Z s | 2 + φ s (s, Ȳs ) ds +φ x (s, Ȳs ) Zs dB s , s ∈ [0, T ].

	Proof. Define
			t
		Ȳt := |Y t | +	f s ds and	Zt := Z t sgn(Y t ), t ∈ [0, T ].
			0
	Itô-Tanaka's formula yields
			T
	Ȳt = ȲT +	t (sgn(Y +φ x (s, Ȳs )dL s +	1 2	φ xx (s, Ȳs )| Zs | 2 ds + φ s (s, Ȳs )ds
	≥ φ xx (s, Then, in view of |Y s | ≤ Ȳs , by Proposition 3.3 we have -φ x (s, Ȳs ) β|Y s | + γ|Z s | (ln(e + |Z s |)) λ + 1 2
			dφ(s, Ȳs ) ≥ φ x (s, Ȳs ) Zs dB s , s ∈ [0, T ],	(3.26)
	which means that φ(s, Ȳs ) is a local submartingale.
	For each n ≥ 1 and t ∈ [0, T ], define the following stopping time
	τ t n := inf s ∈ [t, T ] : φ(s, Ȳs ) +	s	φ x (r, Ȳr )	2 | Zr | 2 dr ≥ n ∧ T
					t
	with the convention that inf ∅ = +∞. It follows from the definition of τ t n and (3.26) that for each n ≥ 1
	and t ∈ [0, T ],
			φ(t, Ȳt ) ≤ E φ(τ t n , Ȳτ t n ) F t .	(3.27)
	On the other hand, observe from (3.23) and (3.22) that
	1 2	(k 0

s )g(s, Y s , Z s ) -f s ) ds -T t Zs dB s -T t dL s , t ∈ [0, T ],

where L

60H10 ✩ Partially supported by National Natural Science Foundation of China (Nos. 12171471, 12031009 and 11631004), by ; by Lebesgue Center of Mathematics "Investissements d'avenir" program-ANR-11-LABX-0020-01, by CAESARS-ANR-15-CE05-0024 and by MFG-ANR-16-CE40-0015-01.