Uniqueness of solution to scalar BSDEs with $L\exp{\left(\mu \sqrt{2\log{(1+L)}}\,\right)}$-integrable terminal values

In [4], the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) if the terminal value is $L\exp{\left(\mu \sqrt{2\log{(1+L)}}\,\right)}$-integrable with the positive parameter $\mu$ being bigger than a critical value $\mu\_0$. In this note, we give the uniqueness result for the preceding BSDE.


Introduction
Let {W t , t ≥ 0} be a standard Brownian motion with values in R d defined on some complete probability space (Ω, F, P), and {F t , t ≥ 0} its natural filtration augmented by all P-null sets of F. Let us fix a nonnegative real number T > 0. The σ-field of predictable subsets of Ω × [0, T ] is denoted by P.
For any real p ≥ 1, denote by L p the set of all F T -measurable random variables η such that E|η| p < ∞, by S p the set of (equivalent classes of) all real-valued, adapted and càdlàg processes {Y t , 0 ≤ t ≤ T } such that ||Y || S p := E sup 0≤t≤T |Y t | p 1/p < +∞, by L p the set of (equivalent classes of) all real-valued adapted processes {Y t , 0 ≤ t ≤ T } such that and by M p the set of (equivalent classes of) all predictable processes {Z t , 0 ≤ t ≤ T } with values in R 1×d such that Consider the following Backward Stochastic Differential Equation (BSDE): (1.1) Here, f (hereafter called the generator) is a real valued random function defined on the set Ω × [0, T ] × R × R 1×d , measurable with respect to P ⊗ B(R) ⊗ B(R 1×d ), and continuous in the last two variables with the following linear growth: with f 0 := f (·, 0, 0) ∈ L 1 , β ≥ 0 and γ > 0. ξ is a real F T -measurable random variable, and hereafter called the terminal condition or terminal value.
By BSDE (ξ,f ), we mean the BSDE with generator f and terminal condition ξ.
It is well known that for (ξ, f 0 ) ∈ L p × L p (with p > 1), BSDE (1.1) admits a unique adapted solution (y, z) in the space S p × M p if the generator f is uniformly Lipschitz in the pair of unknown variables. See e.g. [6,3,1] for more details. For (ξ, f 0 ) ∈ L 1 × L 1 , one needs to restrict the generator f to grow sub-linearly with respect to z, i.e., with some q ∈ [0, 1), for BSDE (1.1) to have a unique adapted solution (see [1]) if the generator f is uniformly Lipschitz in the pair of unknown variables.
In [4], the existence of the solution is given for a scalar linearly growing BSDE (1.1) if the terminal value is L exp µ 2 log (1 + L) -integrable with the positive parameter µ being bigger than a critical value µ 0 = γ √ T , and the preceding integrability of the terminal value for a positive parameter µ less than critical value µ 0 is shown to be not sufficient for the existence of a solution. In this note, we give the uniqueness result for the preceding BSDE under the preceding integrability of the terminal value for µ > µ 0 .
We first establish some interesting properties of the function ψ(x, µ) = x exp µ 2 log (1 + x) . We observe that the obtained solution Y in [4] has the nice property: ψ(|Y |, a) belongs to the class (D) for some a > 0, which is used to prove the uniqueness of the solution by dividing the whole interval [0, T ] into a finite number of sufficiently small subintervals.

Lemma 2.1
For any x ∈ R and y ≥ 0, we have .

(2.3)
Proposition 2. 3 We have the following assertions on ψ: Proof. The first assertion has been shown in [4]. It remains to show the Assertions (ii) and (iii). We prove Assertion (ii).
Theorem 2.4 Let f be a generator which is continuous with respect to (y, z) and verifies inequality (2.5), and ξ be a terminal condition. Let us suppose that there exists µ > γ √ T such that ψ(|ξ| + Furthermore, there exists a > 0 such that ψ(Y, a) belongs to the class (D).
By comparison theorem, |Y n,p t | ≤Ȳ n,p t . Letting q n,p s = γ sgn(Z n,p s ) and Since Y n,p is nondecreasing in n and non-increasing in p, then by the localization method in [2], there is some Z ∈ L 2 (0, T ; R 1×d ) almost surely such that (Y := inf p sup n Y n,p , Z) is an adapted solution. Therefore, we have for a > 0, using Jensen's inequality and the convexity of ψ a (·) := ψ(·, a) together with Assertion (ii) of Proposition 2.3, we have For b > γ √ T , applying Lemma 2.1, we have Using Lemma 2.2 and Assertion (iii) of Proposition 2.3, we have for any c > 0, For µ > γ √ T , we can choose a > 0, b > γ √ T , and c > 0 such that a + b + c = µ. Then, we have Letting first n → ∞ and then p → ∞, we have Consequently, we have ψ a (|Y |) belongs to the class (D). Now we state our main result of this note. (y, z), i.e., there are β > 0 and γ > 0 such that for all (y i , z i ) ∈ R × R 1×d , i = 1, 2, we have
Proof. The existence of an adapted solution has been proved in the preceding theorem. It remains to prove the uniqueness.
For i = 1, 2, let (Y i , Z i ) be a solution of BSDE (2.4) such that ψ a i (Y i ) belongs to the class (D) for some a i > 0. Define Then both ψ a (Y 1 ) and ψ a (Y 2 ) are in the class (D), since ψ(x, µ) is nondecreasing in µ, and the pair (δY, δZ) satisfies the following equation By a standard linearization we see that there exists an adapted pair of processes (u, v) such that We define the stopping times with the convention that inf ∅ = ∞.