Existence and uniqueness of solution to scalar BSDEs with $L\exp\left(\mu\sqrt{2\log(1+L)}\right)$-integrable terminal values: the critical case

In \cite{HuTang2018ECP}, the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) when the terminal value is $L\exp\left(\mu\sqrt{2\log(1+L)}\right)$-integrable for a positive parameter $\mu>\mu_0$ with a critical value $\mu_0$, and a counterexample is provided to show that the preceding integrability for $\mu<\mu_0$ is not sufficient to guarantee the existence of the solution. Afterwards, the uniqueness result (with $\mu>\mu_0$) is also given in \cite{BuckdahnHuTang2018ECP} for the preceding BSDE under the uniformly Lipschitz condition of the generator. In this note, we prove that these two results still hold for the critical case: $\mu=\mu_0$.


Introduction
Let us fix a positive integer d and a positive real number T > 0. Let (B t ) t∈[0,T ] be a d-dimensional standard Brownian motion defined on some complete probability space (Ω, F, P), and (F t ) t∈[0,T ] its natural filtration augmented by all P-null sets of F. For any two elements x, y in R d , denote by x · y their scalar inner product. We recall that a real-valued and (F t )-progressively measurable process (X t ) t∈[0,T ] belongs to class (D) if the family of random variables {X τ : τ ∈ Σ T } is uniformly integrable, where and hereafter Σ T denotes the set of all (F t )-stopping times τ valued in [0, T ].
For any real number p ≥ 1, let L p represent the set of (equivalent classes of) all real-valued and F T -measurable random variables ξ such that E[|ξ| p ] < +∞, L p the set of (equivalent classes of) all real-valued and (F t )-progressively measurable processes (X t ) t∈[0,T ] such that S p the set of (equivalent classes of) all real-valued, (F t )-progressively measurable and continuous processes (Y t ) t∈[0,T ] such that and M p the set of (equivalent classes of) all R d -valued and (F t )-progressively measurable processes (Z t ) t∈[0,T ] such that We study the following backward stochastic differential equation (BSDE for short): where ξ is a real-valued and F T -measurable random variable called the terminal condition or terminal value, the function (here called the generator) g(ω, t, y, z) : -progressively measurable for each (y, z) and continuous in (y, z), and the pair of processes is integrable, and verifies (1.1). By BSDE(ξ, g), we mean the BSDE with terminal value ξ and generator g.
The following two assumptions with respect to the generator g will be used in this note. The first one is called the linear growth condition, and the second one is called the uniformly Lipschitz condition, which is obviously stronger than the linear growth condition.
Recently, by applying the dual representation of solution to BSDE with convex generator, see for instance [5,11,4], to establish some a priori estimate and the localization procedure, the authors in [8] proved the existence of a solution to BSDE(ξ, g) when the generator g satisfies (H1) and the terminal value (ξ, g 0 ) is L exp µ 2 log(1 + L)integrable for a positive parameter µ > µ 0 with a critical value µ 0 = γ √ T , and showed by a counterexample that the conventionally expected L log L integrability and even the preceding integrability for a positive parameter µ < µ 0 is not enough for the existence of a solution to a BSDE with the generator g satisfying (H1). Furthermore, by establishing some interesting properties of the function ψ(x, µ) = x exp µ 2 log(1 + x) and observing the nice property of the obtained solution Y that ψ(|Y |, a) belongs to class (D) for some a > 0, the authors in [3] divided the whole interval [0, T ] into a finite number of subintervals and proved the uniqueness of the solution to the preceding BSDE(ξ, g) with the generator g satisfying (H2) and µ > µ 0 .
In this note, we prove that the existence and uniqueness result obtained respectively in [8] and [3] is still true under the critical value case: µ = γ √ T , see Theorem 2.1 in next section.
For the existence, in order to apply the localization procedure put forward initially in [2], the key is always to establish some uniform a priori estimate for the first process Y n · in the solution of the approximated BSDEs. For this, instead of applying the dual representation of solution to BSDE with convex generator, our whole idea consists in searching for an appropriate function φ(s, x; t) in order to apply Itô-Tanaka's formula More specifically, we need to find a positive, continuous, strictly increasing and strictly where and hereafter, for each t ∈ (0, T ], φ s (·, ·; t) denotes the first-order partial derivative of φ(·, ·; t) with respect to the first variable, and φ x (·, ·; t) and φ xx (·, ·; t) respectively the first-order and second order partial derivative of φ(·, ·; t) with respect to the second variable. Observe from the basic inequality 2ab Hence, it suffices if for each t ∈ (0, T ], the function φ(·, ·; t) satisfies the following condition: Inspired by the investigation in [8] and [3], we can choose the following function, for For the uniqueness of the solution to BSDE(ξ, g), by virtue of two useful inequalities obtained in [8], we use a similar idea to that in [3] to divide the whole interval [0, T ] into some sufficiently small subintervals and show successively the uniqueness of the solution in these subintervals. However, different from [3], in our case the number of these subin- , · · · , is infinite. Fortunately, observing that the left end points of these subintervals tend to 0 as n → ∞ and in view of the continuity of the first process in the solution with respect to the time variable, we can obtain the uniqueness of the solution on the whole interval [0, T ] by taking the limit.

Existence and uniqueness
Define the function ψ: which is introduced in [8] and [3].
The following existence and uniqueness theorem is the main result of this note. Theorem 2.1. Let ξ be a terminal condition and g be a generator which is continuous in (y, z). If g satisfies assumption (H1) with parameters β and γ, and T ] belongs to class (D), and P − a.s., for each t ∈ [0, T ], where C is a positive constant depending only on (β, γ, T ). Furthermore, if g satisfies assumption (H2), then BSDE(ξ, g) admits a unique solution In order to prove the above theorem, we need the following lemmas and propositions. First, the following lemma have been proved, see Proposition 2.3 and the proof of Theorem 2.5 in [3].
(iv) For all x 1 , x 2 , µ ≥ 0, we have ψ( For each t ∈ [0, T ], we define the following function ϕ, which will be applied by Itô-Tanaka's formula later. Moreover, we have the following proposition.
The two functions ψ and ϕ defined respectively on (2.1) and (2.3) has the following connection.
In particular, by letting s = t, we have Proof. The first inequality in (2.4) is clear, and (2.5) is a direct corollary of (2.4). We now prove the second inequality in (2.4). In fact, for each t ∈ [0, T ] and (s, And, in the case of x ∈ [0, 1], Hence, for all x ∈ [0, +∞), we have With inequality (2.6) in hand and in view of the fact that the function H 1 (x, γ, T ) is continuous on [1, +∞) and tends to exp as x → +∞, we obtain the second inequality in (2.4). The proof is complete.
The following Proposition 2.5 establish some a priori estimate for the solution to a BSDE with an L p (p > 1) terminal value and a linear-growth generator.
Proposition 2.5. Let ξ be a terminal condition and g be a generator which is continuous in (y, z). If g satisfies assumption (H1) with parameters β and γ, (ξ, g(t, 0, 0)) ∈ L p × L p for some p > 1, and (Y t , Z t ) t∈[0,T ] is a solution in S p × M p to BSDE(ξ, g), then P − a.s., where C is a positive constant depending only on (β, γ, T ), and ψ is defined in (2.1).
Proof. Note first that if (ξ, g(t, 0, 0)) ∈ L p × L p for some p > 1, then for any µ ≥ 0, which has been shown in Remark 1.2 of [8]. Definē = e βs ϕ x (s,Ȳ s ; t) (−sgn(Y s )g(s, Y s , Z s ) + β|Y s | + |g(s, 0, 0)|) ds + ϕ x (s,Ȳ s ; t)Z s · dB s +e βs ϕ x (s,Ȳ s ; t)dL s + 1 2 e 2βs ϕ xx (s,Ȳ s ; t)|Z s | 2 ds + ϕ s (s,Ȳ s ; t)ds Furthermore, by letting x =Ȳ s and z = e βs Z s in Assertion (ii) of Proposition 2.3 we get that dϕ(s,Ȳ s ; t) ≥ ϕ x (s,Ȳ s ; t)Z s · dB s , s ∈ [t, T ]. (2.8) Let us consider, for each integer n ≥ 1, the following stopping time with the convention that inf Φ = +∞. It follows from the inequality (2.8) and the definition of τ n that for each t ∈ (0, T ] and n ≥ 1, Thus, thanks to Proposition 2.4, we know the existence of a positive constant K depending only on γ and T such that And, by virtue of Lemma 2.2 and the fact that |g(s, 0, 0)|ds , we obtain that for each t ∈ (0, T ] and n ≥ 1, from which the inequality (2.7) follows for t ∈ (0, T ] by sending n to infinity. Finally, in view of the continuity of Y · and the martingale in the right side hand of (2.7) with respect to the time variable t, we know that (2.7) holds still true for t = 0. The proposition is then proved.
Remark 2.6. We specially point out that, to the best of our knowledge, under the critical case: µ = γ √ T , the method of the dual representation used in [8] can not be applied to obtain the desired a priori estimate as that in (2.7) at the time t = 0. Now, we give the proof of the existence part of Theorem 2.1.