A type of globally solvable BSDEs with triangularly quadratic generators

The present paper is devoted to the study of the well-posedness of a type of BSDEs with triangularly quadratic generators. This work is motivated by the recent results obtained by Hu and Tang [14] and Xing and \v{Z}itkovi\'{c} [28]. By the contraction mapping argument, we first prove that this type of triangularly quadratic BSDEs admits a unique local solution on a small time interval whenever the terminal value is bounded. Under additional assumptions, we build the global solution on the whole time interval by stitching local solutions. Finally, we give solvability results when the generators have path dependence in value process.


Introduction
Backward stochastic differential equations (BSDEs) are introduced in Bismut [1]. A BSDE is an equation of the form where W is a d-dimensional Brownian motion, the terminal condition ξ is an n-dimensional random variable, and g : Ω × [0, T ] × R n × R n×d → R n is the generator. A solution consists of a pair of predictable processes (Y, Z) with values in R n and R n×d , called the value and control process, respectively. The first existence and uniqueness result for BSDEs with an L 2 -terminal condition and a generator satisfying a Lipschitz growth condition is due to Pardoux and Peng [24]. In case that the generator satisfies a quadratic growth condition in the control z, the situation is more involved and a general existence theory does not exist. Frei and dos Reis [12] and Frei [11] provide counterexamples which show that multidimensional quadratic BSDEs may fail to have a global solution. In the one-dimensional case the existence of quadratic BSDE is shown by Kobylanski [19] for bounded terminal conditions, and by Briand and Hu [4,5] for unbounded terminal conditions. Briand and Elie [3] provide a constructive approach to quadratic BSDEs with and without delay. Solvability results for superquadratic BSDEs are discussed in Delbaen et al. [9], see also Masiero and Richou [23], Richou [25] and Cheridito and Nam [6].
The focus of the present work lies on multidimensional quadratic BSDEs. In case that the terminal condition is small enough the existence and uniqueness of a solution was first shown by Tevzadze [26]. Cheridito and Nam [7] and Hu and Tang [14] obtain local solvability on [T − ε, T ] for some ε > 0 of systems of BSDEs with subquadratic generators and diagonally quadratic generators respectively, which under additional assumptions on the generator can be extended to global solutions. Cheridito and Nam [7] provide solvability for Markovian quadratic BSDEs and projectable quadratic BSDEs. Xing and Žitković [28] obtained the global solvability for a large class of multidimensional quadratic BSDEs in the Markovian setting. Frei [11] introduced the notion of split solution and studied the existence of solution for multidimensional quadratic BSDEs by considering a special kind of terminal condition. Jamneshan et al. [16] provide solutions for multidimensional quadratic BSDEs with separated generators. Using a stability approach, Harter and Richou [13] establish an existence and uniqueness result for a class of multidimensional quadratic BSDEs. In Bahlali et al. [2] existence is shown when the generator g(s, y, z) is strictly subquadratic in z and satisfies some monotonicity condition. Multidimensional quadratic BSDEs appear in many applications, such as market making problems (see Kramkov and Pulido [20]), nonzero-sum risk-sensitive stochastic differential games (see El Karoui and Hamadène [10], Hu and Tang [14]) and non-zero sum differential games of BSDEs (see Hu and Tang [15]).
Our results are motivated by the recent works of Hu and Tang [14] and Xing and Žitković [28]. We focus on the solvability of a type of BSDEs with triangularly quadratic generators. More precisely, we study the coupled system of quadratic BSDEs By borrowing some techniques from Hu and Tang [14], we first prove that this type of triangularly quadratic BSDEs admits a unique local solution on a small time interval whenever the terminal value is bounded using a contraction mapping argument. Under additional assumptions, we show that the value process is uniformly bounded. Therefore we build the global solution on the whole time interval by stitching local solutions. Finally, we give solvability results when the generators have path dependence in value process. The paper is organized as follows. In Section 2, we state the setting and main results. The proofs of our main results are presented in Section 3. We further investigate solvability results for a type of quadratic BSDEs with path dependence in value process in Section 4.
A type of globally solvable BSDEs with triangularly quadratic generators are understood in the P -a.s. and P ⊗ dt-a.e. sense, respectively. The Euclidean norm is denoted by | · | and · ∞ denotes the L ∞ -norm. C T (R n ) denotes the set C([0, T ]; R n ) of continuous functions from [0, T ] to R n . For p > 1, we denote by Let T be the set of all stopping times with values in [0, T ]. For any uniformly integrable martingale M with M 0 = 0 and for p ≥ 1, we set The class {M : M BM Op < ∞} is denoted by BM O p , which is written as BM O(P ) when it is necessary to indicate the underlying probability measure P . In particular, we will denote it by BM O when p = 2. For (α · W ) t := t 0 α s dW s in BM O, the corresponding stochastic exponential is denoted by E t (α · W ).
is adapted for each y ∈ R n and z ∈ R n×d . Moreover, there exists α ∈ [−1, 1) such that for y,ȳ ∈ R n and z,z ∈ R n×d . Remark 2.1. Assumptions (A2) and (A3) imply that l i (t, y, z) and k i (t, z) will only depend on the first i − 1 components of z for i = 2, . . . , n.
Our first main result ensures local existence and uniqueness for the triangularly quadratic BSDE (2.1). The proof is given in Section 3. Theorem 2.2. Assume (A1)-(A4) hold, then there exist constants T η , C 1 and C 2 only depending on α, C and ξ ∞ such that for T ≤ T η , BSDE (2.1) admits a unique solution Next, we would like to find conditions under which Theorem 2.2 can be extended to obtain global solvability. In the present setting, under additional conditions, a pasting method based on the uniform bound of the value process allows to get global existence and uniqueness for the triangularly quadratic BSDE (2.1). The proof is given in Section 3.
The sign of k i is critical to obtain our main results. Indeed, if the restriction on the sign of k i fails to hold true, we are able to construct a counterexample to the existence of solution of the triangularly quadratic BSDE based on the work of Frei and dos Reis [12].
has no solution for some ξ 1 ∈ L ∞ (F T ). The details are given in Section 3.3.

Remark 2.5.
Our results are closely related to Hu and Tang [14] and Xing and Žitković [28]. Compared with Xing and Žitković [28], we work under a non-Markovian setting. Compared with Hu and Tang [14], we allow two additional type of terms in our generator, i.e., Z i l i (·, Y, Z) for i = 1, . . . , n and k i (·, Z) for i = 2, . . . , n. The main difficulty is how to deal with the additional quadratic term k i (·, Z). We overcome this difficulty by providing some critical estimates based on properties of BMO martingales. The detailed estimates are given in Section 3.

Remark 2.6.
It is critical to obtain the uniform bound of the value process in order to obtain global existence. However, due to the additional quadratic term k i (·, Z), the technique used in Hu and Tang [14] fails to work for our system of BSDEs. We provide EJP 25 (2020), paper 112.
a new approach to obtain the uniform estimate for the value process by additionally assuming that h i ≤ 0 and |l i | ≤ C. If the additional quadratic term k i (·, Z) vanishes, the global existence and uniqueness still hold true under the same assumption of h i as in Hu and Tang [14] by slightly modifying the technique in Hu and Tang [14].

Triangularly quadratic BSDEs
Before moving to the proofs of our main results, we would like to recall some classical results on BM O spaces (see [18,Theorem 3.6] and [18, Corollary 2.1]).
given by dP We consider the following function from R into itself defined by It is easy to check that u has the following properties

Proof of Theorem 2.2
Step 1. We first show that for (y, z · W ) ∈ S ∞ (R n ) × BM O, the following BSDE , using Young's inequality, we obtain that for any p ≥ 1, where Q 1 is the equivalent probability measure given by Therefore, combining with the fact that ξ 1 is bounded and |h 1 (t, y, z)| ≤ C(1 +|y|+|z| 1+α ), we have for any p ≥ 1, andẐ 1 uniquely determined via martingale representation theorem through Therefore, it follows from (3.1) with p = 1 and (3.2) that On the other hand, it holds that Using Young's inequality, we have EJP 25 (2020), paper 112.
Hence, it holds that Applying Itô's formula to u(Y 1 t ), we obtain that Therefore, we have Noting the inequality (3.3), it holds that Therefore by denoting ∆ : EJP 25 (2020), paper 112.
Step 2. Similar to the first step, it is easy to check that where Q 2 is the equivalent probability measure given by dQ 2 dP = E T (l 2 (·, y · , Z 1 · ) · W ).
where Q i is the equivalent probability measure given by dQ i dP = E T (l i (·, y · , Z · ) · W ).
Therefore we obtain that Step 3. We will denote ∆ * = ∆(C), Step 1 and Step 2 that the following BSDE admits a unique solution (Y, Z) such that (Y, Z · W ) ∈ S ∞ (R n ) × BM O. Moreover from Lemma 3.1, ∆ and δ can be replaced by ∆ * and δ * respectively since it holds that EJP 25 (2020), paper 112.
Therefore, it holds that Step 4. We will denotē Step 1, Step 2 and Step 3 that the following BSDE Then, we have is a Brownian motion under an equivalent probability measureP 1 defined by For any stopping time τ taking values in [0, T ], we have It follows from Lemma 3.1 and Lemma 3.2 that
Thus, ϕ is a contraction on S ∞ (R n ) so that there exists a unique solution (Y, Z) to BSDE (4.2) such that (Y, Z · W ) ∈ S ∞ (R n ) × BM O.