Multi-dimensional Backward Stochastic Differential Equations of Diagonally Quadratic Generators with a Special Structure

TL;DR

This paper establishes the well-posedness of multi-dimensional backward stochastic differential equations with diagonally quadratic generators, introducing new estimates and solvability conditions for small growth and triangular structures.

Contribution

It provides new a priori estimates and proves existence and uniqueness of solutions for diagonally quadratic BSDEs with small off-diagonal growth and triangular structure.

Findings

Unique solutions exist under small off-diagonal growth conditions.

A new a priori estimate for diagonally quadratic BSDEs.

Solvability results for triangular diagonally quadratic generators.

Abstract

The present paper is devoted to the well-posedness of a type of multi-dimensional backward stochastic differential equations (BSDEs) with a diagonally quadratic generator. We give a new priori estimate, and prove that the BSDE admits a unique solution on a given interval when the generator has a sufficiently small growth of the off-diagonal elements (i.e., for each $i$, the $i$-th component of the generator has a small growth of the $j$-th row $z^j$ of the variable $z$ for each $j \neq i$). Finally, we give a solvability result when the diagonally quadratic generator is triangular.

Equations

$$Y_t=\xi +{\int }_t^{T}f(s,Y_s,Z_s)\mathrm{d}s-{\int }_t^T{Z}_s\mathrm{d}{W}_s,t\in [0,T],$$

$$\Vert Y{\Vert }_{{\mathcal{S}}^p}:={\left(\mathbb{E}[\underset{t\in [0,T]}{\sup }|Y_t{|}^p]\right)}^{1/p}<+\infty .$$

$$\Vert Y{\Vert }_{{\mathcal{S}}^{\infty }}:={\Vert \underset{t\in [0,T]}{\sup }|Y_t|\Vert }_{\infty }<+\infty .$$

$$\Vert Z{\Vert }_{{\mathcal{H}}^p}:={\left\{\mathbb{E}\left[{\left({\int }_0^T|Z_s{|}^2\mathrm{d}s\right)}^{p/2}\right]\right\}}^{1/p}<+\infty .$$

$$\Vert Z{\Vert }_{\mathrm{BMO}}:=\underset{\tau }{\sup }{\Vert {\mathbb{E}}_{\tau }\left[{\int }_{\tau }^T|Z_s{|}^2\mathrm{d}s\right]\Vert }_{\infty }^{1/2}<+\infty .$$

$$f^i(t,y,z)=g^i(t,z^i)+h^i(t,y,z),i=1,\cdots ,n.$$

$$\begin{array}[]{ll}&\displaystyle|g^{i}(t,z)|\leq\frac{\gamma}{2}|z|^{2},\ \ \forall~{}z\in{\mathbb{R}}^{d};\vspace{0.1cm}\\ &\displaystyle|g^{i}(t,z_{1})-g^{i}(t,z_{2})|\leq K(1+|z_{1}|+|z_{2}|)|z_{1}-z_{2}|,\ \ \forall~{}z_{1},z_{2}\in{\mathbb{R}}^{d};\vspace{0.1cm}\\ &\displaystyle|h^{i}(t,0,0)|\leq K;\vspace{0.1cm}\\ &\displaystyle|h^{i}(t,y_{1},z_{1})-h^{i}(t,y_{2},z_{2})|\leq K|y_{1}-y_{2}|+K(1+|z_{1}|^{\delta}+|z_{2}|^{\delta})|z_{1}-z_{2}|,\ \ \forall~{}y_{1},y_{2}\in{\mathbb{R}}^{n},\\ &\displaystyle z_{1},z_{2}\in{\mathbb{R}}^{n\times d}.\end{array}$$

$$\mathrm{sgn}(y^i)h^i(t,y,z)\le {\alpha }_t+{\beta }_t|y|+{\eta }_t\log (|z|+1),\forall y\in {\mathbb{R}}^n,z\in {\mathbb{R}}^{n\times d}.$$

$$\|\xi\|_{\infty}+\int_{0}^{T}\Big{(}\alpha_{t}+\beta_{t}+\eta_{t}\log(1+\eta_{t})\Big{)}{\rm d}t\leq C_{0}$$

$$f^{i}(t,y,z)=|z^{i}|^{2}\sin\big{(}\log(|z^{i}|+1)\big{)}+|y|+\sin(|z|^{1+\delta})+\log(|z|+1),\ \ i=1,\cdots,n.$$

$$C\log (1+x)\le x^{2}y+\frac{y}{3}+\frac{C}{2}\log (1+\frac{C}{y}).\text{vspace}0.2cm$$

$$F(x)=x^2-k\log (1+x).$$

$$F^{{}^{\prime }}(x)=2x-\frac{k}{1+x},F^{{}^{\prime \prime }}(x)=2+\frac{k}{(1+x{)}^2}>0.$$

$$0=F^{{}^{\prime }}(x_0)=2x_0-\frac{k}{1+x_0},2x_0^2+2x_0=k.$$

$$F(x)\ge F(x_0)=x_0^2-k\log (1+x_0),\forall x>0.$$

$$\begin{array}[]{ll}\displaystyle k=2x_{0}^{2}+2x_{0}\leq 3x_{0}^{2}+1,\ \ x_{0}^{2}\geq\frac{k-1}{3};\\ \displaystyle k+1=2x_{0}^{2}+2x_{0}+1\geq(1+x_{0})^{2},\ \ \log(1+x_{0})\leq\frac{1}{2}\log(1+k).\end{array}$$

$$F(x_0)=x_0^2-k\log (1+x_0)\ge \frac{k-1}{3}-\frac{k}{2}\log (1+k).$$

$$x^2-k\log (1+x)=F(x)\ge \frac{k-1}{3}-\frac{k}{2}\log (1+k)\ge -\frac{1}{3}-\frac{k}{2}\log (1+k),\forall x>0.$$

$$k\log (1+x)\le x^2+\frac{1}{3}+\frac{k}{2}\log (1+k),\forall x,k>0.$$

$$k=\frac{C}{y},$$

$$\frac{C}{y}\log (1+x)\le x^2+\frac{1}{3}+\frac{C}{2y}\log (1+\frac{C}{y}),\forall x,y,C>0.$$

$$\Vert Y{\Vert }_{{\mathcal{S}}_{[t_0,T]}^{\infty }}\le \lambda .\text{vspace}0.2cm$$

$$u(x)=\gamma^{-2}\big{(}\exp(\gamma|x|)-\gamma|x|-1\big{)},\ \ x\in{\mathbb{R}}.$$

$$u^{\prime}(x)=\gamma^{-1}\big{(}\exp(\gamma|x|)-1\big{)}{\rm sgn}(x),\ \ u^{\prime\prime}(x)=\exp(\gamma|x|),\ \ u^{\prime\prime}(x)-\gamma|u^{\prime}(x)|=1.$$

$$\begin{array}[]{lll}\displaystyle u(Y_{t}^{i})&=&\displaystyle u(\xi^{i})+\int_{t}^{T}[u^{\prime}(Y_{s}^{i})\big{(}g^{i}(s,Z_{s}^{i})+h^{i}(s,Y_{s},Z_{s})\big{)}-\frac{1}{2}u^{\prime\prime}(Y_{s}^{i})|Z_{s}^{i}|^{2}]{\rm d}s-\int_{t}^{T}u^{\prime}(Y_{s}^{i})Z_{s}^{i}{\rm d}W_{s}\vspace{0.2cm}\\ &\leq&\displaystyle u(\xi^{i})-\int_{t}^{T}u^{\prime}(Y_{s}^{i})Z_{s}^{i}{\rm d}W_{s}-\frac{1}{2}\int_{t}^{T}\exp(\gamma|Y_{s}^{i}|)|Z_{s}^{i}|^{2}{\rm d}s\vspace{0.2cm}\\ &&\displaystyle+\int_{t}^{T}\gamma^{-1}\Big{(}\exp(\gamma|Y_{s}^{i}|)-1\Big{)}\Big{(}\frac{\gamma}{2}|Z_{s}^{i}|^{2}+\alpha_{s}+\beta_{s}|Y_{s}|+\eta_{s}\log(|Z_{s}|+1)\Big{)}{\rm d}s\vspace{0.2cm}\\ &=&\displaystyle u(\xi^{i})-\int_{t}^{T}u^{\prime}(Y_{s}^{i})Z_{s}^{i}{\rm d}W_{s}\vspace{0.2cm}\\ &&\displaystyle+\int_{t}^{T}\Big{[}-\frac{1}{2}|Z_{s}^{i}|^{2}+\gamma^{-1}(\exp(\gamma|Y_{s}^{i}|)-1)\big{(}\alpha_{s}+\beta_{s}|Y_{s}|+\eta_{s}\log(|Z_{s}|+1)\big{)}\Big{]}{\rm d}s\end{array}$$

$$x=|Z_s|,y=\frac{\gamma }{2n}\exp (-\gamma \Vert Y{\Vert }_{{\mathcal{S}}_{[s,T]}^{\infty }}),C={\eta }_s,$$

$$\begin{array}[]{lll}\displaystyle&&\displaystyle\eta_{s}\log(|Z_{s}|+1)\vspace{0.2cm}\\ &\leq&\displaystyle\frac{\gamma}{2n}\exp(-\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}})|Z_{s}|^{2}+\frac{\gamma}{6n}\exp(-\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}})+\frac{\eta_{s}}{2}\log\big{(}1+\frac{2n\eta_{s}}{\gamma}\exp(\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}})\big{)}\vspace{0.2cm}\\ &\leq&\displaystyle\frac{\gamma}{2n}\exp(-\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}})|Z_{s}|^{2}+\frac{\gamma}{6n}+\frac{\eta_{s}}{2}\log\big{(}1+(\frac{2n\eta_{s}}{\gamma}+1)\exp(\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}})\big{)}\end{array}$$

$$\begin{array}[]{lll}\displaystyle\log(1+x)\leq 1+\log x,\ \ \forall x\geq 1;\vspace{0.2cm}\\ \displaystyle\log(1+kx)\leq k+\log(1+x),\ \ \forall k\geq 0,x\geq 0.\end{array}$$

$$\begin{array}[]{lll}&&\displaystyle\log\big{(}1+(\frac{2n\eta_{s}}{\gamma}+1)\exp(\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}})\big{)}\\ &\leq&\displaystyle 1+\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}}+\log(\frac{2n\eta_{s}}{\gamma}+1)\\ &\leq&\displaystyle 1+\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}}+\log(\eta_{s}+1)+\frac{2n}{\gamma}.\end{array}$$

$$\begin{array}[]{lll}\displaystyle&&\displaystyle\eta_{s}\log(|Z_{s}|+1)\vspace{0.2cm}\\ &\leq&\displaystyle\frac{\gamma}{2n}\exp(-\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}})|Z_{s}|^{2}+\frac{\gamma}{6n}+\frac{\eta_{s}}{2}\big{(}1+\gamma\|Y\|_{\mathcal{S}^{\infty}_{[s,T]}}+\log(\eta_{s}+1)+\frac{2n}{\gamma}\big{)}\end{array}$$

Full text

[TABLE]

**2000 MR Subject Classification ** 60H10

††footnotetext:

1

Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China.

E-mail: gyang19@fudan.edu.cn

∗

Research partially supported by National Natural Science Foundation of China (Grants No. 11631004 and No. 12031009).

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sectionIntroduction

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Bismut [2] first introduced Backward stochastic differential equations (BSDEs in short):

[TABLE]

where $(W_{t})_{t\in[0,T]}$ is a $d$-dimensional standard Brownian motion defined on some complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, and $(\mathcal{F}_{t})_{t\in[0,T]}$ is the augmented natural filtration generated by the standard Brownian motion $W$. The terminal value $\xi$ is an $\mathcal{F}_{T}$-measurable $n$-dimensional random vector, the generator function $f(\omega,t,y,z):\Omega\times[0,T]\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n\times d}\rightarrow{\mathbb{R}}^{n}$ is $(\mathcal{F}_{t})$-progressively measurable for each pair $(y,z)$, and the solution $(Y_{t},Z_{t})_{t\in[0,T]}$ is a pair of $(\mathcal{F}_{t})$-progressively measurable processes with values in ${\mathbb{R}}^{n}\times{\mathbb{R}}^{n\times d}$ which almost surely verifies BSDE (1.1). In 1990, Pardoux and Peng [18] established the existence and uniqueness result for BSDEs with an $L^{2}$-terminal value and a generator satisfying a uniformly Lipschitz continuous condition. When the generators have a quadratic growth in the state variable $z$, the situation is more complicated. In the one-dimensional case, Kobylanski [16] established the first existence and uniqueness result for quadratic BSDEs with bounded terminal values, Tevzadze [19] gives a fixed-point argument, Briand and Elie [3] provide a constructive approach to quadratic BSDEs with and without delay. Briand and Hu [4, 5], Delbaen et al. [7, 8], Barrieu and El Karoui [1] and Fan et al. [9] considered the unbounded terminal value case.

For multidimensional quadratic BSDEs, when the terminal value is small enough in the supremum norm, Tevzadze [19] proved a general existence and uniqueness result for multi-dimensional quadratic BSDEs. Frei and Dos Reis [12] provide a counterexample which show that multidimensional quadratic BSDEs with a bounded terminal value may fail to have a global solution. Frei [11] introduced the notion of split solution and studied the existence of solution by considering a special kind of terminal value. Cheridito and Nam [6] and Xing and Žitković [20] obtained the solvability for multidimensional quadratic BSDEs in the Markovian setting. Jamneshan et al. [14] provided solutions for multidimensional quadratic BSDEs with separated generators. Cheridito and Nam [6], Hu and Tang [13] and Luo [17] obtained local solvability of sysetems of BSDEs with subquadratic, diagonally quadratic and triangularly quadratic generators respectively, which under additional assumptions on the generator can be extended to global solutions. When the terminal value is unbounded, Jamneshan et al. [14] provided solutions when the terminal value is small in the BMO-sense, Fan et al. [10] obtained global solutions when the generator is convex or concave.

As a continuation of Hu and Tang [13] and Fan et al. [10], we are devoted to the solvability of multidimensional diagonally quadratic BSDEs when the generator has a logarithmic growth of the off-diagonal elements. The local solution is constructed directly by [13, Theorem 2.2]. Together with the new priori estimate we build, we are able to stitch local solutions to get the global solution. In contrast to [13, Theorem 2.3] and [10, Theorem 2.4], we allow the generator to have a logarithmic growth of the off-diagonal elements. In contrast to [10, Theorem 2.5] and [17], we do not assume that the generator is strictly quadratic. Finally, assuming that for each $i=1,\cdots,n$, the $i$th component $f^{i}$ of the generator $f$ is diagonally quadratic, depends only on the first $i$ components of the state variable $y$ and the first $i$ rows of the state variable $z$, we prove existence and uniqueness of the global solution to the multidimensional diagonally quadratic BSDE with a bounded terminal value.

The rest of the paper is organized as follows. In Section 2, we prepare some notations and state the main results of this paper. In Section 3, we give an estimate and prove our main results. In Section 4, we prove a global solvability result for triangular and diagonally quadratic BSDEs.

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sectionPreliminaries and statement of main results

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2.1 Notations

Let $W=(W_{t})_{t\geq 0}$ be a $d$-dimensional standard Brownian motion defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, and $(\mathcal{F}_{t})_{t\geq 0}$ be the augmented natural filtration generated by $W$. Throughout this paper, we fix a $T\in(0,\infty)$. We endow $\Omega\times[0,T]$ with the predictable $\sigma$-algebra $\mathcal{P}$ and ${\mathbb{R}}^{n}$ with its Borel $\sigma$-algebra $\mathcal{B}({\mathbb{R}}^{n})$. All the processes are assumed to be $(\mathcal{F}_{t})_{t\in[0,T]}$-progressively measurable, and all equalities and inequalities between random variables and processes are understood in the sense of $\mathbb{P}-a.s.$ and ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, respectively. The Euclidean norm is always denoted by $|\cdot|$, and $\|\cdot\|_{\infty}$ denotes the $L^{\infty}$-norm for one-dimensional or multidimensional random variable defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$.

We define the following four Banach spaces of stochastic processes. By $\mathcal{S}^{p}({\mathbb{R}}^{n})$ for $p\geq 1$ , we denote the set of all ${\mathbb{R}}^{n}$-valued continuous adapted processes $(Y_{t})_{t\in[0,T]}$ such that

[TABLE]

By $\mathcal{S}^{\infty}({\mathbb{R}}^{n})$, we denote the set of all ${\mathbb{R}}^{n}$-valued continuous adapted processes $(Y_{t})_{t\in[0,T]}$ such that

[TABLE]

By $\mathcal{H}^{p}({\mathbb{R}}^{n\times d})$ for $p\geq 1$, we denote the set of all ${\mathbb{R}}^{n\times d}$-valued $(\mathcal{F}_{t})_{t\in[0,T]}$-progressively measurable processes $(Z_{t})_{t\in[0,T]}$ such that

[TABLE]

By ${\rm BMO}({\mathbb{R}}^{n\times d})$, we denote the set of all $Z\in\mathcal{H}^{2}({\mathbb{R}}^{n\times d})$ such that

[TABLE]

Here and hereafter the supremum is taken over all $(\mathcal{F}_{t})$-stopping times $\tau$ with values in $[0,T]$, and $\mathbb{E}_{\tau}$ denotes the conditional expectation with respect to $\mathcal{F}_{\tau}$.

The spaces $\mathcal{S}^{p}_{[a,b]}({\mathbb{R}}^{n})$, $\mathcal{S}^{\infty}_{[a,b]}({\mathbb{R}}^{n})$, $\mathcal{H}^{p}_{[a,b]}({\mathbb{R}}^{n\times d})$, and ${\rm BMO}_{[a,b]}({\mathbb{R}}^{n\times d})$ are identically defined for stochastic processes over the time interval $[a,b]$. We note that for $Z\in{\rm BMO}({\mathbb{R}}^{n\times d})$, the process $\int_{0}^{t}Z_{s}{\rm d}B_{s},t\in[0,T]$, is an $n$-dimensional BMO martingale. For the theory of BMO martingales, we refer the reader to Kazamaki [15].

For $i=1,\cdots,n$, denote by $z^{i}$, $y^{i}$ and $f^{i}$ the $i$th row of matrix $z\in{\mathbb{R}}^{n\times d}$, the $i$th component of the vector $y\in{\mathbb{R}}^{n}$ and the generator $f$, respectively.

2.2 Statement of the main results

The main result of this paper concerns global solutions for bounded terminal value case. Consider the multi-dimensional BSDE (1.1) of the following structured quadratic generator:

[TABLE]

We need the following assumptions.

1. (H1)

There exist two positive real constants $\gamma$ and $K$ and a real constant $\delta\in[0,1)$, such that for $i=1,\cdots,n$, $g^{i}:\Omega\times[0,T]\times{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ and $h^{i}:\Omega\times[0,T]\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n\times d}\rightarrow{\mathbb{R}}$ satisfy the following inequalities:

[TABLE] 2. (H2)

There exist a three-dimensional positive deterministic vector function $(\alpha_{t},\beta_{t},\eta_{t})_{t\in[0,T]}$ such that for $i=1,\cdots,n$, $h^{i}$ satisfies:

[TABLE] 3. (H3)

There exists a positive constant $C_{0}$ such that

[TABLE]

Our main result ensures existence and uniqueness for the diagonally quadratic BSDE(1.1).

Theorem 2.1

Let Assumptions (H1)-(H3) be satisfied. Then BSDE (1.1) has a unique solution $(Y,Z)\in\mathcal{S}^{\infty}({\mathbb{R}}^{n})\times{\rm BMO}({\mathbb{R}}^{n\times d})$ on $[0,T]$.

The proof is given in Section 3.

Remark 2.1

Assumptions (H1)-(H3) of Theorem 2.1 are different from those of [13, Theorem 2.3, p. 1072] and [10, Theorem 2.4]. We allow the generator to have a logarithmic growth of the off-diagonal elements. They are different from those of [10, Theorem 2.5] in that the generator is not required to be strictly quadratic. For example, the following generator $f$ satisfies Theorem 2.1 rather than the others:

[TABLE]

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sectionDiagonally quadratic BSDEs with a logarithmic growth

@xsect

2.3ex plus.2ex

We first give an inequality.

Lemma 3.1

For $x,y,C>0$, we have

[TABLE]

Proof  For a given $k>0$, define

[TABLE]

Then we have

[TABLE]

Hence there exists a unique $x_{0}>0$ such that

[TABLE]

$F$ is decreasing on $[0,x_{0}]$ and increasing on $[x_{0},+\infty)$, therefore we have

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From (3.2), we have

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Hence

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From (3.3), we have

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Therefore

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Let

[TABLE]

we have

[TABLE]

Multiply both sides by $y$, we get (3.1). The proof is complete.

Now we give a priori estimate.

Lemma 3.2

Let assumptions (H1)-(H3) hold, $(Y,Z)\in\mathcal{S}^{\infty}_{[t_{0},T]}({\mathbb{R}}^{n})\times{\mathcal{H}^{2}}_{[t_{0},T]}({\mathbb{R}}^{n\times d})$ is a solution of BSDE (1.1) on $[t_{0},T]$, then there exist a positive constant $\lambda$ (depending on the vector of parameters $(n,\gamma,C_{0},T)$) such that

[TABLE]

Proof  Define

[TABLE]

Then we have for $x\in{\mathbb{R}}$,

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Using Itô’s formula to compute $u(Y_{t}^{i})$, we have

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Using (3.1) and taking

[TABLE]

we have

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It is easy to check that

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Hence we have

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From (3.6), we have

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Let

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From (3.5) and (3.7), we have

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Hence it holds that

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Noting that

[TABLE]

We have

[TABLE]

Taking conditional expectation with respect to $\mathcal{F}_{\tau}$ for $\tau\in[t_{0},t]$, we show that

[TABLE]

Using Gronwall’s inequality, we get

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Setting $\tau=t$, we have

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Using Jensen’s inequality, we obtain that

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Hence we have

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It is easy to check that

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Hence we have

[TABLE]

From the definition of $k_{s}$, we have

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Let

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From (3.13), we have

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Using Gronwall’s inequality, we get

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Let

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We get (3.4). The proof is complete.

Proof of Theorem 2.1  For the number $\lambda$ given in Lemma 3.2, we have

[TABLE]

From [13, Theorem 2.2, p. 1072], there exists $t_{\lambda}>0$ which depends on constants $(n,K,\gamma,\delta,\lambda)$, such that BSDE (1.1) has a local solution $(Y,Z)\in\mathcal{S}^{\infty}({\mathbb{R}}^{n})\times{\rm BMO}({\mathbb{R}}^{n\times d})$ on $[T-t_{\lambda},T]$. From Lemma 3.2, we obtain that

[TABLE]

Taking $T-t_{\lambda}$ as the terminal time and $Y_{T-t_{\lambda}}$ as the terminal value, BSDE (1.1) has a local solution $(Y,Z)\in\mathcal{S}^{\infty}({\mathbb{R}}^{n})\times{\rm BMO}({\mathbb{R}}^{n\times d})$ on $[T-2t_{\lambda},T-t_{\lambda}]$. Stitching the solutions we have a solution $(Y,Z)\in\mathcal{S}^{\infty}({\mathbb{R}}^{n})\times{\mathcal{H}^{2}}({\mathbb{R}}^{n\times d})$ on $[T-2t_{\lambda},T]$ and $\|Y_{T-2t_{\lambda}}\|_{\infty}\leq\lambda$. Repeating the preceding process, we can extend the pair $(Y,Z)$ to the whole interval $[0,T]$ within finite steps such that $Y$ is uniformly bounded by $\lambda$ and $Z\in{\mathcal{H}^{2}}({\mathbb{R}}^{n\times d})$. We now show that $Z\in{\rm BMO}({\mathbb{R}}^{n\times d})$. Identical to the proof of inequality (3.7) and (3.8), we have

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and

[TABLE]

where

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Summing $i$ from $1$ to $n$ and taking conditional expectation with respect to $\mathcal{F}_{t}$, we have

[TABLE]

Hence $Z\in{\rm BMO}({\mathbb{R}}^{n\times d})$. Finally, the uniqueness on the given interval $[0,T]$ is a consequence of [13, Theorem 2.2, p. 1072] via a pasting technique.

Remark 3.1

From [10, Theorem 2.1], (H1) and (H2) can be replaced with the following in Theorem 2.1:

1. (H1’)

There exist a deterministic scalar-valued positive function $(\alpha_{t})_{t\in[0,T]}$, a deterministic nondecreasing continuous function $\phi(\cdot):[0,+\infty)\rightarrow[0,+\infty)$ with $\phi(0)=0$ and several real constants $\gamma>0$, $K\geq 0$, $\delta\in[0,1)$ such that for $i=1,\cdots,n$ and each $(y,\bar{y},z,\bar{z})\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n\times d}\times{\mathbb{R}}^{n\times d}$, $f^{i}$ satisfies the following inequalities:

[TABLE] 2. (H2’)

There exist a two-dimensional positive deterministic vector function $(\beta_{t},\eta_{t})_{t\in[0,T]}$ such that for $i=1,\cdots,n$ and $(y,z)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n\times d}$, the function $f^{i}$ satisfies:

[TABLE]

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sectionDiagonally quadratic and triangular BSDEs

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In this section, we consider a special type of diagonally quadratic BSDEs as follows:

[TABLE]

For each $i=1,\cdots,n$, $H\in{\mathbb{R}}^{n\times d}$, $z\in{\mathbb{R}}^{1\times d}$, $Y\in{\mathbb{R}}^{n}$ and $y\in{\mathbb{R}}$, define by $H(z;i)$ the matrix in ${\mathbb{R}}^{n\times d}$ whose $i$th row is $z$ and whose $j$th row is $H^{j}$ for any $j\neq i$, define by $Y(y;i)$ the vector in ${\mathbb{R}}^{n}$ whose $i$th component is $y$ and whose $j$th component is $Y^{j}$ for any $j\neq i$. We make the following assumptions.

1. (A1)

There exist a constant $\alpha\in(-1,1)$ and a positive constant $C_{1}$ such that for $i=1,\cdots,n$, the function $k^{i}:\Omega\times[0,T]\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n\times d}\rightarrow{\mathbb{R}}$ depends only on the first $i$ components of $y$ and the first $i$ rows of $z$, and:

[TABLE] 2. (A2)

There exist a non-negative constant $\beta$ and a positive constant $C_{2}$ such that for $i=1,\cdots,n$ and each $(Y,Z,y_{1},y_{2},z_{1},z_{2})\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n\times d}\times{\mathbb{R}}\times{\mathbb{R}}\times{\mathbb{R}}^{1\times d}\times{\mathbb{R}}^{1\times d}$, the function $k^{i}$ satisfies:

[TABLE] 3. (A3)

There exists a non-negative constant $C_{3}$ such that $\xi=(\xi^{1},\cdots,\xi^{n})^{*}$ satisfies

[TABLE]

We have the following result.

Theorem 4.1

Let Assumptions (A1)-(A3) be satisfied. Then BSDE (4.1) has a unique solution $(Y,Z)\in\mathcal{S}^{\infty}({\mathbb{R}}^{n})\times{\rm BMO}({\mathbb{R}}^{n\times d})$ on $[0,T]$.

To prove Theorem 4.1, we need the following lemma.

Lemma 4.1

We consider the following one-dimensional BSDE:

[TABLE]

The terminal value $\eta$ and the generator $l$ satisfy the following assumptions:

1. (B1)

$\forall~{}K>0$, the function $l:\Omega\times[0,T]\times{\mathbb{R}}\times{\mathbb{R}}^{1\times d}\rightarrow{\mathbb{R}}$ satisfies:

[TABLE] 2. (B2)

There exist a non-negative constant $\beta$ and a positive constant $C$ such that for each $(y,\bar{y},z,\bar{z})\in{\mathbb{R}}\times{\mathbb{R}}\times{\mathbb{R}}^{1\times d}\times{\mathbb{R}}^{1\times d}$, the function $l$ satisfies:

[TABLE] 3. (B3)

There exists a non-negative constant $C_{1}$ such that $\eta$ satisfies:

[TABLE]

Then BSDE (4.2) has a unique solution $(Y,Z)\in\mathcal{S}^{\infty}({\mathbb{R}})\times{\rm BMO}({\mathbb{R}}^{1\times d})$ on $[0,T]$.

Proof  When $\beta=0$, $l$ is independent of $y$. From [13, Lemma 2.1], we know the result holds. When $\beta>0$, for $y\in\mathcal{S}^{\infty}({\mathbb{R}})$, we define a map $\varphi(y)=Y$, where $Y$ is given by:

[TABLE]

From the preceding result, we know $\varphi$ is well-defined and maps $\mathcal{S}^{\infty}({\mathbb{R}})$ to itself. For $y,\bar{y}\in\mathcal{S}^{\infty}({\mathbb{R}})$, let $Y=\varphi(y),\bar{Y}=\varphi(\bar{y})$. Denote $\delta Y=Y-\bar{Y},\delta y=y-\bar{y},\delta Z=Z-\bar{Z}$. We have

[TABLE]

Here $|\beta_{s}|\leq\beta,~{}|\alpha_{s}|\leq C(1+|Z_{s}|+|\bar{Z}_{s}|)$, therefore $\alpha\cdot W$ is a BMO martingale. Define

[TABLE]

Then, $\widetilde{\mathbb{P}}$ is a new probability equivalent to $\mathbb{P}$, and $\widetilde{W}$ is a Brownian motion with respect to $\widetilde{\mathbb{P}}$. We have

[TABLE]

Taking the conditional expectation with respect to $\widetilde{\mathbb{P}}$, we have

[TABLE]

When $T\leq\frac{1}{2\beta}$, $\varphi$ is a contraction map and and the statement follows from the Banach fixed point theorem. For general $T$, we can repeat the preceding process and get the result within finite steps. The proof is complete.

Proof of Theorem 4.1  We will solve BSDE (4.1) in order. For the first equation, notice that $\|\xi^{1}\|_{\infty}\leq C_{3}$, $|k^{1}(s,0,0)|\leq C_{1}$, from [16] we know it has a unique solution $(Y^{1},Z^{1})\in\mathcal{S}^{\infty}({\mathbb{R}})\times{\rm BMO}({\mathbb{R}}^{1\times d})$. Suppose that we already solve the first $(i-1)$ equations with $(Y^{j},Z^{j})\in\mathcal{S}^{\infty}({\mathbb{R}})\times{\rm BMO}({\mathbb{R}}^{1\times d}),j=1,\cdots,i-1$. For the $i$-th equation, we have $\forall K>0$,

[TABLE]

Using Hölder’s inequality, we get

[TABLE]

Using Young’s inequality, we get

[TABLE]

Let $\varepsilon$ be sufficiently small such that $\frac{1+\alpha}{2}\varepsilon\|Z^{j}\|_{\rm BMO}^{2}\leq\frac{1}{2}$. From John-Nirenberg inequality ([15, Theorem 2.2]), we have for $1\leq j\leq i-1$,

[TABLE]

Combining (4.3),(4.4) and (4.5), we obtain that $\forall~{}K>0$,

[TABLE]

Therefore, we can apply Lemma 4.1 to see the $i$-th equation admits a unique solution $(Y^{i},Z^{i})\in\mathcal{S}^{\infty}({\mathbb{R}})\times{\rm BMO}({\mathbb{R}}^{1\times d})$ on $[0,T]$. The proof is complete.