Multidimensional Quadratic BSDEs

• Multidimensional quadratic BSDEs are backward stochastic systems with quadratic growth in the control variable that present complex coupling between equations.
• They are fundamental in applications like stochastic control, financial mathematics, and the analysis of nonlinear PDEs, relying on tools such as BMO martingales and fixed-point methods.
• Existence and uniqueness depend on structural conditions including diagonal or weak coupling, small interaction effects, and the integrability of terminal data.

A multidimensional quadratic backward stochastic differential equation (BSDE) is a system of stochastic integral equations in which the unknown process evolves backward in time, and the generator exhibits quadratic (or more generally superlinear) growth in its "control" variable. The multidimensional context introduces additional complexity due to coupling between equations, complicating the existence, uniqueness, and stability analysis compared to the one-dimensional setting. These systems are fundamental in stochastic control, financial mathematics, mean-field games, and nonlinear PDE theory.

1. Mathematical Formulation and Quadratic Structure

A prototypical multidimensional quadratic BSDE takes the form

$$Y_t = \xi + \int_t^T f(s, Y_s, Z_s)\,ds - \int_t^T Z_s \, dW_s, \quad t\in[0,T],$$

where $Y_t\in\mathbb{R}^n$, $Z_t\in\mathbb{R}^{n\times d}$, $W_t$ is a $d$-dimensional Brownian motion, $\xi$ is an $\mathcal{F}_T$-measurable terminal condition, and the generator $f$ is allowed to have quadratic growth in $Z$.

Quadratic growth refers to bounds of the form

$$|f(t,y,z)| \leq \alpha_t + \phi(|y|) + \frac{\gamma}{2} |z|^2,$$

possibly with additional subquadratic or structured off-diagonal terms. Key distinctions in the multidimensional theory are captured by how the generator's quadratic terms couple the various components of $Z$:

• Diagonally quadratic: $f^i$ depends quadratically only on $z^i$.
• Interactively quadratic: $f^i$ may grow quadratically in both $z^i$ and $z^j$ for $j\neq i$ (involving terms like $z^j(z^i)^\top$).
• Triangularly quadratic: $f^i$ depends quadratically on $z^j$ for $j\leq i$ (lower/upper triangular structure).
• Skew subquadratic: Off-diagonal elements are permitted subquadratic growth $|z^j|^{1+\delta}$.

General results require boundedness or integrability conditions on $\xi$ and the generator, and, frequently, BMO (bounded mean oscillation) martingale structure for solutions.

2. Existence and Uniqueness: Structural Conditions

Existence and uniqueness of solutions in the multidimensional quadratic regime typically require one or more of the following:

1. Diagonal or weak coupling structure: When each $f^i$ has at most quadratic growth in $z^i$ and only strictly controlled (e.g., logarithmic or subquadratic) growth in off-diagonal terms, global well-posedness is established for bounded $\xi$ (Hu et al., 2014, Yang, 2023, Fan et al., 2020). For more general off-diagonal growth, local well-posedness can often be obtained.
2. Smallness/weak dependence: If the terminal data or generator's coupling is sufficiently small, existence can be shown for general quadratic or superquadratic systems (Jamneshan et al., 2016, Nam, 2019).
3. Mean-field and anticipated structures: For mean-field BSDEs where $f$ depends on the distribution of $(Y_t, Z_t)$, solvability persists under diagonal structure and additional control on the law dependence (Jiang et al., 2023, Tang et al., 2023).
4. Componentwise convexity/concavity and exponential integrability: If the generator is (componentwise) convex or concave in $z^i$ and $\xi$ admits (arbitrary order) exponential moments, unique global solutions can be constructed even for unbounded $\xi$ (Fan et al., 2020, Hu et al., 26 Mar 2025).
5. Special structures (product, lower-triangular, Markovian/FBSDE): Existence and uniqueness are available for tailored product generators, triangular forms, or where an associated FBSDE admits a strong solution (Qian et al., 2017, Cheridito et al., 2013).

The following table summarizes key structural scenarios:

Structure	Off-diagonal growth	Solution Type	Reference
Diagonally quadratic	Subquadratic/logarithmic	Global	(Hu et al., 2014, Yang, 2023, Fan et al., 2020)
Diagonally quadratic	Subquadratic	Local	(Hu et al., 2014, Yang, 2023)
Interactively quadratic	Small interaction	Global/Local	(Fan et al., 11 Oct 2024)
Triangularly diagonal	Arbitrary	Global	(Yang, 2023)
General quadratic (unbounded $\xi$)	Componentwise convex	Global	(Fan et al., 2020, Hu et al., 26 Mar 2025)
Special product/Markovian	Product/negligible	Global (bounded $\xi$)	(Qian et al., 2017, Cheridito et al., 2013)

3. Analytical Techniques and A Priori Estimates

Solving multidimensional quadratic BSDEs relies on advanced stochastic analysis techniques:

• BMO Martingale Framework: BMO spaces control the exponential moments arising from quadratic growth and are essential for establishing uniqueness, a priori bounds, and integrability (Hu et al., 2014, Harter et al., 2016, Fan et al., 2020, Yang, 2023).
• Fixed-Point/Contraction Mapping: For both local and global solvability, the contraction mapping principle is applied in Banach spaces of bounded processes and BMO martingales, leveraging structural decoupling (Hu et al., 2014, Tang et al., 2023, Jiang et al., 2023).
• Stepwise/Backward Iteration: Global existence is often recovered by partitioning $[0,T]$ into subintervals where local solvability applies and iteratively extending the solution backward in time (Nam, 2019, Fan et al., 2020).
• A Priori Exponential Bounds: Sharp a priori estimates using exponential transformations (via Itô’s formula) allow control over the process norms for quadratic and even singular generators (Yang, 2023, Wang et al., 4 Jul 2025).
• Girsanov Transformations: Change of measure techniques are employed in settings with particular generator structure to reduce to more tractable equations (Qian et al., 2017).

4. Extensions: Mean-field, Anticipated, Singular, and Superquadratic Systems

The multidimensional theory is extended to several advanced settings:

• Mean-field BSDEs: Generators depending on the law of $(Y_t, Z_t)$ and exhibiting diagonally quadratic (or even quadratic-in-law) growth in $Z$ are solvable under fixed-point and a priori control, including Volterra-type BSDEs (Jiang et al., 2023, Tang et al., 2023).
• Anticipated BSDEs: Generators that depend on the future values $(Y_{t+\delta(t)}, Z_{t+\zeta(t)})$ (anticipated terms) complicate analysis; solvability is ensured either for small terminal data (general growth) or under diagonal/triangular structure (Hu et al., 2019, Hu et al., 26 Mar 2025).
• Singular Generators: BSDEs with singular diagonal terms such as $\frac{\delta}{2 y^i}|z^i|^2$ (arising in utility maximization) admit unique positive solutions under careful integrability and convexity assumptions (Wang et al., 4 Jul 2025).
• Superquadratic Growth: For generator growth $|z|^q$, $q>2$, a local-to-global solution procedure based on backward iteration and pathwise a priori bounds is applicable, but requires smallness/weak coupling (Nam, 2019).

5. Applications and Connections to PDEs and Control

Multidimensional quadratic BSDEs are intimately connected to fully nonlinear PDEs, stochastic games, and optimal control:

• PDEs with Quadratic Gradient Nonlinearity: Via Feynman-Kac representation, solutions correspond to viscosity or mild solutions of quasilinear (possibly path-dependent) PDEs with superquadratic terms in the gradient (Nam, 2019, Addona et al., 2019).
• Stochastic Differential Games: Coupled quadratic BSDE systems encode value processes and feedback equilibria in zero-sum games, especially under regime switching and random coefficients (Zhang et al., 2023).
• Utility Maximization and Financial Mathematics: Singular quadratic generators model power utility optimization in regime-switching markets, with explicit feedback forms for optimal strategies (Wang et al., 4 Jul 2025).
• Banach Space and Rough-Path Contexts: Quadratic BSDEs in infinite-dimensional settings and with low regularity drivers expand the framework to semilinear PDEs and rough stochastic evolution (Addona et al., 2019, Diehl et al., 2010).

6. Limitations, Challenges, and Open Problems

Despite recent advances, several challenges persist in the general multidimensional quadratic BSDE setting:

• Failure of Comparison and Girsanov Principles: The multidimensional context generally lacks a comparison theorem, precluding monotone approximation and complicating uniqueness proofs, especially for fully coupled and non-diagonal systems (Fan et al., 2020, Fan et al., 11 Oct 2024).
• Necessity of Structure/Sobriety: Global solvability without explicit diagonal, triangular, or decoupling structure is typically impossible—known counterexamples and blow-up mechanisms exist for fully coupled systems (Yang, 2023, Hu et al., 2014).
• BMO-norm Smallness Barrier: Stability and approximation methods require uniform BMO bounds, but multidimensional extensions of reverse Hölder inequalities remain open (Harter et al., 2016).
• Indefinite and Interactively Quadratic Generators: Indefinite stochastic Riccati equations and full interactive quadratic growth in non-Markovian, high-dimensional settings remain challenging, with uniqueness often requiring verification arguments specific to game-theoretic structures (Fan et al., 11 Oct 2024, Zhang et al., 2023).
• Unbounded Data and Singularities: Existence and uniqueness for unbounded terminal data or singular generators require exponential moment methods and often convexity/concavity, limiting generality (Fan et al., 2020, Wang et al., 4 Jul 2025, Hu et al., 26 Mar 2025).

7. References to Foundational and Recent Developments

A selection of key works includes:

• (Hu et al., 2014): Multidimensional BSDEs of diagonally quadratic generators (Hu & Tang).
• (Fan et al., 2020): General solvability for multidimensional BSDEs with diagonally quadratic and convex/concave generators (Fan et al.).
• (Yang, 2023): Global solvability with small off-diagonal quadratic growth.
• (Fan et al., 11 Oct 2024): Interactively quadratic generators and non-Markovian theory.
• (Wang et al., 4 Jul 2025): Multidimensional singular and diagonally quadratic BSDEs.
• (Tang et al., 2023, Jiang et al., 2023): Mean-field and superlinear-in-law extensions.
• (Zhang et al., 2023): Indefinite SREs in stochastic games with regime switching.

The multidimensional quadratic BSDE theory is now a mature field encompassing diagonal, triangular, interactively quadratic, mean-field, singular, and rough settings, with applications permeating modern stochastic analysis, mathematical finance, and PDE theory. However, the full characterization and resolution of general multidimensional, fully quadratic BSDEs remain an outstanding open problem, with ongoing research focused on structural extensions, stability conditions, and control-theoretic implications.