Quadratic Backward Stochastic Differential Equation

• Quadratic BSDE is a stochastic evolution equation with a generator exhibiting quadratic growth in z, widely used to model risk measures and nonlinear PDE obstacles.
• Key results include sharp a priori logarithmic bounds, monotone stability, and uniqueness achieved via convexity, duality, and θ-difference methods.
• Applications span financial indifference pricing, stochastic control, and the probabilistic representation of viscosity solutions for nonlinear PDEs with quadratic gradient terms.

A quadratic backward stochastic differential equation (BSDE) is a stochastic evolution equation where the generator exhibits quadratic growth in the second unknown (often denoted as $z$), and the solution is sought backward in time from a terminal condition. Quadratic BSDEs and their reflected variants are central to modern stochastic analysis, bridging stochastic control, financial mathematics, and nonlinear partial differential equations (PDEs) with quadratic nonlinearities. Their paper involves advanced techniques for existence, uniqueness, a priori estimates, and nonlinear expectation theory, especially in regimes with unbounded obstacles or terminal conditions.

1. Quadratic Growth and Structural Assumptions

Quadratic BSDEs possess generators $f(t, y, z)$ with strict quadratic growth in the $z$-variable and at most linear growth in $y$. The canonical assumption (H1) postulates that, for all $(t, \omega, y, z)$, there exist $\beta \ge 0$, $\gamma > 0$ such that

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

This structure generalizes the classical Lipschitz BSDE theory and is the source of substantial analytic difficulties: quadratic terms defeat standard contraction or Picard iteration arguments. To develop a rigorous theory, additional conditions—such as local Lipschitz continuity in $y$ (H2), and convexity/concavity in $z$ (H3)—are imposed to establish uniqueness and enable comparison arguments. For reflected BSDEs, the equation

$$Y_t = \xi + \int_t^T f(s, Y_s, Z_s) ds + K_T - K_t - \int_t^T Z_s dB_s,$$

is coupled with the constraint $Y_t \ge L_t$ (an obstacle process), with $K$ a nondecreasing process increasing only when $Y_t = L_t$.

2. Existence, A Priori Estimates, and Stability

Existence even with unbounded data is established via a combination of a priori estimates and monotone stability. Key results include:

• Upper/Lower Bounds: Proposition 2.2 provides logarithmic bounds for the process $Y$, e.g.,

$$Y_t \leq c_0 + \frac{1}{\gamma}\ln\,E[\exp(e^{\beta T} (\xi^+ \vee L^+_*) | \mathcal{F}_t )],$$

where $L^+_*$ is the running sup of $L^+$, and $\xi^+$ is the positive part of the terminal data.

• Monotone Stability: Theorem 3.1 asserts that, if the terminal value $\xi^n$ and obstacle $L^n$ form converging monotone sequences (and $f^n$ converges locally uniformly), then solutions $(Y^n, Z^n, K^n)$ converge in appropriate spaces ($S^p$, $H^{2,2p}$, $K^p$), and the limit satisfies the limiting quadratic RBSDE.
• Space of Solutions: Under suitable exponential integrability,

$$Y \in \bigcap_{p=1}^\infty S^p[0,T], \quad Z \in H^{2,2p}([0,T];\mathbb{R}^d), \quad K \in K^p[0,T].$$

Bounds on $Z$ and $K$ are provided up to the $L^p$-level.

These methods bypass the need for boundedness by leveraging exponential moment control, and through localization and compactness extract solutions in global spaces.

3. Uniqueness, Comparison Theorems, and θ-Difference Approach

Quadratic BSDEs admit delicate comparison theorems:

• Convex Case: For convex generators in $z$, uniqueness is established via a θ-difference method: one considers $\theta Y - \widehat{Y}$ for $\theta \in (0,1)$, allowing the convexity in $z$ to control the sign of differences and derive an integral inequality. When the terminal data and obstacle ordering is preserved, so is the solution:

$$\text{If } \xi \leq \widehat{\xi} \text{ and } L \leq \widehat{L},\,\text{ then } Y \leq \widehat{Y}.$$

• Concave Case: The paper employs a Legendre–Fenchel transform to treat generators concave in $z$. Here, the dual generator

$$f^*(t, y, q) = \sup_z \{\langle q, z \rangle + f(t,y,z)\}$$

allows recasting the uniqueness argument in the dual space; uniqueness hence follows for solutions with prescribed exponential integrability.

Comparison principles and duality are central to ensuring not only uniqueness but also monotonicity properties for applications, such as risk measures and control.

4. Quadratic g-Evaluations and Optimal Stopping

The nonlinear expectation (g-evaluation) framework is extended to the quadratic growth setting. For a generator $g$ obeying (H1)-(H3), the solution $Y^\xi_t$ to

$$Y_t^\xi = \xi + \int_t^T g(s, Y_s^\xi, Z_s^\xi) ds - \int_t^T Z_s^\xi dB_s$$

defines the operator

$\hat{g}_\nu[\xi] := Y_\nu^\xi,$

which enjoys monotonicity, time-consistency, and, when $g$ is independent of $y$, translation invariance.

For the reflected case, the solution $Y_t$ possesses a Snell envelope property: $$Y_0 = \sup_{\tau \in \mathcal{T}_{0,T}} \hat{g}_0[R_\tau],$$ where $R_t = L_t$ for $t < T$, $R_T = \xi$, and the optimal stopping time is the first hitting time of $Y$ to $L$. This structure links quadratic BSDEs to dynamic risk measures and nonlinear Markovian optimal stopping, providing a robust extension of the classical linear Snell envelope theory.

5. Probabilistic Representation of Obstacle Problems for Nonlinear PDEs

Through a flow property and a Markovian structure, the quadratic RBSDE admits a probabilistic representation for the viscosity solution of semilinear parabolic PDEs with obstacle and quadratic gradient nonlinearity. Specifically,

$u(t,x) = Y_0^{(t,x)},$

with $(Y^{(t,x)}, Z^{(t,x)}, K^{(t,x)})$ solving the RBSDE with terminal function $h(x)$ and obstacle $l(t,x)$. The corresponding viscosity solution $u$ satisfies

$$\min \Bigl\{ u(t,x) - l(t,x),\; -\partial_t u(t,x) - \mathcal{L}u(t,x) - f\bigl(t,x,u(t,x),D_x u(t,x)\bigr) \Bigr\} = 0,\quad u(T,x) = h(x).$$

Here, $\mathcal{L}$ is the diffusion generator associated with the forward SDE. The comparison theorem at the PDE level (uniqueness) holds under an additional spatial Lipschitz condition on $f$.

This connection supports both existence and uniqueness of viscosity solutions in the presence of a gradient-square (quadratic) nonlinearity and provides a stochastic representation for numerics and qualitative analysis.

6. Methodological Innovations and Applications

• A Priori and Stability Analysis: All main results rest on sharp a priori logarithmic bounds for $Y_t$, exponential moment criteria for the terminal and obstacle data, and explicit stability under monotone perturbations.
• Nonlinear Duality and θ-Difference: Nonstandard comparison and uniqueness results are achieved by exploiting convexity or concavity via duality and θ-difference. These are essential for non-Lipschitz (quadratic) generators and circumvent the failure of strict contractivity.
• Applications: Beyond the general theory, the methods inform quadratic-g-evaluation-based optimal stopping with risk-averse or nonlinearity-aware dynamic criteria, as well as providing probabilistic (BSDE-based) tools for fully nonlinear PDE obstacle problems. In finance, such BSDEs model indifference pricing with risk-sensitive criteria and nonlinear pricing measures; in stochastic control, they characterize value functions where the controller's action affects volatility in a nonlinear way.

7. Broader Impact and Research Directions

The rigorous analysis of quadratic RBSDEs with unbounded obstacles and terminal conditions significantly extends the scope of BSDE theory to nonlinear risk, optimal stopping with unbounded reward/costs, and nonlinear PDEs with nonstandard, nonmonotone terms. The methodology—localization, exponential integrability, monotone stability, and duality—constitutes a robust recipe with potential to generalize further (e.g., to multidimensional systems, path-dependent or non-Markovian settings, or equations driven by rough signals). Current research explores extensions to BSDEs with jumps, mean field interactions, and under nonlinear expectation frameworks (G-Brownian motion), seeking to resolve further open questions on uniqueness under minimal convergence, stability under rough drivers, and explicit numerical approximation in high dimension.