Non-Lipschitz G-BSDEs: Existence & Applications

• Non-Lipschitz G-BSDEs are backward stochastic differential equations driven by G-Brownian motion that relax classical Lipschitz conditions using properties like monotonicity and convexity.
• They are studied using approximation techniques, regularization, and monotone convergence to establish existence, uniqueness, and stability of solutions.
• Applications include robust stochastic control, financial modeling under volatility uncertainty, and probabilistic representations of fully nonlinear PDEs.

Non-Lipschitz G-BSDEs are backward stochastic differential equations driven by G-Brownian motion (in the sense of sublinear expectation theory) whose generators lack the classical global Lipschitz continuity assumption. Instead, the generator may satisfy weaker properties such as monotonicity, uniform continuity, convexity, subquadratic or quadratic growth conditions. This class encompasses equations relevant to robust stochastic control, finance under volatility uncertainty, PDE theory, and path-dependent or rough noise models. The analysis of non-Lipschitz G-BSDEs involves existence, uniqueness, comparison, regularity, convergence, robustness, and connections to nonlinear PDEs and optimization principles.

1. Existence and Uniqueness Theories Under Weak Growth and Continuity

The existence and uniqueness of solutions to non-Lipschitz G-BSDEs are established under various relaxed assumptions on the generator, such as monotonicity, uniform continuity, subquadratic growth in $z$, and local or generalized Mao's conditions. For example, if the generator $f$ satisfies a monotonicity condition in $y$ (i.e., there exists $\beta \geq 0$ such that $-[g(t,y,z)-g(t,y',z)]\sgn(y-y')\leq\beta|y-y'|$) and a uniform continuity condition in $z$, then—via approximation by Lipschitz generators, monotone convergence techniques, or Bihari-type integral bounds—one can prove existence and uniqueness in spaces such as $M_G^2(0,T)$ or more general $L^p$-type spaces for the solution triplet $(Y,Z,K)$ (Wang et al., 2018, He et al., 25 Aug 2025, Wang et al., 31 Jan 2024).

When the generator exhibits convexity in $z$ (with quadratic growth), the existence proof combines convex duality and exponential integrability conditions on the terminal condition. For multidimensional systems, diagonal structure in the generator (each equation's $z$ dependence is restricted to its own row) is often assumed to ensure well-posedness, especially for G–BSDEs (Liu, 2018, Hu et al., 2021).

A typical existence theorem states: If $f$ is uniformly continuous and monotonic in $y$, with linear growth, and the terminal condition $\xi$ is square integrable, then the G–BSDE admits a unique solution $(Y,Z,K)$ in the sublinear expectation space $S_G^2(0,T)$ (He et al., 25 Aug 2025).

2. Comparison Principles, Monotonicity, and Stability

The comparison theorem is a cornerstone in the paper of BSDEs, and its extension to non-Lipschitz G–BSDEs relies on order-preserving properties of solutions. In particular, if two generators $f^1 \leq f^2$ and terminal data $\xi^1 \leq \xi^2$ (almost surely), then $Y_t^1 \leq Y_t^2$ at all times (Wang et al., 2018, Li, 2022, He et al., 25 Aug 2025). These results extend to reflected G–BSDEs under Mao's condition, where the generator is controlled by a nondecreasing, concave function $p(u)$ satisfying $\int du/p(u)=+\infty$ (Li, 2022). The penalization method and Picard iteration provide constructive proofs for comparison results, which in turn yield monotonicity of value functions in stochastic control and risk measure settings.

3. Approximation and Regularization Methodologies

Due to the non-Lipschitz nature of the generator, classical contraction mapping or Picard iteration schemes do not always operate globally in the solution norm. However, by approximating the generator by a sequence of Lipschitz functions (e.g., via inf–sup convolution or penalization—$$\varphi_n(t,y,z)=\inf_{q\in\mathbb{Q}}[\varphi(t,y,q)+n|z-q|]$$), one obtains a sequence of classical G–BSDEs whose solutions approximate the one in the non-Lipschitz case (Wang et al., 2018, He et al., 25 Aug 2025). The monotone convergence theorem under G–expectation (which handles quasi-sure convergence rather than dominated convergence) is essential to pass to the limit in these approximations.

In cases involving nonlinear Young integrals (drivers only Hölder regular), localization and a modified Picard iteration in $p$–variation and BMO–type norms allow the solution map to be a contraction locally in time. Global solutions are then built by patching these local solutions (Song et al., 25 Apr 2025).

4. Fractional Smoothness, Regularity, and Path-Dependent Data

Non-Lipschitz terminal conditions—possibly path-dependent—induce singularities in the solution processes (Y,Z) that must be quantified. The notion of generalized fractional smoothness characterizes the L$_p$-variation of Y and Z, showing that for a terminal condition $\xi$ of the form $g(X_{r_1},...,X_{r_L})$ having local fractional smoothness indices $\Theta=(\theta_1,...,\theta_L)$, one has regularity estimates:

$$\| Y_{r_l} - \mathbb{E}( Y_{r_l}\,|\,\mathcal{F}_s )\|_p \leq c (r_l-s)^{\theta_l/2}.$$

Such control allows for error estimates in numerical schemes (adapted splines, regression-based Monte Carlo), which can be grid-refined near singular points to optimize convergence (Geiss et al., 2011, Geiss et al., 2012).

For BSDEs driven by Lévy processes and Poisson random measures, chaos expansion techniques demonstrate that the structure of irregular (non-Lipschitz, fractionally smooth) terminal data is inherited by the solution through explicit representation and Malliavin calculus (Geiss et al., 2012).

5. Robustness, Convergence, and Numerical Implications

Discrete-time approximations (BSΔEs) with non-Lipschitz drivers provide robust and convergent schemes for BSDEs. If the driver satisfies subquadratic growth in $z$, and the mesh size of the discretization shrinks to zero, the discrete solution converges in $L^2$ to the continuous one (Cheridito et al., 2010). When the driver is convex in $z$, duality methods yield implicit dual representation and reinforce robustness—important for stability under model perturbation or in numerical applications for risk measures and utility maximization (Cheridito et al., 2010).

Monotonicity and uniform continuity of the generator ensure that numerical methods such as penalization, piecewise monotone BSDEs, or splines, remain stable and consistent even when the Lipschitz assumption is violated. This is critical when applying BSDE techniques in financial modeling, insurance, and stochastic control under model uncertainty (Fan et al., 2014, Fan et al., 2014).

6. Connections to PDEs, Control, Utility, and Further Applications

Non-Lipschitz G-BSDEs furnish probabilistic representations of nonlinear PDEs via nonlinear Feynman-Kac formulae. In Markovian frameworks, the solution $Y_t$ provides the unique viscosity solution to fully nonlinear parabolic or elliptic PDEs:

$$\partial_t u + G( H( t, x, u, D_x u, D_x^2 u ) ) + \langle b(t,x), D_x u \rangle + f( t, x, u, \sigma^\top D_x u ) = 0,$$

with terminal/boundary conditions (Wang et al., 2018, Liu, 2018, Hu et al., 2014).

In optimal control and finance, the stochastic recursive utility and value functions determined by non-Lipschitz G–BSDEs (e.g., Epstein–Zin utility under volatility uncertainty) satisfy the dynamic programming principle and solve the associated fully nonlinear Hamilton–Jacobi–BeLLMan equation (He et al., 25 Aug 2025). These connections extend to reflected BSDEs for American option pricing, robust hedging (under Mao's condition), and degenerate G–expectation spaces relevant in regularity theory for PDEs (Hu et al., 2022, Li, 2022).

7. Quadratic, Convex and Young-Driven Extensions

Recent studies address BSDEs with quadratic and convex generators, allowing unbounded terminal conditions subject to exponential integrability. Existence and uniqueness hinge on monotonicity and convex growth conditions (in $z$), with duality (Legendre–Fenchel transform) yielding explicit solution representations and comparison results (Wang et al., 31 Jan 2024, Hu et al., 2021). The nonlinear Young integral variant further relaxes regularity, allowing drivers which are only Hölder in time and unbounded in space, with solution theory built via localization and modified Picard schemes (Song et al., 25 Apr 2025).

Summary Table: Existence and Uniqueness Assumptions

Generator Condition	Terminal Data	Methodology
Monotonic, uniform continuity	$L^2$/$L^p$	Regularization + monotone
Fractional smoothness (path)	Borel/fractional	Fractional/Besov regularity
Quadratic/convex in $z$	Exponential	Duality + exponential est.
Mao's condition, penalization	Obstacle/Bound.	Picard + penalization
Young-Driver (rough)	Bounded/loc. bdd	Modified Picard + localize

Concluding Remarks

Non-Lipschitz G-BSDEs broaden the BSDE landscape by relaxing global regularity conditions on the generator, accommodating monotonicity, convexity, fractional smoothness, and rough/drifted noise. They admit rigorous existence, uniqueness, and comparison theorems under generalized assumptions, robust numerical schemes, and have deep connections to fully nonlinear PDE theory and recursive stochastic control. These advances lay the groundwork for rigorous functional analysis, stochastic optimization, and financial applications where traditional Lipschitz hypotheses are untenable.