Singular Forward-Backward SDEs

• Singular FBSDEs are stochastic systems where forward equations exhibit degenerate or irregular drifts and backward equations feature discontinuous terminal conditions.
• Unified frameworks using decoupling fields and ODE domination techniques ensure existence and uniqueness despite non-standard coefficients and discontinuities.
• Robust time discretization schemes and noise regularization enable efficient numerical approximations for applications in finance, filtering, and optimal control.

Singular forward-backward stochastic differential equations (FBSDEs) encompass systems in which one or both components—the forward SDE and the backward SDE—display degenerate, irregular, or otherwise non-standard features such as non-smooth drifts, distributional coefficients, or discontinuous terminal conditions. These systems frequently arise in nonlinear filtering, stochastic control, financial derivatives pricing (especially emissions markets), and particle systems with absorbing boundaries. The analysis and numerical approximation of singular FBSDEs pose significant theoretical and computational challenges due to the breakdown of classical regularity assumptions. Recent advances have developed unified frameworks, refined existence/uniqueness theory, and robust time discretization techniques for these equations.

1. Structural Features of Singular FBSDEs

Singular FBSDEs are characterized by forward equations with one or more of:

• Degenerate diffusion coefficients or pure drift terms (no noise in certain directions)
• Drifts that are merely bounded, Dini-continuous, distributional, or form-bounded (but not Lipschitz)
• Coupling through hitting times or feedbacks that depend on backward quantities

Backwards equations often involve:

• Non-smooth terminal conditions, e.g., indicator functions of the forward path or step functions
• Quadratic or superlinear generators in $(Y, Z)$
• Coupling through absorption probabilities or moving boundary conditions

For instance, in recent models for particle systems with moving boundaries, the conditional probability of absorption is computed via a backward SDE with terminal condition $Y_T^i = 1_{\{\tau_i \leq T\}}$ where $\tau_i$ is the hitting time of a stochastic boundary determined by other particles' states (Jettkant et al., 3 Oct 2025).

Singular behavior may also arise when distributional drift coefficients (elements of negative Sobolev or Besov spaces) appear in the forward or backward equations (Issoglio et al., 2016, Issoglio et al., 2022).

2. Existence and Uniqueness: Analytical Frameworks

The existence and uniqueness theory for singular FBSDEs must accommodate degenerate forward dynamics, irregular coefficients, and discontinuous boundary data.

A unified approach is built around constructing a so-called "decoupling random field" $u(t, x)$ such that $Y_t = u(t, X_t)$ decouples the FBSDE system. The regularity, typically uniform Lipschitz continuity in $x$, of $u$ is linked to solutions of a "characteristic" Riccati-type BSDE with quadratic growth, which may itself be singular (Ma et al., 2011). The analysis proceeds by dominating the nonlinear terms via ODE comparison and proving that the spatial derivative $u_x$ remains bounded on the time interval.

For equations with binary or indicator terminal data, as in emissions trading models, a relaxed terminal condition is imposed where $Y_T$ must only satisfy bounds between left and right modifications of the discontinuous function: $\phi_-(E_T) \leq Y_T \leq \phi_+(E_T)$ Mollification and uniform estimates enable convergence to a unique solution (Carmona et al., 2012).

When the drift is merely Dini-continuous or distributional, transformations such as the Zvonkin-type are used: the forward SDE solution is regularized by the noise, and the problem is recast in a function space adapted to the singular coefficients. Backward Kolmogorov equations and martingale problem techniques provide the foundation for rigorous well-posedness even when the drift is not a function but a distribution (Zhao, 2020, Issoglio et al., 2022, Kinzebulatov et al., 2023).

Transformations also extend to the virtual and weak solution notions for equations with distributional coefficients, leveraging mild PDE theory in appropriate Sobolev spaces (Issoglio et al., 2016).

3. Time Discretization Schemes and Regularization by Noise

Numerical approximation of singular FBSDEs requires schemes that are robust to non-smooth coefficients. An explicit time discretization in the spirit of Pagès and Sagna, combining a forward Euler–Maruyama approximation and a centered BTZ (Bouchard–Touzi–Zhang) scheme for the backward component, achieves global error close to order $1/2$, even under modest assumptions (Pellat et al., 11 Dec 2024): $$\mathbb{E}\left[\sup_{0 \leq i \leq N} |Y_{t_i} - \mathcal{Y}_i^\pi|^2\right] + \mathbb{E}\left[\sum_{i=0}^{N-1} \int_{t_i}^{t_{i+1}} |Z_t - \mathcal{Z}_i^\pi|^2 dt\right] \leq C h^{1-\gamma}$$ for arbitrarily small $\gamma > 0$.

Regularization by noise plays a central role: the Brownian component imparts sufficient regularity to allow Malliavin and variational derivative estimates for the solution, which are critical for controlling error rates. Key representation results such as $$Z_t = D_t Y_t = (\nabla_x Y_t)^\top (\nabla_x X_t)^{-1}$$ enable the transfer of regularity from the forward to the backward components.

Properties of bounded mean oscillation (BMO) for martingale integrands and reverse Hölder inequalities are used to keep approximation errors stable and uniform across mesh levels.

4. Moving Boundary Problems and Free-Boundary PDEs

A novel class of singular FBSDEs arises in systems where the forward dynamics depend on hitting time conditions determined by the backward conditional probabilities. Each particle's absorption is specified by the first time the forward process crosses a moving threshold:

$$\tau_i = \inf\{ t \in [\varrho, T]: X_t^i \leq \sum_{j=1}^{N} D_{ij} Y_t^j \}$$

The backward conditional probability is computed by solving a tiered moving boundary PDE problem, with the domain itself coupled to the decoupling field from lower-dimensional systems: $$\mathcal{D}_T^I v = \left\{ (t, x): x_i > \sum_{j=1}^{N} D_{ij} v_t^{I\setminus\{i\}, j}(x^{-i}), \forall i \in I \right\}$$ The backward PDE for alive particles reads: $$\begin{cases} \partial_t v_t^{I, i}(x) + \frac{\sigma^2}{2}\Delta v_t^{I, i}(x) = 0, & (t, x) \in \mathcal{D}^I v \ v_t^{I, i}(x) = F_i^I v(t, x), & (t, x) \notin \mathcal{D}_T^I v \ v_T^{I, i}(x) = 0, & (T, x) \in \mathcal{D}_T^I v \end{cases}$$ Classical well-posedness and uniqueness for this free-boundary PDE are essential for establishing uniqueness and stability of the singular FBSDE system (Jettkant et al., 3 Oct 2025).

5. Scalar Conservation Laws, Dirac Mass Formation, and Breakdown of Markovian Structure

When the forward SDE is degenerate (i.e., no noise in certain directions) and the terminal condition is binary or discontinuous, as in CO$_2$ emission markets, the backward field can be interpreted as the entropy solution to a scalar conservation law with stochastic perturbation (Carmona et al., 2012):

• The flow property of the forward process may break down at terminal time due to shock formation and concentration of mass—a Dirac point mass develops at the discontinuity threshold.
• The backward component $Y_T$ cannot be recovered as a deterministic function of the forward state; conditional on $E_T = A$, the law of $Y_T$ fills the entire interval $[0,1]$.

This breakdown of the Markovian property highlights the difference between regular and singular FBSDEs: in the singular case, additional randomness persists in the backward component even when the forward process is fixed at the discontinuity.

6. Stochastic Volterra Equations and Singular Kernels

Singular FBSDEs also feature in infinite-dimensional stochastic Volterra processes, where the convolution kernel $K$ is singular at the origin. The law of the solution is linked to a backward Kolmogorov equation on a reproducing kernel Hilbert space $\mathcal{H}_1$, with unique mild solutions governed by singular directional derivatives along $K$ (Gasteratos et al., 25 Sep 2025). The extra regularity arising from coefficients depending solely on finite-dimensional projections enables classical differentiability even along singular directions, which distinguishes this setting from Nemytskii-type nonlinearities.

7. Applications and Numerical Strategies

Numerical methods adapted for singular FBSDEs combine splitting schemes (diffusion versus nonlinear transport), non-linear regression with deep neural networks, and conservative finite difference approximations of associated conservation laws. Under monotonicity and local Lipschitz conditions, splitting methods yield rigorous convergence rates (typically $1/2$ in mesh size) (Chassagneux et al., 2021).

Significant applications include:

• Carbon market modeling with regulatory mechanisms (banking, borrowing, withdrawal) and multiple compliance periods (Jean-Francois et al., 2020),
• Robust optimal investment strategies for jump-diffusion models, characterized by implicit FBSDEs (Santacroce et al., 2023),
• Turbulence in fluids via FBSDE–PDE links (Ohashi et al., 2016).

Table: Distinguishing Features of Singular FBSDEs

Feature	Regular FBSDEs	Singular FBSDEs
Forward drift	Lipschitz, smooth	Dini-continuous, distributional, form-bounded
Terminal condition (backward)	Smooth, continuous	Indicator, binary, step function
Decoupling field regularity	Uniform Lipschitz	May fail at singularities, only local regularity
Existence/Uniqueness method	Contraction, PDE	ODE domination, Zvonkin transform, mollification
Markovian representation	Holds at terminal time	May break down (Dirac mass formation, persistent randomness)

References to Significant Results

• Unified regularity and ODE domination for characteristic BSDE: (Ma et al., 2011)
• Existence, relaxed uniqueness, and entropy solutions for conservation law coupling: (Carmona et al., 2012)
• Time discretization with nearly $1/2$ error for singular drifts: (Pellat et al., 11 Dec 2024)
• Free-boundary PDE decoupling in particle systems: (Jettkant et al., 3 Oct 2025)
• Distributional drift, martingale problems, and strong solutions: (Kinzebulatov et al., 2023, Issoglio et al., 2016, Issoglio et al., 2022)
• Multiperiod models for emissions markets, explicit banking/borrowing: (Jean-Francois et al., 2020)
• Splitting schemes for degenerate/entropy coupled FBSDEs: (Chassagneux et al., 2021)

Singular forward-backward SDEs thus constitute a vibrant intersection of stochastic analysis, PDE theory, and numerical methods, enabling robust modeling and computation in complex systems where classical regularity is unattainable.