Constrained BSDEs: Theory and Applications

Updated 24 November 2025

Constrained BSDEs are backward stochastic differential equations subject to additional pathwise, mean, or distributional constraints, vital for modeling complex control and financial systems.
The penalization and reflection methods ensure convergence to minimal (or maximal) solutions by embedding the original problem into a broader unconstrained framework.
Applications span optimal control, portfolio management, and obstacle PDEs, where such constraints help model risk, regulatory requirements, and singular terminal conditions.

A constrained backward stochastic differential equation (BSDE) is a backward SDE whose solution is required to satisfy pathwise, integral, or distributional constraints on its components (typically the primary solution, the “gain” or “martingale” coefficient, or in the law of the process), either pointwise or in some average sense. These equations bridge stochastic control, partial differential equations with obstacles, and probabilistic representations of singular and constrained optimization. Constrained BSDEs encompass classical reflected BSDEs, BSDEs with state, law, or gain-process constraints, and mean-reflected BSDEs, with an elaborate taxonomy depending on the nature of the imposed constraints (hard, soft, singular, law-based).

1. Foundational Formulations and Types of Constraints

Constrained BSDEs arise whenever the solution triple $(Y, Z, K)$ of a standard backward SDE

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

is further required to satisfy a constraint, imposed either:

Pathwise: e.g., $Y_t \geq S_t$ a.s. or $Z_t \in C$ , a convex set, almost everywhere.
Integral/Mean-Value: e.g., $\mathbb{E}[L(Y_t)] \leq 0$ , or more generally, that the distribution $\mu_t = \mathcal{L}(Y_t)$ stays in a prescribed subset of measures.
Singular Terminal: e.g., $Y_T = +\infty$ , encoding hard state constraints for associated control problems.

Notable examples include:

Singular terminal BSDEs: A key instantiation (Ankirchner et al., 2013), used in state-constrained stochastic control, where the backward equation has terminal blow-up ( $Y_T = \infty$ ), encoding finite-horizon constraints on forward dynamics.
Convex gain-process constraint: BSDEs that require the martingale term (the "gain process" $Z$ ) to take values in a prescribed convex subset at almost every time (Bouchard et al., 2014, Kharroubi et al., 2020).
Mean/law reflection: Equations in which the expectation, or the law, of the solution component must remain in a set, leading to mean-reflected or law-constrained BSDEs (Li, 2023, Li et al., 19 Jan 2025, Briand et al., 2019).
Gradient/delta-gamma constraints: BSDEs where the control variable is restricted in a "delta-gamma" sense (continuous semimartingale structure) (Heyne et al., 2013).

Each class demands specific existence, uniqueness, and approximation theory tailored to the mode of the constraint.

2. Existence and Uniqueness: Minimal (or Maximal) Solutions

Constrained BSDEs typically admit a minimal (or maximal) solution in a suitable sense. The main structural approach is to embed the original solution class into a broader supersolution (or subsolution) class, and invoke monotonicity and stability arguments.

Monotone Approximation (Penalization): One introduces a family of unconstrained or penalized BSDEs, whose solutions, as the penalty parameter increases, converge monotonically (from above or below) to the minimal solution of the constrained problem (Wu et al., 2010, Ankirchner et al., 2013, Kharroubi et al., 2020, Bouchard et al., 2014). For instance, penalizing the constraint yields solutions $y^n$ to unconstrained BSDEs with penalized driver, and by taking $n \to \infty$ , the limit process satisfies the constraint.
Singular Terminal Condition: In the setting of hard terminal constraints, existence is proven by approximating the singular terminal with increasing finite values and passing to the monotone limit (Ankirchner et al., 2013).
Skorokhod Problem (Reflection): For mean or law-reflected BSDEs, solution theory is based on backward Skorokhod problems for the mean value process, yielding reflected (flat-off) solutions with prescribed minimality properties (Li, 2023, Li et al., 19 Jan 2025, He et al., 15 May 2024, He et al., 26 Aug 2025).
Convex Analysis/Fenchel Duality: When the constraint set and the driver are convex, existence and uniqueness are established by convex duality, with the minimal solution corresponding to a saddle point of an associated convex-concave functional (Heyne et al., 2013, Wu et al., 2010).

In all cases, uniqueness is tied to the minimal (or maximal) property, and the uniqueness of the limiting solution within the broader supersolution (or subsolution) class.

3. Analytical Properties and Structural Results

Key analytical properties have been established for minimal (super)solutions of constrained BSDEs:

Stability and Semicontinuity: The mapping from terminal data $\xi$ to the initial solution value $\mathcal{E}^{g, \phi}(\xi)$ is lower semicontinuous in general (with only one-sided continuity in nonconvex cases), and fully continuous in the convex case in the interior of the effective domain (Wu et al., 2010, Heyne et al., 2013).
Comparison Principles: For properly structured constraints and drivers, pointwise comparison and monotonicity hold, often limited to expectations when the constraint is nonconvex or acts on the law (Li, 21 Nov 2025, Li et al., 19 Jan 2025).
Facelift Phenomenon: In BSDEs with gain-process constraints, the minimal solution at any $t<T$ is the "facelift" (a convex conjugate-type regularization) of the minimal solution map itself (Bouchard et al., 2014, Kharroubi et al., 2020).
Regularity: Under boundedness and Lipschitz hypotheses, spatial Lipschitz and $1/2$-Hölder in time regularity for the value process can be established (Bouchard et al., 2014).
Flat-off Reflection and Skorokhod Conditions: In law-constrained and mean-reflection settings, the solution’s reflection process is constructed so that the corresponding constraint becomes active only when violated, and remains minimal in the sense of additive Skorokhod reflection (Li, 2023, Li et al., 19 Jan 2025).

4. Approximation Schemes and Computation

The solution of constrained BSDEs with possibly singular or distributional constraints often requires specialized computational strategies:

Penalization and Empirical Approximation: Penalized drivers, with parameter tending to infinity, provide converging approximations to the constrained solution. Particle system approaches approximate mean/law-reflected BSDEs by large-agent interacting systems whose empirical law converges to the distributional constraint (Li, 2023, Briand et al., 2019).
Discrete-time Constraint via Facelift Operators: For BSDEs with gain-process constraints, a discretization is performed by applying a facelift operator to the value process at a grid of times, producing a sequence of discretely constrained BSDEs converging to the original problem (Kharroubi et al., 2020). Further, such facelift applications can be efficiently approximated through neural-network-based regression algorithms under gradient constraints.
Fixed-Point and Contraction Mapping: Skorokhod/backward Skorokhod problems naturally admit fixed-point (Picard) schemes, particularly effective for mean-reflected or law-reflected BSDEs, including those in a G-expectation setting (Li, 2023, He et al., 15 May 2024, He et al., 26 Aug 2025).
BMO-martingale Techniques: For quadratic-growth BSDEs with double mean reflection (e.g., under G-Brownian motion), BMO-martingale theory and θ-method provide constructions and a-priori estimates (He et al., 26 Aug 2025).

These approaches offer quantitative rates of convergence and sometimes facilitate dimension-independent implementations in high-dimensional models with convex constraints.

5. Applications in Stochastic Control and Nonlinear PDEs

Constrained BSDEs serve as a probabilistic representation of a range of stochastic control and partial differential equations with constraints or obstacles:

Optimal Control with State Constraints: Hard state constraints in control problems reduce to BSDEs with singular terminal conditions, encoding terminal objectives such as $X_T = 0$ (Ankirchner et al., 2013).
Gradient-constrained/Stochastic Control Problems: Combined singular and regular stochastic control yields BSDEs with constraints on the gain process, connecting to variational inequalities for Hamilton-Jacobi-Bellman equations with gradient obstacles (Bouchard et al., 2018).
Hamilton-Jacobi(-Bellman) Equations with Obstacles: BSDEs with gain-process constraints or reflection correspond to viscosity solutions of Hamilton-Jacobi equations with gradient constraints or double obstacles, in both local and nonlocal (integro-PDE) settings (Cosso et al., 2015, Perninge, 27 Feb 2024).
Portfolio Optimization and Superhedging: Minimal supersolutions of constrained BSDEs provide superhedging prices in incomplete markets or under portfolio constraints; mean/law constraints encode risk management objectives (e.g., VaR/expected shortfall bounds) (Heyne et al., 2013, Li, 21 Nov 2025, Li, 2023).

6. Law, Mean, and Nonlinear Expectation Constraints

Recent developments have focused on BSDEs where the constraint is imposed not on the path of the solution, but on its law or its expected value:

Mean/Distributional Reflection: The solution is required to satisfy $\mathbb{E}[L(Y_t)] \leq 0$ , or more generally, $\mathcal{L}(Y_t)$ must lie in a prescribed set, with the reflection process acting only when the law constraint is violated (Li, 2023, Li et al., 19 Jan 2025, Briand et al., 2019).
BSDEs with Normal Reflection in Law: Incorporating Lions' derivative structure, these equations enforce the constraint via reflection along the normal to the constraint set in the Wasserstein space, yielding connections to obstacle problems for PDEs on the space of probability measures (Briand et al., 2019).
Nonlinear Expectation and Risk Measure Reflection: The constraint is formulated as $\mathcal{E}[l(Y_t)] \ge 0$ under a nonlinear expectation (possibly a risk measure), which subsumes both law and pathwise constraints (Li, 21 Nov 2025).
Conditional and Partial Filtration Constraints: Reflection in conditional expectations generalizes both pathwise and mean reflection, interpolating between classical and law-constrained BSDEs (Hu et al., 2022).

Contraction mappings, monotonic Skorokhod-type fixed-point arguments, and backward Skorokhod problems underpin the existence theory in these contexts. Such constraints naturally encode regulatory or risk-management restrictions and can be used to represent control and stopping problems with partial information.

7. Duality, Representation, and PDE Correspondence

Duality-based approaches, notably in convex and dynamic programming settings, provide strong characterizations and connect BSDE solution theory to viscosity or variational solutions of obstacle/gradient-constrained PDEs:

Dual Representation: Under convexity assumptions, the initial value of the minimal supersolution can be expressed as a saddle point (Fenchel-Legendre duality) over equivalent martingale measures and admissible control decompositions (Heyne et al., 2013). The explicit optimizer structure is available for quadratic generators.
Probabilistic Feynman-Kac Correspondences: The constrained (often mean/law reflected) BSDE solution yields the unique viscosity solution of the associated HJB, variational inequality, or PDE with obstacles/gradient constraints, with equivalence established by limit, comparison, and monotonicity arguments (Cosso et al., 2015, Choukroun et al., 2014, Perninge, 27 Feb 2024).
Viscosity and Dynamic Programming Principles: Existence and regularity at the level of BSDEs furnish probabilistic proofs of comparison and existence results for nonlocal PDEs with constraints, sometimes unattainable by classical analytic methods in the presence of unbounded coefficients or singular data.

The theory of constrained BSDEs integrates probabilistic, convex analytic, and PDE perspectives, yielding robust solution concepts for broad classes of stochastic control, financial mathematics, and nonlinear PDE problems with hard or soft constraints. The minimal (or maximal) solution framework, penalization approximations, Skorokhod-based reflections, and duality techniques constitute the core analytic pillars of this rapidly evolving area (Ankirchner et al., 2013, Wu et al., 2010, Heyne et al., 2013, Bouchard et al., 2014, Li, 2023, Li et al., 19 Jan 2025, Li, 21 Nov 2025, Kharroubi et al., 2020).