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Deep Learning for Reflected BSDEs: Regularization and Error Analysis
Ruimeng Hu, Yihan Zou
TL;DR
High-dimensional RBSDEs remain difficult to solve because the reflection process complicates numerical approximation. The paper regularizes the reflection constraint and develops DFS and DBS solvers, with DBS error controlled by training loss for fixed ε. Experiments report accurate high-dimensional American option pricing while directly addressing continuous-time exercise.
Problem
High-dimensional RBSDE computation is limited by the implicit reflection process and the difficulty of extending standard deep BSDE analyses to reflected problems.
Method
The paper replaces reflection with regularized BSDEs and applies deep forward and deep backward neural-network schemes to the regularized problems.
Results
Both DFS and DBS achieve high accuracy on high-dimensional American-style option pricing problems, while the DBS admits an explicit fixed-ε error bound controlled by training loss.
Takeaways & Limitations
The reflected formulation enables direct treatment of continuous-time American option exercise rather than only Bermudan approximations.
Takeaways & Limitations
The error analysis is for fixed ε because its constants are not uniform as ε decreases, and exact forward simulation is assumed for isolating backward errors.
Abstract
from arXiv · showhide
Reflected backward stochastic differential equations (RBSDEs) provide a probabilistic formulation for obstacle constrained problems, but existing deep learning methods for their high dimensional solution remain limited. In this paper, we propose two deep learning schemes for RBSDEs, a deep forward scheme (DFS) and a deep backward scheme (DBS), by first reducing the reflected problem to a family of regularized BSDEs. Our main theoretical contribution concerns the DBS: we establish an explicit error bound showing that, for each fixed regularization parameter $\varepsilon>0$, the approximation error between the DBS solution and the solution to the regularized BSDE is controlled by the associated training loss. We prove that this training loss can be controlled by the universal approximation capability of neural networks. Together, these results yield a theoretical foundation for the deep learning-based solution and complement existing analysis for forward type methods. We illustrate the framework on high dimensional American option pricing, where the reflected formulation allows us to address the continuous time exercise feature directly rather than through a Bermudan approximation. Numerical experiments demonstrate that both DFS and DBS deliver accurate solutions in high dimensions.
1 Introduction
RBSDEs are difficult to solve numerically in high dimensions because reflection introduces an implicitly defined process. The paper addresses this challenge with regularization-based deep learning schemes and analyzes their accuracy.
- Motivation: RBSDEs model constraints arising in stochastic control, optimal stopping, variational inequalities, and mathematical finance, including American option exercise.In American-style option pricing, reflection enforces the early-exercise constraint.
- Challenges: The reflection process K depends on the entire trajectory of Y and is defined implicitly by the Skorokhod condition, complicating numerical approximation.This prevents standard BSDE conditional-expectation methods from extending directly to the reflected setting.
- Existing methods: Classical regression and penalization methods can be effective in low or moderate dimensions but deteriorate in high dimensions.Regression approximates conditional expectations on finite-dimensional function spaces, while penalization adds increasingly stiff driver terms.
- Deep learning context: Deep learning methods have shown strong performance for standard high-dimensional BSDEs, but the reflection process remains a major obstacle for reflected problems.Existing reflected approaches include local backward optimization and obstacle enforcement through continuation-value maxima.
- Approach: The paper replaces reflection with a family of regularized BSDEs and proposes deep forward and deep backward schemes for the resulting problems.The regularized formulation removes the explicit reflection process and becomes more amenable to deep learning solvers.
- Contributions: For the DBS, the paper derives an explicit error bound controlled by training loss and shows that neural-network universal approximation can make this loss small.The analysis does not require smallness assumptions on the driver’s Lipschitz coefficient with respect to solution variables.
- Applications: The framework directly targets continuous-time American option pricing through the reflected formulation, and both DFS and DBS achieve high accuracy in high-dimensional experiments.This avoids restricting the formulation to Bermudan approximations with finitely many exercise dates.
2 Deep Learning Schemes for RBSDEs via Regularization
The paper regularizes the reflection constraint into a smooth penalty term, then applies neural-network schemes to the regularized BSDE. It combines convergence of the regularization with deep-solver analysis and practical time-discretized training.
- Regularization: The mollifier ϕ is smooth, non-increasing, equals one for inputs at most zero, and equals zero for inputs at least one.Its regularity yields the bound |ϕε(x+h)−ϕε(x)| ≤ Cϕε^-1h.
- Regularization: The regularized formulation replaces the singular reflection process K with a smooth penalty term involving ϕε(y−Φ(x))κt.As ε decreases to zero, the regularized solution converges to the original RBSDE solution.
- Regularization: Under standing coefficient and obstacle assumptions, the regularized RBSDE admits bounded solution processes and converges to the original reflected solution as ε vanishes.The convergence result supplies the regularization component of the overall approximation argument.
- Training: Both schemes train through stochastic optimization of loss functions evaluated on simulated trajectories and Brownian increments.In practice, population expectations are replaced by empirical mini-batch averages, although the theoretical analysis uses population loss.
- Deep learning schemes: The paper proposes a deep forward scheme and a deep backward scheme, both using neural networks to approximate the regularized backward component on a time grid.The DFS parameterizes the initial value with a trainable scalar, while the DBS propagates approximations backward from the terminal condition.
- Assumptions: The analysis assumes exact simulation of the forward process on the time grid to isolate backward approximation and discretization errors.If exact simulation is unavailable, Euler discretization introduces strong error of order h1/2 and weak error of order h.
3 Error Analysis
The DBS error analysis separates the discrepancy from the regularized RBSDE into projection, discretization, and training-loss components, then combines these with regularization error estimates. For fixed regularization, neural-network approximation controls the training objective, while the full bound exposes restrictions on choosing the regularization parameter.
- DBS approximation error: The DBS analysis addresses non-adapted backward approximations by introducing an F_ti-adapted projection and bounding its residual through the training loss and time-discretization error.This restores the martingale-orthogonality structure needed for the error analysis.
- DBS approximation error: Theorem 7 combines the DBS-to-regularized-solution estimate with regularization error to bound the approximation error relative to the original RBSDE solution.The resulting estimate is stated under the assumptions of Theorem 5 and includes the regularization and discretization contributions.
- Theoretical scope: The analysis does not impose an a priori smallness assumption on the driver’s Lipschitz constants with respect to y and z.This distinguishes the estimate from the cited non-reflected deep backward analysis and applies it to a broader class of drivers.
- Parameter trade-off: The regularization parameter must balance approximation and regularization errors because the bound contains an ε-dependent term behaving like exp(C1T + C3T)ε^-4h.Decreasing ε reduces regularization error but can exponentially amplify the DBS discretization error.
- Training-objective bound: For fixed ε and sufficiently small h, the DBS training objective is controlled by neural-network approximation error for the target map σ(ti, x)∂xvε(ti, x), plus a discretization-regularization remainder.The representation Zε_t = σ(t, Xt)^⊤∂xvε(t, Xt) links the training objective to approximation of the regularized solution’s Markovian representation.
4 Numerical Experiments on the Pricing of American-type Options
The experiments evaluate DFS and DBS for high-dimensional American-style option pricing across geometric put, basket call, and max call settings. Both methods show high accuracy, with results compared against binomial-tree or Monte Carlo references.
- Experimental setup: The experiments assess deep learning solvers for RBSDEs in high-dimensional American-type option pricing.The setup uses a d-dimensional geometric Brownian-motion asset process and the American option value formulation characterized by an RBSDE.
- Experimental setup: The numerical implementation uses a three-hidden-layer feedforward network with d + 10 neurons per layer.DFS and DBS use the same network architecture, while their training iterations and learning rates differ.
- Geometric put: The geometric put problem uses a one-dimensional American put benchmark based on a binomial tree with 10^4 time steps.The geometric basket structure permits reduction to a one-dimensional American put problem, providing a highly accurate reference solution.
- Geometric put: Both DFS and DBS achieve consistently high accuracy across tested dimensions for geometric put pricing, with small relative errors and RMSEs.Table 1 reports mean estimates and standard deviations over 50 independent runs.
- Basket call: For American basket call options, DFS and DBS remain closely aligned with each other and with Monte Carlo estimates across all dimensions.The Monte Carlo reference uses 10^6 sample paths and antithetic variates for the corresponding European option; under the stated nonnegative-rate, zero-dividend conditions, American and European prices coincide.
- Max call: For American max call options, DFS and DBS remain close to each other and exhibit good agreement with Monte Carlo estimates across all dimensions.The reported relative differences are slightly larger than in the basket call case, while the methods still show good agreement with the estimates.
5 Conclusion
The paper combines regularization with deep BSDE methods to solve high-dimensional RBSDEs using DFS and DBS. It provides DBS error analysis and reports accurate American-style option pricing results.
- Conclusion: The paper combines a regularization framework with deep BSDE methods to develop DFS and DBS solvers for RBSDEs.Both schemes target the regularized problem rather than the reflected equation directly.
- Conclusion: The DBS approximation error is bounded by its training loss, which can be made arbitrarily small through neural-network universal approximation.This establishes an error-analysis foundation for the deep backward solver in the reflected setting.
- Conclusion: Numerical experiments on high-dimensional American-style option pricing demonstrate the accuracy and effectiveness of both proposed methods.The conclusion summarizes the empirical performance across the reported option-pricing experiments.