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Convergence of the Deep BSDE Method for Coupled FBSDEs
Jiequn Han, Jihao Long
TL;DR
High-dimensional coupled FBSDEs remain difficult for conventional numerical methods because of exponential complexity. This paper extends the deep BSDE method, proves objective-based error bounds under stated assumptions, and shows that neural-network approximation can yield accurate solutions, while the conditions for avoiding dimensionality growth remain open.
Problem
Conventional numerical methods face exponential complexity in high-dimensional FBSDEs, and general quasilinear problems lack a proven curse-of-dimensionality-free algorithm.
Method
The paper directly simulates coupled FBSDEs using neural-network parameterizations and stochastic optimization, with error analysis under weak-coupling or monotonicity assumptions.
Results
Theorems relate simulation error to the objective-function value and show that universal neural-network approximation can make the optimized objective, and resulting solution error, small.
Takeaways & Limitations
The objective function provides a practical accuracy indicator for the deep BSDE method, and numerical experiments demonstrate accurate high-dimensional coupled-FBSDE solutions.
Takeaways & Limitations
The conditions under which neural networks represent general high-dimensional FBSDEs without the curse of dimensionality remain beyond the paper’s scope.
Abstract
from arXiv · showhide
The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs.
1 Introduction
The paper extends the deep BSDE method to coupled FBSDEs and quasilinear parabolic PDEs, addressing high-dimensional settings where conventional numerical methods face exponential complexity. It provides error analysis and numerical validation for the proposed scheme.
- Motivation: High-dimensional PDE and FBSDE algorithms face the curse of dimensionality because computational complexity grows exponentially with dimension.Mesh-based PDE methods require a mesh of size O(N^d), while several conventional FBSDE methods also have exponential complexity.
- Motivation: Existing branching-diffusion and multilevel Picard methods avoid the curse only in specific settings, leaving general quasilinear problems unresolved.The paper states that no numerical algorithm had been proved to overcome this issue for general quasilinear parabolic PDEs and corresponding FBSDEs.
- Approach: The deep BSDE method uses neural networks to approximate unknown gradients and reformulates equation solving as stochastic optimization.Its parsimonious parameterization and universal approximation capability support optimization in high-dimensional cases.
- Contribution: The paper extends the deep BSDE method from decoupled to coupled FBSDEs and a broader class of quasilinear parabolic PDEs.The proposed scheme includes decoupled FBSDEs as a special case.
- Contribution: Theoretical results provide a posteriori error estimation and show that the numerical error converges to zero when neural networks have universal approximation capability.Theorem 1 links accuracy to the optimized objective function, while Theorem 2 relates its infimum to neural-network expressivity.
- Evaluation: The paper presents numerical experiments with the proposed scheme after stating assumptions, proving the main theorems, and discussing related results.The article organization places experiments in section 6 after the scheme, assumptions, and proofs.
2 A Numerical Scheme for Coupled FBSDEs and Main Results
The paper formulates a direct probabilistic neural-network scheme for coupled FBSDEs and connects its objective function to approximation error. Under regularity and weak-coupling assumptions, its theorems relate objective values to solution accuracy and neural-network expressivity.
- Numerical Scheme: The scheme models Y0 as a function of X0 and Zti as a function of Xt_i and Yt_i for a time partition of [0,T].It uses Euler discretization with h = T/N and Brownian increments ΔWi.
- Numerical Scheme: The proposed method solves coupled FBSDEs directly through stochastic optimization rather than spatial discretization of an associated PDE.This is presented as a purely probabilistic scheme for the coupled case.
- Objective Function: The continuous variational problem has the FBSDE solution as a minimizer because the loss reaches zero there, with well-posedness ensuring uniqueness.The discretized objective function is intended to provide a good approximate solution when optimized near its minimum.
- Assumptions: The analysis assumes weak coupling or monotonicity conditions in addition to regularity conditions on the coefficients and terminal function.The paper notes that the main theorems rely on these conditions, while a PDE-based result requires stronger smoothness and ellipticity assumptions.
- Main Results: Theorem 1 bounds simulation error through the objective-function value, including time-discretization error and terminal distance.This provides an a posteriori accuracy indicator for the numerical solution.
- Main Results: Theorem 2 relates the optimal objective value to the approximation capability of the neural-network function spaces.Universal approximation in the L2 sense implies the existence of parameters yielding an approximately accurate numerical solution.
- Scope: The paper leaves open the conditions under which neural networks represent general high-dimensional FBSDEs without the curse of dimensionality.This question is explicitly stated to remain beyond the scope of the work.
- PDE Connection: The nonlinear Feynman–Kac formula transfers the scheme from FBSDEs to quasilinear parabolic PDEs by interpreting the initial-value error as an approximation error for u(0, ξ).The associated PDE solution is linked to the FBSDE through Yt = u(t, Xt).
3 Preliminaries
The preliminaries specify Lipschitz, boundedness, growth, and weak-coupling or monotonicity assumptions for coupled FBSDEs. Under these assumptions, the system connects to a PDE, has a unique solution representation, and admits a convergent implicit discretization.
- Assumptions: The basic assumptions require uniform Lipschitz continuity of b, σ, f, and g, together with bounded values at the origin.The terminal function also satisfies a linear-growth bound.
- Weak coupling and monotonicity: Existence and uniqueness rely on one of five regimes: small T, weak coupling, or sufficiently strong monotonicity in f or b.The five cases are summarized as weak coupling and monotonicity conditions.
- Weak coupling and monotonicity: The quantitative Assumption 3 formalizes these regimes through two inequalities, and any condition holding sufficiently strongly implies them.The paper uses these inequalities as Assumption 3 thereafter.
- FBSDE–PDE connection: Under Assumptions 1–3, a function u exists, is a viscosity solution of the PDE, and represents the unique coupled FBSDE solution through Y_t = u(t, X_t).The associated solution also has stated time and path regularity.
- Discretization: Under the same assumptions and sufficiently small h, the implicit time-discrete scheme is available and converges with constants depending on the Lipschitz and time-horizon parameters.Earlier work established the analogous explicit-scheme result.
- Neural-network spaces: The parametric neural-network spaces are restricted to ensure the discrete system is well-defined; common ReLU and sigmoid networks satisfy the stated requirement.The resulting functions have linear growth bounds.
4 A Posteriori Estimation of the Simulation Error
The paper estimates simulation error by comparing discrete FBSDE solutions through stability inequalities and martingale representations. Under quantitative coupling or monotonicity conditions, the resulting error bounds hold for sufficiently small time steps and imply coercivity of the deep BSDE objective.
- Proof strategy: The proof compares two solutions of an auxiliary discrete FBSDE system and bounds their differences through the objective-function discrepancy.The argument then combines these estimates with convergence of the implicit scheme.
- Proof strategy: Martingale representation and conditional-expectation estimates control the Z component and the integral terms in the difference equations.The proof applies Lemma 2 and inequalities including Cauchy and RMS-GM bounds.
- A posteriori estimate: For sufficiently small h and suitable λ1 > 0 and λ2 ≥ fz, Theorem 1′ provides a constant error bound depending on E|ξ|2, L, T, λ1, and λ2.The admissible parameters exist when one weak-coupling or monotonicity condition holds sufficiently strongly.
- Implication for deep BSDE: The theorem also implies coercivity of the deep BSDE objective function used by the method.This links small objective values to control of the solution differences within the analyzed setting.
5 An Upper Bound for the Minimized Objective Function
This section proves an upper bound for the minimized objective function under weak coupling or monotonicity conditions, supporting convergence of the deep BSDE approximation. It also discusses neural-network approximation and remaining regularity concerns.
- Proof strategy: The upper-bound argument combines estimates for terminal distance, forward-process differences, backward-process differences, and the objective function under sufficiently small time steps.The proof repeatedly applies intermediate inequalities and chooses auxiliary parameters so the resulting coefficient remains below one.
- Proof strategy: The proof connects the discrete stochastic process to deterministic functions and controls differences between backward processes driven by different forward processes.Lemma 5 provides the deterministic-function link, while Lemma 4 estimates the difference between the corresponding backward processes.
- Upper-bound theorem: Theorem 2′ bounds the minimized objective function when the stated assumptions and parameter conditions hold for sufficiently small h.The conditions include Assumptions 1–4, λ1, λ3 > 0, λ2 ≥ fz, λ7 > fz, and B0 < 1.
- Neural-network approximation: Universal approximation by neural networks can make the objective function small when the relevant target functions are approximated accurately in L2.The discussion specifically considers continuous piecewise-linear functions represented by ReLU networks, with depth at most ⌈1 + log2(m + 1)⌉ and potentially increasing width.
- Limitations: The approximation argument leaves concerns because conditional-expectation target functions may depend on earlier backward variables, and their regularity is not known a priori.Even in the decoupled case, the text states that the relevant property of the target function is not known beforehand.
- Stronger regularity result: Under stronger PDE assumptions, Theorem 6 replaces the conditional-expectation issue with a regularity-based estimate involving the Lipschitz constant of σT(t, x, u(t, x))∇xu(t, x).The resulting constant depends on the initial-data moment, T, the relevant Lipschitz constants, and auxiliary parameters; the estimate may depend on dimension d.
6 Numerical Examples
Two 100-dimensional coupled FBSDE examples evaluate the proposed deep BSDE method, with experiments examining iteration convergence and time-discretization effects.
- The experiments use d = m = 100, with deterministic terminal data and relative approximation error measured for the deterministic scalar Y0.
- Example 1: Example 1 uses ξ = (1, 1, . . . , 1), T = 5, and N = 160; all runs converged with an average final relative error of 0.39%.The experiment trained for 25000 steps from an initial Y0 guess uniformly sampled on [2, 4].
- Example 2: Example 2 uses ξ = (π/2, π/2, . . . , π/2), T = 1, r = 0.1, σ = 0.3, and D = 0.1 in 100 dimensions.The true value used for comparison is Y0 ≈ 9.04837.
- Example 2: Example 2 achieves a relative approximation error of 0.09% with N = 200 and h = 0.005.The initial Y0 guess is sampled uniformly from [0, 1], and training uses 5000 steps.
- Example 2: In Example 2, the relative error decreases as the number of time steps N increases and the time step h decreases.The reported errors are means over the tested time partitions.