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arxiv: 2606.24271 · v1 · pith:BLBGRTMQnew · submitted 2026-06-23 · 🧮 math.NA · cs.LG· cs.NA· math.PR

Deep numerical schemes for systems of Ergodic BSDEs with applications to regime-switching forward utilities

Guillaume Broux-Quemerais (LMM) , Sarah Kaakai (LAGA) , Anis Matoussi (LMM) , Wissal Sabbagh (LMM) This is my paper

Pith reviewed 2026-06-25 23:35 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NAmath.PR
keywords ergodic BSDEsneural networksdeep learningregime switchingforward utilitiesstochastic factorsnumerical schemesbackward SDEs
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The pith

Two neural-network schemes solve systems of coupled ergodic BSDEs for regime-switching forward utilities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops two deep learning schemes to solve systems of coupled ergodic backward stochastic differential equations. These systems arise when computing optimal strategies for investors whose preferences switch between economic regimes driven by a stochastic factor. The schemes first connect the ergodic equations to a standard BSDE stopped at the random time when the factor hits a level, then train networks to minimize either local approximation errors or the residual of the associated ergodic partial differential equation system. Application to homothetic forward utilities yields a consistency equation and allows numerical assessment of how regime changes alter investment behavior. Tests demonstrate that the methods capture the effects of switches on preferences.

Core claim

We introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs). We establish a link between these equations and an associated multidimensional BSDE with random terminal time given by the hitting time of the positive recurrent stochastic factor. We present a locally additive deep learning scheme minimizing aggregated local error terms and a Deep Galerkin Method inspired algorithm minimizing the residual of the ergodic PDE system using an ergodic cost representation. We apply the framework to regime-switching forward utilities, deriving a general consistency SPDE and retrieving the eBSDE representation in the h

What carries the argument

The link between the system of ergodic BSDEs and the multidimensional BSDE with random terminal time at the hitting time of the stochastic factor, which enables the neural network training procedures.

If this is right

  • The schemes provide numerical approximations of optimal strategies in forward utility models with regime switches.
  • In the homothetic case, forward utilities admit a representation via systems of ergodic BSDEs.
  • Regime switches affect forward preferences in ways that can be quantified by solving the eBSDE systems.
  • The consistency of regime-switching forward utilities is characterized by a stochastic partial differential equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The neural schemes could be applied to other problems involving ergodic BSDEs outside of utility maximization.
  • Accuracy in high-dimensional cases may require adjustments to the network architecture or training procedure.
  • Extensions to non-recurrent factors would need alternative representations beyond hitting times.

Load-bearing premise

The stochastic factor is positive recurrent, which ensures the hitting time is finite and connects the ergodic system to a BSDE with random terminal time.

What would settle it

Solve a simple one-dimensional regime-switching model with known closed-form ergodic BSDE solution using the proposed schemes and verify that the numerical outputs converge to the exact values.

Figures

Figures reproduced from arXiv: 2606.24271 by Anis Matoussi (LMM), Guillaume Broux-Quemerais (LMM), Sarah Kaakai (LAGA), Wissal Sabbagh (LMM).

Figure 1
Figure 1. Figure 1: Error convergence for the hyperbolic tangent example (H). The training behavior of both solvers is reported in Figure 1a. Both methods show a fast initial decrease of the L 2 (ν)-errors, which stabilize around 10−2 after 4000 epochs. The LAeBSDE approxi￾mation provides the smallest final errors for both y and z. Figure 1b illustrates the error convergence 28 [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: compares the approximated solutions y¯ i and z i produced by the two solvers against the exact solution, over the main support of the invariant law. Both approaches recover the qualitative structure of the solution, the two regimes crossing near v = 0 where the normalization y 1 (0) = 1 is enforced, while the gradients z 1 and z 2 are of opposite sign, small, and nearly flat over the relevant range. Overal… view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of empirical loss and λ¯ estimation for both the LAeBSDE and DGM. This offers a good framework for the study of the associated regime-switching forward utility (3.5) generated by a family of utility random fields in power form as (3.14), which takes the form U(t, x) = x δ δ e y αt (v)−λt, yαt (v) = X i∈I y i (v)1{αt=i} . (4.13) Simulating the jump times of the two Poisson processes N1,2 and N2,… view at source ↗
Figure 4
Figure 4. Figure 4: Trajectory of the regime-switching forward utility and solution while accelerating the chain. The corresponding PDE residuals and normalization errors are reported in [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Optimal portfolio allocation for different risk aversion parameters. Interpretation of the ergodic cost. For δ = 0.5, the ergodic cost is estimated at λ¯ = 8.02 × 10−2 . In the representation U i (t, x) = x δ δ e y i (Vt)−λt, the constant λ plays the role of an endogenous time￾preference rate: a positive λ exponentially discounts future utility and thus reflects a preference for the present, whereas a nega… view at source ↗
read the original abstract

In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces two neural-network schemes for systems of coupled ergodic BSDEs arising from regime-switching forward utility maximization. It first establishes a representation linking the eBSDE system to a multidimensional BSDE with random terminal time equal to the hitting time of a positive-recurrent stochastic factor process. Building on this, it proposes a locally additive deep-learning scheme that minimizes aggregated local errors and a DGM-style algorithm that minimizes the residual of the associated ergodic PDE system (using an ergodic-cost representation). The framework is applied to derive a consistency SPDE for regime-switching forward utilities (recovering the eBSDE representation in the homothetic case) and is tested numerically on the forward-utility problem, with emphasis on the effect of regime switches.

Significance. If the representation result and the numerical schemes hold with supporting analysis, the work supplies practical tools for long-horizon stochastic control problems that incorporate regime switches, an area of growing interest in mathematical finance. The link between ergodic BSDEs and stopped BSDEs under positive recurrence is a useful structural contribution that enables the neural methods; the forward-utility application further illustrates relevance.

major comments (2)
  1. [Numerical experiments] The numerical experiments section reports performance on the forward-utility example but provides no quantitative error measures, convergence rates, or comparisons against known solutions or alternative discretizations. This absence undermines the claim that the schemes demonstrate reliable performance, especially given that the central motivation is numerical approximation of optimal strategies.
  2. [Representation result and scheme construction] The representation result (linking the eBSDE system to the BSDE stopped at the hitting time) is load-bearing for both proposed schemes, yet the manuscript supplies no error analysis or stability estimates for the neural approximations of this stopped BSDE. Without such analysis the practical utility of the locally additive and DGM schemes remains unquantified.
minor comments (2)
  1. Notation for the regime-switching process and the associated generator should be introduced once and used consistently; occasional redefinition of symbols across sections reduces readability.
  2. [Introduction] The abstract and introduction cite [HLT20] for the eBSDE representation; a brief self-contained recap of the key assumptions from that reference would help readers who have not consulted the prior work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive suggestions. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Numerical experiments] The numerical experiments section reports performance on the forward-utility example but provides no quantitative error measures, convergence rates, or comparisons against known solutions or alternative discretizations. This absence undermines the claim that the schemes demonstrate reliable performance, especially given that the central motivation is numerical approximation of optimal strategies.

    Authors: We agree that the numerical section would benefit from quantitative support. The current experiments emphasize qualitative behavior under regime switches, but in the revision we will add error tables against available benchmark solutions (where the ergodic BSDE admits an explicit form), convergence plots with respect to network width/depth and training iterations, and direct comparisons to a standard Euler-type discretization of the stopped BSDE. These additions will be placed in a new subsection of the numerical experiments. revision: yes

  2. Referee: [Representation result and scheme construction] The representation result (linking the eBSDE system to the BSDE stopped at the hitting time) is load-bearing for both proposed schemes, yet the manuscript supplies no error analysis or stability estimates for the neural approximations of this stopped BSDE. Without such analysis the practical utility of the locally additive and DGM schemes remains unquantified.

    Authors: The representation theorem is used to recast the ergodic system as a stopped BSDE whose terminal time is the hitting time of the recurrent factor process; both neural schemes are then applied to this equivalent problem. While the manuscript does not contain a full a-priori error analysis for the neural approximation of the stopped BSDE (consistent with the largely empirical nature of deep-learning BSDE solvers in the literature), the locally additive scheme explicitly minimizes aggregated local residuals and the DGM scheme minimizes the ergodic PDE residual. In the revision we will insert a short discussion of the Lipschitz stability of the stopped BSDE with respect to the neural-network approximation error and will report additional numerical diagnostics (e.g., sensitivity to the hitting-time truncation) that quantify practical stability. A complete theoretical convergence rate lies outside the present scope but can be noted as future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper derives a link between ergodic BSDE systems and multidimensional BSDEs with random terminal time under the positive-recurrence assumption, then introduces two new neural schemes (locally additive and DGM-style) that minimize local errors or PDE residuals on that representation. These steps are presented as original constructions, not reductions to fitted parameters or self-citations. The application to regime-switching forward utilities derives a consistency SPDE and retrieves the eBSDE representation in the homothetic case, again without the central claims collapsing to prior inputs by definition. Reliance on [HLT20] for the initial representation is external and does not create load-bearing circularity under the stated rules, as the numerical methods and consistency results are independently developed and tested.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper depends on the prior domain assumption from [HLT20] for the BSDE representation and the positive recurrence assumption for the stochastic factor to enable the random terminal time; no free parameters or new entities are specified.

axioms (1)
  • domain assumption The models admit a representation through systems of eBSDEs as per [HLT20]
    This is the foundational link used to develop the numerical schemes and the application to forward utilities.

pith-pipeline@v0.9.1-grok · 5784 in / 1231 out tokens · 42862 ms · 2026-06-25T23:35:32.358338+00:00 · methodology

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