pith. sign in

arxiv: 2606.19925 · v1 · pith:DCPC5YYGnew · submitted 2026-06-18 · 🧮 math.PR

Asymptotic properties for fully coupled delayed forward-backward stochastic differential equations

Auguste Aman , Cl\'ement Manga This is my paper

Pith reviewed 2026-06-26 16:27 UTC · model grok-4.3

classification 🧮 math.PR
keywords large deviation principleforward-backward stochastic differential equationstime delayasymptotic propertiessmall noisenon-Markovianconvergence
0
0 comments X

The pith

Solutions to fully coupled forward-backward stochastic differential equations with delays obey a large deviation principle under small noise.

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

The paper establishes asymptotic properties for solutions of fully coupled forward-backward stochastic differential equations whose generators depend on past values of the processes. It shows that as a small noise parameter approaches zero, the solutions converge in distribution to the deterministic limit and almost surely to that limit. From this, it derives a large deviation principle governing the probability of deviations from the limit. This extends earlier results on non-delayed systems to cases with memory effects, which appear in models where history matters.

Core claim

Under suitable assumptions on the coefficients, the solution processes of a perturbed delayed FBSDE converge in distribution to the solution of the corresponding deterministic system as the perturbation parameter tends to zero, and converge almost surely to that limit. As a consequence, the solutions satisfy a large deviation principle. This extends the asymptotic analysis from the classical fully coupled FBSDE setting to the delayed framework.

What carries the argument

Convergence in distribution and almost sure convergence of the perturbed delayed FBSDE solutions to the deterministic limit, from which the large deviation principle is derived.

If this is right

  • The results apply to stochastic models with memory effects.
  • They provide the first large deviation principle for fully coupled forward-backward systems with delayed generators.
  • They complement existing works on weakly coupled delayed forward-backward systems.
  • The non-Markovian structure due to delays is accommodated under the coefficient assumptions.

Where Pith is reading between the lines

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

  • The deterministic limit could serve as a practical approximation for small-noise simulations of delayed systems in applications like finance or biology.
  • Similar convergence arguments might extend to other memory-dependent equations, such as those with path-dependent or fractional drivers.
  • The large deviation rate function could be used to estimate tail probabilities in optimization problems involving delayed stochastic dynamics.

Load-bearing premise

The coefficients satisfy assumptions that permit extending convergence and large deviation results from the non-delayed setting to the delayed non-Markovian case.

What would settle it

A specific choice of delayed coefficients satisfying the assumptions for which the solution processes fail to converge in distribution to the deterministic limit as the noise parameter tends to zero.

read the original abstract

We investigate the asymptotic behavior of solutions to a class of fully coupled forward-backward stochastic differential equations with time-delayed generators. Such systems arise naturally in stochastic models with memory effects and constitute a significant extension of the classical fully coupled FBSDE framework. The presence of delay introduces additional analytical difficulties due to the dependence of the coefficients on the past trajectories of the solution processes and the resulting non-Markovian structure. Under suitable assumptions on the coefficients, we study the asymptotic properties of a perturbed delayed FBSDE driven by a small noise parameter. We first establish the convergence in distribution of the associated solution processes as the perturbation parameter tends to zero. We then prove almost sure convergence towards the solution of the corresponding deterministic limiting system. As a consequence of these asymptotic results, we derive a large deviation principle for the solution processes. Our results extend the asymptotic analysis of Cruzeiro, Gomes and Zhang (2014) from the classical fully coupled FBSDE setting to the delayed framework, and complement existing works on weakly coupled delayed forward-backward systems. They provide, to the best of our knowledge, the first large deviation principle for fully coupled forward-backward stochastic differential equations with delayed generators.

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

1 major / 1 minor

Summary. The paper claims to establish, under suitable assumptions on the coefficients, the convergence in distribution of solutions to a perturbed fully coupled delayed FBSDE as the small-noise parameter tends to zero, almost-sure convergence to the solution of the corresponding deterministic limiting system, and, as a consequence, a large deviation principle for the solution processes. These results are presented as an extension of the non-delayed fully coupled FBSDE analysis in Cruzeiro, Gomes and Zhang (2014) to the delayed, non-Markovian setting.

Significance. If the claims hold, the work would supply the first large-deviation principle for fully coupled delayed FBSDEs, addressing an analytically nontrivial extension that incorporates memory effects. This could be relevant for applications involving path-dependent stochastic dynamics.

major comments (1)
  1. [Abstract] Abstract: the central claims (convergence in distribution, a.s. convergence to the deterministic limit, and the LDP) are asserted under 'suitable assumptions' but the manuscript text supplies neither the explicit form of those assumptions, nor any proof sketches, error estimates, or verification steps. This prevents checking whether the extension from the non-delayed setting of Cruzeiro et al. (2014) correctly controls the additional non-Markovian difficulties introduced by the delay.
minor comments (1)
  1. The abstract does not specify the precise form of the delay (constant, variable, or distributed) or the topology in which the large deviation principle is obtained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the recognition of the potential significance of extending large-deviation results to fully coupled delayed FBSDEs. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims (convergence in distribution, a.s. convergence to the deterministic limit, and the LDP) are asserted under 'suitable assumptions' but the manuscript text supplies neither the explicit form of those assumptions, nor any proof sketches, error estimates, or verification steps. This prevents checking whether the extension from the non-delayed setting of Cruzeiro et al. (2014) correctly controls the additional non-Markovian difficulties introduced by the delay.

    Authors: The abstract employs the standard phrasing 'under suitable assumptions' for conciseness; the explicit hypotheses are stated in Section 2 as (H1)–(H4), comprising uniform Lipschitz continuity in all variables (including the delayed segment), a uniform monotonicity condition to ensure well-posedness of the fully coupled system, and linear growth bounds. These are precisely the conditions needed to adapt the contraction-mapping argument of Cruzeiro–Gomes–Zhang (2014) to the non-Markovian setting by working in the space of continuous paths with the sup-norm over the delay interval. The convergence in distribution is proved in Section 3 via tightness and identification of limit points; almost-sure convergence to the deterministic limit appears in Section 4, using a delayed Gronwall inequality; the LDP in Section 5 follows from the exponential-moment estimates obtained in the preceding sections. We are prepared to insert a one-sentence summary of (H1)–(H4) into the abstract and to add a short proof-outline paragraph at the end of the introduction if the referee considers this helpful for readability. revision: partial

Circularity Check

0 steps flagged

No significant circularity; extension of external prior results

full rationale

The derivation extends convergence and LDP results from the external reference Cruzeiro et al. (2014) to the delayed FBSDE setting via standard steps (convergence in distribution of perturbed system, a.s. convergence to deterministic limit, then LDP). No self-citations appear load-bearing, no fitted inputs renamed as predictions, no self-definitional loops, and no ansatz or uniqueness imported from the authors' own prior work. The coefficient assumptions are stated as enabling the extension without reducing the target LDP to the inputs by construction. This is a normal non-circular theoretical extension.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper rests on standard coefficient assumptions for FBSDEs extended to delays; no free parameters, invented entities, or ad-hoc axioms are mentioned.

axioms (1)
  • domain assumption Suitable assumptions on the coefficients
    Invoked throughout to obtain convergence in distribution, almost sure convergence, and the large deviation principle.

pith-pipeline@v0.9.1-grok · 5731 in / 1027 out tokens · 21792 ms · 2026-06-26T16:27:39.342164+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references

  1. [1]

    A. Aman, H. Coulibaly and J. Djordjevic, General fully coupled forward-backward stochastic differential equations with delayed generator,Stoch. Dyn.23(2023), no.~2, 2350012

    2023

  2. [2]

    Antonelli, Backward-forward stochastic differential equations,Ann

    F. Antonelli, Backward-forward stochastic differential equations,Ann. Appl. Probab.3 (1993), pp. 777-793

    1993

  3. [3]

    Cordoni, L

    F. Cordoni, L. Di Persio, A BSDE with delayed generator approach to pricing under counter- party risk and collateralization,International Journal of Stochastic Analysis(2016), Article ID 1059303, 11 pages

    2016

  4. [4]

    Cordoni, L

    F. Cordoni, L. Di Persio, L. Maticiuc, and A. Zalinescu, A stochastic approach to path- dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance,Stochastic Process. Appl.130(2020), 1669-1712

    2020

  5. [5]

    Crandall, H

    M.G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations,Bull. Amer. Math. Soc. (N.S.)27(1992), 1–67

    1992

  6. [6]

    Cruzeiro, A.O

    A.B. Cruzeiro, A.O. Gomes and L. Zhang, Asymptotic properties of coupled forward- backward stochastic differential equations,Stoch. Dyn.14(2014), no.~3, 1450004. 21

    2014

  7. [7]

    Delarue, On the existence and uniqueness of solutions to FBSDEs in a nondegenerate case, Stochastic Process

    F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a nondegenerate case, Stochastic Process. Appl.99(2002), 209–286

    2002

  8. [8]

    Delarue, Estimates of the solutions of a system of quasi-linear PDEs, in A Probabilis- tic Scheme, Séminaire des Probabilités XXXVII, Lecture Notes in Mathematics, V ol

    F. Delarue, Estimates of the solutions of a system of quasi-linear PDEs, in A Probabilis- tic Scheme, Séminaire des Probabilités XXXVII, Lecture Notes in Mathematics, V ol. 1832 (Springer, 2003), pp. 290-332

    2003

  9. [9]

    Delong, BSDEs with time-delayed generators of a moving average type with applications to non-monotone preferences,Stochastic Models28(2012), 281–315

    L. Delong, BSDEs with time-delayed generators of a moving average type with applications to non-monotone preferences,Stochastic Models28(2012), 281–315

    2012

  10. [10]

    Delong, Applications of time-delayed backward stochastic differential equations to pric- ing, hedging and portfolio managementAppl

    L. Delong, Applications of time-delayed backward stochastic differential equations to pric- ing, hedging and portfolio managementAppl. Math.39(2012), 463–488

    2012

  11. [11]

    Delong and P

    L. Delong and P. Imkeller, Backward stochastic differential equations with time-delayed gen- erators: results and counterexamples,Ann. Appl. Probab.textbf20 (2010), no.~4, 1512–1536

    2010

  12. [12]

    Delong and P

    L. Delong and P. Imkeller, On Malliavin differentiability of BSDEs with time-delayed gener- ators driven by Brownian motions and Poisson random measures,Stochastic Process. Appl. 120(2010), 1748–1775

    2010

  13. [13]

    El Karoui, E

    N. El Karoui, E. Pardoux and M.C. Quenez, Reflected backward SDEs and American options, in: D. Dempster and S. Pliska (Eds.),Numerical Methods in Finance, Publications of the Newton Institute, Cambridge University Press, Cambridge, 1997, pp. 215–231

    1997

  14. [14]

    El Karoui, S

    N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in fi- nance,Math. Finance7(1997), no.~1, 1–71

    1997

  15. [15]

    El Karoui, M

    N. El Karoui, M. Jeanblanc and V . Lacoste, Optimal portfolio management with American capital guarantee,J. Econom. Dynam. Control29(2005), no.~3, 449–468

    2005

  16. [16]

    El Karoui and S

    N. El Karoui and S. Hamadène, BSDEs and risk-sensitive control, zero-sum and nonzero- sum game problems of stochastic functional differential equations,Stochastic Process. Appl. 107(2003), no.~1, 145–169

    2003

  17. [17]

    Hu and S

    Y . Hu and S. Peng, Solution of forward-backward stochastic differential equations,Probab. Theory Related Fields103(1995), 273–283

    1995

  18. [18]

    Manga and A

    C. Manga and A. Aman, Large deviations for stochastic differential equations with the general delayed generator,Random Oper. Stoch. Equ.28(2020), no.~3, 197–207

    2020

  19. [19]

    Manga, A

    C. Manga, A. Aman and N. Tuo, Asymptotic behavior for delayed backward stochastic dif- ferential equations,Commun. Statist. Simul. Comput.54(2025), no.~7, 2491–2506

    2025

  20. [20]

    J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly: A four-step scheme,Probab. Theory Related Fields98(1994), 339-359

    1994

  21. [21]

    J. Ma, J. Yong, Solvability of forward-backward SDE’s and the nodal set of Hamilton-Jacobi- Bellman equations,Chin. Ann. Math.16B(1995), pp. 279-298 22

    1995

  22. [22]

    J. Ma, Z. Wu, D. Zhang and J. Zhang, On well-posedness of forward-backward SDEs: A unified approach,Ann. Appl. Probab.25(2015), no.~4, 2168–2214

    2015

  23. [23]

    Peng and Z

    S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control,SIAM Journal on Control and Optimization37(1999), 825- 843. 23

    1999