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

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.

• The paper demonstrates the first rigorous large deviation principle for fully coupled delayed FBSDEs with memory effects.
• It establishes quantitative stability and convergence results, showing almost sure and distribution convergence as the noise parameter vanishes.
• The analysis leverages advanced techniques such as uniform estimates and viscosity solutions to handle the non-Markovian, infinite-dimensional system.

Asymptotic Properties for Fully Coupled Delayed Forward-Backward Stochastic Differential Equations

Introduction

The paper addresses the asymptotic analysis of fully coupled forward-backward stochastic differential equations (FBSDEs) driven by generators with delay, i.e., path-dependent coefficients reflecting memory effects. The study frames these systems in the context of stochastic analysis, emphasizing applications across mathematical finance, stochastic control, and related fields. The analysis extends classical FBSDE theory to encompass non-Markovian dynamics introduced by time delay, rendering the system infinite-dimensional and invalidating standard decoupling and PDE arguments.

Mathematical Framework

The central object of study is a perturbed fully coupled delayed FBSDE system. For a small noise parameter $\varepsilon > 0$, the system is characterized by:

$$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$

where $$(X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s)$$ denotes the trajectory segment with delay, and coefficients depend on the past values over $[-T, 0]$. The delay induces a path-dependent and thus infinite-dimensional system. The deterministic limiting system is realized as $\varepsilon \to 0$:

$$\begin{cases} \mathcal{X}(t) = x + \int_0^t b(s, \mathcal{X}_s, \mathcal{Y}_s, 0) ds \ \mathcal{Y}(t) = g(\mathcal{X}_T) + \int_t^T f(s, \mathcal{X}_s, \mathcal{Y}_s, 0) ds \end{cases}$$

The paper establishes existence and uniqueness results for both the stochastic and deterministic systems under appropriate delayed Lipschitz conditions and integrability assumptions.

Main Results

Stability and Perturbation Estimates

The paper rigorously develops quantitative stability estimates for the solutions with respect to perturbations in initial conditions, the noise parameter $\varepsilon$, and the initial time. For sufficiently small time horizon or Lipschitz constant, a uniform bound independent of $\varepsilon$ is established:

$$\sup_{t\leq s \leq T} e^{\beta s} |X^{t,x,\varepsilon}(s) - X^{t,y,\varepsilon}(s)|^2 + \cdots \leq C|x-y|^2$$

and continuity with respect to $\varepsilon$:

$$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$0

Asymptotic Behavior and Convergence

The paper proves two strong convergence results for the solution trajectories:

• In Distribution: As $$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$1, $$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$2 converges in distribution to the corresponding deterministic solution $$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$3, with the convergence uniform in the supremum norm.
• Almost Sure Convergence: For each fixed $$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$4, the triplet $$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$5 converges (in appropriate Banach spaces) almost surely to $$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$6 as $$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$7.

The limiting backward component is proven to satisfy the viscosity solution property for a path-dependent deterministic PDE of the form:

$$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$8

with $$\begin{cases} X^{\varepsilon}(t) = x + \int_0^t b(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds + \sqrt{\varepsilon} \int_0^t \sigma(s, X^{\varepsilon}_s, Y^{\varepsilon}_s) dW(s) \ Y^{\varepsilon}(t) = g(X^{\varepsilon}_T) + \int_t^T f(s, X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s) ds - \int_t^T Z^{\varepsilon}(s) dW(s) \end{cases}$$9. The regularity and continuity properties of the limiting field are explicitly characterized.

Large Deviation Principle

The primary novel result is the derivation of a Freidlin-Wentzell-type large deviation principle (LDP) for fully coupled delayed FBSDEs, including both the forward and backward components. For the family $$(X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s)$$0, the paper shows

• Forward Component LDP: The random variables $$(X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s)$$1 satisfy an LDP with a good rate function specified by deterministic controls, i.e.,

$$(X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s)$$2

• Backward Component LDP: Using the contraction principle, an LDP for $$(X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s)$$3 is obtained with rate function

$$(X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s)$$4

where $$(X^{\varepsilon}_s, Y^{\varepsilon}_s, Z^{\varepsilon}_s)$$5 reflects the deterministic solution mapping of the backward PDE.

This constitutes the first rigorous LDP for fully coupled FBSDEs with delayed generators.

Technical Implications

The analytical difficulties stem from the loss of Markovianity and the infinite-dimensional nature of the system induced by delayed coefficients. The strong coupling between forward and backward equations precludes decoupling via PDE methods as in classical FBSDEs. The methodology relies on uniform estimates, equicontinuity, and Arzela-Ascoli arguments, as well as viscosity solution theory for the limiting PDEs. The coupling of stability estimates with weak convergence techniques provides the foundation for the large deviation results.

The reduction to the classical (finite-dimensional) case is discussed; when delay coefficients depend only on constant trajectory segments, the infinite-dimensional PDE reduces to the classical semilinear form. The fully infinite-dimensional case is noted as requiring substantially different methodology.

Practical and Theoretical Implications

Practically, the results furnish analytic guarantees for stochastic control and mathematical finance models exhibiting memory effects. The existence of an LDP for delayed FBSDEs enables precise characterization of rare event probabilities for non-Markovian systems, impacting risk assessment and robust optimization under uncertainty.

Theoretically, the extension of stability, convergence, and LDP results beyond classical FBSDEs broadens stochastic analysis to infinite-dimensional, path-dependent settings. Future avenues necessitate the development of decoupling and PDE techniques for genuinely infinite-dimensional coefficients, potentially leveraging recent advances in functional analysis and stochastic calculus on spaces of trajectories.

Conclusion

The paper provides a comprehensive asymptotic theory for fully coupled delayed forward-backward SDEs, including existence, uniqueness, convergence in distribution and almost surely, and the first large deviation principle for such systems. The methods generalize classical FBSDE analysis to accommodate memory effects, establishing foundational results for path-dependent stochastic systems with strong coupling. The implications span both theoretical developments in infinite-dimensional stochastic analysis and practical modeling in areas requiring memory-aware probabilistic models.