Delayed Forward-Backward Stochastic Control

• Delayed forward-backward stochastic control systems are optimal control models characterized by coupled FBSDEs with explicit delay and memory effects.
• They incorporate delayed state, cost, and control factors using SDDEs and BSDEs under strong Lipschitz and monotonicity conditions to ensure well-posedness.
• Advanced numerical schemes like discretization and Volterra transformations enable explicit feedback laws and optimality analysis in complex delay scenarios.

A delayed forward-backward stochastic control system is a class of stochastic optimal control models in which the system dynamics and/or cost functionals depend on both current and delayed (history-dependent) values of the state and control processes, with the system jointly described by coupled forward and backward stochastic differential equations (FBSDEs) with delays. Such systems are motivated by applications where hereditary effects, after-effects, or memory (e.g., economic delays, engineering lags) play a structural role and are rigorously modeled as stochastic delay differential equations (SDDEs) interfaced with recursive or terminal functionals defined by backward SDEs (BSDEs) with delay, anticipated, or advanced arguments.

1. System Formulation, Notation, and Mathematical Setting

Let $T>0$ be a finite time horizon, and fix $\delta>0$ (delay length). On a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, P)$ supporting a $d$-dimensional Brownian motion $W(t)$, and possibly additional orthogonal Teugels martingales $H^{(i)}$ (associated to a Lévy process), the general delayed forward-backward stochastic control system reads: $$\begin{aligned} &\text{Forward (state) equation:}\ &\quad dx(t) = b\bigl(t,\Theta(t),\Theta(t-\delta),\Theta(t+\delta)\bigr)\,dt + \sigma\bigl(t,\Theta(t),\Theta(t-\delta),\Theta(t+\delta)\bigr)\,dW(t) + \sum_{i=1}^\infty g^{(i)}(\cdots)\,dH^{(i)}(t), \ &\text{Backward (costate/adjoint) equation:}\ &\quad dy(t) = -f\bigl(t,\Theta(t),\Theta(t-\delta),\Theta(t+\delta)\bigr)\,dt + z(t)\,dW(t) + \sum_{i=1}^\infty k^{(i)}(t)\,dH^{(i)}(t), \end{aligned}$$ where $\Theta(t) = (x(t), y(t), z(t), k(t))$, and delays may appear in both the drift, diffusion, and noise terms. The control $u(t)$ is adapted and may influence both forward and backward equations, and its delayed versions $u(t-\delta)$ and distributed versions (e.g., convolutions with kernels) may appear as arguments.

The performance index (cost functional) typically involves both running and terminal costs, possibly with explicit path-dependence or terminal constraints (e.g. $x(T) \in K$ for a convex set $K$), and is optimized over admissible controls.

2. Structural Properties, Assumptions, and Main Existence Results

To ensure unique solvability of the delayed FBSDE system and well-posedness of the stochastic optimal control problem, several key structural conditions are imposed:

• Lipschitz and boundedness: All coefficients and their state/control derivatives are globally Lipschitz and of at most linear growth in their arguments. In mean-field/measure-dependent problems, this includes smoothness (up to order two) in the Wasserstein sense for the law dependence (Hao et al., 2017).
• Domination-monotonicity: For fully coupled, possibly nonlinear delayed systems—especially under nonconvexity or presence of Lévy noise—existence and uniqueness is established under joint domination and monotonicity constraints in the forward and backward drivers (Xu et al., 2023). Specifically, for differences of two solutions, the drivers are required to satisfy one of two classes of inequalities involving preconditioning operators in the increments.
• Terminal and history constraints: When path-dependence or convex closed terminal constraints are present, suitable boundary conditions and anticipated/advanced arguments for the backward component must be imposed (Wen et al., 2017).
• Stochastic Volterra/Nugatory operators: For extended mixed delays (distributed and noisy memory), the stochastic Fubini theorem is used to transform state equations into stochastic Volterra integral equations, removing direct delay from the system (Li et al., 16 Jan 2026).

Under these hypotheses, one establishes existence, uniqueness, and regularity of the (possibly vector-valued) quadruple $(x(t), y(t), z(t), k(t))$ in suitable Banach spaces of square-integrable, adapted processes.

3. Stochastic Maximum Principle and Adjoint Equations

The stochastic maximum principle (SMP) provides necessary (and under additional conditions, sufficient) optimality conditions. The forward–backward delayed SMP augments classical Hamiltonian analysis with novel structures:

• Hamiltonian: $$\mathcal{H}(t, x, x_{-\delta}, x_{+\delta}, y, z, k, u, \ldots)$$ includes all arguments with possible delay and anticipation (including laws in mean-field cases), as well as all adjoint variables and second-order derivatives in nonconvex cases.
• Adjoint system: The costate dynamics typically include anticipated BSDEs, advanced SDEs, or, for mixed delay, backward stochastic Volterra integral equations with Malliavin derivatives (Hao et al., 2017, Chen et al., 2011, Li et al., 16 Jan 2026, 2002.03953). For instance, in mean-field and nonconvex cases the first-order adjoint is an anticipated mean-field BSDE, while the second-order adjoint is a matrix-valued anticipated BSDE.
• Variational inequalities: Using spike variation or convex analysis, the SMP yields for all $u$ in the control set,

$$\mathcal{H}_u^*(t) + \mathbb{E}_t[ \mathcal{H}_{u(\cdot)}^*(t+\delta) ] \cdot (v-u^*(t)) \ge 0,$$

with the specific anticipated (or advanced) terms reflecting the delay structure (Li, 20 Jan 2026, Cheng, 19 Dec 2025, Chen et al., 2011, Li et al., 2024).

In settings with general, possibly mixed, distributed, or noisy memory delays, the first-order variation is recast as a Volterra integral equation. The corresponding adjoint (costate) system is then a backward SVIE driven by Malliavin derivatives or, after applying Clark-Ocone, anticipated BSDEs (Li et al., 16 Jan 2026).

4. Decoupling, Numerical Schemes, and Explicit Solution Methods

Delayed FBSDEs are inherently infinite-dimensional due to their dependence on past trajectories. Several techniques have been developed:

• Discretization and limit: The continuous-time system is first discretized, yielding a difference equation with finite 'delay memory', solved by backward induction. As the mesh is refined ($\delta\to0$), one recovers the continuous-time solution, including delayed Riccati equations for linear-quadratic cases (Ma et al., 2020).
• Stochastic Volterra transformation: For general time-varying or distributed delays, one constructs an augmented state that absorbs all dependencies, transforming the delayed problem into a delay-free stochastic Volterra integral system, for which explicit closed-loop Riccati equations and feedback kernels can be derived (Meng et al., 3 Oct 2025).
• Continuation/homotopy method: In nonlinear settings with Lévy noise or full coupling, unique solvability is achieved via a one-parameter homotopy between a trivial linear system and the original, with contraction mapping iteration (Xu et al., 2023).

Such methods not only establish constructive solvability but also enable explicit computation of optimal controls for a wide range of delay models.

5. Representative Applications and Explicit LQ Laws

Delayed forward-backward stochastic control systems are central in:

• Finance and recursive utility: Models with delayed wealth or delayed controls (e.g., mean-field games, pension fund optimization, recursive utility maximization) require delayed FBSDEs with mean-field arguments and anticipated/backward drivers (Hao et al., 2017, Shi et al., 2013, Agram et al., 2014).
• Engineering and economics: LQ regulators, stochastic integro-differential systems with pointwise and distributed delays, and production-consumption problems with terminal constraints and after-effects (Xu et al., 2023, Meng et al., 3 Oct 2025, Wen et al., 2017).
• Games and partially observed systems: Nash equilibria in delayed, partially observed stochastic differential games, under both information delays and anticipated cost structure (Zhuang, 2017).

Explicit closed-form laws for admissible classes (notably LQ) take the form

$$u^*(t) = -[R_1(t)+\mathbb{E}[R_1(t+\delta)]]^{-1}\left(\cdots\right),$$

where the $(\cdots)$ depend on adjoint variables and delayed/anticipated state and costate values (Xu et al., 2023). Feedback gains are determined by delayed Riccati or Volterra systems, with uniqueness ensured under positive-definiteness and structural monotonicity.

6. Extensions, Generalizations, and Methodological Trends

Recent advancements encompass:

• Extended mixed delays: Joint presence of pointwise, distributed, and noisy memory in state and control, handled by backward SVIEs with Malliavin derivatives and duality via the Clark-Ocone formula (Li et al., 16 Jan 2026).
• Nonconvexity and global SMP: Global optimality principles with auxiliary first- and second-order backward equations, enabling SMP derivation without convexity, using duality arguments in stochastic Volterra integral settings (Li, 20 Jan 2026).
• Mean-field, infinite horizon, and stochastic games: Incorporation of mean-field terms, partial/anticipative information, or infinite horizon (with associated transversality conditions), and applications to stochastic differential games (Agram et al., 2014, Agram et al., 2013, Cheng, 19 Dec 2025).

These contributions show a trend toward unifying classical (delay-free), anticipated, and extended-memory stochastic control in both theory and computable practice, under increasingly weak or general technical assumptions, and with explicit feedback controls in paradigmatic cases.

7. Table: Key Delay Structures in Forward-Backward Stochastic Control

Paper/Approach	Delay Type(s)	Adjoint Structure
(Hao et al., 2017)	Pointwise, mean-field	Anticipated mean-field BSDE, second-order matrix BSDE
(Xu et al., 2023)	Pointwise, anticipation, Lévy	Fully coupled FBSDELDA, domination-monotonicity
(Chen et al., 2011)	State/control, moving average	Advanced SDE (ASDE) as adjoint
(Meng et al., 3 Oct 2025)	State/control/distributed	Transformation to SVIE, Riccati-Volterra system
(Li et al., 16 Jan 2026)	Point, distributed, noisy memory	Backward SVIE (with Malliavin derivatives), anticipated BSDEs
(2002.03953)	State/control, full path dependence	Linear anticipated BSDE as adjoint

This taxonomy encapsulates the diversity of delay structures and associated mathematical methodologies for analysis and optimality characterization in contemporary delayed forward-backward stochastic control systems.