New approach to optimal control of delayed stochastic Volterra integral equations

Abstract: We address the optimal control of stochastic Volterra integral equations with delay through the lens of Hida-Malliavin calculus. We show that the corresponding adjoint processes satisfy an anticipated backward stochastic Volterra integral equation (ABSVIE), and, exploiting this structure, we establish both necessary and sufficient stochastic maximum principles. Our results provide a comprehensive and rigorous framework for characterizing optimal controls in delayed stochastic systems.

• The paper develops an anticipated backward stochastic Volterra integral equation (ABSVIE) adjoint using Hida–Malliavin calculus to manage delayed, non-Markovian state variations.
• The paper proves a sufficient maximum principle under Hamiltonian and terminal-cost concavity, including partial-information controls and conditional maximization.
• The paper establishes a necessary first-order condition without concavity, linking directional stationarity of the objective to conditional Hamiltonian stationarity, while leaving numerical methods and broader solvability open.

Problem setting and contribution

The paper studies optimal control of stochastic Volterra integral equations (SVIEs) with a fixed delay $\delta > 0$. The state process $X^u$ solves

$$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$

with prescribed history on $[-\delta,0]$, and the objective is to maximize

$$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$

The delay term $X(s-\delta)$ makes the system non-Markovian, so finite-dimensional dynamic programming is unavailable; the authors instead develop a Pontryagin–Bismut–Bensoussan type stochastic maximum principle. The central technical device is Hida–Malliavin calculus: because the Volterra structure prevents isolating the first-order variation process $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$ directly, the Malliavin derivative and a generalized Clark–Ocone duality formula are used to handle anticipative integrals arising in the adjoint derivation. The main contributions are (i) an explicit characterization of the adjoint pair $(p,q)$ as the solution of an anticipated backward stochastic Volterra integral equation (ABSVIE), and (ii) both sufficient and necessary maximum principles for optimality.

Preliminaries: Malliavin calculus and representation results

The paper works on a filtered probability space carrying a Brownian motion $B$, with admissible controls valued in a set $\mathcal{U}$. Two function spaces are used: $X^u$0 for adapted continuous processes with finite $X^u$1, and $X^u$2 for adapted processes square-integrable in time.

The key structural result is a representation theorem for BSVIEs of the form

$X^u$3

which asserts that under regularity assumptions the second component satisfies

$X^u$4

where $X^u$5 denotes the Hida–Malliavin derivative. This identity, combined with the generalized duality formula $X^u$6, is what allows the cross-variation and martingale terms in the Itô product expansions to be rewritten entirely in terms of the adjoint processes — the step that replaces the classical BSDE duality argument, which fails in the Volterra setting.

The adjoint equation as an anticipated BSVIE

Under assumptions (A1)–(A3) — boundedness and smoothness of the coefficients, $X^u$7 terminal cost, and a $X^u$8 regularity condition on $X^u$9 with square-integrable partial derivative — the paper derives the adjoint equation

$$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$0

where the driver $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$1 collects the delayed running-cost gradient $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$2, the first-derivative terms $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$3 evaluated at shifted arguments, and integral terms involving the mixed derivatives $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$4 and $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$5 weighted by $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$6 and $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$7. The proof proceeds by perturbing the optimal control along $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$8, applying Itô's formula to $$X^{u}(t)= x_{0}(t)+ \int_{0}^{t}b(t,s,X^{u}(s-\delta),u(s))\,ds + \int_{0}^{t}\sigma(t,s,X^{u}(s-\delta),u(s))\,dB(s),$$9, eliminating all terms involving the variation process $[-\delta,0]$0 via Fubini's theorem, the duality formula, and the representation theorem, and then choosing $[-\delta,0]$1 to annihilate the remaining $[-\delta,0]$2-dependent coefficients.

The resulting equation is genuinely anticipated: the driver at time $[-\delta,0]$3 depends on $[-\delta,0]$4 and $[-\delta,0]$5, i.e., on future values of the adjoint. The authors note that such equations were first studied by Wen and Shi, who established existence and uniqueness under global Lipschitz conditions; the present work relies on that solvability theory rather than proving well-posedness itself. This dependence is a substantive assumption inherited from the literature rather than verified within the paper.

Sufficient maximum principle under partial information

The sufficiency result is formulated under partial information: the controller observes only a sub-filtration $[-\delta,0]$6, and admissible controls are càdlàg, $[-\delta,0]$7-valued, $[-\delta,0]$8-adapted processes with convex control set $[-\delta,0]$9. If (i) $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$0 and the Hamiltonian $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$1 are concave in $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$2 almost surely for each $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$3, (ii) assumption (A3) holds for all admissible controls, and (iii) the conditional maximum condition

$$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$4

holds, then $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$5 is optimal. The proof follows the standard concavity-duality route: decompose $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$6 into running and terminal parts, bound each by first-order expansions using concavity, apply Itô's formula to $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$7, and use the same Malliavin/Fubini machinery to cancel all state-difference terms. The cancellation is exact — the delayed gradient term $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$8 appearing through the adjoint dynamics precisely offsets the shifted concavity term — leaving only $$J(x_0,u) = \mathbb{E}\left[\int_0^T f(t, X^{u}(t-\delta), u(t))\,dt + g(X^{u}(T))\right].$$9, which vanishes by the conditional maximum condition and $X(s-\delta)$0-adaptedness. A notable feature is that the entire argument remains valid under partial information without additional technical cost.

Necessary maximum principle

Recognizing that the concavity hypothesis of the sufficient principle "is not always satisfied in practical cases," the paper establishes a converse result: directional stationarity of the performance functional is equivalent to stationarity of the conditional Hamiltonian. Specifically, under a convexity-of-admissible-set assumption (perturbations $X(s-\delta)$1 remain admissible), a spike-variation construction $X(s-\delta)$2 with bounded $X(s-\delta)$3-measurable $X(s-\delta)$4, and existence of the derivative process $X(s-\delta)$5 in $X(s-\delta)$6, the following are equivalent:

1. For every bounded perturbation $X(s-\delta)$7, $X(s-\delta)$8;
2. $X(s-\delta)$9 a.s. for all $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$0.

The proof again computes the variational derivative, applies Itô's formula to $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$1, and shows that all $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$2-dependent contributions cancel identically against the adjoint dynamics, reducing the derivative to $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$3. Differentiating at $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$4 and arbitrariness of $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$5 yield the conditional stationarity condition; the reverse implication follows by approximating general bounded $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$6 by linear combinations of spike controls. This equivalence gives a sharp first-order characterization of candidate optima that does not require concavity, complementing the sufficient principle.

Limitations and open questions

Several restrictions should be noted. First, the adjoint equation is anticipated, and its solvability is invoked from existing Lipschitz theory (Wen–Shi) rather than established here; whether the framework extends to non-Lipschitz or quadratic drivers remains open. Second, the sufficient principle requires joint concavity of the Hamiltonian and terminal cost, which the authors themselves flag as restrictive; no relaxation (e.g., via second-order conditions) is provided. Third, the analysis assumes a single fixed delay $$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$7, scalar-valued state and control ($$Y(t) = \frac{d}{d\varepsilon}X^{u^\varepsilon}(t)|_{\varepsilon=0}$$8), and continuous (Brownian) noise; extension to multiple delays, jumps, or infinite-dimensional state spaces is not addressed. Fourth, the necessary principle characterizes only first-order critical points and does not distinguish local maxima from saddle points. Finally, the paper provides no numerical scheme or example illustrating how the ABSVIE-based characterization would be computed in practice.

Conclusion

The paper extends the Malliavin-calculus approach to stochastic Volterra control problems to the delayed case by showing that the natural adjoint object is an anticipated BSVIE whose driver involves future values of the adjoint pair. Building on this structure, it delivers a complete pair of maximum principles — a sufficient one valid under partial information and concavity, and a necessary one establishing equivalence between variational stationarity and Hamiltonian stationarity without concavity. The results place delayed SVIE control within the anticipated-BSVIE framework and reduce the open questions chiefly to solvability of the adjoint equation beyond the Lipschitz regime and to computational exploitation of the characterization.