A Novel Approach to Peng's Maximum Principle for McKean-Vlasov Stochastic Differential Equations

We present a novel approach to the proof of Peng’s maximum principle for McKean-Vlasov stochastic differential equations (SDE). The main step is the introduction of a third adjoint equation, a conditional McKean-Vlasov backward SDE, to accommodate the dualization of quadratic terms containing two independent copies of the first-order variational process. This is an intrinsic extension of the maximum principle from Peng for standard SDE and gives a conceptually consistent proof. Our approach will be useful in further extensions to the common noise setting and the infinite dimensional setting.

💡 Research Summary

The paper tackles the optimal control problem for stochastic differential equations (SDEs) of McKean‑Vlasov type, where the dynamics depend not only on the state but also on its probability distribution. The classical Peng’s maximum principle provides necessary optimality conditions for standard SDEs, but extending it to mean‑field (McKean‑Vlasov) systems encounters serious difficulties. In particular, the second‑order Taylor expansion of the cost functional generates quadratic terms that involve mixed Lions derivatives (∂_μ∂_x, ∂_μ∂_μ). These terms cannot be dualized by the usual first‑ and second‑order adjoint backward SDEs (BSDEs) employed in Peng’s original proof.

The authors propose a fundamentally new approach: they introduce a third adjoint equation, a conditional McKean‑Vlasov BSDE, which is specifically designed to dualize the quadratic terms that contain two independent copies of the first‑order variational process. This third adjoint does not appear in the final optimality condition; it serves only as a technical device to control the higher‑order error terms. By doing so, the need for the “sharper estimates” used in earlier works (e.g., Buckdahn et al., Proposition 4.3) is eliminated. Those estimates fail in settings with common noise or infinite‑dimensional state spaces, limiting the applicability of previous proofs.

The paper is organized as follows. Section 2 introduces the necessary functional‑analytic background on Lions differentiability on the 2‑Wasserstein space, and defines the regularity class C^{2,1}_b for functions of state and measure. Section 3 constructs the spike variation of the control, the first‑order variational process Y^ε (solution of a linearized McKean‑Vlasov SDE), and the second‑order variational process Z^ε (solution of a more complex SDE containing second‑order derivatives). Precise estimates are proved: Y^ε = O(ε) and Z^ε = O(ε^2) in L^{2k}‑norm, and the remainder K^ε = X^ε−X−Y^ε−Z^ε satisfies K^ε = o(ε^2).

Section 4 defines three adjoint equations. The first adjoint (p,q) is the standard first‑order BSDE from Peng’s theory, incorporating the gradients of the drift, diffusion, and running cost with respect to both state and measure. The second adjoint is a conventional second‑order BSDE handling the homogeneous part of the first variational equation. The novel third adjoint (r,s) is a conditional McKean‑Vlasov BSDE; its driver contains conditional expectations of the mixed Lions derivatives and the variational processes. Existence and uniqueness are established via a fixed‑point argument that combines classical BSDE theory with conditional expectation operators.

Section 5 derives the duality relations between the variational processes and the adjoint processes. By pairing Y^ε and Z^ε with (p,q) and (r,s) respectively, the authors obtain an exact representation of the cost increment J(α^ε)−J(α). The first‑order term yields the familiar Hamiltonian gradient condition, while the second‑order terms are shown to be of order o(ε) thanks to the third adjoint. Consequently, the optimal control α* must satisfy the pointwise maximization of the Hamiltonian: \