Peng's Maximum Principle for Stochastic Delay Differential Equations of Mean-Field Type
Giuseppina Guatteri, Federica Masiero, Lukas Wessels
TL;DR
This work extends Peng's stochastic maximum principle to stochastic delay differential equations of mean-field type by lifting the delay into an infinite-dimensional Hilbert space, enabling a Markovian formulation. A key innovation is a refined asymptotic for the mean of the first-order variational process, which allows a second-order expansion even with control entering the diffusion and non-convex control domains. The authors develop a framework of operator-valued and McKean–Vlasov BSDEs in infinite dimensions, including a tensor-product variational term and a second-order adjoint equation, to derive a stochastic maximum principle with a nonconvex control condition. The results establish well-posedness of the infinite-dimensional mean-field BSDEs and provide a comprehensive set of adjoint equations and dualities that yield the necessary optimality condition for a broad class of mean-field delay control problems.
Abstract
We extend Peng's maximum principle to the case of stochastic delay differential equations of mean-field type. More precisely, the coefficients of our control problem depend on the state, on the past trajectory and on its expected value. Moreover, the control enters the noise coefficient and the control domain may be non-convex. Our approach is based on a lifting of the state equation to an infinite dimensional Hilbert space that removes the explicit delay in the state equation. The main ingredient in the proof of the maximum principle is a precise asymptotic for the expectation of the first order variational process, which allows us to neglect the corresponding second order terms in the expansion of the cost functional.