Partial derivative with respect to the measure and its application to general controlled mean-field systems

Abstract

Let (E,E) be an arbitrary measurable space. The paper first focuses on studying the partial derivative of a function f:P2,0(Rd×E)→R defined on the space of probability measures μ over (Rd×E,B(Rd)⊗E) whose first marginal μ1≔μ(⋅×E) has a finite second order moment. This partial derivative is taken with respect to q(dx,z), where μ has the disintegration μ(dxdz)=q(dx,z)μ2(dz) with respect to its second marginal μ2(⋅)=μ(Rd×⋅). Simplifying the language, we will speak of the derivative with respect to the law μ conditioned to its second marginal. Our results extend those of the derivative of a function g:P2(Rd)→R over the space of probability measures with finite second order moment by P.L. Lions (see Lions (2013)) but cover also as a particular case recent approaches considering E=Rk and supposing the differentiability of f over P2(Rd×Rk), in order to use the derivative ∂μf to define the partial derivative (∂μf)1. The second part of the paper focuses on investigating a stochastic maximum principle, where the controlled state process is driven by a general mean-field stochastic differential equation with partial information. The control set is just supposed to be a measurable space, and the coefficients of the controlled system, i.e., those of the dynamics as well as of the cost functional, depend on the controlled state process X, the control v, a partial information on X, as well as on the joint law of (X,v). Through considering a new second-order variational equation and the corresponding second-order adjoint equation, and a totally new method to prove the estimate for the solution of the first-order variational equation, the optimal principle is proved through spike variation of an optimal control and with the help of the tailor-made form of second-order expansion. We emphasize that in our assumptions we do not need any regularity of the coefficients neither in the control variable nor with respect to the law of the control process.