A Nonlocal Stochastic Optimal Control with Elliptic-Type Smoothing

We study a nonlocal stochastic optimal control problem in which the control acts through the elliptic smoothing operator $\mathcal{S}=(I-\mathrm{\Delta }{)}^{-1}$ on ${\mathbb{R}}^d$. The state is described by the law of a controlled diffusion, equivalently by a controlled Fokker-Planck equation with coefficients depending on $\mathcal{S}u$. We prove the existence of an optimal control by the direct method in the calculus of variations. We then derive a Pontryagin-type maximum principle by spike variations, relying on sharp properties of the operator $\mathcal{S}$. The result yields a pointwise minimisation rule for the optimal control. We discuss the meaning of the optimal control problem in the case of population dynamics.