A Stochastic Maximum Principle for Volterra SDEs

We establish a stochastic maximum principle for finite-horizon optimal control of stochastic Volterra integral equations with the fractional kernel $k(t) = t^{H-1/2}/\Gamma(H+1/2)$, Hurst index $H \in (0, 1/2]$, Lipschitz coefficients and a convex compact control set. The adjoint process solves a backward stochastic Volterra integral equation (BSVIE) in which the forward kernel is read over later times and the gradient of the terminal cost enters with the singular weight $k(T-t)$. An optimal control satisfies the first-order form of the maximum condition, which is pointwise maximisation of the Hamiltonian when the Hamiltonian is concave in the control; when the terminal cost is convex and the Hamiltonian is jointly concave in state and control, the condition is also sufficient. A linear–quadratic example has a closed-form adjoint in Mittag-Leffler functions, which we check against direct numerical solutions.