A Stochastic Maximum Principle for Volterra SDEs
Adjoint Equations and Pontryagin Necessary Conditions under Rough Drivers
Abstract
We establish a stochastic maximum principle for finite-horizon optimal control of stochastic Volterra integral equations driven by rough kernels with Hurst index $H \in (0, 1/2)$. The adjoint process is shown to satisfy a backward stochastic Volterra integral equation (BSVIE) whose kernel inherits the fractional regularity of the forward driver. Under standard convexity assumptions on the running cost and admissible-control set, the Pontryagin necessary conditions hold; under additional concavity assumptions they are sufficient. The framework specialises to rough-volatility option-pricing and rough-Heston hedging problems, where classical Markovian HJB methods fail.