Necessary conditions of Pontraygin’s type for general controlled stochastic Volterra integral equations

This article is addressed to giving a solution to a unsolved problem, i.e. , to establish the necessary optimality conditions of Pontraygin’s type for controlled stochastic Volterra integral equations (SVIEs) when the control region is non-convex and the control variable enters into the diffusion. This problem has been open since [J. Yong, Stochastic Process Appl. 116 (2006) 779–795] obtained the analogue result for the case of convex control region. The key is to introduce a pair of suitable second-order adjoint processes (SOAPs). It is found that the usual way of using only one SOAP in the maximum condition for the classical setting of controlled stochastic differential equations does not work here.