Mean-field stochastic Volterra optimal singular control with Poisson jumps∗

This paper considers an infinite horizon mean-field type of stochastic Volterra singular control problem. The dynamical system is governed by a Itô-Lévy processes and a standard one dimensional independent Brownian motion. Stochastic Volterra controlled system is difficult to manipulate by standard methods such as dynamic programming and classical maximum principle, because the presence of memory terms in the dynamics of the system. In this study, Malliavin calculus is a useful tool to overcome these difficulties. Necessary and sufficient condition for the aforementioned system is established by using Malliavin calculus, convex perturbation technique and transversality condition. Finally an example is given to show that the application of the theoretical study.