Existence of optimal controls for stochastic Volterra equations

We provide sufficient conditions that guarantee the existence of relaxed optimal controls in the weak formulation of control problems for stochastic Volterra equations (SVEs). Our study can be applied when the kernel appearing in the controlled SVE is singular at zero. The existence of relaxed optimal policies relies on the interaction between integrability hypotheses on the kernel and growth conditions on the running cost functional and the coefficients of the controlled SVEs. Under classical convexity assumptions, we can also deduce the existence of optimal strict controls.