  We provide sufficient conditions that guarantee the existence of relaxed
optimal controls in the weak formulation of stochastic control problems for
stochastic Volterra equations (SVEs). Our study can be applied to rough
processes that arise 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.
