  In this paper, we focus on solving the optimal control problem for integral
stochastic Volterra equations in a finite dimensional setting. In our setting,
the noise term is driven by a pure jump L\'evy noise and the control acts on
the intensity of the jumps.
  We use recent techniques proposed by Hamaguchi, where a crucial requirement
is that the convolution kernel should be a completely monotone function. This
allows us to use Bernstein's representation and the machinery of Laplace
transform to obtain a Markovian lift.
  It is natural that the Markovian lift, in whatever form constructed,
transforms the state equation into a stochastic differential equation in an
infinite-dimensional space. This space should be large enough to contain all
the information about the history of the process. Hence, although the original
equation is taken in a finite dimensional space, the resulting lift is always
infinite dimensional.
  We solve the problem by using the forward-backward approach in the
infinite-dimensional setting and prove the existence of the optimal control for
the original problem. Under additional assumptions on the coefficients, we see
that a control in closed-loop form can be achieved.
