  We study a stochastic control problem for a Volterra-type controlled forward
equation with past dependence obtained via convolution with a deterministic
kernel. To be able to apply dynamic programming to solve the problem, we lift
it to infinite dimensions and we formulate a UMD Banach-valued Markovian
problem, which is shown to be equivalent to the original finite-dimensional
non-Markovian one. We characterize the optimal control for the infinite
dimensional problem and show that this also characterizes the optimal control
for the finite dimensional problem.
