  We consider a general class of stochastic optimal control problems, where the
state process lives in a real separable Hilbert space and is driven by a
cylindrical Brownian motion and a Poisson random measure; no special structure
is imposed on the coefficients, which are also allowed to be path-dependent; in
addition, the diffusion coefficient can be degenerate. For such a class of
stochastic control problems, we prove, by means of purely probabilistic
techniques based on the so-called randomization method, that the value of the
control problem admits a probabilistic representation formula (known as
non-linear Feynman-Kac formula) in terms of a suitable backward stochastic
differential equation. This probabilistic representation considerably extends
current results in the literature on the infinite-dimensional case, and it is
also relevant in finite dimension. Such a representation allows to show, in the
non-path-dependent (or Markovian) case, that the value function satisfies the
so-called randomized dynamic programming principle. As a consequence, we are
able to prove that the value function is a viscosity solution of the
corresponding Hamilton-Jacobi-Bellman equation, which turns out to be a
second-order fully non-linear integro-differential equation in Hilbert space.
