We prove existence and uniqueness of Crandall-Lions viscosity solutions of
Hamilton-Jacobi-Bellman equations in the space of continuous paths, associated
to the optimal control of path-dependent SDEs. This seems the first uniqueness
result in such a context. More precisely, similarly to the seminal paper of
P.L. Lions, the proof of our core result, that is the comparison theorem, is
based on the fact that the value function is bigger than any viscosity
subsolution and smaller than any viscosity supersolution. Such a result,
coupled with the proof that the value function is a viscosity solution (based
on the dynamic programming principle, which we prove), implies that the value
function is the unique viscosity solution to the Hamilton-Jacobi-Bellman
equation. The proof of the comparison theorem in P.L. Lions' paper, relies on
regularity results which are missing in the present infinite-dimensional
context, as well as on the local compactness of the finite-dimensional
underlying space. We overcome such non-trivial technical difficulties
introducing a suitable approximating procedure and a smooth gauge-type
function, which allows to generate maxima and minima through an appropriate
version of the Borwein-Preiss generalization of Ekeland's variational principle
on the space of continuous paths