Viscosity Solutions for HJB Equations on the Process Space

In this paper we investigate a path dependent optimal control problem on the process space with both drift and volatility controls, with possibly degenerate volatility. The dynamic value function is characterized by a fully nonlinear second order path dependent HJB equation on the process space, which is by nature infinite dimensional. In particular, our model covers mean field control problems with common noise as a special case. We shall introduce a new notion of viscosity solutions and establish both the existence and the comparison principle, under merely Lipschitz/Holder continuity assumptions. The main feature of our notion is that, besides the standard smooth part, the test function consists of an extra singular component which allows us to handle the second order derivatives of the smooth test functions without invoking the Crandall-Ishii lemma. We shall use the doubling variable arguments, combined with the Ekeland-Borwein-Preiss variational principle in order to overcome the noncompactness of the state space. A smooth gauge-type function on the path space is crucial for our estimates.