We recall the standard theory of viscosity solutions for HJB equations on $\R^d$ with a focus on the doubling variable technique. Motivated by recent applications in optimal control of path-dependent and/or interacting stochastic systems, we investigate a path-dependent optimal control problem on the process space with both drift and diffusion 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 existence and comparison principle through the doubling variable technique only, and without invoking the Ishii's lemma.
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