Before revision:  These equations represent analogs of Hamilton-Jacobi-Bellman equations for the value functions of optimal control problems in non-Markovian setting arising in financial modelling.
After revision:  These equations represent analogs of Hamilton-Jacobi-Bellman equations for the value functions of optimal control problems with unspecified dynamics where the underlying processes do not necessarily satisfy stochastic differential equations of a known kind with a given structure. The problems considered arise in financial modelling.