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In this paper, we consider the functional It\^o calculus framework to find apath-dependent version of the Hamilton-Jacobi-Bellman equation for stochasticcontrol problems that feature dynamics and running cost that depend on the pathof the control. We also prove a Dynamic Programming Principle for suchproblems. We apply our results to path-dependence of the delay type. We furtherstudy Stochastic Differential Games in this context.

Stochastic Control and Differential Games with Path-Dependent Influence of Controls on Dynamics and Running Cost