This paper investigates an equilibrium feedback control for time-inconsistent reward functionals when the state variable follows a Volterra process. As Volterra processes are non-Markovian and non-semimartingale in general, we develop an extended path-dependent Hamilton-Jacobi-Bellman (PHJB) equation system and offer a verification theorem to the solution of the PHJB equation. We apply the theory to three time-inconsistent problems when the risky asset price follows the Volterra Heston model, a typical rough volatility model. Analytical solutions are derived for the three problems: mean-variance portfolio problem (MVP) with constant risk aversion, MVP with a state-dependent risk aversion, and an investment/consumption problem with non-exponential discounting. Through these examples, we address the effects of roughness on equilibrium strategies.
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