Before revision:  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 problemswhen 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 .
After revision:  As Volterra processes are non-Markovian and non-semimartingale in general , we develop an extended path-dependent Hamilton-Jacobi-Bellman (PHJB) equation system . A verification theorem is provided. By deriving explicit solutions to three problems, including 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 .