  Control problems not admitting the dynamic programming principle are known as
time-inconsistent. The game-theoretic approach is to interpret such problems as
intrapersonal dynamic games and look for subgame perfect Nash equilibria. A
fundamental result of time-inconsistent stochastic control is a verification
theorem saying that solving the extended HJB system is a sufficient condition
for equilibrium. We show that solving the extended HJB system is a necessary
condition for equilibrium, under regularity assumptions. The controlled process
is a general It\^o diffusion.
