  We study time-inconsistent recursive stochastic control problems. Since, for this class of problems, classical optimal controls may fail to exist or to be relevant in practice, we focus on sub-game perfect equilibrium policies. The approach followed in our work relies on the stochastic maximum principle and, indeed, we adapt the classical spike variation technique to obtain a characterisation of equilibrium strategies in terms of a generalised Hamiltonian function of second-order defined through a flow of pairs of BSDEs. We deal then time-inconsistent recursive stochastic control problems under a state constraint, defined by means of an additional recursive utility, by adapting the Ekeland variational principle to this more tricky situation. The theoretical results are applied to the financial field on finite horizon investment-consumption policies with non-exponential actualisation. The possibility that the constraint is a risk constraint is covered. 
Subgame-perfect equilibrium strategies in state-constrained recursive   stochastic control problems