  We study time-inconsistent recursive stochastic control problems, i.e., for
which the Bellman principle of optimality does not hold. For this class of
problems classical optimal controls may fail to exist, or to be relevant in
practice, and dynamic programming is not easily applicable. Therefore, the
notion of optimality is defined through a game-theoretic framework by means of
subgame perfect equilibrium: we interpret our preference changes which,
realistically, are inconsistent over time, as players in a game for which we
want to find a Nash equilibrium. The approach followed in our work relies on
the stochastic (Pontryagin) maximum principle: we adapt the classical spike
variation technique to obtain a characterisation of equilibrium strategies in
terms of a generalised second-order Hamiltonian function defined through pairs
of backward stochastic differential equations, even in the multidimensional
case. The theoretical results are applied in the financial field to finite
horizon investment-consumption policies with non-exponential actualisation.
Here the existence of non-trivial equilibrium policies is also ascertained.
