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We study time inconsistent recursive stochastic control problems with
constraint, where the notion of optimality is defined by means of subgame
perfect equilibrium. The foundation of our work consists in the stochastic
(Pontryagin) maximum principle which, in a context devoid of any constraints,
provides a characterisation of equilibrium strategies in terms of a generalised
second order Hamiltonian function defined through a pair of backward stochastic
differential equations. The class of constraints we analyse is therefore
defined through a further recursive utility system and a description of Nash
equilibria in this situation is obtained by adapting the Ekeland variational
principle. All this finds application in the financial field of finite horizon
investment-consumption policies with non-exponential actualisation, where the
possibility that the constraint is a risk constraint is covered.
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