Abstract
This paper deals with the discrete-time stochastic optimal control problems with recursive utilities under weakened convexity assumption. A new stochastic maximum principle is established. Moreover, by constructing two new adjoint equations and two new variational equations for the backward stochastic difference equation (BS <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta$</tex-math></inline-formula> E), we obtain the second-order necessary optimality condition of quasi-singular control. Finally, as an illustration, a discrete-time mean-variance portfolio selection mixed with a recursive utility functional optimization problem is solved.
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