Two-level Optimization in Stochastic Control Problems

This paper is devoted to a stochastic optimal control problem with time inconsistency arising from the presence of an extremal measure in the quality functional. Using the methodology described in K. Miller’s paper, the initial formulation is reduced to a two-level optimization task, in which the inner problem is classical and the outer problem is solved using proximal gradient methods. This study proves a theorem that strengthens a theorem previously formulated and proven in the Miller’s work. This strengthening of the new result stems from the fact that in the current formulation, the performance functional consists of the discounted sum of the performance functional obtained from the classical formulation and the extremal measure, rather than just the extremal measure. This paper also examines control dynamics containing jumps.