Stochastic representation under filtration-consistent nonlinear expectations

In this paper, we investigate the stochastic representation problem under filtration-consistent nonlinear expectations. We establish the existence and uniqueness of the solutions for the discrete-time case. Besides, we provide a characterization of the solution, which is helpful to derive the uniqueness of solutions for the continuous time case. The stochastic representation can be applied to the variant Skorokhod problem, the optimal stopping problem and the intertemporal utility optimization problem.