Fixed-point characterization of the original value function and verification of the constructed optimal portfolio

Develop a fixed-point argument to characterize the original infinite-horizon value function associated with the ratio-type periodic evaluation in the incomplete market model with stochastic factor Y, and provide a verification proof that the constructed portfolio process is optimal over the infinite horizon.

Background

After reducing the infinite-horizon problem to an auxiliary one-period problem via dynamic programming, a fixed-point equation characterizing the value function arises. In stochastic factor models, the fixed point depends on the factor state y rather than being a scalar, which complicates both existence/uniqueness arguments and the verification of optimality.

The verification step must reconcile the duality arguments in an incomplete market with the concatenation of one-period solutions over an infinite sequence of evaluation times, ensuring admissibility and optimality of the resulting portfolio strategy.