Existence and characterization of the optimal portfolio for the auxiliary one-period problem

Establish the existence of, and characterize, the optimal portfolio for the auxiliary one-period portfolio optimization problem that arises from reformulating the infinite-horizon ratio-type periodic evaluation objective (based on X_Ti/(X_{T_{i-1}})^γ with γ in (0,1]) via dynamic programming in the incomplete market model where the risky asset S follows the diffusion (2.1) and the stochastic factor Y follows (2.2).

Background

The paper studies optimal portfolio selection under a ratio-type periodic evaluation criterion over an infinite horizon in an incomplete market where the risky asset’s drift and volatility are driven by an external stochastic factor Y. By the dynamic programming principle, the infinite-horizon problem is reduced to an auxiliary one-period optimization problem, which in turn requires proving the existence and structure of an optimal portfolio in an incomplete setting with unhedgeable risk.

The authors highlight that, compared to the Black–Scholes case, the presence of stochastic factors necessitates handling infinitely many dual processes and complicates identification of a dual optimizer, particularly because the induced utility deviates from standard forms such as the pure power utility.