Optimal portfolio under ratio-type periodic evaluation in stochastic factor models under convex trading constraints

TL;DR

This paper develops a mathematical framework for optimal portfolio management with periodic evaluation and convex trading constraints in stochastic factor models, transforming the problem into auxiliary unconstrained problems and establishing duality results.

Contribution

It introduces a novel approach to solve infinite horizon portfolio optimization with ratio-based periodic evaluation under convex constraints using duality and fixed point methods.

Findings

Derived explicit optimal portfolio strategies under the model.

Established duality results connecting constrained and unconstrained problems.

Proved existence and characterization of optimal solutions in the periodic setting.

Abstract

This paper studies a type of periodic utility maximization problems for portfolio management in incomplete stochastic factor models with convex trading constraints. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon, featuring the dynamic adjustments in portfolio decision according to past achievements. Under power utility, we transform the original infinite horizon optimal control problem into an auxiliary terminal wealth optimization problem under a modified utility function. To cope with the convex trading constraints, we further introduce an auxiliary unconstrained optimization problem in a modified market model and develop the martingale duality approach to establish the existence of the dual minimizer such that the optimal unconstrained wealth process can be obtained using the dual representation. With the help of the duality results in the auxiliary problems, the relationship between the constrained and unconstrained models as well as some fixed point arguments, we finally derive and verify the optimal constrained portfolio process in a periodic manner for the original problem over an infinite horizon.