Robust Asymptotic Growth in Stochastic Portfolio Theory under Long-Only Constraints

TL;DR

This paper develops a framework for optimizing long-only portfolios in stochastic portfolio theory, proving existence and uniqueness of solutions, and providing numerical methods for practical implementation under model uncertainty.

Contribution

It introduces a finite-dimensional approximation for the optimization problem and extends the theory to a broad class of calibratable volatility models.

Findings

Proved existence and uniqueness of optimal portfolios under constraints.

Developed a numerical approximation method for portfolio optimization.

Derived explicit formulas for optimal portfolios in certain models.

Abstract

We consider the problem of maximizing the asymptotic growth rate of an investor under drift uncertainty in the setting of stochastic portfolio theory (SPT). As in the work of Kardaras and Robertson we take as inputs (i) a Markovian volatility matrix $c(x)$ and (ii) an invariant density $p(x)$ for the market weights, but we additionally impose long-only constraints on the investor. Our principal contribution is proving a uniqueness and existence result for the class of concave functionally generated portfolios and developing a finite dimensional approximation, which can be used to numerically find the optimum. In addition to the general results outlined above, we propose the use of a broad class of models for the volatility matrix $c(x)$, which can be calibrated to data and, under which, we obtain explicit formulas of the optimal unconstrained portfolio for any invariant density.