# Polynomial processes in stochastic portfolio theory

## TL;DR

This paper introduces polynomial processes into stochastic portfolio theory to model company market capitalizations and weights, allowing for correlation, tractability, and practical estimation in high-dimensional equity markets.

## Contribution

It extends volatility stabilized models by incorporating correlation and polynomial structures, providing explicit conditions for arbitrage and boundary behavior.

## Key findings

- Polynomial processes model market capitalizations and weights with correlation.
- Explicit conditions for local martingale deflators and arbitrage strategies.
- Extensions to jump models and optimal arbitrage computation.

## Abstract

We introduce polynomial processes in the sense of [8] in the context of stochastic portfolio theory to model simultaneously companies' market capitalizations and the corresponding market weights. These models substantially extend volatility stabilized market models considered by Robert Fernholz and Ioannis Karatzas in [18], in particular they allow for correlation between the individual stocks. At the same time they remain remarkably tractable which makes them applicable in practice, especially for estimation and calibration to high dimensional equity index data. In the diffusion case we characterize the joint polynomial property of the market capitalizations and the corresponding weights, exploiting the fact that the transformation between absolute and relative quantities perfectly fits the structural properties of polynomial processes. Explicit parameter conditions assuring the existence of a local martingale deflator and relative arbitrages with respect to the market portfolio are given and the connection to non-attainment of the boundary of the unit simplex is discussed. We also consider extensions to models with jumps and the computation of optimal relative arbitrage strategies.