UC Santa Barbara

The relative arbitrage portfolio introduced in Stochastic Portfolio Theory (SPT), outperforms a benchmark portfolio over a time-horizon with probability one. Following this concept, when an investor competes with both market and peers, does relative arbitrage opportunity exist as well? What is the best performance one can achieve? What is the impact on market dynamics and investors when a large group competes in this way?

This thesis constructs a framework of multi-agent optimization under SPT to tackle these questions. With a market model depending on stock capitalizations and targeted investors, we analyze the market behavior and optimal investment strategies to attain relative arbitrage in a large population regime under some market conditions.

% By creating a finite dynamical system of the equity market, an investor uses a benchmark of market and peer investors, expecting to outperform the benchmark and minimizing the initial capital. The objective can be characterized by the smallest nonnegative continuous solution of a Cauchy problem. We show a unique equilibrium for relative arbitrage in $N$-player and mean field games (MFG) with mild conditions on the equity market, by modifying extended MFG with common noise and its notion of the uniqueness in Nash equilibrium. The optimal arbitrage can be decomposed and generated using the idea of functionally generated portfolios. In this way, the constraints on relative return and investment time horizon can be specified.

The second part of the thesis studies numerical aspects of solving high dimensional PDEs with multiple solutions, and learning relative arbitrage opportunities. A grid based solution for relative arbitrage is derived in volatility stabilized market models. We then study deep learning schemes for non-unique solutions of PDEs. Experiments on solving the non-negative minimal solution of a Cauchy problem is provided.