Game-Theoretic Optimal Portfolios for Jump Diffusions

Abstract

This paper studies a two-person trading game in continuous time that generalizes Garivaltis (2018) to allow for stock prices that both jump and diffuse. Analogous to Bell and Cover (1988) in discrete time, the players start by choosing fair randomizations of the initial dollar, by exchanging it for a random wealth whose mean is at most 1. Each player then deposits the resulting capital into some continuously-rebalanced portfolio that must be adhered to over $[0,t]$. We solve the corresponding `investment $\phi$-game,' namely the zero-sum game with payoff kernel $\mathbb{E}[\phi\{\textbf{W}_1V_t(b)/(\textbf{W}_2V_t(c))\}]$, where $\textbf{W}_i$ is player $i$'s fair randomization, $V_t(b)$ is the final wealth that accrues to a one dollar deposit into the rebalancing rule $b$, and $\phi(\bullet)$ is any increasing function meant to measure relative performance. We show that the unique saddle point is for both players to use the (leveraged) Kelly rule for jump diffusions, which is ordinarily defined by maximizing the asymptotic almost-sure continuously-compounded capital growth rate. Thus, the Kelly rule for jump diffusions is the correct behavior for practically anybody who wants to outperform other traders (on any time frame) with respect to practically any measure of relative performance.