# Game-Theoretic Optimal Portfolios in Continuous Time

## TL;DR

This paper models a continuous-time trading game where two players choose rebalancing rules, revealing that the Nash equilibrium involves both adopting the Kelly rule, aligning with discrete-time results.

## Contribution

It demonstrates that in a continuous-time game, the Nash equilibrium involves both players using the Kelly rule, extending discrete-time game results to continuous time.

## Key findings

- Nash equilibrium involves both players using Kelly rule.
- The Kelly rule is optimal even over short time intervals.
- Results align with Bell and Cover's discrete-time findings.

## Abstract

We consider a two-person trading game in continuous time whereby each player chooses a constant rebalancing rule $b$ that he must adhere to over $[0,t]$. If $V_t(b)$ denotes the final wealth of the rebalancing rule $b$, then Player 1 (the `numerator player') picks $b$ so as to maximize $\mathbb{E}[V_t(b)/V_t(c)]$, while Player 2 (the `denominator player') picks $c$ so as to minimize it. In the unique Nash equilibrium, both players use the continuous-time Kelly rule $b^*=c^*=\Sigma^{-1}(\mu-r\textbf{1})$, where $\Sigma$ is the covariance of instantaneous returns per unit time, $\mu$ is the drift vector of the stock market, and $\textbf{1}$ is a vector of ones. Thus, even over very short intervals of time $[0,t]$, the desire to perform well relative to other traders leads one to adopt the Kelly rule, which is ordinarily derived by maximizing the asymptotic exponential growth rate of wealth. Hence, we find agreement with Bell and Cover's (1988) result in discrete time.