Game-theoretic optimal portfolios in continuous time

We consider a two-person trading game in continuous time where each player chooses a constant rebalancing rule b that he must adhere to over [0, t]. If Vt(b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_t(b)$$\end{document} denotes the final wealth of the rebalancing rule b, then Player 1 (the “numerator player”) picks b so as to maximize E[Vt(b)/Vt(c)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[V_t(b)/V_t(c)]$$\end{document}, 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∗=Σ-1(μ-r1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^*=c^*=\varSigma ^{-1}(\mu -r\mathbf 1 )$$\end{document}, where Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varSigma $$\end{document} is the covariance of instantaneous returns per unit time, μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is the drift vector, and 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf 1 $$\end{document} 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 ( Manag Sci 34(6):724–733, 1988) result in discrete time.