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 ϕ -game”, namely the zero-sum game with payoff kernel E [ ϕ { W 1 V t ( b ) / ( W 2 V t ( c ) ) } ] , where 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 ϕ ( • ) 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.
Highlights
We show that the unique saddle point is for both players to use the Kelly rule for jump diffusions, which is ordinarily defined by maximizing the asymptotic almost-sure continuously compounded capital growth rate
Cover [1,3], we solved a leveraged “investment φ-game” where the object is to outperform the other investor with respect to some more or less arbitrary criterion φ() of relative performance
We showed that the unique saddle point of the expected final wealth ratio is for both players to use the Kelly rule for jump diffusions, in conjunction with appropriate fair randomizations that are completely determined by the criterion φ()
Summary
Bell and Cover [1] studied a static, zero-sum competitive investment game whose payoff kernel is the probability that Player 1 has more wealth than Player 2 after a single period’s fluctuation of the stock market They found it necessary to introduce the device of “fair randomization” of the initial dollar for a random capital whose mean is at most 1. Garivaltis [5] wanted to see whether or not Bell and Cover’s results hold up in the context of a stochastic differential investment φ-game for several Itô processes with state-dependent drift and diffusion They do; the correct behavior is to use Bell and Cover’s [3] randomizations Wi [φ] in conjunction with the feedback control policy b(S, t) := Σ−1 [μ(S, t) − r1], which is the local version of the multivariate Kelly rule in continuous time. Μ(S, t) is the local drift vector, Σ(S, t) is the local covariance matrix of instantaneous returns per unit time, and 1 is a vector of ones
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