For sequential betting games, Kelly’s theory, aimed at maximization of the logarithmic growth of one’s account value, involves optimization of the so-called betting fraction $K$. In this paper, we extend the classical formulation to allow for temporal correlation among bets. For the example of successive coin flips with even-money payoff, used throughout the paper as an example demonstrating the potential for this new research direction, we formulate and solve a problem with *memory depth* $m$. By this, we mean that the outcomes of coin flips are no longer assumed to be i.i.d. random variables. Instead, the probability of heads on flip $k$ depends on previous flips $k-1,k-2,…,k-m$. For the simplest case of $n$ flips, even-money payoffs and~$m = 1$, we obtain a closed form solution for the optimal betting fraction, denoted $K_n$, which generalizes the classical result for the memoryless case. That is, instead of fraction $K = 2p-1$ which pervades the literature for a coin with probability of heads $p \ge 1/2$, our new fraction $K_n$ depends on both $n$ and the parameters associated with the temporal correlation model. Subsequently, we obtain a generalization of these results for cases when $m > 1$ and illustrate the use of the theory in the context of numerical simulations.

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Publication

IEEE Control Systems Letters