the theory of asymptotic capital growth in discrete and continuous time, respectively. Thorp (cf. his 2017 biography) demonstrated the practical effectiveness of the Kelly Criterion when he used it to size his Blackjack bets in certain favorable situations that are identifiable via his trademark (1966) theory of card counting. In this connection, the correct behavior is to bet the fraction $b^*:=p-q$ of your net worth on a given hand for which $p$ is the chance of winning and $q$ is the chance of losing.

For growth opportunities in the stock market, the analog of Kelly’s fixed fraction betting scheme is a certain constant-rebalanced portfolio $b^*$ that trades continuously so as to maintain a target growth-optimal fraction of wealth in each risk asset. For instance, rather than bet $b:=2\%$ of wealth on a (favorable) hand of Blackjack, one could bet $2\%$ of wealth (or even $b:=200\%$ of wealth) on the S&P $500$ index. In theory, if stock market returns are *iid* across (discrete) time then one can calculate the corresponding log-optimal portfolio directly from the return distribution. But in practice, equity investors must get along without complete knowledge of the return distribution. Thus, a real-world investor cannot measure the exact regret 