We study the problem of optimizing the betting frequency in a dynamic game setting using Kelly's celebrated expected logarithmic growth criterion as the performance metric. The game is defined by a sequence of bets with independent and identically distributed returns X(k). The bettor selects the fraction of wealth K wagered at k = 0 and waits n steps before updating the bet size. Between updates, the proceeds from the previous bets remain at risk in the spirit of "buy and hold." Within this context, the main questions we consider are as follows: How does the optimal performance, we call it g * n , change with n? Does the highfrequency case, n = 1, always lead to the best performance? What are the effects of accrued interest and transaction costs? First, we provide rather complete answers to these questions for the important special case when X(k) ∈ {−1, 1} is a Bernoulli random variable with probability p that X(k) = 1. This serves as an entry point for future research using a binomial lattice model for stock trading. The latter sections focus on more general probability distributions for X(k) and two conjectures. The first conjecture is simple to state: Absent transaction costs, g * n is non-increasing in n. The second conjecture involves the technical condition which we call the sufficient attractiveness inequality. We first prove that satisfaction of this inequality is sufficient to guarantee that the low-frequency bettor using large n can match the performance of the high-frequency bettor using n = 1. Subsequently, we conjecture, and provide supporting evidence that this condition is also necessary.

Kelly betting is a prescription for optimal resource allocation among a set of gambles which are typically repeated in an independent and identically distributed manner. In this setting, there is a large body of literature which includes arguments that the theory often leads to bets which are "too aggressive" with respect to various risk metrics. To remedy this problem, many papers include prescriptions for scaling down the bet size. Such schemes are referred to as Fractional Kelly Betting. In this paper, we take the opposite tack. That is, we show that in many cases, the theoretical Kelly-based results may lead to bets which are "too conservative" rather than too aggressive. To make this argument, we consider a random vector X with its assumed probability distribution and draw m samples to obtain an empirically-derived counterpartX. Subsequently, we derive and compare the resulting Kelly bets for both X andX with consideration of sample size m as part of the analysis. This leads to identification of many cases which have the following salient feature: The resulting bet size using the true theoretical distribution for X is much smaller than that forX. If instead the bet is based on empirical data, "golden" opportunities are identified which are essentially rejected when the purely theoretical model is used. To formalize these ideas, we provide a result which we call the Restricted Betting Theorem. An extreme case of the theorem is obtained when X has unbounded support. In this situation, using X, the Kelly theory can lead to no betting at all.

Control of drawdown, that is, the control of the drops in wealth over time from peaks to subsequent lows, is of great concern from a risk management perspective. With this motivation in mind, the focal point of this paper is to address the drawdown issue in a stock trading context. Although our analysis can be carried out without reference to control theory, to make the work accessible to this community, we use the language of feedback systems. The takeoff point for the results to follow, which we call the Drawdown Modulation Lemma, characterizes any investment which guarantees that the percentage drawdown is no greater than a prespecified level with probability one. With the aid of this lemma, we introduce a new scheme which we call the drawdown-modulated feedback control. To illustrate the power of the theory, we consider a drawdown-constrained version of the well-known Kelly Optimization Problem which involves maximizing the expected logarithmic growth of the trader's account value. As the drawdown parameter dmax in our new formulation tends to one, we recover existing results as a special case. This new theory leads to an optimal investment strategy whose application is illustrated via an example with historical stock-price data.Comment: Proc. 20th IFAC World Congress (IFAC WC 2017), Toulouse, France, July 9-14, 2017 (accepted

In this paper, motivated by the celebrated work of Kelly, we consider the problem of portfolio weight selection to maximize expected logarithmic growth. Going beyond existing literature, our focal point here is the rebalancing frequency which we include as an additional parameter in our analysis. The problem is first set in a control-theoretic framework, and then, the main question we address is as follows: In the absence of transaction costs, does high-frequency trading always lead to the best performance? Related to this is our prior work on betting, also in the Kelly context, which examines the impact of making a wager and letting it ride. Our results on betting frequency can be interpreted in the context of weight selection for a two-asset portfolio consisting of one risky asset and one riskless asset. With regard to the question above, our prior results indicate that it is often the case that there are no performance benefits associated with high-frequency trading. In the present paper, we generalize the analysis to portfolios with multiple risky assets. We show that if there is an asset satisfying a new condition which we call dominance, then an optimal portfolio consists of this asset alone; i.e., the trader has "all eggs in one basket" and performance becomes a constant function of rebalancing frequency. Said another way, the problem of rebalancing is rendered moot. The paper also includes simulations which address practical considerations associated with real stock prices and the dominant asset condition.

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