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The takeoff point for this paper is the voluminous body of literature addressing recursive betting games with expected logarithmic growth of wealth being the performance criterion. Whereas almost all existing papers involve use of linear feedback, the use of nonlinear control is conspicuously absent. This is epitomized by the large subset of this literature dealing with Kelly Betting. With this as the high-level motivation, we study the potential for use of nonlinear control in this framework. To this end, we consider a ``demonstration case'' which is one of the simplest scenarios encountered in this line of research: repeated flips of a biased coin with probability of heads~$p$, and even-money payoff on each flip. First, we formulate a new robust nonlinear control problem which we believe is both simple to understand and apropos for dealing with concerns about distributional robustness; i.e., instead of assuming that~$p$ is perfectly known as in the case of the classical Kelly formulation, we begin with a bounding set ~${\cal P} \subseteq [0,1]$ for this probability. Then, we provide a theorem, our main result, which gives a closed-form description of the optimal robust nonlinear controller and a corollary which establishes that it robustly outperforms linear controllers such as those found in the literature. A second, less significant, contribution of this paper bears upon the computability of our solution. For an $n$-flip game, whereas an admissible controller has~$2^n-1$ parameters, at the optimum only~$O(n^2)$ of them turn out to be distinct. Finally, it is noted that the initial assumptions on payoffs and the use of the uniform distribution on~$p$ are made solely for simplicity of the exposition and compliance with length requirements for a Letter. Accordingly, the paper also includes a new section with a discussion indicating how these assumptions can be relaxed.

The Simultaneous Long-Short(SLS) controller for trading a single stock is known to guarantee positive expected value of the resulting gain-loss function with respect to a large class of stock price dynamics. In the literature, this is known as the Robust Positive Expectation(RPE)property. An obvious way to extend this theory to the trading of two stocks is to trade each one of them using its own independent SLS controller. Motivated by the fact that such a scheme does not exploit any correlation between the two stocks, we study the case when the relative sign between the drifts of the two stocks is known. The main contributions of this paper are three-fold: First, we put forward a novel architecture in which we cross-couple two SLS controllers for the two-stock case. Second, we derive a closed-form expression for the expected value of the gain-loss function. Third, we use this closed-form expression to prove that the RPE property is guaranteed with respect to a large class of stock-price dynamics. When more information over and above the relative sign is assumed, additional benefits of the new architecture are seen. For example, when bounds or precise values for the means and covariances of the stock returns are included in the model, numerical simulations suggest that our new controller can achieve lower trading risk than a pair of decoupled SLS controllers for the same level of expected trading gain., Comment: Presented at IFAC World Congress, 2020. Will appear in IFAC-PapersOnline

In this paper, we introduce a new control-theoretic paradigm for mitigating the spread of a virus. To this end, our discrete-time controller, aims to reduce the number of new daily deaths, and consequently, the cumulative number of deaths. In contrast to much of the existing literature, we do not rely on a potentially complex virus transmission model whose equations must be customized to the "particulars" of the pandemic at hand. For new viruses such as Covid-19, the epidemiology driving the modelling process may not be well known and model estimation with limited data may be unreliable. With this motivation in mind, the new paradigm described here is data-driven and, to a large extent, we avoid modelling difficulties by concentrating on just two key quantities which are common to pandemics: the doubling time, denoted by $d(k)$ and the peak day denoted by $\theta(k)$. Our numerical studies to date suggest that our appealingly simple model can provide a reasonable fit to real data. Given that time is of the essence during the ongoing global health crisis, the intent of this paper is to introduce this new paradigm to control practitioners and describe a number of new research directions suggested by our current results., Comment: 14 pages, 6 figures

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 Letter, we extend the classical formulation to allow for temporal correlation among bets. To demonstrate the potential of this new paradigm, for simplicity of exposition, we mainly address the case of a coin-flipping game with even-money payoff. To this end, we 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, with $m = 1$, we obtain a closed form solution $K_n$ for the optimal betting fraction. This 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 $p\geq 1/2$, our new fraction $K_n$ depends on both $n$ and the parameters associated with the temporal correlation. Generalizations of these results for $m > 1$ and numerical simulations are also included. Finally, we indicate how the theory extends to time-varying feedback and alternative payoff distributions., Comment: 6 pages, 3 figures. Accepted for publication in IEEE Control Systems Letters

Stock trading based on Kelly's celebrated Expected Logarithmic Growth (ELG) criterion, a well-known prescription for optimal resource allocation, has received considerable attention in the literature. Using ELG as the performance metric, we compare the impact of trade execution delay on the relative performance of high-frequency trading versus buy and hold. While it is intuitively obvious and straightforward to prove that in the presence of sufficiently high transaction costs, buy and hold is the better strategy, is it possible that with no transaction costs, buy and hold can still be the better strategy? When there is no delay in trade execution, we prove a theorem saying that the answer is ``no.'' However, when there is delay in trade execution, we present simulation results using a binary lattice stock model to show that the answer can be ``yes.'' This is seen to be true whether self-financing is imposed or not., Comment: Has been accepted to the IEEE Conference on Decision and Control, 2019

We consider a discrete-time, linear state equation with delay which arises as a model for a trader's account value when buying and selling a risky asset in a financial market. The state equation includes a nonnegative feedback gain $\alpha$ and a sequence $v(k)$ which models asset returns which are within known bounds but otherwise arbitrary. We introduce two thresholds, $\alpha_-$ and $\alpha_+$, depending on these bounds, and prove that for $\alpha < \alpha_-$, state positivity is guaranteed for all time and all asset-return sequences; i.e., bankruptcy is ruled out and positive solutions of the state equation are continuable indefinitely. On the other hand, for $\alpha > \alpha_+$, we show that there is always a sequence of asset returns for which the state fails to be positive for all time; i.e., along this sequence, bankruptcy is certain and the solution of the state equation ceases to be meaningful after some finite time. Finally, this paper also includes a conjecture which says that for the "gap" interval $\alpha_- \leq \alpha \leq \alpha_+,$ state positivity is also guaranteed for all time. Support for the conjecture, both theoretical and computational, is provided., Comment: Accepted to IEEE Transactions on Automatic Control

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., Comment: To appear in the Proceedings of the IEEE Conference on Decision and Control, Miami Beach, FL, 2018

The starting point of this paper is the so-called Robust Positive Expectation (RPE) Theorem, a result which appears in literature in the context of Simultaneous Long-Short stock trading. This theorem states that using a combination of two specially-constructed linear feedback trading controllers, one long and one short, the expected value of the resulting gain-loss function is guaranteed to be robustly positive with respect to a large class of stochastic processes for the stock price. The main result of this paper is a generalization of this theorem. Whereas previous work applies to a single stock, in this paper, we consider a pair of stocks. To this end, we make two assumptions on their expected returns. The first assumption involves price correlation between the two stocks and the second involves a bounded non-zero momentum condition. With known uncertainty bounds on the parameters associated with these assumptions, our new version of the RPE Theorem provides necessary and sufficient conditions on the positive feedback parameter K of the controller under which robust positive expectation is assured. We also demonstrate that our result generalizes the one existing for the single-stock case. Finally, it is noted that our results also can be interpreted in the context of pairs trading., Comment: accepted at the European Control Conference 2018

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 gn*, change with n? Does the high-frequency 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) in {-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, gn* 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., Comment: To appear in Proceedings of the IEEE American Control Conference (ACC), 2018

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 counterpart Xhat. Subsequently, we derive and compare the resulting Kelly bets for both X and Xhat 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 for Xhat. 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., Comment: Accepted in 2016 IEEE 55th Conference on Decision and Control (CDC)

The focal point of this paper is the so-called Kelly Criterion, a prescription for optimal resource allocation among a set of gambles which are repeated over time. The criterion calls for maximization of the expected value of the logarithmic growth of wealth. While significant literature exists providing the rationale for such an optimization, this paper concentrates on the limitations of the Kelly-based theory. To this end, we fill a void in published results by providing specific examples quantifying what difficulties are encountered when Taylor-style approximations are used and when wealth drawdowns are considered. For the case of drawdown, we describe some research directions which we feel are promising for improvement of the theory., Comment: Accepted in 53rd Annual Allerton Conference on Communication, Control, and Computing, 2015

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)

The focal point of this paper is the issue of "drawdown" which arises in recursive betting scenarios and related applications in the stock market. Roughly speaking, drawdown is understood to mean drops in wealth over time from peaks to subsequent lows. Motivated by the fact that this issue is of paramount concern to conservative investors, we dispense with the classical variance as the risk metric and work with drawdown and mean return as the risk-reward pair. In this setting, the main results in this paper address the so-called "efficiency" of linear time-invariant (LTI) investment feedback strategies which correspond to Markowitz-style schemes in the finance literature. Our analysis begins with the following principle which is widely used in finance: Given two investment opportunities, if one of them has higher risk and lower return, it will be deemed to be inefficient or strictly dominated and generally rejected in the marketplace. In this framework, with risk-reward pair as described above, our main result is that classical Markowitz-style strategies are inefficient. To establish this, we use a new investment strategy which involves a time-varying linear feedback block K(k), called the drawdown modulator. Using this instead of the original LTI feedback block K in the Markowitz scheme, the desired domination is obtained. As a bonus, it is also seen that the modulator assures a worst-case level of drawdown protection with probability one., Comment: This paper has been published in Proceedings of 56th IEEE Conference on Decision and Control (CDC) 2017

Pairs trading is a market-neutral strategy that exploits historical correlation between stocks to achieve statistical arbitrage. Existing pairs-trading algorithms in the literature require rather restrictive assumptions on the underlying stochastic stock-price processes and the so-called spread function. In contrast to existing literature, we consider an algorithm for pairs trading which requires less restrictive assumptions than heretofore considered. Since our point of view is control-theoretic in nature, the analysis and results are straightforward to follow by a non-expert in finance. To this end, we describe a general pairs-trading algorithm which allows the user to define a rather arbitrary spread function which is used in a feedback context to modify the investment levels dynamically over time. When this function, in combination with the price process, satisfies a certain mean-reversion condition, we deem the stocks to be a tradeable pair. For such a case, we prove that our control-inspired trading algorithm results in positive expected growth in account value. Finally, we describe tests of our algorithm on historical trading data by fitting stock price pairs to a popular spread function used in literature. Simulation results from these tests demonstrate robust growth while avoiding huge drawdowns., Comment: Accepted at IEEE Multi Systems Conference, 2016