Kelly Betting Can Be Too Conservative

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

Rebalancing Frequency Considerations for Kelly-Optimal Stock Portfolios in a Control-Theoretic Framework

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, 201

The Impact of Execution Delay on Kelly-Based Stock Trading: High-Frequency Versus Buy and Hold

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, 201

Avoidance Control on Time Scales

We consider dynamic systems on time scales under the control of two agents. One of the agents desires to keep the state of the system out of a given set regardless of the other agent's actions. Leitmann's avoidance conditions are proved to be valid for dynamic systems evolving on an arbitrary time scale.Comment: Revised edition in JOTA format. To appear in J. Optim. Theory Appl. 145 (2010), no. 3. In Pres

Investigation of robust stability for fractional-order LTI systems with multilinear structure of ellipsoidal parametric uncertainty

The contribution focuses on the investigation of robust stability for fractional-order linear time-invariant (LTI) systems with the multilinear structure of ellipsoidal parametric uncertainty, i.e., the analyzed family of fractional-order polynomials has the multilinear uncertainty structure and an ellipsoid-shaped uncertainty bounding set. The robust stability test is based on the numerical calculation and subsequent plot of the value sets, and the application of the zero exclusion condition. Unlike the previously published works, this contribution shows that, contrary to the case of a two-dimensional ellipse of parameters, the internal points of a three-dimensional ellipsoid of parameters cannot create the boundary of the value set in the complex plane even under more complicated uncertainty structures, such as the multilinear one. © 2020, The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

Predictors of Change Following Cognitive-Behavioral Treatment of Children with Anxiety Problems: A Preliminary Investigation on Negative Automatic Thoughts and Anxiety Control

The purpose of the present study was to evaluate negative automatic thoughts and anxiety control as predictors of change produced by cognitive-behavioral treatment of youths with anxiety disorders. Forty-five high-anxious children aged between 9 and 12 years who were selected from the primary school population, received a standardized CBT intervention that was provided in a group format. Before and after the intervention, children completed scales of negative automatic thoughts and perceived control over anxiety-related events as well as a questionnaire for measuring DSM-defined anxiety disorders symptoms, which was the outcome measure. Results indicated that CBT was effective in reducing children’s anxiety symptoms. Most importantly, the reduction of anxiety disorders symptoms was significantly associated with a decrease in negative automatic thoughts and an increase of anxiety control, which provides support for the notion that these variables are candidate mediators of CBT in anxious youths

The direction of effects between perceived parental behavioral control and psychological control and adolescents’ self-reported GAD and SAD symptoms

This study examined the direction of effects and age and sex differences between adolescents’ perceptions of parental behavioral and psychological control and adolescents’ self-reports of generalized anxiety disorder (GAD) and separation anxiety disorder (SAD) symptoms. The study focused on 1,313 Dutch adolescents (early-to-middle cohort n = 923, 70.3%; middle-to-late cohort n = 390, 29.7%) from the general population. A multi-group, structural equation model was employed to analyze the direction of the effects between behavioral control, psychological control and GAD and SAD symptoms for the adolescent cohorts. The current study demonstrated that a unidirectional child effect model of the adolescents’ GAD and SAD symptoms predicting parental control best described the data. Additionally, adolescent GAD and SAD symptoms were stronger and more systematically related to psychological control than to behavioral control. With regard to age–sex differences, anxiety symptoms almost systematically predicted parental control over time for the early adolescent boys, whereas no significant differences were found between the late adolescent boys and girls

Cognitive Behavior Therapy for Anxious Adolescents: Developmental Influences on Treatment Design and Delivery

Anxiety disorders in adolescence are common and disruptive, pointing to a need for effective treatments for this age group. Cognitive behavior therapy (CBT) is one of the most popular interventions for adolescent anxiety, and there is empirical support for its application. However, a significant proportion of adolescent clients continue to report anxiety symptoms post-treatment. This paper underscores the need to attend to the unique developmental characteristics of the adolescent period when designing and delivering treatment, in an effort to enhance treatment effectiveness. Informed by the literature from developmental psychology, developmental psychopathology, and clinical child and adolescent psychology, we review the ‘why’ and the ‘how’ of developmentally appropriate CBT for anxious adolescents. ‘Why’ it is important to consider developmental factors in designing and delivering CBT for anxious adolescents is addressed by examining the age-related findings of treatment outcome studies and exploring the influence of developmental factors, including cognitive capacities, on engagement in CBT. ‘How’ clinicians can developmentally tailor CBT for anxious adolescents in six key domains of treatment design and delivery is illustrated with suggestions drawn from both clinically and research-oriented literature. Finally, recommendations are made for research into developmentally appropriate CBT for anxious adolescents