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

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

Basic Properties of Power Series

Abstract These notes provide a quick introduction (with proofs) to the basic properties of power series, including the exponential function and the fact that power series can be differentiated term by term. It is assumed that the reader is familiar with the following facts and concepts from analysis [4]: • The triangle inequality [4, pp. 14-15, Theorem 1.13 and p. 23, Problem 13]: For complex a and b, |a| − |b| ≤ |a + b| ≤ |a| + |b|. • The binomial theorem: For complex a and b

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

Upper Bounds for the Average Error Probability of a Time-Hopping Wideband System

Abstract Ultra-wideband technology has been proposed as a viable solution for highspeed indoor short-range wireless communication systems because of its robustness to severe multipath and multi-user conditions, low cost, and low power implementation. Time-hopping combined with pulse position modulation was the original proposal for ultra-wideband systems. This paper proposes and analyzes a multiuser time-hopping system in which symbols are sent multiple times. A single-user receiver structure is proposed in which collisions with interfering users are discarded. Two formulas for the probability of error are derived. The first formula is a finite sum in which the number of terms grows with the number of symbol repetitions. The second formula is an integral that is well-suited to Chebyshev-Gauss quadrature as well as for allowing the derivation of bounds on the probability of error. Asymptotic formulas and upper bounds for the error probability are derived by letting various system parameters go to infinity. We evaluate the bounds numerically for some reasonable parameters and study their interplay with other parameters of interest

Impulsivity in disorders of food and drug misuse.

BACKGROUND: Evidence suggests some overlap between the pathological use of food and drugs, yet how impulsivity compares across these different clinical disorders remains unclear. Substance use disorders are commonly characterized by elevated impulsivity, and impulsivity subtypes may show commonalities and differences in various conditions. We hypothesized that obese subjects with binge-eating disorder (BED) and abstinent alcohol-dependent cohorts would have relatively more impulsive profiles compared to obese subjects without BED. We also predicted decision impulsivity impairment in obesity with and without BED. METHOD: Thirty obese subjects with BED, 30 without BED and 30 abstinent alcohol-dependent subjects and age- and gender-matched controls were tested on delay discounting (preference for a smaller immediate reward over a larger delayed reward), reflection impulsivity (rapid decision making prior to evidence accumulation) and motor response inhibition (action cancellation of a prepotent response). RESULTS: All three groups had greater delay discounting relative to healthy volunteers. Both obese subjects without BED and alcohol-dependent subjects had impaired motor response inhibition. Only obese subjects without BED had impaired integration of available information to optimize outcomes over later trials with a cost condition. CONCLUSIONS: Delay discounting appears to be a common core impairment across disorders of food and drug intake. Unexpectedly, obese subjects without BED showed greater impulsivity than obese subjects with BED. We highlight the dissociability and heterogeneity of impulsivity subtypes and add to the understanding of neurocognitive profiles across disorders involving food and drugs. Our results have therapeutic implications suggesting that disorder-specific patterns of impulsivity could be targeted.V.V. is a Wellcome Trust Intermediate Fellow in Clinical Neurosciences (Wellcome Trust grant no. WT093705MA). Y.W. is supported by the Fyssen Fondation. N.A.H. is a Wellcome Trust Intermediate Fellow in Clinical Neurosciences. The BCNI is supported by both the Wellcome Trust and Medical Research Council.This is the accepted manuscript. The final version is available from CUP at http://dx.doi.org/10.1017/S003329171400183

MEMS Deformable Mirrors for Space-Based High-Contrast Imaging

Micro-Electro-Mechanical Systems (MEMS) Deformable Mirrors (DMs) enable precise wavefront control for optical systems. This technology can be used to meet the extreme wavefront control requirements for high contrast imaging of exoplanets with coronagraph instruments. MEMS DM technology is being demonstrated and developed in preparation for future exoplanet high contrast imaging space telescopes, including the Wide Field Infrared Survey Telescope (WFIRST) mission which supported the development of a 2040 actuator MEMS DM. In this paper, we discuss ground testing results and several projects which demonstrate the operation of MEMS DMs in the space environment. The missions include the Planet Imaging Concept Testbed Using a Recoverable Experiment (PICTURE) sounding rocket (launched 2011), the Planet Imaging Coronagraphic Technology Using a Reconfigurable Experimental Base (PICTURE-B) sounding rocket (launched 2015), the Planetary Imaging Concept Testbed Using a Recoverable Experiment - Coronagraph (PICTURE-C) high altitude balloon (expected launch 2019), the High Contrast Imaging Balloon System (HiCIBaS) high altitude balloon (launched 2018), and the Deformable Mirror Demonstration Mission (DeMi) CubeSat mission (expected launch late 2019). We summarize results from the previously flown missions and objectives for the missions that are next on the pad. PICTURE had technical difficulties with the sounding rocket telemetry system. PICTURE-B demonstrated functionality at >100 km altitude after the payload experienced 12-g RMS (Vehicle Level 2) test and sounding rocket launch loads. The PICTURE-C balloon aims to demonstrate 10(-7) contrast using a vector vortex coronagraph, image plane wavefront sensor, and a 952 actuator MEMS DM. The HiClBaS flight experienced a DM cabling issue, but the 37-segment hexagonal piston-tip-tilt DM is operational post-flight. The DeMi mission aims to demonstrate wavefront control to a precision of less than 100 nm RMS in space with a 140 actuator MEMS DM.DARPA; NASA Space Technology Research FellowshipOpen Access JournalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]