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

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

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

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

Deterministic Codes for Arbitrarily Varying Multiple-Access Channels.

The arbitrarily varying multiple-access channel (AVMAC) is a model of a multiple access channel with unknown parameters. In 1981, Jahn characterized the capacity region of the AVMAC, assuming that the region had a nonempty interior; however, he did not address the problem of deciding whether or not the capacity region had a nonempty interior. Using the method of types and an approach completely different from Jahn's, we have partially solved this problem. We begin by introducing the simple but crucial notion of symmetrizability for the two-user AVMAC. We show that if an AVMAC is symmetrizable, then its capacity region has an empty interior. For the two-user AVMAC, this means that at least one (and perhaps both) users cannot reliably transmit information across the channel. More importantly, we show that if the channel is suitably nonsymmetrizable, then the capacity region has a nonempty interior, and both users can reliably transmit information across the channel. In light of these results, it is indeed fortunate that to test a channel for symmetrizability, one simply solves a system of linear equations whose coefficients are the channel transition probabilities. Our proofs rely heavily on a rather complicated decoding rule. This leads us to seek conditions under which simpler multiple-message decoding techniques might suffice. In particular, we give conditions under which the universal mazimum mutual informatzon decoding rule will be effective. We then consider the situation in which a constraint is imposed on the sequence of "states" in which the channel can reside. We extend our approach to show that in the presence of a state constraint, the capacity region can increase dramatically. A striking example of this effect occurs with the adder channel . This channel is symmetrizable, and without a state constraint, neither user can reliably transmit information across the channel. However, if a suitable state constraint is imposed, each user can reliably transmit more than 0.4 bits of information per channel use

On the capacity region of the discrete additive multiple-access arbitrarily varying channel

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Distributed estimation and quantization

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On the Deterministic-Code Capacity of the Multiple-Access Arbitrarily Varying Channel.

The capacity region of the multiple-access arbitrarily varying channel (AVC) was characterized by Jahn, assuming that the region had a nonempty interior; however, he did not indicate how one could decide whether or not the capacity region had a nonempty interior. Using the method of types and an approach different from Jahn's, we have partially solved this problem. We begin by considering the notion of symmetrizability for the two-user AVC as an extension of the same notion for the single-user AVC. We show that if a multiple-access AVC is symmetrizable, then its capacity region has an empty interior. For the two-user AVC, this means that at least one (and perhaps both) users cannot reliably transmit information across the channel. More importantly, we show that if the channel is suitably nonsymmetrizable, then the capacity region has a nonempty interior, and both users can reliably transmit information across the channel. Our proofs rely heavily on a rather complicated decoding rule. This leads us to seek conditions under which simpler multiple-message decoding techniques might suffice. In particular, we give conditions under which the universal mazimum mutual information decoding rule will be effective