On the Probability of Correct Selection in Ordinal Comparison over Dynamic Networks

We consider distributed ordinal comparison of selecting the best option which maximizes the average of local reward function values among available options in a dynamic network. Each node in the network knows only his reward function, and edge-connectivity across the nodes changes over time by Calafiore??s model. To estimate each option??s global reward function value, local samples for each option are generated at each node and those are iteratively mixed over the network by a weighted average of local estimates of instantaneous neighbors. Each node selects an option that achieves the maximum of the current global estimates as an estimate of the best option. We establish a lower bound on the probability of correct local-selection at any node, which uniformly converges over the nodes to a lower bound on the probability of correct global-selection by a virtual centralized node with the total available samples.     

Accusation probabilities in Tardos codes: beyond the Gaussian approximation

We study the probability distribution of user accusations in the q-ary Tardos fingerprinting system under the Marking Assumption, in the restricted digit model. In particular, we look at the applicability of the so-called Gaussian approximation, which states that accusation probabilities tend to the normal distribution when the fingerprinting code is long. We introduce a novel parametrization of the attack strategy which enables a significant speedup of numerical evaluations. We set up a method, based on power series expansions, to systematically compute the probability of accusing innocent users. The ‘small parameter’ in the power series is 1/m, where m is the code length. We use our method to semi-analytically study the performance of the Tardos code against majority voting and interleaving attacks. The bias function ‘shape’ parameter k{{\kappa}} strongly influences the distance between the actual probabilities and the asymptotic Gaussian curve. The impact on the collusion-resilience of the code is shown. For some realistic parameter values, the false accusation probability is even lower than the Gaussian approximation predicts.     

Asymptotic properties of digit sequences of random numbers

We consider several aspects of the relationship between a [0, 1]‐valued random variable X and the random sequence of digits given by its m‐ary expansion. We present results for three cases: (a) independent and identically distributed digit sequences; (b) random variables X with smooth densities; (c) stationary digit sequences. In the case of i.i.d. an integral limit thorem is proved which applies for example to relative frequencies, yielding asymptotic moment identities. We deal with occurrence probabilities of digit groups in the case that X has an analytic Lebesgue density. In the case of stationary digits we determine the distribution of X in terms of their transition functions. We study an associated [0, 1]‐valued Markov chain, in particular its ergodicity, and give conditions for the existence of stationary digit sequences with prespecified transition functions. It is shown that all probability measures induced on [0, 1] by such sequences are purely singular except for the uniform distribution. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fuzzy play,matching devices and coordination failures

We revisit n-player coordination games with Pareto-ranked Nash equilibria. As a novelty, we introduce fuzzy play and a matching device. By fuzzy play we mean that each player does not choose which pure strategy to play, but instead chooses a nonempty subset of his strategy set that he submits to the matching device. The matching device is a very simple one. It randomly selects a match if possible, and it selects randomly some strategy belonging to the strategy set sent by each player otherwise. That is, it does not impose that the best alternatives are matched. Using the concepts of perfect Nash equilibrium and of trembling-hand perfect rationalizability, we show that players coordinate directly on the Pareto optimal outcome. This implies that they neither use the option of fuzzy play, nor make use of the matching device.We thank an anonymous referee and an Associate Editor for valuable comments. Jean-Jacques Herings would like to thank the Netherlands Organisation for Scientific Research (NWO) for financial support. Vincent Vannetelbosch is Chercheur Qualifié at the Fonds National de la Recherche Scientifique. The research of Ana Mauleon has been made possible by a fellowship of the Fonds Européen du Développement Economique Régional (FEDER). Financial support from the Belgian French Communitys program Action de Recherches Concertée 99/04-235 (IRES, Université catholique de Louvain) is gratefully acknowledged.     

Positional number systems with digits forming an arithmetic progression

A novel digit system that arises in a natural way in a graph-theoretical problem is studied. It is defined by a set of positive digits forming an arithmetic progression and, necessarily, a complete residue system modulo the base b. Since this is not enough to guarantee existence of a digital representation, the most significant digit is allowed to come from an extended set. We provide explicit formul? for the j th digit in such a representation as well as for the length. Furthermore, we study digit frequencies and average lengths, thus generalising classical results for the base-b representation. For this purpose, an appropriately adapted form of the Mellin-Perron approach is employed.     

A two-sample procedure for selecting the population with the largest mean fromk normal populations

Summary A two-sample procedure is given for selecting the population with the largest mean fromk normal populations with known variances. Thek populations are screened through the first sample for possible elimination of those populations which have considerably smaller means. The second sample is drawn from the non-eliminated populations only. The two-sample procedure is compared with the fixed sample procedure of Bechhofer [2]. The ratio of the expected total number of observations required in the first case to the total number of observations required in the second case for the same level of the probability of a correct selection measures the relative advantage of experimenting in two stages. Fork=2 it is shown that the ratio is bounded below by 1/4. Fork=2 the probability of a correct selection can be computed from the tables of the bivariate normal distribution function. Fork≧3 a lower bound on the probability of a correct selection is derived which can be computed with the help of available tables. An upper bound is also given for the expected sample size. The author's work was partly supported by the National Science Foundation Grant No. GP-7496 at Indiana University.     

A graphical method for valuing switching options

This paper introduces a graphical method for valuing options on real asset investments that allow the investor to switch between different operating modes at a single point-in-time. The technique uses mixtures of truncated exponential functions to approximate both the probability density function for project value and the expressions for option value of each alternative. The distribution for project value is transformed into an expected cash flow function for the option under each mode of operation. After determining an optimal exercise strategy, these functions are used to determine the option value. The graphical method allows the option exercise strategy to be communicated effectively through a graphical representation of the expected cash flow functions. A comparison of this approach to the existing binomial lattice method is presented. The efficiency of the graphical method is comparable to the binomial lattice and in some cases accurate solutions can be obtained with less CPU time.     

Correlated equilibrium payoffs and public signalling in absorbing games

An absorbing game is a repeated game where some action combinations are absorbing, in the sense that whenever they are played, there is a positive probability that the game terminates, and the players receive some terminal payoff at every future stage.  We prove that every multi-player absorbing game admits a correlated equilibrium payoff. In other words, for every ε>0 there exists a probability distribution p _ε over the space of pure strategy profiles that satisfies the following. With probability at least 1−ε, if a pure strategy profile is chosen according to p _ε and each player is informed of his pure strategy, no player can profit more than ε in any sufficiently long game by deviating from the recommended strategy. Received: April 2001/Revised: June 4, 2002     

The survival value of assuming others to be rational

I study the evolution of rationality, using an indirect evolutionary approach, in which nature selects a decision-making procedure, and the procedure chooses actions in matching-games. The main result is that in order for (knowledge of) rationality to survive, it is necessary and sufficient that the rational procedure respects the attraction principle. That is, when a rational agent eliminates a strictly dominated action A, he should only increase the choice probability of the actions actually dominating A and not change the choice probability of other undominated actions. The attraction principle sharpens gametheoretic predictions. Attraction effects have been verified in psychological experiments. Received: November 1997/Final version: January 2000     

Passport options with stochastic volatility

A passport option is a call option on the profits of a trading account. In this article, the robustness of passport option pricing is investigated by incorporating stochastic volatility. The key feature of a passport option is the holders' optimal strategy. It is known that in the case of exponential Brownian motion the strategy is to be long if the trading account is below zero and short if the account is above zero. Here this result is extended to models with stochastic volatility where the volatility is defined via an autonomous SDE. It is shown that if the Brownian motions driving the underlying asset and the volatility are independent then the form of the optimal strategy remains unchanged. This means that the strategy is robust to misspecification of the underlying model. A second aim of this article is to investigate some of the biases which become apparent in a stochastic volatility regime. Using an analytic approximation, comparisons are obtained for passport option prices using the exponential Brownian motion model and some well-known stochastic volatility models. This is illustrated with numerical examples. One conclusion is that if volatility and price are uncorrelated, then prices are sometimes lower in a model with stochastic volatility than in a model with constant volatility.     

Efficient smoothing of Crank-Nicolson method for pricing barrier options under stochastic volatility

Most of the option pricing problems have nonsmooth payoff. In barrier options certain aspects of the option are triggered if the asset price becomes too high or too low. Standard smoothing schemes used to solve problems with nonsmooth payoff do not work well for the barrier option because a discontinuity is introduced in the time domain each time a barrier is applied. An improved smoothing strategy is introduced for smoothing the A -stable Cranck-Nicolson scheme at each time when a barrier is applied. A partial differential equation (PDE) approach is utilized for the evaluation of complex option pricing models under stochastic volatility which brings major mathematical and computational challenges for estimation and stability of the estimates. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quadratic Hedging Methods for Defaultable Claims

We apply the local risk-minimization approach to defaultable claims and we compare it with intensity-based evaluation formulas and the mean-variance hedging. We solve analytically the problem of finding respectively the hedging strategy and the associated portfolio for the three methods in the case of a default put option with random recovery at maturity.     

Prime numbers and the first digit phenomenon

The set of primes which have lead digit 1 does not have relative natural density in the prime numbers. However, Bombieri has shown that this set does have relative Zeta density equal to log₁₀ 2. This means that a prime chosen at random (w.r.t. the Zeta distribution) will have lead digit 1 with the determined probability. Here the question, Is this a special property of Zeta density or a more universal property of primes? is answered. It is shown that for any generalization of relative natural density (obeying a few basic assumptions) if a value is assigned to the relative density of primes of lead digit 1 then this value is always log₁₀ 2. Another density which does converge on this set is also exhibited. Additionally the relative densities of primes beginning with any specified string of digits are found.     

On the indifference zone approach to selection—A consistency result

Summary The problem of selecting thet best amongk populations is considered. The concept of ψ-correct selection is defined and it is shown that the indifference zone approach is consistent in the following sense. The minimum probability of ψ-correct selection over the entire parameter space is no less than the minimum probability of correct selection over the preference zone.     

An integrated formulation for selecting the most probable multinomial cell

We refer to the two classical approaches to multinomial selection as the indifference zone approach and the subset selection approach. This paper integrates these two approaches by separating the parameter space into two disjoint subspaces: the preference zone (PZ) and the indifference zone (IZ). In the PZ we insist on selecting the best (most probable) cell for a correct selection (CS ₁) but in the IZ we define any selected subset to be correct (CS ₂) if it contains the best cell. We then propose a single stage procedure R to achieve the selection goals CS ₁ and CS ₂ simultaneously with certain probability requirements. It is shown that both the probability of a correct selection under IZ, P(CS ₂ |PZ), and the probability of a correct selection under IZ, P(CS ₂ |IZ), satisfy some monotonicity properties and the least favorable configuration in PZ and the worst configuration in IZ can be found by these properties.     

Bayesian Inference for Extremes: Accounting for the Three Extremal Types

The Extremal Types Theorem identifies three distinct types of extremal behaviour. Two different strategies for statistical inference for extreme values have been developed to exploit this asymptotic representation. One strategy uses a model for which the three types are combined into a unified parametric family with the shape parameter of the family determining the type: positive (Fréchet), zero (Gumbel), and negative (negative Weibull). This form of approach never selects the Gumbel type as that type is reduced to a single point in a continuous parameter space. The other strategy first selects the extremal type, based on hypothesis tests, and then estimates the best fitting model within the selected type. Such approaches ignore the uncertainty of the choice of extremal type on the subsequent inference. We overcome these deficiencies by applying the Bayesian inferential framework to an extended model which explicitly allocates a non-zero probability to the Gumbel type. Application of our procedure suggests that the effect of incorporating the knowledge of the Extremal Types Theorem into the inference for extreme values is to reduce uncertainty, with the degree of reduction depending on the shape parameter of the true extremal distribution and the prior weight given to the Gumbel type.     

Confidence sets for discrete stochastic optimization

Consider the problem of finding the points of maximum of an expectation functional over a finite setS. Based on statistical estimates at each point ofS, confidence sets for theargmax-set are constructed which guarantee a prespecified probability of correct selection. We review known selection methods and propose a new two-stage procedure that works well for largeS and few global maxima. The performance is compared in a simulation study.     

Analysis of the Herlestam and Johannesson discrete logarithm scheme inGF(2 N ) for largeN

The Herlestam and Johannesson algorithm for computing discrete logarithms inGF(2 ⁿ ) requires the precomputation of logarithms for a target set consisting of all field elements of Hamming weight less than some predetermined value. The procedure, both in precomputation and at run-time, selects elements of lowest weight from large sets of elements. These sets are not randomly chosen but their minimum weight statistics parallel those for sets of equal size chosen entirely at random. By analyzing the statistics for randomly chosen sets, we show that the target set must contain all elements up to about weightn/3–8. This is clearly impractical for even moderately large values ofn.This work was supported by MITRE Corp. IR & D funds.Dr. Berkovits was on leave from the University of Lowell, Lowell, MA, 01854.     

Optimal investment to minimize the probability of drawdown

We determine the optimal investment strategy in a Black–Scholes financial market to minimize the so-called probability of drawdown, namely, the probability that the value of an investment portfolio reaches some fixed proportion of its maximum value to date. We assume that the portfolio is subject to a payout that is a deterministic function of its value, as might be the case for an endowment fund paying at a specified rate, for example, at a constant rate or at a rate that is proportional to the fund’s value.     

支付连续红利的欧式和美式期权定价问题的研究

本文从投资策略的角度出发,针对支付连续红利欧式和美式期权,通过构造等价鞅测度,进而构造出最小保值策略即复制策略,由此得到相应的期权的一般定价公式,并在此基础上运用概率求期望和方程代换这两种方法推导出带红利标准欧式看涨期权的定价B-S公式.