Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the...
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Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. This model was introduced by Benaim, Benjamini, Chen, and Lima. When G is not balanced bipartite, Chen and Lucas proved that the proportion of balls in the bins converges to a point w(G) almost surely. We prove almost sure convergence for balanced bipartite graphs: the possible limit is either a single point w(G) or a closed interval J(G).
This paper introduces a post-iteration averaging algorithm to achieve asymptotic optimality in convergence rates of stochastic approximation algorithms for consensus control with structural constraints. The algorithm ...
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This paper introduces a post-iteration averaging algorithm to achieve asymptotic optimality in convergence rates of stochastic approximation algorithms for consensus control with structural constraints. The algorithm involves two stages. The first stage is a coarse approximation obtained using a sequence of large stepsizes. Then, the second stage provides a refinement by averaging the iterates from the first stage. We show that the new algorithm is asymptotically efficient and gives the optimal convergence rates in the sense of the best scaling factor and ＇smallest＇ possible asymptotic variance.
We apply the stochasticapproximation method to construct a large class of recursive kernel estimators of a distribution function. We study the properties of these estimators and compare them with Nadaraya's distr...
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We apply the stochasticapproximation method to construct a large class of recursive kernel estimators of a distribution function. We study the properties of these estimators and compare them with Nadaraya's distribution estimator. It turns out that, with an adequate choice of the stepsize of the proposed algorithm, the MWISE (Mean Weighted Integrated Squared Error) of the proposed estimator is smaller than that of Nadaraya's estimator. We corroborate these theoretical results by simulations and a real dataset.
We propose a simulation-based algorithm for computing the optimal pricing policy for a product under uncertain demand dynamics. We consider a parameterized stochastic differential equation (SDE) model for the uncertai...
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We propose a simulation-based algorithm for computing the optimal pricing policy for a product under uncertain demand dynamics. We consider a parameterized stochastic differential equation (SDE) model for the uncertain demand dynamics of the product over the planning horizon. In particular, we consider a dynamic model that is an extension of the Bass model. The performance of our algorithm is compared to that of a myopic pricing policy and is shown to give better results. Two significant advantages with our algorithm are as follows: (a) it does not require information on the system model parameters if the SDE system state is known via either a simulation device or real data, and (b) as it works efficiently even for high-dimensional parameters, it uses the efficient smoothed functional gradient estimator.
In this paper, we study theoretical and computational aspects of risk minimization in financial market models operating in discrete time. To define the risk, we consider a class of convex risk measures defined on L-p(...
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In this paper, we study theoretical and computational aspects of risk minimization in financial market models operating in discrete time. To define the risk, we consider a class of convex risk measures defined on L-p(P) in terms of shortfall risk. Under mild assumptions, namely, the absence of arbitrage opportunity and the nondegeneracy of the price process, we prove the existence of an optimal strategy by performing a dynamic programming argument in a non-Markovian framework. In a Markovian framework, the shortfall risk and optimal dynamic strategies are estimated using three main tools: Newton-Raphson Monte Carlo-based procedure, stochastic approximation algorithm, and Markovian quantization scheme. Finally, we illustrate our approach by considering several shortfall risk measures and portfolios inspired by energy and financial markets.
Longitudinal continuous proportional data. is common in many fields such as biomedical research, psychological research and so on, e.g., the percent decrease in glomerular filtration rate at different follow-up times ...
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Longitudinal continuous proportional data. is common in many fields such as biomedical research, psychological research and so on, e.g., the percent decrease in glomerular filtration rate at different follow-up times from the baseline. As shown in Song and Tan [16] such data can be fitted with simplex models. However, the original models of [16] for such longitudinal continuous proportional data assumed a fixed effect for every subject. This paper extends the models of Song and Tan [16] by adding random effects, and proposes simplex distribution nonlinear mixed models which are one kind of nonlinear reproductive dispersion mixed model. By treating random effects in the models as hypothetical missing data and applying the Metropolis-Hastings (M-H) algorithm, this paper develops the stochasticapproximation (SA) algorithm with Markov chain Monte-Carlo (MCMC) method for maximum likelihood estimation in the models. Finally, for ease of comparison, the method is illustrated with the same data from an ophthalmology study on the use of intraocular gas in retinal surgeries in [16].
A stochasticalgorithm for the recursive approximation of the location theta of a maximum of a regression function was introduced by Kiefer and Wolfowitz [Ann. Math. Statist. 23 (1952) 462-466] in the univariate frame...
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A stochasticalgorithm for the recursive approximation of the location theta of a maximum of a regression function was introduced by Kiefer and Wolfowitz [Ann. Math. Statist. 23 (1952) 462-466] in the univariate framework, and by Blum [Ann. Math. Statist. 25 (1954) 737-744] in the multivariate case. The aim of this paper is to provide a companion algorithm to the Kiefer-Wolfowitz-Blum algorithm, which allows one to simultaneously recursively approximate the size p of the maximum of the regression function. A precise study of the joint weak convergence rate of both algorithms is given;it turns out that, unlike the location of the maximum, the size of the maximum can be approximated by an algorithm which converges at the parametric rate. Moreover, averaging leads to an asymptotically efficient algorithm for the approximation of the couple (theta, mu).
Ruppert [Handbook of Sequential Analysis, B. K. Ghosh and P. K. Sen, eds., Marcel Dekker, New York, 1991] and Polyak [Automat. Remote Control, 51 (1990), pp. 937-946] independently introduced the averaging principle f...
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Ruppert [Handbook of Sequential Analysis, B. K. Ghosh and P. K. Sen, eds., Marcel Dekker, New York, 1991] and Polyak [Automat. Remote Control, 51 (1990), pp. 937-946] independently introduced the averaging principle for Robbins-Monro's algorithm in order to construct an algorithm that is asymptotically efficient, and that is quickly updated. Their averaging principle has been extended to other algorithms (including Kiefer-Wolfowitz's algorithm), leading to algorithms that are asymptotically efficient, but whose updatings are less straightforward. We introduce the use of two-time-scale stochastic approximation algorithms to construct a large class of asymptotically efficient algorithms. This method generalizes the averaging principle. Moreover, for the algorithms that do not behave as Robbins-Monro's algorithm, it allows one to provide algorithms whose updating is as straightforward as that of the averaged Robbins-Monro's algorithm.
Designing decision, control and information systems is motivated, in part, by the need to support the deployment of multiple aircraft, such as combat vehicles, unmanned combat air vehicles, unmanned aerial vehicles, a...
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Designing decision, control and information systems is motivated, in part, by the need to support the deployment of multiple aircraft, such as combat vehicles, unmanned combat air vehicles, unmanned aerial vehicles, and weapons, in missions taking place in a dynamic, although uncertain, environment. Such systems aim at ensuring mission success without overloading the operating crew, the pilots, and the commanders. One of the main design challenges lies in obtaining some sort of coherent behaviour of the fleet, by means of solutions to potentially NP-hard problems, given incomplete and imperfect information, and despite limited computational and communication capabilities. In this context, this article proposes a hierarchical decision and information system aiming at providing, in real-time, coordinated aircraft path planning and deceptive engagement assignments. The blue-red engagement policy is obtained by minimizing, and balancing, the energy expenditure among the vehicles while constraining information exchanges to a minimum defined by a risk of inconsistency. The proposed system relies on dynamic programming, online heuristic techniques and stochastic, consistency-checking methods. Numerical simulations show that the proposed approach compares advantageously to a random process and to a law that seeks to minimize the cost of the confrontation at a given time regardless of past moves. However, there is a trade-off between increasing the level of deception and the level of energy consumption.
To enhance efficiency in Monte Carlo simulations, we develop an adaptive stratified sampling algorithm for allocation of sampling effort within each stratum, in which an adaptive variance reduction technique is applie...
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To enhance efficiency in Monte Carlo simulations, we develop an adaptive stratified sampling algorithm for allocation of sampling effort within each stratum, in which an adaptive variance reduction technique is applied. Given the number of replications in each batch, our algorithm updates allocation fractions to minimize the work-normalized variance of the stratified estimator of the mean. We establish the asymptotic normality of the stratified estimator of the mean as the number of batches tends to infinity. Although implementation of the proposed algorithm requires a small amount of initial work, the algorithm has the potential to yield substantial improvements in estimator efficiency. Equally important is that the adaptive framework avoids the need for frequent recalibration of the parameters of the variance reduction methods applied within each stratum when changes occur in the experimental conditions governing system performance. To illustrate the applicability and effectiveness of our algorithm, we provide numerical results for a Black-Scholes option pricing, where we stratify the underlying Brownian motion with respect to its terminal value and apply an importance sampling method to normal random variables filling in the Brownian path. Relative to the estimator variance with proportional allocation, the proposed algorithm achieved a fourfold reduction in estimator variance with a negligible increase in computing time.
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