As more people move back into densely populated cities, bike sharing is emerging as an important mode of urban mobility. In a typical bike-sharing system (BSS), riders arrive at a station and take a bike if it is available. After retrieving a bike, they ride it for a while, then return it to a station near their final destinations. Since space is limited in cities, each station has a finite capacity of docks, which cannot hold more bikes than its capacity. In this paper, we study BSSs with stations having a finite capacity. By an appropriate scaling of our stochastic model, we prove a mean-field limit and a central limit theorem for an empirical process of the number of stations with k bikes. The mean-field limit and the central limit theorem provide insight on the mean, variance, and sample path dynamics of large-scale BSSs. We also leverage our results to estimate confidence intervals for various performance measures such as the proportion of empty stations, the proportion of full stations, and the number of bikes in circulation. These performance measures have the potential to inform the operations and design of future BSSs.

In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

Many cryptocurrencies including Bitcoin are susceptible to a so-called double-spend attack, where someone dishonestly attempts to reverse a recently confirmed transaction. The duration and likelihood of success of such an attack depends on the recency of the transaction and the computational power of the attacker, and these can be related to the distribution of time for counts from one Poisson process to exceed counts from another by some desired amount. We derive an exact expression for this distribution and show how it can be used to obtain efficient simulation estimators. We also give closed-form analytic approximations and illustrate their accuracy.

The present paper is concerned with reliability economics, considering a certain performance-per-cost criterion for coherent and mixed systems, as introduced in [Dugas, M.R. & Samaniego, F.J. (2007). On optimal system designs in reliability-economics frameworks. Naval Research Logistics 54, 568–582]. We first present a new comparison result for performance-per-cost of systems with independent and identically distributed component lifetimes under certain stochastic orderings. We then consider optimization of the performance-per-cost criterion, first reconsidering and refining results from the above cited paper, and then considering mixtures of given subsets of coherent systems.

The reliability study of k-out-of-n systems is of interest both from theoretical and practical points of view. Applications of such models can be seen in many real-world phenomena, including telecommunication, transmission, transportation, manufacturing, and services. A probabilistic study of a real-world k-out-of-n system often helps to develop an optimal strategy for maintaining high system-level reliability. There are many investigations devoted to the reliability-centric analysis of such systems. We consider a mathematical model of a repairable k-out-of-n system that works until k of its n components have failed. During the system's life cycle, its components are repaired with the help of a single repair facility. It is supposed that the components' lifetimes have an exponential distribution and their repair times have a general distribution. The proposed model is intended to be applied to the description of operation of unmanned rotorcraft high-altitude platforms and to be validated with the help of an experimental prototype. For the considered system, we propose an algorithm for calculation of the reliability function, and for special cases, k = 2 and k = 3, its closed-form representation is given. A numerical investigation is performed for special cases. The obtained results are a first step toward the sensitivity analysis of reliability characteristics of k-out-of-n systems to the shape of the repair time distributions of their components.

In this paper, we review different definitions that multi-state k-out-of-n systems have received along the literature and study them in a unified way using the algebra of monomial ideals. We thus obtain formulas and algorithms to compute their reliability and bounds for it. We provide formulas and computer experiments for simple and generalized multi-state k-out-of-n systems and for binary k-out-of-n systems with multi-state components.

This paper investigates the volatility in regime-switching models formulated based on the geometric Brownian motion with its drift and volatility factors randomized with Markov chains. By developing explicit formulas about occupation time of Markov chains, we analysis the difference between global volatility of this model and the volatility caused by Brownian randomness, in order to measure the volatility caused by regime-switching after justifying its existence. Utilizing this structure of volatility, we optimize the methods of volatility parameters estimation.

We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler–Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of a large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in this regime. Using this representation, we obtain analytical expansions of the implied volatility for small maturity and extreme strikes, which reproduce at leading order known asymptotic results for the continuous time model.

This paper studies the parameter estimation for Ornstein–Uhlenbeck stochastic volatility models driven by Lévy processes. We propose computationally efficient estimators based on the method of moments that are robust to model misspecification. We develop an analytical framework that enables closed-form representation of model parameters in terms of the moments and autocorrelations of observed underlying processes. Under moderate assumptions, which are typically much weaker than those for likelihood methods, we prove large-sample behaviors for our proposed estimators, including strong consistency and asymptotic normality. Our estimators obtain the canonical square-root convergence rate and are shown through numerical experiments to outperform likelihood-based methods.