In this article, we analyze the large deviations bounds for the nonergodic face-homogeneous random walk in the positive quadrant. Under some condition the value of the local rate function for the path identically equal to zero is found, and an explicit expression is derived for it. This makes the computation of its value possible for specific stochastic networks. Some numerical examples are given.

Simultaneous estimation problems that deal with the estimation of several means for normal distributions are considered under the squared-error loss. Heuristic procedures are presented which can be applied to parameter estimation problems for a wide class of distributions specified only by their means and variances. Explicit results are obtained for the heuristic shrinkage estimators in the normal distribution case. Limiting behavior of the relative risk savings of these estimators is studied. The performances of the proposed estimators for normal distribution means are compared with other existing estimators by a computer simulation.

Gauss-Seidel is a general method for solving a system of equations (possibly nonlinear). It makes repeated sweeps through the variables; within a sweep as each new estimate for a variable is computed, the current estimate for that variable is replaced with the new estimate immediately, instead of on completion of the sweep. The idea is to use new data as soon as it is computed. Gauss- Seidel is often efficient for computing the invariant measure of a Markov chain (especially if the transition matrix is sparse), and for computing the value function in optimal control problems. In many applications the computation can be significantly improved by appropriately ordering the variables within each sweep. A simple heuristic is presented here for computing an ordering that quickens convergence. In parallel processing, several variables must be computed simultaneously, which appears to work against Gauss-Seidel. Simple asynchronous parallel Gauss-Seidel methods are presented here. Experiments indicate that the methods retain the benefit of a good ordering, while further speeding up convergence by a factor of P if P processors participate.

In this paper, we focus on the optimal stopping problem. A probabilistic interpretation of the Gauss-Seidel (and the Jacobi) method for computing the value function is given, which motivates our ordering heuristic. However, the ordering heuristic and parallel processing methods apply in a broader context, in particular, to the important problem of computing the invariant measure of a Markov chain.

We give a new characterization of the ruin probability of the classical insurance risk model and use it to obtain bounds on and a computational approach for evaluating this probability.

We introduce the concept of “strong interchangeability” of random vectors. Strongly interchangeable random vectors arise naturally in packetized voice channels, M/G/1 queues, symmetric queueing networks, and other standard symmetric distributions. We study some properties of strongly interchangeable random vectors. We show that if (X1, …, XN) is a strongly interchangeable random vector, then even though there is no Markov property, taboo probabilities can be used to compute the joint distribution of ŽN = min1≤n≤N σnk=IXk and ZN = max1≤n≤N σnk=1Xk. For a special instance of this problem that arises in packetized voice communication, it is shown that the resulting algorithm essentially has a complexity of order N4. When ( σnk=1Xk, n = 1,… N) is an associated random vector bound for the joint distribution of ŽN and ZN are obtained and applied to the packetized voice communication problem.

In this paper, we develop the lower–upper-bound approximation in the space of Laplace transforms for pricing American options. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, e.g. a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the corresponding “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures in the Laplace space. Finally, we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option and further utilize the early exercise premium representation in the Laplace space to obtain the upper bound. Numerical examples are conducted to compare the method with a variety of existing methods in the literature as benchmark to demonstrate the accuracy and efficiency.

Capital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

Extremal arrival processes, in the sense of increasing convex order of waiting time of queueing systems, are investigated. Two types of extremal processes are proposed: one in the class of processes that have identical marginal distributions and the other in the class of bounded stochastic processes that have the same mean and covariance structure. The worst performance with regard to waiting time in the sense of increasing convex order is guaranteed when these extremal processes are fed into a first in-first out single-server queue.

Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$, $a > 0$. As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

This paper investigates the effects of measurement and parameter errors on the calculation of power transfer distribution factors (PTDFs). Calculation of PTDFs depends mainly on two factors: the operating conditions and the network topology. Both of these factors change in real time and are monitored through the use of a state estimator. The role of the state estimator in providing not only the state information but also the real-time network model is shown to influence power market operations via PTDF-based decisions such as congestion management and pricing. The IEEE 14 bus test system is used for the illustration of these influences.

We consider a communication network that can support both wideband video calls and narrowband data traffic. First we consider a single link and prove a weak convergence result to justify a piecewise-deterministic Markov process approximation to the system. We then generalize this approximation to allow priorities and more than one link. This second approximation is a generalization of the Erlang fixed-point approximation for loss networks and is justified via a diverse routing limit theorem.

This article is concerned with the problem of controlling a simple immigration process, which represents a pest population, by the introduction of a predator. It is assumed that the cost rate caused by the pests is an increasing function of their population size and that the cost rate of the controlling action is constant. The existence of a control-limit policy that minimizes the expected long-run average cost per unit time is established. The proof is based on the variation of a fictitious parameter over the entire real line.

Motivated by applications in areas such as cloud computing and information technology services, we consider GI/GI/1 queueing systems under workloads (arrival and service processes) that vary according to one discrete time scale and under controls (server capacity) that vary according to another discrete time scale. We take a stochastic optimal control approach and formulate the corresponding optimal dynamic control problem as a stochastic dynamic program. Under general assumptions for the queueing system, we derive structural properties for the optimal dynamic control policy, establishing that the optimal policy can be obtained through a sequence of convex programs. We also derive fluid and diffusion approximations for the problem and propose analytical and computational approaches in these settings. Computational experiments demonstrate the benefits of our theoretical results over standard heuristics.

In light of recent empirical research on jump activity, this article study the calibration of a new class of stochastic volatility models that include both jumps in return and volatility. Specifically, we consider correlated jump sizes and both contemporaneous and independent arrival of jumps in return and volatility. Based on the specifications of this model, we derive a closed-form relationship between the VIX index and latent volatility. Also, we propose a closed-form logarithmic likelihood formula by using the link to the VIX index. By estimating alternative models, we find that the general counting processes setting lead to better capturing of return jump behaviors. That is, the part where the return and volatility jump simultaneously and the part that jump independently can both be captured. In addition, the size of the jumps in volatility is, on average, positive for both contemporaneous and independent arrivals. However, contemporaneous jumps in the return are negative, but independent return jumps are positive. The sub-period analysis further supports above insight, and we find that the jumps in return and volatility increased significantly during the two recent economic crises.

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.

We study the degree profile of random hierarchical lattice networks. At every step, each edge is either serialized (with probability p) or parallelized (with probability 1−p). We establish an asymptotic Gaussian law for the number of nodes of outdegree 1, and show how to extend the derivations to encompass asymptotic limit laws for higher outdegrees. The asymptotic joint distribution of the number of nodes of outdegrees 1 and 2 is shown to be bivariate normal. No phase transition with p is detected in these asymptotic laws.

For the limit laws, we use ideas from the contraction method. The recursive equations which we get involve coefficients and toll terms depending on the recursion variable and thus are not in the standard form of the contraction method. Yet, an adaptation of the contraction method goes through, showing that the method has promise for a wider range of random structures and algorithms.

This paper proposes a novel stochastic volatility model with a flexible jump structure. This model allows both contemporaneous and independent arrival of jumps in return and volatility. Moreover, time-varying jump intensities are used to capture jump clustering. In the proposed framework, we provide a semi-analytical solution for the pricing problem of VIX futures and options. Through numerical experiments, we verify the accuracy of our pricing formula and explore the impact of the jump structure on the pricing of VIX derivatives. We find that the correct identification of the market jump structure is crucial for pricing VIX derivatives, and misspecified model setting can yield large errors in pricing.

Two-part framework and the Tweedie generalized linear model (GLM) have traditionally been used to model loss costs for short-term insurance contracts. For most portfolios of insurance claims, there is typically a large proportion of zero claims that leads to imbalances, resulting in lower prediction accuracy of these traditional approaches. In this article, we propose the use of tree-based methods with a hybrid structure that involves a two-step algorithm as an alternative approach. For example, the first step is the construction of a classification tree to build the probability model for claim frequency. The second step is the application of elastic net regression models at each terminal node from the classification tree to build the distribution models for claim severity. This hybrid structure captures the benefits of tuning hyperparameters at each step of the algorithm; this allows for improved prediction accuracy, and tuning can be performed to meet specific business objectives. An obvious major advantage of this hybrid structure is improved model interpretability. We examine and compare the predictive performance of this hybrid structure relative to the traditional Tweedie GLM using both simulated and real datasets. Our empirical results show that these hybrid tree-based methods produce more accurate and informative predictions.

Log-concavity [log-convexity] and their various properties play an increasingly important role in probability, statistics, operations research and other fields. In this paper, we first establish general preservation theorems of log-concavity and log-convexity under operator $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$, θ ∈ Θ, where Θ is an interval of real numbers or an interval of integers, and the random variable $X_\theta$ has a distribution function belonging to the family $\{F_\theta , \theta \in \Theta \}$ possessing the semi-group property. The proofs are based on the theory of stochastic comparisons and weighted distributions. The main results are applied to some special operators, for example, operators occurring in reliability, Bernstein-type operators and Beta-type operators. Several known results in the literature are recovered.

In this paper, we consider a d-node GI|GI|1 tandem queue with i.i.d. inter-arrival process and service processes that are independent of each other. Our main interest is to estimate the probability to reach a high level N in a busy cycle of the system using simulation. As crude simulation does not give a sufficient precision in reasonable time, we use importance sampling. We introduce a method to find a state-independent change of measure and we show that this is equivalent to a change of measure that was earlier, but implicitly, described by Parekh and Walrand [8]. We also show that this change of measure is the only exponential state-independent change of measure that may result in an asymptotically efficient estimator. Lastly, we provide necessary conditions for this state-independent change of measure to give an asymptotically efficient estimator.