In quantitative finance, it is often necessary to analyze the distribution of the sum of specific functions of observed values at discrete points of an underlying process. Examples include the probability density function, the hedging error, the Asian option, and statistical hypothesis testing. We propose a method to calculate such a distribution, utilizing a recursive method, and examine it using various examples. The results of the numerical experiment show that our proposed method has high accuracy.

Expressions for the distribution, density, and percentiles of weighted sums of Rayleigh random variables are given, including the tilted Edgeworth expansion.

In this paper, we discuss stochastic orderings of lifetimes of two heterogeneous parallel and series systems with heterogeneous dependent components having generalized Birnbaum–Saunders distributions. The comparisons presented here are based on the vector majorization of parameters. The ordering results are established in some special cases for the generalized Birnbaum–Saunders distribution based on the multivariate elliptical, normal, t, logistic, and skew-normal kernels. Further, we use these results by considering Archimedean copulas to model the dependence structure among systems with generalized Birnbaum–Saunders components. These results have been used to derive some upper and lower bounds for survival functions of lifetimes of parallel and series systems.

We introduce a broad class of multi-hooking networks, wherein multiple copies of a seed are hooked at each step at random locations, and the number of copies follows a predetermined building sequence of numbers. We analyze the degree profile in random multi-hooking networks by tracking two kinds of node degrees—the local average degree of a specific node over time and the global overall average degree in the graph. The former experiences phases and the latter is invariant with respect to the type of building sequence and is somewhat similar to the average degree in the initial seed. We also discuss the expected number of nodes of the smallest degree. Additionally, we study distances in the network through the lens of the average total path length, the average depth of a node, the eccentricity of a node, and the diameter of the graph.

For the GI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.

We consider an optimal stopping problem for a general discrete-time process X1, X2, …, Xn, … on a common measurable space. Stopping at time n (n = 1, 2, …) yields a reward Rn(X1, …, Xn) ≥ 0, while if we do not stop, we pay cn(X1, …, Xn) ≥ 0 and keep observing the process. The problem is to characterize all the optimal stopping times τ, i.e., such that maximize the mean net gain:

$$E(R_\tau(X_1,\dots,X_\tau)-\sum_{n=1}^{\tau-1}c_n(X_1,\dots,X_n)).$$

In the particular case of Markov sequence X1, X2, … we estimate the stability of the optimal stopping problem under perturbations of transition probabilities.

In this article we investigate conditions by a unified method under which the covariances of functions of two adjacent ordered random variables are nonnegative. The main structural results are applied to several kinds of ordered random variable, such as delayed record values, continuous and discrete ℓ∞⩽-spherical order statistics, epoch times of mixed Poisson processes, generalized order statistics, discrete weak record values, and epoch times of modified geometric processes. These applications extend the main results for ordinary order statistics in Qi [28] and for usual record values in Nagaraja and Nevzorov [25].

We study tandem ordering of constant-service stations with unlimited buffers where service at each station adds a certain value to the job. With the goal of minimizing the total expect value of the jobs in the system, we provide conditions under which some particular orderings are optimal and describe a heuristic that finds a near-optimal order for stations of arbitrary service lengths and added values.

The purpose of this paper is to provide further understanding into the structure of the sequential allocation (“stochastic multi-armed bandit”) problem by establishing probability one finite horizon bounds and convergence rates for the sample regret associated with two simple classes of allocation policies. For any slowly increasing function g, subject to mild regularity constraints, we construct two policies (the g-Forcing, and the g-Inflated Sample Mean) that achieve a measure of regret of order O(g(n)) almost surely as n → ∞, bound from above and below. Additionally, almost sure upper and lower bounds on the remainder term are established. In the constructions herein, the function g effectively controls the “exploration” of the classical “exploration/exploitation” tradeoff.

We consider a hierarchical linear regression model where the regression parameters for the units have a multivariate normal distribution whose parameters are unknown. Several replications are available for each unit. The design matrices for the units need not be the same. A complicating feature of the model is that each observation is subject to measurement error. The objective of the paper is to derive the predictive distribution of the “true” value of the response at a given design point. A Bayesian treatment is given to the problem. In addition to standard prior distributions, other prior distributions are considered. The calculations are done with the Gibbs sampler. An example is discussed in detail.

In our article [3] we have found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order.

A stochastic scheduling model with linear waiting costs and unknown routing probabilities is considered. Using a Bayesian approach and methods of Bayesian dynamic programming, we investigate the finite-horizon stochastic scheduling problem with incomplete information. In particular, we study an equivalent nonstationary bandit model and show the monotonicity of the total expected reward and of the Gittins index. We derive the monotonicity and well-known structural properties of the (greatest) maximizers, the so-called stay-on-a-winnerproperty and the stopping-property. The monotonicity results are based on a special partial ordering on .

A software release game was formulated by Zeephongsekul and Chiera [Zeephongsekul, P. & Chiera, C. (1995). Optimal software release policy based on a two-person game of timing. Journal of Applied Probability 32: 470–481] and was reconsidered by Dohi et al. [Dohi, T., Teraoka, Y., & Osaki, S. (2000). Software release games. Journal of Optimization Theory and Applications 105(2): 325–346] in a framework of two-person nonzero-sum games. In this paper, we further point out the faults in the above literature and revisit the Nash equilibrium strategies in the software release games from the viewpoints of both silent and noisy type of games. It is shown that the Nash equilibrium strategies in the silent and noisy of software release games exist under some parametric conditions.

We extend the classic sequential stochastic assignment problem to include arrivals of workers. When workers are all of the same type, we show that the socially optimal policy is the same as the individually optimal policy for which workers are given priority according to last come–first served. This result also holds under several variants in the model assumptions. When workers have different types, we show that the socially optimal policy is determined by thresholds such that more valuable jobs are given to more valuable workers, but now the individually optimal policy is no longer socially optimal. We also show that the overall value increases when worker or job values become more variable.

Two stochastic knapsack problem (SKP) models are considered: the static broken knapsack problem (BKP) and the SKP with simple recourse and penalty cost problem. For both models, we assume: the knapsack has a constant capacity; there are n types of items and each type has an infinite supply; a type i item has a deterministic reward vi and a random weight with known distribution Fi. Both models have the same objective to maximize expected total return by finding the optimal combination of items, that is, quantities of items of each type to be put in knapsack. The difference between the two models is: if knapsack is broken when total weights of items put in knapsack exceed the knapsack's capacity, for the static BKP model, all existing rewards would be wiped out, while for the latter model, we could still keep the existing rewards in knapsack but have to pay a fixed penalty plus a variant cost proportional to the overcapacity amount. This paper also discusses the special case when knapsack has an exponentially distributed capacity.

We study layered queueing systems comprised two interlacing finite M/M/• type queues, where users of each layer are the servers of the other layer. Examples can be found in file sharing programs, SETI@home project, etc. Let Li denote the number of users in layer i, i=1, 2. We consider the following operating modes: (i) All users present in layer i join forces together to form a single server for the users in layer j (j≠i), with overall service rate μjLi (that changes dynamically as a function of the state of layer i). (ii) Each of the users present in layer i individually acts as a server for the users in layer j, with service rate μj.

These operating modes lead to three different models which we analyze by formulating them as finite level-dependent quasi birth-and-death processes. We derive a procedure based on Matrix Analytic methods to derive the steady state probabilities of the two dimensional system state. Numerical examples, including mean queue sizes, mean waiting times, covariances, and loss probabilities, are presented. The models are compared and their differences are discussed.

In a classical chess round-robin tournament, each of $n$ players wins, draws, or loses a game against each of the other $n-1$ players. A win rewards a player with 1 points, a draw with 1/2 point, and a loss with 0 points. We are interested in the distribution of the scores associated with ranks of $n$ players after ${{n \choose 2}}$ games, that is, the distribution of the maximal score, second maximum, and so on. The exact distribution for a general $n$ seems impossible to obtain; we obtain a limit distribution.

Alternating renewal processes have been widely used to model social and scientific phenomenal where independent “on” and “off” states alternate. In this paper, we study a model where the value of a process cumulates and declines according to two modes of compound Poisson processes with respect to an underlying alternating renewal process. The model discussed in the present paper can be used as a revenue management model applied to inventory or to finance. The exact distribution of the process is derived as well as the double Laplace transform with respect to the level and time of the process.

The present paper studies the likelihood ratio order of posterior distributions of parameter when the same order exists between the corresponding prior of the parameter, or when the observed values of the sufficient statistic for the parameter differ. The established likelihood order allows one to compare the Bayesian estimators associated with many common and general error loss functions analytically. It can also enable one to compare the Bayes factor in hypothesis testing without using numerical computation. Moreover, using the likelihood ratio (LR) order of the posterior distributions can yield the LR order between marginal predictive distributions, and posterior predictive distributions.

Digital-to-analog converters (DAC) transform signals from the abstract digital domain to the real analog world. In many applications, DACs play a crucial role. Due to variability in the production, various errors arise that influence the performance of the DAC. We focus on the current errors, which describe the fluctuations in the currents of the various unit current elements in the DAC. A key performance measure of the DAC is the Integrated Nonlinearity (INL), which we study in this article. There are several DAC architectures. The most widely used architectures are the thermometer and the binary and the segmented architectures. We study the two extreme architectures, namely the thermometer and the binary architectures. We assume that the current errors are independent and identically normally distributed and reformulate the INL as a functional of a Brownian bridge. We then proceed by investigating these functionals. For the thermometer case, the functional is the maximal absolute value of the Brownian bridge, which has been investigated in the literature. For the binary case, we investigate properties of the functional, such as its mean, variance, and density.