In this paper we discuss a nondecreasing lower bound for the Poisson cumulative distribution function (CDF) at z standard deviations above the mean λ, where z and λ are parameters. This is important because the normal distribution as an approximation for the Poisson CDF may overestimate or underestimate its value. A sharp nondecreasing lower bound in the form of a step function is constructed. As a corollary of the bound's properties, for a given percent α and parameter λ, the minimal z is obtained such that, for any Poisson random variable with the mean greater or equal to λ, its αth percentile is at most z standard deviations above its mean. For Poisson distributed control parameters, the corollary allows simple policies measuring performance in terms of standard deviations from a benchmark.

When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable. In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (k x k) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.

In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

We make a connection between the continuous time and lazy discrete time Markov chains through the comparison of cutoffs and mixing time in total variation distance. For illustration, we consider finite birth and death chains and provide a criterion on cutoffs using eigenvalues of the transition matrix.

We consider a general homogeneous continuous-time Markov process with restarts. The process is forced to restart from a given distribution at time moments generated by an independent Poisson process. The motivation to study such processes comes from modeling human and animal mobility patterns, restart processes in communication protocols, and from application of restarting random walks in information retrieval. We provide a connection between the transition probability functions of the original Markov process and the modified process with restarts. We give closed-form expressions for the invariant probability measure of the modified process. When the process evolves on the Euclidean space, there is also a closed-form expression for the moments of the modified process. We show that the modified process is always positive Harris recurrent and exponentially ergodic with the index equal to (or greater than) the rate of restarts. Finally, we illustrate the general results by the standard and geometric Brownian motions.

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation X_t = A_tX_t-1 + B_t, t ∈ Z, where ((A_t, B_t))_t∈Z is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X₀|^p, p ∈ R. Special attention is given to the case when B_t has an Erlang distribution. We provide various approximations to the moments E|X₀|^p and show their performance in a small numerical study.

Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

Self-exciting point processes (SEPPs), or Hawkes processes, have found applications in a wide range of fields, such as epidemiology, seismology, neuroscience, engineering, and more recently financial econometrics and social interactions. In the traditional SEPP models, the baseline intensity is assumed to be a constant. This has restricted the application of SEPPs to situations where there is clearly a self-exciting phenomenon, but a constant baseline intensity is inappropriate. In this paper, to model point processes with varying baseline intensity, we introduce SEPP models with time-varying background intensities (SEPPVB, for short). We show that SEPPVB models are competitive with autoregressive conditional SEPP models (Engle and Russell 1998) for modeling ultra-high frequency data. We also develop asymptotic theory for maximum likelihood estimation based inference of parametric SEPP models, including SEPPVB. We illustrate applications to ultra-high frequency financial data analysis, and we compare performance with the autoregressive conditional duration models.

We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain, explicitly, the optimal portfolio strategy and the value function in some parameter settings. In particular, we do this when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is to identify when the solution of the Hamilton-Jacobi-Bellman equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

We propose an estimator for the cumulative distribution function G of the sojourn time in a steady-state M/G/∞ queueing system, when the available data consists of the arrival and departure epochs alone, without knowing which arrival corresponds to which departure. The estimator generalizes an estimator proposed in Brown (1970), and is based on a functional relationship between G and the distribution function of the time between a departure and the rth latest arrival preceding it. The estimator is shown to outperform Brown's estimator, especially when the system is heavily loaded.

A continuously monitored system is considered, which is subject to accumulating deterioration modelled as a gamma process. The system fails when its degradation level exceeds a limit threshold. At failure, a delayed replacement is performed. To shorten the down period, a condition-based maintenance strategy is applied, with imperfect repair. Mimicking virtual age models used for recurrent events, imperfect repair actions are assumed to lower the system degradation through a first-order arithmetic reduction of age model. Under these assumptions, Markov renewal equations are obtained for several reliability indicators. Numerical examples illustrate the behaviour of the system.

We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

The distributions of discrete, continuous and conditional multiple window scan statistics are studied. The finite Markov chain imbedding technique has been applied to obtain the distributions of fixed window scan statistics defined from a sequence of Bernoulli trials. In this manuscript the technique is extended to compute the distributions of multiple window scan statistics and the exact powers for multiple pulse and Markov dependent alternatives. An application in blood component quality monitoring is provided. Numerical results are also given to illustrate our theoretical results.

This paper provides an analysis on a discrete version of the Ornstein-Uhlenbeck (OU) process which reflects the small discrete movements caused by the tick size effect. This discrete OU process is derived from matching the first two moments to those of the standard OU process in an infinitesimal sense. We discuss the distributional convergence from the discrete to the continuous processes, and show that the convergence speed is in the second order of the step (tick) size. We also provide some analytical results for the proposed discrete OU process itself, including the closed-form formula of the moment generating function and a full characterisation of the steady state distribution. These results enable us to examine the convergence order explicitly.

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z₂^d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on K_n^d, where K_n is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.

Consider a random walk S_n = ∑_i=0ⁿX_i with negative drift. This paper deals with upper bounds for the maximum M = max_n≥1S_n of this random walk in different settings of power moment existences. As is usual for deriving upper bounds, we truncate summands. Therefore, we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby, we can reduce the problem of finding upper bounds for M to the problem of finding upper bounds for M_τ = max_n≤τS_n. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

We consider a stochastic SIR (susceptible → infective → removed) epidemic on a random graph with specified degree distribution, constructed using the configuration model, and investigate the `acquaintance vaccination' method for targeting individuals of high degree for vaccination. Branching process approximations are developed which yield a post-vaccination threshold parameter, and the asymptotic (large population) probability and final size of a major outbreak. We find that introducing an imperfect vaccine response into the present model for acquaintance vaccination leads to sibling dependence in the approximating branching processes, which may then require infinite type spaces for their analysis and are generally not amenable to numerical calculation. Thus, we propose and analyse an alternative model for acquaintance vaccination, which avoids these difficulties. The theory is illustrated by a brief numerical study, which suggests that the two models for acquaintance vaccination yield quantitatively very similar disease properties.

We study an urn model introduced in the paper of Chen and Wei (2005), where at each discrete time step m balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m = 1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.

Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

Formulae for ζ(2n) and L_χ4(2n + 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π² / 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.