For general risk processes, we introduce and study the expected time-integrated negative part of the process on a fixed time interval. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below 0, studied by Dos Reis, and the probability of ruin. We carry out differentiation of other functionals of one-dimensional and multidimensional risk processes with respect to the initial reserve level. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve that minimizes one of these risk measures, illustrate the variety of fields of application and the benefits deriving from an efficient and effective use of such tools.

Competing patterns are compound patterns that compete to be the first to occur pattern-specific numbers of times. They represent a generalisation of the sooner waiting time problem and of start-up demonstration tests with both acceptance and rejection criteria. Through the use of finite Markov chain imbedding, the waiting time distribution of competing patterns in multistate trials that are Markovian of a general order is derived. Also obtained are probabilities that each particular competing pattern will be the first to occur its respective prescribed number of times, both in finite time and in the limit.

Berry-Esseen-type bounds to the normal, based on zero- and size-bias couplings, are derived using Stein's method. The zero biasing bounds are illustrated in an application to combinatorial central limit theorems in which the random permutation has either the uniform distribution or one that is constant over permutations with the same cycle type, with no fixed points. The size biasing bounds are applied to the occurrences of fixed, relatively ordered subsequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs in finite graphs.

The first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear barrier is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties.

We generalize a theorem due to Keilson on the approximate exponentiality of waiting times for rare events in regenerative processes. We use the result to investigate the limit distribution for a family of first entrance times in a sequence of Ehrenfest urn models. As a second application, we consider approximate pattern matching, a problem arising in molecular biology and other areas.

We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation of the boundary crossing probabilities is feasible. In particular, we partially generalize and improve results obtained by Pötzelberger and Wang in the case when the approximating boundaries are piecewise linear. Applications to barrier option pricing are also discussed.

We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

In this note, we consider discrete-time finite Markov chains and assume that they are only partly observed. We obtain finite-dimensional normalized filters for basic statistics associated with such processes. Recursive equations for these filters are derived by means of simple computations involving conditional expectations. An application to the estimation of parameters of the so-called discrete-time batch Markovian arrival process is outlined.

For uniformly ergodic Markov chains, we obtain new perturbation bounds which relate the sensitivity of the chain under perturbation to its rate of convergence to stationarity. In particular, we derive sensitivity bounds in terms of the ergodicity coefficient of the iterated transition kernel, which improve upon the bounds obtained by other authors. We discuss convergence bounds that hold in the case of finite state space, and consider numerical examples to compare the accuracy of different perturbation bounds.

In this work, we study some geometrical properties of a stationary Gaussian field modeling the sea surface, using the energy spectrum. We consider the length of a crest and the mean speed of contours, which can be expressed as integrals over level sets. We also give central limit theorems for some of these quantities, using chaos expansions.

In practical situations, we observe the number of claims to an insurance portfolio but not the claim intensity. It is therefore of interest to try to solve the ‘filtering problem’; that is, to obtain the best estimate of the claim intensity on the basis of reported claims. In order to use the Kalman-Bucy filter, based on the Cox process incorporating a shot noise process as claim intensity, we need to approximate it by a Gaussian process. We demonstrate that, if the primary-event arrival rate of the shot noise process is reasonably large, we can then approximate the intensity, claim arrival, and aggregate loss processes by a three-dimensional Gaussian process. We establish weak-convergence results. We then use the Kalman-Bucy filter and we obtain the price of reinsurance contracts involving high-frequency events.

Let (Φt)t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function Pt(x, ·) to π; specifically, we find conditions under which r(t)||Pt(x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

Let S be a countable set and let Q = (qij, i, j ∈ S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number μ ≥ 0 and a strictly positive probability measure m = (mj, j ∈ S) such that ∑i∈Smiqij = −μmj, j ≠ b. We prove that there exists a Q-process P(t) = (pij(t), i, j ∈ S) for which m is a μ-invariant measure, that is ∑i∈Smipij(t) = e−μtmj, j∈S. We illustrate our results with reference to the Kolmogorov ‘K1’ chain and a birth-death process with catastrophes and instantaneous resurrection.

In this paper, we investigate the geometric growth of homogeneous multitype Markov chains whose states have nonnegative integer coordinates. Such models are considered in a situation similar to the supercritical case for branching processes. Finally, our general theoretical results are applied to a class of controlled multitype branching process in which the control is random.

Let X1, X2,…, Xn be independent, identically distributed random variables, uniform on [0,1]. We observe the Xk sequentially and must stop on exactly one of them. No recollection of the preceding observations is permitted. What stopping rule τ minimizes the expected rank of the selected observation? This full-information expected-rank problem is known as Robbins' problem. The general solution is still unknown, and only some bounds are known for the limiting value as n tends to infinity. After a short discussion of the history and background of this problem, we summarize what is known. We then try to present, in an easily accessible form, what the author believes should be seen as the essence of the more difficult remaining questions. The aim of this article is to evoke interest in this problem and so, simply by viewing it from what are possibly new angles, to increase the probability that a reader may see what seems to evade probabilistic intuition.

We study the asymptotic tail behavior of the conditional probability distributions of rt+k and rt+1+⋯+rt+k when (rt)t∈ℕ is a GARCH(1, 1) process. As an application, we examine the relation between the extreme lower quantiles of these random variables.

We describe a stochastic susceptible–infective–susceptible (SIS) model for transmission of infectious disease through a population, incorporating both direct host–host transmission and indirect transmission via free-living infectious stages (e.g. environmental bacteria). Existence of a quasi-stationary distribution conditional upon nonextinction of infection is established. A bivariate Ornstein–Uhlenbeck approximation is used to investigate the long-term behaviour of the process conditional upon nonextinction of infection. We show that indirect transmission leads to lower variability in the number of infected hosts present in quasi-stationarity and, consequently, to a greater tendency of infection to persist, compared with a model with direct transmission only and the same average individual infectivity. Some numerical work illustrating these results is presented.

A multitype branching process is presented in the framework of marked trees and its structure is studied by applying the strong branching property. In particular, the Markov property and the expression for the generator are derived for the process whose components are the numbers of particles of each type. The filtering of the whole population, observing the number of particles of a given type, is discussed. Weak uniqueness for the filtering equation and a recursive structure for the linearized filtering equation are proved under a suitable assumption on the reproduction law.

Mr. G owes $100 000 to a loan shark, and will be killed at dawn if the loan is not repaid in full. Mr. G has $20 000, but partial payments are not accepted, and he has no other source of income or credit. The loan shark owns a primitive casino where one can stake any amount in one's possession, gaining r times the stake with probability w and losing the stake with probability 1 - w (r > 0, 0 < w < 1). Mr. G is permitted to gamble at the casino, but each time he places a bet, the amount of his debt is increased by a factor of 1 + α (α ≥ 0). How should Mr. G gamble to maximize his chance of reaching his (moving) target and thereby surviving? Dubins and Savage showed that an optimal strategy is to stake boldly if the primitive casino is subfair or fair (i.e. w(1 + r) ≤ 1) and the inflation rate α is 0. Intuitively, a positive inflation rate would motivate Mr. G to try to reach his goal as quickly as possible, so it seems plausible that the bold strategy is optimal. However, Chen, Shepp, and Zame found that, surprisingly, the bold strategy is no longer optimal for subfair primitive casinos with inflation if both r > 1 and α satisfies 1/r ≤ α < r. They also conjectured that the bold strategy is optimal for subfair primitive casinos with inflation if r < 1. It is shown in the present paper that this conjecture is true provided that w ≤ ½. Furthermore, by introducing an interesting notion of sharp strategy, additional results are obtained on optimality of the bold strategy.

A simple, widely applicable method is described for determining factorial moments of N̂t, the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂t, and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.