In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

We consider a discrete-time risk process driven by proportional reinsurance and an interest rate process. We assume that the interest rate process behaves as a Markov chain. To reduce the risk of ruin, we may reinsure a part or even all of the reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a stationary policy. To illustrate these results, a numerical example is included.

Let X n (Λ) be the number of nonoverlapping occurrences of a simple pattern Λ in a sequence of independent and identically distributed (i.i.d.) multistate trials. For fixed k, the exact tail probability P{Xn (∧) < k} is difficult to compute and tends to 0 exponentially as n → ∞. In this paper we use the finite Markov chain imbedding technique and standard matrix theory results to obtain an approximation for this tail probability. The result is extended to compound patterns, Markov-dependent multistate trials, and overlapping occurrences of Λ. Numerical comparisons with Poisson and normal approximations are provided. Results indicate that the proposed approximations perform very well and do significantly better than the Poisson and normal approximations in many cases.

We consider a modified version of the classical optimal dividends problem of de Finetti in which the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Lévy process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Lévy measure has a completely monotone density. As a prerequisite for the proof, we show that, under the aforementioned condition on the Lévy measure, the q-scale function of the spectrally negative Lévy process has a derivative which is strictly log-convex.

This work is concerned with a class of mean-field models given by a switching diffusion with a continuous-state-dependent switching process. Focusing on asymptotic properties, the regularity or nonexplosiveness, Feller continuity, and strong Feller continuity are established by means of introducing certain auxiliary processes and by making use of the truncations. Based on these results, exponential ergodicity is obtained under the Foster–Lyapunov drift conditions. By virtue of the coupling methods, the strong ergodicity or uniform ergodicity in the sense of convergence in the variation norm is established for the mean-field model with a Markovian switching process. Besides this, several examples are presented for demonstration and illustration.

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

Suppose that n identical particles evolve according to the same marginal Markov chain. In this setting we study chains such as the Ehrenfest chain that move a prescribed number of randomly chosen particles at each epoch. The product chain constructed by this device inherits its eigenstructure from the marginal chain. There is a further chain derived from the product chain called the composition chain that ignores particle labels and tracks the numbers of particles in the various states. The composition chain in turn inherits its eigenstructure and various properties such as reversibility from the product chain. The equilibrium distribution of the composition chain is multinomial. The current paper proves these facts in the well-known framework of state lumping and identifies the column eigenvectors of the composition chain with the multivariate Krawtchouk polynomials of Griffiths. The advantages of knowing the full spectral decomposition of the composition chain include (a) detailed estimates of the rate of convergence to equilibrium, (b) construction of martingales that allow calculation of the moments of the particle counts, and (c) explicit expressions for mean coalescence times in multi-person random walks. These possibilities are illustrated by applications to Ehrenfest chains, the Hoare and Rahman chain, Kimura's continuous-time chain for DNA evolution, a light bulb chain, and random walks on some specific graphs.

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.

We study defaultable bond prices in the Black–Cox model with jumps in the asset value. The jump-size distribution is arbitrary, and following Longstaff and Schwartz (1995) and Zhou (2001) we assume that, if default occurs, the recovery at maturity depends on the ‘severity of default’. Under this general setting, the vehicle for our analysis is an integral equation. With the aid of this, we prove some properties of the bond price which are consistent numerically and empirically with earlier works. In particular, the limiting credit spread as time to maturity tends to 0 is nonzero. As a byproduct, we show that the integral equation implies an infinite-series expansion for the bond price.

We study a process where balls are repeatedly thrown into n boxes independently according to some probability distribution p . We start with n balls, and at each step, all balls landing in the same box are fused into a single ball; the process terminates when there is only one ball left (coalescence). Let c := ∑j p j 2, the collision probability of two fixed balls. We show that the expected coalescence time is asymptotically 2c −1, under two constraints on p that exclude a thin set of distributions p . One of the constraints is c = o(ln−2 n). This ln−2 n is shown to be a threshold value: for c = ω(ln−2 n), there exists p with c(p ) = c such that the expected coalescence time far exceeds c −1. Connections to coalescent processes in population biology and theoretical computer science are discussed.

Consider a sequence (X k : k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (S n : n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of S n in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

We consider stochastic replicator processes for games that are composed of finitely many trials. Several general results on the relation between Nash equilibria and the long-run behaviour of the stochastic processes are proved. In particular, a sufficient condition is given for almost sure convergence to a state where everyone plays in every trial a strict Nash equilibrium. The results are applied to multiple-trial conflicts based on wars of attrition and on sperm competition games with fair raffles, respectively.

The upper tail behaviour is explored for a stopped random product ∏j=1 N X j , where the factors are positive and independent and identically distributed, and N is the first time one of the factors occupies a subset of the positive reals. This structure is motivated by a heavy-tailed analogue of the factorial n!, called the factoid of n. Properties of the factoid suggested by computer explorations are shown to be valid. Two topics about the determination of the Zipf exponent in the rank-size law for city sizes are discussed.

In this paper, a fourth moment bound for partial sums of functionals of strongly ergodic Markov chains is established. This type of inequality plays an important role in the study of the empirical process invariance principle. This inequality is specially adapted to the technique of Dehling, Durieu, and Volný (2008). The same moment bound can be proved for dynamical systems whose transfer operator has some spectral properties. Examples of applications are given.

This paper deals with censoring of infinite-state banded Markov chains. Censoring involves reducing the time spent in states outside a certain set of states to 0 without affecting the number of visits within this set. We show that, if all states are transient, there is, besides the standard censored Markov chain, a nonstandard censored Markov chain which is stochastic. Both the stochastic and the substochastic solutions are found by censoring a sequence of finite transition matrices. If all matrices in the sequence are stochastic, the stochastic solution arises in the limit, whereas the substochastic solution arises if the matrices in the sequence are substochastic. We also show that, if the Markov chain is recurrent, the only solution is the stochastic solution. Censoring is particularly fruitful when applied to quasi-birth-and-death (QBD) processes. It turns out that key matrices in such processes are not unique, a fact that has been observed by several authors. We note that the stochastic solution is important for the analysis of finite queues.

We present two new stochastic volatility models in which option prices for European plain-vanilla options have closed-form expressions. The models are motivated by the well-known SABR model, but use modified dynamics of the underlying asset. The asset process is modelled as a product of functions of two independent stochastic processes: a Cox-Ingersoll-Ross process and a geometric Brownian motion. An application of the models to options written on foreign currencies is studied.

We build a family of Markov chains on a sphere using distance-based long-range connection probabilities to model the decentralized message-passing problem that has recently gained significant attention in the small-world literature. Starting at an arbitrary source point on the sphere, the expected message delivery time to an arbitrary target on the sphere is characterized by a particular expected hitting time of our Markov chains. We prove that, within this family, there is a unique efficient Markov chain whose expected hitting time is polylogarithmic in the relative size of the sphere. For all other chains, this expected hitting time is at least polynomial. We conclude by defining two structural properties, called scale invariance and steady improvement, of the probability density function of long-range connections and prove that they are sufficient and necessary for efficient decentralized message delivery.

We study investment and disinvestment decisions in situations where there is a time lag d > 0 from the time t when the decision is taken to the time t + d when the decision is implemented. In this paper we apply the probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay. In particular, we prove the independence between Parisian stopping times and a general Brownian motion with drift stopped at the stopping time. Relying on this result, we solve the constrained maximization problem, obtaining an analytic solution to the optimal ‘starting’ and ‘stopping’ levels. We compare our results with the instantaneous entry and exit situation, and show that an increase in the uncertainty of the underlying process hastens the decision to invest or disinvest, extending a result of Bar-Ilan and Strange (1996).

We consider a piecewise-deterministic Markov process (Xt ) governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. The paper deals with the point process of upcrossings of some level b by (Xt ). We prove a version of Rice's formula relating the stationary density of (Xt ) to level crossing intensities and show that, for a wide class of processes (Xt ), as b → ∞, the scaled point process where ν+(b) denotes the intensity of upcrossings of b, converges weakly to a geometrically compound Poisson process.

For exponential open and closed queueing networks, we investigate the internal dependence structure, compare the internal dependence for different networks, and discuss the relation of correlation formulae to the existence of spectral gaps and comparison of asymptotic variances. A central prerequisite for the derived theorems is stochastic monotonicity of the networks. The dependence structure of network processes is described by concordance order with respect to various classes of functions. Different networks with the same first-order characteristics are compared with respect to their second-order properties. If a network is perturbed by changing the routeing in a way which holds the routeing equilibrium fixed, the resulting perturbations of the network processes are evaluated.