In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

Let {(Xn , S n ), n ≥ 0} be a Markov random walk in which X n takes values in a general state space and S n takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

The problem of optimal dividends paid until absorbtion at zero is considered for a rather general diffusion model. With each dividend payment there is a proportional cost and a fixed cost. It is shown that there can be essentially three different solutions depending on the model parameters and the costs. (i) Whenever assets reach a barrier y *, they are reduced to y * - δ* through a dividend payment, and the process continues. (ii) Whenever assets reach a barrier y *, everything is paid out as dividends and the process terminates. (iii) There is no optimal policy, but the value function is approximated by policies of one of the two above forms for increasing barriers. A method to numerically find the optimal policy (if it exists) is presented and numerical examples are given.

In this article a collection of random self-similar fractal dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have relied on geometrical properties of an underlying metric space or the scaling factors being bounded uniformly away from 0. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that conditions such as these are not necessary. The scaling factors of the recursively defined structures in consideration form what is known as a multiplicative cascade, and results about the height of this random object are also obtained.

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

In this paper we propose a class of sequential urn designs based on generalized Pólya urn (GPU) models for balancing the allocations of two treatments in sequential clinical trials. In particular, we consider a GPU model characterized by a 2 x 2 random addition matrix with null balance (i.e. null row sums) and replacement rule depending upon the urn composition. Under this scheme, the urn process has a Markovian structure and can be regarded as a random extension of the classical Ehrenfest model. We establish almost sure convergence and asymptotic normality for the frequency of treatment allocations and show that in some peculiar cases the asymptotic variance of the design admits a natural representation based on the set of orthogonal polynomials associated with the corresponding Markov process.

This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. We give conditions on the game's primitive data under which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using the extended infinitesimal operator associated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide a recursive way of computing (or at least approximating) the value of the game. Moreover, we present a ‘martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.

A simple derivation of the explicit form of the transition density of a planar random motion at finite speed, based on some specific properties of the wave propagation on the plane R 2, is given.

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

We consider basic ergodicity properties of adaptive Markov chain Monte Carlo algorithms under minimal assumptions, using coupling constructions. We prove convergence in distribution and a weak law of large numbers. We also give counterexamples to demonstrate that the assumptions we make are not redundant.

We present bounds on the decay parameter for absorbing birth–death processes adapted from results of Chen (2000), (2001). We address numerical issues associated with computing these bounds, and assess their accuracy for several models, including the stochastic logistic model, for which estimates of the decay parameter have been obtained previously by Nåsell (2001).

In this paper we present counter-intuitive examples for the multiclass queueing network, where each station may serve more than one job class with differentiated service priority and each job may require service sequentially by more than one service station. In our examples, the network performance is improved even when more jobs are admitted for service.

We introduce a new class of spatial-temporal point processes based on Voronoi tessellations. At each step of such a process, a point is chosen at random according to a distribution determined by the associated Voronoi cells. The point is then removed, and a new random point is added to the configuration. The dynamics are simple and intuitive and could be applied to modelling natural phenomena. We prove ergodicity of these processes under wide conditions.

This paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

Based on the concept of multipower variation we establish a class of easily computable and robust estimators for the integrated volatility, especially including the squared integrated volatility, in Lévy-type stochastic volatility models. We derive consistency and feasible distributional results for the estimators. Furthermore, we discuss the applications to time-changed CGMY, normal inverse Gaussian, and hyperbolic models with and without leverage, where the time-changes are based on integrated Cox-Ingersoll-Ross or Ornstein-Uhlenbeck-type processes. We deduce which type of market microstructure does not affect the estimates.

In this paper, we discuss the problem of the pricing of American-style options in the exponential Lévy security market model. This model is typically incomplete, and we derive the explicit bounds of the interval of no arbitrage prices and the related optimal stopping moments for American put options and American call options in both finite and infinite horizon time. We consider a large class of Lévy processes.

Subcritical Markov branching processes {Z t } die out sooner or later, say at time T < ∞. We give results for the path to extinction {Z uT , 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of x u−1 Z uT , as Z 0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by Z T-u , u > 0, and show that as Z 0= x → ∞ this process converges to a Markov process {Y u }, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.

We consider a birth, death and catastrophe process where the transition rates are allowed to depend on the population size. We obtain an explicit expression for the expected time to extinction, which is valid in all cases where extinction occurs with probability 1.