Correlated random walks provide an elementary model for processes that exhibit directional reinforcement behavior. This paper develops optimal multiple stopping strategies - buy/sell rules - for correlated random walks. The work extends previous results given in Allaart and Monticino (2001) by considering random step sizes and allowing possibly negative reinforcement of the walk's current direction. The optimal strategies fall into two general classes - cases where conservative buy-and-hold type strategies are optimal and cases for which it is optimal to follow aggressive trading strategies of successively buying and selling the commodity depending on whether the price goes up or down. Simulation examples are given based on a stock index fund to illustrate the variation in return possible using the theoretically optimal stop rules compared to simpler buy-and-hold strategies.

We consider a risk model with threshold strategy, where the insurance company pays off a certain percentage of the income as dividend whenever the current surplus is larger than a given threshold. We investigate the ruin time, ruin probability, and the total dividend, using methods and results from queueing theory.

In a two-person red-and-black game, each player holds an integral amount of chips. At each stage of the game, each player can bet any integral amount in his possession, winning the chips of his opponent with a probability which is a function of the ratio of his bet to the sum of both players' bets and is called a win probability function. Both players seek to maximize the probability of winning the entire fortune of his opponent. In this paper we propose two new models. In the first model, at each stage, there is a positive probability that two players exchange their bets. In the second model, the win probability functions are stage dependent. In both models, we obtain suitable conditions on the win probability functions such that it is a Nash equilibrium for the subfair player to play boldly and for the superfair player to play timidly.

We give a criterion for extinction or local extinction of branching symmetric α-stable processes in terms of the principal eigenvalue for time-changed processes of symmetric α-stable processes. Here the branching rate and the branching mechanism are spatially dependent. In particular, the branching rate is allowed to be singular with respect to the Lebesgue measure. We apply this criterion to some branching processes.

The stochastic sequential assignment problem assigns distinct workers to sequentially arriving tasks with stochastic parameters. In this paper the assignments are performed so as to minimize the threshold probability, which is the probability of the long-run reward per task failing to achieve a target value (threshold). As the number of tasks approaches infinity, the problem is studied for independent and identically distributed (i.i.d.) tasks with a known distribution function and also for tasks that are derived from r distinct unobservable distributions (governed by a Markov chain). Stationary optimal policies are presented, which simultaneously minimize the threshold probability and achieve the optimal long-run expected reward per task.

We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

A number of stochastic queueing systems exhibit an interesting phenomenon known as the cut-off phenomenon. A properly scaled version of the distance between the transient process and the stationary one converges to a step function as the initial load converges to infinity. The purpose of this paper is to promote the idea that this phenomenon is a direct consequence of the coupling between the two processes, being thus generalizable to systems lacking any kind of Markovian structure.

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.

In this paper the behaviour of the failure rate and reversed failure rate of an n-component coherent system is studied, where it is assumed that the lifetimes of the components are independent and have a common cumulative distribution function F. Sufficient conditions are provided under which the system failure rate is increasing and the corresponding reversed failure rate is decreasing. We also study the stochastic and ageing properties of doubly truncated random variables for coherent systems.

From a random sample X1, X2, …, XN there can be constructed N – K + 1 successive sample means of the form for 0 ≦ n ≦ N − K, where Erdös and Rényi (1970) studied the maximum Σ(N, K) of these N – K + 1 sample means. Under appropriate conditions, they showed that for a wide interval of λ's there exist constants C(λ), depending only on λ and the distribution from which the sample was selected, such that Σ(N, [C(λ) log N])→ λ a.s. as N→∞. In the present article, analogous results are developed for the maximum of the N – K + 1 successive sample medians and, more generally, for all sample quantiles.

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

A model in which a foraging animal can reproduce if its energy reserves reach a critical level is presented. The animal's reserves are modelled as a finite-state Markov chain. The animal has the choice between foraging options which may have the same mean net gain but differ in their variances. In addition to starvation, animals are subject to death because of predation, bad weather, and so on. We focus on the case where the rate of mortality due to these sources is the same under all options. We investigate policies that maximise the expected lifetime reproductive success. It is found that the optimal value function is concave at low reserves (low-variance action region) and convex at high reserves (high-variance action region). The value function under the low-variance action has also the same shape and the same inflexion point. This result allows us to compute optimal policies just by looking at the low-variance value function. The result is also used to show that increasing the mortality rate increases the high-variance region under the optimal policy. The pattern of risk-sensitive behaviour predicted by this model is in contrast to that predicted by a similar model in which no reproduction occurs and the optimality criterion is to minimise the probability of death (McNamara (1990)).

The paper takes a closer look at some insensitivity results which are not contained in the class of insensitive generalised semi-Markov schemes. Based on a study of a special insensitive queue, a conjecture is proposed which covers all known results on insensitive equilibrium distributions. When insensitive events are also included, a study of the G/G/1 queue shows that the conjecture does not always hold.

Suppose that N is a ℤ+-valued random variable and that X,X1,X2,… is a sequence of independent and identically distributed ℤ+ random variables independent of N. In this paper we are interested in properties of the conditional variable In particular, we want to know the asymptotic behavior of the conditional mean ENk or the conditional variance varNk as k→∞. We consider the cases when X is Poisson and when X is mixed Poisson. The problem is motivated by modeling loss reserves in nonlife insurance.

This paper considers the undershoot of a general continuous-time risk process with dependent increments under a certain initial level. The increments are given by the locations and amounts of claims which are described by a stationary marked point process. Under a certain balance condition, it is shown that the distribution of the undershoot depends only on the mark distribution and on the intensity of the underlying point process, but not on the form of its distribution. In this way an insensitivity property is extended which has been proved in Björk and Grandell [3] for the ruin probability, i.e. for the probability that after a finite time interval the initial level will be crossed from above.

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.

The comparison of optimal strategies in a simple stochastic replacement model for two types of machines with identical cost characteristics when one of them is more reliable than the other is conducted. It is proven that the scheduled replacement period for the more reliable type is always less than for the less reliable one. An example is presented when even the expected period of use of a more reliable machine is less than the expected period for the less reliable one. Some related problems are briefly discussed.

We consider a system of particles in birth and death on a stochastic flow. The system includes a particle process tracking the spatial configuration of live particles on the flow. The particle process is a Markov process on the space of bounded counting measures. We show that its transition semigroup is a Feller semigroup and exhibit its pregenerator. The pregenerator defines a martingale problem. We show that the particle process solves the problem uniquely.

Linear and non-linear secondary Poisson process models are developed for the level of carboxyhemoglobin in cigarette smoking. The moments of the time of first exit above a critical threshold are expanded as power series in the initial state of the process. Recurrence relations for the coefficients in the moment series are solved numerically. An explicit formula involving Poisson probability functions is obtained for the mean exit time of shot noise.