This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.

A new Markov chain is introduced which can be used to describe the family relationships among n individuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain.

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation used to describe a number of processes including radioactive decay and population growth.

In this paper, we develop general repair models for a repairable system by using the idea of the virtual age process of the system. If the system has the virtual age Vn –1 = y immediately after the (n – l)th repair, the nth failure-time Xn is assumed to have the survival function where is the survival function of the failure-time of a new system. A general repair is represented as a sequence of random variables An taking a value between 0 and 1, where An denotes the degree of the nth repair. For the extremal values 0 and 1, An = 1 means a minimal repair and An= 0 a perfect repair. Two models are constructed depending on how the repair affects the virtual age process: Vn = Vn– 1+ AnXn as Model 1 and Vn = An(Vn– 1 + Xn) as Model II. Various monotonicity properties of the process with respect to stochastic orderings of general repairs are obtained. Using a result, an upper bound for E[Sn] when a general repair is used is derived.

In this article we shall present a summary of the various stochastic approaches and applications to chemical reaction kinetics, but before discussing these we first briefly introduce the basic ideas and definitions of classical or deterministic chemical kinetics. One of the basic questions to which chemists address themselves is the rate of chemical reactions, or in other words, how long it takes for a chemical reaction to attain completion, or equilibrium. Apparently the first significant quantitative investigation was made in 1850 by L. Wilhelmy [93]. He studied the inversion of sucrose (cane sugar) in aqueous solutions of acids, whose reaction is He found empirically that the rate of decrease of concentration of sucrose was simply proportional to the concentration remaining unconverted, i.e., if S(t) is the concentration of sucrose, then The constant of proportionality is called the rate constant of the reaction. If So is the initial concentration of sucrose, then Since then an enormous number of reactions has been studied and the field of chemical kinetics is now one of the largest areas of chemical research. The importance of the field lies in the fact that it yields concise expressions for the time dependence of reactions, predicts yields, optimum economic conditions, and gives one much insight into the actual molecular processes involved. The detailed molecular picture of a reaction process is called the mechanism of the reaction.

A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

We consider a population of n projects which in general continue to evolve whether in operation or not (although by different rules). It is desired to choose the projects in operation at each instant of time so as to maximise the expected rate of reward, under a constraint upon the expected number of projects in operation. The Lagrange multiplier associated with this constraint defines an index which reduces to the Gittins index when projects not being operated are static. If one is constrained to operate m projects exactly then arguments are advanced to support the conjecture that, for m and n large in constant ratio, the policy of operating the m projects of largest current index is nearly optimal. The index is evaluated for some particular projects.

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.

Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

A device is repaired at failure. With probability p, it is returned to the ‘good-as-new' state (perfect repair), with probability 1 – p, it is returned to the functioning state, but it is only as good as a device of age equal to its age at failure (imperfect repair). Repair takes negligible time. We obtain the distribution Fp of the interval between successive good-as-new states in terms of the underlying life distribution F. We show that if F is in any of the life distribution classes IFR, DFR, IFRA, DFRA, NBU, NWU, DMRL, or IMRL, then Fp is in the same class. Finally, we obtain a number of monotonicity properties for various parameters and random variables of the stochastic process. The results obtained are of interest in the context of stochastic processes in general, as well as being useful in the particular imperfect repair model studied.

The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numéraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numéraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself.

The key theorem of general numéraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.

Population genetics is that branch of genetics, whose object is the study of the genetical make-up of natural populations. By investigating the laws which govern the genetic structure of natural populations, we intend to clarify the mechanism of evolution.

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.

In [3] this author gave conditions under which a sequence of jump Markov processes Xn(t) will converge to the solution X(t) of a system of first order ordinary differential equations, in the sense that for every δ > 0.

First consider two particles diffusing on the same line, with positions at time t given by y1(t) and y2(t) respectively. We suppose that they cannot pass one another, so that if initially y1(0) < y2(0), then . As long as the two particles are not in contact, we suppose that each moves, independently of the other, according to the Wiener process. We must also prescribe what happens when collisions occur. There seems to be no unique natural way to do this, and we shall adopt a method recommended by its mathematical convenience. However, we shall see that there are two possible motivations for the prescription, one in terms of random walks, the other in terms of a more mechanical model.

A stochastic model is developed to describe the operation in time of the following maintained system setting. A piece of equipment is put in operation at time 0. Each time it fails, a maintenance action is taken which, with probability p(t), is a complete repair or, with probability q(t)=1– p(t), is a minimal repair, where t is the age of the equipment in use at the failure time. It is assumed that complete repair restores the equipment to its good as new condition, that minimal repair restores the equipment to its condition just prior to failure and that both maintenance actions take negligible time.

If the equipment's life distribution F is a continuous function, the successive complete repair times are shown to be a renewal process with interarrival distribution for t ≧ 0. Preservation and monotone properties of the model extending the results of Brown and Proschan (1983) are obtained.

This paper approaches the problem of computing the price of an Asian option in two different ways. Firstly, exploiting a scaling property, we reduce the problem to the problem of solving a parabolic PDE in two variables. Secondly, we provide a lower bound which is so accurate that it is essentially the true price.

The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More important, the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

We consider a queuing system consisting of a finite number of identical exponential servers. Each server has its own queue, and upon arrival each customer must be assigned to some server's queue. Under the assumption that no jockeying between queues is permitted, it is shown that the intuitively satisfying rule of assigning each arrival to the shortest line maximizes, with respect to stochastic order, the discounted number of customers to complete their service in any time t.

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

We introduce a new class of queueing networks in which customers are either ‘negative' or ‘positive'. A negative customer arriving to a queue reduces the total customer count in that queue by 1 if the queue length is positive; it has no effect at all if the queue length is empty. Negative customers do not receive service. Customers leaving a queue for another one can either become negative or remain positive. Positive customers behave as ordinary queueing network customers and receive service. We show that this model with exponential service times, Poisson external arrivals, with the usual independence assumptions for service times, and Markovian customer movements between queues, has product form. It is quasi-reversible in the usual sense, but not in a broader sense which includes all destructions of customers in the set of departures. The existence and uniqueness of the solutions to the (nonlinear) customer flow equations, and hence of the product form solution, is discussed.