A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

A threshold AR(1) process with boundary width 2δ > 0 was defined by Brockwell and Hyndman [5] in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold AR(1) process with δ = 0 in terms of the weak solution of a certain stochastic differential equation. Two characterizations of the distributions of the process are investigated. Both express the characteristic function of the transition probability distribution as an explicit functional of standard Brownian motion. It is shown that the joint distributions of this solution with δ = 0 are the weak limits as δ ↓ 0 of the distributions of the solution with δ > 0. The sense in which an approximating sequence of processes used by Brockwell and Hyndman [5] converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron–Martin–Girsanov formula and results of Engelbert and Schmidt [9]. We also derive the stationary distribution (under appropriate assumptions) and investigate stability of these processes.

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfy λ0 = 0, λ j > 0 for each j > 0, and . Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λ j = jλ, µj = jμ) with catastrophes of several different types.

The distribution of the extinction time for a linear birth and death process subject to catastrophes is determined. The catastrophes occur at a rate proportional to the population size and their magnitudes are random variables having an arbitrary distribution with generating function d(·). The asymptotic behaviour (for large initial population size) of the expected time to extinction is found under the assumption that d(.) has radius of convergence greater than 1. Corresponding results are derived for a related class of diffusion processes interrupted by catastrophes with sizes having an arbitrary distribution function.

We consider Markov models for growth of populations subject to catastrophes. Emphasis is placed on discrete-state models where immigration is possible and the catastrophe rate is population-dependent. Explicit formulas for descriptive quantities of interest are derived when catastrophes reduce population size by a random amount which is either geometrically, binomially or uniformly distributed. Comparison is made with continuous-state Markov models in the literature in which population size evolves continuously and deterministically upwards between random jumps downward.

The long-run average cost per unit time of operating a finite dam controlled by a PlM policy (Faddy (1974), Zuckerman (1977)) is determined when the cumulative input process is (a) a Wiener process with drift and (b) the integral of a Markov chain. It is shown how the cost for (a) can be obtained as the limit of the costs associated with a sequence of input processes of the type (b).

A construction is given of a process in which X(t) represents the content at time t of a dam whose cumulative input process is a Lévy process with measure v and whose release rate at time t is r(X(t)). It is assumed only that r(0) = 0 and that r is strictly positive and left-continuous with strictly positive finite right limits on (0,∞). The sample-paths of X are shown to satisfy the storage equation

The process X is analyzed using renewal theory and stochastic comparison techniques, and necessary and sufficient conditions are found in terms of v and r for X to have a stationary distributionπ. These generalize previous results which were obtained under the assumption that v is finite. Conditions for Πto have an atom at 0 are considered in some detail, and related results on the positivity of the expected occupation time of level 0 are given.

Necessary and sufficient conditions for the existence of Πare expressed in terms of the existence of non-negative integrable solutions of certain integral equations and conditions are given under which such solutions are necessarily stationary densities for X. A simple sufficient condition for X to have a stationary distribution is found in terms of and in the case when r is non-decreasing the condition is shown to be also necessary. Finally some examples are considered; these show that the results described above unify various known conditions in special cases, and confirm several conjectures in the related literature.

A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

The method of invariant imbedding, used extensively in astrophysics and in neutron transport theory, is applied to the analysis of dams whose input rate is a finite state-space continuous-parameter Markov chain. Passage-time distributions are considered as functions of the dam capacity. When the release rate is constant the method yields first-order non-linear differential equations with initial conditions obtained from the easily derived behaviour of a dam with zero capacity. Solutions of these equations are given. Some possible correlation functions for the increments of the cumulative input process are examined when the underlying Markov chain is stationary. Finally the analysis is extended to deal with dams having content-dependent release-rate of the form

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt} of an infinite dam whose cumulative input {At} is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = xα and {At} is stable with index β ∊ (0, 1). In general if EAt, < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA1 is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA1 it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt} as t → ∞. For {At} stable with index β and r(x) = xα, α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {Xt} in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.

We investigate the nature of the set of emptiness times of a dam whose release rate depends on the content and whose cumulative input process is a pure-jump Lévy process. Detailed results are obtained for stable input processes and release functions of the form r(x) = xβ I(o,∞)(x).

If X(t) is a Wiener process with EX(t) = vt and var X(t) = σ2t (where v > 0) and if M(t) = max0≦τ≦tX(τ) and Ta is the time of first passage through level a (where a > 0) we show that where is the coefficient of variation of Ta. Applications of the result to the “maturity-time” representation of cell-growth and to queues with heavy traffic are discussed.

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x1, x2, x3, x4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z(4) (a0,…, an, t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y01,…, Y04,…, Yi1,…, Yi4,…, Yn4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij, i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).

Starting from a characterization of radiative transfer in terms of a collision rate λ and a single-collision transition probability Ψ, we study the distribution of the generalized state ζ(t) of a radiation particle at time t conditional on a specified initial state at time t= 0. The generalized state is a vector consisting of the state ω(t) at time t and the states ω1, ω2, …, ωn of the particle immediately after the collisions it experiences in the time interval (0, t]. The variable ζ(t) takes values in a population space and can be studied conveniently with the aid of a certain generating functional G. The first-collision integral equation and the backward integro-differential equation for G are derived. Simultaneous consideration of the first-collision and last-collision equations lead to a generalized reciprocity principle for G. First-passage problems are also considered. Finally a number of illustrative examples are given.

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

Consider a set of n elements α1, … , αn (denoted by S) and the set of all possible multiplication tables on these elements. The total number of such tables is clearly and each table can be represented by a square matrix [μij] where μij is the product αiαj (μij ∈ S, i = 1, … , n; j = 1, … , n). The triple (αi, αj, αk) is said to be associative if the following equation is satisfied:

1.1

A particle travels along the half-line [0,∞) in such a way that it has probability λδu + o(δu) of generating an event in any small element (u,u + δu) of its track. The particle is observed only in the line segment 0 ≦ u ≦ x and successive events occur at X1, X2, …, Xn (0 ≦ X1 ≦ X2 ≦ … ≦ Xn ≦ x) where X1, …,Xn are random variables and the number n of events in [0, x] is also random. The distances constitute a finite univariate population process as defined by Moyal [1], the individuals being the distances Yi with state space [0, x].