We study an intertemporal consumption and portfolio choice problem under Knightian uncertainty in which agent’s preferences exhibit local intertemporal substitution. We also allow for market frictions in the sense that the pricing functional is nonlinear. We prove existence and uniqueness of the optimal consumption plan, and we derive a set of sufficient first-order conditions for optimality. With the help of a backward equation, we are able to determine the structure of optimal consumption plans. We obtain explicit solutions in a stationary setting in which the financial market has different risk premia for short and long positions.

An expression for the asymptotic or steady-state performance function is derived for a multistate coherent system when each component changes states in time according to a semi-Markov process, the stochastic processes being mutually independent. This generalizes the expression for system availability of a binary coherent system when the components are governed by mutually independent alternating renewal processes.

A new Markov process is introduced, describing growth or spread in two dimensions, via the aggregation of particles or the filling of cells. States of the process are configurations of part of the boundary of the growing aggregate, and transitions are captures or escapes of single particles. For suitably chosen transition rates, the process is dynamically reversible, leading to an explicit stationary distribution and a statistical description of the boundary. The growth rate is calculated and growth behaviour described. Different asymptotic relations between transition rates lead to different growth patterns or regimes. Besides the regimes familiar in polymer crystal growth, several new ones are described. The aggregate can have a porous structure resembling thin solid films deposited from vapour. Two measures of porosity, one for the boundary and one for the bulk, are calculated. The process is relevant to growing colonies of bacteria or the like, to the spread of epidemics and grass or forest fires, and to voter models.

This meeting was envisaged by the organisers as the first of a continuing series of short, informal conferences within the general area of applied stochastic processes. The twenty-seven participants were drawn mostly from statistics and mathematics departments of British Universities, but also included a small number of biologists and geographers. The meeting was notable for its pleasantly informal atmosphere, and the consequent lively interchange of ideas, although for future meetings in the series a better balance between statisticians and applied scientists should, we feel, be effected.

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.

The paper deals with large trunk line systems of the type appearing in telephone networks. There are many nodes or input sources, each pair of which is connected by a trunk line containing many individual circuits. Traffic arriving at either end of a trunk line wishes to communicate to the node at the other end. If the direct route is full, a rerouting might be attempted via an alternative route containing several trunks and connecting the same endpoints. The basic questions concern whether to reroute, and if so how to choose the alternative path. If the network is ‘large’ and fully connected, then the overflow traffic which is offered for rerouting to any trunk comes from many other trunks in the network with no one dominating. In this case one expects that some sort of averaging method can be used to approximate the rerouting requests and hence simplify the analysis. Essentially, the overflow traffic that a trunk offers the network for rerouting is in some average sense similar to the overflow traffic offered to that trunk. Indeed, a formalization of this idea involves the widely used (but generally heuristic) ‘fixed point' approximation method. One sets up the fixed point equations for appropriate rerouting strategies and then solves them to obtain an approximation to the system loss. In this paper we work in the heavy traffic regime, where the external offered traffic to any trunk is close to the service capacity of that trunk. It is shown that, as the number of links and circuits within each link go to infinity and for a variety of rerouting strategies, the system can be represented by an averaged limit. This limit is a reflected diffusion of the McKean–Vlasov (propagation of chaos) type, where the driving terms depend on the mean values of the solution of the equation. The averages occur due to the symmetry of the network and the averaging effects of the many interactions. This provides a partial justification for the fixed point method. The concrete dynamical systems flavor of the approach and the representations of the limit processes provide a useful way of visualizing the system and promise to be useful for the development of numerical methods and further analysis.

Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb{X}$ . Let $ S_n = \sum_{k=1}^n \ln f_{X_k}^{\prime}(1)$ be the Markov walk associated to $(X_n)_{n\geq 0}$ , where $f_i$ is the offspring generating function when the environment is $i \in \mathbb{X}$ . Conditioned on the event $\{ Z_n>0\}$ , we show the nondegeneracy of the limit law of the normalized number of particles ${Z_n}/{e^{S_n}}$ and determine the limit of the law of $\frac{S_n}{\sqrt{n}} $ jointly with $X_n$ . Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of $ \log Z_n$ and $X_n$ given $Z_n>0$ .

We examine necessary and sufficient conditions for recurrence and positive recurrence of a class of irreducible, level-dependent quasi-birth-and-death (LDQBD) processes with a block tridiagonal structure that exhibits asymptotic convergence in the rows as the level tends to infinity. These conditions are obtained by exploiting a multi-dimensional Lyapunov drift approach, along with the theory of generalized Markov group inverses. Additionally, we highlight analogies to well-known average drift results for level-independent quasi-birth-and-death (QBD) processes.

The authors prove that certain reliability formulas which link asymptotic availability, mean normal operation time, mean time between failures, mean number of failures over a period of time and asymptotic Vesely rate, and which are well known in the case of modelling using a Markov jump process or an alternating renewal process, are also true in the context of more general modelling.