A dynamic co-operative two-person game with imperfect information is considered, in which the conflicting aspects of an optimal policy (choice of actions to minimise ‘own cost’, and choice of actions to optimise communication of information) can to a large extent be decoupled and solved separately. In deriving a strategy which is asymptotically optimal (in a sense explained) we derive formulae for the capacity of a vector Gaussian channel and the rate-distortion of a vector Gaussian source.

It is shown that the optimal strategy is in general non-linear in observables, but a numerical example shows that, at least in one rather special but non-trivial case, the best linear strategy may be very nearly optimal. This conclusion is made explicit by the analysis of Section 7.

One way of analysing complicated Markov population processes is to approximate them by a diffusion about the deterministic path. This approximation alone may not, however, answer all the questions which might reasonably be asked. Many processes have phases, for example near boundaries, where a different approximation is required; such processes are better described by a succession of diffusion and special approximations alternately.

This paper looks at the special treatment required near a point where the deterministic equations are in equilibrium. When the equilibrium is unstable, the process will eventually wander off, and possibly follow a diffusion around another deterministic path. If the equilibrium is stable, the process will behave as if in stable equilibrium about it for a very much longer time, but may at last be trapped away from it, for instance in an absorbing state. The results presented describe the distributions of the time and place of leaving neighbourhoods of the equilibrium point. The neighbourhoods considered are large enough, in the unstable case, to make possible the link with the next phase of motion. In the stable case, exit times are shown to be so long that the possibility of exit can often be ignored in practice, and the quasi-equilibrium distribution treated as a true equilibrium. A more detailed result, showing how closely the normal approximation holds in this situation, is also provided.

A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.