We consider a controlled i.i.d. process, where several i.i.d. sources are sampled sequentially. At each time instant, a controller determines from which source to obtain the next sample. Any causal sampling policy, possibly history-dependent, may be employed. The purpose is to characterize the extremal large deviations of the sample mean, namely to obtain asymptotic rate bounds (similar to and extending Cramér's theorem) which hold uniformly over all sampling policies. Lower and upper bounds are obtained, and it is shown that in many (but not all) cases stationary sampling policies are sufficient to obtain the extremal large deviations rates. These results are applied to a hypothesis testing problem, where data samples may be sequentially chosen from several i.i.d. sources (representing different types of experiments). The analysis provides asymptotic estimates for the error probabilities, corresponding both to optimal and to worst-case sampling policies.

In this article we examine an R & D project in which the project status changes according to a diffusion process. The decision variables include a resource expenditure strategy and a stopping policy which determines when the project should be terminated. The drift and the diffusion parameters of the project status process are assumed to be functions of the resource expenditure rate.

The terminal reward from the project is a non-decreasing function of the project status. Our purpose is to select optimal investment strategies under the discounted return criterion.

The value of the project is shown to be a solution of a second order, non-linear differential equation. Finally, we derive the optimal investment strategies for an R & D project in which the project status changes according to a non-homogeneous compound Poisson process by using diffusion approximation.