The clock errors are modeled by stochastic differential equations (SDE) and the relationships between the diffusion coefficients used in SDE and the Allan variance, a typical tool used to estimate clock noise, are derived. This relationship is fundamental when a mathematical clock model is used, for example in Kalman filter, noise estimation, and clock prediction activities.

The inverse first-passage problem for a Wiener process (Wt)t≥0 seeks to determine a function b: ℝ+→ℝ such that τ=inf{t>0|Wt≥b(t)} has a given law. In this paper two methods for approximating the unknown function b are presented. The errors of the two methods are studied. A set of examples illustrates the methods. Possible applications are enlighted.

We extend the mathematical model based on stochastic differential equations describing the error gained by an atomic clock to the cases of anomalous behavior including jumps and an increase of instability. We prove an exact iterative solution that can be useful for clock simulation, prediction, and interpretation, as well as for the understanding of the impact of clock error in the overall system in which clocks may be inserted as, for example, the Global Satellite Navigation Systems.