2003/06/01 by Lorenzo Galleani, L Galleani, Laura Sacerdote +5 · 2 citations
Decision Sciences · Economics, Econometrics and Finance · Physics and Astronomy · #Advanced Frequency and Time Standards #Complex Systems and Time Series Analysis #Scientific Measurement and Uncertainty Evaluation
paper · doi:10.1088/0026-1394/40/3/305
crossref issued 2003/06/01 · crossref published 2003/06/01 · crossref published-print 2003/06/01 · openalex publication_date 2003/06/01 · crossref published-online 2003/06/06 · crossref created 2004/01/09 · crossref deposited 2020/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30 · crossref indexed 2026/08/01
A mathematical model for the clock phase and frequency deviation based on the theory of stochastic differential equations (SDEs) is discussed. In particular, we consider a model that includes what are called the `white and random walk frequency noises' in time metrology, which give rise in a mathematical context to a Wiener and an integrated Wiener process on the clock phase. Due to the particular simple expression of the functions involved an exact solution exists, and we determine it by considering the model in a wider theoretical context, which is suitable for generalizations to more complex instances. Moreover, we determine an iterative form for the solution, useful for simulation and further processing of clock data, such as filtering and prediction. The Euler method, generally applied in literature to approximate the solution of SDEs, is examined and compared to the exact solution, and the magnitude of the approximation is evaluated. The possible extension of the model to other noise sources, such as the flicker and white phase noises, is also sketched.