2024/08/16 by Elvis Han Cui, Cui, Elvis Han
Mathematics · #Applications (stat.AP) #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #FOS: Mathematics #Probability (math.PR) #Statistical Methods in Clinical Trials #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2408.16011
openalex publication_date 2024/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Brownian motion is a compact mathematical language for continuous-time uncertainty in biostatistics. This tutorial develops the process from construction and path properties to tools that recur in applied biomedical work: the Markov and strong Markov properties, the Karhunen-Loeve expansion, functional principal component analysis (Functional PCA), reflection principles, local time, stochastic differential equations (SDEs), Brownian bridges, and empirical-process limits. The applications emphasize longitudinal biomarkers, degradation modelling, first-passage endpoints, dynamic frailty, group-sequential monitoring, calibration diagnostics, recurrent-event processes, electronic health records, and wearable streams. A short cross-domain section uses literary and historical archives to make Brownian-bridge thinking concrete without shifting the paper away from biostatistics, and includes a reproducible chapter-level experiment on Frankenstein. The Black-Merton-Scholes model is included as a solved SDE template, not as a finance application in its own right. The aim is to connect rigorous probability with modelling decisions faced by biostatisticians when biological processes evolve between noisy observation times.