  Stochastic volatility often implies increasing risks that are difficult to
capture given the dynamic nature of real-world applications. We propose using
arc length, a mathematical concept, to quantify cumulative variations (the
total variability over time) to more fully characterize stochastic volatility.
The hazard rate, as defined by the Cox proportional hazards model in survival
analysis, is assumed to be impacted by the instantaneous value of a
longitudinal variable. However, when cumulative variations pose a significant
impact on the hazard, this assumption is questionable. Our proposed Bayesian
Arc Length Survival Analysis Model (BALSAM) infuses arc length into a united
statistical framework by synthesizing three parallel components (joint models,
distributed lag models, and arc length). We illustrate the use of BALSAM in
simulation studies and also apply it to an HIV/AIDS clinical trial to assess
the impact of cumulative variations of CD4 count (a critical longitudinal
biomarker) on mortality while accounting for measurement errors and relevant
variables.
