Dynamic functional connectivity (DFC) analysis involves measuring correlated neural activity over time across multiple brain regions. Significant regional correlations among neural signals, such as those obtained from resting-state functional magnetic resonance imaging (fMRI), may represent neural circuits associated with rest. The conventional approach of estimating the correlation dynamics as a sequence of static correlations from sliding time-windows has statistical limitations. To address this issue, we propose a multivariate stochastic volatility model for estimating DFC inspired by recent work in econometrics research. This model assumes a state-space framework where the correlation dynamics of a multivariate normal observation sequence is governed by a positive-definite matrix-variate latent process. Using this statistical model within a sequential Bayesian estimation framework, we use blood oxygenation level dependent activity from multiple brain regions to estimate posterior distributions on the correlation trajectory. We demonstrate the utility of this DFC estimation framework by analyzing its performance on simulated data, and by estimating correlation dynamics in resting state fMRI data from a patient with a disorder of consciousness (DoC). Our work advances the state-of-the-art in DFC analysis and its principled use in DoC biomarker exploration.