A robust estimation framework for expectation-maximization in financial time series modeling

Financial time series are often non-stationary and unstable, making likelihood-based estimation challenging. Standard Expectation-Maximization (EM) estimation maximizes the observed-data likelihood, but in financial time series with local instability it may assign unacceptably low likelihood to some observed segments. This motivates a robust estimation objective that focuses on low-likelihood observed segments, since segments that have actually been observed should not be treated as small-probability events. To address this estimation problem, we formulate a robust objective and develop a robust EM algorithm for probabilistic financial time-series models. This algorithm preserves the basic EM structure and modifies only the Maximization (M)-step into a robust M-step to improve robustness to local non-stationarity. To demonstrate its practical value, we develop a robust version of the Baum-Welch (BW) algorithm for hidden Markov models, a classic model in quantitative finance. We formulate the problem and solve it using Lagrangian duality, resulting in weighted parameter updates. We also show monotonic non-decrease of the original robust log-likelihood and provide a convergence result. Moreover, we propose a novel, data-driven proxy derived from the robust BW (RBW) algorithm to characterize periods of high data heterogeneity. Experiments on synthetic and real-world financial datasets show that the RBW algorithm provides more stable calibration for 5% Value-at-Risk, with corresponding improvements in risk-adjusted trading performance.