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We develop algorithms to automate discovery of stochastic dynamical system models from noisy, vector-valued time series. By discovery, we mean learning both a nonlinear drift vector field and a diagonal diffusion matrix for an Itô stochastic differential equation in Rd . We parameterize the vector field using tensor products of Hermite polynomials, enabling the model to capture highly nonlinear and/or coupled dynamics. We solve the resulting estimation problem using expectation maximization (EM). This involves two steps. We augment the data via diffusion bridge sampling, with the goal of producing time series observed at a higher frequency than the original data. With this augmented data, the resulting expected log likelihood maximization problem reduces to a least squares problem. We provide an open-source implementation of this algorithm. Through experiments on systems with dimensions one through eight, we show that this EM approach enables accurate estimation for multiple time series with possibly irregular observation times. We study how the EM method performs as a function of the amount of data augmentation, as well as the volume and noisiness of the data.
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This content will become publicly available on March 1, 2026
Maximum trimmed likelihood estimation for discrete multivariate Vasicek processes
The multivariate Vasicek model is commonly used to capture mean-reverting dynamics typical for short rates, asset price stochastic log-volatilities, etc. Reparametrizing the discretized problem as a VAR(1) model, the parameters are oftentimes estimated using the multivariate least squares (MLS) method, which can be susceptible to outliers. To account for potential model violations, a maximum trimmed likelihood estimation (MTLE) approach is utilized to derive a system of nonlinear estimating equations, and an iterative procedure is developed to solve the latter. In addition to robustness, our new technique allows for reliable recovery of the long-term mean, unlike existing methodologies. A set of simulation studies across multiple dimensions, sample sizes and robustness configurations are performed. MTLE outcomes are compared to those of multivariate least trimmed squares (MLTS), MLE and MLS. Empirical results suggest that MTLE not only maintains good relative efficiency for uncontaminated data but significantly improves overall estimation quality in the presence of data irregularities. Additionally, real data examples containing daily log-volatilities of six common assets (commodities and currencies) and US/Euro short rates are also analyzed. The results indicate that MTLE provides an attractive instrument for interest rate forecasting, stochastic volatility modeling, risk management and other applications requiring statistical robustness in complex economic and financial environments.
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- PAR ID:
- 10657306
- Publisher / Repository:
- MDPI
- Date Published:
- Journal Name:
- Economies
- Volume:
- 13
- Issue:
- 3
- ISSN:
- 2227-7099
- Page Range / eLocation ID:
- 68
- Subject(s) / Keyword(s):
- econometric modeling autoregressive models times series analysis maximum trimmed likelihood estimation statistical robustness outliers
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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