A simple variational approach to the estimation of timeseries is studied in detail and mathematical rigor. The functional in question is a complexity penalized sum of squares. The results include existence and uniqueness of the statistical estimate, as well as continuous dependence and stability in dependence on parameters and data.
Wittich, O., Kempe, A., Winkler, G., & Liebscher, V. (2008). Complexity penalized least squares estimators : Analytical results. Mathematische Nachrichten, 281(4), 582-595. https://doi.org/10.1002/mana.200510627