Robust filtering and propagation of uncertainty in hidden Markov models (Q2042836)

The basic problem in filtering of continuous-time finite-state hidden Markov models is to derive an optimal online estimator for an unobserved signal process \(\{X_t\}\) from discrete observations of the path of another process \(\{Y_t\}\) whose dynamics depend on the current state of the signal. Filters are usually sensitive to both uncertainty of the model parameters and to errors in the observed data. One may therefore desire a filter to be robust in both of these distinct senses, namely robust with respect to parameter uncertainty, and continuous with respect to the observation path \(t\to\{Y_t\}\). The author proposes a fully pathwise approach to the problem of filtering and constructs resulting filters which are robust in both of these senses based on new results of the optimal control of rough differential equations introduced by \textit{T. J. Lyons} [Rev. Mat. Iberoam. 14, No. 2, 215--310 (1998; Zbl 0923.34056)] and developed by \textit{J. Diehl} et al. [Appl. Math. Optim. 75, No. 2, 285--315 (2017; Zbl 1373.60099)]. The author's approach is an extension of the one described by himself and \textit{S. N. Cohen} [Ann. Appl. Probab. 30, No. 5, 2274--2310 (2020; Zbl 1471.93265); SIAM J. Control Optim. 57, No. 3, 1646--1671 (2019; Zbl 1420.62407)].