Abstract We introduce a new definition of bubbles in discrete-time models based on the discounted stock price losing mass under an equivalent martingale measure at some finite drawdown. provide probabilistic characterisations this and give examples martingales that are others not. In Markovian case, we sufficient analytic conditions for presence bubbles. also show existence is directly linked t...

The construction of score-driven filters for nonlinear time series models is described, and they are shown to apply over a wide range disciplines. Their theoretical practical advantages other methods highlighted. Topics covered include robust modeling, conditional heteroscedasticity, count data, dynamic correlation association, censoring, circular switching regimes.