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Extreme events are ubiquitous in nature and social society, including natural disasters, accident disasters, crises in public health (such as Ebola and the COVID-19 pandemic), and social security incidents (wars, conflicts, and social unrest). These extreme events will heavily impact financial markets and lead to the appearance of extreme fluctuations in financial time series. Such extreme events lack statistics and are thus hard to predict. Recurrence interval analysis provides a feasible solution for risk assessment and forecasting. This Element aims to provide a systemic description of the techniques and research framework of recurrence interval analysis of financial time series. The authors also provide perspectives on future topics in this direction.

Interval analysis (Mathematics) --- Time-series analysis. --- Recurrent sequences (Mathematics) --- Business mathematics. --- Financial risk management --- Mathematical models.