Arthur Thomas (LEDa, Université Paris Dauphine - PSL).

Titre : Forecasting extreme trajectories using semi-norm representations

Abstract : For a two-sided stable moving average, this paper studies the conditional distribution of future paths given a piece of observed trajectory when the process is far from its central values. Under this framework, vectors of the form Xt =(Xt-m, . . . ,Xt, Xt+1, . . . ,Xt+h), m >0, h>1, are multivariate stable and the dependence between the past and future components is encoded in their spectral measures. A new representation of stable random vectors on unit cylinder sets for an adequate semi-norm is proposed to describe the tail behaviour of vectors Xt when only the first m+1 components are assumed to be observed and large in norm. Not all stable vectors admit such a representation and (Xt) must be anticipative enough for Xt to admit one. The conditional distribution of future paths can then be explicitly derived using the regularly varying tails property of stable vectors and has a natural interpretation in terms of pattern identification. Through Monte Carlo simulations we develop procedures to forecast crash probabilities and crash dates and demonstrate their finite sample performances. We estimate probabilities and reversal dates of El Nino and La Nina occurrences as an empirical illustration.

Download slides