On Max-Semistable Laws and Extremes for Dynamical Systems

Suppose (f,X,μ) is a measure preserving dynamical system and ϕ:X→R a measurable observable. Let Xi=ϕ∘fi−1 denote the time series of observations on the system, and consider the maxima process Mn:=max{X1,…,Xn}. Under linear scaling of Mn, its asymptotic statistics are usually captured by a three-parameter generalised extreme value distribution. This assumes certain regularity conditions on the measure density and the observable. We explore an alternative parametric distribution that can be used to model the extreme behaviour when the observables (or measure density) lack certain regular variation assumptions. The relevant distribution we study arises naturally as the limit for max-semistable processes. For piecewise uniformly expanding dynamical systems, we show that a max-semistable limit holds for the (linear) scaled maxima process.

Associating Climatic Trends with Stochastic Modelling of Flow Sequences

Water is essential to all lifeforms including various ecological, geological, hydrological, and climatic processes/activities. With the changing climate, associated El Niño/Southern Oscillation (ENSO) events appear to stimulate highly uncertain patterns of precipitation (P) and evapotranspiration (E) processes across the globe. Changes in P and EV patterns are highly sensitive to temperature (T) variation and thus also affect natural streamflow processes. This paper presents a novel suite of stochastic modelling approaches for associating streamflow sequences with climatic trends. The present work is built upon a stochastic modelling framework (HMM_GP) that integrates a hidden Markov model (HMM) with a generalised Pareto (GP) distribution for simulating synthetic flow sequences. The GP distribution within the HMM_GP model aims to improve the model’s efficiency in effectively simulating extreme events. This paper further investigated the potential of generalised extreme value distribution (GEV) coupled with an HMM model within a regression-based scheme for associating the impacts of precipitation and evapotranspiration processes on streamflow. The statistical characteristic of the pioneering modelling schematic was thoroughly assessed for its suitability to generate and predict synthetic river flow sequences for a set of future climatic projections, specifically during ENSO events. The new modelling schematic can be adapted for a range of applications in hydrology, agriculture, and climate change.

Analysing Extreme Risk in the South African Financial Index (J580) using the Generalised Extreme Value Distribution

The aim of this study is to model the probabilistic behaviour of unusually large financial losses (extreme-risk)and gains of the South African Financial Index (J580). Risk is defined as uncertainty in return in this paper. This study makes use of Extreme Value Theory (EVT) for the period years: 1995-2018 to build models that are used to estimate extreme losses and gains. The quarterly block maxima/minima of monthly returns are tted to the Generalised Extreme Value Distribution (GEVD). Return levels (maximum loss/gain) based on the parameters from the GEVD are estimated. A comparative analysis with the Generalised Pareto Distribution (GPD) is carried out. The study reveals that EVT provides an efficient method of forecasting potentially high risks in advance. The conclusion is that analysing extreme risk in the South African Financial Index helps investors understand its riskness better and manage to reduce the risk exposure in this portfolio.