2019/03/10 by Ioannis Papastathopoulos, Papastathopoulos, Ioannis, Casey, Adrian +1
Economics, Econometrics and Finance · Mathematics · Decision Sciences · #Financial Risk and Volatility Modeling #Stochastic processes and statistical mechanics #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.1903.04059
We derive some key extremal features for kth order Markov chains that can\nbe used to understand how the process moves between an extreme state and the\nbody of the process. The chains are studied given that there is an exceedance\nof a threshold, as the threshold tends to the upper endpoint of the\ndistribution. Unlike previous studies with k>1, we consider processes where\nstandard limit theory describes each extreme event as a single observation\nwithout any information about the transition to and from the body of the\ndistribution. Our work uses different asymptotic theory which results in\nnon-degenerate limit laws for such processes. We study the extremal properties\nof the initial distribution and the transition probability kernel of the Markov\nchain under weak assumptions for broad classes of extremal dependence\nstructures that cover both asymptotically dependent and asymptotically\nindependent Markov chains. For chains with k>1, the transition of the chain\naway from the exceedance involves novel functions of the k previous states,\nin comparison to just the single value, when k=1. This leads to an increase\nin the complexity of determining the form of this class of functions, their\nproperties and the method of their derivation in applications. We find that it\nis possible to derive an affine normalization, dependent on the threshold\nexcess, such that non-degenerate limiting behaviour of the process is assured\nfor all lags. These normalization functions have an attractive structure that\nhas parallels to the Yule-Walker equations. Furthermore, the limiting process\nis always linear in the innovations. We illustrate the results with the study\nof kth order stationary Markov chains with exponential margins based on\nwidely studied families of copula dependence structures.\n