Levy Approximation of Impulsive Recurrent Process with Semi-Markov Switching

In this paper, the weak convergence of impulsive recurrent process with semi-Markov switching in the scheme of Levy approximation is proved. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness

Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

We consider an evolutionary system switched by a semi-Markov process. For this system, we obtain an inhomogeneous diffusion approximation results where the initial process is compensated by the averaging function in the average approximation scheme.Для систем, що перемикаються иапівмарковськими процесами, одержано результати про неоднорідну дифузійну апроксимацію, де вихідний процес компенсується усередненою функцією в апроксимаційній схемі усереднення

A semi-Markov model with memory for price changes

We study the high frequency price dynamics of traded stocks by a model of returns using a semi-Markov approach. More precisely we assume that the intraday returns are described by a discrete time homogeneous semi-Markov which depends also on a memory index. The index is introduced to take into account periods of high and low volatility in the market. First of all we derive the equations governing the process and then theoretical results have been compared with empirical findings from real data. In particular we analyzed high frequency data from the Italian stock market from first of January 2007 until end of December 2010

SEMI-MARKOV MODELS FOR SEISMIC HAZARD ASSESSMENT IN CERTAIN AREAS OF GREECE

The long-term probabilistic seismic hazard is studied through the application of semi-Markov model. In this model a sequence of earthquakes is considered as a Markov process and the waiting time distributions depend only on the type of the last and the next event. The principal hypothesis of the model is the property of one-step memory, according to which the probability of moving to any future state depends only on the present state. The model under consideration defines a continuous-time, discrete-state stationary process in which successive state occupancies are governed by the transition probabilities of the Markov process. The space of states is considered to be finite and the process started far in the past has achieved stationarity. Firstly, a non-parametric method is applied in order to determine the waiting times. Then, the waiting times derived by means of the exponential and Weibull distributions will be compared to each other, as well as with the actual waiting times. Thus, the probability of occurrence of the anticipated earthquakes of a specific magnitude scale is calculated. The models are applied to an historical catalogue for Northern Aegean Sea

Diffusion approximation algorithms in merging phase space

Diffusion approximation algorithms for stochastic systems in split and merging phase space are represented in servey form. The main mathematical tools of such algorithms are described in our book ”Stochastic Systems in Merging Phase Space” (World Scientific Publishing, 2005)

A Markov model for seismic hazard analysis along the Hellenic subduction Zone (Greece)

Εφαρμόζεται  ένα  ομογενές  Μαρκοβιανό  μοντέλο  διακριτού  χρόνου  και  χώρου καταστάσεων για τη γένεση σεισμών στο Ελληνικό Τόξο, περιοχή υψηλής σεισμικής δραστηριότητας  και  ιδιαίτερης  σημασίας  από  σεισμοτεκτονική  άποψη.  Το  μοντέλο παρέχει μια στοχαστική αναπαράσταση της γένεσης των σεισμών συμβάλλοντας στην εκτίμηση της σεισμικής επικινδυνότητας για την περιοχή μελέτης. Τα δεδομένα που χρησιμοποιούνται  λήφθηκαν  από  τον  κατάλογο  του  Τομέα  Γεωφυσικής  του Αριστοτελείου  Πανεπιστημίου  Θεσσαλονίκης,  ο  οποίος  θεωρείται  ομογενής  και  πλήρης  για  σεισμούς  με  από  το  1911.  Ο  συνεχής  χώρος  καταστάσεων χωρίζεται  σε  κλάσεις  μεγεθών  καθορίζοντας  με  αυτό  τον  τρόπο  τον  χώρο καταστάσεων  του  μοντέλου.  Η  στοχαστική  συμπεριφορά  του  μοντέλου  καθορίζεται XLVII, No 3 - 1376από  τον  πίνακα  πιθανοτήτων  μετάβασής  του,  του  οποίου  υπολογίζεται  αρχικά ο εκτιμητής   μέγιστης   πιθανοφάνειας.   Στη  συνέχεια   εκτιμώνται   σημαντικά χαρακτηριστικά της Μαρκοβιανής αλυσίδας, παρέχοντας προγνωστικά αποτελέσματα σχετικά  με  την  πιθανότητα  γένεσης  ενός  επερχόμενου  ισχυρού  σεισμού.  Οι υπολογισμοί  περιλαμβάνουν  την  εκτίμηση  της  μέσης  τιμής,  της  διασποράς  και  του 95% διαστήματος εμπιστοσύνης του πλήθους των βημάτων που απαιτούνται ώστε η Μαρκοβιανή αλυσίδα να μεταβεί για πρώτη φορά σε μια ορισμένη κατάσταση (που σχετίζεται με τη γένεση ενός επερχόμενου ισχυρού σεισμού).A homogeneous finite–state discrete–time Markov model is applied for the earthquake occurrence in the Hellenic Subduction Zone (Greece), a region accommodating high seismic activity, being a key structure from a seismotectonic point of view. An attempt is made to provide a stochastic representation of the earthquake process and to assess the seismic hazard through the application of the Markov model. The model is applied on a complete data sample comprising strong () eart  h-quakes that occurred in the study area since 1911 up to present. The continuous magnitude scale is divided into appropriate intervals to specify discrete states of the model. As the stochastic behavior of the model is governed by its transition probability matrix, we firstly estimate its well–known maximum likelihood estimator. The estimation of the transition probability matrix leads to the estimation of important indicators of the Markov chain, including hitting times and failure rate functions. The   mean number of steps for the first occurrence of an anticipated earthquake (belonging to the class with the stronger events, which we are more interested in) is estimated along with its variance. In a next step, we calculate the confidence interval of the   aforementioned estimators