Title: On Fractional Counting Processes
Speaker: Dr. Kuldeep Kataria, Assistant Professor, IIT Bhilai.
Day, Date and Time: Thursday, November 11, 2021, 9:30 AM to 10:30 AM.
Abstract:
Advances in various fields of modern studies have shown the limitations of traditional probabilistic models. The one such example is that of the Poisson process which fails to model the data traffic of bursty nature, especially on multiple time scales. The empirical studies have shown that the power law decay of inter-arrival times in the network connection session offers a better model than exponential decay. The quest to improve Poisson model led to the formulations of new processes such as non-homogeneous Poisson process, Cox point process, higher dimensional Poisson process, etc. The fractional generalizations of the Poisson process has drawn the attention of many researchers since the last decade. Recent works on fractional extensions of the Poisson process, commonly known as the fractional Poisson processes, lead to some interesting connections between the areas of fractional calculus, stochastic subordination and renewal theory. The state probabilities of such processes are governed by the systems of fractional differential equations which display a slowly decreasing memory. It seems a characteristic feature of all real systems. Here, we discuss some recently introduced generalized counting processes and their fractional variants. The system of differential equations that governs their state probabilities are discussed.
Brief biography: Dr. Kuldeep Kumar Kataria received his BSc (Hons) degree in Mathematics from St. Stephen’s College, University of Delhi. He received his MSc degree in Mathematics from IIT Kanpur. In 2018, he received his PhD in Mathematics from IIT Bombay. Later, he joined IISc, Bangalore as a NBHM Post-Doctoral Fellow. He is currently working as an Assistant Professor in the Department of Mathematics at IIT Bhilai. The research interest of Dr. Kataria lies in the area of fractional stochastic processes and subordinated (time-changed) versions of certain counting processes. He deals with stable subordinators and space-time fractional versions of the Poisson process. In his PhD thesis, he has studied the applications of Adomian Decomposition Method to certain fractional stochastic processes. So far, Dr. Kataria has published 17 research articles in international journals of repute like Journal of Theoretical Probability, ALEA. Latin American Journal of Probability and Mathematical Statistics, Journal of Mathematical Analysis and Applications, Comptes Rendus Math ́ematique, Statistics and Probability Letters, Stochastic Analysis and Applications, etc. Also, he has published several expository articles in reputed mathematical magazine like American Mathematical Monthly, Mathematics Magazine, etc. For his research contributions he has been honoured with the Award of Excellence in Thesis Work for the year 2016-2018 by Hon’ble Prime Minister of India at the 56th Convocation of the Institute (IIT Bombay).
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