AIS Stochastic Processes - Level II
Most Indian Universities however do not have a rigorous study on probability. The aim of this school is to give a comprehensive training to students in a undergraduate/PhD programme on probability and stochastic processes. Also to give them a chance to interact with researchers in these topics. This is a follow up program of last year AIS on stochastic process held here at NISER, Bhubaneswar.
We propose to run three courses – 1) Measure theoretic probability, 2) Conditional probability and Martingale, 3) Brownian motion. There will be around 30 hours of lectures including tutorials per topic over a 3 weeks period.
Organizers:
a) Nabin Kumar Jana, Assistant Professor, NISER, Bhubaneswar
b) Rahul Roy, Professor, ISI, Delhi
Funded by: IASc, Bengaluru and NCM, Mumbai
Target audience: Those who have attended Advanced Instructional School on stochastic processes 2018.
Speakers:
a) B V Rao, CMI, Chennai
b) Rahul Roy, ISI, Delhi
c) Parthanil Roy, ISI, Bangalore
d) Arijit Chakrabarty, ISI, Kolkata
e) Srikanth Iyar, IISc, Bangalore
f) Manjunath Krishnapur, IISc, Bangalore
Syllabus: We plan to cover the following topics in this AIS.
1. Measure theoretic probability: Caratheodory extension theorem , Monotone class theorem, Dynkin’s pi-lambda theorem, MCT, Fatou’s Lemma, DCT, Fubini’s theorem. Probability spaces, random variables and random vectors, expected value and its properties. Independence. Various modes of convergence and their relation. The Borel-Cantelli lemmas. Weak Law of large numbers for i.i.d. finite mean case. Kolmogorov 0-1 law, Kolmogorov’s maximal inequality. Statement of Kolmogorov’s three-Series theorem (proof if time permits). Strong law of large numbers for i.i.d. case. Characteristic functions and its basic properties, inversion formula, Levy’s continuity theorem. Lindeberg CLT, CLT for i.i.d. finite variance case, Lyapunov CLT.
2. Conditional probability and Martingale: Absolute continuity and singularity of measures. Hahn-Jordon decomposition, Radon-Nikodym Theorem, Lebesgue decomposition. Conditional expectation – Definition and Properties. Regular conditional probability, proper RCP. Regular conditional distribution. Discrete parameter martingales, sub-and super-martingales. Doob’s Maximal Inequality, Upcrossing inequality, martingale convergence theorem, Lp inequality, uniformly integrable martingales, reverse martingales, Levy’s upward and downward theorems. Stopping times, Doob’s optional sampling theorem. Discrete martingale transform, Doob’s Decomposition Theorem. Applications of martingale theory: SLLN for i.i.d. random variables.
3. Brownian motion: Introduction to Brownian Motion, Kolmogorov Consistency theorem, Kolmogorov Continuity theorem, Construction of BM. Basic Martingale Properties and path properties – including Holder continuity and non-differentiability. Quadratic variation. Markov Property and strong Markov property of BM, reflection principle, Blumenthal’s 0-1 law. Distributions of first passage time and of running maximum of BM.
Interested participant can fill the following google form till 19th June 2019:
https://forms.gle/KHr4WYjxbk9gHT569
Program Schedule:
24th June 2019 | |
08:00-08:50 | Breakfast at M5 of SMS Building |
8:30-09:00 | Registration at SMS Seminar room |
09:00-11:00 | Measure Theoretic Probability By B V Rao |
11:00-11:30 | Tea Break |
11:30-12:30 | Tutorial |
12:40-14:30 | Lunch Break |
14:30-16:30 | Measure Theoretic Probability By S Iyer |
16:30:17:00 | Tea & Snacks |
17:00-18:00 | Tutorial |
20:00-21:00 | Dinner at M5 of SMS Building |
25th -- 28th June 2019 | |
08:00-08:50 | Breakfast |
09:00-11:00 | Measure Theoretic Probability By B V Rao |
11:00-11:30 | Tea Break |
11:30-12:30 | Tutorial |
12:30-14:30 | Lunch Break |
14:30-16:30 | Measure Theoretic Probability By S Iyer |
16:30:17:00 | Tea & Snacks |
17:00-18:00 | Tutorial |
20:00-21:00 | Dinner |
29th June 2019 | |
08:00-08:50 | Breakfast |
9:00-16:00 | Heritage Tour to Bhubaneswar |
20:00-21:00 | Dinner |
01st July -- 05th July 2019 | |
08:00-08:50 | Breakfast |
09:00-11:00 | Conditional Probability and Martingales by A Chakrabarty |
11:00-11:30 | Tea Break |
11:30-12:30 | Tutorial |
12:30-14:30 | Lunch Break |
14:30-16:30 | Brownian Motion by R Roy |
16:30:17:00 | Tea & Snacks |
17:00-18:00 | Tutorial |
20:00-21:00 | Dinner |
06th July 2019 | |
08:00-08:50 | Breakfast |
09:00-12:30 | Group Discussions |
11:00-11:30 | Tea Break |
12:30-14:30 | Lunch Break |
14:30-16:30 | Group Discussions |
16:30:17:00 | Tea & Snacks |
20:00-21:00 Dinner | |
08th July -- 11th July 2019 | |
08:00-08:50 | Breakfast |
09:00-11:00 | Brownian Motion by M Krishnapur |
11:00-11:30 | Tea Break |
11:30-12:30 | Tutorial |
12:30-14:30 | Lunch Break |
14:30-16:30 | Conditional Probability and Martingales by P Roy |
16:30:17:00 | Tea & Snacks |
17:00-18:00 | Tutorial |
20:00-21:00 | Dinner |
12th July 2019 | |
08:00-08:50 | Breakfast |
09:00-11:00 | Brownian Motion by M Krishnapur |
11:00-11:30 | Tea Break |
11:30-12:30 | Tutorial |
12:30-14:00 | Lunch Break |
14:00-16:00 | Conditional Probability and Martingales by P Roy |
16:00-16:30 | Feedback |
16:30:17:00 | Tea & Snacks |
17:00-17:30 | Conclusion |
20:00-21:00 | Dinner |
Confirmed Participants:
Name | Affiliation |
Shivam Dhama | IIT Gandhinagar |
Ekta | IIT Gandhinagar |
Debashish Bose | Shiv Nadar University |
MOHAMMED HISHAM M | Pondicherry University |
Akhil Kumar Sahoo | IISER Kolkata |
Saikat Patra | University of Calcutta |
MOSTAFIZAR KHANDAKAR | IIT BHILAI |
Soham Ghosh | University Of Kalyani |
Barniit Adhikary | University of kalyani |
JEWEL MAHAJAN | Iiser pune |
Suvadip Sana | Indian statistical Institute,Bangalore. |
Monika Singh Dhull | Indian Institute of Technology, Ropar |
Ghodekar Shivani Shankarrao | Pondicherry University |
Aniket Datta | University Of Kalyani |
Anewsha Basu | University Of Kalyani |
DEBOLENA BASAK | UNIVERSITY OF KALYANI |
poushali sengupta | university of kalyani |
Arunodaya Bhattacharya | university of Kalyani |
Susmita Ghosh | University of Kalyani |
Aishani Barman Roy | University of Kalyani |