+91-674-249-4082

Submitted by nabinjana on 14 May, 2019 - 20:26

Date/Time:

Monday, June 24, 2019 (All day) to Friday, July 12, 2019 (All day)

Venue:

SMS Seminar Hall

Title:

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:

**School of Mathematical Sciences**

NISER, PO- Bhimpur-Padanpur, Via- Jatni, District- Khurda, Odisha, India, PIN- 752050

Tel: +91-674-249-4081

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