Discrete stochastic models (2025)

Convener(s)

 
Name: Anup Biswas Moumanti Podder
Mailing Address: Associate Professor
IISER Pune		 Assistant Professor
IISER Pune
Email: anup at iiserpune.ac.in moumanti at iiserpune.ac.in

Mini courses on three fascinating topics in the area of probability and stochastic processes will be offered in this workshop, namely Martingales, Percolation and Heavy-tailed Distributions. Martingales, one of the most fundamental concepts in probability, finds myriad applications in financial mathematics such as in arbitrage-free pricing, in martingale optimal transport, in probabilistic methods implemented for solving various problems in graph theory, in the swiftly expanding subfield of elephant random walks in probability. Percolation, an area of immense interest to researchers in both mathematics and statistical physics, finds wide applications elsewhere as well, such as in ecology (modeling the spread of infections, understanding the impact of environment fragmentation on animal habitats, modeling the spread of forest fires etc.), in biochemistry and physical virology (predicting fragmentation thresholds for biological virus shells), to name just a few. Heavy-tailed probability distributions are indispensable in the area of risk analysis in financial mathematics (such as for understanding ruin probabilities in insurance risk models). Apart from a thorough introduction to each of these areas via lectures and tutorial sessions, certain topics deemed as prerequisites, such as Markov chains, simple random walks, branching processes and the basics of measure theory (including the notion of conditional expectation), will also be discussed as needed.

Prerequisites: This meeting is aimed for 1st year PhD students and final year MS students. The first three days of the workshop will be utilized to revisit topics such as the basics of measure theory and conditional expectations, Markov chains, simple random walks, Galton-Watson branching processes etc., which will be important to understand the mini courses. Participants are expected to know measure theoretic probability.

Dates: 

Thursday, July 10, 2025 - 19:00 to Friday, July 18, 2025 - 19:00

Venue: 

Venue Address: 

Indian Institute of Science Education and Research (IISER) Pune
Dr. Homi Bhabha Road,
Pashan, Pune 411008

Venue State: 

Venue City: 

PIN: 

411008

Syllabus: 

Name of the Speakers with their affiliation.

No. of Lectures

Detailed Syllabus

Moumanti Podder
IISER Pune

4
(each 1.5 hrs)

Introduction to discrete and continuous time Markov chains and their various applications, random walks (including simple symmetric random walks on the infinite d-dimensional integer lattice), irreducibility and equivalence classes, transience and recurrence, branching processes (with emphasis on Galton-Watson branching
processes) and their applications..

Anup Biswas
IISER Pune

5
(each 1.5 hrs)

 Conditional expectation, definition, existence and uniqueness, properties of conditional expectation, relation with the classical conditional probability, conditional inequalities and limit theorems, conditional Bayes’ formula, regular conditional probability

Anish Sarkar
ISI Delhi

5
(each1.5 hrs)

 Martingales: Definitions & Examples, Submartingales, Functions of Martingales, Inequalities and some applications, Convergence Theorems, Reversed Martingales, Central Limit Theorem (time permitting).

Krishanu Maullik
ISI Kolkata

5
(each 1.5 hrs)

Heavy Tailed Distributions

Definitions, properties and examples of heavy-tailed, subexponential and regularly varying distributions, Karamata properties of regular variation, one large jump principle for subexponential distributions, tails of random sums, generalized central limit theorem, maximum of random walk with negative drift, degree distribution of preferential attachment graphs, record processes.

Rahul Roy
ISI Delhi

5
(each 1.5 hrs)

Percolation: Percolation on trees, bond percolation on the square lattice, oriented percolation in 2 dimensions. Introduction to percolation theory and why is it different from standard stochastic processes (here the randomness is in the medium and not in the motion), the notion of phase transition, using branching process techniques to explicitly calculate the phase transition point for percolation on trees. Showing that the phase transition point for d-dimensional lattice is non-trivial. Discussing Correlation inequalities (i.e., the FKG and BK inequalities), proving the subadditive lemma. Discussion of the notion of ergodicity, invariant sigma field and state the ergodic theorem. Proving the uniqueness of the unbounded cluster (i.e. the Burton-Keane theorem). Showing that p_c = 1/2 for bond percolation on the square lattice. If time permits, discussion of site percolation on the square lattice.

References:

  1. Patrick Billingsley. Probability and measure. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1995.

  2. Sheldon M. Ross. Stochastic Processes. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1995.

  3. Samuel Karlin, Howard E. Taylor. A First Course in Stochastic Processes. Academic Press, San Diego, CA, 1975.

  4. Krishna B. Athreya and Peter E. Ney. Branching Processes. Springer-Verlag Berlin, Heidelberg, 1972.

  5. William Feller. An Introduction to Probability Theory and its Applications. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1957.

  6. David Williams. Probability with Martingales. Statistical Laboratory, University of Cambridge, 1991.

  7. Ronald Meester, Rahul Roy. Continuum Percolation. Cambridge University Press, 1996.

  8. Geoffrey Grimmett. Percolation. Springer-Verlag Berlin, Heidelberg, 1999.

  9. Paul Embrechts, Claudia Kluppelberg, Thomas Mikosch. Modelling Extremal Events: for Insurance and Finance. Vol 33, Springer Science and Business Media, Berlin, 2013.

  10. Jayakrishnan Nair, Adam Wierman, Bert Zwart. The Fundamentals of Heavy Tails: Properties, Emergence and Identification. Cambridge University Press, 2022.

Names of the tutors with their affiliation:

  1. Dhruv Bhasin, IISER Pune

  2. Archi Roy, IISER Pune

  3. Tamojit Sadhukhan, ISI Kolkata

  4. Aritra Majumdar, ISI Kolkata

 

 

Time Table: 

 

 

Tentative time-table, mentioning names of the speakers and tutors with their affiliation:

Day

Date

Lecture 1
9.30
to
11.00

Tea
11.05
to
11.25

Lecture 2
11.30
to
1.00

Lunch
1.05
to
1.55

Lecture 3
2.00
to
3.30

Tea
3.35
to
3.55

Discussion
4.00
to
5.00

Snacks
5.05
to
5.35

 

 

(name of the speaker)

 

(name of the speaker)

 

(name of the speaker)

 

(Name of the tutor)

 

Thu

July 10

AB

 

AB

 

MP

 

AB

 

Fri

July 11

AB

 

MP

 

MP

 

MP

 

Sat

July 12

AB

 

AB

 

MP

 

AB

 

Mon

July 14

AS

 

AS

 

KM

 

MA+AS

 

Tues

July 15

KM

 

KM

 

RR

 

KM

 

Wed

July 16

AS

 

KM

 

RR

 

KM

 

Thu

July 17

AS

 

KM

 

RR

 

RR

 

Fri

July 18

AS

 

RR

 

RR

 

MA+AS

 

Full forms for the abbreviations of speakers and tutors:

  • MP: Moumanti Podder
  • AS: Anish Sarkar
  • KM: Krishanu Maulik
  • RR: Rahul Roy
  • AB: Anup Biswas
  • MA: Maruf Alam

Selected Applicants: 

 

Sr.n SID Full Name Gender Affiliation Position in College/ University University/ Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Degree Date
1 61947 Mr. Nayandeep Singh Male National Institute of Technology Calicut Msc student National Institute of Technology Calicut 2025  
2 62138 Ms. Roopanshi Mann Female Indian Institute of Technology, Indore MSc Student IIT Indore Awaiting Result  
3 62143 Mr. Ashish Jain Male IIT Indore Ph.D. student Doctor Harisingh gour central university sagar 2022  
4 62153 Mr Bikram Podder Male University of Delhi Ph.D Student University of Delhi 2023  
5 62175 Mr. Pankaj Saha Male IIT Indore MSc student IIT Indore Awaiting Result  
6 62291 Mr. Satya Sai Aditya Duggaraju Male Indian Institute of Technology, Bombay PhD Indian Institute of Technolgoy Bhilai 2024  
7 62422 Mr. Pradeep Vishwakarma Male Indian Institute of Technology Bhilai PhD student IISER Kolkata 2022  
8 62451 Ms. Manisha . Female Indian Institute of Technology Bhilai PhD student Central University of Haryana 2020  
9 62542 Mr. Pradyumna Kumar Dhar Male IIT TIRUPATI PhD IIT Tirupati 2023  
10 62543 Mr Harsh Jaiswal Male INDIAN INSTITUTE OF TECHNOLOGY TIRUPATI PhD IIT BOMBAY 2023  
11 62919 Mr. Pankit Male PANJAB UNIVERSITY PhD PANJAB UNIVERSITY 2021  
12 63073 Mr Sripati Mazumder Male IIT MADRAS PhD IIT MADRAS 2023  
13 63176 Ms. Sumana Bhattacharjee Female Indian Statistical Institute MSTAT student Indian Statistical Institute    
14 63296 Mr. Rahul Sachan Male Central university of south bihar, Gaya, bihar Ph.D. Central University of Punjab 2023  
15 63580 Mr Arif Ali Male Indian Institute of Science Education and Research, Thiruvananthapuram (IISER TVM) MSc Student (final year) Indian Institute of Science Education and Research, Thiruvananthapuram (IISER TVM) 2025  
16 63959 Mr. Pratim Dey Male Indian Institute of Technology Guwahati Ph.D. Student Indian Institute of Technology Guwahati 2020  
17 64027 Mr. Ashutosh Kasaudhan Male IIT BHU (Varanasi) PhD IIT Bhubaneswar 2024  
18 64090 Mr Kuldeep Pal Male IIT Madras PhD student University of Delhi 2023  
19 64161 Ms. Sneha Patra Female IIT Kanpur MSc Student IIT Kanpur    
20 64275 Mr. Kousik Pramanik Male Indian Institute of Technology Kanpur MSc Student Indian Institute of Technology Kanpur    
21 64323 Mr. Aninda Kumar Nanda Male Indian Statistical Institute PhD Babasaheb Bhimrao Ambedkar University 2024  
22 64343 Mr. Shatrughna Chaurasia Male Indian institute of technology Ropar, Punjab PhD Pondicherry University, Puducherry 2021  
23 64468 Mr Husna Moab Male IIT, Patna PhD IIT Kanpur 2020  
24 64475 Ms Sadaf Fatma Female Indian Institute of Information Technology, Allahabad PhD Madan Mohan Malaviya University of Technology, Gorakhpur 2022  
25 64945 Mr. Dipto Kumar Haldar Male IISER Berhampur MSc Student IISER Berhampur 2025  
26 65232 Ms Jyoti Purba Female BITS Pilani, Pilani Campus PhD The Maharaja Sayajirao University of Baroda 2021  
27 65237 Mr. Akhil Vijayan R Male Central University of Kerala PhD Central University of Kerala 2019  
28 65241 Mr Srishak Dash Male SOUTH ASIAN UNIVERSITY,DELHI Msc Student South Asian University Awaiting Result  
29 65274 Ms. Geetika Female Indraprastha Institute of Information Technology- Delhi PhD Jamia Millia Islamia 2022  
30 65318 Ms. Shinjini Paul Female IISER Pune MSc Student IISER Pune 2025  

How to Reach: 

TBA

School Short Name: 

dsc

Last Date Application: 

Monday, August 15, 2022

School Type: 

NCMW

Separate faculty form: 

0