Swayam Central

Stochastic Processes

By Prof. S. Dharmaraja   |   IIT Delhi

This course explanations and expositions of stochastic processes concepts which they need for their experiments and research. It also covers theoretical concepts pertaining to handling various stochastic modeling. This course provides classification and properties of stochastic processes, discrete and continuous time Markov chains, simple Markovian queueing models, applications of CTMC, martingales, Brownian motion, renewal processes, branching processes, stationary and autoregressive processes.

INTENDED AUDIENCE: Under-graduate, Post-graduate and PhD students of mathematics, electrical engineering, computer engineering
PRE REQUISITES :        A basic course on Probability
INDUSTRY SUPPORT:  Goldman Sachs, FinMachenics, Deutsche Bank and other finance companies.

Learners enrolled: 1104


Course Status : Ongoing
Course Type : Elective
Duration : 12 weeks
Start Date : 29 Jul 2019
End Date : 18 Oct 2019
Exam Date : 17 Nov 2019
Category :
  • Mathematics
  • Level : Postgraduate
    This is an AICTE approved FDP course


    Week 1:Probability theory refresher
    1. Introduction to stochastic process
    2. Introduction to stochastic process (contd.)
    Week 2:Probability theory refresher (contd.)
    1. Problems in random variables and distributions
    2. Problems in Sequence of random variables
    Week 3:Definition and simple stochastic process 
    1. Definition, classification and Examples
    2. Simple stochastic processes
    Week 4:Discrete-time Markov chains
    1. Introduction, Definition and Transition Probability Matrix
    2. Chapman-Kolmogorov Equations
    3. Classification of States and Limiting Distributions
    Week 5:Discrete-time Markov chains (contd.)
    1. Limiting and Stationary Distributions
    2. Limiting Distributions, Ergodicity and stationary distributions
    3. Time Reversible Markov Chain, Application of Irreducible Markov chains in Queueing Models
    4. Reducible Markov Chains
    Week 6:Continuous-time Markov chains
    1. Definition, Kolmogrov Differential Equation and Infinitesimal Generator Matrix
    2. Limiting and Stationary Distributions, Birth Death Processes
    3. Poisson processes
    Week 7:Continuous-time Markov Chains (contd.)
    1. M/M/1 Queueing model
    2. Simple Markovian Queueing Models
    Week 8:Applications of CTMC
    1. Queueing networks
    2. Communication systems
    3. Stochastic Petri Nets
    Week 9:Martingales
    1. Conditional Expectation and filteration
    2. Definition and simple examples
    Week 10:Brownian Motion
    1. Definition and Properties
    2. Processes Derived from Brownian Motion
    3. Stochastic Differential Equation
    Week 11:Renewal Processes
    1. Renewal Function and Equation
    2. Generalized Renewal Processes and Renewal Limit Theorems
    3. Markov Renewal and Markov Regenerative Processes
    4. Non Markovian Queues
    5. Application of Markov Regenerative Processes
    Week 12:Branching Processes, Stationary and Autoregressive Processes


    1. J Medhi, Stochastic Processes, 3rd edition, New Age International Publishers, 2009
    2. Liliana Blanco Castaneda, Viswanathan Arunachalam, Selvamuthu Dharmaraja, Introduction to Probability and Stochastic Processes with Applications, Wiley, 2012.
    3. Kishor S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Edition, Wiley, 2002.


    Prof.S. Dharmaraja earned his M.Sc. degree in Applied Mathematics from Anna University, Madras, India, in 1994 and Ph.D. degree in Mathematics from the Indian Institute of Technology Madras, in 1999. From 1999 to 2002, he was a post-doctoral fellow at the Department of Electrical and Computer Engineering, Duke University, USA. From 2002 to 2003, he was a research associate at the TRLabs, Winnipeg, Canada.
    He has been with the Department of Mathematics, IIT Delhi, since 2003, where he is currently a Professor and Head, Department of Mathematics and joint faculty of Bharti School of Telecommunication Technology and Management. He appointed as 'Jaswinder & Tarvinder Chadha Chair Professor' for teaching and research in the area of Operations Research from May 2010 to July 2015. He has held visiting positions at the Duke University, USA, University of Calgary, Canada, University of Los Andes, Bogota, Colombia, National University of Colombia, Bogota, Colombia, University of Verona, Verona, Italy, Sungkyunkwan University, Suwon, Korea and Universita degli Studi di Salerno, Fisciano, Italy.
    His research interests include applied probability, queueing theory, stochastic modeling, performance analysis of computer and communication systems and financial mathematics. He has published over 30 papers in refereed international journals and over 20 papers in refereed international conferences in these areas. He is an Associate Editor of International Journal of Communication Systems. Recently, he is co-author of a text book entitled "Introduction to Probability and Stochastic Processes with Applications" in John Wiley and co-author of a text book entitled "Financial Mathematics: An Introduction" in Narosa.


    • The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centers.
    • The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
    • Date and Time of Exams: 17th November 2019, Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
    • Registration url: Announcements will be made when the registration form is open for registrations.
    • The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
    • Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

    • Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course. 
    • Exam score = 75% of the proctored certification exam score out of 100
    • Final score = Average assignment score + Exam score

    • If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.
    • Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras. It will be e-verifiable at nptel.ac.in/noc.
    • Only the e-certificate will be made available. Hard copies are being discontinued from July 2019 semester and will not be dispatched.