Stochastic Processes 2026

Logistics

Timings: Wednesday and Friday, 11:30 AM–1:00 PM

Venue: Cognizant Lab, COE Building

Teaching Assistant: Aman Ray

Tutorial: Monday, 9:30–11:00 AM

The course follows the core development in the lecture notes, drawing primarily on Brzeźniak and Zastawniak, with complementary material from Ross.

Probability spaces: sigma-algebras, measurable sets, probability measures, continuity of probability, random variables, distribution functions, and the Strong Law of Large Numbers.
Conditional expectation: finite sigma-algebras, information generated by a random variable, conditional expectation, and worked examples.
Stochastic processes: sample paths, finite-dimensional distributions, examples of stochastic processes, stationarity, ergodicity, discrete- and continuous-time processes, the Markov property, martingales, and the Kolmogorov Extension Theorem.
Poisson and Brownian motion: Poisson processes and their interarrival times, the Lévy-process viewpoint, the Brownian/Wiener process, Gaussian increments and covariance, continuity and path properties, the Markov and martingale properties, quadratic variation, a proof that Brownian motion is almost surely not differentiable at zero, the reflection principle, and the connection with random-walk limits through Donsker’s theorem.
Markov chains: transition probabilities, the Chapman-Kolmogorov equation, invariant distributions, harmonic functions, stationarity, stopping times, the strong Markov property, first-passage probabilities, recurrence and transience, communicating classes, and long-run visit rates.

The course introduces undergraduates to probability theory and random processes. Its objectives are twofold:

To equip students with the mathematical tools needed to identify and characterize the essential properties of stochastic processes.
To train students to compute and interpret quantities such as expectations, distributions, transition probabilities, and limiting behavior.

Emphasis is placed on precise definitions, proofs, examples, and problem-solving.

Sheldon M. Ross; “Introduction to Probability Models”, Academic Press, Ninth Edition, 2007.
Anurag Kumar; “Discrete Event Stochastic Processes: Lecture Notes for an Engineering Curriculum”, Indian Institute of Science.
Brzezniak and Zastawniak; “Basic Stochastic Processes”, Springer, Second Indian Reprint, 2009.