Stochastic Processes (MT4254)  

Summary of all lectures 

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Post-mid semester lectures:

• Lec.  17-18   02/03/2026-06/02/2026:  Gaussian distributions on R^n and their properties, Definition of Wiener process in terms of its finite dimensional distributions, 

• Lec.  19-21   09/03/2026-13/02/2026: Symmetries of Wiener process, Markov property, Levy's Construction of  Wiener process (brief outline), Stopping times and strong Markov property

Tentative plan for next few lectures: 

• Lec.  22-27   16/03/2026-27/02/2026: Reflection principle, Distribution of maximum of a Brownian motion, Non-differentiability of paths of Brownian motion, Martinagles associated with Brownian motion

• Lec. 28 onwards Stochastic Calculus and stochastic differential equations

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Pre-mid semester lectures:

• Lec.  01-03   05/01/ to 09/01/2026:  We had an interactive discussion about probabilistic aspects of snake and ladder game and their variations. We came across various notions like conditional probabilities, time homogeneity, average number of turns to win the game, fair/unfair dice that relate this example to Stochastic Processes in general. I also gave a short outline of the course contents in the end. We defined some notions like filtration, stopping times, distributions about stochastic processes rigorously and saw more examples: Random walks, Poisson process in brief without all technical details. In the end we recalled the definition of conditional expectation. We defined the Markov property, the definition and properties of Markov processes, Transition probability kernels. The non-random example of uniform motion was also discussed. We saw how the Random walk on integers satisfies the Markov property.

• Lec.  04-06  12/01 to 16/01/2026: We discussed more about probability kernels, Chapman-Kolmogorov equation, semigroup of operators, defined time homogenous Markov chains and transition probability matrix and how Chapman-Kolmogorov equation looks in this case. Markov property, Stopping times and strong Markov property for discrete time Markov chains, Proof of strong Markov property, Classification of states, distribution of holding times

• Lec.  07-09  19/01 to 23/01/2026: Classification of states, Mean recurrence times, Example of two state chain, Convergence theorem when 1 is simple eigenvalue of P and all other eigenvalues have absolute value < 1, Stationary distributions for DTMC, Limit theorems: Uniqueness and Convergence to stationary distribution for finite state space DTMC (statement)

                                                                 No lecture on 26/10/2026 (Republic day)

• Lec.  10-11  29/01/ to 30/01/2026: Continuous time Markov chains: jump time and holding time, Embedded discrete time chain, Infinitesimal generator (rate matrix), Backward and forward equations

No lecture on Monday 02/02/2026

• Lec. 12-16  05/02/2026-06/02/2026: Poisson process, Characterisation of explosion, Backward and forward equations, Stationary distributions and limit theorems for Continuous time Markov chains with finite state space, 

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   Mid semester exam

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