Topics in Applied Stochastic Processes

  • Class-Notes: The consolidated notes of the class can be found here (125MB). The notes are also archived as the topics were covered every below.

    • Week 1 - Week3: January 25th to February 12th.

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    • Topics Covered
      • Weighted Graphs, Product Graphs
      • Stochastic Matrix : Eigenvalues, Eigenfunctions
      • Laplacian
      • Harmonic Functions
      • Poincare Inequality
      • Dirichlet Problem
      • Maximum Principle
      • Review of Discrete time Markov Chains

    • Week 4 : February 16th, and February 19th, 2021.

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    • Topics Covered
      • Review of Discrete time Markov Chains
      • Example: Reflected Random walk on m vertices, on m-cycle
      • Result on Convergence to Stationarity;Rate of convergence;Mixing time
      • Example: Gambler's ruin chain
      • Results on Hitting times; Properties: Harmonic and Solution to Dirichlet Problem

    • Week 5 : February 23rd, and February 25th, 2021.

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    • Topics Covered
      • Sufficient criteria for Recurrence on weighted graphs
      • Examples of graphs with Exponential volume growth
      • Simple Random walk on ${\mathbb Z}^d$.
      • Foster's Criteria/Lyapunov Function
      • Queue Chain

    • Week 6 : March 2nd and March 5th, 2021.

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    • Topics Covered
      • Simple Random walk of finite length $N$: Construction and Properties
      • Observable events and Filtration
      • Stopping times
      • Zero expected profit
      • Reflection Principle

    • Week 7 : March 9th and March 12th, 2021.

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    • Topics Covered
      • Last visit to origin
      • Hitting distribution.
      • Infinite time horizon Walk
      • Discrete time Martingale
      • Conditional Expectation

    • Week 8 : March 16th and March 19th, 2021.

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    • Topics Covered
      • Tower Property
      • Conditioning Martingale on its past.
      • Exponential Martingale
      • Conditional Expectation for discrete random Variables
      • Conditional Expectation with respect to Observable events/-filtration.
      • Stopping times and Stopped Processes

    • Week 9 : March 23rd and March 26th, 2021.

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    • Topics Covered
      • Optional Stopping Theorem
      • Sub-Martingale
      • Kolmogorov Maximal Inequalities

    • Week 10 : March 30th, and April 2nd 2021.

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    • Week 11 : April 6th, 2021 and April 9th, 2021.

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    • Week 12 : April 13th, 2021 and April 16th, 2021.

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    • Topics Covered
      • Martingale Convergence Theorem (non-negative)
      • Expected hitting time of a finite pattern
      • Mixing time of Simple random walk on the $n$-cycle.

    • Week 13 : April 20th, 2021 and April 23rd, 2021.

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    • Topics Covered
      • Chernoff Bounds Re-visited
      • Galton-Watson Process
      • Moran Model
      • Liouville Property for Harmonic functions on $\Z^d$.

    • Week 14 : April 27th, 2021 and April 30th, 2021

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    • Topics Covered
      • Binomial Model for Option Pricing
      • Harnack Inequality Statement
      • Strong Liouville Property
    Last Modified: April 30th, 2021. Courses Page Teaching Page