Intro to Martingales

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Course Description

On Countable or finite state space-:

• Basic definition of Probability Expectation
• Random Variable
• Conditional Probability, Conditional Expectation, Properties of Expectation and Conditional Expectation
• Martingales
• Super Martingales
• Sub Martingales
• Stopping Time
• Doob’s Maximal inequality
• Martingale Transform
• Up crossing Inequality

Then we went to arbitrary set there we covered-:

• Sigma field
• Probability Measure
• Random Variable
• Integral of Random variables
• Monotone Convergence theorem
• Dominated Convergence theorem
• Conditional Expectation

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Notes

• Pre reqs
  • Probability Theory and Gambling
  • Probability
  • Chebyshev’s Inequality
  • Dice Example
  • Conditional Expectation
• Filtrations (Measure Theory)
  • Filtered Probability Space
• Repeated Events
  • Coin Example
• Martingales
  • Convergence of a Sequence of Martingles
  • Martingale Transform
• Doob Decomposition Theorem
  • Predicable Process
• Stopping Time
  • Theorems about Stopping Times
• Doob’s Maximal Inequality
• Doob’s Upcrossings Inequality
• Doob’s Martingale Convergence Theorem
• Radon-Nikodym Theorem for Countably Generated Sigma Field

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MOCs

• Measure Theoretic Probability

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Practicle Information

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References