MATH 280C: PROBABILITY (SPRING 2011)
Copyright c2011, R. J. Williams
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Lecture 1. Doob's inequalities, L^p convergence
theorem for martingales.
Convergence of conditional expectations, backwards submartingale convergence theorem.
Lecture 3. Proof of backwards submartingale convergence theorem, convergence of more conditional expectations, start of continuous time submartingale theory.
Lecture 4. Path regularity for submartingales.
Lecture 5. Usual conditions, right continuous modifications
of submartingales with right continuous expectations, Doob's inequalities and submartingale convergence theorem
for right continuous submartingales.
Lecture 6. Stopping times, properties of stopping times,
right continuous, adapted processes evaluated at stopping times.
Lecture 7. Stopping theorem, Doob-Meyer decomposition
theorem, Brownian motion - definition, some properties.
Lecture 8. Gaussian processes,
time inversion of Brownian motion.
Lecture 9. Existence of Brownian motion via
Kolmogorov extension theorem and Kolmogorov's continuity criterion.
Lecture 10. Completion of proof of K's continuity criterion. Path properties of Brownian motion:
not Holder continuous of exponent 1/2, nowhere differentiability.
Lecture 11. Introduction to Levy processes.
Markov property of Brownian motion.
Lecture 12. Proof of the Markov property for Brownian motion.
Lecture 13. Blumenthal's zero-one law and applications;
strong Markov property of Brownian motion.
Lecture 14. A process that is Markov but not strong
Markov; reflection principle; multidimensional Brownian motion.
Lecture 15. Poisson random measures,
definition, Laplace functional, example, existence for Sigma finite mean.
Lecture 16. Examples
of Poisson random measures: Poisson process, pure jump Levy process with finite Levy measure.
Lecture 17. Construction of Levy process with
general Levy measure.
Last updated March 30, 2011.