Advanced Stochastic Processes

Graduate · Historical course

Fall 2013

This class covers the analysis and modeling of stochastic processes. Topics include measure theoretic probability, martingales, filtration, and stopping theorems, elements of large deviations theory, Brownian motion and reflected Brownian motion, stochastic integration and Ito calculus and functional limit theorems. In addition, the class will go over some applications to finance theory, insurance, queueing and inventory models.

Readings

[Billingsley] = Billingsley, Patrick. Weak Convergence of Measures: Applications in Probability . Society for Industrial and Applied Mathematics, 1987. ISBN: 9780898711769. [Preview with Google Books ] ↗
Listed reading
[Chen and Yao] = Chen, H., and D. Yao. Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization . Springer-Verlag, 2001. ISBN: 9780387951669. [Preview with Google Books ] ↗
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[Dembo and Zeitouni] = Dembo, Amir, and Ofer, Zeitouni. Large Deviations Techniques and Applications . 2nd ed. Springer, 2009. ISBN: 9783642033100. [Preview with Google Books ] ↗
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[Durrett] = Durrett, Rick. Probability: Theory and Examples . 4th ed. Cambridge University Press, 2010. ISBN: 9780521765398. [Preview with Google Books ] ↗
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[Karatzas and Shreve] = Karatzas, Ioannis and Steven, Shreve. Brownian Motion and Stochastic Calculus . 2nd ed. Springer-Verlag, 1991. ISBN: 9780387976556. [Preview with Google Books ] ↗
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[Øksendal] = Øksendal, B. Stochastic Differential Equations: An Introduction with Applications . Springer, 2010. ISBN: 9783540047582. [Preview with Google books ] ↗
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[Resnick] = Resnick, Sydney. Adventures in Stochastic Processes . 1st ed. Birkhauser Verlag, 1992. ISBN: 9780817635916. [Preview with Google Books ] ↗
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[Shwartz and Weiss] = Shwartz, Adam, and Alan Weiss. Large Deviations for Performance Analysis: Queues, Communication and Computing . Chapman and Hall/CRC, 1995. ISBN: 9780412063114. [Preview with Google Books ] ↗
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LEC #: 1 · TOPICS: Metric spaces and topology · READINGS: [Billingsley]: Appendix M1-M10.
Reading assignment
[Shwartz and Weiss]: Chapter 0. This is non-technical introduction to the field which describes motivation and various applications of the large deviations theory.
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[Dembo and Zeitouni]: Chapter 2.2.
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[Resnick]: Sections 6.1, and 6.4 from chapter 6.
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[Billingsley]: Chapter 8.
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[Resnick]: Sections 6.5, and 6.8 from chapter 6.
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LEC #: 8 · TOPICS: Quadratic variation property of Brownian motion · READINGS: [Resnick]: Sections 6.11, and 6.12 from chapter 6.
Reading assignment
LEC #: 9 · TOPICS: Conditional expectations, filtration and martingales · READINGS: [Durrett]: Section 4.1, and 4.2.
Reading assignment
LEC #: 10 · TOPICS: Martingales and stopping times I · READINGS: [Durrett]: Chapter 4.
Reading assignment
[Durrett]: Chapter 4.
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Grimmett, Geoffrey R., and David R. Stirzaker. Section 7.8 in Probability and Random Processes . 3rd ed. Oxford University Press, 2001. ISBN: 9780198572220.
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LEC #: 14 · TOPICS: Introduction to Ito calculus · READINGS: [Karatzas and Shreve]: Chapter I.
Reading assignment
[Øksendal]: Chapter III.
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LEC #: 18 · TOPICS: Integration with respect to martingales · READINGS: [Øksendal]: Chapters III, IV, and VIII
Reading assignment
LEC #: 19 · TOPICS: Applications of Ito calculus to financial economics · READINGS: Duffie, Darrell. Dynamic Asset Pricing Theory. Princeton University Press, 2001. ISBN: 9780691090221. [Preview with Google Books ] ↗
Reading assignment
LEC #: 20 · TOPICS: Introduction to the theory of weak convergence · READINGS: [Billingsley]: Chapter 1. Section 2.
Reading assignment
LEC #: 21 · TOPICS: Functional law of large numbers Construction of the Wiener measure · READINGS: [Billingsley]: Chapter 2. Section 8.
Reading assignment
Dinwoodie, I. H. “ A Note on the Upper Bound for I.I.D Large Deviations .” The Annals of Probability 19 , no. 4 (1991): 1732–6. ↗
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Lancaster, Peter, and Miron Tismenetsky. The Theory of Matrices: With Applications . Vol. 2. Academic Press, 1985. ISBN: 9780124355606. [Preview with Google Books ] ↗
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Seneta, E. Non-Negative Matrices and Markov Chains . 2nd ed. Springer-Verlag, 2006. ISBN: 9780387905983.
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Slaby, M. “ On the Upper Bound for Large Deviations of Sums of I.I.D Random Vectors .” The Annals of Probability 16, no. 3 (1988): 925-1388. ↗
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Sources

Course catalogue ↗Historical OpenCourseWare
MIT OCW syllabus ↗
MIT OCW readings ↗
Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.