Advanced stochastic processes

Lectures: 2 sessions / week, 1.5 hours / session

6.431 Applied Probability, 15.085J Fundamentals of Probability, or 18.100 Real Analysis (18.100A, 18.100B, or 18.100C).

The 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.

Your grade is based on the in-class midterm exam, take home final exam, and homework problem sets.

Large deviations theory

Cramér's theorem

Extension of LD to ℝd and dependent process

Gärtner-Ellis theorem

The reflection principle

The distribution of the maximum

Brownian motion with drift

Martingales and stopping times II

Martingale convergence theorem

Ito process

Ito formula

Functional law of large numbers

Construction of the Wiener measure

Skorokhod mapping theorem

Reflected Brownian motion

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