# Syllabus for Math 452, Section G1

Text: Øksendal, Stochastic Differential Equations, 5th ed., 1998, Springer-Verlag.

Roughly, this is a course about stochastic differential equations and the associated ideas of Ito calculus. We hope to understand both of these subjects and understand some of their structure. By structure, we mean some different perspectives and applications which reveal different perspectives on the material. Our goal will be to present this knowledge in a way which will be accessible to engineers who possess a certain level of mathematical maturity and at the same time to be precise enough that the mathematics student should have little trouble filling in the details. While Math 451 is not a prerequisite, you should be willing to dig through the notes for that class (which are online) to fill in as much detail as you need. Similarly, while you are not required per se to know measure theory, you should be able to become comfortable with the usage of measure theory (I will give a brief summary of the relevant portions of measure theory as needed).

Grades will be determined on the basis of homework questions and a final exam.

## Provisional Schedule

• Lecture 1: Brownian motion
• Review of Brownian motion
• Scaling properties
• Lecture 2: Stochastic integration and differentiation
• Construction and basic properties
• Ito's formula
• Burkholder-Davis-Gundy inequalities
• Lecture 3: Stochastic differential equations
• Existence and Uniqueness
• 1-dimensional SDE's
• Girsanov's formula
• Feynman-Kac
• Lecture 4: Markov processes
• Markov property
• Strong Markov property
• Generators
• Martingale problem
• Jump processes
• Chapman-Kolmogorov
• Dirichlet forms
• Lecture 5: Asymptotics
• Large deviations
• Diffusion limits
• Separation of scales
• Karatzas and Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., 1994, Springer-Verlag.

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