#
Probability III (Stat GR6303) - Fall 2017

This course is for PhD students only.

*Instructor:* Marcel Nutz
*Meeting time:* Tuesdays and Thursdays 8:40-9:55, Room 520 Math
- Office
hours: Thursdays 11:25-12:15, Rooms 520 Math/910 SSW

This
course is an advanced introduction to stochastic control, dynamic
programming, viscosity solutions of nonlinear parabolic PDEs and
backward stochastic differential equations (BSDEs) with examples from
mathematical finance. It is mostly based on the following lecture notes:

- N. Touzi (2013), Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs No. 29, Springer Verlag, New York. DOI: 10.1007/978-1-4614-4286-8.

You can download the eBook from the publisher's website (licensed through Columbia's library services).

The course also follows excepts from

- W. Fleming and H. M. Soner (2005), Controlled Markov Processes and Viscosity Solutions, Springer Verlag, New York (link)
- I. Karatzas and S. E. Shreve (1998), Methods of Mathematical Finance, Springer Verlag, New York (link)
- H. Pham (2009), Continuous-time Stochastic Control and Optimization with Financial Applications, Springer Verlag, New York (link)

Prerequisites: Probability I
and II. This will be a fast-paced lecture; students are expected to
have some maturity in real analysis, probability theory and stochastic
analysis (Ito calculus). Knowledge of PDEs is not required but
some prior exposure may be helpful.

## Contents

- Conditional Expectation and Linear Parabolic PDEs
- Controlled Diffusions and Dynamic Programming
- Solving Control Problems by Verification
- Viscosity Solutions
- Dynamic Programming Equation in the Viscosity Sense
- Optimal Stopping: Snell Envelope and Markovian Theory, Free Boundary Problem

- Backward SDEs and Stochastic Control
- Quadratic Backward SDEs
- Stochastic Target Problems
- Singular Control; Applications to Transaction Costs and de Finetti's Dividend Problem