## Stochastic Control with Applications to Finance

 Unit code: MATH69122 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

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#### Aims

The aim of the course is to provide students a fundamental background in the optimization of stochastic processes. This is split into three components: 1. Markov Decision Theory, 2. Multi-armed Bandit Problems and the exploration-exploitation trade-off, and 3. Queueing and game-theory in market design. In each area we focus on financial applications.

#### Overview

Dynamic programming and its extension to Markov processes, Markov Decision Processes, is one of the fundamental algorithmic contributions of the last century. Here one can sequentially optimize a process over time. The theory is extended to diffusion processes and thus is a key tool in optimal investment.

Further, within this framework one can consider aspects of the so-called exploration-exploitation trade-off. Here one can either choose to optimize decisions with all available information or can choose to explore further potential options. This is the Multi-armed bandit framework and literature in this domain has grown rapidly over recent years. We consider aspects of this theory in application to portfolio selection and revenue management.

Further we develop queueing models and consider their application in inventory models and the limit order book model -- a model studied in the context of high frequency trading. Finally, we consider aspect of optimal market design covering the famed VCG-mechanism and Myerson's optimal auction framework.

#### Learning outcomes

On successful completion of this course unit students will

- Acquire knowledge required to implement Markov Decision Theory.
- Understand the fundamental limits of exploration-exploitation trade-off.
- Be able to anticipate game-theoretic aspects in market design.
- Have a close working knowledge of how each theoretic component is applied in financial situations.

#### Assessment methods

• Written exam - 100%

#### Syllabus

1. Dynamic Programming & Markov Decision Processes. Applications including shortest-paths, secretary problem, optimal stopping, (s,S) inventory control. [8]
2. Control of Diffusions. Hamilton-Jacobi-Bellman formulation. Applications including optimal consumption and investment [6]
3. Multi-armed Bandits. Gitten's Indices. Lai Robbins, optimal estimation. Applications including revenue management. [5]
4. Portfolio selection and sequential investment. Introduction to Hannan consistent algorithms, Kelly Market Vectors, universal portfolio selection, the EG investment strategies. [4]
5. Introduction to queueing models and applications: the G/G/1 queue, Cramer-Lundberg model, Economic Order Quantity model, limit order book modeling. [6]
6. Auction Theory. Bayes-Nash equilibrium. Revenue Equivalence Theorem. Mechanism design. Vickery-Clark-Groves mechanism. Myerson's revenue optimal auctions. [4]

- Dynamic Programming and Optimal Control, Dimitri P. Bertsekas, 2nd Edition, Athena, (2012), ISBN: 1-886529-08-6

- Arbitrage Theory in Continuous Time, Thomas Bjork, Oxford University Press (2009) ISBN: 9780199574742

- Multi-armed Bandit Allocation Indices, John Gittins, Kevin Glazebrook and Richard Weber, 2nd edition, Wiley, (2011)  ISBN: 978-0-470-67002-6

- Prediction, Learning, and Games, Nicolo Cesa-Bianchi and Gabor Lugosi, Cambridge University Press, (2006),  ISBN: 9780521841085

- Applied Probability and Queues, Soeren Asmussen, Springer (2003) ISBN 978-0-387-21525-9.

- Putting Auction Theory to Work, Paul Milgrom, Cambridge University Press, (2004) ISBN: 9780521536721

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 33 hours
• Tutorials - 11 hours
• Independent study hours - 106 hours

#### Teaching staff

Neil Walton - Unit coordinator