MSc Mathematical Finance

Dynamic programming and its extension to Markov Decision Processes is one of the fundamental algorithmic contributions of the last century. Here one can sequentially optimise a process over time. The theory extends to diffusion processes and thus has become a tool for optimal investment.

The unit aims to:
provide students a fundamental background in the optimisation of stochastic processes and to introduce some of reinforcement learning techniques. That includes identifying different types of optimisation problems and working with the Bellman and Hamilton–Jacobi–Bellman equation, solving approximately some of dynamic programming problems. 
 

• perform dynamic programming in discrete time and space
   
• recognise the mathematical structure of control problems to formulate appropriate Bellman and Hamilton-Jacobi-Bellman equations in practical applications
   
• predict and verify solutions to control problems, including numerical approaches and educated guesses for PDEs in typical situations.
   
• apply knowledge about Markov Decision Processes and Diffusions, including probability theory and Ito's calculus and the derivation of Bellman equations, to address mathematical issues arising in control problems
   
• Apply some of reinforcement learning algorithms including numerical experimentation.

Syllabus:

-Dynamic Programming. Bellman equation. Markov chains and Markov Decision processes. [6] 
-Dynamic Programming Algorithms. Optimal Stopping Problems [4]
-Continuous Time and Diffusion Control Problems. HJB equation. Merton's Portfolio [6] 
-Multi-armed bandits. Reinforcement learning algorithms [6]
 

Lectures and tutorials are given to the entire cohort in one room together.  
 

Coursework: weekly problem sheet

Feedback provided through coursework marking and final grade with written overall feedback for unit for exam

-Dynamic Programming and Optimal Control, Dimitri P. Bertsekas, 2nd Edition, Athena, (2012) 
-Arbitrage Theory in Continuous Time, Thomas Bjork, Oxford University Press (2009) 
- Reinforcement Learning; An introduction. Richard S. Sutton and Andrew G. Barto 2nd Edition, MIT Press (2018)
- Control Systems and Reinforcement Learning. Sean Meyn. Cambridge University Press (2022)