MA6000D: Stochastic Optimal Control and Applications in Finance

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• Alex Cox

Notes

• Motivating Examples (Lecture 1)
• Stochastic Integral and related results (Lectures 2 & 3)
• Theory of Stochastic Optimal Control (Maren Eckhoff, Lecture 4)
• Complete Financial Markets (Marion Hesse, Lecture 5)
• Incomplete Financial Markets (Christoph Höggerl, Lecture 6)
• Utility Maximisation (Alex Watson, Lecture 7)
• Optimal Consumption and Investment with Transaction Costs (Curdin Ott, Lecture 8) Slides
• Inverse Optimal Consumption (Lecture 9)

This graduate course will aim to cover some of the fundamental probabilistic tools for the understanding of Stochastic Optimal Control problems, and give an overview of how these tools are applied in solving particular problems. Since many of the important applications of Stochastic Control are in financial applications, we will concentrate on applications in this field.

This course will be suitable for students with a strong undergraduate background in probability and stochastic processes, but we will provide a brief introduction to important underpinning theoretical ideas such as stochastic integration, Itô's Lemma, martingale representation theorem, stochastic differential equations, diffusions and the Feynman-Kac formula, however we will try to cover material quickly, and so the presentation of these ideas will be a bit informal.

The course will roughly break into two parts: after some motivation and discussion of introductory problems, we will review much of the background theory: in particular, we will provide an overview of stochastic integration in a Brownian filtration, and some SDE theory and key results, following the presentation in Øksendal's book . We will then review some of the key results in Stochastic optimal control, following the presentation in Chapter 11 of this book. To see some of the important applications in Finance, we will use Karatzas and Shreve , "Methods of Mathematical Finance" and in some circumstances, directly refer to research papers.

The course is timetabled at 10:15-12.05 on Wednesdays in 4W1.7 . An approximate outline of the lectures is as follows:

Lecture 1 (10/10/12)

Introduction; Motivating examples: a simple control problem with details; quadratic regulator; utility maximisation; option pricing.

Lecture 2 (17/10/12)

Introduction to the stochastic integral. Basic properties of the stochastic integral; Itô's Formula.

Lecture 3 (24/10/12)

Martingale representation theorem; SDEs; weak & strong solutions. Diffusions; strong Markov property; Generators; Dynkin’s formula; Feynman-Kac formula; Girsanov’s theorem; Dirichlet-Poisson problem.

Lecture 4 (31/10/12: Maren Eckhoff)

Stochastic Optimal Control `Theory’: Problem statement; Markov controls; value function; dynamic programming principle; characterisation of an optimal control: HJB equation; verification theorem; `Guess and Verify'.

Lecture 5 (7/11/12: Marion Hesse)

Option pricing in complete markets; trading and arbitrage; fundamental theorem.

Lecture 6 (14/11/12: Christoph Höggerl)

Option pricing in incomplete markets: upper and lower hedging price; dual representation.

Lecture 7 (21/11/12: Alex Watson)

Utility maximisation: investment and consumption; Merton problem; dual representation of solutions.

Lecture 8 (28/11/12: Curdin Ott)

Utility maximisation under transaction costs (Davis & Norman).

Lecture 9 (5/12/12: AC)

Utility maximisation: an inverse problem.

It is expected/hoped that some volunteers will prepare some of the later lectures, and the list will be updated as the semester progresses.

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