Continuous Time Optimisation

**This information is for the 2019/20 session.**

**Teacher responsible**

Prof Adam Ostoja-Ostaszewski

**Availability**

This course is available on the MSc in Applicable Mathematics, MSc in Operations Research & Analytics and MSc in Quantitative Methods for Risk Management. This course is available as an outside option to students on other programmes where regulations permit.

**Pre-requisites**

Students will need adequate background in linear algebra (facility with diagonalization of matrices for the purposes of solving simultaneous first-order differential equations is key here; knowledge of the relation between the range of a matrix transformation and the kernel of its transpose would be helpful), and in advanced calculus (manipulation of Riemann integrals such as `differentiation under the integral’ and the associated Leibniz Rule). Students unsure whether their background is appropriate should seek advice from the lecturer before starting the course.

**Course content**

This is a course in optimisation theory using the methods of the Calculus of Variations. No specific knowledge of functional analysis will be assumed and the emphasis will be on examples. It introduces key methods of continuous time optimisation in a deterministic context, and later under uncertainty. Calculus of variations and the Euler-Lagrange Equations. Sufficiency conditions. Pontryagin Maximum Principle. Extremal controls. Transversality conditions. Linear time-invariant state equations. Bang-bang control and switching functions. Dynamical programming. Control under uncertainty. Itô's Lemma. Hamilton-Jacobi-Bellman equation. If time allows: Applications to Economics and Finance: Economic Growth models, Consumption and investment, Optimal Abandonment, Black-Scholes model, Singular control, Verification lemma.

**Teaching**

24 hours of lectures, 4 hours of seminars and 8 hours of seminars in the LT. 2 hours of seminars in the ST.

Background review of (i) elementary methods for solving differential equations, and (ii) pertinent linear algebra (diagonalization) will be included in the the seminars of Weeks 1 and 2.

Four of the 24 lecture hours are dedicated to Exam Revision

**Indicative reading**

A full set of lecture notes will be provided. Reference will be made to the following books: E R Pinch, Optimal Control and the Calculus of Variations, Oxford Science Publications; G Leitmann, Calculus of Variations and Optimal Control, Plenum; A K Dixit & R S Pindyck, Investment under Uncertainty, Princeton University Press; D Duffie, Security Markets, Academic Press; D J Bell & D H Jacobsen, Singular Optimal Control, Academic Press; J L Troutman, Variational Calculus and Optimal Control, Springer; W H Fleming & R W Rishel, Deterministic and Stochastic Optimal Control, Springer; W H Fleming; H M Soner Controlled Markov Processes & Viscosity Solutions, Springer; G Hadley; M C Kemp, Variational Methods in Economics, North Holland; D Burghes; A Graham Control and Optimal Control Theories with Applications,Horwood; A Sasane, Optimization in Function Spaces, Dover.

**Assessment**

Exam (100%, duration: 2 hours) in the summer exam period.

**
Key facts
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Department: Mathematics

Total students 2018/19: 10

Average class size 2018/19: 10

Controlled access 2018/19: No

Value: Half Unit

**Personal development skills**

- Self-management
- Problem solving
- Application of information skills
- Communication
- Application of numeracy skills
- Specialist skills