Professor Adam Ostaszewski, COL 4.06

Intended for students on MSc Applicable Mathematics, MSc Financial Mathematics and MSc Risk and Stochastics and other suitably qualified students.

Students should have attended a course in Mathematical Methods, similar to MA200 Further Mathematical Methods (Calculus), and should have experience with proofs and proof techniques used in pure mathematics.

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. Singular control. Dynamical programming. Control under uncertainty. Itô's Lemma. Hamilton-Jacobi-Bellman equation. Verification lemma. Applications to Economics and Finance: Economic Growth models, Consumption and investment, Optimal Abandonment. If time allows: Black-Scholes model.

20 lectures and 10 seminars in MT. Revision lectures will be arranged in the ST.

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.

There will be a two-hour written examination in the ST.