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Prof Adam Ostoja-Ostaszewski

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.

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.

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.

This course is delivered through a combination of classes and lectures totalling a minimum of 34 hours across Lent Term. This year, some or all of this teaching will be delivered through a combination of virtual classes and lectures delivered as online videos.

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

Four of the 34 hours are dedicated to Exam Revision which will also be delivered through a combination of virtual classes and lectures delivered as online videos.

A full set of lecture notes will be provided. Reference will be made to the following essential books: D Burghes & A Graham, Control and Optimal Control Theories with Applications, Horwood; E R Pinch, Optimal Control and the Calculus of Variations, Oxford Science Publications; A. Sasane, Optimization in Function Spaces, Dover; J L Troutman, Variational Calculus and Optimal Control, Springer; and occassionally to: D G Luenberger, Optimization by Vector Space Methods, Wiley.

Further Reading and Advanced Literature: 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; 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; 

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