The course presents methods of stochastic optimisation for the study of a general consumption-investment problem in a complete and and incomplete market. For this some background knowledge in stochastic analysis is useful. This includes: Brownian motion, stochastic integration, Ito formula, stochastic differential equations driven by Brownian motion, martingales, Ito representation theorem and the Girsanov theorem. Stochastic integration extended to the case of local martingales will be used.
After a short revision of the elements of stochastic calculus, the core of the course starts with the introduction of the market model used in the course. This is based on Brownian dynamics. Within this introductory part a short revision about pricing of financial derivatives in a complete market will be given.
The first part of the course deals with the single agent problem to find an optimal consumption-investment scheme in a complete market. This is based on the so-called martingale method via Lagrance multipliers.
The second part of the course deals with incomplete markets. The incompleteness is determined by constrains on the portfolio choices (e.g. constrains on short-selling). In this framework we study the pricing of financial derivatives and we address the single agent problem of finding an optimal consumption-investment scheme. For this the methods above cannot be directly applied because of the incompleteness. Hence some new theory has to be devised.
Reference book is:
[KS] I.Karatzas and S.Shreve: Methods of Mathematical Finance, 1998. Springer. ISBN: 0-387-94839-2.
The following book will be used as reference on occasional topics.
[K] I. Karatzas: Lectures on the Mathematics of Finance, 1997. Amer.Math.Soc.
As for background material in stochastic calculus we will refer mostly to:
[?] B. ?ksendal: Stochastic Differential Equations, 2007, 6th Edition. Springer. ISBN: 978-3-540-04758-2.
[KS0] I.Karatzas and S.Shreve: Brownian Motion and Stochastic Calculus, 1998, 2nd Edition. Springer. ISBN: 0-387-97655-8.
Though any book in stochastic calculus can be equally be valid.
All books are available at the University Library!