These Lecture Notes are based on a course given in June 2001 at the
Cattedra Galileiana of Scuola Normale Superiore di Pisa. The course
consisted of a short introduction into the basic concepts of
Mathematical Finance, focusing on the notion of “no arbitrage”,
and subsequently applying these concepts to portfolio optimization.
To avoid technical difficulties I mainly dealt with the situation
where the underlying probability space is finite and only sketched
the difficulties arising in the general case. We then pass to the
scheme of utility optimisation for general semi-martingale models.
Some topics of this course are not standard: for example, in the
treatment of the general existence theorem for the optimal
portfolio, we give a direct proof which is not relying on duality
theory. Similarly, the treatment of the asymptotic elasticity of
utility functions and a related counter-example are original to
these notes.
General
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