1) Stochastic processes with applications to Stochastic Finance
     (Stokastiset prosessit ja niiden rahoitusteoreettiset sovellukset)

     Time and Place : Monday and Wednesday, 10-12, MaD 381
     First lecture       : 10/09/2001

     Description: The famous Brownian motion was introduced in 1828 by the
     botanist Robert Brown and used later in 1900 by Louis Bachelier to model
     derivatives in Financial markets. However, it took a long time the mathematical
     foundation of this particular stochastic process was complete. The Brownian
     motion became a central object in Stochastics and its applications, and is still
     of great interest in our days. Other examples of stochastic processes were
     arising from applications in Physics, Biology, or Finance in the same way and
     gave rise to develop mathematical tools important for mathematics itself as well
     as for the applications.The course will develop some of these parts of the theory
     of stochastic processes.

     0. Introduction
     1. Brownian motion
     2. Poisson and Poisson point processes
     3. Levy processes and, in particular, stable processses
     4. Stochastic Models based on stable processes
         and jump-diffusions

     Exercises:
     Problems 17/09/2001
     Problems 24/09/2001
     Problems 01/10/2001
     Problems 08/10/2001
     Problems 22/10/2001
     Problems 29/10/2001
     Problems 05/11/2001
     Problems 12/11/2001
     Problems 19/11/2001
     Problems 26/11/2001
     Problems 03/12/2001
 

     Literature:
     [1] J. Bertoin: Levy Processes. Cambridge 1996.
     [2] D. Lamberton and B. Lapeyre: Stochastic Calculus Applied to Finance.
           Chapman & Hall 1996.
     [3] K.-I. Sato: Levy Processes and Infinitely Divisible Distributions. Cambridge 1999.
     [4] A.N. Sirjaev: Essentials of Stochastic Finance. World Scientific 1999.
     [5] A.N. Sirjaev: Probability. Springer.

     The references [1] and [3] are only for advanced reading.

2) Seminar on Stochastic Calculus (Malliavin Calculus)

     Time and Place :  Monday, 16:15-18:00, MaA 104
     First Seminar    : 17/09/2001
     The timetable of the talks You find here.

      Description: The stochastic calculus of variations is a modern and
      powerful extension of  classical  stochastic calculus with applications
      in different areas of Stochastics. It has been developed in the last
      decades and introduced by Malliavin in order to give a probabilistic
      proof of Hörmander's hypoellipticity theorem.The aim of the seminar
      consists in a basic and elementary introduction to this theory. Up to
      now the following topics are planed:
      -) The derivative operator and the Skorohod integral: How to
           differentiate certain random variables and how to integrate
           very general stochastic processes?
      -) Wiener chaos expansion and the representation of the derivative
           operator and the Skorohod integral in terms of this expansion.
      -) Basic Sobolev type-inequalities.
      -) Applications to Stochastic Differential Equations. For example:
           +) Smoothness properties of the solutions (Hörmander's Theorem).
           +) Quantitative properties of the solutions (Theorem of Kusuoka and
                Strook).

      Literature:
      [1] N. Ikeda and S. Watanabe: Stochastic Differential Equations and Diffusion
           Processes. Second Edition. North Holland 1989.
      [2] D. Nualart: Analysis on Wiener space and anticipating stochastic calculus.
            Lect. Notes Math. 1690 (1998).
      [3] D. Nualart: The Malliavin Calculus and Related Topics. Springer 1995.
      [4] P. Malliavin: Stochastic Analysis. Springer 1997

      The seminar is based on [2] and [3].