Matstat seminarium fredagen 18 maj 2001

Martin Jacobsen, Köpenhamn
SOME EXAMPLES OF MULTI-DIMENSIONAL DIFFUSIONS

Abstract:

The talk will focus on examples of multi-dimensional diffusions that
have at least some decent and tractable properties. In the one-dimensional
case it is perfectly understood how to determine e.g. the range of the
diffusion, whether it is transient or recurrent, and when it has an invariant
distribution, which can then be given explicitly. In higher dimensions the
situation is quite obscure with the finite-dimensional Ornstein-Uhlenbeck
processes (homogeneous Gaussian diffusions) one of the very few reasonably
well understood examples. One example to be given in the talk is a
mult-idimensional Cox-Ingersoll-Ross type process. One problem here is to
control the range - all coordinates must stay strictly positive at all
times. As it turns out, this class of processes has some additional attractive
structural properties.
Other examples include reversible diffusions: consider a multi-dimensional
diffusion with an invariant distribution. Finding the density for this appears
quite hopeless in general, but if the diffusion is time reversible explicit
expressions can sometimes be given.