Title: Learning how to stop -on data driven optimal decision making

Abstract: We discuss different classes of classical stochastic control problems under the assumption that the dynamics of the underlying stochastic process is unknown. More precisely, we consider problems of impulse- and singular-control type with an ergodic criterion and assume that the decision maker just knows the class of underlying processes (e.g., diffusions or Lévy processes), but has to learn the dynamics while controlling the process as good as possible. Therefore, the decision maker is usually faced with an exploration-vs.-exploitation dilemma. We present strategies whose values become optimal in the long run and analyze their speed of convergence. As a main tool, we establish nonparametric statistical results for stochastic processes.

Title: Sticky Feller diffusions

Abstract: The question of finding the transition density function of a Feller branching diffusion process whose boundary point zero is slowly reflecting (sticky) is considered. The method employed reveals a 'convolution identity for sticky laws' which shows that the entire 'stickiness' is embodied in a single multiplication factor of the Green function. Applying Laplace inversion makes the transition density function expressible by means of a convolution integral involving a new special function (which reduces to the Mittag-Leffler function when the branching drift is zero) and a modified Bessel function of the second kind. The result is applicable to 'sticky Cox-Ingersoll-Ross processes' and 'sticky reflecting Vasicek processes' that can be used to model slowly reflecting (sticky) interest rates. [This is joint work with David Roodman.]