Financial/Actuarial Mathematics Seminar

Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their practicability suffers from the assumption of known dynamics of the underlying stochastic process, raising the statistical challenge of developing purely data-driven controls in a nonparametric framework.
In this talk, we will mainly concentrate on long-term average singular control problems for general Lévy processes on the real line. First, we present a method for solving such problems for known underlying dynamics in terms of the ladder height process. To construct a data-driven procedure, the fundamental observation is that this solution can be represented using an auxiliary function involving the stationary distribution of the overshoot process. This leads to the statistical question of finding rate-optimal estimators with respect to the sup-norm risk for such functionals. As a result, we present a fully data-driven strategy that is optimal on the long run and show that the regret per time unit is of order $1/\sqrt(T)$. Speaker(s): Soren Christensen (Christian-Albrechts-University Kiel)