Advances in Stochastic Control Theory and Applications

The problem to make optimal decisions under uncertainty can in many cases be handleded within the mathematical theory of optimal stochastic control. There exists today a powerful mathematical theory for such standard problems.

But there are still many important nonstandard stochastic control situations which are beyong reach for these established techniques. Therefore, there is a need for new solution approaches, and this is the motivation for my project.

For example, I want to develop solution methods for the optimal control for nonstandard systems such as stochastic partial differential equations with space dependence and memory, Volterra stochastic partial integral equations, systems with dynamics involving not just the value of the state in the coefficients, but also the history of its probability distribution and modelling neuro network via mea-field stochastic differential euqations.These chanllenging optimal stochastic control problems are of fundamental importance because of their use in the modelling of optimal decisions within many applications, including population dynamics and epidemics, physics and finance.

I will develop solution methods that can handle such systems by combining, and developing further, results from stochastic analysis, white noise theory, Hida-Malliavin calculus, functional analysis and deep learning.As demonstrated inmy research publications I have already obtained substantial results in this direction, but much work remains.