Dynamic uncertainty modeling in Finance

• Applicant: Professor Dr. Thorsten Schmidt
• Subject Area: Mathematics
• Term: from 2018 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 403615786

• Final Report Year: 2025

Final Report Abstract

Since the 2007 financial crisis, ensuring the stability of financial markets has become a major concern in finance and economics. Many models start from a single probability measure P and assume that this probability measureremains constant over time. In financial applications, P is certainly not known - it has to be estimated. Moreover, the modern financial world faces fast changing environments and the assumption that P remains unchanged even over a relatively short time horizon turns out to be too strong. In mathematical finance, this has led to the emergence of a field known as robust finance, which systematically addresses uncertainty in financial modelling. Uncertainty, following the ideas from Frank Knight, is captured by replacing the single probability measure P with a set of probability measures P. Intuitively one does not fix a single model but works with a collection of possible model. Frank Knight coined the notion uncertainty for this setting. A modern risk manager is then interested in the worst-case evaluation rule, i.e. what is the supremum over the expectations in all considered models P. While already well understood in static settings and in simpler dynamic models, a good understanding in a flexible large class of models was lacking. Consequently, the modelling of dynamic phenomena under Knightian uncertainty has recently become a rapidly growing research field. In this project, through a sequence of works, we were able to develop a general framework for Markov processes under Knightian uncertainty, laying a solid foundation for a deeper understanding of risk in uncertain environments, with applications extending beyond financial markets. A central object is the study of the non-linear conditional expectation Et (X) := sup P∈Pt EP [X], where the class Pt = Pt (ω) depends the current information - in a Markovian environment this would be the current state of the considered process. The goal is to establish a dynamic programming principle, which allows to derive a non-linear version of the Kolmogorov equation. The latter is the key tool for numerical evaluations of the non-linear conditional expectation. While the project started with the study of continuous semimartingales, the setting could be generalised to Markov processes with jumps. A further extension includes path dependence. The results establish a precise setting for well-founded modelling in a dynamic context that accounts for uncertainty, while also opening the field to numerous unresolved questions.

Link to the final report

https://oa.tib.eu/renate/items/f8c2ab0d-e678-4db2-bcbd-d277cdf49e1c

Publications

• Nonlinear continuous semimartingales. Electronic Journal of Probability, 28(none).
  Criens, David & Niemann, Lars
  
• Robust utility maximization with nonlinear continuous semimartingales. Mathematics and Financial Economics, 17(3), 499-536.
  Criens, David & Niemann, Lars
  
• A class of multidimensional nonlinear diffusions with the Feller property. Statistics & Probability Letters, 208, 110057.
  Criens, David & Niemann, Lars
  
• A conditional Version of the second fundamental theorem of asset pricing in discrete time. Frontiers of Mathematical Finance, 3(2), 239-269.
  Niemann, Lars & Schmidt, Thorsten
  
• Markov selections and Feller properties of nonlinear diffusions. Stochastic Processes and their Applications, 173, 104354.
  Criens, David & Niemann, Lars
  
• Nonlinear semimartingales and Markov processes with jumps. Journal of Evolution Equations, 25(1).
  Criens, David & Niemann, Lars