Research
Limit theorems in Probability Theory
Limit theorems in Probaility Theory
Many models in applied probability theory and statistical physics are too complicated to derive explicit formulas for important model variables. Weak and strong limit theorems allow such models to be approximated by “continuous” models, which are often easier to analyze. We study limit theorems for stochastic processes, for example for martingales, branching processes and shot noise processes with deterministic or random response function as well as for the test statistics in a problem of asymptotic test theory.
Mathematical Finance
Mathematical Finance
Key motivations for stochastic analysis—and, at the same time, its prominent applications—include the modeling of financial markets, the valuation of financial products such as options and derivatives, and the associated risk theory. In particular, we consider models with changing underlying regimes and models with affine structures. On the more applied side, the focus is on credit risks and energy markets. This also includes estimating the respective parameters using statistical methods from the fields of artificial intelligence and machine learning.
Stochastical Analysis
Stochastical Analysis
Stochastic analysis deals with stochastic processes. These can also be viewed as random functions. Due to the element of “randomness,” these functions exhibit less regularity but prove to be more realistic for many applications. As a rule, the functions are not differentiable and, like chaotic and fractal phenomena, have no length; that is, they are of infinite variation. In particular, we study infinite-dimensional processes such as measure-valued diffusions, (affine) path processes, and rough processes.
Branching processes
Branching processes
Branching processes are a class of stochastic processes. They serve as models of biological populations such as humans, cells or plants, as models for tumor growth, but also for neutron chain reactions or fragmentation. Branching processes are also important tools within related fields of applied probability or theoretical computer science, more specifically, for the probabilistic algorithms of the divide-and-conquer type. We investigate martingales from branching models and with their help the fluctuations of branching processes themselves.
External Funding
- DAAD project "Probability, Computation and Applied Stochastic Analysis" with the AIMS-Center in Kigali, Ruanda. 3 PhD students and 1 Postdoc (2024-2026).
- DFG project ME 3625/6-1 "Multivariate smoothing equations with coefficients from the general linear group". 1 Postdoc (2025-2027).
- DFG project ME 3625/5-1 "Non-Malthusian supercritical Crump-Mode-Jagers processes". 1 PhD student (2024-2027).
Selected Publications
- Iksanov, A., Kolesko, K., and Meiners, M.: Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes
Ann. Probab. 52 (2024), no. 4, 1538-1606 article arxiv - Alfeus, M., Nikitopoulos, C., and Overbeck, L.: Implied roughness in the term structure of oil markets
Quantitative Finance 24 (2024), 347-362 - Ackermann, J., Kruse, T., and Overbeck, L.: Inhomogeneous affine Volterra Processes
Stochastic Process. Appl. 150 (2022), p. 250-279. - Meiners, M., and Mentemeier, S.: Solutions to complex smoothing equations
Probab. Theory Related Fields 168 (2017), 199–268. arxiv springerlink - Alsmeyer, G., and Meiners, M.: Fixed points of the smoothing transform: two-sided solutions
Probab. Theory Related Fields 155 (2013), no. 1-2, 165–199. arxiv springerlink - Alsmeyer, G., Biggins, J.D., and Meiners, M.: The functional equation of the smoothing transform
Ann. Probab. 40 (2012), no. 5, 2069-2105. pdf arxiv projecteuclid - Bluhm, C., Overbeck, L. and Wagner, C.: Introduction to Credit Risk Modeling
Chapman & Hall. Boca Raton, Fl, USA. 2nd edition 2010 - Overbeck, L.: Non-linear superprocesses
Annals of Probability 24 (1996), 743-760.