I have been teaching courses at all levels from the large mathematical fields of analysis, probability theory and statistics, algebra, and discrete mathematics. At the time being, I am almost entirely teaching courses of the Master Programme in Mathematics and Modelling, specialisation in Mathematical Statistics and Financial Mathematics, a specialisation for which I am responsible.
I have been working at the borderline between probability theory and functional analysis, in particular operator theory. These fields also constitute the basic mathematical toolboxes for quantum mechanical models. More precisely, my area of research is stochastic dynamic systems driven by stochastic processes, e.g. degenerate Brownian motion as starting point and more general Lévy and non-Markovian noises like fractional Brownian motion and non-linear processes involving a mean-field contribution. The existence of solutions is studied while the focus is on qualitative properties of the system like transience, ergodicity, asymptotic dependence on parameters like time, scaling parameters, number of particles/players.
Due to their importance in applications, optimization problems were investigated recently, i.e. optimal filtering and stochastic optimal control as part of differential games. Motivated by applications and applicability, my field of research was completed by time series models, better accessible for statistical inference. My focuses are functional data analysis and computational statistics for high frequency and large data sets.
There are numerous applications in the natural sciences as well as in economy, e.g. to hydrological problems in nuclear waste management, or the prediction of prices, e.g. electricity prices, inter-bank monetary transfer, appropriate reinsurance levels, solvency of insurance business, and optimal evacuation of large buildings.
My research groups and projects
Stochastic Analysis and Stochastic Processes The researchers of this group focus on analytical and numerical aspects of solutions to stochastic differential equations (SDE’s and SPDE’s) and stochastic …
International Center for Mathematical Modeling ICMM is a centre for research in mathematical modelling, applicable within a number of different subjects and fields. Since 2000, the research group has a…
Article in journal (Refereed)
- Basna, R., Hilbert, A., Kolokoltsov, V. (2017). An Approximate Nash Equilibrium for Pure Jump Markov Games of Mean-field-type on Continuous State Space. Stochastics : An International Journal of Probablitiy and Stochastic Processes. 89. 967-993.
- Doerr, B., Fischer, P., Hilbert, A., Witt, C. (2017). Detecting Structural Breaks in Time Series via Genetic Algorithms. Soft Computing - A Fusion of Foundations, Methodologies and Applications. 21. 4707-4720.
- Djechiche, B., Hilbert, A., Nassar, H. (2016). On the Functional Hodrick-Prescott Filter with Non-compact Operators. Random Operators and Stochastic Equations. 24. 33-42.
- Basna, R., Hilbert, A., Kolokoltsov, V. (2014). An Epsilon Nash Equilibrium For Non-Linear Markov Games of Mean-Field-Type on Finite Spaces. Communications on Stochastic Analysis. 8. 449-468.
- Albeverio, S., Hilbert, A., Kolokoltsov, V. (2012). Uniform Asymptotic Bounds for the Heat Kernel and the Trace of a Stochastic Geodesic Flow. Stochastics : An International Journal of Probablitiy and Stochastic Processes. 84. 315-333.
- Albeverio, S., Hilbert, A., Kolokoltsov, V. (1999). Estimates uniform in time for the transition probability ofdiusions with small drift and for stochastically perturbed Newton equations. Journal of theoretical probability. 12. 293-300.
- Albeverio, S., Hilbert, A., Kolokoltsov, V. (1997). Transcience of stochastically perturbed classical Hamiltonian systems and random wave operators. Stochastics and Stochastics Reports. 41-55.
Article in journal (Other academic)
- Albeverio, S., Hilbert, A., Kolokoltsov, V. (2009). The small time asymptotics of the transition density of some degenerate diffusions on the cotangent bundle. arXiv:0912.4683v2.
- Hilbert, A. (2000). Sur le comportement asymptotique du noyau associé à une diffusion dégénérée. Reviews Canadian Math. 151-159.
Conference paper (Refereed)
- Fischer, P., Hilbert, A. (2014). Fast Detection of Structural Breaks. Proceedings of 21th International Conference on Computational Statistics, Geneva, August 19-22, 2014.
- Doerr, B., Fischer, P., Hilbert, A., Witt, C. (2013). Evolutionary Algorithms for the Detection of Structural Breaks in Time Series : extended abstract. Proceedings of the 15th annual conference on Genetic and evolutionary computation. 119-120.
- Fischer, P., Hilbert, A. (2012). Visual time series analysis. Proceedings of COMPSTAT 2012 : 20th International Conference on Computational Statistics. 225-234.
- Avis, D., Fischer, P., Hilbert, A., Khrennikov, A. (2009). Single, Complete, Probability Spaces Consistent With EPR-Bohm-Bell Experimental Data. Foundations of Probability and Physics-5. 294-301.
- Hilbert, A. (2004). Some features of regular Hamiltonian dynamics corresponding to degenerate diffusions on a cotangent bundle.
- Hilbert, A., Léandre, R. (2001). Nagel-Stein-Wainger estimates for balls associated with the Bismut condition. Stochastic Processes, Physics and Geometry: New Interplays II. 269-284.
Conference paper (Other academic)
- Al-Talibi, H., Hilbert, A., Kolokoltsov, V. (2010). Nelson-type Limit for a Particular Class of Lévy Processes. AIP Conference Proceedings; 1232. 189-193.
Manuscript (preprint) (Other academic)
- Al-Talibi, H., Hilbert, A. Differentiable Approximation by Solutions of Newton Equations Driven by Fractional Brownian Motion..
- Al-Talibi, H., Hilbert, A., Kolokoltsov, V. Smoluchowski-Kramers Limit for a System Subject to a Mean-Field Drift.
- Assing, S., Hilbert, A. A time change method for second order SDEs.
- Assing, S., Hilbert, A. On the collapse of a wave functionsatisfying a damped driven non-linearSchr ̈odinger equation.