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GRK 1845:  Stochastic Analysis with Applications in Biology, Finance and Physics

Subject Area Mathematics
Term from 2012 to 2017
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 208415830
 

Final Report Abstract

The RTG research in the area stochastic finance had its highlights in the following topics. Model free arbitrage pricing, for example for basket options, was treated with novel methods based on a game theoretic formulation of mathematical finance, paracontrolled distributions and rough paths. It produced scaling limits for superreplication prices under market friction. Studies of hedging and pricing under model uncertainty were done via representation results for increasing convex functions. Illiquid financial market models or multiplicative transient price impact models were investigated, in particular for optimal order execution problems. Methods employed focussed on optimal control, asymptotic methods therefore, and convex analysis. A constructive solution for the variational HJB inequality for two-dimensional free boundary in a non-convex stochastic singular control problem for random liquidity was derived. For backward stochastic differential equations (BSDE), a crucial tool for stochastic optimization, RTG students dealt with discrete and continuous time BSDE driven by general semimartingales (with jumps), the stability of solutions and their approximation by regression schemes. New approaches of convex BSDE went via duality of minimal supersolutions. Second order BSDE and G-Lévy processes were investigated. Utility optimization was studied in financial market models with asymmetric information. In the research area stochastic analysis and statistical inference for stochastic dynamics one of the main highlights was work with high impact on the foundations of rough paths, paracontrolled distributions and regularity structures, and a deep link between the two, that led to novel insights on (singular) stochastic partial differential equations (SPDE) and ordinary differential equations (ODE) with rough signals. It gave access to a systematic analysis of singular SPDE, and a characterization of SPDE as universal scaling limits, for instance in quantum field theory. (Forward) backward stochastic differential equations ((F)BSDE) were treated by the efficient tool of decoupling fields, leading to a new approach of Skorokhod’s embedding for diffusion processes. Work on the asymptotics of random dynamical systems highlighted in a study on the connectedness of random attractors, dealt primarily with the strong completeness of flows, and the long-time behavior of Brownian flows. Ergodic behavior was tackled by means of generalized asymptotic couplings, e. g. for stochastic delay equations. Stabilization by noise was on the agenda, and work with impact done on the novel topic of synchronization of random dynamical systems. The research on statistical inference for diffusion processes and SDE emphasized adaptation and efficiency in statistics. One focus was on the statistics for integrated volatility estimation for semi-martingales, another one on high-dimensional covariance matrix estimation, and statistical inverse problems. In the area of stochastic processes in biology and physics one of the foci was given by random walks and diffusions in random environments and effective interface models. Remarkable work was on random bridges of jump processes in connection with Schrödinger’s problem and optimal transport. RTG students dealt with spectral concentration of two types of random operators (random Schrödinger and random Laplace operator), did an extreme-value analysis of concentration sites, and studied Anderson localization at the edge of the spectrum. A methodological highlight was a combination of large deviations techniques with homogenization, where a transition between homogenized and localized behavior was studied. Duality formulas on path spaces were investigated, skew Brownian motion with several barriers, and a generalized rejection sampling method. Work on stochastic processes in neuroscience dealt with the analysis of neural activity in the brain on all scales, the numerical approximation of stochastic nerve axon equations, and stochastic neural field equations. One highlight was the multiscale analysis of stochastic neural field equations and stochastic reaction diffusion equations w.r.t. travelling waves as well as diffusion limits of discrete neural population models. Data assimilation problems in neuroscience and related stochastic filtering equations were treated. Successful work was done on stochastic models in population genetics and evolution. It highlighted in results on the duality theory for generalized Wright-Fisher models with frequency dependent selection as well as with geometric seed bank component. SPDE arising in interacting species models, in particular the symbiotic branching model, were studied. Many of the RTG students obtained their PhD degrees with summa cum laude, several were awarded local and national prizes (e. g. Tiburtius prize 2014, DMV student conference prize 2015, Michelson prize of U Potsdam 2016, publication prize of Leibniz-Kolleg Potsdam 2017, Förderpreis der Fachgruppe Stochastik 2016, Rollo-Davidson prize 2018, Heinz-Maier-Leibnitz prize 2019), and many obtained very good international positions in- and outside academia.

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