Stochastic Algorithms and Nonparametric Statistics

The group contributes to the following mathematical research topics of WIAS:

Optimal transport (OT) distances between probability measures or histograms, including the Earth Mover's Distance and Monge-Kantorovich or Wasserstein distance, have an increasing number of applications in statistics, such as unsupervised learning, semi-supervised learning, clustering, text classification, as well as in image retrieval, clustering, segmentation, and classification, and other fields, e.g., economics and finance or condensed matter physics. [>> more]

An ordinary differential equation is often used to model the movement of a particle. Similarly, partial differential equation can be used to describe the evolution of a total of trajectories of particles. It is natural to add randomness to such models: sometimes because this is a more realistic description which takes into account random noise, sometimes because this randomness is fundamental to the model itself as is the case for financial markets. [>> more]

When developing financial models for a practical description of a financial process, such as stock prices or the EurIBOR interest rate curve, the classical Black-Scholes model is known to be insufficient. In particular, volatility profiles observed in the market are not explained by the Black-Scholes model. For this reason, financial markets are typically modeled by processes with stochastic volatility or jumps, that is, by Ito-Levy processes. Crucial to the practical relevance of such models is the ability to accurately and efficiently compute the prices of derivatives, and to calibrate the models to market data (be it time series or derivative prices), i.e., to solve inverse problems. [>> more]

Stochastic numerical algorithms for optimal stopping and control problems are required for the evaluation of usually high-dimensional callable or cancelable products, or the determination of optimal decision strategies in systems involving high-dimensional underlying quantities. In this respect, primal methods provide suboptimal exercise strategies, hence lower estimations of the target value (e.g. price), while dual methods provide upper estimations. Naturally, the gap between lower and upper bounds due to these approaches should be as small as possible. [>> more]

The theory of Optimal Transport has been immensely influential in connecting partial differential equations, geometry, and probability. On the one hand, research at WIAS is focused on applying methods and tools from Optimal Transportation Theory to problems in statistics, such as semi-supervised and unsupervised learning, clustering, text classification, as well as in image retrieval, clustering, segmentation, and classification by developing and analyzing new numerical algorithms and schemes. On the other hand, the theory of optimal transport is extended, e.g., towards unbalanced optimal transport and connections to evolutionary partial differential equations via gradient systems. [>> more]

The term statistical inference summerizes methods to extract information from observed data in order to characterize properties in populations. This involves statistical modeling, estimation and uncertainty assessment of parameters, testing of hypothesis. [>> more]

In many applications the quantities of interest can be observed only indirectly, or they must be derived from other measurements. Often the measurements are noisy and the reconstruction of the quantities of interest from noisy measurements is unstable. [>> more]

Stochastic Optimization in the widest sense is concerned with optimization problems influenced by random parameters in the objective or constraints. [>> more]