# Mathematical Methods

"When it is not in our power to follow what is true, we ought to follow what is most probable."
Descartes (1596-1650)

Our world, unfortunately, is not deterministic, but we are constantly confronted with uncertainty and randomness. Therefore, probability theory and its time dependent generalization, stochastic processes, are required to model complex, real world problems. They are a central part of our work and our projects. To give some examples: queuing theory is applied to modeling IP networks, renewal theory is used to analyze failure rates, and time series analysis and Kalman filtering are the basis for optimal prediction and decision making.

Our approach is characterized by a subjective perception of the concept of probability. Using a priori hypotheses, new data are used to re-evaluate these hypotheses with probabilities. Thus previous knowledge and new data can be combined in an inductive way. Recently this brilliant idea of Bayes was used to map probability distributions with many variables to graphs. Currently these Bayes’ networks are applied to a multitude of problems. We show this method in detail in an introductory article and another, more technical paper.

The other focus of our work is optimization theory for analyzing complex dynamic systems. In projects for “planning optimal paths” dynamic programming plays a central role. This method is an improvement over the classical calculus of variation.

Probability theory and stochastic elements also play a central role in optimization theory. Examples for technical applications are the calculation of paths and the control of autonomous vehicles subject to uncertainty. Modeling financial markets is an example of a non-technical problem. A financial market is mapped to a dynamic system, and its hidden state has to be estimated in an optimal way from observable but noisy prices . By optimizing a utility function it is possible to find an optimal investment strategy. Thus, portfolio management is equivalent to a stochastic optimal control problem.

To demonstrate the inter-disciplinary character of our work we extend the mathematical modeling to sport topics like climbing and mountaineering.