The New Role of Mathematical Risk Modeling and Its Importance for Society

Abstract

This book on risk and security is an example for the new role of mathematical modeling in science. In Newtonian times, mathematical models were mainly applied to physics and astronomy (e.g., planetary systems) as definitive mappings of reality. They aimed at explanations of past events and predictions of future events. Models and theories were empirically corroborated or falsified by observations, measurements and lab experiments. Mathematical predictions were reduced to uniquely determined solutions of equations and the strong belief in one model as mapping of reality. In probabilistic models, extreme events were underestimated as improbable risks according to normal distribution. The adjective “normal” indicates the problematic assumption that the Gaussian curve indicates a kind of “natural” distribution of risks ignoring the fat tails of extreme events. The remaining risks are trivialized. The last financial crisis as well as the nuclear disaster in Japan are examples of extreme events which need new approaches of modeling.

Mathematical models are interdisciplinary tools used in natural and engineering sciences as well as in financial, economic and social sciences. Is there a universal methodology for turbulence and the emergence of risks in nature and financial markets? Risks which cannot be reduced to single causes, but emerge from complex interactions in the whole system, are called systemic risk. They play a dominant role in a globalized world. What is the difference between microscopic interactions of molecules and microeconomic behavior of people? Obviously, we cannot do experiments with people and markets in labs. Here, the new role of computer simulations and data mining comes in.

These models are mainly stochastic and probabilistic and can no longer be considered as definitive mappings of reality. The reason is that, for example, a financial crisis cannot be predicted like a planetary position. With this methodic misunderstanding, the political public blamed financial mathematics for failing anticipations. Actually, probabilistic models should serve as stress tests. Model ambiguity does not allow to distinguish a single model as definitive mapping of reality. We have to consider a whole class of possible stochastic models with different weights. In this way, we can overcome the old philosophical skepticism against mathematical predictions from David Hume to Nassim Taleb. They are right in their skepticism against classical axiomatization of human rationality. But they forget the extreme usefulness of robust stochastic tools if they are used with sensibility for the permanent model ambiguity. It is the task of philosophy of science to evaluate risk modeling and to consider their interdisciplinary possibilities and limits.