Risk Modelling: Analysis, Simulation and Optimization

This project aims at studying different aspects of risk models. Such models try to capture the evolution of wealth processes of an economic agent in a risky environment. Subsequently, one is interested in the determination of associated stability criteria and risk measures, as a mean to make decisions under risk. The first risk models goes back to the beginning of the 20 th century. It describes the surplus process of an insurance portfolio with a simple stochastic process. Naturally, such a simple model comprises various basic assumptions which are hard to verify in reality. In a first part of the project the focus is put on the development and analysis of general model extensions. A broadly applicable risk model should be able to capture for example: different economic background scenarios, links to the financial market, several lines of business and various risk sharing mechanisms. Fortunately, the class of so-called piecewise-deterministic Markov processes is rich enough to fulfil several of these requests and also allows for controllable modifications. Therefore, these processes are central for the project. For being able to determine and analyse risk measures and stability criteria in such a general framework, we need to develop new methods and techniques. Subsequently, these novel mathematical tools will be used to solve stochastic optimization problems such as the minimization of a risk measure in this risk theoretic context. One needs to mention that this particular class of processes is broadly applicable and can be used to model biological as well as physical phenomena. Therefore, the projects aimed for results will be of broad interest. A second part of the projects future research is devoted to statistical issues of the aforementioned risk models. Here, we aim at the development of estimation procedures for the models ingredients and plan to formulate them in a dynamic way. With this procedure we will try to give a complete description of general risk models which comprises analytical, numerical and statistical features. This complete characterization will allow us to study the effects of estimation errors on risk measures. Consequently, we will be able to quantify model robustness and compare sophisticated with naive statistical approaches.

Within the scope of this project, a general class of risk models was developed and mathematically analyzed. These stochastic models are primarily used in the insurance and financial sectors to understand the long-term effects of incurred risks and, consequently, to control their impact. Furthermore, they are also relevant to problems in queuing theory and operations research. A first set of results obtained relate to the precise growth behavior of so-called ruin probabilities in models based on stochastic intensities - the probability that available funds will be insufficient to meet payment obligations. This class of intensities also allows for the description of temporal clusters of events, which is particularly important for modeling natural disasters and cybersecurity incidents. Our findings can now be used to determine risk measures and to analyze situations in which an exceptional temporal cluster of events or individual, but catastrophic, events - in the sense of financial claims - pose a particular threat. This leads to a second research focus within this project: the control of previously developed risk models. Among other things, this allowed us to answer the question of how two competing insurance companies can control their risk exposure for succeeding in a competition for market shares. A more individual perspective was analyzed in another optimization problem. An individual with an income is exposed to certain risks and has access to an insurance policy that potentially covers damages in exchange for premiums. The question is how to utilize the policy to maximize available assets, knowing that any risk sharing increases premiums over a certain time interval. The mathematical analysis of the aforementioned models and derived stochastic optimization problems was complemented by the design of supplementary numerical methods to illustrate theoretical results using exemplary scenarios. These diverse results clearly demonstrate that the presented models can be effectively used to address different problems from various perspectives.