Wider research context and theoretical framework:
Risk theory constitutes a branch of applied probability. The goal is the description of risks - affecting an economic agent - and their evolution by means of stochastic models. Based on these models one is trying to quantify them by determination of risk measures. Classical models have been developed to answer questions on long term stability in insurance mathematics. In this proposal the focus is put on the study of generalized risk models and related questions of optimization and parameter uncertainty. The studied models are quite flexible and applicable in many other fields such as reliability and queuing theory, or mathematical finance.
Hypotheses, research questions and objectives:
Classical risk models are based on restrictive assumptions, the most significant being a static parameter structure. In this way current phenomena can hardly be accurately described. In this project we will investigate models which are able to incorporate varying parameters, i.e., capture changing economic environments. In particular, we aim at analyzing models based on piecewise-deterministic Markov processes. This class is ideally suited for this goal and allows for multi-variate extensions. Due to the lack of broadly applicable mathematical methods this class of processes is hardly used in the actual literature. In a first part of research we will develop analytical and numerical tools for the determination of functionals of such processes. Based on these developments we will turn our attention to related optimization problems. The second part will study the problem of parameter uncertainty and the effect of estimated parameters on risk measures and their computability.
Approach and methods:
It is planned to use a variety of mathematical methods. Since many risk measures can be written in terms of expected values which can be characterized as solutions to integro-differential equations, there is need to analyse them. Lack of sufficiently smooth solutions will ask for the concept of viscosity solutions. From a numerical perspective these equations are a-typical and new approaches need to be developed. A mixture of (quasi-) Monte Carlo and traditional methods seems to be promising. For dealing with parameter uncertainty it is intended to use and extend techniques from stochastic filtering and survival analysis.
Level of originality and innovation:
The proposed research topics give new insights and non-standard views on risk models.
The envisaged techniques rely on novel combinations of methods from different mathematical fields (probability theory, (numerical) analysis and QMC integration based on number theoretic results). We believe that new results in the proposed directions will create substantial interest in the research community.
Primary researchers involved:
Assoc. Prof. Dr. Stefan Thonhauser, Graz University of Technology, and two doctoral students (mathematics).