Brief description and motivation

The objective of the research is the analysis and modelling of large systems of individual agents in the framework of incomplete or biased information. Both industrial and economic systems will constitute the object of our studies. The primary goal is to develop new methods and techniques allowing to quantify and control risk that arises in the context of random perturbations of complex systems. For this purpose, it is necessary to provide a solid scientific foundation for the analysis, a better understanding and a proper design for various categories of risk. In particular, it is planned to extend the risk management methods put forward in the qualitative and quantitative analysis of huge financial markets to general investment models.

The complexity and the size of data do not allow to study the above mentioned issues using traditional methodologies. Randomness of nature makes uncertainty ever present; there are various sources of uncertainty, however. They include: a large number of objects (agents), their permanent (frequently instantaneous) fluctuations and random interactions, as well as the influence of external objective forces. Consequently, the uncertainty often comes from the external environment, which exhibits a random behaviour that is hard to predict and control. All decisions made under uncertainty are therefore subject to non-negligible risk factors. Let us now describe in more detail specific objectives of joint research activities.

Equilibria in complex systems. The notion of equilibrium has a fundamental importance for the study of large systems. It is well known that the concept of an equilibrium is closely related to the existence of so-called invariant measures for the system. In the analysis of complex systems, one is typically interested not only in finding the equilibrium state, but also in properties in equilibrium of some functionals defined for states of the system. On the other hand, mathematical finance offers a different point of view on equilibrium. In the financial set-up, one deals with a number of observables (prices) and adjoint processes (investment strategies). It appears that a regularity of adjoint processes is related to the existence of invariant measures for observables. This behaviour is called the absence of arbitrage. Hence, the arbitrage property is the mechanism which pushes the system out of equilibrium. This situation is well understood for simple situations like discrete state space and finite time horizon. The objective of the network is to solve the challenging problem of general state space and/or infinite horizon.

Optimization under uncertainty. An important aspect of the study of complex systems is uncertainty, which appears because of randomness, chaotic behaviour of complex systems and the dependence of the coefficients in the system’s dynamics on exogenous random parameters, called frequently factors. Risk-sensitive control problems are closely associated with optimal investment decisions. Risk-sensitive dynamic asset management theory considers portfolio optimization in the framework of several macroeconomic and financial factors. Instead of maximizing the expected utility of terminal wealth, the objective is to maximize the portfolio’s long-run growth rate adjusted by a measure of the portfolio’s average volatility. Although there is a number of papers extending the classical model from linear dependencies on the factors and linear dynamics of the factors, the complete approach to the subject, including partial observation of various factors, remains an open problem. The solution to this issue will constitute an important objective of the network.

Study of large data systems. It should be stressed that the essential part of the project shall concern analysis of large data systems, which describe a collective behaviour of various risk factors. Therefore, the research will be interdisciplinary: from computer science (computational techniques), applied mathematics, stochastics, statistics, control and system theory, game theory to mathematics of finance, econometrics and mathematical economy. Building on the extended and new developed theory, it is planned to construct algorithms for computational techniques that would be helpful in predicting and controlling risk. Although risk studied in economical setting will be an important part of the network’s activities, the research will cover the modelling and analysis of risks appearing in various kinds of social and industrial activities.

Information driven systems. One of the main objectives of mathematical modelling of large complex random systems is to make it possible to control systems or results of functioning of the systems. In a typical situation, due to random fluctuation and large dimension of the system, we have only partial observation, that is, we have to control under uncertainty of the flow of information. This shows an important role played by the flow in the managing of complex systems. First, due to the system complexity, only a partial information is available. Hence, it is important to learn how to optimize the flow of information within a given complex system. Second, in many complex systems, especially those dealing with human activities, the knowledge about a system is heavily influencing the system itself. Therefore, in the study of a complex system it is vital to identify these features, which are robust in the sense that they do not depend on the information about the system, and these features, which are information driven. The issue of modelling of the information flow is also essential in the analysis of systems with default risk. Some previous studies were done using an initial enlargement of the filtration. The so-called progressive enlargements are also used in the default risk setting. It is expected that they can be used in a more general approach to the risk management in large systems. The verification of this conjecture will be another objective of the joint research.

Large markets. One of the most striking feature of today’s global markets is the diversity of investment possibilities: one can be present in several economies at the same time. This provides unprecedented opportunities for the risk diversification, i.e., for reducing risk via trading in a large number of assets, investing in various enterprises. The mathematical theory of large markets provides a convenient framework for the study of such situations. As in the statistical approach, in which one analyzes large samples using asymptotic distributions for infinitely many observations, large financial markets can also be modeled as infinite sequences of asset price processes. By investigating their “asymptotic” behavior, one hopes to find some general qualitative and quantitative results, which can be subsequently tested empirically.

Decisions under uncertainty. Two basic interpretations of risk are possible. On one hand, it is a possibility to suffer a loss due to unexpected and unfavourable behaviour of nature. On the other hand, the risk considered as unpredictable behaviour may cause not only negative, but also positive effects, which in turn give a chance for possible gains. From the practical point of view, the decision makers are interested to know when and in what form the risk appears. For instance, the typical kinds of risk involved in investments are: interest rate risk, foreign exchange risk, inflation risk, market risk, management risk, business risk, financial risk, bankruptcy risk, legal and political risks. Implementable decision making procedures under uncertainty will be an essential objective of the research.

Foreseen benefits of project

Development of existing and creation of new tools and methods for managing risk, which would allow to understand, predict and control (whenever possible and desirable) the business and investment risk. Risk is constantly present in all activities of the society. The long-term objective of is to develop a new innovative approach to complex management of risks. It is expected that since it will never be possible to completely eliminate the risk from all kinds of social activities, one of the benefits of the project would be to organise a durable network of European experts and their laboratories providing expertise in various aspects of risk management on a long term basis. As a consequence, the project will have a strong potential socio-economical impact through the improvement of competitiveness of European industry and the increase of efficiency of European resources. Results of the research should be readily applicable, though due to the necessity to work with huge amount of data and complexity of the models, certain stages of work will require additional prolonged efforts to be implemented.