oraz grupy niezależnych ekspertów z innych jednostek akademickich i badawczych.

Poniżej załączamy informację o jednostkach naukowych wchodzacych w skład CDMF.

Centrum Zastosowań Matematyki IMPAN zostało powołane w dniu 28 listopada 2003. Celem Centrum jest konsolidacja działań środowiska matematycznego na rzecz zastosowań matematyki. Rada Naukowa Centrum Zastosowań składa się z 22 specjalistów z różnych stosowanych działów matematyki i dziedzin pokrewnych.

Kierownikiem Centrum Zastosowań Matematyki jest prof. Ł. Stettner, zaś jego zastępcami są prof. J. Grabowski i R. Rudnicki.

Centrum Zastosowań obejmuje grupy naukowe pracujące w następujących dziedzinach:

• Współpraca zagraniczna: członkowie FMG prowadzą wspólne badania w zakresie matematyki finansowej i ubezpieczeniowej oraz teorii ryzyka m.in. z następującymi naukowcami:

Prof. Monique Jeanblanc, Universite d'Evry, Francja (credit risk),

Prof. Tomasz Bielecki, Illinois Institute of Technology, USA (credit risk),

Prof. Ben Gołdys, UNSW, Australia (stochastic volatility),

Prof. Marek Musiela, BNP Paribas, Wielka Brytania (term structure modelling),

Prof. Renata Mansini, Universita di Brescia, Włochy (portfolio optimization),

Prof. Andrzej Ruszczyński, Rutgers University, USA (risk measures).

Zgodnie z założeniami zdecydowano się na wybór ściśle określonej problematyki badawczej:

profesor Aleksander Weron z Centrum Steinhausa,

Program badawczo-wdrożeniowy

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.

The potential impact offered by this multidisciplinary study of markets is due to the fact that the problem is relevant to an actual and in a sense typical complex system for which the global behaviour cannot simply and directly be deduced neither from the individual behaviours nor from average values connected to such behaviours. In the context of financial markets, a rather evident role is played by phenomena linked to the adaptation of individuals to the market as well as by mutual influence of the various random components which drive both the individual behaviours and the collective ones. These systems consist of a very large amount of heterogeneous and competitive agents and, at least in the long run, they prove to be robust with respect to external shocks and to suitably adapt to a changing dynamic environment. Therefore, it can be reasonably expected that the above considerations could constitute the starting point for the setting up of a new conceptual and intellectual framework for the proposed study. In this framework concepts connected to complexity as a discipline should play a primary role. The various specific theories connected with markets methodologies and financial problems will enter in a certainly not marginal way in the proposed construction. The latter, however, will not proceed in accordance to a purely hierarchic top-down logical procedure, but a particular emphasis will be dedicated to the study of the mechanisms that determine the behavioural and physical evolution of the system. In the framework of the study of financial markets it is rather apparent, as already mentioned, that the evolution at the subsystems level can be determinantly influenced in view of the interactions at the level of global system.

For the proposed research there is the necessity of the contribution, set forth in a rationally coordinated way, of people involved in the fields of economics, finance, system theory and mathematics, connected with academic environments but with strong and meaningful scientific links with researchers as well as practitioners related to the economic, financial and banking systems. In such a way the particular knowledge at a subsystem level (economics, finance and market practice) and the general one (mathematics and system theory) will be able to favour the growth and development of a new approach for the analysis, identification, design and control of the global system constituted by the so called economic markets.

Let us now examine in a little more detail the characteristics and properties of economic markets that make them particularly suited to the framework of complex adaptive systems and in connection with which the potential impact of the proposed research can be mainly recognized. However, for a deeper and more detailed discussion of such aspects, one should refer to the program of the proposed activity. Here we shall limit ourselves to point out the following relevant points:

- The frequently encountered non-equilibrium situations which characterize large economic markets. Indeed the most conspicuous example of a large system is perhaps today’s global economy. The global market has become a kind of all-encompassing entity, which connects phisically very distant economies. A previously unimaginable diversity of investment possibilities has emerged. On one hand this is an ideal situation as demand and supply find each other more easily, hence an equilibrium might be realized. On the other hand, however, shocks and unforeseen events seem to have more widespread, possibly more devastating effects. A serious crisis in any of the subsystems might result in overall recession due to the high complexity and the often non-detectable relationships between system components. The same remarks can be applied to the EU economic system and its submarkets: as more and more countries become members of the unified Europen economy, not only its size, but also its interior complexity increases. An effective theory describing such large systems has to provide adequate tools for recognizing risk factors, classifying and evaluating risk. If by risk we mean “unpredictable behaviour”, then the theory also has to account for the detection of positive effects: e.g. arbitrage possibilities, growth trends. The proposed research aims at laying the foundations of such a theory. Relying on a firm scientific basis (methods of financial mathematics, system theory and operational research), we hope to gain a better understanding of equilibrium in large economic systems. To this end we wish to introduce satisfactory notions of equilibrium and obtain empirically testable criteria for deciding if the market is in equilibrium. This is intimately related to the absence or presence of long-term arbitrage opportunities. Another important direction is the analysis of how phenomena in smaller subsystems propagate to the whole market and also inversely; how global trends are reflected in the interacting individual components. This may contribute to a novel, global and more realistic view of large economic systems.

- The situations in which the system (or the strategies employed for its control) adapts to the environment and in particular to the overall system. Such problems had strong impact on the formation of the basis for an active new area of applied mathematics, the theory of systems, control and filtering during the second half of the 20th century. Mathematical modelling of systems and their operating environment together with open loop behaviour and feedback control stimulates a very successful approach to the design of modern engineering systems such as circuit and aircraft design, as well as computers, communication networks and finance. In the proposal we restrict ourselves to the modelling and control of complex random systems. Such systems consist of heterogeneous interacting components which result in robust functionality. A classical example for such evolution, which is frequently referred to in our proposal, is financial market. The asset prices, bonds, and financial derivatives evolve accordingly to extremely random and unpredictable dynamics. We model it using various approaches. Since our usual purpose is not modelling itself but the possibility to form portfolios which would allow us to hedge (minimize) the risk involved in the randomness, or maximize a given utility function we have to use an adaptive methodology. We estimate time varying parameters describing our models and choose control appropriate for our specific needs. Important factor in our study is complexity of the model and its increasing high dimensionality. Very often we are able to find a satisfactory (explicit) solutions for simple models consisting of bond and one asset investment possibilities. In multidimensional cases we obtain only existence theorems which characterize optimal strategies but the problem to find nearly optimal strategy is open. In Workpackages 3 and 4 described below we study more specific examples of modeling and control of complex systems. Namely, in Workpackage 3, in particular, we consider the risk-sensitive control under uncertainty (with partial observation) which, besides the expected value, also measures the variance and higher moments of the cost functional. Workpackage 4 is devoted to the study of complexity of control aspects which arise, e.g., in pricing when we incorporate in our model transaction costs. Such a natural assumption leads to significant difficulties: we loose completeness of the market, a number of models trivialize themselves, and even in the case of fixed costs we encounter computational difficulties because of the dimension of the model. Information is an important factor in modelling and control. Its influence on complex systems (in particular financial markets) is studied in Workpackage 5.

- In the statistical analysis of random systems one usually supposes that the samples have Gaussian distribution. This assumption proved to be satisfied by a large number of real phenomena. In the description of price processes most often Brownian motion based models appear; these are widely employed in actual applications. Investigations show, however, that fluctuations of real market prices do not fit into such models. Firstly, they show long-range dependence, i.e. even the (relatively) distant past may influence today’s price movements. Secondly, the hypothesis of Gaussian distribution has been refused. One is looking for alternative processes with non-Gaussian distributions. Economic considerations as well as empiric evidence suggest that so-called “stable” laws have to be used when modelling an economic environment near equilibrium. Also the problem of a suitable scaling should be considered in the context of such complex data sets. These facts prompt the use of a generalization of Brownian motion: namely, the so-called fractional Brownian motions. These appear in the mathematical theory of other physical problems, but the justification of their use in market modelling has not yet been settled in a satisfactory way. Our objective is to extend classical theory of pricing to this model class. We hope that the progress we make in this direction will help to clarify the structural properties of such complex economic systems and that the results will also be applicable to time series analysis and will change the current market practice of pricing.

- An often observable characteristics of large, complex systems is the presence of risk, where by risk we mean “unpredictable behaviour”. This is particularly true in the case where there are several competitive agents acting. Disasters of various nature can lead to the instabillity or even destruction of the system. Hence much effort is made in order to evaluate and, to certain extent, reduce risk. This is particularly spectacular in economic operations. It is also vital to identify individual risk factors and understand their correlations. Our starting points are risk measuring methods, which are currently employed in the stock exchange, insurance companies, etc. Relying on recent developments of the theory, we will concentrate on finding appropriate measures of correlation and risk. This is related to the previous point, as several problems arise because the data are not Gaussian. Products of our research activity will hopefully be applicable to analyse a wide range of economic, industrial and non-economic (e.g. weather) risks, and they will provide efficient tools to separate and classify risk sources.

- In the identification and control of complex random systems an adequate picture of the fluctuation sizes (volatilities) is necessary. The qualitative and quantitative properties of these fluctuations change following the system dynamics (e.g. growth). Prevailing models of financial markets aim at finding a suitable dynamics for volatility. Data analysis gives evidence that volatility has to be modelled as random. The first problem is to retrieve from available data how this volatility process behaves. One should investigate the feedback mechanism between the process and its volatility, i.e. the correlation between them; and then calibrate parameters in an appropriate way. There are also theoretical issues connected to volatility: is it theoretically (and also practically) possible to reconstruct the volatility knowing the original process only? Solution to these problems will enhance calibration methods in engineering and, in particular, in financial market analysis. It will refine methods for determining frequency and amplitude of perturbations and extremal events which may destabilize the system.

- The availability of only limited local information, especially in connection with not traditional financial products, and the consequent necessity of its modelization. As a matter of fact, in recent years new products appeared in the markets, which can be formally treated as derivative products, but with underlying products which are not typical market commodities. Either they are not traded and have no market price (weather, investment opportunities) or they are not standard (energy which cannot be stored). These new products require new approach to their pricing and risk management. A complex system point of view will be used as the right way to deal with these problems as we have to analyze a large set of data (weather data, investment data, energy market data) and extract meaning from these sets. In dealing with such a huge set of elements we shall look for the qualitative characteristics of the whole set of factors and not to analyse each factor separately. Our aim is to construct relevant models in terms of macroscopic characteristics of the whole set of data. A new challenge in dealing with these products is due to the heterogeneity of information of different agents. Some agents (that we might call insiders) can profit from privileged information that is disclosed to the others only at the end of the trading interval, and by which they may even be able to influence the market development. We are planning to investigate models with agents on different information levels and to use methods based on the stochastic calculus of variations that allows an explicit description of the information drift by which the additional utility of better informed agents can be expressed. In this analysis we shall investigate the problem how the additional information may be blurred to rule out lack of equilibrium. All that research will add substantially to present knowledge of complex systems. First, we shall create new ways of extracting meaning from huge data sets by constructing relevant models. Second, we shall incorporate to our analysis heterogeneity of information of different agents. Third, we shall show by examples how these new ideas can be used to real problems of risk management of new products.

It can be anticipated that the proposed research will result in a new approach to the study of such complex adaptive systems, which will be the prototype of a general theory of complex systems. As it will be explained below, the network will also allow the penetration of this new point of view and of the relative concrete techniques into real market practice. This system theoretic point of view will produce a completely different attitude towards the analysis of economic systems and the connected concrete activities. To mention a few simple examples, price fluctuations will no longer be interpreted as exogenously produced data, but as the result of the interaction and feedback of the various agents acting on the market. Furthermore, the role of information flows and the related concept of "price" of additional information will be drastically reviewed as a consequence of the fact that an increase of information, e.g. on prices, will not simply be a matter of a deeper and more detailed statistical analysis; indeed, the construction of a system theoretic model of such complex systems will broaden the understanding of the relevant information both from the viewpoint of its methodological aspect and from that of market practice.

The proposed research is a strategic need in the context of the increasing integration of markets at an international level. In fact, there is a flourishing of initiatives, outside Europe too, in this direction and it is then clear that there is a strong need to reinforce the existing research capacities, which are of very good quality at a national level, by means of a network that will be able to link together the various competences, thus producing as a result the scientific jump necessary for the development of the proposed new approach to complex economic systems. The ways in which the network will achieve the spreading of excellence will be realized along three main lines. The first is constituted by the presence (inside the network) of people coming from the world of market practice and/or with deep and strong connections with such people. This will not only reach the goal of spreading excellence, but also of exploiting the results of the research in the context of the economic, financial and banking systems of Europe. The second line is connected to the numerous and diverse conferences and workshops which are expected to take place under the auspices of the network, but with active participation of people all over Europe working in connection to actual applications as well as involved in research activities in the field of economic markets. The presence of these people will be greatly influential for the development of a complex-system theoretic attitude in the field. The third line of development is offered by the direct involvement of numerous PhD students in the activities of the proposed research. They will be the most precious and effective ambassadors of the new approach to be developed both in the context of academic research and of actual applications. They will also be the guarantee of a durable structuring impact of the research for the next future. Under this respect, it is however worth mentioning that the new approch to complex economic systems is being developed at a national level and that the involved institutions as well as individuals are strongly motivated to the continuation of the proposed research, especially if European funds will allow the mentioned scientific jump, both qualitative and quantitative, which will initiate the construction of a solid foundation for such innovative fields.

9. 9. Measures of risk

10. Ruin probability and operational risk

11. Fairness and equity in systems design and operation

16. New commodity markets

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Współpraca z przedsiębiorcami CDMF była realizowana przez Grupę Matematyki Finansowej (FMG) z Ośrodka Promocji Badań Wydziału Matematyki i Informatyki Politechniki Warszawskiej oraz Centrum Steinhausa

Analiza opłacalności działania wybranych placówek bankowych dostosowanych do potrzeb klienta - PKO BP (1998) – statystyczna analiza działalności placówek PKO BP uwzględniająca rozkład liczby klientów w ciągu dnia, optymalizację liczby pracowników obsługujących klientów indywidualnych i korporacyjnych; analiza strategii marketingowych i portfela usług

Rozwój algorytmów prognozowania zapotrzebowania na energię elektryczną dla ZE Opole S.A. (2002) – statystyczna analiza danych dotyczących zapotrzebowania oraz warunków pogodowych; oprogramowanie algorytmu prognostycznego współdziałającego z systemem komputerowym ORL w celu wspomagania decyzji rynkowych ZE na wysoko zmiennym rynku bulansującym

Zintegrowany system zarządzania ryzykiem w Elektrowni Opole S.A. (2003) – opracowanie założeń systemu zarządzania ryzykiem na rynku energii elektrycznej z uwzględnieniem segmentu bilansującego, giełdowego i pozagiełdowego dla wytwórcy energii elektrycznej; koncepcja systemu; metodologia (analiza, modelowanie i mapowanie czynników ryzyka, wyznaczenie wartości zagrożonej – CFaR, EaR)

Tools for efficient risk management on German and Polish electric energy markets – DAAD/KBN (2002-2003) –