Research Program
Central Research Idea
The central conceptional idea of the research trainig group in the shape of a "red thread" is reflected also in the research program. We want to investigate relevant questions from areas of economy by means of mathematical modelling, analysis and simulation. For this we first isolate the economical problems which shall be examined within the research trainig group. They derive in particular from finance and insurance. We differ 5 areas which are combined in Table 1. The described fields are imported into the research trainig group from the involved scientists as well as from the partners from outside the university.
Economical Subject Areas:
- A.1 Valuation of complex financial products
- A.2 Risk analysis and management
- A.3 Optimal strategies
- A.4 Econometric analysis and strategies
- A.5 Knowledge Discovery and Data Mining
In the following we first describe the several economical subject areas. At this we show the relevant questions and describe to which mathematical problems this leads and respectively which mathematical techniques meaningfull can be used. These again are represented in the research trainig group through 6 mathematical research emphases which mainly cooperate in the areas of data analysis and simulation closely with the Applied Information Processing. These research emphases are summarized here:
Mathematical Research Emphases of the research trainig group:
- M.1 Stochastical models, their analysis and simulation
- M.2 Statistical inference and data analysis
- M.3 Stochastische Steuerungen und Optimierung
- M.4 Partial differential equations and functional analysis
- M.5 Financial mathematical modelling and analysis
- M.6 Numerical analysis and simulation
- I Methods of software development and of software quality management
After the depiction of the economical questions we describe explicitly the mathematical research emphases which are relevant for the research trainig group.
Economical Subject Areas
At first we describe the economial questions, which shall be examined with mathematical methods within the research trainig group. Here we restrict to the desription of the problems and the depiction of the relevant mathematical disciplines.
A.1 Valuation of complex financial products
The progression on the financial markets during the last years is characterized through a rapid growth of the amount of complex financial products such as interest and credit derivates, structured financial products (like Collaterized Debt Obligations (CDO), Mortgage-backed Securities, Securitization of insurance risk, Convertible Bonds and much more). The rating of these securities and derivates requires an exact modelling of all important risk circumstances together with an explicit data analysis and an efficient numerical realization. We will examine especially credit derivates and asset-backed securities (with the special difficulty of modelling the structures of subjection). The econometric analysis of the structures of subjection represents a central aspect of the area A.4. One more very current field of research in this area es the application of financial mathematical techniques for rating the current values of insurance policies (especially in the area of life and pension insurances). Here it is in particular important to rate optionalities and guarantees of forms of contracts.
A.2 Risk Analysis and Management
Risk management plays an outstanding role with banks and insurers an at a progressive rate also at industrial enterprises. The present discussion about suggestions for a regulating demand of capital (Basel II) against credit institutions prove the topically of this battery of questions. In the context of this main focus the follwoing projects shall be followed up: furhter development of existing models for risk measurement; design of risk management systems, parameter esitmation for risk models (in particular in relation to default and migration probabilities, subjection structures); optimal policy towards risk based on quantitative risk management systems. Furthermore the steering of risk in the context of Asset-Liability Models shall be analysed by insurance companies. Here in particular the analysis of dynamical strategies and criteria (so called management rules) and their measurement processes for evaluating the consequences of such strategies on the risk of enterprise are concerend. This methods shall be also applied to analog questions of operational retirement provisions in particular to
- pension fonds which exist only for a short time in Germany and on which it is allowed to embed interest guarantees and
- capital cover of direct contracts on which the financing of the entrepreneur can be made without any regulation for the capital investment.
A.3 Optimal Strategies
The definition of optimal portfolio strategies is a central function of the finance and insurance industry. The problems of optimization that appear thereby afford an accurate modelling of the underlying stochastical processes, an exact data analysis as well as an efficient solution and simulation methods. We want to examine especially portfolio problems with discrete asset processes. Benchmark and utility criteria are in the front besides game-theoretic solution approach. One furhter emphasis will be the analysis of asset problems. Of special interest here are problems with incomplete data (e.g. unknown driftrate or jump height) and with different assumptions of information infrastructure of the involved economical agents (e.g. insider knowledge, modelling of negotiations for bond and equityholder in the context of rating of corporate bonds). At last asset-liability models for insurance companies and the optimal design of old-age provision insurances shall be analysed.
A.4 Econometric Analysis and Strategies
One central aspect of the econometric analysis of financial markets is the examination of the characteristics of time series of capital market data. Recently panel data - combined cross-section and time series data - are used more and more. In particular at non stationary panel data there are a number of methodical problems becaus the distribution of the estimated coefficents can not be defined exactly. For this generally simulation methods are used. In this area methodical bases for the ecometric analysis of panel data shall be researched for capital market variables. For al lot of econometric questions aggregated time series are used which also exist as spacial detached data. Methods of modelling temporal dependencies (time series models) and procedures of the spacial statistics (see M.1.2) shall be combined to enable a broader analysis. The resulting methods are interesting for much more applications. For mortality study at insurance companies scoring methods and life expectancy prospects shall be conceived and examined. For this new methods of non parametric statisitcs must be developed. One more field of application is the analysis of modification of ratings which results are important for the optimal use of ratings in risk management. Besides panel structure (a lot of enterprises over a period) the econometric analysis must also satsify the categorial data and possible thresholds of the revaluation of ratings. One important question in the context of credit risk management is the estimation of cancellation probabilities and transition matrices. Tybically cancellation probabilities (transition matrices) were estimated on base of generalized linear models, mostly (ordered) probit. In the context of this project results from pure parametrical methods shall be compared with flexible non-parametric methods. There in particular non-parametric link functions an a generalization of independent variables are introduced. A widely spreaded assumption in the analysis of international financial markets is the increase of the subjection of the return in phases of a economic slump (contagion effects in bear markets). This proposition shall be examined with a new developed tail-dependence estimator. Here the explicit dependendy of time is to be modelled.
A.5 Knowledge Discovery and Data Mining
In the front of the rating of financial and insurance risks there are traditional questions wich either concern only the analysis of the present actual state of the risks or predominantly refer to the temporally development of the related risks. In particular with insurance risks the question is of great interest if and in which way these risks depend on their relative (geographical) location and respectively if (and if yes how) this dependency of location chances in the course of time (e.g. local tariffs in the vehicle insurance, rating of risks of elementary loss). In the context of this thematical emphasis we want to deal with the development of the mathematical analysis of the morhpology, i.e. the spatial structure of financial or insurance (image) data in view of improving the efficiency of risk management systems. One furhter important aspect is the development of statistical significance tests in relation of such spatial structures. in particular this shall be inserted for combined space time analysis of financial and insurance data to record and quantify in principle geometrical characteristics of structures of the data. With this it becomes possible to diagnose chances in structures even if there are only "short time series" available. In this connection necessary transformations shall be examined which can be needed to obtain a homogeneous distribution all over the whole "image". This in particular shcall be consulted for the development of "early warning systems" to be able to recognize variances in the business development of financial service providers in time. There we will also analyse to which extent these methods can be also used for similar structured (virtural) "maps" of bivariate data that have no geographical relation. So these methods become interesting too for more central questions like multidimensional approaches of the customer value analysis (in shape of generalized scoring methods) or the cancellation prophylaxis. More substantial aspects which are examined in this subject area of the research trainig group are questions of applicable structures of information technology as well as the integration of statsitical methods in organsation and contolling processes of companies. At the latter question the CRISP-DM-Model in context of the KDE shall be generalized in particular.
Mathematical Emphases of Research
M.1 Stochastical models, their analysis and simulation
In this emphasis of research ideas of the theory of stochastical processes and fields, of the extremum theory and modelling of subjection (Copulas) are used. Furthermore techniques especially from of the stochstical geometry or Markov-Chain-Monte-Carlo are picked up and developed.
M.1.1 Theory of extreme value and modelling of subjections
In the field of risk estimation of asset and first of all credit portfolios one wants to control the likelihood for simultaneous big losses. For this one assumes that the related corporate distribution is in the adduction of a multivariate distribution of extreme value and is able to figure out the so-called tail probability. In contrast to one dimensional distributions of the extreme value one has a semi parametric model in the multi dimensional case. Our basis problem is high dimensional and so the "curse of dimensionality" and the "sparseness of data" come together in the tails an make the problem for statistical methods sophisticated. We want to track the following strategies: On the one hand one tries to find rich families of distribution which enable dependencies in the tails and at the same time are variable and parametrical. With suitable classes of couple and boundary distibutions one can adapt a parametrical model. On the other hand the dimension reduction approaches which are also known in other fields of the stastic are interesting in the tails. There one has to develop methods for summarizing the components into groups in the tails to reduce the dimension systematically and to adapt this into the model.
M.1.2 Stochastical Geometry and Monte-Carlo Simulation
In this area of the project problems shall be stochastically modelled, analysed and simulated that are connected with the economical questions which are described in A.4 and A.5. The models and methods which are to develop will pick up ideas of the theory of stochastical processes and fields, of stochastical geometry and of the Markov-Chain-Monte-Carlo (MCMC) and develop them. nn particular stochastical space-time models shall be developed for financial and actuarial data which are shown through irregular but geometrical structured dot patterns, mosaics wicht convex cells or Keim-Korn-configuration and in the same tiome can be liable to a temporal dynamic. Furhtermore "rated" random sets that can act as models for multiphase picture data (grayscale pictures) shall be examined too. The interesting questions and also the adequate mathematical models are typically that complex that they can not be examined only with analytical formulas. The MCMC-simulation of the considered models then is a useful alternative analysis tool. Especially time inverse algorithms of coupling of the perfect MCMC-simulation shall be developed and validated.
M.2. Statistical inference and data analysis
The intention of this emphasis of research is the development of methods for spatial statistics and non parametric statistics as well as their application on risk problems.
M.2.1 Non parametric statistics
Here modelling of loss and transition probabilities between rating classes is the topic. In financial mathematical models they depend e.g. on indices which describe the economic progression. Because of this it is obvious to use generlized linear models (logit or probit models). The variables of influence are modelled now as a finite vector. But the underlying data follow stochastical processes which are included in probabilities as linear predictors. Here it shall be examined systematically how the prameter function can be estimated in the linear predictor and how the asymptotical behavior of the estimator looks like, to carry out further statistical analyses. these models also play a role in bio statistics. The next question is how one really has to choose the so-called score function this means the statistical problem deals with the non parametric estimation of this function on simultaneous estimation of the parameter function. In the next step on the statistical aspect there are not only models of interest which are as constrained variables of a real value but where the purpose value is a function. These questions shall be approached in different projects. At al lot of data reports from the world of finance first a dynamical component has to be removed. Mostly this tried to be done with GARCH models although one knows that this approach not always goes with the data. As we know, systematical adaption tests do not exist. Thereon models with non parametric volatility functions shall be examined. In time contiunuous data outsiede influences shall be discriminated with covariance structure and functional data analysis.
M.2.2 Spatial Statistics
In this subproject methods shall be developed to adapt stochastical models for dynamical rated random sets to spatial place detached (picture)data by financial or actuarial circumstances. Because here the data typically can be collected only through selctive measurements often the data have to be prprocessed at first, e.g. through extrapolytion with krigin techniques. Thus a plane or spatial continuous description of the structure characteristc of the data becomes possible which then can be modelle stochastical-geometrical. One further probelm is that the data in a lot of cases are explicit anisotropic, this means they havedirection subjected covariates. The statistical evaluation of extrapolated picture data shall be carried out with extracted quantitative picture characteristics. There an (as small as possible) amount of morphological picture characteristics shall be particulized which are qualified for estimating reliably the model parameter and for describing the essential structural characteristics of the observed picture data. For this purpose e.g. the Minkowski functionals of rated random sets are suitable (area or volume parts of the idndividual phases, specific boundary extensions or specific surface of phase combinations, et al.). Furhtermore the parameter of furhter morphological funtions (for example of contact distribution functions) shall be regarded. And in this context methods of solution shall be developed for classification problems by means of asymptotic tests of simulation tests.
M.3. Stochastical controls and optimization
In this emphasis of research stochastical optimization problems are examined. The controls base on stochastical processes with constant time parameter, e.g. diffusions, Lévy-processes or piecewise deterministical Markov-processes. Important questions concern the existence and construction of optimal controls. Both questions are analysed with stochastical maximum principles, the Hamilton-Jacobi-Bellmann equation or the martingal method. We want to examine especially the questions from the portfolio optimization. Dynamical portfolio problems need an exact modelling of the underlying stochastical processes an also of the infromation structure. Asset processes with leaps shall be admitted and different aim criteria shall be examined: Benchmark optimization and risk sensitiv profit maximization. In particular play theoretical approaches shall be tracked, where especially differential plays play a important role. Of big interest are also asset problems with partial information or insider knowledge. Such problems appear when e.g. the drift rate or the step height are not known completely. by means of statistidal mehtods (e.g. Kalman filter, Bayes method) such portfolio problems shall be analysed. At insider problems there appear magnifications of filtrations which leads to profound questions in stochastical analysis. If there don't exist closed solutions one is always relient on numerical mehtods for solution for appropriate optimization problems. Their development and implementations is worked out in close cooperation with M.6.
M.4 Partial differential equations and functional analysis
Numerous models in physics, technology and economical mathematics are expressed in the shape of partial differential equations. Here the functional analysis yields important methods and structures. A large class of problems are linear parabolical and elliptical equations of second order. Important questions concern the asymptotic, regularity, perturbation theory and spectral theory. In the economical mathematics there appear also equations of the same type in the modelling. But typically the are degenerated like for instance the Black--Scholes equation.
M.4.1 Equations of evolution: asymptotic and regularity
Purpose of this project is to examine elliptic operators of the form Gl1 with measureable coefficients. Here different boundary conditions in a region are regarded. Spectral theoretical characterisitcs shall be studied and the related parabolic equations Gl2 be examined. It describes e.g. diffusion processes but also appears at the rating of portfolios. Gauss estimations are of major importance. For Nasch e.g. they served as proof for his famous theorem about the consistency of weak solutions. But they have also consequences for the regularity and for this reason for the solution of non linear equations and numerical solving. So far such estimations are unknown for Robin boundary conditions.
M.4.2 Partial differential equations in financial mathematics
There is a number of elliptic equations which are modeling the rating of a portfolio. This is also about characteristics of regularity which ar important e.g in numerical analysis. The temporal asymptotical behavior shall be examined too. One particular difficulty ist that the coefficients generally are degenerated. The newer functional analytical approaches are of special interest.
M.5 Financial mathematical modelling and analysis.
Financial mathematical research has the intention to improve the appreciation of the development on the financial markets through formulation of mathematical models. In particular new efficient methods shall be found for pricing and hedging complex financial derivates, investment decisions and the management of stock portfolios shall be emproved as well as the risk of financial situations shall be appreciated explicit. The needed techniques are out of numerous fields of the applied and abstract mathematics like e.g. stochastics, statistics, stochastical analysis, numerical analysis, analysis, (partial) differential equations and functional alanysis.
M.5.1 Valuation problems
In this sub-project credit derivates and asset-backed securities especially colleralized debt obligation (cdos) shall be rated and hedging strategies for risk control shall be developed. Furthermore contracts of insurance (portfolios of insurers) shall be analysed with financial mathematical methods and hedging strategies for insurance companies shall be developed. A further problem is the rating and hedging of derivates in general semiparametric models in which the pricings of bonds are modelled as Lévy processes. In all cases the complexity of the question leads to models of partial finacial markets, i.e. non definite martingale measures and incomplete hedgable risks. An aim of this sub-project iwil be to develop valuating principles and optimal (i.e. at the best possible rato of a given criterion) hedging strategies for such situations.
M.5.2 Risk management, regulating aspects
In the context of this sub-project the analysis and controlling of portfolios comes to the fore. As a central question we regard the modelling of portfolios of credit risky bonds, where the empirical analysis of the control factor (model parameter) plays an important role. At the modelling essential aspects are design of rating systems, lifelike transformations of dependenca structures, estimations of the loss-given-default and quantification of individual risk componts in correlation with the analysis of credit spreads. These control factors shall be also analysed empirical. So far ratings for example are mostly analysed for their outlook potency and predictability. What is missing are structural analyses of the underlying decision processes. In the filed of credit-scoring in correlation with the design of rating methods different statistical methods present themselves. Open questions are suitable valuation measures for scoring mthods, handling of data problems or the necessity of qualitative information. A starting point for this is the use of bayesian methods in particular to soften problems with a lack of data. For estimating dependency structures like in M.1.2, techniques of the extremum theory and copula estimation must be extended and refined. The empirical analysis of credit spreads includes the validation and analysis of structural default models (Merton models). With such models the valuation of different credit risk addicted bonds of an underlying (of a company e.g.) can be examined for consistency (absence of arbitrage) and the specific non-diversible risk of companies can be analysed. Furthermore models of portfolio control for risk management (like CreditMetrics, CreditRisk+) shall be analysed. In many cases it is yet unsettled if these models are adequate, how the shall be parameterized and intergrated in the sense-making process. This does not hold only for kinds of risk which became interesting recently (credit risks, operational risks, profit risks) but also for risks of the market. Such regulatory aspects of the question are the center of the present activity of the Basel committee and are examined in the context of the project in the cooperation with the BuBa and the BaFin. One further problem is the analysis of asset-liability models. At this general classes of stocastical processes (like Semi-Markov processes, Lévy processes) which require new rating and hedging methods (from M.5.1) shall be used as basis of the modelling. Optimal portfolio allocations must be anaysed within the framework of stochastical problems of control. Typically these lead to equations of Hamilton-Jacobi-Bellman type or ( using the martingale approach) to non-linear integro-differential equations. For the numerical implementation the simulation methods from subproject M.1.1 shall be used as well as the methods from M.6.
M.6 Numerical analysis and simulation
Even if in some cases explicit solutions for model equations of the economical mathematics (e.g. Black-Scholes equations in simplest form) are known, the relevance of numerical analysis and simulation in economical studies increases enormously. J.P. Lions says it like this: "Numerical knowledge desperaely neede in economics". The intention of this emphasis of research ist not the pure application of known numerical methods to economical problems but the construction, analysis and implementation of new optimal methods which are adapted to the problems.
M.6.1 Numerical methods for the solution of operator equations
In this subproject first of all the operator equations which are emerged from this modelling shall be solved numericaly by means of recent multiscale and wavelet methods. For this first of all accordant methods are constructed and analysed (convergence, order of convergence, effort, efficiency), and furthermore numerical experiments realized. Like described above (see A.1-A.5) operator equations appear in the form of elliptical partial differential equations, integral and integro differential equations as well as variation inequations. Discrete and continuous optimization problems also play an important role. In the litrature so far financial derivates are primarily regarded in dependency of a basic value. At more complex products and portfolios this is no longer given. We want to develop and realize appropriate numerical methods for higher dimensional cases. In particular valuation problems with sophisticated dependency structures concerning multiple basis values shall be examined.
M.6.2 Numerical methods in signal and picture processing
Numerous subject areas of the research trainig group lead to the problem of the analysis of signals and pictures. This is the case e.g. at time series of financial (market) data or the geographical influences at finance and insurance products. Requirements on the analysis can be among other things the diminishing of noise, pattern and trend recognition, compression or criterion extraction. Because these data analysis often is combined with another algorithmic processing (e.g. partial differential equations), a connection of signal and picture processing with the numerical solution of the evolving operator equations (see M.6.1) makes sense. Thus we want to include wavelet concepts in the mentioned areas of problems to work out a continuous algorithmic access which also admits a rigorous analysis (convergence, efficiency).
I. Methods of the software development and the software quality management.
The idea of research of the research trainig group can only be realized with recent and efficient methods of the software engineering. Thus Prof. Dr. Schweiggert head of the Department of Applied Information Processing in the Faculty of Mathematics and Economics as well as in person coopted by the Computer Science Faculty is involved in the research trainig group. The transformation of mathematical methods has to use efficiently methods of object navigation with the reusability, the flexibilty (e.g. the utilization of the plugin-concept), the verifiabiltiy and especially the possibility of care in mind. This focusses not only on the software as a product but also and first of all on the process of software development and of software care.
I.1 Object-oriented software library for adaptive numerical methods
The adaptive wavelet mehtods described in M.6 require completely new algorithmic ingridients and other realization concepts as the classical (adaptive) methods). Conventional matrix-vector -structures are not helpful in these applications. Thus it is the aim to create a object-oriented software library which on the one hand realizes the mentioned adaptive methods in an efficient way to enable substantial comparisons to conventional adaptive methods and on the other hand is structured in a way that it can be applied and extended easyly in research and teaching.
I.2 Improvement of quality of scientific software
Scientific software development, this means development of software which takes place in the scientific environment, is largely affected by modifications. Short membership in projects because of short-lived graduations and diploma thesises, because of strong heterogeneity of the computer science and software engineering knowledge of members of the team - which typiclly are no computer scientists - and the rapid change of standards due to new results of research are common problems with which one is faced up at the development of scientific software. Aim is the (further) development and combination of techniques which arise the so generated scientific software as well as the testing and application within the limitis of graduations and diploma thesises.