Project 1834

Discrete and continuous stochastic models with applications

Leader: dr Milan Merkle


Subject of research

Convexity, stochastic majorization, special functions with their applications in stochastic models. Probability distributions and estimation of parameters. Stochastic differential equations. Time series analysis. Design of experiments. Continuous time stochastic processes. Reliability theory with applications. Kalman filters and neural networks with applications. Stochastic systems. Applications of stochastic models in finance.

Description of the work

Logarithmic concavity of probability distributions, in particular the distribution of a sum of independent random variables. The Gamma function and Gamma distribution, related distributions. Schur convexity and applications in stochastic majorization problems.

Robust estimation, in particular Pitman’s estimation and related Monte Carlo approximations. Median theory as an alternative to mathematical expectations. Weak convergence of finitely additive probability measures. Some new characterisations of uniform distribution and their stability.

Generalized moments in Karamata’s sense and the research related to Karamata’s slowly varying functions.

Investigation of existence, uniqueness and stability of solutions of various stochastic differential equations of Ito’s type. Analytic and numeric approximations to solutions of perturbed stochastic differential equations. Applications in mechanics and economy are expected.

Time series with random coefficients and various marginals: conditions for existence, estimation of parameters, prediction, applications in economy.

For continuous time second order processes in a separable Hilbert space, the problems related to spectral types and multiplicity will be researched.

In reliability theory, we shall investigate the reliability of systems of two and three elements with repair and with preventve maintenance. A part of this investigation will include a Monte Carlo simulation study.

Extended Kalman filter based training of artificial neural networks. Aplication of nonlinear/non-Gaussian filters in parameter and structure sequential adaptation.

of neural networks.

In systems theory we will investigate various concepts of causality and their applications to stochastic differential equations.

All members of the proposed project team will try to apply their reasearch and knowledge in models of financial mathematics: pricing of assets, stocks, bonds and derivatives, research in portfolio optimization, risk theory in finance.

Originality of research

The originality of proposed research will be verified by publishing results in scientific journals. The originality of the project as a whole is implied by its multidisciplinarity.

Research Goal

In the realization of the proposed project, the research team will publish at least 35 papers in international journals and at least four scientific monographs. Five members of the team will obtain their M.Sc and two of them will complete their doctoral theses. Within three years, the team will become recognisible for its contributions to mathematics of finance in Serbia, as wel as for its contributions to the world science in the field of stochastic models.

State of the Art in Scientific Field

World-wide Situation

Stochastic models belong to a mainstream of world mathematics. All topics which are included in the proposal are being developed in an international scale and papers in these topics appear in leading journals and monographs. In last decade, the applications in finance are in the very top of interests of leading mathematicians of the world. We give a selection of important references for the topics covered by this proposal.

Domestic Situation

The theory of stochastic differential equation is being developed in Serbia within last 15 years.Recently, this is one of main topics in the Seminar for stochastics at the Mathematical Institute.Convexity and special function have a long tradition here. A team of mathematicians led by late Professor D.S.Mitrinović of the Faculty of Electrical Engineering,Belgrade,used to be a top world center for convexity and special functions. The role of the proposed project is to continue the tradition,applying convexity in stochastic models,which is a relative novelty with us, but fits in the international trends. A similar argument holds for the role of Karamata theory of slowly varying functions. Chiefly,the most of existing results in the areas of the proposed project have been obtained by members of our team,and the references are attached in individual lists of references.The applications in finance have not been studied so far,except that our team realized a six-month working seminar.

Planned Project Results

It is a well known fact that stochastic models may find applications in practically all sciences. The research which will be conducted in the proposed project have a range of potential applications, among which we mention the most immediate ones:

  1. Stochastic differential equations in models of automatic control; possible applications in Mechanics.
  2. Time series in Economy;
  3. Second order processes in Telecomunications;
  4. Reliability theory in maintenance of compound systems (electrical power plants, industry, traffic) ;
  5. Probability distributions and estimation of parameters in all areas where stochastic models can be applied. In particular, some of proposed research could have an application in Economy.

We emphasise that the common task of all participants in the project is to investigate possible applications in Finance. Among the proposed activities in the project, some can be potentially and some immediately applied in finance, most concretely in stock market transactions. Belgrade Stock Exchange is recently established and in the future it will have to trade in world markets, the results in the field of financial applications will be very important for Stock Exchange, as well as for other financial institutions. In recent years the number of mathematicians employed in leading world financial institutions has been rapidly increasing. Their task is to deal with modelling prices of assets, stocks, bonds and derivatives, mimimising the risk and other activities in finance.

Some of research in the proposed project will have a direct applicability in finance. All financial institutions, for instance, Belgrade Stoch Exchange (recently established) have a direct need for a concrete mathematical knowledge and its implementation in the form of a software or concrete instructions in decision making in trading stocks and bonds, or estimation of value of assets and derivatives.

Since in this area there are few routine recipes, activities of leading world financial institutions are based on investigations of their own teams of mathematicians. Having in mind that our stock market will probably work under different assumptions from those in other world markets, the results of the proposed project will certainly have a direct applicability provided that the proposed cooperation will be realized. The proof of interest of Belgrade Stock Exchange for a cooperation with us is attached in the appendix.