Project 144021

Stochastic with applications

Leader: Slobodanka JankoviŠ

Abstract

Generalization of the notion of median in multivariate distributions, concepts analogous to moments (iterated medians). Convexity with applications in probability and statistics. Stability of characterization of the uniform distribution.

Asymptotic behavior of moments of power series distributions, generated by entire functions of finite order and having a regularly varying logarithm of maximum of the modulus.

Reliability of three-unit system with repair and with different kinds of preventive maintenance, simulation, limit theorems for quick repair and quick preventive maintenance.

Continuous stochastic processes of second order in a separable Hilbert space. Conditions for equivalence, spectral type and multiplicity. Stochastic models. Wiener process, martingales, stochastic differential equations and applications in financial mathematics and biology.

Causality relations between stochastic processes and families of sigma algebras.

Statistical algorithms of machine learning: nonlinear Bayesian filters, neural networks, Gaussian Mixture Model estimation using Expectation Maximization, incremental hierarchical classification: Decision Trees, hidden Markov Model, hierarchical Hidden Markov Model, orthogonal polynomials in dynamic systems identification.

Expected results: 2 monographs, 30 articles in journals, software implementation of developed algorithms, 1 masters thesis and 1 doctoral thesis.

Applications:
A) Identification and control of nonlinear dynamic systems
B) Nonstationary time series prediction
C) Operational Research and Optimization
D) Risk assesment

Subject, description and importance of research

The subject of investigations is various stochastic models studied worldwide and being not only of theoretical, but also of practical importance. The range of potential applications are the following ones: Economy (financial mathematics), Telecommunications (second order processes), maintenance of compound systems - electrical power plants, industry, traffic (reliability theory), applied statistics (distribution theory, parameter estimation, stability of characterization), Mechanics (Stochastic differential equations), Identification and control of nonlinear dynamical systems, Non-stationary time series prediction, Text analysis, Operational Research and Optimization, Risk assessment.

The planned investigations are the following:
Generalization of the notion of median in multivariate distributions. Investigations connected to median: concepts analogous to moments (iterated medians). Convexity with applications in probability and statistics. Investigation of stability of characterization of probability distributions, in particular of the uniform distribution.

Asymptotic behavior of moments of power series distributions, generated by entire functions of finite order and having a regularly varying logarithm of the maximum of the modulus.

Reliability analysis of three-unit system with repair and with different kinds of preventive maintenance, simulation of the work of this system on computer, limit theorems under conditions that repair and preventive maintenance are quick.

Continuous stochastic processes of second order in a separable Hilbert space. In particular, Gaussian continuous stochastic processes of second order and conditions for their equivalence. The investigation of their spectral type and multiplicity. Finding conditions for the processes to have a prescribed spectral type and multiplicity. Especially, the processes with multiplicity one are investigated. The conditions of equivalence of Gaussian processes of second order are analyzed and compared with conditions of having prescribed spectral type and multiplicity.

Various stochastic models. Wiener process and martingales in financial mathematics. Stochastic differential equations and their applications in financial mathematics and biology. Mathematical models based on stochastic differential equations in determining the market value of options. Analyzing versions of the well-known Black-Scholes equation and their application. The use of stochastic differential equations in construction of mathematical models for solving other problems concerned with prediction in biology.

The investigation of different causality relations between stochastic processes and between families of sigma algebras. The relationship between causal-consequential relations with stochastic dynamical systems and with various types of stochastic differential equations (s.d.e. with ordinary Brownian motion, s.d.e. with fractal Brownian motion, s.d.e. with martingales).

Statistical algorithms of machine learning: nonlinear Bayesian filters as learning algorithms for parameter and structure adaptation of artificial neural networks, Gaussian Mixture Model estimation using Expectation Maximization, incremental hierarchical classification: Decision Trees, Winnow, Support Vector Machine, hidden Markov Model, hierarchical Hidden Markov Model, orthogonal polynomials in dynamic systems identification.

Research Goal

As the result of the proposed investigations, the original contributions published in journals and monographs are expected. The publication of at least two monographs, and at least 30 articles in journals, software implementation of the developed algorithms, one master's thesis and one doctoral thesis are planed.