On the Affine Maps of En

According to well-known methods of standard calculus any smooth vector-field of the Euclidean n-space is often approximated by one of its Taylor polynomials of first degree (i.e. by a corresponding affine map of the space). Such an affine map is given in general by its linear part and by the translation of an origin. However, the number of input parameters characterizing this affine map can be reduced considerably by using a suitably chosen new coordinate system. The paper gives a straightforward method for finding the position of the most convenient Cartesian coordinate system and answers also the question how to find the minimal translation part of any affine map

An improved bound for the exponential stability of predictive filters of hidden Markov models

We consider hidden Markov processes in discrete time with a finite state space X and a general observation or read-out space Y, which is assumed to be a Polish space. It is well-known that in the statistical analysis of HMMs the so-called predictive filter plays a fundamental role. A useful result establishing the exponential stability of the predictive filter with respect to perturbations of its initial condition was given in the paper of LeGland and Mevel, MCSS, 2000, in the case, when the assumed transition probability matrix was primitive. The main technical result of the present paper is the extension of the cited result by showing that the random constant and the deterministic positive exponent showing up in the inequality stating exponential stability can be chosen so that for any prescribed s exceeding 1 the s-th exponential moment of the random constant is finite. An application of this result to the estimation of HMMs with primitive transition probability densities will be also briefly presented

Sztochasztikus Rendszerek és Pénzügyi Piacok Modellezése = Stochastic Systems and Modelling of Financial Markets

A kutatások célja a sztochasztikus rendszerek legkorszerűbb módszereinek az alkalmazása a pénzügyi piacok modellezésében és maguknak a módszereknek a továbbfejlesztése. A pénzügyi matematika ma egyik legnagyobb kihívása jó fedezeti stratégiák kialakítása nem-teljes piacokon. Ez matematikailag egy sajátos sztochasztikus adaptív kontrol problémát jelent, ahol a dinamikát egy sokdimenziós switching diffúziós folyamat írja le. Ehhez az általános problémához számos részprobléma köthető. . Kutatásaink javarészt PhD hallgatók által is megoldható módszertani problémákhoz kötődnek. A fő területek: rejtett Markov-folyamatok, tőzsdemodellek, sztochasztikus volatilitás, valamint a kontroll elmélet és az opcióárazás kapcsolata. Ezen túl munkáinkban a sztochasztikus adaptív kontrol néhány alapvető kérdését is vizsgáltuk. | The objective of this research was to apply and develop advanced methods of stochastic systems for modelling financial markets. A current challenge in financial mathematics is the development of reliable hedging strategies for incomplete markets. Mathematically this is a stochastic asptive control problem in which the dynamics of the system is described by a multivariable switching diffusion process. This major problem could be related to a number of simpler problems. Most of our research topics were releated to technical problems that could be handled within the framework of a PhD program. The main areas were: hidden Markov models, models for a stock exchange, stochastic volatility and the relationship between stochastic control and option pricing. In addition we have studied a few fundamental problems of stochastic adaptiv control

A PRACTICAL APPROACH TO THE AFFINE TRANSFORMATIONS OF THE EUCLIDEAN PLANE

The aim of this paper is to give an elementary treatment of a classical item which plays central role in the applied geometry. The actual need for a more or less new presentation of such a well-known subject is explained by the fact that the use of computers easily allows us to work with any given affine transformation in a suitably (canonically) chosen coordinate-system. The choice of that new orthonormal coordinate-system is based on the diagonalization process of the Gram matrix belonging to the linear part of the transformation, and on the change of the origin for an eventual fixpoint of the given affine transformation. Nevertheless, the entire classification of the considered transformations could be given here by elementary algebraic tools. To simplify our discussion we omit too technical denotations and details, and restrict ourselves to plane geometry. For other aspects we refer e.g. to [4] in this volume

Identification of hidden Markov models - uniform LLN-s

We consider hidden Markov processes in discrete time with a finite state space X and a general observation or read-out space Y. The identification of the unknown dynamics is carried out by the conditional maximum-likelihood method. The normalized log-likelihood function is shown to satisfy a uniform law of large numbers over certain compact subsets of the parameter space. Two cases are covered: first, when the running value of the transition probability matrix, denoted by Q is positive, second, when Q is primitive, but the read-out densities are strictly positive