Application of the Multifractional Brownian Motion Process in Spatial Analyses

Adrianna Mastalerz-Kodzis

Abstract


The article combines methodology applied for time series with elements of spatial econometrics. Its aim is to present a modified method of spatial modelling using selected stochastic processes and the application of that method in economics and other fields of science. The research hypothesis verified in this work can be described as follows: generalised to a multivariate case, Brownian motion processes are a useful tool in econometrics modelling as well as in the analysis of variability and correlation in space. The multifractional Brownian motion process is applied to conduct an analysis of the degree and variability of environmental pollution. The article comprises an introduction, a theoretical part in which concepts connected with the class of stochastic processes in question are clarified, and an empirical part, where selected applications of the aforementioned method are discussed.



Keywords


stochastic process, Hölder function, spatial modelling, variability analysis

Full Text:

PDF

References


Ayache, A. and Lévy Véhel, J. (1999) “Generalized Multifractional Brownian Motion: Definition and Preliminary Results” in M. Dekking, J. Lévy Véhel, E. Lutton and C. Tricot (eds) Fractals: Theory and Applications in Engineering. New York: Springer-Verlag.

Ayache, A. and Taqqu, M. S. (2004) “Multifractional Processes with Random Exponent”. Stochastic Processes and their Applications 111 (1): 119–56.

Barrière, O. (2007) Synthèse et estimation de mouvements browniens multifractionnaires et autres processus à régularité prescrite. Définition du processus autorégulé multifractionnaire et applications. PhD thesis, IRCCyN.

Daoudi, K., Lévy Véhel, J. and Meyer, Y. (1998) “Construction of Continuous Functions with Prescribed Local Regularity”. Constructive Approximation 14 (03): 349–85, https://doi.org/10.1007/s003659900078.

Echelard, A., Lévy Véhel, J. and Barrière, O. (2010) “Terrain Modelling with Multifractional Brownian Motion and Self-regulating Processes”. Lecture Notes in Computer Science, p. 342–51, https://doi.org/10.1007/978-3-642-15910-7_39.

Falconer, K. J. and Lévy-Véhel, J. (2008) “Multifractional Multistable and Other Processes with Prescribed Local Form”. Journal of Theoretical Probability 22 (2): 375–401, https://doi.org/10.1007/s10959-008-0147-9.

Hagerstrand, T. (1952) “The Propagation and Innovation Waves”. Lund Studies in Geography 4. Lund: Gleerup.

Krugman, P. R. (1991) Geography and Trade. Cambridge, Massachusetts: The MIT Press.

Lévy Véhel, J. and Mendivil, F. (2011) “Multifractal and Higher-dimensional Zeta Functions”. Nonlinearity 24 (1): 259–76, https://doi.org/10.1088/0951-7715/24/1/013.

Mastalerz-Kodzis, A. (2003) Modelowanie procesów na rynku kapitałowym za pomocą multifraktali [Modelling processes on the capital market with multifractals]. Prace Naukowe. Katowice: Akademia Ekonomiczna im. Karola Adamieckiego w Katowicach.

Mastalerz-Kodzis, A. (2016a) “Risk Analysis of Foreign Currency Bank Loans Offered on the Polish Capital Market” in Miroslav Čulík (ed.) Managing and Modelling of Financial Risks. VŠB-Technická Univerzita Ostrava.

Mastalerz-Kodzis, A. (2016b) “Algorytm modelowania danych przestrzennych o zadanej lokalnej regularności” [Spatial data modelling algorithm with a given local regularity]. Metody i modele analiz ilościowych w ekonomii i zarządzaniu. Katowice: Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach.

Mastalerz-Kodzis, A., Pośpiech, E. (2017) Application of Hölder Function to Expansion Intensity of Spatial Phenomena Analysis. Folia Oeconomica Lodziensis (in print).

Paelinck, J. H. P. and Klaassen, L. H. (1983) Ekonometria przestrzenna [Spatial econometrics]. Warsaw: PWN.

Peltier, R. F. and Lévy Véhel, J. (1995) “Multifractional Brownian Motion: Definition and Preliminary Results”. Rapport de Recherche No. 2645. INRIA Recquencourt.

Suchecki, B. (2010) Ekonometria przestrzenna [Spatial econometrics]. Warsaw: C.H. Beck.




DOI: https://doi.org/10.15678/AOC.2018.1805