The main advantage of these methods is that we do not need to find the inverse images of subsets under the transformations of the random map $T$.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:We consider position random maps $T=\{\tau_1(x),\tau_2(x),\ldots, \tau_K(x); p_1(x),p_2(x),\ldots,p_K(x)\}$ on $I=[0, 1],$ where $\tau_k, k=1, 2, \dots, K$ is non-singular map on $[0,1]$ into $[0, 1]$ and $\{p_1(x),p_2(x),\ldots,p_K(x)\}$ is a set of position dependent probabilities on $[0, 1]$.
📊
Temel Bulgular & Veri:We assume that the random map $T$ posses a density function $f^*$ of the unique absolutely continuous invariant measure (acim) $\mu^*$.
💡
Sonuç & Katkı:The main advantage of these methods is that we do not need to find the inverse images of subsets under the transformations of the random map $T$.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Dynamicalsystems · Invariant measures · Invariant density
Abstract
We consider position random maps $T=\{\tau_1(x),\tau_2(x),\ldots, \tau_K(x); p_1(x),p_2(x),\ldots,p_K(x)\}$ on $I=[0, 1],$ where $\tau_k, k=1, 2, \dots, K$ is non-singular map on $[0,1]$ into $[0, 1]$ and $\{p_1(x),p_2(x),\ldots,p_K(x)\}$ is a set of position dependent probabilities on $[0, 1]$. We assume that the random map $T$ posses a density function $f^*$ of the unique absolutely continuous invariant measure (acim) $\mu^*$. In this paper, first, we present a general numerical algorithm for the approximation of the density function $f^*.$ Moreover, we show that Ulam's method is a special case of the general method. Finally, we describe a Monte-Carlo and a Quasi Monte Carlo implementations of Ulam's method for the approximation of $f^*$. The main advantage of these methods is that we do not need to find the inverse images of subsets under the transformations of the random map $T$.
ISLAM, M. S. (2020). Monte Carlo and Quasi Monte Carlo Approach to Ulam's Method for Position Dependent Random Maps. Communications in Advanced Mathematical Sciences, Cilt: 3 - Sayı: 4, 173–185. https://ajindex.com/articles/748248
P. Gora, A. Boyarsky, Absolutely continuous invariant measures for random maps with position dependent probabilities, Math. Anal. and Appl., 278 (2003), 225 – 242
M. Barnsley, Fractals Everywhere, Academic Press, London. 1998
A. Boyarsky, P. Gora, A dynamical model for interference effects and two slit experiment of quantum physics, Phys, Lett. A., 168 (1992), 103 – 112
W. Slomczynski, J. Kwapien, K. Zyczkowski, Entropy computing via integration over fractal measures, Chaos, 10 (2000), 180-188
W. Bahsoun, P. Gora, S. Mayoral, M. Morales, Random dynamics and finance: constructing implied binomial trees from a predetermined stationary density, Appl. Stochastic Models Bus. Ind., 23 (2007), 181– 212
K.R. Schenk-Hoppe, Random Dynamical Systems in Economics, working paper series, Institute of empirical research in economics, University of Zurich, ISSN 1424-0459 (2000)
M.S. Islam, Existence, approximation and properties of absolutely continuous invariant measures for random maps, PhD thesis, Concordia University, 2004
S. Pelikan, Invariant densities for random maps of the interval, Proc. Amer. Math. Soc., 281 (1984), 813 – 825
A. Lasota, J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481– 488
S.M. Ulam, A collection of Mathematical problems Interscience Tracts in pure and Applied Math., 8, Interscience, New York, 1960
T-Y. Li, Finite approximation for the Frobenius-Perron operator: A solution to Ulam’s conjecture, J. Approx. Theory. 17 (1976), 177 – 186
J. Ding, Z. Wang, Parallel Computation of Invariant Measures,Annals of Operations Research, 103 (2001), 283 – 290
W. Bahsoun, P. Gora, Position Dependent Random Maps in One and Higher Dimensions, Studia Math., 166 (2005), 271 – 286
A. Boyarsky, P. Gora, Laws of Chaos: Invariant Measures And Dynamical Systems In One Dimension, Birkhauser, 1997
F.Y. Hunt, A Monte Carlo approach to the approximation of invariant measure, Random Comput. Dynam., 2(1) (1994), 111 – 133.