Monte Carlo and Quasi Monte Carlo Approach to Ulam's Method for Position Dependent Random Maps

Year 2020, Volume: 3 Issue: 4, 173 - 185, 22.12.2020

Md Shafiqul Islam

https://doi.org/10.33434/cams.725619

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 $T$.

Keywords

Dynamicalsystems, Invariant measures, Invariant density, Position dependent random maps, Monte Carlo approach, Quasi Monte Carlo approach, Ulam's method

References

• [1] P. Gora, A. Boyarsky, Absolutely continuous invariant measures for random maps with position dependent probabilities, Math. Anal. and Appl., 278 (2003), 225 – 242.
• [2] M. Barnsley, Fractals Everywhere, Academic Press, London. 1998.
• [3] A. Boyarsky, P. Gora, A dynamical model for interference effects and two slit experiment of quantum physics, Phys, Lett. A., 168 (1992), 103 – 112.
• [4] W. Slomczynski, J. Kwapien, K. Zyczkowski, Entropy computing via integration over fractal measures, Chaos, 10 (2000), 180-188.
• [5] 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.
• [6] 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).
• [7] M.S. Islam, Existence, approximation and properties of absolutely continuous invariant measures for random maps, PhD thesis, Concordia University, 2004.
• [8] S. Pelikan, Invariant densities for random maps of the interval, Proc. Amer. Math. Soc., 281 (1984), 813 – 825.
• [9] A. Lasota, J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973) , 481– 488.
• [10] S.M. Ulam, A collection of Mathematical problems Interscience Tracts in pure and Applied Math., 8, Interscience, New York, 1960.
• [11] T-Y. Li, Finite approximation for the Frobenius-Perron operator: A solution to Ulam’s conjecture, J. Approx. Theory. 17 (1976), 177 – 186.
• [12] J. Ding, Z. Wang, Parallel Computation of Invariant Measures,Annals of Operations Research, 103 (2001), 283 – 290.
• [13] W. Bahsoun, P. Gora, Position Dependent Random Maps in One and Higher Dimensions, Studia Math., 166 (2005), 271 – 286.
• [14] A. Boyarsky, P. Gora, Laws of Chaos: Invariant Measures And Dynamical Systems In One Dimension, Birkhauser, 1997.
• [15] F.Y. Hunt, A Monte Carlo approach to the approximation of invariant measure, Random Comput. Dynam., 2(1) (1994), 111 – 133.