Position Dependent Random Map Research Articles

Stochastic Perturbations and Invariant Measures of Position Dependent Random Maps via Fourier Approximations

Let T = {τ1(x), τ2(x),…, τK(x); p1(x), p2(x),…, pK(x)} be a position dependent random map which possesses a unique absolutely continuous invariant measure [Formula: see text] with probability density function [Formula: see text]. We consider a family {TN}N≥1 of stochastic perturbations TN of the random map T. Each TN is a Markov process with the transition density [Formula: see text], where qN(x, ⋅) is a doubly stochastic periodic and separable kernel. Using Fourier approximation, we construct a finite dimensional approximation PN to a perturbed Perron–Frobenius operator. Let [Formula: see text] be a fixed point of PN. We show that [Formula: see text] converges in L1 to [Formula: see text].

Toward a Mathematical Holographic Principle

In work started in [17] and continued in this paper our objective is to study selectors of multivalued functions which have interesting dynamical properties, such as possessing absolutely continuous invariant measures. We specify the graph of a multivalued function by means of lower and upper boundary maps $$\tau _{1}$$ and $$\tau _{2}.$$ On these boundary maps we define a position dependent random map $$R_{p}=\{\tau _{1},\tau _{2};p,1-p\},$$ which, at each time step, moves the point $$x$$ to $$\tau _{1}(x)$$ with probability $$p(x)$$ and to $$\tau _{2}(x)$$ with probability $$1-p(x)$$ . Under general conditions, for each choice of $$p$$ , $$R_{p}$$ possesses an absolutely continuous invariant measure with invariant density $$f_{p}.$$ Let $$\varvec{\tau }$$ be a selector which has invariant density function $$f.$$ One of our objectives is to study conditions under which $$p(x)$$ exists such that $$R_{p}$$ has $$f$$ as its invariant density function. When this is the case, the long term statistical dynamical behavior of a selector can be represented by the long term statistical behavior of a random map on the boundaries of $$G.$$ We refer to such a result as a mathematical holographic principle. We present examples and study the relationship between the invariant densities attainable by classes of selectors and the random maps based on the boundaries and show that, under certain conditions, the extreme points of the invariant densities for selectors are achieved by bang-bang random maps, that is, random maps for which $$p(x)\in \{0,1\}.$$

Lyapunov Exponents for Position Dependent Random Maps: Formulae and Applications

In this article, we provide formulae for Lyapunov exponents of position dependent random maps of the interval. We then apply our results to a financial market model with short-lived assets.

INVARIANT MEASURES FOR RANDOM MAPS VIA INTERPOLATION

Let T = {τ1, τ2, …, τK; p1, p2, …, pK} be a position dependent random map on [0, 1], where {τ1, τ2, …, τK} is a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2, …, pK} is a collection of position dependent probabilities on [0, 1]. We assume that the random map T has a unique absolutely continuous invariant measure μ with density f*. Based on interpolation, a piecewise linear approximation method for f* is developed and a proof of convergence of the piecewise linear method is presented. A numerical example for a position dependent random map is presented.

METASTABLE SYSTEMS AS RANDOM MAPS

Metastable dynamical systems were recently studied [González-Tokman et al., 2011] in the framework of one-dimensional piecewise expanding maps on two disjoint invariant sets, each possessing its own ergodic absolutely continuous invariant measure (acim). Under small deterministic perturbations, holes between the two disjoint systems are created, and the two ergodic systems merge into one. The long term dynamics of the newly formed metastable system is defined by the unique acim on the combined ergodic sets. The main result of [González-Tokman et al., 2011] proves that this combined acim can be approximated by a convex combination of the disjoint acims with weights depending on the ratio of the respective measures of the holes. In this note we present an entirely different approach to metastable systems. We consider two piecewise expanding maps: one is the original map, τ1, defined on two disjoint invariant sets of ℝN and the other is a deterministically perturbed version of τ1, τ2, which allows passage between the two disjoint invariant sets of τ1. We model this system by a position dependent random map based on τ1 and τ2, to which we associate position dependent probabilities that reflect the switching between the maps. A typical orbit spends a long time in one of the ergodic sets but eventually switches to the other. Such behavior can be attributed to physical holes as between adjoining billiard tables or more abstract situations where balls can "leap" from one table to the other. Using results for random maps, a result similar to the one-dimensional main result of [González-Tokman et al., 2011] is proved in N dimensions. We also consider holes in more than two invariant sets. A number of examples are presented.

The Geometric Markov Renewal Processes with Application to Finance

We introduce the geometric Markov renewal processes as a model for a security market and study this processes in a series scheme. We consider its approximations in the form of averaged, merged and double averaged geometric Markov renewal processes. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes are presented. Martingale properties, infinitesimal operators of geometric Markov renewal processes are presented and a Markov renewal equation for expectation is derived. As an application, we consider the case of two ergodic classes. Moreover, we consider a generalized binomial model for a security market induced by a position dependent random map as a special case of a geometric Markov renewal process.

MAXIMUM ENTROPY METHOD FOR POSITION DEPENDENT RANDOM MAPS

Let {τ1, τ2,…,τK} be a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2,…,pK} be a collection of position dependent probabilities on [0, 1]. We consider position dependent random maps T = {τ1,τ2,…,τK;p1,p2,…,pK} such that T preserves an absolutely continuous invariant measure with density f*. A maximum entropy method for approximating f* is developed. We present a proof of convergence of the maximum entropy method for random maps.

INVARIANT MEASURES FOR HIGHER DIMENSIONAL MARKOV SWITCHING POSITION DEPENDENT RANDOM MAPS

A higher dimensional Markov switching position dependent random map is a random map where the probabilities of switching from one higher dimension transformation to another are the entries of a stochastic matrix and the entries of stochastic matrix are functions of positions. In this note, we prove sufficient conditions for the existence of absolutely continuous measures for a class of higher dimensional Markov switching position dependent random maps. Our result is a generalization of the result in [Bahsoun & Góra, 2005; Bahsoun et al., 2005].

Deterministic representation for position dependent random maps

We give a deterministic representation for position dependent random maps and describe the structure of its set of invariant measures. Our construction generalizes the skew product representation of random maps with constant probabilities. In particular, we establish one-to-one correspondence between eigenfunctions corresponding to eigenvalues of unit modulus for the Frobenius-Perron (transfer) operator of the random map and for those of the skew. An immediate consequence is one-to-one correspondence between absolutely continuous invariant measures (acims) for the position dependent random map and acims for its deterministic representation.

CONDITIONALLY INVARIANT MEASURES FOR POSITION DEPENDENT RANDOM MAPS OF AN INTERVAL WITH HOLES

We study position dependent random maps on the unit interval with holes where the possible laws of motion are piecewise monotonic transformations. The main result of this note is proving the existence of absolutely continuous conditionally invariant measures.

Position dependent random maps in one and higher dimensions

A random map is a discrete-time dynamical system in which one of a number of transformations is randomly selected and applied on each iteration of the process. We study random maps with position dependent probabilities on the interval and on a bounded dom

Markov Switching for Position Dependent Random Maps with Application to Forecasting

A Markov switching random map consists of a collection of transformations and a controlling stochastic matrix. In this process, at each time step, one transformation is selected randomly and applied. The selection of the transformations is controlled by the stochastic matrix of the process. In this note, we first prove the existence of absolutely continuous invariant measures (acims) for random maps, whose underlying transformations are piecewise monotonic, controlled by a position dependent stochastic matrix and study the ergodic properties of the acim. In particular, we prove a Birkhoff type ergodic theorem. Then we prove the existence of an acim for another class of Markov switching random maps based on geometric properties of the underlying transformations. We apply these results to forecasting in financial markets.

Small Stochastic Perturbations of Random Maps with Position Dependent Probabilities

A position dependent random map is a dynamical system consisting of a collection of maps such that, at each iteration, a selection of a map is made randomly by means of probabilities which are functions of position. Let f* be an invariant density of the position dependent random map T. We consider a model of small random perturbations 𝔗ϵ of the random map T. For each ϵ > 0, 𝔗ϵ has an invariant density function f ϵ. We prove that f ϵ → f* as ϵ → 0.

Absolutely continuous invariant measures for random maps with position dependent probabilities

A random map is discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. Usually the map τ k is chosen from a finite collection of maps with constant probability p k . In this note we allow the p k 's to be functions of position. In this case, the random map cannot be considered to be a skew product. The main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on [0,1]. Geometrical and topological properties of sets of absolutely continuous invariant measures, attainable by means of position dependent random maps, are studied theoretically and numerically.