markov switching for position dependent random maps with application to forecasting

SIAM J. APPLIED DYNAMICAL SYSTEMS c© 2005 Society for Industrial and Applied MathematicsVol. 4, No. 2, pp. 391–406

Markov Switching for Position Dependent Random Maps with Application toForecasting∗

Wael Bahsoun†, Pawe�l Gora‡, and Abraham Boyarsky‡

Abstract. A Markov switching random map consists of a collection of transformations and a controlling stochas-tic 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 thisnote, we first prove the existence of absolutely continuous invariant measures (acims) for randommaps, whose underlying transformations are piecewise monotonic, controlled by a position depen-dent stochastic matrix and study the ergodic properties of the acim. In particular, we prove aBirkhoff type ergodic theorem. Then we prove the existence of an acim for another class of Markovswitching random maps based on geometric properties of the underlying transformations. We applythese results to forecasting in financial markets.

Key words. random map, Markov switching, absolutely continuous invariant measure, Frobenius–Perron op-erator

AMS subject classifications. 37A05, 37E05

DOI. 10.1137/040604042

1. Introduction. Random dynamical systems provide a useful framework for modelingand analyzing various phenomena [14]. A random dynamical system of special interest is arandom map where the process switches from one map to another according to fixed proba-bilities or, more generally, position dependent probabilities. Hamilton [10] observed that aneconomy may be in either a fast or a slow growth phase, with the switch between the twogoverned by the outcome of a Markov process.

In this paper, we build on this motivation to study a very general random map controlledby a position dependent stochastic matrix, i.e., a discrete-time dynamical system in which ateach step one of K transformations is randomly selected and applied to the previous iterationof the process. The selection is controlled by a matrix W which is a K×K position dependentstochastic irreducible matrix such that Wkl, the klth entry of W , is the probability of the switchfrom transformation k to transformation l. After the first step, choosing a transformationdepends only on the position and the transformation applied at the previous time step. Sucha random map model is more general than the models considered in [11, 8, 9, 1]. One canthink of the transformations as corresponding to and representing the dynamics of differentregimes, and the matrix W controls the probabilistic switching between the regimes.

∗Received by the editors February 10, 2004; accepted for publication (in revised form) by L.-S. Young October 29,2004; published electronically April 22, 2005. The research of the authors was supported by NSERC grant.

http://www.siam.org/journals/siads/4-2/60404.html†Department of Mathematics and Statistics, University of Victoria, PO BOX 3045 STN CSC, Victoria, B.C.,

V8W 3P4, Canada ([email protected]). This author is a recipient of a PIMS postdoctoral fellowship.‡Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal,

Quebec H4B 1R6, Canada ([email protected], [email protected]).

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http://www.siam.org/journals/siads/4-2/60404.html

mailto:[email protected]



392 WAEL BAHSOUN, PAWE�L GORA, AND ABRAHAM BOYARSKY

In this note, we prove the existence of absolutely continuous invariant measures (acims)for random maps controlled by a position dependent stochastic matrix and study the ergodicproperties of the acims. In particular, we prove a Birkhoff-type ergodic theorem. We alsoapply these results to forecasting in financial markets by introducing a generalization of thebinomial model where we allow regime switching between trends in the market.

In section 2 we present the notation and the Frobenius–Perron operator framework inwhich we study the existence of acims for random maps controlled by a position dependentstochastic irreducible matrix. In section 3 sufficient conditions are presented for the existenceof an acim using bounded variation techniques. In section 4 we prove an ergodic theorem. Insection 5, under certain geometric conditions on the transformations, we prove the existenceof acims. The class of transformations that we consider and the techniques that we use insection 5 are different from those of section 3. In section 6 we introduce a generalization ofthe binomial model where we allow regime switching between the trends in the market.

2. Preliminaries. Let (X,B, λ) be a measure space, and let τ1, τ2, . . . , τK , τk : X → X bepiecewise continuous nonsingular transformations. We define a Markov process T as follows:at time n = 1 we select a transformation τk randomly according to initial probabilities wk,k = 1, 2, . . . ,K. Then we define a K ×K position dependent stochastic irreducible matrix Wsuch that Wk,l, the (k, l)th entry of W , is the probability of switching from τk to τl. Thatis, after choosing τkN−1

, choosing the transformation τkN at time N , N = 2, 3, . . . , dependsonly on the transformation applied at the previous time step and the position at the previoustime step. Therefore, if we choose τk1 at time n = 1, where we are at position x, the Markovprocess at time N is given by the composition

τkN ◦ τkN−1◦ · · · ◦ τk2 ◦ τk1(x)

with probability

WkN−1kN (τkN−2◦ · · · ◦ τk1(x)) ·WkN−2kN−1

(τkN−3◦ · · · ◦ τk1(x)) · · ·Wk1k2(x).

We define the τk’s on a common partition P of X: P = {I1, . . . , Iq} and τk,i = τk |Ii ,i = 1, . . . , q, k = 1, . . . ,K (if the τk’s are not piecewise monotonic on the same P, we can finda finer partition so that all the τk’s are piecewise monotonic on it). Let Ω = {1, 2, . . . ,K}.We define the transition function of the Markov process on Ω ×X as follows:

P((k, x), {l} ×A) = Wk,l(x)χA(τk(x)),(2.1)

where A is any measurable set and χA denotes the characteristic function of the set A. Therandom map T is the projection the process we defined on the space X. The transitionfunction P induces an operator P∗ on measures μ on Ω ×X as follows:

P∗μ({l} ×A) =

∫Ω×X

P((k, x), {l} ×A)dμ(k, x)

=

∫Ω×X

Wk,l(x)χA(τk(x))dμ(k, x).

(2.2)

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MARKOV SWITCHING FOR RANDOM MAPS 393

Let ν be the measure on Ω×X such that ν({s}×A) = λ(A). If μ has density f with respectto ν, f(s, x) =

∑Kk=1 fk(x)χ{k}×X(s, x), where

∑Kk=1

∫X fk(x)dλ(x) = 1, then P∗μ also has a

density which we denote by PT f . By a change of variables, we obtain∫{l}×A

PT f(s, x)dν(s, x) =

K∑k=1

∫XWk,l(x)χA(τk(x))fk(x)dλ(x)

=

K∑k=1

∫τ−1k (A)

Wk,l(x)fk(x)dλ(x).

(2.3)

Using the definition of Pτk , the Frobenius–Perron operator associated with the transforma-tion τk [5], and (2.3), we obtain∫

Afl(x)dλ(x) =

K∑k=1

∫APτk(Wk,lfk)(x)dλ(x),(2.4)

where PT f(s, x) =∑K

l=1 flχ{l}×X(s, x). Since (2.4) is true for any A ∈ B, we obtain ana.e. equality

fl(x) =

K∑k=1

Pτk(Wk,lfk)(x).(2.5)

Thus, the density f∗(s, x) =∑K

l=1 f∗l (x)χ{l}×X(s, x) is T -invariant if

f∗l (x) =

K∑k=1

Pτk(Wk,lf∗k )(x)(2.6)

for l = 1, . . . ,K. If we denote

wl =

∫Xf∗l (x)dλ(x), l = 1, . . . ,K,

then, integrating (2.6) with respect to λ, we obtain

wl =

K∑k=1

wk

∫XWk,l(x)

f∗k (x)∫

X f∗k (x)dλ(x)

dλ(x).(2.7)

Note that, in the special case when the Wk,l’s are constants, (2.7) reduces to wl =∑

k=1 wkWk,l,i.e., to the case when (w1 . . . wK) is a left invariant eigenvector of the matrix W .

If the density f∗(s, x) =∑K

k=1 f∗k (x)χ{k}×X(s, x) is a fixed point of PT , we call

f∗ =

K∑k=1

f∗k =

K∑k=1

wkf∗k∫

I f∗kdλ

an invariant density of the Markov random map. This is motivated by Theorem 4.1.

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If the controlling Markov chain is actually a sequence of independent identically distributedrandom variables independent of position x, then Wk,l = wl, and the system (2.6) reduces to

the equations f∗l = wl

∑Kk=1 Pτkf

∗k , l = 1, 2, . . . ,K, which shows that all f∗

l ’s are proportional.Then, we have

f∗ =

K∑l=1

f∗l =

K∑l=1

wl

K∑k=1

Pτkf∗k

=

K∑k=1

wkPτk

(K∑l=1

wl

wkf∗k

)=

K∑k=1

wkPτk

(K∑l=1

f∗l

)=

K∑k=1

wkPτkf∗,

(2.8)

the known equation for an invariant density of a random map.

2.1. The natural model. In our representation, we have used a very general Markovswitching model. In this model, choosing a transformation depends only on the transformationapplied at the previous time step and the position at the previous time step (see the transitionfunction (2.1)). We call this model general because all previous models of random maps arespecial cases of this model (see (2.8)).

However, it is more natural to consider a model where choosing a transformation dependsonly on the transformation applied at the previous time step and the current position. Thenwe modify the transition function in (2.1) slightly, and it becomes

P((k, x), {l} ×A) = Wk,l(τk(x))χA(τk(x)).(2.9)

Consequently, we modify (2.6) slightly, and it becomes

f∗l (x) =

K∑k=1

Pτk((Wk,l ◦ τk)f∗k )(x)(2.10)

for l = 1, . . . ,K.The existence results of sections 3 and 5 hold verbatim, with the proper slight modification,

using the natural model whose transition function and invariant density equation are givenby (2.9) and (2.10), respectively.

We call this model natural because it is more intuitive and maybe more realistic forapplications.

3. The existence of acims. In this section we will prove that the operator PT has aninvariant density, i.e., that the random map T has an acim. We will use the result of Ruelle [13];see also [4] for more detailed treatment of the method. This method of proof is much shorterand the result obtained is more general than the traditional bounded variation technique.

Let S = {1, 2, . . . ,K} and X = I = [a, b] be a bounded interval of real line. Let ([a, b],B, λ)be a measure space, where λ is Lebesgue measure on [a, b]. We represent the space S × I as aunion of disjoint intervals S× I =

⋃Kk=1{k}× I. τk : [a, b] → [a, b], k = 1, . . . ,K, are piecewise

one-to-one continuous nonsingular transformations on a partition P of [a, b]: P = {I1, . . . , Iq}and τk,i = τk |Ii , i = 1, . . . , q, k = 1, . . . ,K. We will also assume that the probabilities Wk,l(x)are defined on the same partition P. Denote by V (·) the standard one dimensional variation of

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a function (modulo functions equal almost everywhere) and by BV (I) the space of functionsof bounded variations on I = [a, b] equipped with the norm || · ||BV = V (·) + || · ||1, where|| · ||1 denotes the norm on L1(I,B, λ) [5].

We consider a space BV of functions of bounded variation on S×X, where f =∑K

k=1 fkχ{k}×I

is of bounded variation if fk ∈ BV (I), k = 1, . . . ,K, and V (f) =∑K

k=1 V (fk). Let ‖f‖1 =∑Kk=1 ‖fk‖1 and ‖f‖

BV= ‖f‖1 + V (f).

For k, l ∈ {1, . . . ,K} and i ∈ {1, . . . , q} we define

V(k,l,i) = {l} × τk(Ii);

ψ(k,l,i)(l, x) = (k, τ−1k,i (x));

φ(k,l,i)(l, x) = |(τ−1k,i )′(x)| ·Wk,l(τ

−1k,i (x)) · χV(k,l,i)

(l, x).

Now we will check that the operator PT coincides with Ruelle’s transport operator M definedusing the foregoing ψ’s and φ’s:

(Mf)(l, x) =∑

(k,l,i)

f(ψ(k,l,i)(l, x)) · φ(k,l,i)(l, x)

=∑k,i

f(k, τ−1k,i (x)) · |(τ−1

k,i )′(x)| ·Wk,l(τ−1k,i (x))

=

K∑k=1

Pτk(fkWk,l)(l, x) = PT (f)(l, x).

Following Ruelle, we define a “formal dual” operator

(M′f)(k, x) =∑

(k,l,i)

f(ψ−1(k,l,i)(k, x)) · φ(k,l,i)(ψ

−1(k,l,i)(k, x))

=∑

(k,l,i)

f(l, τk,i(x)) · |(τ−1k,i )′(τk,i(x))| ·Wk,l(x) · χV(k,l,i)

(l, τk,i(x))

=

K∑l=1

fl(τk(x))Wk,l(x)

|τ ′k(x)| .

The main result of [13] states that if V (∑

(k,l,i) φ(k,l,i)) < ∞, then the essential spectral radiusof M = PT is equal to

R = limn→∞

(sup(k,x)

(M′)n(1))1/n.

We have

(M′(1))(k, x) =1

|τ ′k(x)| ;

((M′)2(1))(k, x) =

K∑l=1

1

|τ ′l (τk(x))|Wk,l(x)

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and in general

((M′)n(1))(k0, x) =

K∑kn−1=1

· · ·K∑

k1=1

1

|τ ′kn−1(τkn−2 ◦ · · · ◦ τk0(x))|

Wkn−2,kn−1(τkn−3 ◦ · · · ◦ τk0(x))

|τ ′kn−2(τkn−3 ◦ · · · ◦ τk0(x))| · · · Wk1,k2(τk0(x))

|τ ′k1(τk0(x))|

Wk0,k1(x)

|τ ′k0(x)| .

(3.1)

Since the operator PT has spectral radius equal to 1, PT is quasi-compact whenever R < 1.In particular, if R < 1, then T has an acim. It is easy to see that if there exists an n suchthat the supremum of the expression (3.1) is less than 1, then R < 1. In particular, if all τk’sare piecewise expanding, then R < 1.

We have proved the following theorem.

Theorem 3.1. Let τk : [a, b] → [a, b], k = 1, . . . ,K, be piecewise one-to-one, continuous,nonsingular transformations and Wk,l, 1 ≤ k, l ≤ K, be a position dependent stochastic matrix.If ∑

(k,l,i)

V[|(τ−1

k,i )′(x)| ·Wk,l(τ−1k,i (x)) · χV(k,l,i)

(l, x)]< +∞,

and for some n ≥ 1 the quantity in (3.1) is smaller than some α < 1, then the operator PT isquasi-compact. In particular, the Markov process T and the Markov random map induced byit have acims.

The lemma below gives alternative conditions for Theorem 3.1 to hold. It can be gener-alized to more complicated conditions involving more iterations of the Markov process.

Lemma 3.2. If∑K

l=1 supxWk,l(x)

|τ ′l (τk(x))| ≤ α and supx1

|τ ′k(x)| < β, β < ∞, for all k = 1, . . . ,K,

then ((M′)n(1))(k0, x) ≤ βαn−1. Thus,

limn→∞

(supk,x

((M′)n(1))(k, x)

)1/n

≤ α.

In particular, for α < 1 the Markov random map admits an acim.

Proof. We prove the lemma by induction on n. For n = 2, we have

K∑k1=1

Wk0,k1(x)

|τ ′k1(τk0(x))|

1

|τ ′k0(x)| < β · α

by assumption. Now, assume that

K∑kn−2=1

· · ·K∑

k1=1

supx

Wkn−3,kn−2(τkn−4 ◦ · · · ◦ τk0(x))

|τ ′kn−2(τkn−3 ◦ · · · ◦ τk0(x))| · · · 1

|τ ′k0(x)| < β · αn−2.(3.2)D

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Then,

K∑kn−1=1

· · ·K∑

k1=1

supx

1

|τ ′kn−1(τkn−2 ◦ · · · ◦ τk0(x))|

Wkn−2,kn−1(τkn−3 ◦ · · · ◦ τk0(x))

|τ ′kn−2(τkn−3 ◦ · · · ◦ τk0(x))| · · · Wk1,k2(τk0(x))

|τ ′k1(τk0(x))|

Wk0,k1(x)

|τ ′k0(x)|

≤K∑

kn−1=1

· · ·K∑

k1=1

supx

Wkn−2,kn−1(τkn−3 ◦ · · · ◦ τk0(x))

|τ ′kn−1(τkn−2 ◦ · · · ◦ τk0(x))| sup

x

Wkn−3,kn−2(τkn−4 ◦ · · · ◦ τk0(x))

|τ ′kn−2(τkn−3 ◦ · · · ◦ τk0(x))| · · · 1

|τ ′k0(x)|

=

K∑kn−1=1

supx

Wkn−2,kn−1(τkn−3 ◦ · · · ◦ τk0(x))

|τ ′kn−1(τkn−2 ◦ · · · ◦ τk0(x))|

K∑kn−2=1

· · ·K∑

k1=1

supx

Wkn−3,kn−2(τkn−4 ◦ · · · ◦ τk0(x))

|τ ′kn−2(τkn−3 ◦ · · · ◦ τk0(x))| · · · 1

|τ ′k0(x)|

≤ β · αn−2K∑

kn−1=1

supx

Wkn−2,kn−1(τkn−3 ◦ · · · ◦ τk0(x))

|τ ′kn−1(τkn−2 ◦ · · · ◦ τk0(x))| ≤ βαn−1.

(3.3)

Using (3.3), we conclude that ((M′)n(1))(k0, x) ≤ βαn−1 for all n.Now, we present an example of a random map which admits an acim even though one of

its transformations is nonexpanding.Example 3.3. Let T = {τ1, τ2} be a random map controlled by a stochastic matrix W ,

where

τ1(x) =

{2x for 0 ≤ x < 1/2,

2x− 2 for 1/2 ≤ x ≤ 1,(3.4)

τ2(x) =

{x + 1/2 for 0 ≤ x < 1/2,

1 − x for 1/2 ≤ x ≤ 1,(3.5)

and

W =

[1/4x + 1/2 1/2 − 1/4x

1/2 1/2

].(3.6)

We have

2∑l=1

supx

W1,l(x)

|τ ′l (τ1(x))| =3/4

2+

1/2

1=

7

8< 1

and

2∑l=1

supx

W2,l(x)

|τ ′l (τ2(x))| =1/2

2+

1/2

1=

3

4< 1,

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and supx1

|τ ′k(x)| < 1 for all k. Thus, T admits an acim.

Now, we construct an example of a random map where a complete switch over is allowed,i.e., at a certain point, a certain map is selected with probability one. An example like thiscannot be constructed for a Markov random map with constant switching probabilities [8].In fact, the following example shows that we can have some control over our process. Thus,contrary to the case discussed in [8], our process may not be completely random.

Example 3.4. Let T = {τ1, τ2} be a random map controlled by a stochastic matrix W ,where

τ1(x) =

{2x for 0 ≤ x < 1/2,

2x− 2 for 1/2 ≤ x ≤ 1,(3.7)

τ2(x) =

{−2x + 1 for 0 ≤ x < 1/2,

2x− 1 for 1/2 ≤ x ≤ 1,(3.8)

and

W =

[1/2x 1 − 1/2x

1/2 1/2

].(3.9)

We have

2∑l=1

supx

W1,l(x)

|τ ′l (τ1(x))| =1/2

2+

1

2=

3

4< 1

and

2∑l=1

supx

W2,l(x)

|τ ′l (τ2(x))| =1/2

2+

1/2

2=

1

2< 1,

and supx1

|τ ′k(x)| < 1/2 for all k. Thus, T admits an acim.

4. Ergodic theorem. We consider the space Z = (Ω×X)N∪{0} with the measure P definedby the initial probability μ on Ω×X and the transition function P defined in (2.1). For eachx ∈ X it induces a measure Px on ΩN∪{0} = {(k0, k1, . . .) : 1 ≤ ki ≤ K, i = 0, 1, . . .} which, onthe cylinder C(k0, k1, . . . , kN ) = {(s0, s1, . . .) : si = ki, i = 0, 1, . . . , N}, has the value

Px(C(k0, k1, . . . , kN )) = WkN−1kN (τkN−1◦ · · · ◦ τk0(x))

·WkN−2kN−1(τkN−2

◦ · · · ◦ τk0(x)) · · ·Wk1k2(τk0(x))Wk0k1(x)wk0 .

Px can also be understood as a measure on Zx = {(Z0, Z1, . . .) ∈ Z : Z0 = (·, x)}. Then, forany measurable subset B of Z, we have

P (B) =

K∑k=1

∫XPx(Bk,x)f

∗k (x)dλ(x),

where Bk,x = {(Z0, Z1, . . .) ∈ B : Z0 = (k, x)}.

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Theorem 4.1. Suppose that f∗(s, x) =∑K

k=1 wkf∗k (x) · χ{k}×X(s, x), where

∑Kk=1 wk = 1

and∫X fkdλ = 1, k = 1, 2, . . . ,K. Assume that the measure μ = f∗ν is T -invariant and

unique. Let g(s, x) = g(x) ∈ L1(X,μ), where μ =∑K

k=1 wkf∗kλ. Then, for λ-a.e. x with

Px probability 1,

1

n

n−1∑m=0

g(Tm(s, x)) →∫

g(x)dμ(x).(4.1)

Proof. Let {Zn}∞n=0 be the Markov process on (Ω × X)N∪{0} corresponding to the mea-sure P . Then, if Z0 = (·, x), then Zm = Tm(x). The process {Zn}∞n=0 is stationary since μ isa T -invariant measure and, in view of Theorem 3.1, we have finite number of ergodic acims,and since μ is unique it is ergodic. Let g ∈ L1(X,μ) and g(s, x) = g(x). By Theorem 6.1of [6],

1

n

n−1∑m=0

g(Zm) →∫

gdμ(4.2)

for {Zn} ∈ Λ, with P (Λ) = 1. Note that

∫gdμ =

K∑k=1

wk

∫Xg(k, x)f∗

k (x)dλ =

∫Xgdμ.

On the other hand,

1 = P (Λ) =

K∑k=1

∫XPx(Λk,x)f

∗k (x)dλ(x).(4.3)

Thus, the convergence in (4.2) takes place for λ-a.e. x ∈ X and for any such x with probabilityPx equal to 1 since, by (4.3), we must have Px(Λk,x) = wk and then Px(Λx) =

∑Kk=1 Px(Λk,x) =

1.

5. The existence of acims using geometric conditions. In this section, following theideas of [2], we prove the existence of acims for a different class of random maps. We donot assume that the transformations satisfy an expanding condition and we do not use thebounded variation technique. On the other hand, we require the transformations to satisfysome geometric conditions.

First, we present the setting for this section. Let ([a, b],B, λ) be a measure space, whereλ is Lebesgue measure on [a, b]; let τk : [a, b] → [a, b], k = 1, . . . ,K, be piecewise one-to-onecontinuous nonsingular transformations on a partition P of [a, b]: P = {I1, . . . , Iq}, and letτk,i = τk |Ii , i = 1, . . . , q, k = 1, . . . ,K. We will also assume that the probabilities Wk,l(x)are defined on the same partition P. We denote the L1(I,B, λ) norm by || · ||1. We sayf = (f1, . . . , fK) is nonincreasing and positive if fk is nonincreasing and positive for all k anddefine ||f1, . . . , fK ||1 =

∑Kk=1 ||fk||1 and ||f1, . . . , fK ||∞ = maxk ||fk||∞.

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We define an operator PT : ||·, . . . , ·||1 → ||·, . . . , ·||1 by

PT (f1, . . . , fK) =

(K∑k=1

Pτk(Wk,1fk), . . . ,

K∑k=1

Pτk(Wk,Kfk)

).(5.1)

If (f∗1 , . . . , f

∗K) is a fixed point of PT , we call

f∗ =

K∑k=1

f∗k =

K∑k=1

wkf∗k∫

I f∗kdλ

an invariant density of the Markov random map.

The existence of acims. Let ([0, 1],B, λ) be a measure space, where λ is Lebesgue measureon [0, 1]. Let τk : [0, 1] → [0, 1], k = 1, . . . ,K, be piecewise continuous increasing transforma-tions on a common partition P of [0, 1]: P = {I1, . . . , Iq}, Ii = [ai−1, ai], and τk,i = τk |Ii ,i = 1, . . . , q, k = 1, . . . ,K. Define Fk,l,i =

Wk,l(τ−1k,i (x))

τ ′k(τ−1k,i (x))

. We assume that

(A) τk,i(ai−1) = 0, 1 ≤ i ≤ q, for all 1 ≤ k ≤ K.(B)

∑ri=1 Fk,l,i, 1 ≤ r ≤ q, is decreasing for all 1 ≤ k ≤ K.

(C) maxl∑K

k=1Wk,l(0)τ ′k(0)

≤ α < 1.

The following lemma is a modification of Lemma 2.2 of [3].Lemma 5.1. Let T be a random map which satisfies (A), (B), and (C). Let f = (f1, . . . , fK)

be nonincreasing and positive. Then (PT f)l is nonincreasing and so is PT f .Proof. Let f = (f1, . . . , fK) be a nonincreasing positive function. We prove the lemma for

(PT f)l for any l. Define τ−1k,i (x) = xk,i. Let x < y. Since τk,i is increasing and τk(ai−1) = 0,

if χτk(Ii)(x) = 0, then χτk(Ii)(y) = 0. Thus, we consider the case when they are both nonzero,and we have

(PT f)l(x) − (PT f)l(y) =

K∑k=1

(PτkWk,lfk)(x) − (PτkWk,lfk)(y)

=

K∑k=1

q∑i=1

(Fk,l,i(x)fk(xk,i) − Fk,l,i(y)fk(yk,i))

=K∑k=1

q∑i=1

(Fk,l,i(x)fk(xk,i) − Fk,l,i(y)fk(xk,i)

+ Fk,l,i(y)fk(xk,i) − Fk,l,i(y)fk(yk,i))

=K∑k=1

q∑i=1

(Fk,l,i(x) − Fk,l,i(y))fk(xk,i)

+

K∑k=1

q∑i=1

(fk(xk,i) − fk(yk,i))Fk,l,i(y)

≥K∑k=1

q∑i=1

(Fk,l,i(x) − Fk,l,i(y))fk(xk,i)

(5.2)

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since fk is nonincreasing and Fk,l,i > 0. Equation (5.2) implies that (PT f)l is nonincreasing,if∑q

i=1(Fk,l,i(x) − Fk,l,i(y))fk(xk,i) ≥ 0 for all k. Define the q-dimensional vectors

→Fk = 〈Fk,l,i(x) − Fk,l,i(y)〉,→fk = 〈fk(xk,i)〉,

and

→bj = 〈1, . . . , 1︸︷︷︸

j times

, 0, . . . , 0〉; j = 1, 2, . . . , q.

Using this notation, we can rewrite∑q

i=1(Fk,l,i(x) − Fk,l,i(y))fk(xk,i) as

→Fk ·

→fk.(5.3)

Condition (B) implies

→Fk ·

→bj ≥ 0; j = 1, 2, . . . , q.(5.4)

Moreover,

→fk =

q−1∑j=1

(fk(xkj ) − fk(xkj+1))

→bj + fk(xkq)

→bq.(5.5)

Then

→Fk ·

→fk =

q−1∑j=1

(fk(xkj ) − fk(xkj+1))

→Fk ·

→bj + fq(xkq)

→Fk ·

→bq ≥ 0(5.6)

by (5.4) and the fact that fk is positive and nonincreasing for all k.Lemma 5.2. If fk is positive and nonincreasing, then fk(x) ≤ 1

xλ(fk) for any x ∈ [0, 1],where

λ(fk) =

∫Ifkdλ.

Proof. For any 0 < x ≤ 1, we have

λ(fk) ≥∫ x

0fk(t)dλ(t) ≥ x · fk(x).

Lemma 5.3. Let T satisfy conditions (A), (B), and (C). If f is nonincreasing, then

||PT f ||∞ ≤ α||f ||∞ + β||f ||1,(5.7)

where β = maxk,l∑q

i=2Wk,l(ai−1)

ai−1·τ ′k(ai−1).

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Proof. By Lemma 5.1, since f is nonincreasing, (PT f)l is nonincreasing for all l. It followsthat

||PT f ||∞ = maxl

||(PT f)l||∞ ≤ maxl

(PT f)l(0)

= maxl

K∑k=1

Wk,l(0)

τ ′k(0)fk(0) + max

l

K∑k=1

q∑i=2

Wk,l(τ−1k,i (0))

τ ′k(τ−1k,i (0))

fk(τ−1k,i (0))

≤ maxk

fk(0) · maxl

K∑k=1

Wk,l(0)

τ ′k(0)+ max

l

K∑k=1

q∑i=2

Wk,l(τ−1k,i (0))

τ ′k(τ−1k,i (0))

fk(τ−1k,i (0))

≤ αfk(0) +

K∑k=1

q∑i=2

Wk,l(ai−1)

τ ′k(ai−1)fk(ai−1)

≤ αfk(0) +

K∑k=1

q∑i=2

Wk,l(ai−1)

τ ′k(ai−1)

λ(fk)

ai−1

≤ α||f ||∞ + β∑k

||fk||1

= α||f ||∞ + β||f ||1.

(5.8)

Theorem 5.4. Suppose that T satisfies conditions (A), (B), and (C). Then T admits anacim, μ = f∗λ, and the density f∗ is nonincreasing.

Proof. Let fk ≡ 1K for all k. Then f is positive and nonincreasing. Applying inequality

(5.8) iteratively, we obtain

||PnT f ||∞ ≤ αn||f ||∞ + β(1 + α + · · · + αn−1)||f ||1 ≤ 1 + β

1

1 − α.

Thus, the sequence {PnT f}∞n=1 is uniformly bounded and thus weakly compact in ||·, . . . , ·||1.

By the Yosida–Kakutani theorem (see [5, Theorem 2.2.14]), the sequence 1n

∑n−1i=1 P i

T f con-

verges in ||·, . . . , ·||1 to a PT -invariant function f∗. It is nonincreasing since it is the limit ofnonincreasing functions.

Remark 5.5. Similar results can be obtained using the following conditions:

(A′) τk,i(ai) = 1, 1 ≤ i ≤ q, for all 1 ≤ k ≤ K.(B′)

∑ri=1 Fk,l,i, 1 ≤ r ≤ q, is increasing, for all 1 ≤ k ≤ K.

(C′) maxl∑K

k=1Wk,l(1)τ ′k(1)

≤ α < 1.

We now present an example of a random map T which satisfies conditions (A), (B), and (C)and thus preserves an acim.

Example 5.6. Let T be a random map which is given by {τ1, τ2} and controlled by thestochastic matrix W , where

τ1(x) =

{x

1−x for 0 ≤ x < 12 ,

2x− 1 for 12 ≤ x ≤ 1,

(5.9)

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τ2(x) =

{3x

2−x for 0 ≤ x < 12 ,

x− 12 for 1

2 ≤ x ≤ 1,(5.10)

and

W =

[W11 W12

W21 W22

],(5.11)

where

W21(x) =

{12 for 0 ≤ x < 1

2 ,14 for 1

2 ≤ x ≤ 1,(5.12)

W22(x) =

{12 for 0 ≤ x < 1

2 ,34 for 1

2 ≤ x ≤ 1,(5.13)

W11 = 7/12, and W12 = 5/12. Observe that

τk,i(0) = τk,i

(1

2

)= 0

for k, l = 1, 2 and i = 1, 2. Also, since τk, k = 1, 2, is piecewise convex and Wkl, k, l = 1, 2, ispiecewise constant, it is easy to see that Fk,l,i is nonincreasing for all k, l, i. Moreover,

W11(0)

τ ′1(0)+

W21(0)

τ ′2(0)=

7

12+

1

3=

11

12< 1

and

W12(0)

τ ′1(0)+

W22(0)

τ ′2(0)=

5

12+

1

3=

9

12< 1.

Therefore, T satisfies conditions (A), (B), and (C) and thus preserves an acim.Remark 5.7. The above example does not satisfy the conditions of Lemma 3.2. In partic-

ular,

W21(0)

τ ′1(0)+

W22(1)

τ ′2(1/2)=

1

2+

3

4=

5

4> 1.

6. A binomial model with a switch in the trends. In this section, we propose a discrete-time model which consists of L risky security processes sl = (s1(n), . . . , sL(n)), n = 0, 1, . . . , N ,where sl is a nonnegative discrete-time stochastic process for l = 1, . . . , L. sl(n) is the priceof the risky security l at time n. For example, sl is the price of one share of common stock ofa particular corporation. Discrete-time models which are described by a Markov process areimportant models of security prices because they are often realistic [12].

We assume that the price of the sl risky security is an adapted stochastic process; i.e.,the investors have full knowledge of the past and present prices. For instance, sl(n) will beknown only at time n.

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The prices of the securities are assumed to be smaller than a finite number; i.e., the priceshave an upper bound M ∈ R, 0 < M < ∞, such that 0 < sl(n) < M . We normalize theprices over M so that

0 < sl(n) < 1

for 1 ≤ l ≤ L and n = 0, . . . , N . We consider the binomial model which is a simple but veryimportant model for the price of a single risky security. It is commonly used by practitioners,for example, to determine the price of various kinds of stock options. In this model we assumeone risky security price s1. At each time period there are two possibilities: the securityprice may go up by the factor u or it may go down by a factor d; i.e., s1(n) = u · s1(n − 1)or s1(n) = d · s1(n − 1). The probability of an up move during a period is equal to theparameter pu, and the probability of going down is pd = 1 − pu. The moves over time areindependent of each other.

We now introduce a generalization of the binomial model. We assume that the factors uand d are functions of the prices, u(x) : (0, 1) → (1,∞) and d : (0, 1) → (0, 1); i.e., at time n,u and d depend on the price of the risky security s1 at time n − 1. u and d can be constanton subsets of (0, 1) or more general functions. Similarly, the probabilities pu and pd canbe constant or price dependent. Price dependent probabilities are more realistic because inpractice the probability for the market to go up or down is not constant in time. We also allowthe u function to have two regimes uf and us. uf drives the price up faster than us. Similarly,we allow the d function to have two regimes df and ds. df drives the price down faster than ds.The switch between the regimes is governed by a position dependent stochastic matrix W .

We believe that this generalization of the binomial model is more realistic since the realmarket does not go up always with the same factor and its trends may switch between fastand slow regimes.

Let T1 be a random map which represents a “trend” in the binomial model; i.e., twosimilar transformations τf and τs. The transformations of T1 drive the price up if the priceis smaller than 1/2 and drive the price down when the price is greater than or equal to 1/2.However, τf is “faster” than τs.

Consequently, let T2 be a random map, which represents another trend, which drives theprice down if the price is smaller than 1/2 and drives the price up when the price is greaterthan or equal to 1/2. Thus, the price has the possibility of going up or down at any giventime. Here, we discuss the outcome of a single trend—for instance, T1.

At time n = 0, we estimate the functions uf , us, df , and ds. For example, we can estimatethe functions from historical data. Once we are given the functions uf (x), us(x), df (x), andds(x) and the matrix W at time n = 0, we can construct the trends that drive the prices. Inparticular, we can construct T1:

τf (x) =

{uf (x) · x for 0 < x < 0.5,

df (x) · x for 0.5 ≤ x < 1,(6.1)

and

τs(x) =

{us(x) · x for 0 < x < 0.5,

ds(x) · x for 0.5 ≤ x < 1.(6.2)D

ownl

oade

d 11

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Moreover, we extend τf and τs from (0, 1) to the closed interval [0, 1] continuously. We usethe same matrix W to control the switching between τf and τs.

In the following example we present a trend and the switch between its regimes, whichadmits an acim.

Example 6.1. Let T = {τf , τs} be a random map controlled by a stochastic matrix W ,where

τf (x) =

{2x for 0 ≤ x < 1/2,

2x− 1 for 1/2 ≤ x ≤ 1,(6.3)

τs(x) =

{1.9x for 0 ≤ x < 1/2,

x for 1/2 ≤ x ≤ 1,(6.4)

and

W =

[1/4x + 1/2 1/2 − 1/4x

1/2 1/2

].(6.5)

We have ∑l∈{f,s}

supx

Wf,l(x)

|τ ′l (τf (x))| =3/4

2+

1/2

1=

7

8< 1

and ∑l∈{f,s}

supx

Ws,l(x)

|τ ′l (τs(x))| =1/2

2+

1/2

1=

3

4< 1,

and supx1

|τ ′l (x)| < 1 for all l. Thus, T satisfies the assumptions of Lemma 3.2 and admits an

acim.

Acknowledgments. We would like to thank an anonymous reviewer who brought to ourattention Ruelle’s paper [13], which allowed us to give a shorter proof for the existence resultof section 3.

REFERENCES

[1] W. Bahsoun, P. Gora, and A. Boyarsky, Stochastic perturbations for position dependent randommaps, Stoch. Dyn., 3 (2003), pp. 545–557.

[2] W. Bahsoun and P. Gora, Weakly convex and concave random maps with position dependent probabil-ities, Stochastic Anal. Appl., 21 (2003), pp. 983–994.

[3] C. Bose, V. Maume-Deschamps, B. Schmitt, and S. Shin, Invariant measures for piecewise convextransformations of an interval, Studia Math., 152 (2002), pp. 263–297.

[4] V. Baladi, Positive Transfer Operators and Decay of Correlations, Adv. Ser. Nonlinear Dynam. 16,World Scientific Publishing, River Edge, NJ, 2000.

[5] A. Boyarsky and P. Gora, Laws of Chaos, Birkhauser Boston, Boston, 1997.[6] J. Doob, Stochastic Processes, Wiley, New York, 1953.

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[7] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York,1964.

[8] G. Froyland, Ulam’s method for random interval maps, Nonlinearity, 12 (1999), pp. 1029–1052.[9] P. Gora and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding C2 trans-

formations in RN , Israel J. Math., 67 (1989), pp. 272–286.

[10] J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the businesscycle, Econometrica, 57 (1989), pp. 357–384.

[11] S. Pelikan, Invariant densities for random maps of the interval, Trans. Amer. Math. Soc., 281 (1984),pp. 813–825.

[12] S. R. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell Publishers, Oxford,UK, 1997.

[13] D. Ruelle, Spectral properties of a class of operators associated with maps in one dimension, ErgodicTheory Dynam. Systems, 11 (1991), pp. 757–767.

[14] K. R. Schenk-Hoppe, Random dynamical systems in economics, Stoch. Dyn., 1 (2001), pp. 63–83.

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markov switching for position dependent random maps with application to forecasting

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