Scaling and Multi Scaling in Financial Indexes - A Simple Model

8/9/2019 Scaling and Multi Scaling in Financial Indexes - A Simple Model

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SCALING AND MULTISCALING IN FINANCIAL INDEXES:A SIMPLE MODEL

ALESSANDRO ANDREOLI, FRANCESCO CARAVENNA, PAOLO DAI PRA, AND GUSTAVO POSTA

Abstract. We propose a simple stochastic model for time series which is analyticallytractable, easy to simulate and which captures some relevant stylized facts of financialindexes, including scaling properties. We show that the model fits the Dow Jones IndustrialAverage time series in the period 1935-2009 with a remarkable accuracy.

Despite its simplicity, the model has several interesting features. The volatility is notconstant and displays high peaks. The empirical distribution of the log-returns (incrementsof the logarithm of the index) is non-Gaussian and may exhibit heavy tails. Log-returns

corresponding to disjoint time intervals are uncorrelated but not independent: the corre-lation of their absolute values decays exponentially fast in the distance between the timeintervals for large distances, while it has a slower decay for moderate distances. Finally, thedistribution of the log-returns obeys scaling relations that are detected on real time series,but are not satisfied by most available models.

1. Introduction

1.1. Modeling financial indexes. Recent developments in stochastic modelling of timeseries have been strongly influenced by the analysis of financial indexes. The basic model,that has given rise to the celebrated Black & Scholes formula [11, 16], assumes that the

logarithm Xt of the price of the underlying index, after subtracting the trend, is given by(1.1) dXt = dWt,

where (the volatility) is a constant and (Wt)t0 is a standard Brownian motion.Despite its success, this model is not consistent with a number of stylized facts that are

empirically detected in many real time series. Some of these facts are the following:

the volatility is not constant: in particular, it may have high peaks, that may beinterpreted as shocks in the market (see Figure 6(c) below);

the empirical distribution of the increments Xt+h Xt of the logarithm of the price(the log-returns) has tails heavier than Gaussian (see Figure 4(b) below);

log-returns corresponding to disjoint time-interval are uncorrelated, but not indepen-

dent: in fact, the correlation between the absolute values |Xt+h Xt| and |Xs+h Xs|has a slow decay in |ts|, up to moderate values for |ts|. This phenomenon is knownas clustering of volatility (see Figure 3 below).

In order to have a better fit with real data, many different models have been proposed todescribe the volatility and the price process. In discrete-time, autoregressive models suchas ARCH and GARCH [9, 6] have been widely used. In continuous time, the basic model(1.1) has been modified by letting = t be a stochastic process, often the solution of a

Date: June 2, 2010.2010 Mathematics Subject Classification. 60G44, 91B25, 91G70.Key words and phrases. Financial Index, Scaling, Multiscaling, Brownian Motion, Stochastic Volatility.We gratefully acknowledge the support of the University of Padova under grant CPDA082105/08.

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2 ALESSANDRO ANDREOLI, FRANCESCO CARAVENNA, PAOLO DAI PRA, AND GUSTAVO POSTA

stochastic differential equation driven by a general Levy process. A systematic account ofthese stochastic volatility models can be found in [3]. More recent developments along theselines include the generalized Ornstein-Uhlenbeck processes and the COGARCH (GARCH in

continuous time) [13, 14]. All these models involve several parameters, whose estimation onreal data is itself a subject of research.

More recently (see [8, 5, 18]), other stylized facts of financial indexes have been pointedout, concerning the scaling properties of the empirical distribution of the log-returns. Con-sider the time series of an index (si)1iT over a period of T 1 days and denote by ph theempirical distribution of the (detrended) log-returns corresponding to an interval of h days:

(1.2) ph() := 1T h

Thi=1

xi+hxi() , xi := log(si) di ,

where di is the local rate of linear growth of log(si) (see section 3 for details) and x()denotes the Dirac measure at x

R. The statistical analysis of various indexes, such as theDow Jones Industrial Average (DJIA) or the Nikkei 225, shows that, for h within a suitable

time scale, ph obeys approximately a diffusive scaling relation (cf. Figure 1(a)):

(1.3) ph(dr) 1h

g

rh

dr,

where g is a probability density with tails heavier than Gaussian. If one considers the q-thempirical moment mq(h), defined by

(1.4) mq(h) :=1

T hThi=1

|xi+h xi|q =

|r|qph(dr) ,

from relation (1.3) it is natural to guess that mq(h) should scale as hq/2

. This is indeed whatone observes for moments of small order q q (with q 3 for the DJIA). However, formoments of higher order q > q, the different scaling relation hA(q), with A(q) < q/2, takesplace, cf. Figure 1(b) (see also [8]). This is the so-called multiscaling of moments.

Despite of the variety of models that can be found in the literature, it is quite nontrivialto identify a good one which agrees with all mentioned stylized facts. For example, the cele-brated and widely used GARCH [2] exhibits non-constant volatility, clustering of volatilityand non-Gaussian distribution of log-returns, but a closer analysis shows that multiscalingof moments is not present, at least for the range of values of the parameters that most oftenoccur in practice. More refined version of GARCH (for instance FIGARCH, see [ 2, 7]) havebeen proposed. It should be stressed that although models with sufficiently many parameterscan be adapted to data, the soundness of the procedure of statistical inference may be veryweak. We shall make more comments on this issue in section 3.

1.2. Baldovin and Stellas approach. Motivated by renormalization group argumentsfrom statistical physics, Baldovin and Stella [5, 18] have recently proposed a new model inwhich scaling considerations play a major role. Their construction may be summarized as fol-lows. They first introduce a process (Yt)t0 which satisfies the scaling relation (1.3) for a givenfunction g, that is assumed to be even, so that its Fourier transform g(u) :=

R

eiuxg(x)dxis real (and even). The process (Yt)t0 is defined by specifying its finite dimensional laws:for t1 < t2 < < tn the joint density of Yt1, Yt2, . . . , Y tn is given by

(1.5) p(x1, t1; x2, t2; . . . ; xn, tn) = hx1

t1,

x2 x1t2

t1

, . . . ,xn xn1

tn

tn1,


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SCALING AND MULTISCALING IN FINANCIAL INDEXES 3

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(a) Diffusive scaling of log-returns.

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(b) Multiscaling of moments.

Figure 1. Scaling properties of the DJIA time series (opening prices 1935-2009).(a) The empirical densities of the log-returns over 1, 2, 5, 10, 25 days show a re-markable overlap under diffusive scaling.(b) The scaling exponent A(q) as a function of q, defined by the relation mq(h) hA(q) (cf. (1.4)), bends down from the Gaussian behavior q/2 (red line) for q q 3. The quantity A(q) is evaluated empirically through a linear interpolation of(log mq(h)) versus (log h) for h {1, . . . , 5} (cf. section 3 for more details).

where h is the function whose Fourier transform h is given by

(1.6) h(u1, u2, . . . , un) := gu21 + . . . + u2n.Note that if g is the standard Gaussian density, then (Yt)t0 is the ordinary Brownianmotion. For a non Gaussian g, the expression in (1.6) is not necessarily the Fourier transformof a probability on Rn, so that some care is needed. We will see in section 2 for exactlywhat gs relations (1.5)-(1.6) define a consistent family of finite dimensional distributions,hence a stochastic process (Yt)t0, and it turns out that (Yt)t0 is always a mixture ofBrownian motions. However, it is already clear from (1.5) that the increments of the process(Yt)t0 corresponding to time intervals of the same length (that is, for fixed ti+1 ti) havea permutation invariant distribution and therefore cannot exhibit any decay of correlations.

For this reason, Baldovin and Stella introduce what we believe is the most remarkableingredient of their construction, namely a special form of time-inhomogeneity in the process

(Yt)t0. Although they define it in terms of finite dimensional distributions, we find it simplerto give an equivalent pathwise construction. Given a sequence of times 0 < 1 < 2 < 0 such that |f(x)| M|g(x)| for x in a neighborhood ofx0, while we writef = o(g) if f(x)/g(x) 0 as x x0; in particular, O(1) (resp. o(1)) is a bounded (resp. avanishing) quantity. The standard exponential and Poisson laws are denoted by Exp() andP o(), for > 0: X Exp() means that P(X x) = (1 ex)1[0,)(x) for all x Rwhile Y P o() means that P(Y = n) = en/n! for all n {0, 1, 2, . . .}. We sometimeswrite (const.) to denote a positive constant, whose value may change from place to place.


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2. The model and the main results

If a stochastic process (Yt)t0 is to satisfy relations (1.5)(1.6), it must necessarily have

exchangeable increments.1

If we make the (very mild) assumption that (Yt)t0 has no fixedpoint of discontinuity, then a continuous-time version of the celebrated de Finettis theoremensures that (Yt)t0 is a mixture of Levy processes, see Theorem 3 in [10] (cf. also [1]).Actually, more can be said: since by (1.3) the distribution of the increments of (Yt)t0 isisotropic, i.e., it has spherical symmetry in Rn, by Theorem 4 in [10] the process (Yt)t0 isa mixture of Brownian motions. This means that we have the following representation:

(2.1) Yt = Wt,

where (Wt)t0 is a standard Brownian motion and is a random variable (interpreted as arandom volatility) taking values in (0, ) and independent of (Wt)t0. Viceversa, if a process(Yt)t0 satisfies (2.1), then, denoting by the law of , relations (1.5)(1.6) hold with

(2.2) g(x) = R

12 e

x2

22 (d) ,

or, equivalently,

g(u) =

R

e2u2

2 (d) ,

as one easily checks. This shows that the functions g for which (1.5)(1.6) provide a consistentfamily of finite dimensional distributions are exactly those that may be expressed as in (2.2)for some probability on (0, +).

A sample path of (2.1) is obtained by sampling independently from and (Wt)t0from the Wiener measure. Note that this path cannot be distinguished from the path ofa Brownian motion with constant volatility, and therefore correlations of increment cannotbe detected empirically. The same observation applies to the time-inhomogeneous process(Xt)t0 obtained by (Yt)t0 through (1.7)(1.8). The reason why Baldovin and Stella mea-sure nonzero correlations from their samples is that they do not simulate (Xt)t0 but ratheran autoregressive approximation of it.

2.1. Definition of the model. We propose here a different mechanism, that replaces theautoregressive scheme by a random update of the volatility. Given two real numbers D (0, 1/2], (0, ) and a probability on (0, ) (these may be viewed as our parameters),our model is defined upon the following three sources of alea:

a standard Brownian motion W = (Wt)t0; a Poisson point process T = (n)nZ on R with intensity ;

a sequence = (n)n0 of independent and identically distributed positive randomvariables. The marginal law of the sequence will be denoted by (so that n forall n) and for conciseness we denote by a variable with the same law .

We assume that W,T, are defined on some probability space (, F, P) and that they areindependent. By convention, we label the points of T so that 0 < 0 < 1. We will actuallyneed only the points ofT [0, ), that is the variables (n)n0. We recall that the randomvariables 0, 1, (n+1 n)n1 are independent and identically distributed Exp(), so that1/ is the mean distance between the points in T. Although some of our results would hold

1By this we mean precisely the following: setting Y(a,b) := Yb Ya for short, the distribution of therandom vector (YI1+y1 , . . . , YIn+yn) where the Ij s are intervals and yjs real numbers does notdepend on y1, . . . , yn, as long as the intervals y1 + I1, . . . , yn + In are disjoint.


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for more general distributions of T, we stick for simplicity to the (rather natural) choice ofa Poisson process.

We are now ready to define our model X = (Xt)t0. For t

[0, 1] we set

(2.3) Xt := 0W(t0)2D W(0)2D ,while for t [n, n+1] (with n 1) we set(2.4) Xt := Xn + n

W(tn)2D+

nk=1(kk1)

2D Wnk=1(kk1)

2D

.

In words: at the epochs n the time inhomogeneity t t2D is refreshed and the volatilityis randomly updated: n1 ; n. A possible financial interpretation of this mechanism isthat jumps in the volatility correspond to shocks in the market. The reaction of the marketis not homogeneous in time: if D < 1/2, the dynamics is fast immediately after the shock,and tends to slow down later, until a new jump occurs. For D = 1/2 our model reducesto a simple random volatility model dXt = t dWt, where t :=

k=0 k 1[k,k+1)(t) is a

(random) piecewise constant process.Note that the pieces of the Brownian motion W used in each interval [n, n+1) (cf. (2.4))

are independent. Also observe that in (2.3) the instant t = 0 is not the beginning of thetime inhomogeneity, as it was in (1.7): this is to ensure that the process X has stationaryincrements (see below).

Using the scale invariance of Brownian motion, we now give an alternative definition ofour model X, that is equivalent in law with (2.4) but more convenient for the proofs in thefollowing sections. For t 0, define(2.5) i(t) := sup{n 0 : n t} = #{T (0, t]} ,so that i(t) is the location of the last point in T before t. Now we introduce the process

I = (It)t0 by

(2.6) It := 2i(t)

t i(t)

2D+

i(t)k=1

2k1 (k k1)2D 20 (0)2D ,

with the agreement that the sum in the right hand side is zero if i(t) = 0. We can thenredefine our basic process X = (Xt)t0 by setting

(2.7) Xt := WIt .

Note that I is a strictly increasing process with absolutely continuous paths, and it isindependent of the Brownian motion W. Thus our model may be viewed as an independentrandom time change of a Brownian motion.

2.2. Basic properties. Some basic properties of our model can be easily established.

(A) The process X has stationary increments.

(B) The process X can be represented as a stochastic volatility process:

(2.8) dXt = vt dBt ,

where (Bt)t0 is a standard Brownian motion. More precisely, denoting I(s) := dds I(s),

the variables Bt and vt are defined by

(2.9) Bt :=

It0

1I(I1(u))

dWu , vt :=

I(t) =

2D i(t)

t i(t)D12 .

Note that, whenever D < 12 , the volatility vt has singularities at the random times n.


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(C) The process X is a zero-mean, square-integrable martingale (provided E(2) < ).The proof of these properties is given in section 4, where we also show some relations between

the distribution of the ks and the law of the random variable Xt. Let us mention inparticular that, for any q > 0,

(2.10) E(|Xt|q) < for some (hence any) t > 0 E(q) < .As for most real financial indexes (like the DJIA, see section 3) there is empirical evidencethat E(|Xt|q) < for all q > 0, the main interest of our model is when has finite momentsof all orders. In this case, a link between the exponential moments of and those of Xt isgiven in Proposition 14 in section 4 below.

Remark 1. If we look at the process X for a fixed realization of the variables T and ,averaging only on W that is, if we work under the conditional probability P( |T, ) the increments ofX are no longer stationary, as it is clear from (2.4). We stress however thatthe properties (B) and (C) above continue to hold also under P(

|T, ), as it will be clear

in section 4. Of course, the condition E(2) < in (C) is not needed under P( |T, ).Another important property of the process X is that the distribution of its increments

is ergodic, as we show in section 4. This entails in particular that for every > 0, k Nand for every choice of the intervals (a1, b1), . . . , (ak, bk) (0, ) and of the measurablefunction F : Rk R, we have almost surely

(2.11) limN

1

N

N1n=0

F(Xn+b1 Xn+a1, . . . , X n+bk Xn+ak)

= E

F(Xb1 Xa1, . . . , X bk Xak)

,

provided the expectation appearing in the right hand side is well defined. In words: theempirical average of a function of the increments of the process over a long time period isclose to its expected value.

Thanks to this property, our main results concerning the moments and the correlation ofthe increments of the process X, that we state in the next subsection, are of direct relevancefor the comparison of our model with real data series. Some care is needed, however, becausethe accessible time length N in (2.11) may not be large enough to ensure that the empiricalaverages are close to their limit. We elaborate more on this issue in section 3, where wecompare our model with the DJIA data from a numerical viewpoint.

2.3. The main results. We now state our main results concerning the process X. Theycorrespond to the basic stylized facts that we have mentioned in the introduction: diffu-

sive scaling of the distributions of log-returns (Theorem 2 below); multiscaling of moments(Theorem 3 and Corollary 4); clustering of volatility (Theorem 5 and Corollary 6).

The first result, proved in section 5, states that for small h the increments (Xt+h Xt)have an approximate diffusive scaling, in agreement with (1.3).

Theorem 2. As h 0 we have the convergence in distribution(Xt+h Xt)

h

dh0

f(x) dx ,(2.12)

where f is a mixture of centered Gaussian densities, namely

(2.13) f(x) =

0(d)

0dt et

t1/2D

4Dexp

t

12Dx2

4D2 .


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We stress that the function f appearing in (2.12)(2.13), which describes the asymptoticrescaled law of the increment (Xt+h Xt) in the limit of small h, has a different tail behaviorfrom the density of (Xt+h

Xt) for fixed h. For instance, when has finite moments of all

orders, it follows by (2.10) that the same holds for (Xt+h Xt). However, from (2.13) anda simple change of variables we get

R

|x|q f(x) dx = (2D)q/2E(q)

0es s(D1/2)q ds ,

which is finite if and only if q < q := (1/2 D)1. Therefore, independently of the law of , the density f has always polynomial tails :

R

|x|qf(x) dx = for q q.This feature off has striking consequences on the scaling behavior of the moments of the

increments our model. If we set for q (0, )(2.14) mq(h) := E(|Xt+h Xt|q) ,from the convergence result (2.12) it would be natural to guess that mq(h)

hq/2 as h

0, in

analogy with the Brownian motion case. However, this turns out to be true only for q < q.For q q, the faster scaling mq(h) hDq+1 holds instead, the reason being precisely thefact that the q-moment of f is infinite for q q. This transition in the scaling behaviorof mq(h) goes under the name of multiscaling of moments and is discussed in detail, e.g.,in [8]. Let us now state our result, that we prove in section 5.

Theorem 3 (Multiscaling of moments). Let q > 0, and assume E(q) < +. Then thequantity mq(h) := E(|Xt+h Xt|q) = E(|Xh|q) is finite and has the following asymptoticbehavior as h 0:

mq(h)

Cq hq2 if q < q

Cq hq2 log( 1h ) if q = q

Cq hDq+1 if q > q

, where q :=1

(12 D)

.

The constant Cq (0, ) is given by

(2.15) Cq :=

E(|W1|q) E(q) q/q (2D)q/2 (1 q/q) if q < qE(|W1|q) E(q) (2D)q/2 if q = qE(|W1|q) E(q)

0 ((1 + x)

2D x2D) q2 dx + 1Dq+1

if q > q

,

where () :=

0 x1exdx denotes Eulers Gamma function.

Corollary 4. The following relation holds true:

(2.16) A(q) := limh0

log mq(h)log h

= q

2 if q q

Dq + 1 if q q.

The explicit form (2.15) of the multiplicative constant Cq will be used in section 3 for theestimation of the parameters of our model on the DJIA time series.

Our last theoretical result, proved in section 6, concerns the correlations of the absolutevalue of two increments, a quantity which is usually called volatility autocorrelation. Westart determining the behavior of the covariance.

Theorem 5. Assume that E(2) < . The following relation holds as h 0, for alls, t > 0:(2.17) Cov(

|Xs+h

Xs

|,

|Xt+h

Xt

|) =

4D

12D e|ts| (|t s|) h + o(h) ,


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where

(2.18) (x) := Cov

SD1/2 ,

S+ x

D1/2

and S Exp(1) is independent of .We recall that (Y, Z) := Cov(Y, Z)/

V ar(Y)V ar(Z) is the correlation coefficient of two

random variables Y, Z. As Theorem 3 yields

limh0

1

hV ar(|Xt+h Xt|) = (2D) 12D V ar( |W1| SD1/2) ,

where S Exp(1) is independent of , W1, we easily obtain the following result.Corollary 6 (Volatility autocorrelation). Assume that E(2) < . The correlation of theincrements of the process X has the following asymptotic behavior as h 0:(2.19)

limh0

(|Xs+h Xs|, |Xt+h Xt|) = (t s) := 2 V a r( |W1| SD1/2)

e|ts| (|t s|) ,

where () is defined in (2.18) and S Exp(1) is independent of and W1.

This shows that the volatility autocorrelation of our process decays exponentially fast fortime scales greater than the mean distance 1/ between the epochs k. For shorter timescales, a relevant contribution is given by the function (). Note that by (2.18) we can write(2.20) (x) = V ar() E(SD1/2 (S+ x)D1/2) + E()2 Cov(SD1/2, (S+ x)D1/2) ,

where S Exp(1). As x , the two terms in the right hand side decay as

E(SD1/2 (S+ x)D1/2) xD1/2 , Cov(SD1/2, (S+ x)D1/2) xD3/2 ,while for x = O(1) the decay of both terms is faster than polynomial but slower thanexponential (see Figures 3(a) and 3(b) below).

3. Estimation and data analysis

We now consider some aspects of our model from a numerical viewpoint. We comparethe theoretical predictions and the simulated data of our model with the time series of theDow Jones Industrial Average (DJIA) over a period of 75 years: more precisely, we haveconsidered the DJIA opening prices from 2 Jan 1935 to 31 Dec 2009, for a total of 18849

daily data. Of course, other indexes would be interesting as well, but we have decided tostick to the DJIA both for its importance and for its considerable time length.The data analysis, the simulations and the plots have been obtained with the software

R [17]. The code we have used is available on the web page http://www.math.unipd.it/

~fcaraven/c.html.

3.1. Preliminary considerations. For the numerical comparison of our process (Xt)t0with the DJIA time series, we have decided to focus on the following quantities:

(a) The multiscaling of moments, cf. Corollary 4.

(b) The volatility autocorrelation decay, cf. Corollary 6.

(c) The distribution of Xt.
http://www.math.unipd.it/~fcaraven/c.htmlhttp://www.math.unipd.it/~fcaraven/c.htmlhttp://www.math.unipd.it/~fcaraven/c.htmlhttp://www.math.unipd.it/~fcaraven/c.htmlhttp://www.math.unipd.it/~fcaraven/c.html


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Roughly speaking, the idea is to compute empirically these quantities on the DJIA timeseries and then to compare the results with the theoretical predictions of our model. This is

justified by the ergodicity of the increments of our process (Xt)t0, cf. equation (2.11).

The first problem that one faces is the estimation of the parameters of our model: thetwo scalars (0, ), D (0, 12 ] and the distribution of . This in principle belongs toan infinite dimensional space, but in a first time we focus on the moments E() and E(2).

In order to estimate (D,,E(), E(2)), we take into account four significant quantitiesthat depend only on these parameters: the multiscaling coefficients C1 and C2 (see (2.15)),the multiscaling exponent A(q) (see (2.16)) and the volatility autocorrelation function (t)(see (2.19)). We then consider a natural loss functional L = L(D,,E(), E(2)), see (3.3)below, which describes some distance between these theoretical quantities and the corre-sponding empirical ones (evaluated on the DJIA time series), and we define the estimatorfor (D ,,E(), E(2)) as the point at which L attains its overall minimum, subject to theconstraint E(2)

(E())2.

Quite surprisingly, it turns out that the estimated values are such that E(2) (E())2,that is is nearly constant. The constraint E(2) (E())2 is not playing a relevant role:the unconstrained minimum nearly coincides with the constrained one. Thus, the problemof determining the distribution of beyond its moments E() and E(2) does not appearin the case of the DJIA. If this was not the case, as it may be for other indexes, the law of should be estimated by requiring a good fit of the distribution of the log-return Xt with theempirical one. For the DJIA, all we can do is to verify that there is indeed a (remarkably)good fit between these distributions, with no further parameter to be estimated.

Before describing in detail the procedure we have just sketched, let us fix some notation:the DJIA time series will be denoted by (si)0iN (where N = 18848) and the correspondingdetrended log-DJIA time series will be denoted by (xi)0iN:

xi := log(si) d(i) ,where d(i) := 1250

i1k=i250 log(si) is the mean log-DJIA price on the previous 250 days. Of

course, other reasonable choices for d(i) are possible, but they affect the analysis only in aminor way.

3.2. Estimation of D,,E(), E(2). The theoretical scaling exponent A(q) is defined in(2.16) while the multiscaling constants C1 and C2 are given by (2.15) for q = 1 and q = 2.Since q = ( 12 D)1 > 2 (we recall that 0 D 12 ), we can write more explicitly

(3.1) C1 =2

D ( 12 + D) E() 1/2D

, C2 = 2D (2D) E(2) 12D .

Defining the corresponding empirical quantities requires some care, because the DJIA dataare in discrete-time and therefore no h 0 limit is possible. We first evaluate the empiricalq-moment mq(h) of the DJIA log-returns over h days, namely

mq(h) := 1N + 1 h

Nhi=0

|xi+h xi|q .

By Theorem 3, the relation log mq(h) A(q)(log h) + log(Cq) should hold for h small.By plotting (log mq(h)) versus (log h) one finds indeed an approximate linear behavior, formoderate values of h and when q is not too large (q 5). By a standard linear regressionof (log

mq(h)) versus (log h) for h = 1, 2, 3, 4, 5 days we therefore determine the empirical

values of A(q) and Cq on the DJIA time series, that we call A(q) and Cq.


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(a) Multiscaling in the DJIA(1935-2009).

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19792008

(b) Multiscaling in subperi-ods of 30 years.

Figure 2. Multiscaling of moments in the DJIA time series (1935-2009).

(a) The empirical scaling exponent A(q) in the period 1935-2009 (circles) and thetheoretical scaling exponent A(q) (line) as a function of q.

(b) The empirical scaling exponent A(q) evaluated in subperiods of 30 years (seelegend) and the theoretical scaling exponent A(q) (line) as a function of q.Note. The theoretical scaling exponent A(q) is evaluated for D = 0.16, cf. (3.4).

For what concerns the theoretical volatility autocorrelation, Corollary 6 and the station-

arity of the increments of our process (Xt)t0 yield

(3.2) (t) := limh0

(|Xh|, |Xt+h Xt|) = 2 V a r( |W1| SD1/2)

et (t) ,

where S Exp(1) is independent of and W1 and where the function () is given by(x) = V ar() E(SD1/2 (S+ x)D1/2) + E()2 Cov(SD1/2, (S+ x)D1/2) ,

cf. (2.20). Note that, although () does not admit an explicit expression, it can be quiteeasily evaluated numerically. For the analogous empirical quantity, we define the empiricalDJIA volatility autocorrelation

h(t) over h-days as the sample correlation coefficient of the

two sequences (|xi+h xi|)0iNht and (|xi+h+t xi+t|)0iNht. Since no h 0 limitcan be taken on discrete data, we are going to compare (t) with h(t) for h = 1 day.We can now define a loss functionalL as follows:

L(D ,,E(), E(2)) = 12

C1C1

12

+

C2C2

12

+1

20

20k=1

A(k/4)A(k/4)

12

+400n=1

en/T400m=1 e

m/T1(n)

(n) 12

,

(3.3)

where the constant T controls a discount factor in long-range correlations. Of course, differentweights for the four terms appearing in the functional could be assigned. We fix T = 40 (days)

and we define the estimator (D,,E(),E(2)) of the parameters of our model as the point


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(d) Volatility autocorrelationin subperiods of 30 years.

Figure 3. Volatility autocorrelation decay in the DJIA time series (1935-2009).(a) Log plot for the empirical 1-day volatility autocorrelation

1(t) in the period

1935-2009 (circles) and the theoretical prediction (t) (line), as functions oft (days).

For clarity, only one data out of two is plotted.(b) Same as (a), but log-log plot instead of log plot. For clarity, for t 50 only onedata out of two is plotted.(c) The empirical h-day volatility autocorrelation h(t) in the period 1935-2009 forh = 2 and h = 3 days (see legend) and the theoretical prediction (t) (line) as afunction of t (days).(d) The empirical 1-day volatility autocorrelation 1(t) evaluated in subperiods of30 years (see legend) and the theoretical prediction (t) (line) as a function of t(days). For clarity, only one data out of two is plotted.Note. The theoretical curve (t) is evaluated for the parameters values given in (3.4).


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where the functional L attains its overall minimum, that is(

D,

,E(),E(2)) := arg min

D(0, 12

], ,E(), E(2)(0,)

such that E(2)(E())2 L(D,,E(), E(2))

,

where the constraint E(2) (E())2 is due to V ar() = E(2) (E())2 0. We expectthat such an estimator has good properties, such as asymptotic consistency and normality(we omit a proof of these properties, as it goes beyond the spirit of this paper).

We have then proceeded to the numerical study of the functional L, which appears tobe quite regular. With the help of the software Mathematica [15], we have obtained thefollowing estimates for the parameters:

(3.4) D 0.16, 0.00097, E() 0.108, E(2) 0.117 E()2Using these values of the parameters, we find a very good agreement between the empiricaland the theoretical quantities we have considered:

the empirical values of the multiscaling constants are C1 6.62103, C2 9.23105,while the theoretical values are C1 6.31 103, C2 9.38 105;

the theoretical and empirical multiscaling exponents A(q) and A(q) are remarkablyclose, see Figure 2(a) for a graphical comparison;

the fit is very good also for the volatility autocorrelation functions (t) and 1(t), seeFigures 3(a) (log plot) and 3(b) (log-log plot) for a graphical comparison, which alsoshows that the decay is faster than polynomial but slower than exponential. FromFigure 3(c) one sees that the agreement between (t) and h(t) is still good also forh = 2 and h = 3 days.

We point out that, evaluating the empirical multiscaling exponent A(q) and the empiricalvolatility autocorrelation 1(t) over subperiods or 7500 data (roughly 30 years) of the DJIAtime series, one finds a considerable amount of variability, cf. Figure 2(b) and Figure 3(d).This indicates that some uncertainty in the estimation of the parameters is unavoidable, atleast for time series of the given length. We discuss this issue in subsection 3.4 below, wherewe show that a comparable variability is observed in data series simulated from our model.

Remark 7. Baldovin and Stella [5, 18] find D 0.24 for their (different) model.3.3. The distribution of log-returns. Having found that E(2) (E())2 for the es-timated parameters, cf. (3.4), the estimated variance of is equal to zero, that is is aconstant. In particular, the model is now completely specified.

We have then compared the theoretical distribution pt(

) := P(Xt

) = P(Xt

X0

)

of our model for t = 1 (daily log-return) with the analogous quantity evaluated on the DJIAtime series, i.e., the empirical distribution pt() of the sequence (xi+t xi)0iNt:

pt() := 1N + 1 t

Nti=0

xi+txi() .

Although we do not have an analytic expression for the theoretical distribution pt(), it can beevaluated numerically via Monte Carlo simulations. In Figure 4(a) we have plotted the bulkof the distributions p1() and p1() or, more precisely, the corresponding densities in the range[3s, +3s], where s 0.0095 is the standard deviation of p1() (i.e., the empirical standarddeviation of the daily log returns evaluated on the DJIA time series). In Figure 4(b) wehave plotted the integrated tail of p1(

), that is the function z

P(X1 > z) = P(X1 z}N ,in the range z [s, 6s]. As one can see, the agreement between the empirical and theo-retical distributions is remarkably good for both figures, especially if one considers that noparameter has been estimated using these curves.

Figures 4(c) and 4(d) show that some amount of variability less pronounced thanfor the multiscaling of moments and for the correlations is present also in the empiricaldistribution p1() evaluated over subperiods of 7500 data (roughly 30 years).Remark 8. At this point, a fundamental difference between our model an that proposedby Baldovin and Stella is fully revealed. Baldovin and Stellas starting point is the non-Gaussian density g(

) of the log-return distribution: as a first step, they estimate g(

) on

the DJIA time series and build a basic process having g() as marginal density, cf. (1.5)and (1.6); afterwards, they build a new process via nonlinear time change, in order to breakexchangeability and produce decay of correlations.

On the other hand, when is constant (as for the DJIA), the basic process on which ourmodel is built is just Brownian motion with constant volatility (Wt)t0. Thus, the non-Gaussianity of the law of X1 in our model is not imposed a priori, but is produced by therandomness of the time change, i.e. by the distribution of the shocks times T. Having madethe most natural choice for this distribution, namely that of a Poisson process, the fact thata posteriori we get an excellent fit with the empirical law of X1 is quite remarkable.

Remark 9. It should be stressed that, even if we had found

V ar() := E(2)(E())2 > 0

(as could happen for other indexes), detailed properties of the distribution of are notexpected to be detectable from data. Indeed, the estimated mean distance between thesuccessive epochs (n)n0 of the Poisson process T is 1/ 1031 days, cf. (3.4). Therefore, ina time period of the length of the DJIA time series we are considering, only 18849/1031 18variables k are expected to be sampled, which is certainly not enough to allow more than arough estimation of the distribution of . This should be viewed more as a robustness than

a limitation of our model: even when V ar() > 0, the first two moments of contain theinformation which is relevant for application to real data.

Remark 10. Our model predicts an even distribution for Xt. On the other hand, it isknown that DJIA data exhibit a small but nonzero skewness. At the discrete-time level, andtherefore at the level of simulations, skewness could be easily introduced in the framework

of our model. In fact, recalling that conditionally on T the increments X of our modelare centered Gaussian random variables, it would suffice to replace the distribution of theincrements which follow closely the events of T those that are most likely to be large by a skew deformation of a Gaussian.

3.4. Variability of estimators. Generally speaking, it would be desirable to have goodestimators, whose value does not change much if we compute them in sufficiently long butdifferent time windows of the time series. On the contrary, we have already noticed thatconsiderable fluctuations in different time windows of the DJIA time series are observed for

the estimators we have used, namely the empirical scaling exponent A(q), cf. Figure 2(a),the empirical volatility autocorrelation

1(t), cf. Figure 3(d), and to a less extent the

empirical distribution p1() of the 1-day log-return, cf. Figure 4(c) and 4(d). This is linked


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Figure 5. Simulations over 75 years: variability in subperiods of 30 years.Empirical evaluation of some observables in subperiods or 30 years for a 75-years-long time series sampled from our model (Xt)t0 (with parameters fixed as in (3.4)):

the scaling exponent

A(q) (a), the volatility autocorrelation

1(t) (b), the density

(c) and the integrated tails (d) of the daily log-return distribution p1().to the fact that relation (2.11) is an equality only for N , therefore there is a priorino guarantee that the DJIA time series over a few decades is close to the ergodic limit, i.e.,long enough to guarantee a reasonably approximate equality.

It is relevant to show that also this aspect is consistent with our model. In fact, it couldhappen that, unlike the DJIA, samples from our model over a few decades are long enoughto damp out fluctuations. We have therefore simulated time series distributed accordingto our model, of the same length as the DJIA time series we are considering (18849 daily

data), and we have evaluated the relevant quantities

A(q),

1(t) and

p1() over subperiods

of 7500 days. Figure 5 shows that the variability displayed in such a sample is qualitatively


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Figure 6. Some features non completely caught by our model: tails and variance. Integrated right and left tails (see legend) of the log-return empirical distributionph() for the DJIA time series in the period 1935-2009 and the theoretical prediction

ph() (line) for h = 5 days (a) and h = 400 days (b). Empirical volatility for the daily log-returns of the DJIA time series in the period1935-2009 (c) and for a simulation of our model over 75 years (d). The volatility at

day i is evaluated through the formula

12R+1

i+Rj=iR(xj xj1)2

1/2with R = 50.

The parameters of our model are fixed as in (3.4)

consistent with what is observed on the DJIA time series. We believe that this is a crucialpoint in showing the agreement of our model with the DJIA time series.

Remark 11. The multiscaling of empirical moments has been observed in several financialindexes in [8], where it is claimed that data provide solid arguments against model withlinear or piecewise linear scaling exponents. Note that the theoretical scaling exponent A(q)


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of our model is indeed piecewise linear, cf. (2.16). However, Figure 5(a) shows that the

empirical scaling exponent

A(q) evaluated on data simulated from our model smooths out

the change of slope, yielding graphs that are analogous to those obtained for the DJIA time

series, cf. Figure 2(b). This shows that the objection against models with piecewise linearA(q), raised in [8], cannot apply to the model we have proposed.

Remark 12. We point out that, among the different quantities that we have considered,

the scaling exponent A(q) appears to be the most sensitive. For instance, if instead of theopening prices one took the closing prices of the DJIA time series (over the same time period

1935-2009), one would obtain a different (though qualitatively similar) graph of A(q).3.5. Final considerations. The above statistical analysis shows that the model (Xt)t0that we propose is capable to explain several relevant features of the DJIA time serieswith a very good degree of accuracy, even though the fine estimation of the parametersis a delicate (and interesting) statistical problem. We close this section pointing out some

empirical quantities that fit less precisely with our model.

While for short range log-returns (over 1-5-10-50 days) the tails of the empirical distri-bution are close to the ones predicted by our model, cf. Figure 6(a), for longer ranges(such as 400 ore more days) some discrepancies emerge, cf. Figure 6(b).

The graphic of the empirical volatility for the DJIA time series shows clearly peaksof various heights and of quite symmetric shape, cf. Figure 6(c). On the other hand,the peaks of the empirical volatility evaluated on time series simulated according toour model are all of about the same height and of sharply asymmetric shape, cf.Figure 6(d). While the asymmetric shape is a direct consequence of the definition ofthe process (It)t0, cf. (2.6), the limited variability in the height of the peaks is mainly

due to the fact that is constant.Numerical investigations lead us think that a higher variability in the heights of thepeaks of the volatility of our model could be obtained through a slight modificationof the model, using the following scaling function f(t) instead of t2D:

f(t) :=

t2D if 0 < t < c

c2D + 2 D c2D1 (t c) if t c ,

with c a constant parameter to be estimated. In fact, it turns out that for such amodified model the correlations decay faster than for the original model (with thesame parameters). As a consequence, a non-constant is likely to be the right oneto fit the DJIA time series, because as the variance of increases the theoretical

volatility autocorrelation decay slows down. Of course, a non constant is then likelyto produce more pronounced peaks.

4. Basic properties of the model

We start proving the properties (A)-(B)-(C) stated in section 2.2. We denote by G the-field generated by the whole process (It)t0, which coincides with the -field generated bythe sequences T = {k}k0 and = {k}k0.Proof of property (A). We first focus on the process (It)t0, defined in (2.6). For h > 0 let

Th :=

T h and denote the points in

Th by hk = k

h. As before, let h

ih(t)be the largest


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point ofTh smaller that t, i.e., ih(t) = i(t + h). Recalling the definition (2.6), we can write

It+h Ih = 2

ih(t) t hih(t)2D +ih(t)

k=ih(0)+1 2k1 hk hk12D

2

ih(0) hih(0)2D ,where we agree that the sum in the right hand side is zero if ih(t) = ih(0). This relationshows that (It+h Ih)t0 and (It)t0 are the same function of the two random sets Th andT. Since Th and T have the same distribution (both are Poisson point processes on R withintensity ), the processes (It+h Ih)t0 and (It)t0 have the same distribution too.

We recall that G is the -field generated by the whole process (It)t0. From the definitionXt = WIt and from the fact that Brownian motion has independent, stationary increments,it follows that for every Borel subset A C([0, +))

P (Xh+ Xh A) = E

P

WIh+ WIh A | G

= E

P

WIh+Ih A | G

= PWIh+Ih A = P (WI A) = P (X A) ,where we have used the stationarity property of the process I. Thus the processes (Xt)t0 and(Xh+t Xh)t0 have the same distribution, which implies stationarity of the increments. Proof of property (B). Observe first that Is :=

dds Is > 0 a.s. and for Lebesguea.e. s 0.

By a change of variable, we can rewrite the process (Bt)t0 defined in (2.9) as

Bt =

It0

1I(I1(u))

dWu =

t0

1Is

dWIs =

t0

1Is

dXs ,

which shows that relation (2.8) holds true. It remains to show that (Bt)t0 is indeed astandard Brownian motion. Note that

Bt = It0 (I1)(u) dWu .Therefore, conditionally onI, (Bt)t0 is a centered Gaussian process (it is a Wiener integral),with conditional covariance given by

Cov(Bs, Bt | I) =min{Is,It}

0(I1)(u) du = min{s, t} .

This shows that, conditionally on I, (Bt)t0 is a Brownian motion. Therefore, it is a fortioria Brownian motion without conditioning. Proof of property (C). The assumption E(2) < ensures that E(|Xt|2) < for all t 0,as we prove in Proposition 13 below. Let us now denote by FXt = (Xs, s t) the naturalfiltration of the process X. We recall that

Gdenotes the -field generated by the whole

process (It)t0 and we denote by FXt G the smallest -field containing FXt and G. SinceE(WIt+h WIt |FXt G) = 0 for all h 0, by the basic properties of Brownian motion,recalling that Xt = WIt we obtain

E(Xt+h|FXt G) = Xt + E(WIt+h WIt |FXt G) = Xt.Taking the conditional expectation with respect to FXt on both sides, we obtain the mar-tingale property for (Xt)t0.

Next we show what was announced in (2.10).

Proposition 13. For any t > 0

E(

|Xt

|q) 2: by Jensens inequality we have

(4.4) i(t)

k=0

2kq/2

= (i(t) + 1)q/2 1

i(t) + 1

i(t)k=0

2kq/2 (i(t) + 1)q/21 i(t)

k=0

qk .

Therefore, if E(q) < , from (4.3) we can write(4.5) E(|It|q/2) tDq E((i(t) + 1)q/2) E(q) < ,because i(t), being a Poisson random variable of mean t, has finite moments of all ordersand is independent of the variables (k)k0. Analogously, since (

k=1 xk)

k=1 xk for (0, 1) and for every non-negative sequence (xn)nN, for q 2 by (4.3) we can write

(4.6) E(|It|q/2) tDq Ei(t)

k=0 qk E((t 0)

Dq) E(i(t) + 1) E(q) < .

The proof of (4.1) is complete.

We now relate the exponential moments of with those of Xt.

Proposition 14. Regardless of the distribution of , for every q > (1 D)1 we have(4.7) E[exp (|Xt|q)] = , t > 0 , > 0 .On the other hand, for all q < (1 D)1 and t > 0 we have(4.8) E[exp(|Xt|q)] < > 0 E

exp

2q2q

< > 0 ,

and the same relation holds for q = (1

D)1 provided D < 12 .


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The proof is given in Appendix A. Note that (1 D)1 (1, 2], because D (0, 12 ], so thatfor D < 12 the distribution of Xt has always tails heavier than Gaussian.

Remark 15. Ifhas a bounded support, Proposition 14 implies that E[exp (|Xt|q

)] < +for q < (1 D)1, and E[exp (|Xt|q)] = + for q > (1 D)1. This means that the tailsof the distribution of Xt are heavier than Gaussian but lighter than exponential, seeminglyin contrast with the red curve in Figure 4(b), that shows a decay slower than exponentialfor the distribution of the daily log-returns of our model (with parameters chosen to fitthe DJIA time series, see (3.4)). As a matter of fact, simulations show that the predictedsuper-exponential decay of the tails can actually be seen:

at a much larger distance from the mean than six standard deviations (which is therange of Figure 4(b)) for daily log-returns;

at a more reasonable distance from the mean for log-returns over longer periods, suchas hundreds of days (cf. the red curve in Figure 6(b)).

As both cases appear to have little interest in applications, the relevant tails are heavierthan exponential, despite of Proposition 14.

We finally show that the distribution of its increments is ergodic. We actually show twostronger results; a 01 law for the tail -field of X, and the fact that the increments ofX are exponentially mixing. In what follows, given a continuous or discrete-time stochasticprocess = (t)t0, we set t() := {s : s t} and we define the so-called tail -field() :=

t0 t(). Also, for an interval I [0, +), we let

FDI := (Xt Xs : s, t I)to denote the -field generated by the increments in I of the process X.

Proposition 16. The following properties hold:

(a) The tail -field (X) is trivial, i.e., its events have probability either zero or one.

(b) Let I = [a, b), J = [c, d), with 0 a < b c < d. Then, for every A FDI andB FDJ

(4.9) |P(A B) P(A)P(B)| 2e(cb).As a consequence, equation (2.11) holds true almost surely and in L1, for every measurable

function F : Rk R such that E[|F(Xb1 Xa1, . . . , X bk Xak)|] < +.The proof of Proposition 16 is given in Appendix B.

5. Scaling and multiscaling: proof of Theorems 2 and 3We observe that for all fixed t, h > 0 we have the equality in law Xt+h Xt

Ih W1,

as it follows by the definition of our model ( Xt)t0 = (WIt)t0. We also observe that i(h) =#{T (0, h]} P o(h), as it follows from (2.5) and the properties of the Poisson process.5.1. Proof of Theorem 2. Since P(i(h) 1) = 1 eh 0 as h 0, we may focus onthe event {i(h) = 0} = {T (0, h] = }, on which we have Ih = 20((h 0)2D (0)2D),with 0 Exp(). Since Xt+h Xt

Ih W1, for every Borel set A we can write as h 0

P(Xt+h Xt A) = P(

Ih W1 A; [0, h] T = ) + o(1)

=

0(d)

0ds es P

(h + s)2D s2D W1 A

+ o(1).


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Now observe that h1((h + s)2D s2D) 2Ds2D1 as h 0, for every s > 0, hence, plainly,1

h(h + s)2D s2D W1

d

h0

2DsD1/2 W1 .

Therefore, if we choose A =

h C, where C is a Borel set whose boundary has zero Lebesguemeasure, as h 0 we have

P

Xt+h Xt

h C

=

0

(d)

0

ds es P

W1 C

2DsD1/2

+ o(1).

Since P(W1 dx) = (2)1 exp(x2/2) dx, the proof is completed.

5.2. Proof of Theorem 3. Since Xt+h Xt

Ih W1, we can write

(5.1) E(|Xt+h Xt|q) = E(|Ih|q/2|W1|q) = E(|W1|q) E(|Ih|q/2) = cq E(|Ih|q/2) ,where we set cq := E(|W1|q). We therefore focus on E(|Ih| q2 ), that we write as the sum ofthree terms, that will be analyzed separately:

(5.2) E(|Ih|q2 ) = E(|Ih|

q2 1{i(h)=0}) + E(|Ih|

q2 1{i(h)=1}) + E(|Ih|

q2 1{i(h)2}) .

For the first term in the right hand side of (5.2), we note that P(i(h) = 0) = eh 1 ash 0 and that Ih = 20((h 0)2D (0)2D) on the event {i(h) = 0}. Setting 0 =: 1Swith S Exp(1), we obtain as h 0(5.3) E(|Ih|

q2 1{i(h)=0}) = E(

q) Dq E

((S+ h)2D S2D) q2 1 + o(1) .Recalling that q := ( 12 D)1, we have

q q q2 Dq + 1 1 D 1

2 q .

As 0 we have 1((S+ )2D S2D) 2D S2D1 and note that ES(D12 )q = (1 q/q)is finite if and only if (D 12 )q > 1, that is q < q. Therefore the monotone convergencetheorem yields

(5.4) for q < q : limh0

E

(S+ h)2D S2D q2 q2 h

q2

= (2D)q/2 (1 q/q) (0, ) .

Next observe that, by the change of variables s = (h)x, we can write

E((S+ h)2D S2D) q2 = 0 ((s + h)2D s2D) q2 es ds= (h)Dq+1

0

((1 + x)2D x2D) q2 ehx dx .(5.5)

Note that ((1 + x)2D x2D) q2 (2D) q2 x(D12 )q as x + and that (D 12 )q < 1 if andonly if q > q. Therefore, again by the monotone convergence theorem, we obtain

(5.6) for q > q : limh0

E

((S+ h)2D S2D) q2 Dq+1 hDq+1

=

0

((1+ x)2D x2D) q2 dx (0, ) .

Finally, in the case q = q we have ((1+ x)2D x2D)q/2 (2D)q/2 x1 as x + and wewant to study the integral in the second line of (5.5). Fix an arbitrary (large) M > 0 and


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note that, integrating by parts and performing a change of variables, as h 0 we have

M

ehx

x

dx =

log M ehM + h

M

(log x) ehx dx = O(1) +

hM

log y

h ey dy= O(1) +

hM

logy

ey dy + log

1

h

hM

ey dy = log

1

h

1 + o(1)

.

From this it is easy to see that as h 00

((1 + x)2D x2D) q

2 ehx dx (2D) q

2 log

1

h

.

Coming back to (5.5), noting that Dq + 1 = q2 for q = q, it follows that

(5.7) limh0

E

((S+ h)2D S2D) q

2

Dq+1 h q

2 log( 1

h)

= (2D)q

2 .

Recalling (5.1) and (5.3), the relations (5.4), (5.6) and (5.7) show that the first term in theright hand side of (5.2) has the same asymptotic behavior as in the statement of the theorem,except for the regime q > q where the constant does not match (the missing contributionwill be obtained in a moment).

We now focus on the second term in the right hand side of ( 5.2). Note that, conditionallyon the event {i(h) = 1} = {1 h, 2 > h}, we have

Ih = 21(h1)2D+20

(10)2D(0)2D

21(hhU)2D +20hU+ S2D

S

2D,

where S Exp(1) and U U(0, 1) (uniformly distributed on the interval (0, 1)) are inde-pendent of 0 and 1. Since P(i(h) = 1) = h + o(h) as h 0, we obtain(5.8) E(|Ih|

q2 1{i(h)=1}) = h

Dq+1 E

21(1 U)2D + 20

U +

S

h

2D

S

h

2D q2.

Since (u + x)2D x2D 0 as x , for every u 0, by the dominated convergencetheorem we have (for every q (0, ))

(5.9) limh0

E(|Ih|q2 1{i(h)=1})

hDq+1= E(q1) E

(1 U)Dq = E(q1) 1Dq + 1 .

This shows that the second term in the right hand side of (5.2) gives a contribution of theorder hDq+1 as h 0. This is relevant only for q > q, because for q q the first term givesa much bigger contribution of the order hq/2 (see (5.4) and (5.7)). Recalling (5.1), it followsfrom (5.9) and (5.6) that the contribution of the first and the second term in the right handside of (5.2) matches the statement of the theorem (including the constant).

It only remains to show that the third term in the right hand side of ( 5.2) gives a neg-ligible contribution. Consider first the case q > 2: from the basic bound (4.3) and Jensensinequality, in analogy with (4.4) and (4.5), we obtain

(5.10) E|Ih|q/2 1{i(h)2} tDq E(q) E(i(h) + 1)q/2 1{i(h)2} .

On the other hand, since for (0, 1) we have (k=1 xk) k=1 xk , when q 2 from(4.3) we can write, in analogy with (4.6),

(5.11) E|Ih|q/2 1{i(h)2} tDq E(q) E(i(h) + 1) 1{i(h)2} .


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For any fixed a > 0, by the Holder inequality with p = 3 and p = 3/2 we can write for h 1E(i(h) + 1)

a 1{i(h)2} E(i(h) + 1)3a

1/3

P(i(h) 2)2/3

E(i(1) + 1)3a1/3 (1 eh ehh)2/3 (const.) h4/3 ,because E

(i(1) + 1)3a

< (recall that i(h) P o()) and (1 eh ehh) 12 (h)2

as h 0. Then it follows from (5.10) and (5.11) thatE|Ih|q/2 1{i(h)2} (const.) hDq+4/3 .

This shows that the contribution of the third term in the right hand side of ( 5.2) is alwaysnegligible with respect to the contribution of the second term (recall (5.9)).

6. Decay of correlations: proof of Theorem 5

Given a Borel set I R, we let GI denote the -algebra generated by the family of randomvariables (k1{kI}, k1{kI})k0. Informally, GI may be viewed as the -algebra generatedby the variables k, k for the values of k such that k I. From the basic property of thePoisson process and from the fact that the variables (k)k0 are independent, it follows thatfor disjoint Borel sets I, I the -algebras GI, GI are independent. We set for short G := GR,which is by definition the -algebra generated by all the variables (k)k0 and (k)k0, whichcoincides with the -algebra generated by the process (It)t0.

We have to prove (2.17). Plainly, by translation invariance we can set s = 0 without lossof generality. We also assume that h < t. We start writing

Cov(|Xh|, |Xt+h Xt|)= CovE|Xh|G , E|Xt+h Xt|G + ECov|Xh|, |Xt+h Xt|G .

(6.1)

We recall that Xt = WIt and the process (It)t0 is G-measurable and independent of theprocess (Wt)t0. It follows that, conditionally on (It)t0, the process (Xt)t0 has indepen-dent increments, hence the second term in the right hand side of (6.1) vanishes, becauseCov(|Xh|, |Xt+h Xt||G) = 0 a.s.. For fixed h, from the equality in law Xh = WIh

Ih W1 it follows that E(|Xh||G) = c1

Ih, where c1 = E(|W1|) =

2/. Analogously

E(|Xt+h Xt||G) =

It+h It and (6.1) reduces to

(6.2) Cov(|Xh|, |Xt+h Xt|) = 2

Cov

Ih,

It+h It

.

Recall the definitions (2.5) and (2.6) of the variabes i(t) and It. We now claim that wecan replace It+h It by It+h It 1{T (h,t]=} in (6.2). In fact from (2.6) we can write

It+h It = 2i(t+h)(t + h i(t+h))2D +i(t+h)

k=i(t)+1

2k1(k k1)2D 2i(t)(t i(t))2D ,

where we agree that the sum in the right hand side is zero if i(t + h) = i(t). This showsthat (It+h It) is a function of the variables k, k with index i(t) k i(t + h).Since {T (h, t] = } = {i(t) > h}, this means that

It+h It 1{T (h,t]=} is G(h,t+h]-

measurable, hence independent of

Ih, which is clearly G(,h]-measurable. This showsthat Cov(

Ih,

It+h It 1{T (h,t]=}) = 0, therefore from (6.2) we can write

(6.3) Cov(

|Xh

|,

|Xt+h

Xt

|) =

2

CovIh,It+h It 1{T (h,t]=} .


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It is convenient to define

j(t) := inf{k Z : k > t} = i(t) + 1 ,and note that the variables j(t), j(t)+1, . . . a r e G(t,)-measurable. We now introduce avariable At,h, which is essentially It+h It with i(t) replaced by i(h). More precisely, we set(6.4) At,h :=

2i(h)

(t + h i(h))2D (t i(h))2D

ifT (t, t + h] = ,

and

At,h := 2i(t+h)(t + h i(t+h))2D +

i(t+h)k=j(t)+1

2k1(k k1)2D

+ 2i(h)

(j(t) i(h))2D (t i(h))2D

ifT (t, t + h] = ,(6.5)

where we agree that the sum in the right hand side of (6.5) is zero unless i(t + h) > j(t),

that is #(T (t, t + h]) 2. We stress that At,h is G(,h](t,t+h]-measurable: in fact, At,his nothing but It+h It when we erase from T the points k (if any) falling in (h, t]. Inparticular, on the event {T (h, t] = } we can replace It+h It by At,h, rewriting (6.3) as

Cov(|Xh|, |Xt+h Xt|) = 2

Cov

Ih,

At,h 1{T (h,t]=}

=2

E

Ih E(

Ih) At,h 1{T (h,t]=}

=2

e(th) E

Ih E(

Ih) At,h ,

(6.6)

where in the last equality we have used the independence of the -algebras G(,h](t,)and G(h,t] and the fact that P(T (h, t] = ) = P(i(t) i(h) = 0) = e(th).

We now focus on(6.7) E

Ih E(

Ih) At,h = CovIh , At,h ,

that we can write as the sum of four terms:

Cov

Ih ,

At,h

= Cov

Ih 1{T [0,h]=} ,

At,h 1{T [t,t+h]=}

+ Cov

Ih 1{T [0,h]=} ,

At,h 1{T [t,t+h]=}

+ Cov

Ih 1{T [0,h]=} ,

At,h 1{T [t,t+h]=}

+ Cov

Ih 1{T [0,h]=} ,

At,h 1{T [t,t+h]=}

.

(6.8)

The first term in the right hand side gives the leading contribution:

CovIh 1{T [0,h]=} , At,h 1{T [t,t+h]=}= Cov

0

(h 0)2D (0)2D 1{T [0,h]=} , 0

(t + h 0)2D (t 0)2D 1{T [t,t+h]=}

= 2D e2h Cov

0

(h + S)2D S2D , 0

(t + h + S)2D (t + S)2D ,

where S := (0) Exp(1) and we have used the fact that the variables 0, 1{T [t,t+h]=}and 1{T [0,h]=} are independent. Since

1(( + x)2D x2D) 2Dx2D1 as 0, bymonotone convergence we obtain

1

hCov

Ih 1{T [0,h]=} ,

At,h 1{T [t,t+h]=}

h0 2D 12D CovS2D1, (S+t)2D1 ,in agreement with (2.17) and (2.18).


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It remains to show that the second, the third and the fourth term in the right hand side of(6.8) give a negligible contribution. Using the relation |Cov(X, Y)|

E(X2)E(Y2), valid

for any couple of random variables X, Y, by the Cauchy-Schwarz inequality we can writeCovIh 1C , At,h 1D E(Ih 1C) E(At,h 1D) E(I2h) E(A2t,h) P(C) P(D)1/4 ,for any choice of the events C, D. We now claim that E(A2t,h) E(I2h). In fact, from (6.4)and (6.5) we see that At,h coincides with It+h It once we replace i(h) by i(t). Note thati(h) appears in At,h in a term of the form (bi(h))2D (ai(h))2D, for suitable a < b, hencereplacing i(h) by i(t) we get something bigger, because i(h) i(t) (we assume that h < t).We have thus shown that At,h It+h It Ih, whence E(A2t,h) E(I2h). By Theorem 3 wehave E(I2h) (const.) h2 (observe that q > 2), therefore

(6.9) Cov

Ih 1C ,

At,h 1D (const.) h

P(C) P(D)

1/4

.

To deal with the second, third and fourth term in the right hand side of ( 6.8), we must takeeither C = {T [0, h] = } or D = {T [t, t + h] = }. In any case P(C)P(D) h andtherefore the right hand side of (6.9) is o(h), uniformly in t R (it does not depend on t).This completes the proof.

Appendix A. Proof of Proposition 14

We first need two simple technical lemmas.

Lemma 17. For 0 < q < 2, consider the function q : [0, +) [0, +) define by

q() := +

e|x|q

12x2 dx.

Then there are constants C1, C2 > 0, that depend on q, such that for all > 0

(A.1) C1eC1

22q q() C2eC2

22q

.

Proof. We begin by observing that it is enough to establish the bounds in (A.1) for large

enough. Consider the function of positive real variable f(r) := erq12 r

2. It is easily checked

that f is increasing for 0 r (q) 12q . Thus

q

()

(q)1

2q

12 (q)

12q

f(r) dr

1

2(q)

12q f12 (q) 12q = 12 (q) 12q exp c(q) 22q ,

with c(q) := 12q qq

2q 18 q2

2q > 0. The lower bound in (A.1) easily follows for large.

For the upper bound, by direct computation one observes that f(r) e14 r2 for r >(4)

12q . We have:

q() |x|(4)

12q

f(|x|) dx +|x|>(4)

12q

e14x

2dx 2(4) 12q f +

+

e14x

2dx .

Since f = f((q)1

2q ) = exp

C(q)2

2q

for a suitable C(q), also the upper bound

follows, for large.


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Lemma 18. Let X1, X2, . . . , X n be independent random variables uniformly distributed in[0, 1], and U1 < U2 < .. . < Un be the associated order statistics. For n 2 and k = 2, . . . , n,set k := Uk

Uk1.Then, for every > 0

limn+

Pk {2, . . . , n} : k > 1n1+ n1 = 1.

Proof. This is a consequence of the following stronger result: for every x > 0, as n wehave the convergence in probability

1

n

k {2, . . . , n} : k > xn

ex ,see [19] for a proof.

Proof of Proposition 14. Since Xt = WIt and

It W1 have the same law, we can write

Ee|Xt|q = EexpIq/2t |W1|q .We begin with the proof of (4.8), hence we work in the regime q < (1 D)1, or q =

(1 D)1 and D < 12 ; in any case, q < 2. We start with the implication. Since It andW1 are independent, it follows by Lemma 17 that

(A.2) E

exp

Iq/2t |W1|q

CE

exp

I

q2q

t

,

for some C, > 0. For the moment we work on the event {i(t) 1}. It follows by the basicbound (4.2) that

(A.3) It

i(t)

k=0 2Dk 2k ,where we set

k :=

1 for k = 0

k+1 k for 1 k i(t) 11 i(t) for k = i(t)

Note thati(t)

k=0 k = 1. It is easily shown by Lagrange multipliers that the function

(x0, x1, . . . , xn) n

k=0 x2Dk

2k subject to the constraint

nk=0 xk = 1 attains its maxi-

mum at (x0, x1, . . . , x

n), with

xk =

212D

knh=0 212Dh .Then it follows from (A.3) that

It i(t)k=0

(xk)2D 2k =

i(t)k=0

2

12D

k

12D .By assumption q 11D , which is the same as (1 2D) q2q 1. Thus

Iq

2q

1

i(t)

k=0

212D

k

(12D) q2q

i(t)

k=0

2q2q

k .


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Now observe that if i(t) = 0 we have It = (t 0)2D (0)2D t2D. Therefore, by (A.2)

Ee|Xt|q CEexpi(t)

k=0

2q2q

k 1{i(t)1} + Eexp t 2Dq2q 1{i(t)=0} C

E

i(t)+1

+ exp

t2Dq2q

,

(A.4)

where we have set

:= E

exp

2q2q

0

.

Therefore, if < , the right hand side of (A.4) is finite, because i(t) P o(t) has finiteexponential moments of all order. This proves the implication in (4.8).

The implication in (4.8) is simpler. By the lower bound in Lemma 17 we have

(A.5) Ee|Xt|q = EexpIq/2t |W1|q CEexpI q2qt ,for suitable C, > 0. We note that

E

exp

I

q2q

t

E

exp

I

q2q

t

1{i(t)=0}

= E

exp

(t 0)2D (0)2D

2q2q

0

P(i(t) = 0) .

(A.6)

Under the condition

Eexp2q2q = + > 0,

the last expectation in (A.6) is infinite, since

(t 0)2D (0)2D

> 0 almost surely andis independent of 0. Looking back at (A.5), we have proved the implication in (4.8).

Next we prove (4.7), hence we assume that q > (1 D)1. Consider first the case q < 2(which may happen only for D < 12 ). By (A.5)

E

e|Xt|q

CE

exp

I

q2q

t

.

We note that, by the definition (2.5) of It, we can write

(A.7) It i(t)k=2

2k1(k k1)2D ,

where we agree that the sum is zero ifi(t) < 2. For n 0, we let Pn to denote the conditionalprobability P( |i(t) = n) and En the corresponding expectation. Note that, under Pn, therandom variables (k k1)nk=2 have the same law of the random variables (k)nk=2 inLemma 18, for n 2. Consider the following events:

An := {2k a, k = 2, . . . , n} , Bn :=k = 2, . . . , n : k > 1n1+

n1 ,where a > 0 is such that ([a, +)) =: > 0 and > 0 will be chosen later. Note thatPn(An) =

n1 while Pn(Bn)

1 as n

+

, by Lemma 18. In particular, there is c > 0


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such that Pn(Bn) c for every n. Plainly, An and Bn are independent under Pn. We have

(A.8) (n) := En expIq

2q

t En expIq

2q

t 1AnBn c n1 exp

a

q2q

1

n1+

2D q2qn(1)

q2q

= c n1 exp

a

q2q n(12D(1+2D))

q2q

Note that q > 11D is equivalent to (1 2D) q2q > 1, therefore can be chosen small enoughso that b := (1 2D (1 + 2D)) q2q > 1. It then follows by (A.8) that (n) d exp(d nb)for every n N, for a suitable d > 0. Therefore

E

exp

I

q2q

t

= E[(i(t))] = + ,

because i(t) P o(t) and hence E[exp(d i(t)b)] = for all d > 0 and b > 1.Next we consider the case q 2. Note that

(A.9) E

e|Xt|q

= E

exp

Iq/2t |W1|q

,

hence if q > 2 we have E

e|Xt|q

= , because E[exp(c|W1|q)] = for every c > 0, It > 0almost surely and It is independent of W1. On the other hand, if q = 2 we must have D (1 D)1) and the steps leading to (A.8) have shownthat in this case It is unbounded. It then follows again from (A.9) that E[e

|Xt|2] = .

Appendix B. Proof of Proposition 16

We first need the following technical Lemma.

Lemma 19. Let and be two independent stochastic processes, such that both () and() are trivial. Then the -field

G :=

s,t0

[t() s()]

is trivial, where t() s() denotes the smallest -field containing t() s()Proof. We introduce the function space E := R[0,+), equipped with the -field generatedby cylinder sets. If Z is a bounded G-measurable random variable, then for every s, t 0we can write

(B.1) Z = fs,t (s+, t+) ,

where s+ denotes the process (s+u)u0, and fs,t : E E R is bounded and measurable.We set f := f0,0 and we denote by , respectively the laws of , on E.

By (B.1), for -a.e. x E we have f(x, ) = f0,t(x, t+), for every t 0. Thus f(x, )is ()-measurable and hence a.s. constant: f(x, ) = f1(x) a.s., where f1(x) := E(f(x, )).Therefore, for every measurable B E we can write(B.2)

E

f(x, y) 1B(y) (dy) = f1(x) (B) , for -a.e. x E .

Observe now that, again by (B.1), we have a.s.

f1() = E(f(, )

|) = E(ft,0(t+, )

|) = E(ft,0(x, ))

|x=t+ .


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This shows that f1() is ()-measurable and hence a.s. constant: f1() = c a.s., wherec := E(f1()) = E(f(, )). Therefore, for every measurable A E we can write

(B.3) E

f1(x) 1A(x) (dx) = c (A) .Putting together (B.2), (B.3) and Fubinis theorem, it follows that for all measurable A, B E

(B.4)

EE

f(x, y) 1AB(x, y) ( )(dx, dy) = c ( )(A B) ,

where denotes of course the product measure of and on E E.Since the sets of the form A B are a basis of the product -field of E E, relation

(B.4) still holds true with A B replaced by any measurable subset C E E. ChoosingC+n = {(x, y) : f(x, y) > c + 1n} and Cn = {(x, y) : f(x, y) < c 1n}, it follows easily thatP(C+n ) = P(C

n ) = 0 for every n N, that is f(x, y) = c for ( )-a.e. (x, y) E E.

Therefore Z = f(, ) = c a.s.. We have proved that every bounded G-measurable randomvariable is almost surely constant, which is equivalent to the triviality of G. Proof of Proposition 16. We begin with the proof of statement (a), the 0 1 law for (X).If Y is a bounded (X)-measurable random variable, by definition for every t 0 it can bewritten in the form

Y = t (Xt+) = t

WIt+

,

where t : C[0, +) R is bounded and measurable. Now, for s 0, setYt,s = t

WIt+

1{It>s}.

Note that Yt,s is s(W) t(I)-measurable. Plainly, In + almost surely as n +,therefore for every s

0

limn+

Yn,s = limn+

n WIn+ = Y.It follows that Y is

s,t0 [s(W) t(I)]-measurable. In other words

(X)

s,t0

[s(W) t(I)] ,

and by Lemma 19 all we need to show is that both (W) and (I) are trivial.The triviality of (W) is well known: it follows, e.g., from Blumenthal 0-1 law (cf. The-

orem 2.7.17 in [12]) applied to the Brownian motion (sW1/s)s0. For the triviality of (I)we proceed as follows. If A (I), for each n N

1A = n(In+) ,

for a suitable measurable function n. Recalling the definition (2.5) of the variable i(t), itis clear that i(t) + as t +, hence for every m N

1A = limn+

n(In+)1{i(n)>m} .

Now observe that the function n(In+)1{i(n)>m} is measurable with respect to the -field(k : k 0), where k := (k, k+1 k), and is invariant under any permutation of(0, . . . , m1). This shows that A is measurable for (k : k 0) and invariant by finitepermutations of the ks. By the Hewitt-Savage 0-1 law, A has probability either zero orone. The proof of (a) is now completed.

We now prove part (b). We recall that T denotes the set {k : k Z} and, for I R, GIdenotes the -algebra generated by the family of random variables (k1{kI}, k1{kI})k0,


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where (k)k0 is the sequence of volatilities. We are considering I = [a, b), J = [c, d), with0 a < b c < d. We introduce the G[b,c)-measurable event

:= {T [b, c) = }.We claim that, for A FDI , B FDJ , we have(B.5) P(A B|) = P(A)P(B).To see this, the key is in the following two remarks.

FDI and FDJ are independent conditionally to G = GR. This follows immediately fromthe independence of W and (It).

Conditionally to G, the process (Xt)tI is a Gaussian process whose covariances areall GI-measurable. It follows that P(A|G) is GI-measurable. Similarly, P(B|G) is GJ-measurable.

Thus

P(A B|) = E(P(A B|G)|) = E(P(A|G)P(B|G)|) = E(P(A|G)1P(B|G))P()

,

where the first of the remarks above has been used. Now, by the second remark, P(A|G)1is G[a,c)-measurable, and P(B|G) is G[c,d)-measurable. Therefore they are independent, andwe obtain

P(A B|) = P(A|)P(B) = P(A)P(B)where, in the last equality, we have used the fact that, since GI and are independent,P(A|) = E(P(A|G)|) = E[P(A|G)] = P(A). To complete the proof of part (b) we observethat, by what just shown,

|P(A B) P(A)P(B)| = |P(A B) P(A B|)|.But, letting C := A B

|P(C) P(C|)| = |E[(1 P())1C]|P()

E[|(1 P())|]P()

= 2(1 P()).

Since 1 P() = e(cb), we obtain|P(A B) P(A)P(B)| 2(1 P()) = 2e(cb)

as desired.

We finally show that equation (2.11) holds true almost surely and in L1

, for every mea-surable function F : Rk R such that E[|F(Xb1 Xa1, . . . , X bk Xak)|] < +. Considerthe Rk-valued stochastic process = (n)nN defined by

n := (Xn+b1 Xn+a1, . . . , X n+bk Xn+ak) ,for fixed > 0, k N and (a1, b1), . . . , (ak, bk) (0, ). The process is stationary,because we have proven in section 4 that X has stationary increments. Furthermore, thetail triviality of X implies that is ergodic, because every invariant event of is clearlycontained in the tail -field of X. The existence of the limit in (2.11), both a.s. and in L1,is then a consequence of the classical Ergodic Theorem, cf. Theorem 24.1 in [4]. Note thatthe ergodicity of also follows form the fact that is mixing, as shown in part (b).


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Acknowledgements

We thank Fulvio Baldovin, Massimiliano Caporin, Wolfgang Runggaldier and Attilio

Stella for fruitful discussions.

References

[1] L. Accardi, Y. G. Lu, A continuous version of de Finettis theorem, Ann. Probab. 21 (1993), 14781493.[2] R. T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometrics 73

(1996), 559.[3] O. E. Barndorff-Nielsen, N. Shephard, Non Gaussian Ornstein-Uhlenbeck based models and some of their

uses in financial economics, J. R. Statist. Soc. B 63 (2001), 167241.[4] P. Billingsley, Probability and Measure, Third Edition, John Wiley and Sons (1995).[5] F. Baldovin, A. Stella, Scaling and efficiency determine the irreversible evolution of a market, PNAS

104, n. 50 (2007), 1974119744.[6] T. Bollerslev, Generalized Autoregressive Conditional Heteroskedasticity, J. Econometrics 31 (1986),

307327.

[7] T. Bollerslev, H.O. Mikkelsen, Modeling and pricing long memory in stock market volatility, J. Econo-metrics 31 (1996), 151184.

[8] T. Di Matteo, T. Aste, M. M. Dacorogna, Long-term memories of developed and emerging markets:Using the scaling analysis to characterize their stage of development, J. Banking Finance 29 (2005),827851.

[9] R. F. Engle, Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United KingdomInflation, Econometrica 50 (1982), 9871008.

[10] D. A. Freedman, Invariants Under Mixing Which Generalize de Finettis Theorem: Continuous TimeParameter, Ann. Math. Statist. 34 (1963), 11941216.

[11] J. C. Hull, Options, Futures and Other Derivatives, Pearson/Prentice Hall (2009).[12] I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer (1988).[13] C. Kluppelberg, A. Lindner, R. A. Maller, A continuous time GARCH process driven by a Levy process:

stationarity and second order behaviour, J. Appl. Probab. 41 (2004) 601622.[14] C. Kluppelberg, A. Lindner, R. A. Maller, Continuous time volatility modelling: COGARCH versus

Ornstein-Uhlenbeck models, in From Stochastic Calculus to Mathematical Finance, Yu. Kabanov, R.Lipster and J. Stoyanov (Eds.), Springer (2007).

[15] Wolfram Research, Inc., Mathematica, Version 7.0, Champaign, IL (2008).[16] B. ksendal, Stochastic Differential Equations, Springer-Verlag (2003).[17] R Development Core Team (2009), R: A language and environment for statistical computing, R Founda-

tion for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0. URL: http://www.R-project.org.[18] A. L. Stella, F. Baldovin, Role of scaling in the statistical modeling of finance, Pramana 71 (2008),

341352.[19] L. Weiss, The Stochastic Convergence of a Function of Sample Successive Differences, Ann. Math. Statist.

26 (1955), 532536.

Dipartimento di Matematica Pura ed Applicata, Universita degli Studi di Padova, via Tri-este 63, I-35121 Padova, Italy

E-mail address: [email protected]





Dipartimento di Matematica, Politecnico di Milano, Piazzale Leonardo da Vinci 32, I-20133Milano, Italy

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Scaling and Multi Scaling in Financial Indexes - A Simple Model

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