A class of multifractal processes constructed using an embedded branching process

A class of multifractal processes constructed using an embedded branching process

[ [    [ [ University of Melbourne Department of Mathematics
 and Statistics
University of Melbourne
Parkville VIC 3010
Australia
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E-mail: \printead*e2
\smonth10 \syear2010\smonth10 \syear2011
\smonth10 \syear2010\smonth10 \syear2011
\smonth10 \syear2010\smonth10 \syear2011
Abstract

We present a new class of multifractal process on , constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton–Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change.

In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step , we can generate step in operations. Detailed pseudo-code for this algorithm is provided.

[
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\doi

10.1214/11-AAP834 \volume22 \issue6 2012 \firstpage2357 \lastpage2387 \newproclaimasspAssumption[section] \newproclaimdefinitionDefinition[section] \newproclaimremarkRemark[section]

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Multifractal process with embedded branching process

{aug}

A]\fnmsGeoffrey \snmDecrouezlabel=e1]dgg@unimelb.edu.au and A]\fnmsOwen Dafydd \snmJones\correflabel=e2]odjones@unimelb.edu.au

class=AMS] \kwd[Primary ]60G18 \kwd[; secondary ]28A80 \kwd60J85 \kwd68U20. Self-similar \kwdmultifractal \kwdbranching process \kwdBrownian motion \kwdtime-change \kwdsimulation.

1 Introduction

Information about the local fluctuations of a process can be obtained using the local exponent , defined as R03 ()

When is constant all along the sample path with probability , is said to be monofractal. In contrast, we can consider a class of processes whose exponents behave erratically with time: each interval of positive length exhibits a range of different exponents. For such processes, it is, in practice, impossible to estimate for all , due to the finite precision of the data. Instead, we use the Hausdorff spectrum , a global description of its local fluctuations. is defined as the Hausdorff dimension of the set of points with a given exponent . For monofractal processes, degenerates to a single point at some [so , and the convention is to set for ]. When the spectrum is nontrivial for a range of values of , the process is said to be multifractal.

The term multifractal is also well defined for measures. Let be a ball centered at with radius . The local dimension of a finite measure at is defined as

The Hausdorff spectrum of a measure at scale is then defined as the Hausdorff dimension of the set of points with a given local dimension . Measures for which the Hausdorff spectrum does not degenerate to a point are called multifractal measures. Constructions of multifractal measures date back to the -ary cascades of Mandelbrot M74 (), and the multifractal spectrum of such measures can be found in, for example, R03 ().

A positive nondecreasing multifractal process can be obtained by integrating a multifractal measure. Other processes with nontrivial multifractal structure can be obtained by using the integrated measure as a multifractal time change, applied to monofractal processes such as fractional Brownian motion. This is the basis of models such as infinitely divisible cascades BM02 (), BM03 (), CRA05 ().

Multifractals have a wide range of applications. For example, the rich structure of network traffic exhibits multifractal patterns ABFRV02 (), as does the stock market M97 (), M99 (). Other applications include turbulence SM88 (), seismology H01 (), TLM04 () and imaging RRCB00 (), to cite but a few.

On-line simulation of multifractal processes is in general difficult, because their correlations typically decay slowly, meaning that to simulate one requires . This is the same problem faced when simulating fractional Brownian motion, where to simulate one needs the whole covariance matrix of . Some simple monofractal processes avoid this problem, for example, -stable or processes C84 (), but it remains a real problem to find flexible multifractal models that can be quickly simulated.

We propose a new class of multifractal processes, called Multifractal Embedded Branching Process (MEBP) processes, which can be efficiently simulated on-line. MEBP are defined using the crossing tree, an ad-hoc space–time description of the process, and are such that the spatial component of their crossing tree is a Galton–Watson branching process. For any suitable branching process, there is a family of processes—identical up to a continuous time change—for which the spatial component of the crossing tree coincides with the branching process. We identify one of these as the Canonical Embedded Branching Process (CEBP), and then construct MEBP from it using a multifractal time change. To allow on-line simulation of the process, the time change is constructed from a multiplicative cascade on the crossing tree. The simulation algorithm presents nice features since it only requires operations and storage to generate steps, and can generate a new step on demand.

To construct the time change we use here, we start by constructing a multiplicative cascade on a multitype Galton–Watson tree. The cascade defines a measure on the boundary of the tree, whose existence follows from known results for multitype branching random walks. (See, e.g., L00 () for the single-type case.) To map the cascade measure onto , we use the so called “branching measure” on the tree, in contrast to the way this is usually done, using a “splitting measure.” See Section 3 for details and further background.

The MEBP processes constructed here include a couple of special cases of interest. We can represent Brownian motion as a CEBP, thus MEBP processes include a subclass of multifractal time changed Brownian motions. Such models are of particular interest in finance M97 (), M99 (). In the special case when the number of subcrossings is constant and equal to two (for the definition see Section 2), the CEBP degenerates to a straight line, and the time change is just the well-known binary cascade (see, e.g., B99 (), KP76 (), Mo96 () and references therein).

Although we do show that MEBP possess a form of discrete multifractal scaling [see the discussion following equation (10)], the multifractal nature of MEBP processes is not studied in this paper. We refer the reader to a coming paper for a full study of the multifactal spectrum of MEBP DHJ (). In particular, it can be shown that CEBP processes are monofractal, and that the multifractal formalism holds for MEBP processes, with a nontrivial spectrum. The monofractal nature of CEBP processes, together with an upper bound of the spectrum of MEBP, was derived in the Ph.D. thesis of the first author Dec09 ().

The paper is organized as follows. First we recall the definition of the crossing tree and then construct the CEBP process. We then construct MEBP processes and give conditions for continuity. Finally we provide an efficient on-line algorithm for simulating MEBP processes. An implementation of the algorithm is available from the second author’s website J ().

2 CEBP and the crossing tree

Let be a continuous process, with . For we define level passage times by putting and

The th level (equivalently scale ) crossing is the sample path from to .

When passing from a coarse scale to a finer one, we decompose each level crossing into a sequence of level crossings. To define the crossing tree, we associate nodes with crossings, and the children of a node are its subcrossings. The crossing tree is illustrated in Figure 1, where the level 3, 4 and 5 crossings of a given sample path are shown.

Figure 1: A section of sample path and levels 3, 4 and 5 of its crossing tree. In the top frame we have joined the points at each level, and in the bottom frame we have identified the th level crossing with the point and linked each crossing to its subcrossings.

The crossing tree is an efficient way of representing a self-similar signal, and can also be used for inference. In JS04 () the crossing tree is used to test for self-similarity and to obtain an asymptotically consistent estimator of the Hurst index of a self-similar process with stationary increments, and in JS05 () it is used to test for stationarity.

In addition to indexing crossings be their level and position within each level, we will also use a tree indexing scheme. Let be the root of the tree, representing the first level 0 crossing. The first generation of children (which are level crossings, of size ) are labeled by , , where is the number of children of . The second generation (which are level crossings, of size ) are then labeled , , where is the number of children of . More generally, a node is an element of and a branch is a couple where and . The length of a node is , and the th element is . If , is the curtailment of after terms. Conventionally , and . A tree is a set of nodes, that is, a subset of , such that:

  • ;

  • if a node belongs to the tree, then every ancestor node , , belongs to the tree;

  • if , then for and for , where is the number of children of .

Let be the th generation of the tree, that is, the set of nodes of length . (These are level crossings, of size .) Define and . The boundary of the tree is given by . Let be the position of node within generation , so that crossing is just . The nodes to the left and right of , namely and , will be denoted and . In general, when we have quantities associated with crossings, we will use tree indexing and level/position indexing interchangeably. So , , etc. At present our tree indexing only applies to crossings contained within the first level 0 crossing; however, in Section 3.3 we will extend this notation to the whole tree.

Let be the orientation of , for up and for down, and let be the vector given by the orientations of the subcrossings of . Let be the duration of . Clearly, to reconstruct the process we only need and for all and . The encode the spatial behavior of the process, and the the temporal behavior. Our definition of an EBP is concerned with the spatial component only. {definition} A continuous process with is called an Embedded Branching Process (EBP) process if for any fixed , conditioned on the crossing orientations , the random variables are all mutually independent, and is conditionally independent of all for . In addition we require that are identically distributed, for .

That is, an EBP process is such that if we take any given crossing, then count the orientations of its subcrossings at successively finer scales, we get a (supercritical) two-type Galton–Watson process, where the types correspond to the orientations.

Subcrossing orientations have a particular structure. A level up crossing is from to , a down crossing is from to . The level subcrossings that make up a level parent crossing consist of excursions (up–down and down–up pairs) followed by a direct crossing (down–down or up–up pairs), whose direction depends on the parent crossing: if the parent crossing is up, then the subcrossings end up–up; otherwise, they end down–down. Let be the length of , that is, the number of subcrossings of . The number of up and down subcrossings will be written and , respectively. Clearly, each of the first entries of comes in pairs, each pair being up–down or down–up. The last two components are either the pair up–up or down–down, depending on . Thus, given , we must have and , and conversely given .

Let be the space of possible orientations. That is, consists of some number of pairs, or , then a single pair or . Given an EBP process, for the offspring type distributions we write and , for . Let , and , then the mean offspring matrix is given by

To proceed we need to make some assumptions about . {assp} and for .

The first of these assumptions ensures that is strictly positive with dominant eigenvalue , and corresponding left eigenvector . The corresponding right eigenvector is . The second assumption is the usual condition for the normed limit of a supercritical Galton–Watson process to be nontrivial.

Theorem 2.1

For any offspring orientation distributions satisfying Assumption 2, there exists a corresponding continuous EBP process defined on .

{pf}

A version of this result can be found as Theorem 1 in J04 (), for particular orientation distributions.

Figure 2: Construction of , and , and the associated crossing tree (see the proof of Theorem 2.1). The subtree rooted at crossing (the 4th crossing of size ) corresponds to the tree inside the dashed box. In the notation of the proof of Theorem 2.1, for this subtree we have . If we go down one level in the subtree, corresponding to level of the original tree, then and count the number of up and down crossings at level 1 of the subtree. Similarly, and count the number of up and down crossings at level 2 of the subtree, and so on. This figure also illustrates other notation used in the proof of Theorem 2.1. For example, one has , since for , the the first crossing time of size corresponds to the fourth crossing time of size .

Step 1. We initially construct a single crossing from 0 to 1, with support . In step 2 we will extend the range to and the support to .  is obtained as the limit as , of a sequence of random walks with steps of size and duration . Put and , so that the coarsest scale is . Given we construct by replacing the th step of by a sequence of steps of size and duration . If , then the orientations of the subcrossings are distributed according to . For a given the are all mutually independent, and, given , is conditionally independent of all , for .

Denote the (random) time that hits 1 by

We define for all by linear interpolation, and set for all . The interpolated have continuous sample paths, and we will show that they converge uniformly on any finite interval, from which the continuity of the limit process follows. For any , let and

If , then set . By construction , for all and . The duration of the th level crossing of is .

A realization of , and is given in Figure 2, with the associated crossing tree.

We use a branching process result to establish that the crossing durations converge. When we defined the crossing tree (see Figure 1) we started with a sample path and then defined generations of crossings: taking the first crossing of size 1 as the root (level or generation 0), its subcrossings of size form the second generation (or level), its subcrossings of size form the third generation, and so on. Each crossing can be up or down, so our tree has two types of nodes. Here we are reversing that process. That is, we are growing a tree using a two-type Galton–Watson process, and from the tree, constructing a sample path. The offspring distributions for our tree are just . Given the tree at generation , we get an approximate sample path by taking a sequence of up and down steps of size and duration , with directions taken from the node types of the tree. We need to show that the sequence of sample paths, obtained as , converges.

Consider the subtree descending from crossing . Let and be the number of up and down crossings of size which are descended from the th crossing of size ; then is a two-type Galton–Watson process. From Athreya and Ney AN72 (), Section V.6, Theorems 1 and 2, we have that as , converges almost surely and in mean to , where is strictly positive, continuous and . Moreover, the distribution of depends only on , and for any fixed the are all independent. Finally, since , we have

Accordingly, let .

Take any , and . To establish the a.s. convergence of the processes , uniformly on compact intervals, we show that we can find a so that with probability ,

(1)

Given , let be such that

For any , the triangle inequality yields

since .

For any let be the smallest such that . As , a.s., so for any we can choose such that

and such that for all ,

which yields

Thus, given we can find such that for all , with probability at least ,

Now, since , , and in three steps can move at most distance , we have

Choosing large enough that , we see that (1) follows from (2). Sending and to shows that converges to some continuous limit process uniformly on all closed intervals , with probability . By construction, the duration of crossing is .

Step 2. Clearly the construction above can be used to generate any crossing from 0 to . Thus, to extend our construction from a single crossing to a process defined for all , we proceed by constructing a nested sequence of processes , such that is a crossing from 0 to , and the first level crossing of is precisely . To make this work, we just need to specify in a consistent manner.

Consider the orientation of the first crossing from 0 to for an EBP process. Let and ; then and are determined by , and

(3)

For , we see that equation (3) has fixed point . Moreover, the only doubly infinite sequence which satisfies (3) and remains in is given by for all . Given this, it follows that , and thus from Bayes’s theorem that and . If , then any is possible, but everything else goes through as before. In this case the are all the same, but may be of either type.

Construct as a crossing from 0 to 1 with probability [the fixed point of (3)], otherwise as a crossing from 0 to . Then, given , construct as follows: first, put with probability if , with probability otherwise; second, generate conditional on and ; third, use as the first level crossing of ; finally construct the remaining level crossings conditional on . Write for the limit of the . To complete our construction we just need to check that the process does not escape to in finite time. By construction, we have , where is strictly positive, continuous, and has a distribution depending only on the orientation . Thus for any , as .

Theorem 2.2

Let be the EBP constructed in Theorem 2.1; then, for each , conditioned on the crossing orientations , the crossing durations are all mutually independent, and is conditionally independent of all for . Also, , and the distribution of depends only on . Moreover, up to finite-dimensional distributions,  is the unique such EBP with offspring orientation distributions . That is, for any other EBP process with offspring orientation distributions and crossing durations as above, we have for any .

Accordingly, we call the Canonical EBP (CEBP) process with these offspring distributions.

We also observe that is discrete scale-invariant: let ; then for all ,

(4)

where denotes equality for finite-dimensional distributions. is known as the Hurst index.

{pf}

We retain the notation of Theorem 2.1.

For the process , the dependence structure of the crossing durations is clear from the construction.

To show uniqueness, let be some other EBP process with offspring orientation distributions , and crossing durations satisfying the conditions of the theorem statement. We will make use of the same notation for the crossing times, durations, orientations, etc. of as for , and rely on the context to distinguish them.

For an EBP, the finite joint distributions of the orientations are determined completely by , and thus are identical for and . For the crossing durations of , note that for any and , we have

(5)

where is such that is the index of the first level subcrossing of . Thus by the strong law of large numbers, sending ,

where the distribution of is completely determined by , and thus is the same for and .

Once we have the crossing orientations and the assumed dependence structure of the crossing durations, the crossing distributions (for up and down types) determine the joint distributions of the crossing times . Thus, for any and , and are identically distributed. Since any can be bracketed by a sequence of hitting times, and are identical up to finite-dimensional distributions.

That is discrete scale-invariant is a direct consequence of its construction, since simultaneously scaling the state space by and time space by does not change the distribution of . {remark} From H92 () it is clear that Brownian motion is an example of a CEBP process, where the offspring of any crossing consist of a geometric () number of excursions, each up–down or down–up with equal probability, followed by either an up–up or down–down direct crossing. That is,

where represents a combination of pairs, each either or . It follows that , independently of .

3 From CEBP to MEBP

In this section we construct Multifractal Embedded Branching processes (MEBP processes) as time changed CEBP processes.

Consider initially a single crossing of a CEBP , from 0 to . We constructed as the limit of a sequence of processes , which take steps of size and duration . The crossing tree gives the number of subcrossings of each crossing. If we add a weight of to each branch of the tree, then truncating the tree at level , the product of the weights down any line of descent is , which is the duration of any single crossing by . We generalize this by allowing the weights to be random, then defining the duration of a crossing to be the product of the random weights down the line of descent of the crossing. The resulting process, say, can be viewed as a time-change of , where the time-change is obtained from a multiplicative cascade defined on a (two-type) Galton–Watson tree.

As for CEBP, we will initially construct a single level 0 crossing of an MEBP, then extend the construction to . We will retain the notation of Section 2, but note that we will prefer the tree indexing scheme to the level/position indexing scheme in what follows. In particular, the number of level up and down subcrossings of node in level are denoted and , and, under Assumption 2, the almost sure limit and mean limit of is . The duration of crossing of the CEBP process is then .

We assign weight to the branch . may depend on , but conditioned on must be independent of other nodes that are not descendants of . For , write for the joint distribution of , conditioned on the crossing orientations . The weight attributed to node is

That is, is the product of all weights on the line of descent from the root down to node . We use the weights to define a measure, , on the boundary of the crossing tree. The measure on is then mapped to a measure on , with which we define a chronometer (a nondecreasing process) by . The MEBP process is then given by , where is the CEBP. The crossing trees of and have the same spatial structure, but have different crossing durations. In Figure 4 we plot a realization of an MEBP process and its associated CEBP.

The literature on multiplicative cascades is rather extensive. For the existence of limit random measures and the study of the properties of certain martingales defined on -ary trees, one can refer, for instance, to the works of Kahane and Peyrière KP76 (), Barral B99 (), Liu and Rouault LR00 () and Peyrière P00 (). For results on random cascades defined on Galton–Watson trees, see, for example, Liu L99 (), L00 (), and Burd and Waymire BW00 ().

To obtain the time-change process explicitly, the random measure defined on the boundary of the tree is mapped to then integrated. Note that this mapping, given explicitly in Section 3.2, differs from random partitions previously considered in the literature. The usual approach is to use a “splitting measure” to map the boundary of the tree to , then use the density of the cascade measure with respect to the splitting measure; see, for example, P77 (), P79 (), R03 (). Our approach can be thought of as using the “branching measure” instead of a splitting measure. A splitting measure is constructed by splitting the mass associated with a given node between its offspring, with no mass lost or gained. The branching measure allocates mass according to the number of offspring, and is only conserved in mean. We have taken the terminology of splitting and branching measures from L00 (), Example 1.3. A multifractal study of the measure we construct on is given in a forth-coming paper DHJ ().

3.1 The measure

To construct , we use a well-known correspondence between branching random walks and random cascades, in which the offspring of individual have types given by and displacements (relative to ) given by . For background on multitype branching random walks, we refer the reader to Kyprianou and Sani KS01 () and Biggins and Sani BS05 ().

Suppose and . Define

and for ,

Let , and write for the entry of the th power . Then it is straight forward to check that . If we take constant weights equal to , then , in the notation of Theorem 2.1.

Let be the largest eigenvalue of . We make the following assumptions about . {assp} We suppose that a.s., in an open neighborhood of 1, and .

In the case where the distribution of (and thus ) does not depend on , we assume in addition that

In the case where there is dependence on the crossing orientation (type), we suppose that for some , and

Note that if the weights are finite and strictly positive, then and from the previous section are just and , and from Assumption 2 we get for all . In the case where does not depend on , the BRW simplifies to a single-type process, and the condition simplifies to , which we recognize as a conservation of mass condition.

Left and right eigenvectors corresponding to will be denoted and , normed so that and . The following lemma is a direct consequence of Biggins and Kyprianou BK04 (), Theorem 7.1, and Biggins and Sani BS05 (), Theorem 4.

Lemma 3.1

Under Assumptions 2 and 3.1, converges almost surely to , for some random variable such that the distribution of depends only on , and . Moreover, for each , conditioned on the crossing orientations , , the are mutually independent, and is conditionally independent of for . For all nodes ,

(6)

Note that in the case where does not depend on , the right eigenvector .

We can now define the measure on . Recall and , so contains all the nodes on the boundary of the tree which have as an ancestor. We define . By Carathéodory’s extension theorem, we can uniquely extend to the sigma algebra generated by these cylinder sets.

3.2 The measure and time change

The measure is a mapping of from to . By analogy with -ary cascades, we call a Galton–Watson cascade measure on .

As above, let denote the th level passage time of the CEBP process , and put

Putting , this gives us for all . For arbitrary , let be such that for all . Noting that is a nonincreasing sequence, we define .

We can now define , and define the MEBP process (on ) as

Here we take , so that it is well defined, even if has jumps or flat spots.

Put . Then , so is the th level crossing time for , and the th level crossing duration. Note that if we take constant weights equal to , then and .

Lemma 3.2

Under Assumptions 2 and 3.1, and are continuous. That is, has neither jumps nor flat spots.

{pf}

To show that has no flat spots, it is enough to show that: {longlist}[(b)]