Über die Autorin bzw. den Autor

Paul Embrechts is Professor of Mathematics at the Swiss Federal Institute of Technology (ETHZ), Zurich, Switzerland. He is the author of numerous scientific papers on stochastic processes and their applications and the coauthor of the influential book on "Modelling of Extremal Events for Insurance and Finance". Makoto Maejima is Professor of Mathematics at Keio University, Yokohama, Japan. He has published extensively on selfsimilarity and stable processes.

Von der hinteren Coverseite

"Authoritative and written by leading experts, this book is a significant contribution to a growing field. Selfsimilar processes crop up in a wide range of subjects from finance to physics, so this book will have a correspondingly wide readership."--Chris Rogers, Bath University

"This is a timely book. Everybody is talking about scaling, and selfsimilar stochastic processes are the basic and the clearest examples of models with scaling. In applications from finance to communication networks, selfsimilar processes are believed to be important. Yet much of what is known about them is folklore; this book fills the void and gives reader access to some hard facts. And because this book requires only modest mathematical sophistication, it is accessible to a wide audience."--Gennady Samorodnitsky, Cornell University

Aus dem Klappentext

"Authoritative and written by leading experts, this book is a significant contribution to a growing field. Selfsimilar processes crop up in a wide range of subjects from finance to physics, so this book will have a correspondingly wide readership."--Chris Rogers, Bath University

"This is a timely book. Everybody is talking about scaling, and selfsimilar stochastic processes are the basic and the clearest examples of models with scaling. In applications from finance to communication networks, selfsimilar processes are believed to be important. Yet much of what is known about them is folklore; this book fills the void and gives reader access to some hard facts. And because this book requires only modest mathematical sophistication, it is accessible to a wide audience."--Gennady Samorodnitsky, Cornell University

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Selfsimilar Processes

PRINCETON UNIVERSITY PRESS

Contents

Preface..........................................................................................................ixChapter 1. Introduction..........................................................................................1Chapter 2. Some Historical Background............................................................................13Chapter 3. Selfsimilar Processes with Stationary Increments......................................................19Chapter 4. Fractional Brownian Motion............................................................................43Chapter 5. Selfsimilar Processes with Independent Increments.....................................................57Chapter 6. Sample Path Properties of Selfsimilar Stable Processes with Stationary Increments.....................63Chapter 7. Simulation of Selfsimilar Processes...................................................................67Chapter 8. Statistical Estimation................................................................................81Chapter 9. Extensions............................................................................................93References.......................................................................................................101Index............................................................................................................109

Chapter One

Selfsimilar processes are stochastic processes that are invariant in distribution under suitable scaling of time and space (see Definition 1.1.1).

It is well known that Brownian motion is selfsimilar (see Theorem 1.2.1). Fractional Brownian motion (see Section 1.3 and Chapter 4), which is a Gaussian selfsimilar process with stationary increments, was first discussed by Kolmogorov [Kol40]. The first paper giving a rigorous treatment of general selfsimilar processes is due to Lamperti [Lam62], where a fundamental limit theorem was proved (see Section 2.1). Later, the study of non-Gaussian selfsimilar processes with stationary increments was initiated by Taqqu [Taq75], who extended a non-Gaussian limit theorem by Rosenblatt [Ros61] (see Sections 2.3 and 3.4).

On the other hand, the works of Sinai [Sin76] and Dobrushin [Dob80] in the field of statistical physics, for instance, appeared around 1976 (see Section 2.2). It seems that similar problems were attacked independently in the fields of probability theory and statistical physics (see [Dob80]). The connection between these developments was made by Dobrushin. An early bibliographical guide is to be found in [Taq86].

1.1 DEFINITION OF SELFSIMILARITY

In the following, by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we denote equality of all joint distributions for R^d-valued stochastic processes {X(t), t ≥ 0} and {Y(t), t ≥ 0} defined on some probability space (Ω,F,P). Occasionally we simply write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes equality of the marginal distributions for fixed t. By [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we denote convergence of all joint distributions as n [right arrow] ∞, and by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the convergence in law of random variables {[xi]_n} to [xi]. L(X) stands for the law of a random variable X. The characteristic function of a probability distribution μ is denoted by μ(θ), θ [member of] R^d. For x [member of] R^d; |x| is the Euclidean norm of x and x' is the transposed vector of x.

Definition 1.1.1 An R^d-valued stochastic process {X(t), t ≥ 0} is said to be "selfsimilar" if for any a > 0, there exists b > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.1)

We say that {X(t), t ≥ 0} is stochastically continuous at t if for any ε > 0, lim_{h[right arrow]0} P{|X(t + h) - X(t)| > ε} = 0. We also say that {X(t), t ≥ 0} is trivial if X(t) is a constant almost surely for every t.

Theorem 1.1.1 [Lam62] If {X(t), t ≥ 0} is nontrivial, stochastically continuous at t = 0 and selfsimilar, then there exists a unique H ≥ 0 such that b in (1.1.1) can be expressed as b = a^H.

As there is some confusion about this result in the more applied literature, we prefer to give a proof. We start with an easy lemma.

Lemma 1.1.1 If X is a nonzero random variable in R^d, and if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with b₁, b₂ > 0, then b₁ = b₂.

Proof. Suppose b₁ [not equal to] b₂. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with some b [member of] (0,1). Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any n [member of] N, and, letting n [right arrow] ∞, we have X = 0 almost surely, which is a contradiction.

Proof of Theorem 1.1.1. Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If X(t) is nonzero for this t, then b₁ = b₂ by Lemma 1.1.1. By the nontriviality of {X(t)}, such a t exists. Thus b₁ = b₂, namely b in (1.1.1) is uniquely determined by a. We write b = b(a). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence we have b(aa') = b(a)b(a'). We next show the monotonicity of b(a). Suppose a < 1 and let n [right arrow] ∞ in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since X(aⁿ) tends to X(0) in probability by the stochastic continuity of {X(t)} at t = 0, we must have that b(a) ≤ 1. Since b(a₁/a_{2b(a1)/b(a2), if a1, a1, then b(a1) ≤ b(a2), and thus b(a) is nondecreasing. We have now concluded that b(a) is nondecreasing and satisfies}

b(aa') = b(a)b(a').

Thus b(a) = a^H for some unique constant H ≥ 0. A

We call H the exponent of selfsimilarity of the process {X(t), t ≥ 0}. We refer to such a process as H-selfsimilar (or H-ss, for short).

Property 1.1.1 If {X(t), t ≥ 0} is H-ss and H > 0, then X(0) = 0 almost surely.

Proof. By Definition 1.1.1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and it is enough to let a [right arrow] 0.

Property 1.1.1 does not hold when H = 0.

Example 1.1.1 [Kon84] Let {Y(s), s [member of] R} be a strictly stationary process, [xi] a random variable independent of {Y(s)}, and define {X(t), t ≤ 0} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For t > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

implying that {X(t), t ≤ 0} is 0-ss. However X(0) [not equal to] 0.

Actually we have the following for H = 0.

Theorem 1.1.2 Under the same assumptions of Theorem 1.1.1, H = 0 if and only if X(t) = X(0) almost surely for every t > 0.

Proof. The "if" part is trivial. For the "only if" part, by the property of 0-ss, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then for each a > 0, the joint distributions at t = 0 and t = s/a are the same:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence for any ε > 0,

P{jX(s) - X(0)| > ε} = P{|X(s/a) - X(0) > ε}:

The right-hand side of the above converges to 0 as a [right arrow] ∞, because of the stochastic continuity of the process at t = 0. Hence for each s > 0

P{|X(s) - X(0)| > ε} = 0; [for all]ε > 0;

so that X(s) = X(0) almost surely.

From the above considerations, it seems natural to consider only selfsimilar processes such that they are stochastically continuous at 0 and their exponents H are positive. Without further explicit mention, selfsimilarity will always be used in conjunction with H > 0 and stochastic continuity at 0.

1.2 BROWNIAN MOTION

An R^d-valued stochastic process {X(t), t ≤ 0} is said to have independent increments, if for any m ≤ 1 and for any partition 0 ≥ t₀, t₁ < ... < t_m, X(t₁) - X(t₀), ..., X(t_m) - X(t_m-1) are independent, and is said to have stationary increments, if any joint distribution of {X(t 1 h) - X(h), t ≤ 0} is independent of h ≤ 0. We will always use the term stationarity for the invariance of joint distributions under time shifts. Usually, this is referred to as strict stationarity. This is distinct from weak stationarity where time shift invariance is only required for the mean and covariance functions.

Definition 1.2.1 If an R^d-valued stochastic process {B(t), t ≤ 0} satisfies

(a) B (0) = 0 almost surely,

(b) it has independent and stationary increments,

(c) for each t > 0, B(t) has a Gaussian distribution with mean zero and covariance matrix tI (where I is the identity matrix), and

(d) its sample paths are continuous almost surely, then it is called (standard) Brownian motion.

Theorem 1.2.1 Brownian motion {B(t), t ≤ 0} is 1/2-ss.

Proof. It is enough to show that for every a > 0, {a^-1/2B(at)} is also Brownian motion. Conditions (a), (b) and (d) follow from the same conditions for {B(t)}. As to (c), Gaussianity and the mean zero property also follow from the properties of {B(t)}. Moreover, E[(a^-1/2B(at))(a^-1/2B(at))'] = tI, thus {a^-1/2B(at)} is Brownian motion.

Theorem 1.2.2 E[B(t)B(s)'] = min{t, s}I.

Proof. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.3 FRACTIONAL BROWNIAN MOTION

The following basic result for general selfsimilar processes with stationary increments leads to a natural definition of fractional Brownian motion.

Theorem 1.3.1 [Taq81] Let {X(t)} be real-valued H-selfsimilar with stationary increments and suppose that E[X(1)²] < ∞. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. By selfsimilarity and stationarity of the increments,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 1.3.1 Let 0 < H ≥ 1. A real-valued Gaussian process {B_H(t), t ≤ 0} is called "fractional Brownian motion" if E[B_H(t) = 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 1.3.1 It is known that the distribution of a Gaussian process is determined by its mean and covariance structure. Indeed, the distribution of a process is determined by all joint distributions and the density of a multidimensional Gaussian distribution is explicitly given through its mean and covariance matrix. Thus, the two conditions in Definition 1.3.1 determine a unique Gaussian process.

Theorem 1.3.2 {B_1/2(t)} is Brownian motion up to a multiplicative constant.

Proof. Equation (1.3.1) with H = 1/2 is the same as in Theorem 1.2.2, and it determines the covariance structure of Brownian motion as mentioned in Remark 1.3.1.

For the formulation of the next result, we need the notion of a Wiener integral. See Section 3.5 for a definition in a general setting.

Theorem 1.3.3 A fractional Brownian motion {B_H(t), t ≥ 0} is H-ss with stationary increments. When 0 < H < 1, it has a stochastic integral representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If H = 1, then B₁(t) = tB₁(1) almost surely. Fractional Brownian motion is unique in the sense that the class of all fractional Brownian motions coincides with that of all Gaussian selfsimilar processes with stationary increments. {B_H(t)} has independent increments if and only if H = 1/2.

Proof.

(i) Selfsimilarity. We have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since all processes here are mean zero Gaussian, this equality in covariance implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) Stationary increments. Again, it is enough to consider only covariances. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

concluding that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) For 0 < H < 1, the Wiener integral in (1.3.2) is well defined and a mean zero Gaussian random variable. Denote the integral in (1.3.2) by X(t). We then have by Theorem 3.5.1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, {X(t)} for 0 < H < 1 is fractional Brownian motion.

(iv) For the case H = 1, first note that because of (1.3.1), E[B₁(t)B₁ (s) = ts E[B₁(1)². Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that B₁(t) = tB₁(1) almost surely.

(v) For the uniqueness, first note that once {X(t)} is H-ss and has stationary increments, then by Theorem 1.3.1 above, it has the same covariance structure as in (1.3.1). Since {X(t)} is mean zero Gaussian, it is the same as {B_H(t)} in law.

(vi) If H = 1/2, then by Theorem 1.3.2, the process is Brownian motion. If {B_H(t)} has independent increments, then for 0 < s < t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The latter however only holds for H = 1/2.

Remark 1.3.2 Fractional Brownian motion is defined through (1.3.2) in [ManVNe68].

The integral representation of fractional Brownian motion in (1.3.2) is popular, but there is another useful representation through a Wiener integral over a finite interval.

Theorem 1.3.4 [NorValVir99, DecUst99] When 0 < H < 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and C is a normalizing constant. For 1/2 < H < 1, a slightly simpler expression for K(t; u) is possible:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.4 STABLE LÉVY PROCESSES

Definition 1.4.1 An R^d-valued stochastic process {X(t), t ≥ 0} is called a Lévy process if

(a) X(0) = 0 almost surely,

(b) it is stochastically continuous at any t ≥ 0,

(c) it has independent and stationary increments, and

(d) its sample paths are right-continuous and have left limits almost surely.

Remark 1.4.1 Excellent references on Lévy processes are [Ber96] and [Sat99]. For an edited volume on the topic with numerous examples, see [BarMikRes01].

Definition 1.4.2 A probability measure μ on R^d is called "strictly stable", if for any a > 0, there exists b > 0 such that [??](θ)^a = [??](bθ), [for all]θ [member of] R^d. In the following, we call such a μ just "stable". If μ is symmetric, it is called "symmetric stable".

Each stable distribution has a unique index as follows.

Theorem 1.4.1 [SamTaq94] If μ on R^d is stable, there exists a unique a [member of] (0, 2] such that b = a^1/a. Such a μ is referred to as a-stable. When a = 2, μ is a mean zero Gaussian probability measure.

In terms of independent and identically distributed random variables X, X₁, X₂, ... with probability distribution μ, strict stability means that for some [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all n.

Non-Gaussian stable distributions are sometimes called Lévy distributions by physicists (see [Tsa97]). The special case a = 1 is called Cauchy distribution (or Lorentz distribution by physicists). A significant difference between Gaussian distributions and non-Gaussian stable ones like the Cauchy distributions is that the latter have heavy tails, i.e. their variances are infinite. Such models were for a long time not accepted by physicists. More recently, the importance of modeling stochastic phenomena with heavy tailed processes is dramatically increasing in many fields. One important such heavy tail property is the following.

(Continues...)

Excerpted from Selfsimilar Processesby Paul Embrechts Makoto Maejima Copyright © 2002 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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