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ARTICLE IN PRESS

Stochastic Processes and their Applications ( ) –www.elsevier.com/locate/spa

On the asymptotic behaviour of Levy processes,Part I: Subexponential and exponential processes

J.M.P. Albin∗, Mattias Sunden

Mathematics, Chalmers University of Technology, 412 96 Gothenburg, Sweden

Received 20 May 2007; received in revised form 15 February 2008; accepted 16 February 2008

Abstract

We study tail probabilities of the suprema of Levy processes with subexponential or exponentialmarginal distributions over compact intervals. Several of the processes for which the asymptotics are studiedhere for the first time have recently become important to model financial time series. Hence our resultsshould be important, for example, in the assessment of financial risk.c© 2008 Elsevier B.V. All rights reserved.

MSC: primary 60E07; 60G51; 60G70; secondary 40E05; 44A10; 60F10

Keywords: CGMY process; Esscher transform; Exponential distribution; Extreme value theory; GH process; GZ process;Infinitely divisible distribution; Levy process; Long-tailed distribution; Semi-heavy-tailed distribution; Subexponentialdistribution

1. Introduction

In the past decade there has been a great interest to use Levy processes in mathematicalfinance, see, e.g., Schoutens [34] for a review. Most of the classes of Levy processes thatfeature here, such as generalized z processes (GZ), CGMY processes and generalized hyperbolicprocesses (GH) have univariate marginal distributions with semi-heavy tails.

Recall that a probability distribution is said to have a semi-heavy (upper) tail if it has aprobability density function (PDF) f such that

f (u) ∼ Cuρe−αu as u →∞ for some constants C, α > 0 and ρ ∈ R. (1.1)

∗ Corresponding author. Tel.: +46 31 7723512; fax: +46 31 7723508.E-mail addresses: [email protected] (J.M.P. Albin), [email protected] (M. Sunden).URLs: http://www.math.chalmers.se/∼palbin (J.M.P. Albin), http://www.math.chalmers.se/∼mattib (M. Sunden).

0304-4149/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.spa.2008.02.004

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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ARTICLE IN PRESS2 J.M.P. Albin, M. Sunden / Stochastic Processes and their Applications ( ) –

Owing to the exponential in (1.1) Levy processes with semi-heavy tails could also be calledexponential processes. This is a custom that we will adopt later.

We study the (upper) tail behaviour of suprema over compact intervals of Levy processes{ξ(t)}t≥0 with semi-heavy tails and other related behaviours of the tails of their marginaldistributions. More specifically, given a constant h > 0 we prove that

supt∈[0,h]

ξ(t) ∈ C ⇔ ξ(h) ∈ C (1.2)

for different classes of distributions (tail behaviours) C together with the implication

ξ(h) ∈ C ⇒ limu→∞

1P{ξ(h) > u}

P

{sup

0≤t≤hξ(t) > u

}= H exists. (1.3)

The classes of distributions C we are most interested in for (1.2) and (1.3) are the exponentialclasses L(α) and S(α) which are well-known from the literature, see Definition 2.5. In particular,L(α) includes all distributions with semi-heavy tails.

The implication (1.3) is known from Braverman and Samorodnitsky [14] for C = S(α).We prove (1.2) and (1.3) for C = L(α) under an additional technical condition that alwaysseems to be met in practice. In particular, as L(α) includes S(α) and distributions in S(α) satisfyour technical condition, we complete the result of Braverman and Samorodnitsky [14] with theequivalency (1.2) for C = S(α).

It turns out that semi-heavy-tailed Levy processes with ρ < −1 belong to S(α) whileprocesses with ρ ≥ −1 belong to L(α) \ S(α). Many of the above-mentioned specific Levyprocesses have ρ ≥ −1, so that (1.2) and (1.3) are established here for them for the first time. Wealso show that H > 1 when ρ < −1 unless ξ is a subordinator (a result which does not followfrom Braverman and Samorodnitsky [14]) and that H = 1 for ρ ≥ −1.

Distributions in the class L ≡ L(0) are called long-tailed. As these distributions have tailsthat decay slower than any exponential they can also be called subexponential. The relations (1.2)and (1.3) are known from Willekens [39] for C = L, see Theorem 4.1. We complete his resultsby establishing a partial converse to (1.3) for C = L.

As there is no converse to (1.3) for C = L(α) when α > 0 we consider the larger O-exponential class OL of Bengtsson [8] and Shimura and Watanabe [37], see Definition 2.5. ForC = OL we prove (1.2) together with the following version of (1.3):

ξ(h) ∈ OL⇒ lim supu→∞

1P{ξ(h) > u}

P

{sup

0≤t≤hξ(t) > u

}<∞. (1.4)

In addition, we establish a partial converse to (1.4).From a practical point of view the implication (1.3) for C = L(α) should be the most

interesting of our results. For example, an asset price process {S(t)}0≤t≤h such as a stock priceis often modelled by an exponential Levy model S(t) = eξ(t) where ξ is a Levy process. Thenthe risk that S falls below a low level ε is given by

P{

inf0≤t≤h

S(t) < ε

}= P

{sup

0≤t≤h−ξ(t) > − ln(ε)

}∼ H P{−ξ(h) > ln(1/ε)}

as ε ↓ 0 provided that −ξ ∈ C for a class C such that (1.3) holds.

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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ARTICLE IN PRESSJ.M.P. Albin, M. Sunden / Stochastic Processes and their Applications ( ) – 3

To establish (1.3) for the class L(α) we develop Tauberian results for infinitely divisibledistributions which should be of substantial interest in their own.

The paper is organized as follows: In Section 2 we review various classes of subexponentialand exponential distributions that feature in the paper.

In Section 3 we develop the mentioned Tauberian results. In particular, they show thatξ(h) ∈ L(α)\S(α) when ξ has a semi-heavy-tailed Levy measure such that ρ ≥ −1 in (1.1). Wealso express the tail behaviour of ξ(h) in terms of the Levy characteristic triple of the process.Note that on a less precise level than our Tauberian results, it is well-known that a Levy measurewith an exponential tail more or less corresponds to an infinitely divisible distribution with anexponential tail with the same exponent, see, e.g., Sato [33], Theorem 25.3.

In Section 4 we prove a partial converse to (1.3) for C = L. This converse is crucial for ourproof that H > 1 in (1.3) for C = S(α).

In Section 5 we prove (1.2) for C = OL as well as the implication (1.4) together with a partialconverse to that implication. The equivalency (1.2) for C = OL is crucial for our proof of (1.2)for C = L(α) in Section 6.

In Section 6 we prove (1.2) and (1.3) for C = L(α). The results from Section 3 are crucial toverify the hypothesis of these results.

In Section 7 we give applications to the process classes GZ, CGMY and GH.In the companion article Albin and Sunden [2] we study the tail behaviour of superexponential

Levy processes with lighter than exponential tails. This rich class of processes includes manyprocesses for which the limit H in (1.3) does not exist.

2. Subexponential and exponential distributions

In this section we review classes of probability distributions that feature in our work.

2.1. Subexponential distributions

Here we discuss distributions with tails that are heavier than exponential ones.The following classes of distributions L and S are well-known from the literature:

Definition 2.1. A cumulative distribution function (CDF) F belongs to the class of long-taileddistributions L if

limu→∞

1− F(u + x)

1− F(u)= 1 for x ∈ R.

A CDF F belongs to the class of subexponential distributions S if

limu→∞

1− F ? F(u)

1− F(u)= 2.

In Definition 2.1 ? means convolution, that is, F ? F(u) =∫R F(u − x)dF(x).

It turns out that S ( L (see Embrechts and Goldie [16], Section 3). It is easy to see that F ∈ Sif 1− F is regularly varying at infinity with a non-positive index:

Definition 2.2. A measurable function g > 0 is regularly varying at infinity with index % ∈ R,g ∈ R(%), if

limu→∞

g(ux)

g(u)= x% for x ∈ (0,∞).

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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ARTICLE IN PRESS4 J.M.P. Albin, M. Sunden / Stochastic Processes and their Applications ( ) –

A measurable function g > 0 is O-regularly varying at infinity, g ∈ OR, if

0 < lim infu→∞

g(ux)

g(u)≤ lim sup

u→∞

g(ux)

g(u)<∞ for x ∈ (0,∞).

Example 2.3. Given constants (parameters) x0, % > 0, the Pareto distribution F(x) = 1 −(x/x0)

−% for x ≥ x0 satisfies 1− F ∈ R(−%), so that F ∈ S ⊆ L.

For the class L we will need the following lemma, the proof of which is elementary:

Lemma 2.4. A CDF F belongs to the class L if and only if

lim infu→∞

1− F(u + x)

1− F(u)≥ 1 for some x > 0.

2.2. Exponential distributions

Here we discuss distributions with exponential tails.The following classes L(α) and S(α) are well-known from the literature. The class OL of

Bengtsson [8] and Shimura and Watanabe [37] is an exponential analogue of OR.

Definition 2.5. Given a constant α ≥ 0, a CDF F belongs to the class L(α) if

limu→∞

1− F(u + x)

1− F(u)= e−αx for x ∈ R. (2.1)

A CDF F belongs to the exponential class S(α) if F ∈ L(α) and

limu→∞

1− F ? F(u)

1− F(u)exists (and is finite). (2.2)

A CDF F belongs to the class of O-exponential distributions OL if

0 < lim infu→∞

1− F(u + x)

1− F(u)≤ lim sup

u→∞

1− F(u + x)

1− F(u)<∞ for x ∈ R.

Pitman [31], p. 338, argued that the class L should be called subexponential rather than S.By his logic the class L(α) should be called exponential instead of S(α). In fact, the exponentialdistribution itself belongs to L(α) \ S(α) by Example 2.11. This is the reason that we talk aboutexponential distributions when dealing with L(α).

Note thatL(0) = L, S(0) = S [as the limit (2.2) is 2 for S(0), see Embrechts and Goldie [17],Section 2], and S(α) ⊆ L(α) ⊆ OL. To illustrate how OL differs from ∪α≥0 L(α) we give thefollowing simple result which is proved in Appendix A.1:

Proposition 2.6. An absolutely continuous CDF F belongs to L(α) if and only if

F(u) = 1− exp{−

∫ u

−∞

(a(x)+ b(x))dx

}for u ∈ R, (2.3)

for some measurable functions a and b with a + b ≥ 0 such that

limx→∞

a(x) = α, limu→∞

∫ u

−∞

a(x)dx = ∞ and limu→∞

∫ u

−∞

b(x)dx exists. (2.4)

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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An absolutely continuous CDF F belongs to OL if and only if (2.3) holds for somemeasurable functions a and b with a + b ≥ 0 such that

lim supx→∞

|a(x)| <∞, lim infu→∞

∫ u

−∞

a(x)dx = ∞ and

lim supu→∞

∣∣∣∣∫ u

−∞

b(x)dx

∣∣∣∣ <∞. (2.5)

Example 2.7. Let a(x) = α 1[0,∞)(x) and b(x) = β cos(ex−1)1[0,∞)(x) in (2.3) where |β| ≤ α

and α > 0 are constants. Then we have F ∈ L(α) since limu→∞∫ u

0 cos(ex−1)dx exists. Instead,

if we take b(x) = β cos(x) 1[0,∞)(x), then (2.5) holds but (2.4) does not unless β = 0, so thatF ∈ OL \ L(α) for β 6= 0.

The following elementary result for the class OL corresponds to Lemma 2.4 for L:

Lemma 2.8. A CDF F belongs to the class OL if and only if

lim infu→∞

1− F(u + x)

1− F(u)> 0 for some x > 0.

2.3. Distributions with semi-heavy tails

In mathematical finance log increments of asset prices are often modelled to have semi-heavytails, see, e.g., Barndorff-Nielsen [6] and Schoutens [34].

The following simple corollary to Proposition 2.6 is proved in Appendix A.2:

Corollary 2.9. A CDF F is semi-heavy tailed satisfying (1.1) if and only if

F(u) = 1− exp{−

∫ u

−∞

c(x)dx

}for u ∈ R, (2.6)

for some measurable function c ≥ 0 that satisfies

limx→∞

c(x) = α and

limu→∞

∫ u

−∞

(c(x)− α 1[0,∞)(x)+

ρ

x1[1,∞)(x)

)dx = ln

(C

α

). (2.7)

If any of these equivalent conditions hold so that both of them hold, then F ∈ L(α).

Corollary 2.9 shows which distributions in Example 2.7 are semi-heavy:

Example 2.10. The distributions in Example 2.7 are semi-heavy only if β = 0 as they havec(x) = α + β cos(ex

− 1) and c(x) = α + β cos(x) for x ≥ 0 in (2.6).

By Corollary 2.9 semi-heavy-tailed CDF’s belong to L(α). However, by the followingexample a semi-heavy-tailed CDF belongs to S(α) only if ρ < −1 in (1.1).

Example 2.11. For a semi-heavy-tailed CDF F with ρ < −1, Pakes [29], Corollary 2.1 ii andLemma 2.3 (see also [30]), show that the limit (2.2) exists with value 2

∫R eαx dF(x) < ∞, so

that F ∈ S(α). For a semi-heavy-tailed CDF F with ρ ≥ −1 we have∫∞

1 eαx dF(x) = ∞.Hence Pakes [29], p. 411 (see also [30]), shows that the ratio in (2.2) goes to infinity as u →∞,so that F ∈ L(α) \ S(α).

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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3. Tauberian theorems for L(α) \ S(α)

In this section we first introduce some standard notation for Levy processes that will beused repeatedly from here on. Then we state and prove the Tauberian results mentioned in theIntroduction that are of fundamental importance in Sections 6 and 7.

An infinitely divisible CDF F is characterized by a characteristic triple (ν, m, s2) as∫R

eiθx dF(x) = exp{

iθm +∫R

(eiθx− 1− iθ1(−1,1)(x) x

)dν(x)−

θ2s2

2

}for θ ∈ R. (3.1)

Here ν is the Levy (Borel) measure on R that satisfies ν({0}) = 0 and∫R(1 ∧ |x |2)dν(x) < ∞

while m ∈ R and s2≥ 0 are constants. This triple is unique.

A Levy process is a continuous in probability and cadlag stochastic process {ξ(t)}t≥0 withξ(0) = 0 that has stationary and independent increments. As the marginal distribution of ξ(t)is infinitely divisible with characteristic triple (νt, mt, s2t) where (ν, m, s2) is the characteristictriple of ξ(1), the latter triple is called the characteristic triple of ξ .

The Tauberian results Theorem 3.3, Corollary 3.5 and Theorem 3.6 establish that if ξ is a Levyprocess with Levy measure ν that satisfies the condition (3.3), possibly together with additionalconditions, then we have the behaviour (3.4) the tails of the Levy process. The claim (3.4) is akey condition in Sections 6 and 7.

Our first Tauberian result is derived from Braverman [12], Lemma 5, together with thefollowing two results from the literature that are stated here for easy reference:

Theorem 3.1 (Embrechts, Goldie and Veraverbeke [18], Sgibnev [36]). Given a constant α ≥ 0and a Levy process ξ with Levy measure ν, we have

ν([1,∞) ∩ ·)

ν([1,∞))∈ S(α)⇔ ξ(t) ∈ S(α) for some t > 0⇔ ξ(t) ∈ S(α) for t > 0.

Moreover, if any of these conditions holds so that all of them hold, then we have

limu→∞

P{ξ(t) > u}

ν((u,∞))= t E{eαξ(t)

} for t > 0.

Theorem 3.2 (Albin [1], Pakes [29,30]). Let α ≥ 0 be a constant and ξ a Levy process withcharacteristic triple (ν, m, s2). Write ξ = ξ1+ξ2 where ξ1 and ξ2 are independent Levy processeswith characteristic triples (ν(·∩[1,∞)), 0, 0) and (ν(·∩(−∞, 1)), m, s2), respectively. We have

ν([1,∞) ∩ ·)

ν([1,∞))∈ L(α)⇒ ξ1(t) ∈ L(α) for t > 0⇒ ξ(t) ∈ L(α) for t > 0.

Moreover, if ξ1(t) ∈ L(α) for t > 0, then we have

P{ξ(t) > u} ∼ P{ξ1(t) > u}∫R

eαx dFξ2(t)(x) as u →∞ for t > 0. (3.2)

Here is our first Tauberian result. This result is due to Braverman [13], Theorem 1, but ourproof is very much shorter. See also Section 8 on priority:

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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ARTICLE IN PRESSJ.M.P. Albin, M. Sunden / Stochastic Processes and their Applications ( ) – 7

Theorem 3.3 (Braverman [13]). Let ξ be a Levy process that satisfies

F(·) ≡ν([1 ∨ ·,∞))

ν([1,∞))∈ L(α) for some α > 0 and lim

u→∞

1− F(u)

1− F ? F(u)= 0. (3.3)

Then it holds that

ξ(t) ∈ L(α) \ S(α) for t > 0 and limu→∞

P{ξ(s) > u}

P{ξ(t) > u}= 0 for 0 < s < t. (3.4)

Proof. Let ξ have triple (ν, m, s2) and write ξ = ξ1 + ξ2 where ξ1 and ξ2 are independentLevy processes with triples (ν(·∩ [1,∞)), 0, 0) and (ν(·∩ (−∞, 1)), m, s2), respectively. FromTheorem 3.2 we have ξ1(t), ξ(t) ∈ L(α). As ξ1 is a compound Poisson process with jumpCDF F , and as the second requirement of (3.3) means that F is light-tailed in the sense ofBraverman [12], Definition 1, it follows from Braverman [12], Lemma 5, that ξ1 satisfies thesecond requirement of (3.4). As tail probabilities for ξ1 are proportional to those of ξ by (3.2) wesee that the second requirement of (3.4) holds also for ξ . Hence ξ(t) 6∈ S(α) by Theorem 3.1,which finishes the proof of all claims of the theorem. �

In Corollary 3.5 and Theorem 3.6 below we specialize Theorem 3.3 to semi-heavy-tailed Levymeasures with ρ > −1 and ρ = −1, respectively (see Example 2.11). We also express the tailprobability P{ξ(t) > u} as u → ∞ in terms of the characteristic triple, which makes possiblymore explicit results in our applications in Section 7.

For the statement of Corollary 3.5 and Theorem 3.6, consider the moment generating function(MGF) of a Levy process ξ given by

φ(t, λ) = E{e−λξ(t)} =

(E{e−λξ(1)

}

)t= φ(1, λ)t

≡ φ(λ)t for t > 0 and λ ∈ R. (3.5)

Writing (ν, m, s2) for the characteristic triple of ξ , Sato [33], Theorem 25.17, shows that

φ(t, λ) <∞ for some t > 0⇔ φ(t, λ) <∞ for t > 0

∫R\(−1,1)

e−λx dν(x) <∞. (3.6)

We will need the following functions µ and V given by [cf. (3.1)]µ(λ) = −

φ′(λ)

φ(λ)=

∫R

(x e−λx

− x 1(−1,1)(x))

dν(x)+ m − λs2

V (λ) = −µ′(λ) =

∫R

x2e−λx dν(x)+ s2(3.7)

for λ ∈ R such that the definition makes sense. We will also need the inverse function µ←(u)

of µ which will be well-defined in all cases we encounter for u ∈ R sufficiently large as we willhave limλ↓−α µ(λ) = ∞ with µ′(λ) = −V (λ) < 0 (see below).

We will need the following well-known Tauberian result for compound Poisson processeswith semi-heavy Levy measures. Our version of the result is stated slightly differently than in theliterature to better suit our purposes, but it is easy to see that it is equivalent to the results in theliterature.

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Theorem 3.4 (Embrechts, Jensen, Maejima and Teugels [19], Homble and McCormick [23,24], Jensen [25]). For a compound Poisson process ξ with Levy measure ν that is absolutelycontinuous sufficiently far out to the right with

dν(u)

du∼ C uρe−αu as u →∞ for some constants C, α > 0 and ρ > −1, (3.8)

we have, with the notation (3.5) and (3.7),

P{ξ(t) > u} ∼euµ←(u/t)φ(µ←(u/t))t

α√

2π t V (µ←(u/t))as u →∞ for t > 0.

The following corollary to Theorems 3.3 and 3.4 addresses processes which have semi-heavy-tailed Levy measures with ρ > −1:

Corollary 3.5. Let ξ be a Levy process with characteristic triple (ν, m, s2) such that ν isabsolutely continuous sufficiently far out to the right and satisfies (3.8). Write ξ = ξ1 + ξ2where ξ1 and ξ2 are independent Levy processes with triplets (ν(· ∩ [1,∞)), 0, 0) and (ν(· ∩

(−∞, 1)), m, s2), respectively, and let φ1, µ1, µ←1 and V1 denote the quantities φ, µ, µ← andV in (3.5) and (3.7) calculated for the process ξ1 instead of ξ . Eq. (3.4) holds and

P{ξ(t) > u} ∼ E{eαξ2(t)}euµ←1 (u/t)φ1(µ

1 (u/t))t

α√

2π t V1(µ←

1 (u/t))as u →∞ for t > 0. (3.9)

Proof. As (3.3) holds by (3.8) and Example 2.11 we get (3.4) from Theorem 3.3. As ξ1 is acompound Poisson process that satisfies (3.8) Theorem 3.4 shows that

P{ξ1(t) > u} ∼euµ←1 (u/t)φ1(µ

1 (u/t))t

α√

2π t V1(µ←

1 (u/t))as u →∞ for t > 0.

Hence (3.9) follows from (3.2) [as ξ1(t) ∈ L(α) by the proof of Theorem 3.3]. �

By further calculations one can phrase (3.9) in terms of φ, µ, µ← and V instead of φ1, µ1,µ←1 and V1 in special cases: See the proof of (7.8) below for an example on this!

In Theorem 3.6 we address processes which have semi-heavy-tailed Levy measures withρ = −1. Now the Tauberian arguments have to be developed in detail from scratch as thereare no suitable results in the literature to take off from. However, the idea of the proof to useEsscher transforms to find tail probabilities is well-known and is also used in the companionpaper Albin and Sunden [2], where we give a bibliography.

Theorem 3.6. If ξ is a Levy process with Levy measure ν that is absolutely continuoussufficiently far out to the right and satisfies (3.8) with ρ = −1, then (3.4) holds. If in additionCt > 1 and

lim infx↓0

x

ln(1/x)

dν(x)

dx= ∞, (3.10)

then we have

P{ξ(t) > u} ∼

√C(Ct)Ct e−Ct

α Γ (Ct + 1)

euµ←(u/t)φ(µ←(u/t))t

σ(µ←(u/t))as u →∞. (3.11)

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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Proof. We get (3.4) from Theorem 3.3 as (3.3) holds by Example 2.11.To prove (3.11) consider the Esscher transform of ξ(t), which with the notation (3.5) and (3.7)

is a random variable Z t,λ with CDF defined by the change of measure

dFZt,λ(x) =e−λx dFξ(t)(x)

φ(λ)t for t > 0 and λ ∈ R such that φ(λ) <∞. (3.12)

By elementary calculations we have E{Z t,λ} = tµ(λ) and Var{Z t,λ} = tV (λ) ≡ t σ(λ)2.Further, (3.1) and (3.7) show that (Z t,λ − tµ(λ))/σ (λ) has CHF gt,λ given by

gt,λ(θ) =φ(λ− iθ/σ(λ))t

φ(λ)t e−itµ(λ)θ/σ(λ)

= exp{

t∫R

(cos

(θx

σ(λ)

)− 1

)e−λx dν(x)

+ it∫R

(sin(

θx

σ(λ)

)−

θx

σ(λ)

)e−λx dν(x)−

θ2s2t

2V (λ)

}. (3.13)

By (3.7), (3.8) and elementary calculations we have

µ(λ) ∼C

α + λand V (λ) ∼

C

(α + λ)2 as λ ↓ −α. (3.14)

Let ΓCt denote a gamma distributed random variable with PDF fΓCt (x) = xCt−1e−x/Γ (Ct) forx > 0. By (3.8), (3.14) and the change of variable x = σ(λ) y in (3.13) we have

gt,λ(θ) → exp{

Ct∫∞

0

cos(θy)− 1y

e−√

C ydy + iCt∫∞

0

sin(θy)− θy

ye−√

C ydy

}= exp

{−

Ct ln(1+ θ2/C)

2+ iCt

(arctan

(θ√

C

)−

θ√

C

)}=

1

(1− iθ/√

C)Cte−i√

C tθ

= E{

exp[iθ(ΓCt − Ct)/

√C]}≡ gt (θ) as λ ↓ −α, (3.15)

cf. Erdelyi, Magnus, Oberhettinger and Tricomi [21], Equations 4.2.1, 4.7.59 and 4.7.82.A key step in the proof is to prove that

lim supλ↓−α

∫|θ |>K

∣∣gt,λ(θ)∣∣ dθ → 0 as K →∞, (3.16)

as this shows that gt,λ is integrable for λ > −α small enough (since gt,λ is bounded by 1), sothat (Z t,λ − tµ(λ))/σ (λ) has a continuous PDF f(Zt,λ−tµ(λ))/σ (λ), which by (3.15) and (3.16)satisfies (using bounded convergence)

lim supλ↓−α

supx∈R

∣∣∣ f(Zt,λ−tµ(λ))/σ (λ)(x)− f(ΓCt−Ct)/√

C (x)

∣∣∣≤ lim sup

K→∞lim supλ↓−α

(∫|θ |≤K

∣∣gt,λ(θ)− gt (θ)∣∣ dθ +

∫|θ |>K

(|gt,λ(θ)| + |gt (θ)|

)dθ

)= 0. (3.17)

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From (3.12) and (3.17) together with (3.14), we get

fξ(t)(tµ(λ)− x/λ) =etλµ(λ)−xφ(λ)t

σ(λ)f(Zt,λ−tµ(λ))/σ (λ)

(−

x

λ σ(λ)

)∼

etλµ(λ)−xφ(λ)t

σ(λ)f(ΓCt−Ct)/

√C (0)

=

√C(Ct)Ct e−Ct

Γ (Ct + 1)

etλµ(λ)−xφ(λ)t

σ(λ)as λ ↓ −α. (3.18)

Note that supx∈R f(Zt,λ−tµ(λ))/σ (λ)(x) is bounded for λ > −α small enough by (3.17) [asCt > 1], so that we may integrate (3.18) using bounded convergence to obtain

P{ξ(t) > tµ(λ)} =

∫∞

0

fξ(t)(tµ(λ)− x/λ)

−λdx

∫∞

0

√C(Ct)Ct e−Ct

(−λ)Γ (Ct + 1)

etλµ(λ)−xφ(λ)t

σ(λ)dx

√C(Ct)Ct e−Ct

α Γ (Ct + 1)

etλµ(λ)φ(λ)t

σ(λ)as λ ↓ −α.

From this in turn we get (3.11) by a change of variable in the limit.In order to finish the proof of the theorem it remains to prove (3.16). To that end pick constants

ε ∈ (0, 1) and A ≥ 1 such that (1−ε)5Ct ≥ 1 and ε tdν(x)/dx ≥ 16 ln(1/x)/x for x ∈ (0, 1/A),cf. (3.10). As 1− cos(x) ≥ 1

4 x2 for |x | ≤ 1 we have [as e−λx≥ 1 for x ≥ 0]

ε t∫R

(1− cos

(θx

σ(λ)

))e−λx dν(x) ≥ ε t

∫ σ(λ)/|θ |

0

x2θ2

4 V (λ)dν(x)

∫ σ(λ)/|θ |

0

4 θ2

V (λ)x ln(1/x)dx

= 1+ ln(

θ2

V (λ)

)for |θ | ≥ Aσ(λ)

and λ > −α small enough. As (3.8) shows that dν(x)/dx ≥ (1 − ε) C e−αx/x for x ≥ B, forsome constant B > 0 large enough to make B + 2π/A ≤ B/(1− ε), we further have

(1− ε) t∫R

(1− cos

(θx

σ(λ)

))e−λx dν(x)

≥ (1− ε)2Ct∞∑

k=0

∫ B/σ(λ)+2π(k+1)/|θ |

B/σ(λ)+2πk/|θ |

(1− cos(θx))e−(α+λ)σ(λ)x

xdx

≥ (1− ε)2Ct∞∑

k=0

|θ |

e−(α+λ)σ(λ)(B/σ(λ)+2π(k+1)/|θ |)

B/σ(λ)+ 2π(k + 1)/|θ |

≥ (1− ε)2Ct∫∞

B/σ(λ)

e−(α+λ)σ(λ)(x+2π/(Aσ(λ)))

x + 2π/(Aσ(λ))dx

≥ (1− ε)2Ct e−2π(α+λ)/A B

B + 2π/A

∫∞

B/σ(λ)

e−(α+λ)σ(λ)x

xdx

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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≥ (1− ε)4Ct∫∞

B(α+λ)

e−x

xdx

≥ (1− ε)5Ct ln(

1B(α + λ)

)for |θ | ≥ Aσ(λ)

and λ > −α small enough, where the last inequality is an elementary calculation.By the estimates of the previous paragraph together with (3.13) and (3.14) we obtain∫

|θ |≥Aσ(λ)

|gt,λ(θ)|dθ =

∫|θ |≥Aσ(λ)

exp{−t∫R

(1− cos

(θx

σ(λ)

))e−λx dν(x)

}dθ

∫|θ |≥Aσ(λ)

e(1−ε)5Ctγ (B(α + λ))(1−ε)5Ct V (λ)

θ2 dθ

= e(1−ε)5Ctγ 2(B(α + λ))(1−ε)5Ctσ(λ)

A→ 0 as λ ↓ −α. (3.19)

Moreover, we have, using Erdelyi, Magnus, Oberhettinger and Tricomi [21], Eq. 4.7.59, togetherwith the inequality 1− cos(x) ≤ x2/2 and (3.14),

t∫R

(1− cos

(θx

σ(λ)

))e−λx dν(x)

≥ (1− ε) Ct∫∞

B

(1− cos

(θx

σ(λ)

))e−(α+λ)x

xdx

≥ (1− ε) Ct∫∞

0

(1− cos

(θx

σ(λ)

))e−(α+λ)x

xdx − Ct

∫ B

0

(1− cos

(θx

σ(λ)

))1x

dx

≥(1− ε) Ct

2ln(

1+θ2

(α + λ)2V (λ)

)− Ct

∫ B

0

θ2x

2 V (λ)dx

≥(1− ε) Ct

2ln(

θ2

2C

)− Ct

A2 B2

4for |θ | ≤ Aσ(λ) and λ > −α small enough.

Hence we have (for λ > −α small enough)∫K≤|θ |≤Aσ(λ)

|gt,λ(θ)|dθ ≤ 2∫∞

Kexp

{Ct

A2 B2

4−

(1− ε) Ct

2ln(

θ2

2C

)}dθ

= 2 exp{

CtA2 B2

4

}(2C)(1−ε)Ct/2

(1− ε)Ct − 1K 1−(1−ε)Ct ,

which goes to 0 as K →∞ since (1− ε) Ct > 1. Recalling (3.19) this gives (3.16). �

4. Subexponential Levy processes

The following result we will later extend from long-tailed processes to exponential ones.

Theorem 4.1 (Berman [9], Marcus [28], Willekens [39]). For a Levy process ξ we have

ξ(h) ∈ L⇔ supt∈[0,h]

ξ(t) ∈ L.

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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Moreover, if any of these memberships holds so that both of them hold, then

limu→∞

1P{ξ(h) > u}

P

{sup

t∈[0,h]ξ(t) > u

}= 1. (4.1)

The following simple converse to Theorem 4.1 is new and will be used in Section 6 to provethat H > 1 in (1.3) for C = S(α).

Theorem 4.2. For a Levy process ξ that satisfies (4.1), but is not a subordinator, one of thefollowing two conditions holds:

1. ξ(h) ∈ L;2.

lim infu→∞

P{ξ(t) > u}

P{ξ(h) > u}= 0 for all t ∈ (0, h).

Proof. Let (4.1) hold and assume that the liminf in Condition 2 takes a value `(t) > 0 for somet ∈ (0, h). To show that Condition 1 holds note that

0 = limu→∞

1P{ξ(h) > u}

(P

{sup

s∈[0,h]ξ(s) > u

}− P{ξ(h) > u}

)

≥ lim supu→∞

P{ξ(h) ≤ u, ξ(t) > u}

P{ξ(h) > u}

≥ P{ξ(h − t) ≤ −ε} lim supu→∞

P{u < ξ(t) ≤ u + ε}

P{ξ(h) > u}

= `(t) P{ξ(h − t) ≤ −ε} lim supu→∞

P{u < ξ(t) ≤ u + ε}

P{ξ(t) > u}for ε > 0. (4.2)

As ξ is not a subordinator we have P{ξ(h− t) ≤ −ε} > 0 for ε small enough, see, e.g., Sato [33],Theorem 24.7. Therefore (4.2) shows that

lim infu→∞

P{ξ(t) > u + ε}

P{ξ(t) > u}= 1.

Hence ξ(t) ∈ L by Lemma 2.4, so that ξ(h) ∈ L by the existence of the limit `(t). �

While subordination of ξ implies (4.1), as does Condition 1 of Theorem 4.2 by Theorem 4.1,Condition 2 does not imply (4.1) as is exemplified by Brownian motion.

5. O-exponential Levy processes

In this section we extend Theorems 4.1 and 4.2 from the class L to OL. The extension ofTheorem 4.1 will be used in Section 6 in the proof of (1.2) and (1.3) for C = L(α).

Theorem 5.1. For a Levy process ξ we have

ξ(h) ∈ OL⇔ supt∈[0,h]

ξ(t) ∈ OL.

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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Moreover, if any of these memberships holds so that both of them hold, then

lim supu→∞

1P{ξ(h) > u}

P

{sup

t∈[0,h]ξ(t) > u

}<∞. (5.1)

Proof. The fact that ξ(h) ∈ OL implies supt∈[0,h] ξ(t) ∈ OL and (5.1) follows as

P{ξ(h) > x} ≤ P

{sup

t∈[0,h]ξ(t) > x

}≤

P{ξ(h) > x − 1}P{ inf

t∈[0,h]ξ(t) > −1}

≤C P{ξ(h) > x}

P{ inft∈[0,h]

ξ(t) > −1}

for x large enough for some constant C > 0, where the middle inequality follows from Sato [33],Remark 45.9. Conversely, if supt∈[0,h] ξ(t) ∈ OL, then by the same inequality

lim infu→∞

P{ξ(h) > u + x}

P{ξ(h) > u}

≥ lim infu→∞

P{

inft∈[0,h]

ξ(t) > −1}

P

{sup

t∈[0,h]ξ(t) > u + x + 1

}/P

{sup

t∈[0,h]ξ(t) > u

}> 0

for x > 0, so that ξ(h) ∈ OL by Lemma 2.8. �

Theorem 5.2. For a Levy process ξ such that −ξ is not a subordinator and (5.1) holds one ofthe following two conditions holds:

1. ξ(h) ∈ OL;2.

lim infu→∞

P{ξ(t) > u}

P{ξ(h) > u}= 0 for t ∈ (0, h).

Proof. If (5.1) holds and the liminf in Condition 2 equals `(t) > 0 for a t ∈ (0, h), then

∞ > lim supu→∞

1P{ξ(h) > u}

P

{sup

t∈[0,h]ξ(t) > u

}

≥ P{ξ(h − t) ≥ ε} lim supu→∞

P{ξ(t) > u − ε}

P{ξ(h) > u}

≥ `(t)P{ξ(h − t) ≥ ε} lim supu→∞

P{ξ(h) > u − ε}

P{ξ(h) > u}for ε > 0.

As −ξ is not a subordinator we get P{ξ(h − t) ≥ ε} > 0 for ε > 0 small enough as in the proofof Theorem 4.2. Hence we see that ξ(h) ∈ OL using Lemma 2.8. �

6. Exponential Levy processes

For Levy processes in S(α) Braverman and Samorodnitsky [14], Theorem 3.1, proved that

limu→∞

1P{ξ(h) > u}

P

{sup

t∈[0,h]ξ(t) > u

}= H exists with value H ∈ [1,∞). (6.1)

Although Braverman [11], Theorem 2.1, expresses H in terms of the characteristic triple, he alsonotes that the expression typically cannot be evaluated except for subordinators.

Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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The next theorem extends Theorem 4.1 from L to L(α) as well as (6.1) from S(α) to L(α)

assuming (6.2) given below. It seems that specific processes in L(α) always satisfy (6.2) (seeSections 3 and 7), making our result very useful in practice, while we are unsure of the realtheoretical significance of (6.2). Our proof is quite short and transparent while in the literaturealready proofs of (6.1) for S(α) are long and difficult.

Theorem 6.1. For a constant α ≥ 0 and a Levy process ξ such that

L(t) = limu→∞

P{ξ(t) > u}

P{ξ(h) > u}exists for t ∈ (0, h) (6.2)

we have

ξ(h) ∈ L(α)⇔ supt∈[0,h]

ξ(t) ∈ L(α) for α ≥ 0.

Moreover, if any of these memberships holds so that both of them hold, then (6.1) holds. In thatcase we have H = 1 if L(t) = 0 for t ∈ (0, h).

Proof. Assume that ξ(h) ∈ L(α). Note that for each t ∈ (0, h) with L(t) > 0 we haveξ(t) ∈ L(α) by (6.2). This in turn by inspection of (2.1) means that

limu→∞

P{ξ(t)− u > x | ξ(t) > u} = e−αx for x ≥ 0. (6.3)

Letting η be an exp(α) distributed random variable that is independent of ξ (6.3) gives

lim infu→∞

1P{ξ(h) > u}

P

{sup

t∈[0,h]ξ(t) > u

}

≥ lim supa↓0

lim infu→∞

1P{ξ(h) > u}

P{

maxk=0,...,bh/ac

ξ(h − ka) > u

}

= lim supa↓0

lim infu→∞

bh/ac∑k=0

P{ξ(h − ka) > u}

P{ξ(h) > u}P

{k−1⋂`=0

{ξ(h − `a) ≤ u}

∣∣∣∣∣ ξ(h − ka) > u

}

= lim supa↓0

bh/ac∑k=0

L(h − ka) P

{k−1⋂`=0

{ξ((k − `)a)+ η ≤ 0}

}(6.4)

(where⋂−1`=0 is the empty intersection, that is, the whole sample space). For a matching upper

bound, note that the strong Markov property gives

P

{sup

t∈[0,h]ξ(t) > u + x

}

≤ P{

maxk=0,...,bh/ac

ξ(h − ka) > u

}+ P

{sup

t∈[0,h]ξ(t) > u + x

}P{

inft∈[0,a]

ξ(t) < −x

}for x > 0.

From this together with (6.4) and the fact that ξ(h) ∈ L(α) we get

lim supu→∞

1P{ξ(h) > u}

P

{sup

t∈[0,h]ξ(t) > u

}Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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= lim supx↓0

lim supu→∞

1P{ξ(h) > u + x}

P

{sup

t∈[0,h]ξ(t) > u + x

}

≤ lim supx↓0

lim infa↓0

lim supu→∞

eαx

P{ξ(h) > u}

×P{

maxk=0,...,bh/ac

ξ(h − ka) > u

}/P{

inft∈[0,a]

ξ(t) ≥ −x

}

≤ lim infa↓0

bh/ac∑k=0

L(h − ka) P

{k−1⋂`=0

{ξ((k − `)a)+ η ≤ 0}

}. (6.5)

From (6.4) together with (6.5) we conclude that (6.1) holds with

H = lima↓0

bh/ac∑k=0

L(h − ka) P

{k−1⋂`=0

{ξ((k − `)a)+ η ≤ 0}

}. (6.6)

Here H ≥ 1 by (6.1) with H = 1 if L(t) = 0 for t ∈ (0, h) by (6.6), while H < ∞ byTheorem 5.1.

Conversely, assume that supt∈[0,h] ξ(t) ∈ L(α). To finish the proof we have to show thatξ(h) ∈ L(α). Assume that α > 0 as we are done otherwise by Theorem 4.1. Observe that it isenough to show that given any x ≥ 0 we have

limn→∞

P{ξ(h) > un + x}

P{ξ(h) > un}= e−αx (6.7)

for any sequence un → ∞ as n → ∞ such that the limit (6.7) really exists. This is so becausethe ratio in (6.7) is bounded so that every subsequence of that ratio has a further subsequencethat converges to the limit e−αx . It follows that (2.1) holds for x ≥ 0, which in turn gives (2.1) ingeneral by an elementary argument.

Consider the distributions supported on [0,∞) with CDF given by

Fn(x) = P{ξ(h) ≤ un + x | ξ(h) > un} = 1−P{ξ(h) > un + x}

P{ξ(h) > un}for x ≥ 0.

For suitable constants N ∈ N and C, ε > 0, Theorem 5.1 gives [use (5.1) in the first step]

lim supx→∞

supn≥N

(1− Fn(x))

≤ lim supx→∞

C supn≥N

P

{sup

t∈[0,h]ξ(t) > un + x

}/P

{sup

t∈[0,h]ξ(t) > un

}

≤ lim supx→∞

C supn≥N

bxc∏k=1

P

{sup

t∈[0,h]ξ(t) > un + k

}

P

{sup

t∈[0,h]ξ(t) > un + k − 1

}

≤ lim supx→∞

C supn≥N

bxc∏k=1

(1+ ε) e−α= 0.

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Hence the sequence {Fn}∞

n=1 is tight in the sense of weak convergence. Therefore Prohorov’s

theorem shows that there exists a weakly convergent subsequence Fnk

d→ F .

Letting η be a random variable with CDF F that is independent of ξ (6.4) gives

lim infk→∞

1P{ξ(h) > unk }

P

{sup

t∈[0,h]ξ(t) > unk

}

≥ lim supa↓0

bh/ac∑k=0

L(h − ka) P

{k−1⋂`=0

{ξ((k − `)a)+ η ≤ 0}

}.

Using that supt∈[0,h] ξ(t) ∈ L(α) we further get the following version of (6.5):

lim supk→∞

1P{ξ(h) > unk }

P

{sup

t∈[0,h]ξ(t) > unk

}

= lim supx↓0

lim supk→∞

eαx

P{ξ(h) > unk }P

{sup

t∈[0,h]ξ(t) > unk + x

}

≤ lim infa↓0

bh/ac∑k=0

L(h − ka) P

{k−1⋂`=0

{ξ((k − `)a)+ η ≤ 0}

}.

With the notation (6.6) we thus obtain the following version of (6.1):

P

{sup

t∈[0,h]ξ(t) > unk

}∼ H P{ξ(h) > unk } as k →∞

with H ∈ [1,∞) as before. This gives the required (6.7) since supt∈[0,h] ξ(t) ∈ L(α). �

We immediately get the following powerful corollary to Theorem 6.1. See also Braverman[13], Theorem 2, and Section 8 below on priority issues:

Corollary 6.2. For a Levy process ξ satisfying (3.4) we have (6.1) with H = 1.

We now complete the result (6.1) of Braverman [11] and Braverman and Samorodnitsky [14]for S(α) to a result in the fashion of Theorem 4.1. We also show that H > 1 unless ξ is asubordinator which does not follow from the mentioned literature.

Corollary 6.3. For a constant α ≥ 0 and a Levy process ξ we have

ν([1,∞) ∩ ·)

ν([1,∞))∈ S(α)⇔ ξ(h) ∈ S(α)⇔ sup

t∈[0,h]ξ(t) ∈ S(α) and (6.2).

Moreover, if any of these memberships holds so that all of them hold, then (6.1) holds. In thatcase we have H > 1 unless α = 0 or ξ is a subordinator.

Proof. The left equivalency in the corollary follows from Theorem 3.1, as does the factthat ξ(h) ∈ S(α) implies (6.2). Hence Theorem 6.1 shows that ξ(h) ∈ S(α) also impliessupt∈[0,h] ξ(t) ∈ S(α) as (6.1) holds. Conversely, the fact that supt∈[0,h] ξ(t) ∈ S(α) and (6.2)imply ξ(h) ∈ S(α) follows from Theorem 6.1 alone, again as (6.1) holds.

As any of the equivalent statements in the corollary implies that (6.2) holds with ξ(h) ∈ L(α),Theorem 6.1 shows that they also imply (6.1).

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If α > 0, then Condition 1 of Theorem 4.2 fails. As ξ(h) ∈ S(α) implies that the limit in(6.2) is strictly positive, by Theorem 3.1, also Condition 2 of Theorem 4.2 fails. Since ξ is not asubordinator Theorem 4.2 thus shows that

lim supu→∞

1P{ξ(h) > u}

P

{sup

t∈[0,h]ξ(t) > u

}> 1,

which combines with (6.1) to show that H > 1. �

7. Applications

We now consider applications of our results to GZ, CGMY and GH processes.

7.1. GZ processes

The GZ process was introduced by Grigelionis [22], thereby generalizing the z processes ofPrentice [32]. The GZ process is a Levy process with characteristic triple(

dν(x)

dx, m, s2

)=

(2δ

|x |(1− e−2π |x |/ζ )

(e2πβ1x 1(−∞,0)(x)+ e−2πβ2x 1(0,∞)(x)

), m, 0

), (7.1)

where β1, β2, δ, ζ > 0 and m ∈ R are parameters.

Theorem 7.1. For a GZ Levy process (6.1) holds with H = 1. If in addition h > 1/(2δ), thenwe have

P{ξ(h) > u} ∼e(A−2δγ ) h

2πβ2Γ (2δh)u2δh−1e−2πβ2u as u →∞, (7.2)

where γ is Euler’s constant and

A = 2πβ2m +∫R

(e2πβ2x 1(−∞,1)(x)− 1− 2πβ21(−1,1)(x) x

)dν(x)

+

∫∞

1

(e2πβ2x dν(x)

dx−

x

)dx .

Proof. By inspection of (7.1) we see that (3.8) holds with C = 2δ, ρ = −1 and α = 2πβ2.Hence (3.4) holds by Theorem 3.6, so that Corollary 6.2 gives (6.1) with H = 1.

If h > 1/(2δ), then the hypothesis of the second part of Theorem 3.6 holds, as (3.10)holds by inspection of (7.1). Hence (3.11) applies. By (3.14) we have µ←(u) + 2πβ2 ∼ 2δ/uas u → ∞, so that by dominated convergence [note that µ←(u/h) + 2πβ2 ≥ 0 and(µ←(u/h)+ 2πβ2) u/(2hδ) ≥ 1/2 for u large enough] and a change of variable∫

1e−µ←(u/h) x dν(x)

=

∫∞

1e−(µ←(u/h)+2πβ2) x

(e2πβ2x dν(x)

dx−

x

)dx + 2δ

∫∞

1e−(µ←(u/h)+2πβ2) x dx

x

=

∫∞

1

(e2πβ2x dν(x)

dx−

x

)dx + 2δ

∫∞

2hδ/u

e−x

xdx + o(1) as u →∞. (7.3)

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Using (3.1) together with (3.11), (3.14) and (7.3), elementary calculations give

P{ξ(h) > u} ∼(2δh)2δheAh

2πβ2Γ (2δh)

e−2πβ2u

uexp

{2δh

∫∞

2δh/u

e−x

xdx

}∼

(2δh)2δheAh

2πβ2Γ (2δh)

e−2πβ2u

u

(e−γ u

2δh

)2δh

as u →∞,

which establishes (7.2): Here the last asymptotic relation follows from, e.g., Erdelyi, Magnus,Oberhettinger and Tricomi [20], Equations 6.9.25 and 6.7.13. �

Example 7.2. The Meixner process of Schoutens and Teugels [35] is a GZ process with β1, β2 >

0 and β1 + β2 = 1. Thus it satisfies (6.1) with H = 1. Further, (7.2) holds when h > 1/(2δ).

Remark 7.3. According to Barndorff-Nielsen, Kent and Sørensen [7], Theorem 5.2, if for aconstant α > 0 and a PDF f the function

Fk(x) =

∫ x

0ykeαy f (y)dy, x > 0,

has an ultimately monotone derivative with a Laplace transform that satisfies∫∞

0xke(α+s)x f (x)dx ∼ C Γ (k − ρ)(−s)−(k+ρ+1) as s ↑ 0, (7.4)

for some constants C > 0, k ∈ N and ρ > −k−1, then f satisfies (1.1). However, in general onecannot tell whether Fk has an ultimately monotone derivative just by inspection of the Laplacetransform. And should such additional information on f be available the Tauberian result shouldtypically not be needed anyway.

For example, Grigelionis [22], Corollary 1, deduces that

fξ(t)(u) ∼

(2π Γ (β1 + β2)

ζ Γ (β1)Γ (β2)

)2δt u2δt−1

Γ (2δt)exp

{−

2πβ2(u − mt)

ζ

}as u →∞

for GZ processes from information like (7.4) only, with the property that Fk has a monotonederivative waived as “standard calculations”: We find this argument incomplete!

7.2. CGMY processes

The CGMY Levy process of Carr, Geman, Madan and Yor [15] has characteristic triple(dν(x)

dx, m, s2

)=

(C−(−x)−1−Y−eGx 1(−∞,0)(x)+ C+x−1−Y+e−Mx 1(0,∞)(x), m, 0

), (7.5)

where C−, C+, G, M > 0, Y−, Y+ < 2 and m ∈ R are parameters.

Theorem 7.4. For a CGMY process (6.1) holds. Further, we have H > 1 and

P{ξ(h) > u} ∼C+h

Mexp

{hMm + h

∫R

(eMx− 1− M1(−1,1)(x) x

)dν(x)

}e−Mu

u1+Y+(7.6)

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as u →∞ for Y+ > 0, while H = 1 and

P{ξ(h) > u} ∼MC+h−1eBh

Γ (C+h)uC+h−1e−Mu (7.7)

as u →∞ for Y+ = 0, where γ is Euler’s constant and

B = Mm +∫R

(eMx 1(−∞,0)(x)− 1− M1(−1,1)(x) x

)dν(x),

and H = 1 and

P{ξ(h) > u} ∼(C+hΓ (1− Y+))−1/(2(1−Y+))

M√

2π(1− Y+) u(1−Y+/2)/(1−Y+)

× exp

{−Mu +

(1+ 1/Γ (1− Y+)) u−Y+/(1−Y+)

(C+hΓ (1− Y+))−1/(1−Y+)+ Bh

}(7.8)

as u →∞ for Y+ < 0.

Proof. For Y+ > 0, (7.5) and Example 2.11 show that ν([1,∞)∩ ·)/ν([1,∞)) ∈ S(M), so thatCorollary 6.3 gives ξ(h) ∈ S(M) and (6.1) with H > 1 (as ξ is not a subordinator). Further, weget (7.6) by insertion of (7.5) in the second part of Theorem 3.1.

For Y+ = 0, (7.5) shows that (3.8) holds with ρ = −1, so that Theorem 3.6 gives (3.4).Hence (6.1) holds with H = 1 by Corollary 6.2. To prove (7.7), write ξ = ξ1 + ξ2 where ξ1and ξ2 are independent Levy processes with triplets (ν((0,∞) ∩ ·), C+(1 − e−M )/M, 0) and(ν((−∞, 0) ∩ ·), m − C+(1− e−M )/M, 0), respectively. Then ξ1 is a gamma subordinator, see,e.g., Schoutens [34], Section 5.3.3, and satisfies

P{ξ1(h) > u} ∼MC+h−1

Γ (C+h)uC+h−1e−Mu as u →∞, (7.9)

so that ξ1(h) ∈ L(M). As E{eβξ2(h)} < ∞ for β ≥ 0 [recall (3.6)] it follows from Pakes [29],

Lemma 2.1 (see also [30]), that [cf. (3.2)]

P{ξ(h) > u} ∼ P{ξ1(h) > u}E{eMξ2(h)} as u →∞. (7.10)

Inserting (7.9) and calculating the MGF in (7.10) using (3.1) and (7.5) we get (7.7).For Y+ < 0, (7.5) shows that (3.8) holds with ρ = −1 − Y+ > −1, α = M and C = C+,

so that Corollary 3.5 gives (3.4). Hence (6.1) holds with H = 1 by Corollary 6.2, with theasymptotic behaviour of P{ξ(h) > u} as u → ∞ in (6.1) given by (3.9), by Corollary 3.5. Toevaluate (3.9) we note that (3.7) and (7.5) give

µ1(λ) = C+

∫∞

1x−Y+e−(M+λ)x dx = C+Γ (1− Y+)(M + λ)Y+−1

−C+

1− Y++ o(1)

V1(λ) = C+

∫∞

1x1−Y+e−(M+λ)x dx ∼ C+Γ (2− Y+)(M + λ)Y+−2

as λ ↓ −M . It follows that

µ←1 (u) = −M +

(u − C+/(1− Y+)

C+Γ (1− Y+)

)1/(Y+−1)

+ o(

u1/(Y+−1)−1)

as u →∞,

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so that

uµ←1 (u/h) ∼ −Mu + (C+h Γ (1− Y+))1/(1−Y+)uY+/(Y+−1)

V1(µ←

1 (u/h)) ∼ C+Γ (2− Y+)(C+h Γ (1− Y+))(Y+−2)/(1−Y+)u(Y+−2)/(Y+−1)

= (1− Y+) h−1(C+h Γ (1− Y+))−1/(1−Y+)u(Y+−2)/(Y+−1)

as u →∞. Moreover, we have

ln(

E{eMξ2(h)}φ1(λ)h

)∼ h

∫∞

0x−1−Y+e−(M+λ)x dx + Bh

= C+h Γ (−Y+)(M + λ)Y+ + Bh

as λ ↓ −M , so that

ln(

E{eMξ2(h)}φ1(µ

1 (u/h))h)∼ C+h(C+h Γ (1− Y+))Y+/(1−Y+)uY+/(Y+−1)

+ Bh

∼ (C+h Γ (1− Y+))1/(1−Y+)Γ (1− Y+)−1uY+/(Y+−1)+ Bh

as u →∞. Inserting the above findings into (3.9) we readily obtain (7.8). �

Braverman [11] and Braverman and Samorodnitsky [14] apply to CGMY for Y+ > 0.CGMY processes with Y−, Y+ < 0 are compound Poisson and light tailed in the sense of

Braverman [12], Definition 1, by Example 2.11. Hence Braverman’s Theorem 1 applies if theprocess has non-negative drift while his Theorem 3 applies if 0 < M < 1 and the drift isnegative, in both cases to yield (6.1) with H = 1.

Example 7.5. As the variance gamma processes of Madan and Seneta [27] are CGMY processeswith C− = C+ and Y− = Y+ = 0 they satisfy (6.1) with H = 1 and the asymptotic behaviourof P{ξ(h) > u} as u →∞ in (6.1) given by (7.7).

Example 7.6. The Kou [26] jump-diffusion Levy process has characteristic triple(dν(x)

dx, m, s2

)=

((1− p)λeλ−x 1(−∞,0)(x)+ pλe−λ+x 1(0,∞)(x), m, s2

), (7.11)

where p ∈ (0, 1), m ∈ R and λ, λ−, λ+, s2 > 0 are parameters.Besides the Gaussian component s2, (7.11) is (7.5) with Y− = Y+ = −1. Hence the proof

of Theorem 7.4 carries over to show that (6.1) holds with H = 1 and the asymptotic behaviourof P{ξ(h) > u} as u → ∞ in (6.1) given by (3.9). The evaluation of (3.9) in the proof ofTheorem 7.4 can be modified in a straightforward manner to include the Gaussian component,but we omit the details.

7.3. GH processes

The GH Levy process introduced by Barndorff-Nielsen [3,4] has characteristic triple(dν(x)

dx, m, s2

)

=

(eβx

|x |

(∫∞

0

exp{−|x |√

2y + %2}

π2 y(J|ζ |(δ√

2y)2 + Y|ζ |(δ√

2y)2)dy + ζ+e−%|x |

), m, 0

), (7.12)

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where β, ζ, m ∈ R, δ > 0 and % > |β| are parameters. Here Jζ denotes the Bessel function andYζ the Bessel function of the second kind, respectively.

Theorem 7.7. For a GH process (6.1) holds where H = 1 if ζ ≥ 0 while H > 1 if ζ < 0.

Proof. Using the facts from Watson [38], Equations 3.1.8, 3.51.1, 3.52.3 and 3.54.1-2, about theasymptotic behaviour of Jζ (y) and Yζ (y) as y ↓ 0 we readily obtain∫

0

exp{−|x |√

2y + %2}

π2 y(Jζ (δ√

2y)2 + Yζ (δ√

2y)2)dy ∼

∫∞

0

exp{−x(% + y/%)}

π2 yYζ (δ√

2y)2dy

∫∞

0

δ2ζ[sin(πζ )Γ (1− ζ )]2 yζ−1 exp{−x(% + y/%)}

π22ζdy for ζ ∈ [0,∞) \ N∫

0

δ2ζ yζ−1 exp{−x(% + y/%)}

2ζΓ (ζ )2 dy for ζ ∈ N \ {0}∫∞

0

exp{−x(% + y/%)}

y ln(y/%)2 dy for ζ = 0

as x →∞, so that by insertion in (7.12)

dν(x)

dx∼

2ζ[sin(πζ )Γ (1+ ζ )]2 exp{−(% − β)x}

π2δ2ζ %ζ x1−ζfor ζ ∈ (−∞, 0] \ N

2ζ exp{−(% − β)x}

δ2ζ %ζΓ (−ζ )2x1−ζfor ζ ∈ (−N) \ {0}

exp{−(% − β)x}

ln(2)xfor ζ = 0

ζe−(%−β)x

xfor ζ > 0.

(7.13)

For ζ ≥ 0, (7.13) shows that (3.8) holds with ρ = −1, so that Theorem 3.6 gives (3.4). HenceCorollary 6.2 shows that (6.1) holds with H = 1. For ζ < 0, (7.13) and Example 2.11 show thatν([1,∞) ∩ ·)/ν([1,∞)) ∈ S(% − β), so that Corollary 6.3 gives (6.1) with H > 1 (as ξ is not asubordinator). �

Braverman [11] and Braverman and Samorodnitsky [14] apply to GH for ζ < 0.

Example 7.8. The normal inverse Gaussian process introduced by Barndorff-Nielsen [4,5] is aGH processes with ζ = − 1

2 . Thus it satisfies (6.1) with H > 1.

Example 7.9. The hyperbolic process introduced by Barndorff-Nielsen [3] is a GH process withζ = 1. Thus is satisfies (6.1) with H = 1.

Remark 7.10. When ζ < 0 the asymptotic behaviour of P{ξ(h) > u} as u → ∞ in (6.1) forGH processes is given by the second part of Theorem 3.1 together with integration of (7.13).When ζ ≥ 0 the asymptotic behaviour of P{ξ(h) > u} can be calculated from (3.11) in thefashion of Theorem 7.1 for h large enough. We have omitted these calculations to avoid additionaltechnicalities.

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8. Two priority issues

Theorem 3.3 is due to Braverman [13], Theorem 1.Braverman showed us his result during Fall 2004, so we were aware of his finding when

submitting our paper (although, frankly, we had forgotten about it for the final version of ourarticle, so we had to be reminded about it).

Braverman [13], Theorem 2, states that the conclusion of Corollary 6.2 holds under thehypothesis of Theorem 3.3. Thus our Corollary 6.2 is more general than Braverman [13],Theorem 2. However, from a practical point of view, one may argue the importance of the addedgenerality.

Our version of Corollary 6.2 is implicit in Chapter 5 of the thesis of Bengtsson [8] (now namedSunden) that already appeared during Fall 2004, which is also acknowledged by Braverman [13],Remark 1.

It should be noted that there is a more or less complete difference between the approach andmethods of proof of Theorem 3.3 and Corollary 6.2 and those of the corresponding results ofBraveman.

Acknowledgements

We are grateful to Søren Asmussen, Michael Braverman, Peter Carr, Ron Doney, CharlesGoldie, Claudia Kluppelberg, Vladimir Piterbarg, Johan Tykesson and Bernt Wennberg for theirkind help and support.

We are grateful to the referees of both the first version and the second version of this paperfor their very valuable advice as well as very accurate technical comments.

The first author’s research was supported by The Swedish Research Council contract no. 621-2003-5214.

Appendix. Technical details of Section 2

Here we prove Proposition 2.6 and Corollary 2.9. We remark that these results are not veryfar from some standard results on regular variation that can be found in e.g., Bingham, Goldieand Teugels [10], as is indeed indicated by their proofs: The results and proofs are just suppliedas a service to the reader not very expert in regular variation.

A.1. Proof of Proposition 2.6

We have F ∈ L(α) if and only if 1 − F(ln(·)) ∈ R(−α) and F ∈ OL if and only if1−F(ln(·)) ∈ OR, see, e.g., Shimura and Watanabe [37], p. 451. By the representation theoremsfor R(−α) and OR (see e.g., Bingham, Goldie and Teugels [10], Theorems 1.3.1 and 2.2.7) afunction 1− F(ln(·)) belongs to R(−α) [OR] if and only if

1− F(ln(u)) = c(u) exp{−

∫ u

0

a(x)

xdx

}for u ∈ R large enough, (A.1)

for some measurable functions a and c such that limx→∞ a(x) = α and limx→∞ c(x) > 0 exists[lim supx→∞ |a(x)| < ∞ and 0 < lim infx→∞ c(x) ≤ lim supx→∞ c(x) < ∞]. Since F is

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absolutely continuous with limu→−∞ F(u) = 0 and limu→∞ F(u) = 1 we can rewrite (A.1) as

1− F(u) = exp

{−

∫ eu

0

a(x)+ b(x)

xdx

}

= exp{−

∫ u

−∞

(a(ex )+ b(ex )

)dx

}for u ∈ R,

where a(x) ≡ a(ex ) and b(x) ≡ b(ex ) satisfies (2.4) and (2.5), respectively, depending onwhether F ∈ L(α) or F ∈ OL. Finally, as F is non-decreasing we have a + b ≥ 0. �

A.2. Proof of Corollary 2.9

Integrating (1.1) we readily obtain

1− F(u) =

∫∞

uf (x)dx ∼

C

αuρe−αu

∼f (u)

αas u →∞. (A.2)

In particular we see that F ∈ L(α), so that (2.3) holds with a + b ≥ 0 as in (2.4), where

limu→∞

∫ u

−∞

(a(x)+ b(x)− α 1[0,∞)(x)+

ρ

x1[1,∞)(x)

)dx = ln

(C

α

)by (A.2). Writing c = a + b, we thus have (2.6) with c ≥ 0 as well as the second part of (2.7).Differentiating both sides of (2.6) and using (A.2) we get

f (u) = c(u) exp{−

∫ u

−∞

c(x)dx

}= c(u) (1− F(u)) ∼ c(u)

f (u)

αas u →∞, (A.3)

which gives the first part of (2.7).Conversely, if (2.6) and (2.7) hold, then it is immediate that (A.2) holds so that F ∈ L(α),

while (1.1) follows from (A.3). �

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Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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Please cite this article in press as: J.M.P. Albin, M. Sunden, On the asymptotic behaviour of Levyprocesses, Part I: Subexponential and exponential processes, Stochastic Processes and their Applications (2008),doi:10.1016/j.spa.2008.02.004

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� J��\� w#H�i ��4/���L��t���t54�� H _ J�P%P%� b H �c�M4���60�!� � &X+/� 60�M��4/ JK�>60�.�e�c� +/ ' 8�!� 4L������%0���X ,6]�6�Y*����s60�5%'���������� 0�M H � ] � h � ����� � !��sa8������� � f �mY \ �� ] ! �bh �=k f i � ��8� � T ���Rk ��!��sa8������� � f H

J��

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J Stat Phys (2008) 130: 295–312DOI 10.1007/s10955-007-9424-8

Brownian Approximation and Monte Carlo Simulationof the Non-Cutoff Kac Equation

Mattias Sundén · Bernt Wennberg

Received: 15 November 2006 / Accepted: 4 September 2007 / Published online: 3 October 2007© Springer Science+Business Media, LLC 2007

Abstract The non-cutoff Boltzmann equation can be simulated using the DSMC method,by a truncation of the collision term. However, even for computing stationary solutions thismay be very time consuming, in particular in situations far from equilibrium. By adding anappropriate diffusion, to the DSMC-method, the rate of convergence when the truncationis removed, may be greatly improved. We illustrate the technique on a toy model, the Kacequation, as well as on the full Boltzmann equation in a special case.

Keywords Kac equation · Direct simulation Monte Carlo · Diffusion approximation ·Thermostat · Non-equilibrium stationary state · Markov jump process

The Boltzmann equation describes the evolution of the phase space density of a gas. It isa nonlinear equation in many dimensions, which makes it difficult to treat by e.g. finitedifference methods. The classical way of solving the Boltzmann equation numerically is bymeans of Monte Carlo simulation. The method was first described by Bird (see the book [4]),but since then many variations on the theme have been published [2, 23, 25, 29].

Very briefly, the DSMC-method can be described as follows: The gas is represented bya finite (although sometimes rather large) number of particles. The time evolution is thencarried out by alternating a transport step, in which the particles move independently withtheir own velocity, and a collision step. The spatial domain of calculation is divided into cellswhich should be large enough that it typically contains a not too small number of particles,but still small enough to take into account the spatial gradients in the problem.

The collision step is then carried out in each cell separately as a jump process in R3n,

where n is the number of particles, and the jumps occur as the velocity change in clas-sical collisions between randomly chosen pairs of particles. The collisions conserve en-

M. Sundén · B. Wennberg (�)Department of Mathematical Sciences, Chalmers University of Technology, 41296 Göteborg, Swedene-mail: [email protected]

M. Sundén · B. WennbergDepartment of Mathematical Sciences, Göteborg University, Göteborg, Sweden

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296 J Stat Phys (2008) 130: 295–312

ergy and momentum, so the jump process actually takes place on {(v1, . . . , vn) ∈ R3n |∑

vk = u ∈ R3,

∑v2

k = W }.The simulation can be understood as a sampling of the solution of a Poisson driven

stochastic differential equation. In the original DSMC methods, the rate of the underlyingPoisson process was always taken to be finite, but there are at least two reasons for con-sidering infinite (or very large) collision rates. The first one occurs when the density of thegas is very large in a cell, i.e. when n is very large. One way of handling that situation isto sample the velocity distribution at the end of a collision step as a suitable mixture of adistribution as the one that results from a moderate number of collisions and the equilibriumdistribution, which is a Maxwellian [25–27].

The other case derives from the fact that many realistic collision models correspondto long-range potentials, which effectively gives rise to an infinite collision rate. The vastmajority of collisions only change the velocities marginally, and so the rate of change ofmomentum due to collisions is finite.

To carry out a Monte Carlo simulation in this situation, one may truncate the jumpprocess so as to obtain a finite collision rate. It has been proven [10, 13, 14] that truncatedMonte Carlo methods converge as the truncation is lifted, and this has also been illustratedby numerical experiments. However, we have found that in certain cases, in particular whenthe stationary solutions are far from equilibrium, the jump rate must be truncated at a veryhigh (and so, costly) level to get an acceptable accuracy.

We propose here a method to replace the small jumps by an appropriate diffusion, andshow by example that this gives an important improvement of the accuracy compared to justignoring the small jumps. This has been inspired by the works [1, 30], where this techniqueis proposed for simulating Lévy processes in R

n.There are, of course, other methods than the Monte Carlo methods, for solving the Boltz-

mann equation, and in particular there are fast methods based on the Fourier transform, thathave been used successfully for the non-cutoff situation [11, 12, 22].

This study was motivated by the difficulty of obtaining accurate estimations of the non-equilibrium stationary state for a non-cutoff collision kernel, and of the theoretical resultsin [5], where the one-dimensional Kac equation with a Gaussian thermostat was studied.Also in this paper, the main part is devoted to the Kac equation. The diffusion term is thena Brownian motion on the sphere Sn−1, and the approximation is very straight forward.We describe the method in some detail for this case. However, from a conceptual point ofview, the method is not restricted to one-dimensional models, and we have also carried outnumerical calculations for the Boltzmann equation with a thermostat and Maxwellian non-cutoff collisions. In that case, the diffusion model is more complicated (it is essentially theBalescu-Prigogine model for Maxwellian molecules [18, 33]), and the actual calculation iscarried out somewhat differently, as described in Sect. 3.

In [34, 36], it is shown that, contrary to the Kac equation, the Boltzmann equation with athermostatted force field only has trivial stationary states, and hence it is only interesting tocompare the evolution of the solutions. At the end of the paper, we present some numericalresults for this model.

We note, finally, that for non-Maxwellian molecules the situation is rather different, andthen it may in some cases be more appropriate to go the other way around and to approximatethe diffusion process by a non-cutoff collision process [7].

1 The Kac Equation, the Master Equation and Monte Carlo Simulations

We consider a system of n particles that are entirely characterized by their one-dimensionalvelocities vi , i = 1, . . . , n. These velocities undergo random jumps,

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J Stat Phys (2008) 130: 295–312 297

(v1, . . . , vj , . . . , vk, . . . , vn)

�→ Rj,k,θ (v1, . . . , vj , . . . , vk, . . . , vn)

= (v1, . . . , vj cos θ − vk sin θ, . . . , vj sin θ + vk cos θ, . . . , vn),

1 ≤ j, k ≤ n, θ ∈] − π,π ]. (1)

These jumps occur independently with a rate proportional to

n−1b(θ)dθ. (2)

This is the Kac model of a dilute gas [17]. In the original paper, b(θ) = (2π)−1, i.e. allrotation angles θ are equally probable.

We note that the jumps are rigid rotations in a plane spanned by a pair of velocities,and hence it is clear that the kinetic energy, W = 1

2

∑n

j=1 v2j is preserved, and therefore this

describes a jump process with values in Sn−1(√

2W). It is convenient to choose 2W = n.Another important thing to notice is the factor n−1 in (2); this implies that for the rate

of jumps that involve a particular velocity, e.g. v1, asymptotically does not depend on thenumber of velocities, n. However, in many physically realistic cases, the rate of jumps de-pend on the rotation angle, approximately as |θ |−(1+α), where α ∈ [0,2[. This implies thatthe total jump rate for the vector v = (v1, . . . , vj , . . . , vk, . . . , vn) is infinite. We speak aboutnon-cutoff models as opposed to the cutoff models where b is replaced by some function b

such that∫ π

−πb(θ) dθ < ∞.

This jump process may equally well be defined by the master equation, which de-scribes the evolution of a phase space density under the process. We let �(t, ·) ∈C([0,∞[,L1(Sn−1(

√n))), and assume that �(0, ·) is a non negative density on Sn−1(

√n).

Then � satisfies the equation

∂t�n(t,v) = 2

n

1≤j<k≤n

∫ π

−π

(�n(t,Rj,k,−θv) − �n(t,v))b(θ) dθ, (3)

which is known as Kac’s master equation. The superscript n denotes the number of variablesin the model, and this corresponds to the number of particles in a cell. Kac proved that if oneconsiders the family of master equations for �n, n = 0, . . . ,∞, with initial data �n

0 that issymmetric with respect to permutation of the variables, and such that the marginal densities�n

k,0(v, . . . , vk) satisfy

limn→∞�n

k,0(v1, . . . , vk) = limn→∞

j=1,...,k

�n1,0(vj ). (4)

Then also the time evolved density �n(t,v) factories into a product of one-particle marginalsf (t, v) ≡ �n

1 (t, v), and that f (t, v) satisfies the so-called Kac equation

∂tf = Q(f,f ), (5)

where the collision operator Q

Q(g,g)(v) =∫

R

∫ π

−π

(g(v′)g(v′∗) − g(v)g(v∗))b(θ) dθ dv∗, (6)

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298 J Stat Phys (2008) 130: 295–312

and where

(v′, v′∗) = (v cos θ − v∗ sin θ, v sin θ + v∗ cos θ). (7)

Compared to the real Boltzmann equation, the Kac equation is very easy to analyze math-ematically, and we refer to [9, 21] for some of the basic results concerning existence anduniqueness of solutions, trend to equilibrium, etc.

From a numerical point of view, the connection between the jump process describedin (1) and (2) and the Kac equation, is that one may regard

1

n

n∑

j=1

δvj (t) (8)

as an approximation of the probability density f (t, v), and it has been proven for manydifferent cases that (8) does indeed converge to f (t, v) (see [3, 28]).

The solutions of (5) converge to equilibrium exponentially as t increases to infinity, theequilibrium solution being a Gaussian function with mean zero. It is of interest to studysituations where the stationary solution is not an equilibrium state. This is the case e.g. inkinetic models for dissipative systems (see e.g. [6, 8, 15]).

Another example comes from molecular dynamics and the introduction of thermostats,and which is used here as a test case for the Monte Carlo method with Brownian approxi-mation.

The basic model is the one described in the beginning of this section, with a vec-tor v ∈ R

n, that jumps according to (1). The difference in the thermostat model is that thevelocity field is accelerated by a constant force field E = E(1, . . . ,1), which is projected onthe tangent plane to the sphere Sn−1(

√n). This means that between jumps, the a velocity

component vj satisfies

d

dtvj (t) = E

(

1 −∑

vk∑

v2k

vj

)

. (9)

The physical interpretation is that each particle is accelerated by the same, constant, forcefield of strength E, but that a force field depending on the whole system of particles keepsthe total kinetic energy fixed. The model is described more in details in [34, 36], wherealso the corresponding Kac equation is derived and analyzed. This Kac equation takes theform

∂tf + E

∂v((1 − ζ(t)v)f ) = Q(f,f ), (10)

where

ζ(t) =∫

R

vf (v, t) dv. (11)

The collision operator, Q is defined as before, in (6). The stationary states to this equationare far from Gaussian. One can show that for an integrable kernel,

∫ π

−πb(θ) dθ , the stationary

state becomes singular for sufficiently large values of E (see [35]), whereas in the non-cutoffcase, the stationary state is C∞ (see [5]). The latter case is really the motivation for the workpresented in this paper, because it proved very difficult to get accurate numerical resultsusing a truncated kernel.

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J Stat Phys (2008) 130: 295–312 299

2 Brownian Approximation

The evolution of the n-particle system (including a thermostatted force field of strength E)can be described by a stochastic differential equation driven by a Poisson random measure:

v(t) = v(0) + E

∫ t

0

(

e − e · v(s)

|v(s)|2 v(s)

)

ds

+∑

1≤j<k≤n

∫ t

0

∫ π

−π

Aj,k(θj,k)v(s−)N(ds, dθj,k). (12)

Here Aj,k(θ) is the n × n-matrix

Aj,k(θ) =

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 · · · 0 · · · 0 · · · 0 00 1 · · · 0 · · · 0 · · · 0 0...

......

......

......

......

0 0 · · · cos θ − 1 · · · − sin θ · · · 0 0...

......

......

......

......

0 0 · · · sin θ · · · cos θ − 1 · · · 0 0...

......

......

......

......

0 0 · · · 0 · · · 0 · · · 1 00 0 · · · 0 · · · 0 · · · 0 1

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (13)

e = (1,1, . . . ,1) ∈ Rn, and N(ds, dθ) is a Poisson random measure with intensity measure

n−1b(θ) dθ dt . In the non-cutoff case,

b(θ) ∼ |θ |−(1+α) (14)

near θ = 0, and with 0 < α < 2, this implies that the total jump rate is infinite.The Brownian approximation consists in replacing N(ds, dθ) in (12) by a truncated mea-

sure Nε(ds, dθ) with intensity measure n−1bε(θ) dθ dt , where b is defined by

bε(θ) = min(b(θ), b(ε)), (15)

and adding a Brownian term to compensate for the truncated part.The explicit form of a stochastic differential equation whose solution is a Brownian

motion on an n-dimensional sphere Sn−1(r) can be found e.g. in [31], or in Øksendal’sbook [24]:

X(t) = X(0) +∫ t

0λrW (X(s)) ds +

∫ t

0

√λσ(X(s)) dW(s), (16)

where {W(t)} is a standard Wiener process in Rn with mean zero and whose covariance is

the identity matrix. The matrix σ projects the dW onto the tangent plane to Sn−1(√

n) at X,its elements being given by

σj,k(x) = δj,k − xjxk

|x|2 . (17)

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300 J Stat Phys (2008) 130: 295–312

The drift term rW is

rW (x) = −n − 1

2

x

|x|2 . (18)

The diffusion rate μ is computed so as to match the second moment of the truncated part ofthe jump measure:

μ = με = 2n

n − 1

1

2

∫ ε

−ε

(b(θ) − b(ε)) θ2 dθ. (19)

When b(θ) = |θ |−(1+α), we find

με = 2n

n − 1

1 + α

3(2 − α)ε2−α. (20)

Details of this calculation will be found in [32], where also the rate of convergence isanalysed. One result is that while the generators of the processes corresponding to (14)with a truncated kernel converges with rate ε2−α , the convergence rate is ε3−α when theBrownian term is added. When α is close to 2, the improvement is significant.

Adding the force term, like in (12), gives

v(t) = v(0) +∫ t

0(rT h(v(s)) + rW (v(s))) ds +

∫ t

0σ(v(s)) dW(s)

+∑

1≤j<k≤N

∫ t

0

∫ π

−π

Aj,k(θj,k)v(s−)Nε(ds, dθj,k), (21)

where rT h(v(t)) = e − e·v(s)

|v(s)|2 v(t) = �(v)e, � being the matrix with elements �j,k =δj,k − vj (t)vk(t)/|v|2, and where Nε is a Poisson random measure with intensity measuren−1bε(θ) dθ dt .

3 The 3n Dimensional Master Equation and the Boltzmann Equation

The master equation that corresponds to the full Boltzmann equation and with an addedthermostatted force term is [36]

∂t�n(t,v) +

n∑

i=1

∂vi

([F − F · j vi]�n(t,v))

= 2

n

1≤j<k≤n

S2(�n(t,Rj,k,ωv) − �n(t,v))b(θ) dω, (22)

where in this case, v ∈ R3n, and where Rj,k,ω is an operator that models the collision of two

particles,

(vj , vk) �→(

vj + vk

2+ |vj − vk|

2ω,

vj + vk

2− |vj − vk|

)

.

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J Stat Phys (2008) 130: 295–312 301

In b(θ), θ = arccos(ω · vj −vk

|vj −vk | ). To keep the notation similar to the one-dimensional case,

we let v = (v1, . . . , vn), where vj ∈ R3, j = 1, . . . , n. The non-cutoff case again corresponds

to allowing∫

S2 b(θ) dω to diverge. The force field is F = (E,0,0), and j = 1n

∑n

j=1 vj . At

the level of the master equation, the Brownian approximation corresponds to truncating thenon-cutoff jump rate, and replacing the truncated part by a suitable diffusion term. For theKac equation, the diffusion is just the one given by the Laplace-Beltrami operator on thesphere Sn−1, but here it is rather the Balescu-Prigogine operator for Maxwellian interactions(see [18, 33]),

μ

n

j �=k

(∂vj− ∂vk

)|vj − vk|2P⊥vj −vk

(∂vj− ∂vk

), (23)

where P⊥z is the 3 × 3-matrix I − zztr

|z|2 , and μ is a constant depending on the level of trunca-tion.

Given this expression, one can write a stochastic equation much like (21). However, froma computational point of view, it does not seem to be efficient, because of the effort neededto compute σ dW . An alternative, that gives good results at much lower computational cost,is to replace (23) by

μ

n

j �=k

∂vj|vj − vk|2P⊥

vj −vk∂vj

, (24)

which corresponds to adding a Brownian motion Rj dWj ∈ R3 to each velocity, where

Rtrj Rj = μ

∑j �=k(|vj − vk|2I − (vj − vk)(vj − vk)

tr). The matrices Rj can be expressed interms of moments and vj , and hence the computational cost is proportional to the number ofparticles. The increments (R1 dW1, . . . ,Rn dWn) still need to be projected onto the tangentspace of the manifold of constant energy and momentum, but that is an operation that canbe carried out in time proportional to the number of particles. Some numerical results aregiven in the following section.

4 Numerical Experiments

The numerical experiments have been carried out in the most direct way, with no large effortto make the code efficient. The large jumps have been simulated by computing exponentiallydistributed time intervals with rate proportional to n

∫ π

−πb(θ)dθ . At the end of such an inter-

val, a random pair (j, k) is chosen and the jump is effectuated by rotating the vector (vj , vk)

by a random angle θ distributed according to b(θ)/∫ π

−πb(θ)dθ .

In the intervals between the large jumps, we solve (21) using a simple explicit Eulermethod with a step size that depends on με . The step size is taken to be a given fraction ofthe typical rate for the truncated jump process. We have not made a rigorous analysis thatwould help in choosing the step size, rather we have numerically tested that the choice givesrelevant answers.

We have computed uniformly distributed pseudo-random variables using the routineDLARAN from the LAPACK package [19] and to compute normally distributed randomvariables, we have used the ziggurat method of Marsaglia and Tsang [20].

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302 J Stat Phys (2008) 130: 295–312

Fig. 1 Simulation results using different values of truncation: the velocity distribution, and an enlargement

As initial data, we have taken 12 (δv=1 + δv=−1).

We have no exact solutions to compare the results with. However, taking the Fouriertransform of the time independent Kac equation, i.e. (10) considered without t , gives

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J Stat Phys (2008) 130: 295–312 303

Fig. 2 Simulation results using different values of truncation: time evolution of some of the moments

equation

f ′(ξ) + i

γ ξf (ξ) = 1

Eγ ξ

∫ π

−π

(f (ξ cos θ)f (ξ sin θ) − f (0)f (ξ))b(θ) dθ, (25)

where γ is the stationary current, which can be explicitly computed. Equation (25) canbe solved accurately numerically using a finite difference method, combined with thebuilt-in ODE-solvers of Matlab™. Numerical results obtained in this way were pre-sented in [5], where also a detailed mathematical analysis of the non-cutoff Kac equa-tion with a thermostat can be found. Although no rigorous error analysis has been car-ried out, we consider this finite-difference solution to be an accurate solution to the sta-tionary problem: a fine discretization was used, and the method was found to convergewell.

Another test for the accuracy is to compare the evolution of moments. Also here wedo not have any exact results to compare with, but as with other Boltzmann like equa-tions of Maxwell type (i.e. models where the collision rate does not depend on the relativevelocity of the colliding particles), one can write a closed system of ordinary differentialequations for the first moments mk = ∫

f (v, t)vk dv, and this system can then be solvedaccurately with a numerical ODE-solver. We have used Matlab™ to solve the following

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304 J Stat Phys (2008) 130: 295–312

Fig. 3 Simulation results using different values of truncation for a stronger singularity: the velocity distrib-ution, and the evolution of moments. Here the effect of the Brownian correction is more evident

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J Stat Phys (2008) 130: 295–312 305

Fig. 4 Simulation results using different values of the number of particles. The simulation is repeated moretimes to get comparable results. As few as 50 particles gives a rather good agreement, but as few as five oreven three particles are clearly not enough

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306 J Stat Phys (2008) 130: 295–312

Fig. 5 The evolution between jumps is computed using a simple forward Euler method. The results hereshow that the time step is not critical

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J Stat Phys (2008) 130: 295–312 307

Fig. 6 Simulation of the Boltzmann equation: the distribution of the first velocity component vx , i.e. anapproximation of

∫R2 f (v, t) dvy dvz at t = 0.06

system:

mk = Ekmk−1 − Ekm1mk − Akmk +k−1∑

j=1

Bkjmk−jmj ,

Ak =∫ π

−π

(1 − cosk θ − sink θ)b(θ) dθ,

Bkj =(

k

j

)∫ π

−π

cosk−j θ sinj θb(θ) dθ,

mk(0) = 1

2(1 + (−1)k). (26)

In the numerical calculations we have used b(θ) = |θ |−1−α for different values of β =α + 1 ∈]1,3[, and different values of the force parameter E ∈ [2,5]. We have also varied n,the number of particles, and the time step used in solving the SDE, (21).

The first series of results, presented in Figs. 1, 2 (an enlargement), and Fig. 3 (for astronger singularity) shows how the results depend on the level of truncation, and comparesthis with the result from using a much truncated model but with a Brownian correction. Theparameters used were E = 3.0, β = 2.0, n = 2000, and ε = 0.2,0.02,0.002. The time dt

used in the Euler method for approximating the SDE was here taken to be 0.0001 × dt0,

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308 J Stat Phys (2008) 130: 295–312

Fig. 7 Evolution of the moment M6,0,0 with truncation at ε = 0.5, with and without the diffusion correction.The moments for the Boltzmann equation are given as a reference. The enlargement shows that although thereference curve and the curve with the diffusion correction almost coincide, they are not identical

where dt0 corresponds to a displacement of order ε from the Brownian motion. This is ex-cessively small, and we will see below that it is far from necessary to get an accurate result.The calculation was then repeated 200 times to reduce noise. We see that ε = 0.2 togetherwith a Brownian approximation compares very well with a simulation using ε = 0.002 with

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J Stat Phys (2008) 130: 295–312 309

Fig. 8 Evolution of the moment M4,0,0 with truncation at ε = 0.5, with and without the diffusion correction.The moments for the Boltzmann equation are given as a reference

no approximation. The estimates in [32] give a convergence rate of ε3−β without the Brown-ian correction, and a rate of ε4−β with the Brownian term added. Hence it is not surprisingto see that, when the singularity in the crossection b(θ) is stronger, the influence of thetruncation and of the Brownian approximation is much more important.

Figure 4 shows results for different values of n, the number of particles used in thesimulation. There are at least two reasons for using a large value of n, when simulatingkinetic equations: first, the Boltzmann equation itself assumes a limit of infinitely manyparticles, and secondly, a large value of n reduces the noise when computing moments orother functions. In this series we have taken n × number of simulated trajectories � const,and very large in order to obtain a noiseless result. The calculations show that in fact it isnot necessary to use a very large number of particles to find a good agreement, n = 50 isquite enough, both to get a reasonable agreement of the distribution functions and of theevolution of moments. However, with a small number of particles, it is necessary to repeatthe calculations many times to avoid excessive noise in the result.

The last example for the Kac model, Fig. 5, shows some simulations that illustrate theinfluence of the time-step in the Euler method for solving the SDE (21). The reference timestep is so large that the mean step is of the order ε, and the figure shows that both forcomputing the distribution function and the evolution of moments, it is not necessary todecrease the step size much below the reference value to get a good result.

The simulation is carried out in very much the same way for velocities in R3, the main

difference being the in which the diffusion is added. To compute the matrices Rj , we have

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310 J Stat Phys (2008) 130: 295–312

Fig. 9 Evolution of the moment M2,0,2 with truncation at ε = 0.5, with and without the diffusion correction.The moments for the Boltzmann equation are given as a reference

used routines from the Gnu Scientific Library [16] to evaluate the square root. This is rathertime consuming, and the code spends a major part of the time doing this; still the methodis faster than just using a smaller truncation, also before any attempts have been made tomaking the code efficient.

In this case we do not have an alternative method for computing the velocity distribution,but rather we compare the velocity distribution with a calculation with very small truncation.Because we are dealing with the case of Maxwellian interactions, there is a closed set ofequations that describe the evolution of moments for the limiting Boltzmann equation alsohere.

All calculations were carried out with a constant force field F = (1,0,0), and the numberof particles was n = 500. The initial data was 1

2 (δ(1,0,0) + δ(−1,0,0)).The first graph, Fig. 6 compares the approximations of

R2f (vx, vy, vz, t) dvy dvz,

with and without the diffusion correction, and a reference solution obtained by carrying outa simulation with a very small truncation of the crossection.

We also compare the evolution of moments of the form

Mjx,jy ,jz (t) = 1

n

n∑

k=1

vjxkxv

jykyv

jzkz,

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J Stat Phys (2008) 130: 295–312 311

with the moments (computed as the solution of the closed ODE-system) for solutions to thelimiting Boltzmann equation,

R3f (vx, vy, vz, t)v

jx0xv

jy

0yvjz0z dvx dvy dvz.

We consider the solutions to the ode’s to be exact. Hence Figs. 7, 8 and 9 show the evolutionof M6,0,0(t), M4,0,0(t), and M2,0,2(t), respectively, for a strong truncation of the collisionterm ε = 0.5, with and without the diffusion correction, and compared with the moments forthe limiting Boltzmann equation.

5 Conclusions

The paper presents a method to compute accurate solutions to non-cutoff Boltzmann equa-tions in the non-cutoff case, by approximating the small jumps by a diffusion. We havepresented numerical examples showing that it works well, and gives accurate results.

There are several open issues that merit being studied. From a numerical point of view,of course one would have to find good means of choosing n, the level of truncation, and anefficient method for solving (21).

Some results that aim at putting the method on a solid theoretical ground will be pre-sented in [32].

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