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TEMPERED STABLE DISTRIBUTIONS AND APPLICATIONS TO FINANCIAL MATHEMATICS UWE K ¨ UCHLER AND STEFAN TAPPE Abstract. We investigate the class of tempered stable distributions and pre- sent applications to financial mathematics. Our analysis of tempered stable distributions includes limit distributions, parameter estimation and the study of their densities. Afterwards, we deal with stock price models driven by tem- pered stable processes and are concerned with the existence of appropriate martingale measures as well as option pricing. Key Words: Tempered stable distributions, limit distributions, martingale measures, option pricing. 91B28, 60G51 1. Introduction Tempered stable distributions form a class of distributions that have attracted the interest of researchers from probability theory as well as financial mathematics. They have first been introduced in [27], where the associated L´ evy processes are called “truncated L´ evy flights”, and have been generalized by several authors. Tem- pered stable distributions form a six parameter family of infinitely divisible distribu- tions, which cover several well-known subclasses like Variance Gamma distributions [34, 35], bilateral Gamma distributions [30] and CGMY distributions [7]. Properties of tempered stable distributions have been investigated, e.g., in [40, 47, 45, 4]. For financial modeling they have been applied, e.g., in [8, 36, 25, 3], see also the recent textbook [39]. In this paper, we contribute to the theory of tempered stable distributions from a probabilistic side as well as to their applications in financial mathematics. In the first part of this paper, we provide limit results for tempered stable distributions, deal with statistical issues and analyze their density functions. Afterwards, we deal with stock price models driven by tempered stable processes. Here, we will be concerned with choices of equivalent martingale measures and with option pricing formulas. Tempered stable distributions cover the class of bilateral Gamma distributions, an analytical tractable class which we have investigated in [30, 31, 32]. Our sub- sequent investigations will show that, in many respects, the properties of bilateral Gamma distributions differ from those of all other tempered stable distributions (for example the properties of their densities, see Section 7, their p-variations, see Section 9, or results concerning the existence of equivalent martingale measures in finance, see Sections 11–14) and that bilateral Gamma distributions can be re- garded as boundary points within the class of tempered stable distributions. In this paper, we are in particular interested in the question, which relevant proper- ties for bilateral Gamma distributions still hold true for general tempered stable distributions. Date : November 14, 2011. The authors thank Gerd Christoph for discussions about the optimal constant in the Berry- Esseen theorem.

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Page 1: TEMPERED STABLE DISTRIBUTIONS AND APPLICATIONS TO ... · TEMPERED STABLE DISTRIBUTIONS AND APPLICATIONS TO FINANCIAL MATHEMATICS UWE KUCHLER AND STEFAN TAPPE Abstract. We investigate

TEMPERED STABLE DISTRIBUTIONS AND APPLICATIONS

TO FINANCIAL MATHEMATICS

UWE KUCHLER AND STEFAN TAPPE

Abstract. We investigate the class of tempered stable distributions and pre-

sent applications to financial mathematics. Our analysis of tempered stable

distributions includes limit distributions, parameter estimation and the studyof their densities. Afterwards, we deal with stock price models driven by tem-

pered stable processes and are concerned with the existence of appropriatemartingale measures as well as option pricing.

Key Words: Tempered stable distributions, limit distributions, martingale

measures, option pricing.

91B28, 60G51

1. Introduction

Tempered stable distributions form a class of distributions that have attractedthe interest of researchers from probability theory as well as financial mathematics.They have first been introduced in [27], where the associated Levy processes arecalled “truncated Levy flights”, and have been generalized by several authors. Tem-pered stable distributions form a six parameter family of infinitely divisible distribu-tions, which cover several well-known subclasses like Variance Gamma distributions[34, 35], bilateral Gamma distributions [30] and CGMY distributions [7]. Propertiesof tempered stable distributions have been investigated, e.g., in [40, 47, 45, 4]. Forfinancial modeling they have been applied, e.g., in [8, 36, 25, 3], see also the recenttextbook [39].

In this paper, we contribute to the theory of tempered stable distributions froma probabilistic side as well as to their applications in financial mathematics. In thefirst part of this paper, we provide limit results for tempered stable distributions,deal with statistical issues and analyze their density functions. Afterwards, we dealwith stock price models driven by tempered stable processes. Here, we will beconcerned with choices of equivalent martingale measures and with option pricingformulas.

Tempered stable distributions cover the class of bilateral Gamma distributions,an analytical tractable class which we have investigated in [30, 31, 32]. Our sub-sequent investigations will show that, in many respects, the properties of bilateralGamma distributions differ from those of all other tempered stable distributions(for example the properties of their densities, see Section 7, their p-variations, seeSection 9, or results concerning the existence of equivalent martingale measuresin finance, see Sections 11–14) and that bilateral Gamma distributions can be re-garded as boundary points within the class of tempered stable distributions. Inthis paper, we are in particular interested in the question, which relevant proper-ties for bilateral Gamma distributions still hold true for general tempered stabledistributions.

Date: November 14, 2011.The authors thank Gerd Christoph for discussions about the optimal constant in the Berry-

Esseen theorem.

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2 UWE KUCHLER AND STEFAN TAPPE

The remainder of this text is organized as follows: In Section 2 we review tem-pered stable distributions and collect some basic properties. In Section 3 we in-vestigate closure properties of tempered stable distributions with respect to weakconvergence. Afterwards, in Section 4 we show convergence of tempered stable dis-tributions to normal distributions and provide a convergence rate. In Section 5 weprove “law of large numbers” results, with a view to parameter estimation fromobservation of a typical sample path, and in Section 6 we perform statistics for a fi-nite number of realizations of tempered stable distributions. In Section 7 we analyzethe densities of tempered stable distributions. In Section 8 we investigate equivalentmeasures under which a tempered stable process remains tempered stable, and inSection 9 we compute the p-variation index of tempered stable processes.

Our applications to finance start in Section 10, where we introduce the stockprice model. In Sections 11, 12 we study Esscher transforms and bilateral Esschertransforms in order to obtain appropriate martingale measures. Section 13 is de-voted to option pricing formulas and in Section 14 we treat the minimal martingalemeasure, which has applications to quadratic hedging.

2. Tempered stable distributions and processes

In this section, we introduce tempered stable distributions and processes andcollect their relevant properties.

We call an infinitely divisible distribution η on (R,B(R)) a one-sided temperedstable distribution, denoted η = TS(α, β, λ), with parameters α, λ ∈ (0,∞) andβ ∈ [0, 1) if its characteristic function is given by

ϕ(z) = exp

(∫R

(eizx − 1

)F (dx)

), z ∈ R(2.1)

where the Levy measure F is

F (dx) =α

x1+βe−λx1(0,∞)(x)dx.(2.2)

We call the Levy process associated to η a tempered stable subordinator.Next, we fix parameters α+, λ+, α−, λ− ∈ (0,∞) and β+, β− ∈ [0, 1). An infin-

itely divisible distribution η on (R,B(R)) is called a tempered stable distribution,denoted

η = TS(α+, β+, λ+;α−, β−, λ−),

if η = η+ ∗ η−, where η+ = TS(α+, β+, λ+) and η− = ν with ν = TS(α−, β−, λ−)and ν denoting the dual of ν given by ν(B) = ν(−B) for B ∈ B(R). Note that ηhas the characteristic function (2.1) with the Levy measure F given by

F (dx) =

(α+

x1+β+ e−λ+x

1(0,∞)(x) +α−

|x|1+β−e−λ

−|x|1(−∞,0)(x)

)dx.(2.3)

We call the Levy process associated to η a tempered stable process.

2.1. Remark. In [8, Sec. 4.5], the authors define generalized tempered stable pro-cesses for parameters α+, λ+, α−, λ− > 0 and β+, β− < 2 as Levy processes withcharacteristic function

ϕ(z) = exp

(izγ +

∫R

(eizx − 1− izx

)F (dx)

), z ∈ R(2.4)

for some constant γ ∈ R and Levy measure F given by (2.3). In the case β+ = β−

they call such a process a tempered stable process. The behaviour of the sample pathsof a generalized tempered stable process X depends on the values of β+, β−:

• For β+, β− < 0 we have F (R) <∞, and hence, X is a compound Poissonprocess and of type A in the terminology of [42, Def. 11.9].

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TEMPERED STABLE DISTRIBUTIONS 3

• For β+, β− ∈ [0, 1), which is the situation that we consider in the present

paper, we have F (R) =∞, but∫ 1

−1|x|F (dx) <∞. Therefore, X is a finite-

variation process making infinitely many jumps in each interval of positivelength, which we can express as Xt =

∑s≤t ∆Xs, and it belongs to type B

in the terminology of [42, Def. 11.9]. In particular, we can decompose X asthe difference of two independent one-sided tempered stable subordinators.

• For β+, β− ∈ [1, 2) we have∫ 1

−1|x|F (dx) = ∞. Therefore, the tempered

stable process X has sample paths of infinite variation and belongs to typeC in the terminology of [42, Def. 11.9].

2.2. Remark. The tempered stable distributions considered in this paper also corres-pond to the generalized tempered stable distributions in [39]. The following particularcases are known in the literature:

• β+ = β− is a KoBol distribution, see [6];• α+ = α− and β+ = β− is a CGMY-distribution, see [7], also called classical

tempered stable distribution in [39];• β+ = β− and λ+ = λ− is the infinitely divisible distribution associated to

a truncated Levy flight, see [27];• β+ = β− = 0 is a bilateral Gamma distribution, see [30];• α+ = α− and β+ = β− = 0 is a Variance Gamma distribution, see [34, 35].

According to [8, Prop. 4.1], a tempered stable process X can be represented as atime changed Brownian motion with drift if and only if X is a CGMY-process.Accordingly, a bilateral Gamma process is a time changed Brownian motion if andonly if it is a Variance Gamma process.

2.3. Remark. In [39], tempered stable distributions are considered as one-dimensionalinfinitely divisible distributions with Levy measure

F (dx) = q(x)Fstable(dx),(2.5)

where

Fstable(dx) =

(α+

x1+β1(0,∞)(x) +

α−

|x|1+β1(−∞,0)(x)

)dx

is the Levy measure of a β-stable distribution and q : R → R+ is a temperingfunction. For example, with

q(x) = e−λ+x1(0,∞)(x) + e−λ

−|x|1(−∞,0)(x)

and α+ = α− we have a CGMY-distribution. Further examples are the modi-fied tempered stable distribution, the normal tempered stable distribution, the Kim-Rachev tempered stable distribution and the rapidly decreasing tempered stable dis-tribution, see [39, Chapter 3.2]. Note that the Levy measures of the tempered stabledistributions considered in our paper are generally not of the form (2.5).

2.4. Remark. One can also consider multi-dimensional tempered stable distri-butions. In [40], a distribution η on (Rd,B(Rd)) is called tempered α-stable forα ∈ (0, 2) if it is infinitely divisible without Gaussian part and Levy measure M ,which in polar coordinates is of the form

M(dr, du) =q(r, u)

rα+1drσ(du),

where σ is a finite measure on the unit sphere Sd−1 and q : (0,∞)× Sd−1 → (0,∞)is a Borel function satisfying certain assumptions.

We shall now collect some basic properties of generalized tempered stable distri-butions which we require for this text. In the sequel, Γ : R \ 0,−1,−2, . . . → Rdenotes the Gamma function.

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4 UWE KUCHLER AND STEFAN TAPPE

2.5. Lemma. Suppose that β ∈ (0, 1). The one-sided tempered stable distribution

η = TS(α, β, λ)

has the characteristic function

ϕ(z) = exp(αΓ(−β)

[(λ− iz)β − λβ

]), z ∈ R(2.6)

where the power stems from the main branch of the complex logarithm.

Proof. Let G ⊂ C be the region G = z ∈ C : Im z > −λ. We define the functionsfi : G→ C for i = 1, 2 as

f1(z) :=

∫R

(eizx − 1

)F (dx) and f2(z) := αΓ(−β)

[(λ− iz)β − λβ

].

Then f1 is analytic, which follows from [11, Satz IV.5.8], and f2 is analytic bythe analyticity of the power function z 7→ zβ on the main branch of the complexlogarithm. Let B ⊂ G be the open ball B = z ∈ C : |z| < λ. Using (2.2) andLebesgue’s dominated convergence theorem, for all z ∈ B we obtain

f1(z) =

∫R

(eizx − 1

)F (dx) = α

∫ ∞0

(eizx − 1

) e−λxx1+β

dx

= α

∫ ∞0

( ∞∑n=1

(izx)n

n!

)e−λx

x1+βdx = α

∞∑n=1

(iz)n

n!

∫ ∞0

xn−β−1e−λxdx

= α

∞∑n=1

(iz)n

n!

∫ ∞0

(x

λ

)n−β−1

e−x1

λdx = α

∞∑n=1

(iz)n

n!λβ−nΓ(n− β)

= α

∞∑n=1

(iz)n

n!λβ−nΓ(−β)

n−1∏k=0

(k − β) = αΓ(−β)λβ∞∑n=1

(iz

λ

)n(−1)n

n∏k=1

β − k + 1

k

= αΓ(−β)λβ∞∑n=1

n

)(− iz

λ

)n= αΓ(−β)λβ

[(1− iz

λ

)β− 1

]= αΓ(−β)

[(λ− iz)β − λβ

]= f2(z).

By the identity theorem for analytic functions we deduce that f1 ≡ f2 on G, whichin particular yields that f1 ≡ f2 on R. In view of (2.1), this proves (2.6).

2.6. Lemma. Suppose that β+, β− ∈ (0, 1). The tempered stable distribution

η = TS(α+, β+, λ+;α−, β−, λ−)

has the characteristic function

(2.7)ϕ(z) = exp

(α+Γ(−β+)

[(λ+ − iz)β

+

− (λ+)β+]

+ α−Γ(−β−)[(λ− + iz)β

−− (λ−)β

−]), z ∈ R

where the powers stem from the main branch of the complex logarithm.

Proof. This is an immediate consequence of Lemma 2.5.

According to equation (2.2) in [30], the characteristic function of a bilateralGamma distribution (i.e. β+ = β− = 0) is given by

ϕ(z) =

(λ+

λ+ − iz

)α+(λ−

λ− + iz

)α−, z ∈ R(2.8)

where the powers stem from the main branch of the complex logarithm.

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TEMPERED STABLE DISTRIBUTIONS 5

Using Lemma 2.6, for β+, β− ∈ (0, 1) the cumulant generating function

Ψ(z) = lnE[ezX ] (where X ∼ TS(α+, β+, λ+;α−, β−, λ−))

exists on [−λ−, λ+] and is given by

(2.9)Ψ(z) = α+Γ(−β+)

[(λ+ − z)β

+

− (λ+)β+]

+ α−Γ(−β−)[(λ− + z)β

−− (λ−)β

−], z ∈ [−λ−, λ+].

For a bilateral Gamma distribution (i.e. β+ = β− = 0), the cumulant generatingfunction exists on (−λ−, λ+) and is given by

Ψ(z) = α+ ln

(λ+

λ+ − z

)+ α− ln

(λ−

λ− + z

), z ∈ (−λ−, λ+)(2.10)

see [30, Sec. 2]. Hence, for all β+, β− ∈ [0, 1) the n-th order cumulant κn =dn

dznΨ(z)|z=0 is given by

κn = Γ(n− β+)α+

(λ+)n−β+ + (−1)nΓ(n− β−)α−

(λ−)n−β−, n ∈ N = 1, 2, . . ..

(2.11)

In particular, for a random variable

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

we can specify:

• The expectation

E[X] = κ1 = Γ(1− β+)α+

(λ+)1−β+ − Γ(1− β−)α−

(λ−)1−β− .(2.12)

• The variance

Var[X] = κ2 = Γ(2− β+)α+

(λ+)2−β+ + Γ(2− β−)α−

(λ−)2−β− .(2.13)

• The Charliers skewness

γ1(X) =κ3

κ3/22

=Γ(3− β+) α+

(λ+)3−β+ − Γ(3− β−) α−

(λ−)3−β−(Γ(2− β+) α+

(λ+)2−β+ + Γ(2− β−) α−

(λ−)2−β−

)3/2.(2.14)

• The kurtosis

γ2(X) = 3 +κ4

κ22

= 3 +Γ(4− β+) α+

(λ+)4−β+ + Γ(4− β−) α−

(λ−)4−β−(Γ(2− β+) α+

(λ+)2−β+ + Γ(2− β−) α−

(λ−)2−β−

)2 .(2.15)

2.7. Remark. Let η = TS(α, β, λ) be a one-sided tempered stable distribution. Forβ ∈ (0, 1) the characteristic function is given by (2.6), see Lemma 2.5, and hence,the cumulant generating function exists on (−∞, λ] and is given by

Ψ(z) = αΓ(−β)[(λ− z)β − λβ

], z ∈ (−∞, λ].(2.16)

For β = 0 the tempered stable distribution is a Gamma distribution η = Γ(α, λ).Therefore, we have the characteristic function

ϕ(z) =

λ− iz

)α, z ∈ R(2.17)

and the cumulant generating function exists on (−∞, λ) and is given by

Ψ(z) = α ln

λ− z

), z ∈ (−∞, λ).(2.18)

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6 UWE KUCHLER AND STEFAN TAPPE

For β ∈ [0, 1) and X ∼ TS(α, β, λ) we obtain the cumulants

κn = Γ(n− β)α

λn−β, n ∈ N,(2.19)

the expectation and the variance

E[X] = Γ(1− β)α

λ1−β ,(2.20)

Var[X] = Γ(2− β)α

λ2−β .(2.21)

2.8. Remark. The characteristic function (2.4) of the tempered stable distribution

η = TS(α+, β+, λ+;α−, β−, λ−)

with β+, β− < 2 is given by

ϕ(z) = exp(izγ + α+Γ(−β+)

[(λ+ − iz)β

+

− (λ+)β+

+ izβ+(λ+)β+−1

]+ α−Γ(−β−)

[(λ− + iz)β

−− (λ−)β

−− izβ−(λ−)β

−−1]), z ∈ R.

Therefore, the cumulant generating function exists on [−λ−, λ+] and is given by

Ψ(z) = γz + α+Γ(−β+)[(λ+ − z)β

+

− (λ+)β+

+ β+(λ+)β+−1z

]+ α−Γ(−β−)

[(λ− + z)β

−− (λ−)β

−− β−(λ−)β

−−1z], z ∈ [−λ−, λ+].

Hence, we have κ1 = γ and the cumulants κn for n ≥ 2 are given by (2.11).

For a tempered stable process X we shall also write

X ∼ TS(α+, β+, λ+;α−, β−, λ−).

Note that we can decompose X = X+ − X− as the difference of two indepen-dent one-sided tempered stable subordinators X+ ∼ TS(α+, β+, λ+) and X− ∼TS(α−, β−, λ−). In view of the characteristic function (2.7) calculated in Lemma2.6, all increments of X have a tempered stable distribution, more precisely

Xt −Xs ∼ TS(α+(t− s), β+, λ+;α−(t− s), β−, λ−) for 0 ≤ s < t.(2.22)

In particular, for any constant ∆ > 0 the process X∆• = (X∆t)t≥0 is a temperedstable process

X∆• ∼ TS(∆α+, β+, λ+; ∆α−, β−, λ−).(2.23)

3. Closure properties of tempered stable distributions

In this section, we shall investigate limit distributions of sequences of temperedstable distributions.

3.1. Proposition. Let sequences

(α+n , β

+n , λ

+n ;α−n , β

−n , λ

−n )n∈N ⊂ ((0,∞)× [0, 1)× (0,∞))2

and real numbers

(α+, β+, λ+;α−, β−, λ−) ∈ ((0,∞)× [0, 1)× (0,∞))2

be given. Then, the following statements are valid:

(1) If we have

(α+n , β

+n , λ

+n ;α−n , β

−n , λ

−n )→ (α+, β+, λ+;α−, β−, λ−),

then we have the weak convergence

TS(α+n , β

+n , λ

+n ;α−n , β

−n , λ

−n )

w→ TS(α+, β+, λ+;α−, β−, λ−).(3.1)

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TEMPERED STABLE DISTRIBUTIONS 7

(2) If we have α−n → 0 and

(α+n , β

+n , λ

+n ;β−n , λ

−n )→ (α+, β+, λ+;β−, λ−),

then we have the weak convergence

TS(α+n , β

+n , λ

+n ;α−n , β

−n , λ

−n )

w→ TS(α+, β+, λ+).(3.2)

(3) If we have α+n , α

−n → 0 and

(β+n , λ

+n ;β−n , λ

−n )→ (β+, λ+;β−, λ−),

then we have the weak convergence

TS(α+n , β

+n , λ

+n ;α−n , β

−n , λ

−n )

w→ δ0.(3.3)

(4) If we have α+n , α

−n , λ

+n , λ

−n →∞ and there are µ+, µ− ≥ 0 such that

Γ(1− β+)α+n

(λ+n )1−β+

→ µ+ and Γ(1− β−)α−n

(λ−n )1−β− → µ−,(3.4)

then we have the weak convergence

TS(α+n , β

+, λ+n ;α−n , β

−, λ−n )w→ δµ,(3.5)

where the number µ ∈ R is the limit of the means

µ = µ+ − µ−.

(5) If we have α+n , α

−n , λ

+n , λ

−n → ∞ and there are µ ∈ R, (σ+)2, (σ−)2 > 0

such that

Γ(1− β+)α+n

(λ+n )1−β+

− Γ(1− β−)α−n

(λ−n )1−β− → µ,(3.6)

Γ(2− β+)α+n

(λ+n )2−β+

→ (σ+)2 and Γ(2− β−)α−n

(λ−n )2−β− → (σ−)2,(3.7)

then we have the weak convergence

TS(α+n , β

+, λ+n ;α−n , β

−, λ−n )w→ N(µ, σ2),(3.8)

where the variance σ2 > 0 is given by

σ2 = (σ+)2 + (σ−)2.

Proof. In order to prove the result, it suffices to show that the respective charac-teristic functions converge. Noting that, by l’Hopital’s rule, for all λ > 0 and z ∈ Rwe have

limβ↓0

Γ(−β)[(λ− iz)β − λβ

]= lim

β↓0Γ(−β)λβ

[(λ− izλ

)β− 1

]

= − limβ↓0

Γ(1− β)λβ(λ−izλ

)β − 1

β= − lim

β↓0

ddβ

((λ−izλ

)β − 1)

ddββ

= − limβ↓0

(λ− izλ

)βln

(λ− izλ

)= ln

λ− iz

),

relations (3.1)–(3.3) follow by taking into account the representations (2.7), (2.8)and (2.6), (2.17) of the characteristic functions.

In the sequel, for n ∈ N we denote by ϕn : R→ C the characteristic function ofthe tempered stable distribution

TS(α+n , β

+, λ+n ;α−n , β

−, λ−n )

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8 UWE KUCHLER AND STEFAN TAPPE

and by (κnj )j∈N its cumulants. Then we have

ϕn(u) = exp

( ∞∑j=1

(iu)j

j!κnj

), u ∈ R.

Suppose that (3.4) is satisfied. Using the estimates

Γ(j − β+) ≤ (j − 1)! and Γ(j − β−) ≤ (j − 1)! for j ≥ 2,(3.9)

by the geometric series and (3.4), for all u ∈ R we obtain

∞∑j=2

∣∣∣∣ (iu)j

j!

(Γ(j − β+)

α+n

(λ+n )j−β+

+ (−1)jΓ(j − β−)α−n

(λ−n )j−β−

)∣∣∣∣≤ α+

n

(λ+n )1−β+

|u|∞∑j=2

(|u|λ+n

)j−1

+α−n

(λ−n )1−β− |u|∞∑j=2

(|u|λ−n

)j−1

=α+n

(λ+n )1−β+

|u|2

λ+n − |u|

+α−n

(λ−n )1−β−|u|2

λ−n − |u|→ 0 as n→∞,

and hence, by taking into account (2.11) and (3.4), we have

ϕn(u)→ eiuµ, u ∈ R

proving (3.5). Now, suppose that (3.6), (3.7) are satisfied. Using the estimates (3.9),by the geometric series and (3.7), for all u ∈ R we obtain

∞∑j=3

∣∣∣∣ (iu)j

j!

(Γ(j − β+)

α+n

(λ+n )j−β+

+ (−1)jΓ(j − β−)α−n

(λ−n )j−β−

)∣∣∣∣≤ α+

n

(λ+n )2−β+

|u|2∞∑j=3

(|u|λ+n

)j−2

+α−n

(λ−n )2−β− |u|2∞∑j=3

(|u|λ−n

)j−2

=α+n

(λ+n )2−β+

|u|3

λ+n − |u|

+α−n

(λ−n )2−β−|u|3

λ−n − |u|→ 0 as n→∞,

and hence, by taking into account (2.11) and (3.6), (3.7), we have

ϕn(u)→ eiuµ−u2σ2/2, u ∈ R

proving (3.8).

By (3.1), the class of tempered stable distributions is closed under weak conver-gence on its domain

((0,∞)× [0, 1)× (0,∞))2.

Note that bilateral Gamma distributions (corresponding to β+ = 0 and β− = 0)belong to this domain and are contained in its boundary. For α+ → 0 or α− → 0we obtain one-sided tempered stable distributions and Dirac measures as bound-ary distributions, see (3.2) and (3.3). If α+, α−, λ+, λ− → ∞ for fixed valued ofβ+, β−, in certain situations we obtain a Dirac measure, see (3.5), or to a normaldistribution, see (3.8), as limit distribution.

4. Convergence of tempered stable distributions to a normaldistribution

In [40, Sec. 3] the long time behaviour of tempered stable processes was studiedand convergence to a Brownian motion was established. Here, we provide a conver-gence rate and show, how close a given tempered stable distribution (or temperedstable process) is to a normal distribution (or Brownian motion).

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TEMPERED STABLE DISTRIBUTIONS 9

4.1. Lemma. The following statements are valid:

(1) Suppose Xi ∼ TS(α+i , β

+, λ+;α−i , β−, λ−), i = 1, 2 are independent. Then

we have

X1 +X2 ∼ TS(α+1 + α+

2 , β+, λ+;α−1 + α−2 , β

−, λ−).(4.1)

(2) For X ∼ TS(α+, β+, λ+;α−, β−, λ−) and a constant ρ > 0 we have

ρX ∼ TS(α+ρβ+

, β+, λ+/ρ;α−ρβ−, β−, λ−/ρ).(4.2)

Proof. For independent Xi ∼ TS(α+i , β

+, λ+;α−i , β−, λ−), i = 1, 2 we have by

Lemma 2.6 the characteristic function

ϕX1+X2(z) = ϕX1(z)ϕX2(z)

= exp(α+

1 Γ(−β+)[(λ+ − iz)β

+

− (λ+)β+]

+ α−1 Γ(−β−)[(λ− + iz)β

−− (λ−)β

−])× exp

(α+

2 Γ(−β+)[(λ+ − iz)β

+

− (λ+)β+]

+ α−2 Γ(−β−)[(λ− + iz)β

−− (λ−)β

−])= exp

((α+

1 + α+2 )Γ(−β+)

[(λ+ − iz)β

+

− (λ+)β+]

+ (α−1 + α−2 )Γ(−β−)[(λ− + iz)β

−− (λ−)β

−]),

showing (4.1). For X ∼ TS(α+, β+, λ+;α−, β−, λ−) and c > 0 we have by Lemma2.6 the characteristic function

ϕρX(z) = ϕX(ρz) = exp(α+Γ(−β+)

[(λ+ − iρz)β

+

− (λ+)β+]

+ α−Γ(−β−)[(λ− + iρz)β

−− (λ−)β

−])= exp

(α+ρβ

+

Γ(−β+)[(λ+/ρ− iz)β

+

− (λ+/ρ)β+]

+ α−ρβ−

Γ(−β−)[(λ−/ρ+ iz)β

−− (λ−/ρ)β

−]),

showing (4.2).

4.2. Lemma. Let X be a random variable

X ∼ TS(α+, β+, λ+;α−, β−, λ−).

We set µ := E[X] and σ2 := Var[X]. Let ρ, τ > 0 and

Y ∼ TS(ρα+/(τ√ρ)β

+

, β+, λ+τ√ρ; ρα−/(τ

√ρ)β

−, β−, λ−τ

√ρ).

Then we have

E[Y ] =

√ρ

τµ and Var[Y ] =

σ2

τ2.

Proof. Note that by (2.12), (2.13) we have

µ = Γ(1− β+)α+

(λ+)1−β+ − Γ(1− β−)α−

(λ−)1−β− ,

σ2 = Γ(2− β+)α+

(λ+)2−β+ + Γ(2− β−)α−

(λ−)2−β− .

Therefore, we obtain

E[Y ] = Γ(1− β+)ρα+

(τ√ρ)β+(λ+τ

√ρ)1−β+ − Γ(1− β−)

ρα−

(τ√ρ)β−(λ−τ

√ρ)1−β−

=

√ρ

τ

(Γ(1− β+)

α+

(λ+)1−β+ − Γ(1− β−)α−

(λ−)1−β−

)=

√ρ

τµ

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10 UWE KUCHLER AND STEFAN TAPPE

as well as

Var[Y ] = Γ(2− β+)ρα+

(τ√ρ)β+(λ+τ

√ρ)2−β+ + Γ(2− β−)

ρα−

(τ√ρ)β−(λ−τ

√ρ)2−β−

=1

τ2

(Γ(2− β+)

α+

(λ+)2−β+ + Γ(2− β−)α−

(λ−)2−β−

)=σ2

τ2,

finishing the proof.

4.3. Lemma. Let X be a random variable

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

and let µ ∈ R and σ2 > 0 be arbitrary. The following statements are equivalent:

(1) We have E[X] = µ and Var[X] = σ2.(2) We have

α+ =(λ+)2−β+

((1− β−)µ+ λ−σ2)

Γ(1− β+)((1− β−)λ+ + (1− β+)λ−),(4.3)

α− =(λ−)2−β−((β+ − 1)µ+ λ+σ2)

Γ(1− β−)((1− β−)λ+ + (1− β+)λ−).(4.4)

Proof. Let A ∈ R2×2 be the matrix

A =

(a11 a12

a21 a22

)=

(Γ(1− β+)/(λ+)1−β+ −Γ(1− β−)/(λ−)1−β−

Γ(2− β+)/(λ+)2−β+

Γ(2− β−)/(λ+)2−β−

).

Then we have

(4.5)

detA =Γ(1− β+)Γ(2− β−)

(λ+)1−β+(λ−)2−β− +Γ(2− β+)Γ(1− β−)

(λ+)2−β+(λ−)1−β−

=Γ(1− β+)Γ(1− β−)

(λ+)2−β+(λ−)2−β−

(λ+(1− β−) + λ−(1− β+)

)> 0.

Using (2.12), (2.13), a straightforward calculation shows that

a11α+ = λ+ (1− β−)µ+ λ−σ2

(1− β−)λ+ + (1− β+)λ−,(4.6)

a12α− = −λ− −(1− β+)µ+ λ+σ2

(1− β−)λ+ + (1− β+)λ−,(4.7)

a21α+ = (1− β+)

(1− β−)µ+ λ−σ2

(1− β−)λ+ + (1− β+)λ−,(4.8)

a22α− = (1− β−)

−(1− β+)µ+ λ+σ2

(1− β−)λ+ + (1− β+)λ−.(4.9)

By (2.12), (2.13), we have E[X] = µ and Var[X] = σ2 if and only if

A ·(α+

α−

)=

(µσ2

).(4.10)

Because of (4.5), the system of linear equations (4.10) has a unique solution. Takinginto account (4.6)–(4.9), the solution for (4.10) is given by (4.3), (4.4).

4.4. Lemma. For X ∼ TS(α, β, λ) we have

E[X3] = Γ(1− β)α

λ3−β

[Γ(1− β)2α2λ2β + 3(1− β)Γ(1− β)αλβ + (1− β)(2− β)

].

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TEMPERED STABLE DISTRIBUTIONS 11

Proof. Denoting by κ1, κ2, κ3 the first three cumulants of X, by [38, p. 346] and(2.19) we obtain the third moment

E[X3] = κ31 + 3κ1κ2 + κ3

=

(Γ(1− β)

α

λ1−β

)3

+ 3Γ(1− β)α

λ1−β Γ(2− β)α

λ2−β + Γ(3− β)α

λ3−β

= Γ(1− β)α

λ3−β

[Γ(1− β)2α2λ2β + 3(1− β)Γ(1− β)αλβ + (1− β)(2− β)

],

completing the proof.

In the sequel, for µ ∈ R and σ2 > 0 the function Φµ,σ2 denotes the distributionfunction of the normal distribution N(µ, σ2). Moreover, c > 0 denotes the constantfrom the Berry-Esseen theorem. The current best estimate is c ≤ 0.4784, see [28,Cor. 1].

4.5. Proposition. There exists a function g : [0, 1)2 × (0,∞)2 × R→ R such thatfor any fixed β+, β− ∈ [0, 1) we have

g(β+, β−, λ+, λ−, µ)→ 0 as λ+, λ−, µ→ 0,(4.11)

and for any random variable

X ∼ TS(α+, β+, λ+;α−, β−, λ−),

all n ∈ N and any random variable

Xn ∼ TS((√n

2−β+

/σ)α+, β+, λ+σ√n; (√n

2−β−/σ)α−, β−, λ−σ

√n)(4.12)

we have

supx∈R|Gn(x)− Φ0,1(x)|

≤ 32c

[(1− β+)(2− β+)

(1− β−)√nµ/σ3 +

√nλ−/σ√

nλ+((1− β−)√nλ+ + (1− β+)

√nλ−)

+ (1− β−)(2− β−)(β+ − 1)

√nµ/σ3 +

√nλ+/σ√

nλ−((1− β−)√nλ+ + (1− β+)

√nλ−)

+g(β+, β−, λ+, λ−, µ)

σ3

],

where µ := E[X], σ2 := Var[X] and Gn denotes the distribution function of therandom variable Xn −

√nµ/σ.

Proof. We define the functions gi : [0, 1)2 × (0,∞)2 × R→ R, i = 1, 2 as

g1(β+, β−, λ+, λ−, µ) :=(1− β−)µ+ λ−σ2

λ+((1− β−)λ+ + (1− β+)λ−)

×[(

(λ+)2((1− β−)µ+ λ−σ2)

(1− β−)λ+ + (1− β+)λ−

)2

+ 3(1− β+)(λ+)2((1− β−)µ+ λ−σ2)

(1− β−)λ+ + (1− β+)λ−

],

g2(β+, β−, λ+, λ−, µ) :=(β+ − 1)µ+ λ+σ2

λ−((1− β−)λ+ + (1− β+)λ−)

×[(

(λ−)2((β+ − 1)µ+ λ+σ2)

(1− β−)λ+ + (1− β+)λ−

)2

+ 3(1− β−)(λ−)2((β+ − 1)µ+ λ+σ2)

(1− β−)λ+ + (1− β+)λ−

],

and let g : [0, 1)2 × (0,∞)2 × R→ R be the function

g := 32(g1 + g2).

Then, for any fixed β+, β− ∈ [0, 1) we have (4.11).

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12 UWE KUCHLER AND STEFAN TAPPE

Now, let (Yj)j∈N be an i.i.d. sequence of random variables with L(Yj) = L(X)for all j ∈ N. We define the sequence (Sn)n∈N as Sn :=

∑nj=1 Yj for n ∈ N. By

Lemma 4.1 we have

L(Snσ√n

)= TS((

√n

2−β+

/σβ+

)α+, β+, λ+σ√n; (√n

2−β−/σβ

−)α−, β−, λ−σ

√n)

= L(Xn) for all n ∈ N,

and therefore

L(Xn −

√nµ

σ

)= L

(Snσ√n−√nµ

σ

)= L

(Sn − nµσ√n

)for all n ∈ N.

By the Berry-Esseen theorem (see, e.g., [10, Thm. 2.4.9]) we have

supx∈R|Gn(x)− Φ0,1(x)| ≤ cE[|X − µ|3]

σ3√n

for all n ∈ N.

We have X = X+−X− with independent random variables X+ ∼ TS(α+, β+, λ+)and X− ∼ TS(α−, β−, λ−), and therefore, using Holder’s inequality and Jensen’sinequality we estimate

E[|X − µ|3] = E[|X+ −X− − (E[X+]− E[X−])|3]

≤ E[(X+ + E[X+] +X− + E[X−])3]

≤ 42E[(X+)3 + E[X+]3 + (X−)3 + E[X−]3]

= 16(E[(X+)3] + E[X+]3 + E[(X−)3] + E[X−]3

)≤ 32

(E[(X+)3] + E[(X−)3]

).

Using Lemma 4.4 we have

E[(X+)3] = Γ(1− β+)α+

(λ+)3−β+

[Γ(1− β+)2(α+)2(λ+)2β+

+ 3(1− β+)Γ(1− β+)α+(λ+)β+

+ (1− β+)(2− β+)],

E[(X−)3] = Γ(1− β−)α−

(λ−)3−β−[Γ(1− β−)2(α−)2(λ−)2β−

+ 3(1− β−)Γ(1− β−)α−(λ−)β−

+ (1− β−)(2− β−)].

Inserting identities (4.3), (4.4) from Lemma 4.3 yields

E[(X+)3] = (1− β+)(2− β+)(1− β−)µ+ λ−σ2

λ+((1− β−)λ+ + (1− β+)λ−)

+ g1(β+, β−, λ+, λ−, µ),

E[(X−)3] = (1− β−)(2− β−)(β+ − 1)µ+ λ+σ2

λ−((1− β−)λ+ + (1− β+)λ−)

+ g2(β+, β−, λ+, λ−, µ).

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TEMPERED STABLE DISTRIBUTIONS 13

Therefore, for all n ∈ N we conclude

supx∈R|Gn(x)− Φ0,1(x)| ≤ 32c

σ3√n

[(1− β+)(2− β+)

(1− β−)µ+ λ−σ2

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)µ+ λ+σ2

λ−((1− β−)λ+ + (1− β+)λ−)+ g(β+, β−, λ+, λ−, µ)

]= 32c

[(1− β+)(2− β+)

(1− β−)√nµ/σ3 +

√nλ−/σ√

nλ+((1− β−)√nλ+ + (1− β+)

√nλ−)

+ (1− β−)(2− β−)(β+ − 1)

√nµ/σ3 +

√nλ+/σ√

nλ−((1− β−)√nλ+ + (1− β+)

√nλ−)

+g(β+, β−, λ+, λ−, µ)

σ3

],

finishing the proof.

4.6. Proposition. For any random variable

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

with Var[X] = 1 we have

supx∈R|GX−µ(x)− Φ0,1(x)| ≤ 32c

[(1− β+)(2− β+)

(1− β−)µ+ λ−

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)µ+ λ+

λ−((1− β−)λ+ + (1− β+)λ−)

],

where µ := E[X].

Proof. Let g : [0, 1)2×(0,∞)2×R→ R the function from Proposition 4.5. It sufficesto show that for each ε > 0 we have(4.13)

supx∈R|GX−µ(x)− Φ0,1(x)| ≤ 32c

[(1− β+)(2− β+)

(1− β−)µ/σ3 + λ−/σ

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)µ/σ3 + λ+/σ

λ−((1− β−)λ+ + (1− β+)λ−)+ ε

].

Let ε > 0 be arbitrary. There exists n ∈ N such that

g(β+, β−, λ+/√n, λ−/

√n, µ/

√n) ≤ ε.(4.14)

Let Y be a random variable

Y ∼ TS(α+/√n

2−β+

, β+, λ+/√n;α−/

√n

2−β−, β−, λ−/

√n).

Applying Lemma 4.2 with ρ = 1/n and τ = 1 we obtain

E[Y ] = µ/√n and Var[Y ] = 1 for all n ∈ N.

Hence, defining the random variable Yn according to (4.12), we have

L(Yn) = TS(√n

2−β+

(α+/√n

2−β+

), β+,√n(λ+/

√n);

√n

2−β−(α−/

√n

2−β−), β−,

√n(λ−/

√n))

= TS(α+, β+, λ+;α−, β−, λ−) = L(X).

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14 UWE KUCHLER AND STEFAN TAPPE

By Proposition 4.5 we deduce

supx∈R|GX−µ(x)− Φ0,1(x)| = sup

x∈R|GYn−µ(x)− Φ0,1(x)|

≤ 32c

[(1− β+)(2− β+)

(1− β−)µ+ λ−

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)µ+ λ+

λ−((1− β−)λ+ + (1− β+)λ−)

+ g(β+, β−, λ+/√n, λ−/

√n, µ/

√n)

],

and, by virtue of estimate (4.14), we arrive at (4.13).

4.7. Theorem. For any random variable

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

we have the estimate

supx∈R|GX(x)− Φµ,σ2(x)| ≤ 32c

[(1− β+)(2− β+)

(1− β−)µ/σ3 + λ−/σ

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)µ/σ3 + λ+/σ

λ−((1− β−)λ+ + (1− β+)λ−)

],

where µ := E[X] and σ2 := Var[X].

Proof. By Lemma 4.1, the random variable X/σ has the distribution

X/σ ∼ TS(α+/σβ+

, β+, λ+σ;α−/σβ−, β−, λ−σ),

and applying Lemma 4.2 with ρ = 1 and τ = σ yields that

E[X/σ] = µ/σ and Var[X/σ] = 1.

Moreover, since

GX(x) = GX/σ−µ/σ

(x− µσ

)and Φµ,σ2(x) = Φ0,1

(x− µσ

)for all x ∈ R,

applying Proposition 4.6 gives us

supx∈R|GX(x)− Φµ,σ2(x)| = sup

x∈R|GX/σ−µ/σ(x)− Φ0,1(x)|

≤ 32c

[(1− β+)(2− β+)

(1− β−)µ/σ + λ−σ

λ+σ((1− β−)λ+σ + (1− β+)λ−σ)

+ (1− β−)(2− β−)(β+ − 1)µ/σ + λ+σ

λ−σ((1− β−)λ+σ + (1− β+)λ−σ)

],

= 32c

[(1− β+)(2− β+)

(1− β−)µ/σ3 + λ−/σ

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)µ/σ3 + λ+/σ

λ−((1− β−)λ+ + (1− β+)λ−)

],

which completes the proof.

Theorem 4.7 tells us, how close the distribution of a tempered stable randomvariable X is to the normal distribution N(µ, σ2) with µ = E[X] and σ2 = Var[X].In the upcoming result, for given values of µ ∈ R and σ2 > 0 we construct a sequenceof tempered stable distributions, which converges weakly to N(µ, σ2), and providea convergence rate.

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TEMPERED STABLE DISTRIBUTIONS 15

4.8. Corollary. Let µ ∈ R, σ2 > 0 and

(β+, λ+;β−, λ−) ∈ ([0, 1)× (0,∞))2

be arbitrary. For each n ∈ N we define

α+n :=

(λ+)2−β+

((1− β−)µ+ λ−σ2√n)

Γ(1− β+)((1− β−)λ+ + (1− β+)λ−)

√n

1−β+

,(4.15)

λ+n := λ+

√n,(4.16)

α−n :=(λ−)2−β−((β+ − 1)µ+ λ+σ2

√n)

Γ(1− β−)((1− β−)λ+ + (1− β+)λ−)

√n

1−β−,(4.17)

λ−n := λ−√n.(4.18)

Then, there exists an index n0 ∈ N with α+n > 0 and α−n > 0 for all integers n ≥ n0,

and for any sequence (Xn)n≥n0of random variables with

Xn ∼ TS(α+n , β

+, λ+n ;α−n , β

−, λ−n ), n ≥ n0

we have the estimate(4.19)supx∈R|GXn(x)− Φµ,σ2(x)|

≤ 32c√n

[(1− β+)(2− β+)

λ+((1− β−)λ+ + (1− β+)λ−)

((1− β−)µ

σ3√n

+λ−

σ

)+

(1− β−)(2− β−)

λ+((1− β−)λ+ + (1− β+)λ−)

((β+ − 1)µ

σ3√n

+λ+

σ

)]→ 0 for n→∞.

Proof. The existence of an index n0 ∈ N with α+n > 0 and α−n > 0 for all n ≥ n0

immediately follows from the Definitions (4.15), (4.17) of α+n , α

−n . By Lemma 4.3

we have E[Xn] = µ and Var[Xn] = σ2 for all n ≥ n0. Applying Theorem 4.7 yieldsthe asserted estimate (4.19).

4.9. Remark. The Definitions (4.15)–(4.18) imply that

Γ(1− β+)α+n

(λ+n )1−β+

− Γ(1− β−)α−n

(λ−n )1−β−

=λ+((1− β−)µ+ λ−σ2

√n)

(1− β−)λ+ + (1− β+)λ−− λ−((β+ − 1)µ+ λ+σ2

√n)

(1− β−)λ+ + (1− β+)λ−= µ

for all n ≥ n0, the convergences

α+n

(λ+n )2−β+

→ λ−σ2

Γ(1− β+)((1− β−)λ+ + (1− β+)λ−)=: (σ+)2,

α−n(λ−n )2−β− →

λ+σ2

Γ(1− β−)((1− β−)λ+ + (1− β+)λ−)=: (σ−)2

for n→∞ as well as

Γ(2− β+)(σ+)2 + Γ(2− β−)(σ−)2 =(1− β+)λ− + (1− β−)λ+

(1− β−)λ+ + (1− β+)λ−σ2 = σ2.

Consequently, conditions (3.6), (3.7) are satisfied, and hence, Proposition 3.1 yieldsthe weak convergence (3.8). In addition, Corollary 4.8 provides the convergence rate(4.19).

4.10. Theorem. For any tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

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16 UWE KUCHLER AND STEFAN TAPPE

we have the estimate(4.20)

supx∈R|GXt(x)−GWt(x)| ≤ 32c√

t

[(1− β+)(2− β+)

(1− β−)µ/σ3 + λ−/σ

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)µ/σ3 + λ+/σ

λ−((1− β−)λ+ + (1− β+)λ−)

]→ 0 for t→∞,

where µ := E[X1], σ2 := Var[X1] and W is a Brownian motion with W1 ∼ N(µ, σ2).

Proof. Noting that by (2.12), (2.13) and (2.22) we have E[Xt] = tµ and Var[Xt] =tσ2 for all t > 0, applying Theorem 4.7 yields

supx∈R|GXt(x)−GWt(x)| = sup

x∈R|GXt(x)− Φµt,σ2t(x)|

≤ 32c

[(1− β+)(2− β+)

(1− β−)tµ/t3/2σ3 + λ−/t1/2σ

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)tµ/t3/2σ3 + λ+/t1/2σ

λ−((1− β−)λ+ + (1− β+)λ−)

]=

32c√t

[(1− β+)(2− β+)

(1− β−)µ/σ3 + λ−/σ

λ+((1− β−)λ+ + (1− β+)λ−)

+ (1− β−)(2− β−)(β+ − 1)µ/σ3 + λ+/σ

λ−((1− β−)λ+ + (1− β+)λ−)

]→ 0 for t→∞,

completing the proof.

4.11. Corollary. If we choose µ ∈ R, σ2 > 0 and

(β+, λ+;β−, λ−) ∈ ([0, 1)× (0,∞))2

and choose

α+ :=(λ+)2−β+

((1− β−)µ+ λ−σ2)

Γ(1− β+)((1− β−)λ+ + (1− β+)λ−)> 0,(4.21)

α− :=(λ−)2−β−((β+ − 1)µ+ λ+σ2)

Γ(1− β−)((1− β−)λ+ + (1− β+)λ−)> 0,(4.22)

then for any tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

estimate (4.20) is valid.

Proof. By Lemma 4.3 we have E[X1] = µ and Var[X1] = σ2. Applying Theorem4.10 yields the desired estimate (4.20).

4.12. Remark. Note that conditions (4.21), (4.22) are always satisfied for µ = 0.

5. Laws of large numbers for tempered stable distributions

In this section, we present laws of large numbers for tempered stable distri-butions. These results will be useful in order to determine parameters from theobservation of a typical trajectory of a tempered stable process.

5.1. Proposition. We have the weak convergence

TS(n1−β+

α+, β+, nλ+;n1−β−α−, β−, nλ−)w→ δµ for n→∞,(5.1)

where the number µ ∈ R equals the mean

µ = Γ(1− β+)α+

(λ+)1−β+ − Γ(1− β−)α−

(λ−)1−β− .

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TEMPERED STABLE DISTRIBUTIONS 17

Proof. Let (Xj)j∈N be an i.i.d. sequence with

Xj ∼ TS(α+, β+, λ+;α−, β−, λ−) for j ∈ N.

By (2.12) we have E[Xj ] = µ for all j ∈ N, and Lemma 4.1 yields that

1

n

n∑j=1

Xj ∼ TS(n1−β+

α+, β+, nλ+;n1−β−α−, β−, nλ−), n ∈ N.

Using the law of large numbers, we deduce the weak convergence (5.1).

5.2. Remark. Note that we can alternatively establish the proof of Proposition 5.1by applying the weak convergence (3.5) from Proposition 3.1.

In the sequel, we shall establish results in order to determine the parametersfrom the observation of one typical sample path of a tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−).

For this, it suffices to treat the case of one-sided tempered stable processes. In-deed, we can decompose X = X+ −X−, where X+ ∼ TS(α+, β+, λ+) and X− ∼TS(α−, β−, λ−) are two independent one-sided tempered stable subordinators. SinceXt =

∑s≤t ∆Xs, the observation of a trajectory of X also provides the respective

trajectories of X+, X−, which are given by

X+t =

∑s≤t

(∆Xs)+ and X−t =

∑s≤t

(∆Xs)−.

5.3. Remark. Note that our standing assumption β+, β− ∈ [0, 1) is crucial, becausein the infinite variation case an observation of X does not provide the trajectoriesof the components X+ and X−.

Now, let X ∼ TS(α, β, λ) be a tempered stable process with β ∈ [0, 1). Supposewe observe the process at discrete time points, say X∆k, k ∈ N for some constant∆ > 0. By (2.23) we may assume, without loss of generality, that ∆ = 1. Setting

mj := E[Xj1 ] for j = 1, 2, 3, by the law of large numbers for n→∞ we have almost

surely

1

n

n∑k=1

(Xk −Xk−1)j → mj , j = 1, 2, 3.

Hence, we obtain the moments m1,m2,m3 by inspecting a typical sample path ofX. According to [38, p. 346], the cumulants (2.19) are given by

κ1 = m1,

κ2 = m2 −m21,

κ3 = m3 − 3m1m2 + 2m31.

By means of the cumulants, we can determine the parameters α, β, λ, as the fol-lowing result shows:

5.4. Proposition. The parameters α, β, λ are given by

β = 1− κ22

κ1κ3 − κ22

,(5.2)

λ = (1− β)κ1

κ2,(5.3)

α =λ1−β

Γ(1− β)κ1.(5.4)

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18 UWE KUCHLER AND STEFAN TAPPE

Proof. According to (2.19) we have

Γ(1− β)α = κ1λ1−β ,(5.5)

(1− β)Γ(1− β)α = κ2λ2−β ,(5.6)

(2− β)(1− β)Γ(1− β)α = κ3λ3−β .(5.7)

Equation (5.5) yields (5.4), and inserting (5.4) into (5.6) gives us (5.3). Note that

κ1κ3 > κ22.(5.8)

Indeed, by (2.19) we have

κ1κ3 = Γ(1− β)α

λ1−β Γ(3− β)α

λ3−β = (2− β)

(Γ(1− β)Γ(2− β)

α

λ2−β

)2

= (2− β)κ22 > κ2

2,

because β ∈ [0, 1). Inserting (5.4) into (5.7) we arrive at (5.2).

5.5. Remark. Note that (5.8) ensures that the right-hand side of (5.2) is well-defined. We emphasize that for β ∈ [1, 2) the parameters α, β, γ cannot be computedby means of the first three cumulants κ1, κ2, κ3, because κ1 does not depend on theparameters, see Remark 2.8. For analogous identities in this case, we would ratherconsider the cumulants κ2, κ3, κ4.

For the next result, we suppose that β ∈ (0, 1). Let ϕβ : R+ → (0,∞) be thestrictly decreasing function

ϕβ(x) = (1 + βx)−1β , x ∈ R+.

Let N be the random measure associated to the jumps of X. Then N is a ho-mogeneous Poisson random measure with compensator dt ⊗ F (dx). We define therandom variables (Yn)n∈N as

Yn := N((0, 1]× (ϕβ(n+ 1), ϕβ(n)]), n ∈ N.

5.6. Proposition. We have almost surely the convergence

1

n

n∑j=1

Yj → α.(5.9)

Proof. The random variables (Yn)n∈N are independent and have a Poisson distri-bution with mean αn = F ((ϕβ(n+ 1), ϕβ(n)]) for n ∈ N. The function ϕβ has thederivative

ϕ′β(x) = − 1

β(1 + βx)−

1β−1β = −(1 + βx)−

1+ββ , x ∈ R+

and thus, by substitution and Lebesgue’s dominated convergence theorem we obtain

αn = F ((ϕβ(n+ 1), ϕβ(n)]) = α

∫ ϕβ(n)

ϕβ(n+1)

e−λx

x1+βdx = α

∫ n

n+1

e−λϕβ(y)

ϕβ(y)1+βϕ′β(y)dy

= α

∫ n+1

n

e−λϕβ(y)

(1 + βy)−1+ββ

(1 + βy)−1+ββ dy = α

∫ n+1

n

e−λϕβ(y)dy

= α

∫ 1

0

e−λϕβ(n+y)dy → α for n→∞.

Consequently, we have∞∑n=1

Var[Yn]

n2=

∞∑n=1

αnn2

<∞.

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TEMPERED STABLE DISTRIBUTIONS 19

Set Sn = Y1 + . . . + Yn for n ∈ N. Applying Kolmogorov’s strong law of largenumbers, see [44, Thm. IV.3.2], yields almost surely

Sn − E[Sn]

n→ 0.

Since we have

E[Sn]

n=α1 + . . .+ αn

n→ α,

we arrive at the almost sure convergence (5.9).

5.7. Remark. Let us consider the situation β = 0. For all x ∈ R+ we have

limβ→0

ϕβ(x) = limβ→0

(1 + βx)−1β = e−x.

This suggests to take the sequence

Yn := N((0, 1]× (e−(n+1), e−n]), n ∈ N

for a Gamma process X ∼ Γ(α, λ), and then, an analogous result is indeed true,see [30, Thm. 7.1].

5.8. Remark. Proposition 5.6 and Remark 5.7 show how we can determine theparameters α, λ > 0 by inspecting a typical sample path of X, provided that β ∈ [0, 1)is known. First, we determine α > 0 according to Proposition 5.6 or Remark 5.7.By the strong law of large numbers we have almost surely Xn/n→ µ, where µ > 0denotes the expectation

µ = Γ(1− β)α

λ1−β .

Now, we obtain the parameter λ > 0 as

λ =

(αΓ(1− β)

µ

) 11−β

.

6. Statistics of tempered stable distributions

This section is devoted to parameter estimation for tempered stable distributions.Let X1, . . . , Xn be an i.i.d. sequence with

Xk ∼ TS(α+, β+, λ+;α−, β−, λ−) for k = 1, . . . , n.

Suppose we observe a realization x1, . . . , xn. We would like to estimate the vectorof parameters

ϑ = (ϑ1, . . . , ϑ6) = (α+, β+, λ+, α−, β−, λ−) ∈ D,

where the parameter domain D is the open set

D = ((0,∞)× (0, 1)× (0,∞))2.

For bilateral Gamma distributions (i.e. β+ = β− = 0) parameter estimation wasprovided in [30]. We perform the method of moments and estimate the k-th mo-

ments mj = E[Xj1 ] as

mj =1

n

n∑k=1

xjk, j = 1, . . . , 6.

According to (2.11) the vector

κ = (κ1, . . . , κ6) ∈ (R× (0,∞))3

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20 UWE KUCHLER AND STEFAN TAPPE

of cumulants is given by

κj = Γ(j − β+)α+

(λ+)j−β+ + (−1)jΓ(j − β−)α−

(λ−)j−β−, j = 1, . . . , 6.(6.1)

Using [38, p. 346] we estimate κ as the vector

κ = (κ1, . . . , κ6) ∈ (R× (0,∞))3

with components

κ1 = m1,

κ2 = m2 − m21,

κ3 = m3 − 3m1m2 + 2m31,

κ4 = m4 − 4m1m3 − 3m22 + 12m2

1m2 − 6m41,

κ5 = m5 − 5m1m4 − 10m2m3 + 20m21m3 + 30m1m

22 − 60m3

1m2 + 24m51,

κ6 = m6 − 6m1m5 − 15m2m4 + 30m21m4 − 10m2

3 + 120m1m2m3

− 120m31m3 + 30m3

2 − 270m21m

22 + 360m4

1m2 − 120m61.

and the function G : (R× (0,∞))3 ×D → R6 as

Gj(c, ϑ) := Γ(j − β+)α+(λ−)j−β−

+ (−1)jΓ(j − β−)α−(λ+)j−β+

− cj(λ+)j−β+

(λ−)j−β−, j = 1, . . . , 6,

where

c = (c1, . . . , c6) and ϑ = (α+, β+, λ+, α−, β−, λ−).

In order to obtain an estimate ϑ for ϑ we solve the equation

G(κ, ϑ) = 0, ϑ ∈ D.(6.2)

Let us have a closer look at equation (6.2) concerning existence and uniquenessof solutions. For the following calculations, we have used the Computer AlgebraSystem “Maxima”.

6.1. Lemma. We have G ∈ C1(D;R6) and for all ϑ ∈ D and κ = κ(ϑ) given by(6.1) we have G(κ, ϑ) = 0 and det ∂G∂ϑ (κ, ϑ) > 0.

Proof. The definition of G shows that G ∈ C1(D;R6). Let ϑ ∈ D be arbitrary andlet κ = κ(ϑ) be given by (6.1). The identity (6.1) yields thatG(κ, ϑ) = 0. Computing

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TEMPERED STABLE DISTRIBUTIONS 21

the partial derivatives of G and inserting the vector (κ, ϑ), for j = 1, . . . , 6 we obtain

∂Gj∂α+

(κ, ϑ) = g1(λ−)j−1

j−1∏k=1

(k − β+),

∂Gj∂α−

(κ, ϑ) = g2(−1)j(λ+)j−1

j−1∏k=1

(k − β−),

∂Gj∂β+

(κ, ϑ) = g3(λ−)j−1

j−1∏k=1

(k − β+)

(lnλ+ − ψ(1− β+)−

j−2∑k=0

1

1− β+ + k

),

∂Gj∂β−

(κ, ϑ) = g4(−1)j(λ+)j−1

j−1∏k=1

(k − β−)

(lnλ− − ψ(1− β−)−

j−2∑k=0

1

1− β− + k

),

∂Gj∂λ+

(κ, ϑ) = g5(λ−)j−1

j∏k=2

(k − β+),

∂Gj∂λ−

(κ, ϑ) = g6(−1)j(λ+)j−1

j∏k=2

(k − β−),

where ψ : (0,∞)→ R denotes the Digamma function

ψ(x) =d

dxln Γ(x) =

Γ′(x)

Γ(x), x ∈ (0,∞)

and where the gi, i = 1, . . . , 6 are given by

g1 = (λ−)1−β−Γ(1− β+),

g2 = (λ+)1−β+

Γ(1− β−),

g3 = α+(λ−)1−β−Γ(1− β+),

g4 = α−(λ+)1−β+

Γ(1− β−),

g5 = −α+(λ+)−1(λ−)1−β−Γ(2− β+),

g6 = −α−(λ−)−1(λ+)1−β+

Γ(2− β−).

Let p1 be the polynomial

p1(β+, β−) = −18(β+)3(β−)2 + 84(β+)3β− + 168(β+)2(β−)2

− 99(β+)3 − 766(β+)2β− − 507β+(β−)2

+ 885(β+)2 + 2243β+β− + 504(β−)2 − 2522β+ − 2160β− + 2356,

let p2 be the polynomial

p2(β+, β−) = −14(β+)4(β−)3 + 98(β+)4(β−)2 + 126(β+)3(β−)3

− 227(β+)4β− − 852(β+)3(β−)2 − 391(β+)2(β−)3

+ 175(β+)4 + 1908(β+)3β− + 2542(β+)2(β−)2 + 499β+(β−)3

− 1422(β+)3 − 5508(β+)2β− − 3076β+(β−)2 − 232(β−)3

+ 3989(β+)2 + 6395β+β− + 1336(β−)2 − 4490β+ − 2628β− + 1772

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22 UWE KUCHLER AND STEFAN TAPPE

and let p3 be the polynomial

p3(β+, β−) = −63(β+)4(β−)3 + 462(β+)4(β−)2 + 609(β+)3(β−)3

− 1092(β+)4β− − 4286(β+)3(β−)2 − 2112(β+)2(β−)3

+ 837(β+)4 + 9686(β+)3β− + 14198(β+)2(β−)2 + 3111β+(β−)3

− 7185(β+)3 − 30373(β+)2β− − 19850β+(β−)2 − 1689(β−)3

+ 21587(β+)2 + 39598β+β− + 10244(β−)2

− 26507β+ − 18923β− + 11748.

A plot with the Computer Algebra System “Maxima” shows that pi > 0 on (0, 1)2

for i = 1, 2, 3, and so we obtain

det∂G

∂ϑ(κ, ϑ) = g1 · · · g6(λ+λ−)3

[(1− β+)2(2− β+)3(3− β+)3(4− β+)(λ−)9

+ (1− β+)2(2− β+)3(3− β+)(4− β+)(7− 3β−)(11− 3β+)λ+(λ−)8

+ 2(1− β+)2(2− β+)(3− β+)p1(β+, β−)(λ+)2(λ−)7

+ 6(2− β+)(3− β+)p2(β+, β−)(λ+)3(λ−)6

+ 2(2− β+)(2− β−)p3(β+, β−)(λ+)4(λ−)5

+ 2(2− β−)(2− β+)p3(β−, β+)(λ+)5(λ−)4

+ 6(2− β−)(3− β−)p2(β−, β+)(λ+)6(λ−)3

+ 2(1− β−)2(2− β−)(3− β−)p1(β−, β+)(λ+)7(λ−)2

+ (1− β−)2(2− β−)3(3− β−)(4− β−)(7− 3β+)(11− 3β−)(λ+)8λ−

+ (1− β−)2(2− β−)3(3− β−)3(4− β−)(λ+)9]> 0,

finishing the proof.

Taking into account Lemma 6.1, by the implicit function theorem (see, e.g., [48,Thm. 8.1]) there exist an open neighborhood Uκ ⊂ (R × (0,∞))3 of κ, an open

neighborhood Uϑ ⊂ D of ϑ and a function g ∈ C1(Uκ;Uϑ) such for all (κ, ϑ) ∈Uκ × Uϑ we have

G(κ, g(κ)) = 0 if and only if ϑ = g(κ).

Recall that n ∈ N denotes the number of observations of the realization. If n is large

enough, then, by the law of large numbers, we have κ ∈ Uκ, and hence ϑ := g(κ)is the unique Uϑ-valued solution for (6.2). This is our estimate for the vector ϑ ofparameters.

7. Analysis of the densities of tempered stable distributions

In this section, we shall derive structural properties of the densities of temperedstable distributions. This concerns unimodality, smoothness of the densities andtheir asymptotic behaviour.

First, we deal with one-sided tempered stable distributions. Let η = TS(α, λ, β)be a one-sided tempered stable distribution with β ∈ (0, 1). Note that for β = 0 wewould have the well-known Gamma distribution.

In view of the Levy measure (2.2), we can express the characteristic function ofη as

ϕ(z) = exp

(∫R

(eizx − 1

) k(x)

xdx

), z ∈ R(7.1)

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TEMPERED STABLE DISTRIBUTIONS 23

where k : R→ R denotes the function

k(x) = αe−λx

xβ1(0,∞)(x), x ∈ R.(7.2)

Note that k > 0 on (0,∞) and that k is strictly decreasing on (0,∞). Furthermore,

we have k(0+) =∞ and∫ 1

0|k(x)|dx <∞.

It is an immediate consequence of [42, Cor. 15.11] that η is selfdecomposable,and hence absolutely continuous according to [42, Example 27.8]. In what follows,we denote by g the density of η.

Note that η is also of class L in the sense of [43]. Indeed, identity (7.1) showsthat the characteristic function is of the form (1.5) in [43] with γ0 = 0 and σ2 = 0.

7.1. Theorem. The density g of the tempered stable distribution η with β ∈ (0, 1)is of class C∞(R) and there exists a point x0 ∈ (0,∞) such that

g′(x) > 0, x ∈ (−∞, x0)(7.3)

g′(x0) = 0,(7.4)

g′(x) < 0, x ∈ (x0,∞).(7.5)

Moreover, we have

(7.6)

max

Γ(1− β)

α

λ1−β −(

3Γ(2− β)α

λ2−β

)1/2

, ξ0

< x0

< min

Γ(1− β)

α

λ1−β ,

1− β

)1/β,

where ξ0 ∈ (0,∞) denotes the unique solution of the fixed point equation

α1/β exp

(− λ

βξ0

)= ξ0.(7.7)

Proof. Since k(0+) + |k(0−)| =∞, it follows from [43, Thm. 1.2] that g ∈ C∞(R).

Furthermore, since∫ 1

0|k(x)|dx <∞, the distribution η is of type I6 in the sense

of [43, page 275]. According to [43, Thm. 1.3.vii] there exists a point x0 ∈ (0,∞)such that (7.3)–(7.5) are satisfied.

Let X be a random variable with L(X) = η. By [43, Thm. 6.1.i] and (2.20) wehave

x0 < E[X] = Γ(1− β)α

λ1−β .

Furthermore, by using [43, Thm. 6.1.ii] we have

α

xβ0 (1− β)=

α

x0

∫ x0

0

1

xβdx ≥ α

x0

∫ x0

0

e−λx

xβdx =

1

x0

∫ x0

0

k(x)dx > 1,

which implies the inequality

x0 <

1− β

)1/β

.

By [43, Thm. 6.1.v] and (2.20), (2.21) we have

x0 > E[X]−(3Var[X]

)1/2= Γ(1− β)

α

λ1−β −(

3Γ(2− β)α

λ2−β

)1/2

.

Moreover, by [43, Thm. 6.1.vi] we have x0 > ξ0, where the point ξ0 ∈ (0,∞) isgiven

ξ0 = supu > 0 : k(u) ≥ 1

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24 UWE KUCHLER AND STEFAN TAPPE

with k being the strictly decreasing function (7.2). Hence, x is the unique solution ofthe fixed point equation (7.7). Summing up, we have established relation (7.6).

The unique point x0 ∈ (0,∞) from Theorem 7.1 is called the mode of η.

7.2. Proposition. We have the asymptotic behaviour

ln g(x) ∼ −1− ββ

(αΓ(1− β))1

1−β x−β

1−β as x ↓ 0.

In particular, g(x)→ 0 as x ↓ 0.

Proof. We have η ∈ I6 in the sense of [43, page 275] and

k(x) ∼ αx−β as x ↓ 0.

Thus, the assertion follows from [43, Thm. 5.2].

We shall now investigate the asymptotic behaviour of the densities for largevalues of x. Our idea is to relate the tail of the infinitely divisible distribution tothe tail of the Levy measure. Results of this kind have been established, e.g., in [12]and [46].

We denote by ν := F the Levy measure of η, which is given by (2.2). Then wehave

ν(r) := ν((r,∞)) > 0 for all r ∈ R,

i.e. η ∈ D+ in the sense of [46]. We define the normalized Levy measure ν(1) on(1,∞) as

ν(1)(dx) :=1

ν((1,∞))1(1,∞)(x)ν(dx)

According to [46] we say that ν(1) ∈ L(γ) for some γ ≥ 0, if for all a ∈ R we have

ν(1)(r + a) ∼ e−aγν(1)(r) as r →∞.

7.3. Lemma. We have ν(1) ∈ L(λ).

Proof. Let a ∈ R be arbitrary. By l’Hopital’s rule we have

limr→∞

ν(1)(r + a)

ν(1)(r)= limr→∞

∫ ∞r+a

e−λx

x1+βdx

/∫ ∞r

e−λx

x1+βdx

= limr→∞

e−λ(r+a)

(r + a)1+β

r1+β

e−λr= e−aλ lim

r→∞

(r

r + a

)1+β

= e−aλ,

showing that ν(1) ∈ L(λ).

We define the quantity d∗ as

d∗ := lim supr→∞

ν(1) ∗ ν(1)(r)

ν(1)(r).

7.4. Lemma. We have the identity

d∗ =2α

βν((1,∞)).

Proof. We denote by g : R→ R the density of the normalized Levy measure

g(x) :=α

ν((1,∞))

e−λx

x1+β1(1,∞)(x).

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TEMPERED STABLE DISTRIBUTIONS 25

Using l’Hopital’s rule, we obtain

limr→∞

ν(1) ∗ ν(1)(r)

ν(1)(r)= limr→∞

∫ r1

∫∞1g(x− y)g(y)dydx∫ r

1g(x)dx

= limr→∞

∫∞1g(r − y)g(y)dy

g(r)

ν((1,∞))limr→∞

r1+β

e−λr

∫ r−1

1

e−λ(r−y)

(r − y)1+β

e−λy

y1+βdy

ν((1,∞))limr→∞

∫ r−1

1

(r

(r − y)y

)1+β

dy.

By symmetry, for all r ∈ (1,∞) we have∫ r−1

1

(r

(r − y)y

)1+β

dy =

∫ r/2

1

(r

(r − y)y

)1+β

dy +

∫ r−1

r/2

(r

(r − y)y

)1+β

dy

= 2

∫ r/2

1

(r

(r − y)y

)1+β

dy.

Using the estimater

r − y≤ 2 for all r ∈ (0,∞) and y ∈ [1, r/2],

by Lebesgue’s dominated convergence theorem we obtain

limr→∞

∫ ∞1

(r

(r − y)y

)1+β

1[1,r/2](y)dy =

∫ ∞1

limr→∞

(r

(r − y)y

)1+β

1[1,r/2](y)dy

=

∫ ∞1

1

y1+βdy =

1

β,

which completes the proof.

For a probability measure ρ on (R,B(R)) denote by ρ the cumulant generatingfunction

ρ(s) :=

∫Resxρ(dx).

7.5. Lemma. The following identity holds:

2ν(1)(λ) =2α

βν((1,∞)).

Proof. The computation

ν(1)(λ) =

∫Reλxν(1)(dx) =

α

ν((1,∞))

∫ ∞1

1

x1+βdx =

α

βν((1,∞))

yields the desired identity.

7.6. Lemma. We have the identity

η(λ) = exp(−αΓ(−β)λβ).

Proof. This is a direct consequence of (2.16).

7.7. Theorem. We have the asymptotic behaviour

g(x) ∼ α exp(−αΓ(−β)λβ)e−λx

x1+βas x→∞.(7.8)

Proof. By Lemma 7.3 we have ν(1) ∈ L(λ) and by Lemmas 7.4, 7.5 we have d∗ =2ν(1)(λ) <∞. Thus, [46, Thm. 2.2.ii] applies and yields C∗ = C∗ = η(λ), where

C∗ := lim infr→∞

η(r)

ν(r)and C∗ := lim sup

r→∞

η(r)

ν(r).

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26 UWE KUCHLER AND STEFAN TAPPE

Using Lemma 7.6 and l’Hopital’s rule we obtain

exp(−αΓ(−β)λβ) = limr→∞

η(r)

ν(r)= limr→∞

∫ ∞r

g(x)dx

∫ ∞r

e−λx

x1+βdx

=1

αlimr→∞

g(r)

/e−λr

r1+β,

and hence, we arrive at (7.8).

Now, we proceed with general two-sided tempered stable distributions. Let

η = TS(α+, λ+, β+;α−, λ−, β−)

be a tempered stable distribution with β+, β− ∈ (0, 1). For bilateral Gamma dis-tributions (i.e. β+ = β− = 0) the behaviour of the densities was treated in [31].

In view of the Levy measure (2.3), we can express the characteristic function ofη as (7.1), where k : R→ R denotes the function

k(x) = α+ e−λ+x

xβ+ 1(0,∞)(x)− α− e−λ−|x|

|x|β−1(−∞,0)(x), x ∈ R.

Note that k ≥ 0 on (0,∞), that k ≤ 0 on (−∞, 0) and that k is strictly decreasingon (0,∞) and (−∞, 0). Furthermore, we have k(0+) = ∞, k(0−) = −∞ and∫ 1

−1|k(x)|dx <∞.

Arguing as above, η is selfdecomposable, absolutely continuous and of class Lin the sense of [43] with characteristic function being of the form (1.5) in [43] withγ0 = 0 and σ2 = 0. In what follows, we denote by g the density of η.

Note that η = η+ ∗ η− with η+ = TS(α+, β+, γ+) and η− = ν with ν =TS(α−, β−, γ−) and ν denoting the dual of ν.

7.8. Theorem. The density g of the tempered stable distribution η is of class C∞(R)and there exists a point x0 ∈ R such that (7.3)–(7.5) are satisfied. Moreover, wehave x0 ∈ (x−0 , x

+0 ), where x+

0 denotes the mode of η+ and x−0 denotes the mode ofη−.

Proof. Since k(0+) + |k(0−)| =∞, it follows from [43, Thm. 1.2] that g ∈ C∞(R).

Furthermore, since∫ 1

−1|k(x)|dx < ∞, the distribution η is of type III6 in the

sense of [43, page 275]. According to [43, Thm. 1.3.xi] there exists a point x0 ∈ Rwith the claimed properties.

Since |k(0−)| > 1, an application of [43, Thm. 4.1.iv] yields x0 ∈ (x−0 , x+0 ).

7.9. Remark. Using Theorem 7.1, the mode x0 from Theorem 7.8 is located as

(7.9)

−min

Γ(1− β−)

α−

(λ−)1−β− ,

(α−

1− β−

)1/β−< x0

< min

Γ(1− β+)

α+

(λ+)1−β+ ,

(α+

1− β+

)1/β+.

We shall now investigate the asymptotic behaviour of the densities for largevalues of x. By symmetry, it suffices to consider the situation x→∞.

7.10. Theorem. Let g be the density of the tempered stable distribution η. Thenwe have the asymptotic behaviour

g(x) ∼ C e−λ+x

x1+β+ as x→∞,(7.10)

where the constant C > 0 is given by

C = α+ exp(− α+Γ(−β+)(λ+)β

+

+ α−Γ(−β−)[(λ+ + λ−)β

−− (λ−)β

−]).

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TEMPERED STABLE DISTRIBUTIONS 27

Proof. Let ν := F be the Levy measure of η given by (2.3). Arguing as in the proofsof Lemma 7.3 and Lemmas 7.4, 7.5 we have ν(1) ∈ L(λ+) and d∗ = 2ν(1)(γ) < ∞.

Thus, [46, Thm. 2.2.ii] applies and yields C∗ = C∗ = η(λ+). By (2.9) we have

η(λ+) =

∫Reλ

+xη(dx) = exp(Ψ(λ+))

= exp(− α+Γ(−β+)(λ+)β

+

+ α−Γ(−β−)[(λ+ + λ−)β

−− (λ−)β

−]),

and hence, proceeding as in the proof of Theorem 7.7 we arrive at (7.10).

7.11. Remark. We shall now compare the densities of general tempered stabledistributions with those of bilateral Gamma distributions (β+ = β− = 0):

• The densities of tempered stable distributions are generally not available inclosed form. For bilateral Gamma distributions we have a representation interms of the Whittaker function, see [31, Section 3].

• Theorem 7.8 and [31, Thm. 5.1] show that both, tempered stable and bilateralGamma distributions, are unimodal.

• The location of the mode x0 is generally not known, but, due to (7.9) wecan determine an interval in which it is located. For some results regardingthe mode of bilateral Gamma densities, we refer to [31, Prop. 5.2].

• According to Theorem 7.8, the densities of tempered stable distributions areof class C∞(R). This is not true for bilateral Gamma distributions, wherethe degree of smoothness depends on the parameters α+, α−. More precisely,we have g ∈ CN (R \ 0) and g ∈ CN−1(R) \CN (R), where N ∈ N0 is theunique integer such that N < α+ + α− ≤ N + 1, see [31, Thm. 4.1].

• For tempered stable distributions the densities have the asymptotic behaviour(7.10). In contrast to this, the densities of bilateral Gamma distributionshave the asymptotic behaviour

g(x) ∼ Cxα+−1e−λ

+x as x→∞,

for some constant C > 0, see [30, Section 6].

8. Density transformations of tempered stable processes

Equivalent changes of measure are important for option pricing in financial math-ematics. As we shall see in the forthcoming Sections 11, 12, under certain measurechanges a tempered stable process X is still tempered stable.

The purpose of the present section is to determine all locally equivalent measurechanges under which X remains tempered stable. This was already outlined in [8,Example 9.1]. Here, we are also interested in the corresponding density process.For the computation of the density process we will use that X = X+ −X− can bedecomposed as the difference of two independent subordinators.

In order to apply the results from Section 33 in [42], we assume that Ω = D(R+)is the space of cadlag functions equipped with the natural filtration Ft = σ(Xs :s ∈ [0, t]) and the σ-algebra F = σ(Xt : t ≥ 0), where X denotes the canonicalprocess Xt(ω) = ω(t). Let P be a probability measure on (Ω,F) such that

X ∼ TS(α+1 , β

+1 , λ

+1 ;α−1 , β

−1 , λ

−1 )

is a tempered stable process. Furthermore, let Q be another probability measureon (Ω,F) such that

X ∼ TS(α+2 , β

+2 , λ

+2 ;α−2 , β

−2 , λ

−2 )

under Q. We shall now investigate, under which conditions the probability measuresP and Q are locally equivalent.

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28 UWE KUCHLER AND STEFAN TAPPE

8.1. Proposition. The following statements are equivalent:

(1) The measures P and Q are locally equivalent.(2) We have α+

1 = α+2 , α−1 = α−2 , β+

1 = β+2 and β−1 = β−2 .

Proof. The Radon-Nikodym derivative Φ = dF2

dF1of the Levy measures is given by

Φ(x) =α+

2

α+1

xβ+1 −β

−1 e−(λ+

2 −λ+1 )x

1(0,∞)(x)

+α−2α−1|x|β

+2 −β

−2 e−(λ−2 −λ

−1 )|x|

1(−∞,0)(x), x ∈ R.

According to [42, Thm. 33.1], the measures P and Q are locally equivalent if andonly if ∫

R

(1−

√Φ(x)

)2

F1(dx) <∞.(8.1)

Since we have∫R

(1−

√Φ(x)

)2

F1(dx)

=

∫ ∞0

(1−

√α+

2

α+1

x(β+1 −β

−1 )/2e−(λ+

2 −λ+1 )x/2

)2α+

1

x1+β+1

e−λ+1 xdx

+

∫ 0

−∞

(1−

√α−2α−1|x|(β

+2 −β

−2 )/2e−(λ−2 −λ

−1 )|x|/2

)2α+

2

|x|1+β+2

e−λ+2 |x|dx

=

∫ ∞0

(√α+

1 x−(1+β+

1 )/2e−(λ+1 /2)x −

√α+

2 x−(1+β+

2 )/2e−(λ+2 /2)x

)2

dx

+

∫ ∞0

(√α−1 x

−(1+β−1 )/2e−(λ−1 /2)x −√α−2 x

−(1+β−2 )/2e−(λ−2 /2)x

)2

dx,

condition (8.1) is satisfied if and only if we have α+1 = α+

2 , α−1 = α−2 , β+1 = β+

2 andβ−1 = β−2 .

Now suppose that α+1 = α+

2 =: α+, α−1 = α−2 =: α−, β+1 = β+

2 =: β+ and

β−1 = β−2 =: β−. We shall determine the Radon-Nikodym derivatives dQdP |Ft for

t ≥ 0.We decompose X = X+ − X− as the difference of two independent one-sided

tempered stable subordinators and denote by Ψ+,Ψ− the respective cumulant gen-erating functions, which can be computed by means of (2.16), (2.18).

8.2. Proposition. The Radon-Nikodym derivatives of the measure transformationare given by

dQdP

∣∣∣∣Ft

= exp(

(λ+1 − λ

+2 )X+

t −Ψ+(λ+1 − λ

+2 )t)

× exp(

(λ−1 − λ−2 )X−t −Ψ−(λ−1 − λ

−2 )t), t ≥ 0.

Proof. The Radon-Nikodym derivative Φ = dF2

dF1of the Levy measures is given by

Φ(x) =dF2

dF1= e−(λ+

2 −λ+1 )x

1(0,∞)(x) + e−(λ−2 −λ−1 )|x|

1(−∞,0)(x).

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TEMPERED STABLE DISTRIBUTIONS 29

A straightforward calculations shows that∑s≤t

ln Φ(∆Xs) =∑s≤t

ln e−(λ+2 −λ

+1 )∆X+

s +∑s≤t

ln e−(λ−2 −λ−1 )∆X−s

= (λ+1 − λ

+2 )∑s≤t

∆X+s + (λ−1 − λ

−2 )∑s≤t

∆X−s

= (λ+1 − λ

+2 )X+

t + (λ−1 − λ−2 )X−t , t ≥ 0

as well as∫R

(Φ(x)− 1)F1(dx)

=

∫R

(e−(λ+

2 −λ+1 )x

1(0,∞)(x) + e−(λ−2 −λ−1 )|x|

1(−∞,0)(x)− 1)F1(dx)

= α+

∫ ∞0

e−λ+2 x − e−λ

+1 x

x1+βdx+ α−

∫ ∞0

e−λ−2 x − e−λ

−1 x

x1+βdx

=

∫R

(e(λ+2 −λ

+1 )x − 1)F+

1 (dx) +

∫R

(e(λ−2 −λ−1 )x − 1)F−1 (dx)

= Ψ+(λ+1 − λ

+2 ) + Ψ−(λ−1 − λ

−2 ).

According to [42, Thm. 33.2] the Radon-Nikodym derivatives are given by

dQdP

∣∣∣∣Ft

= exp

(∑s≤t

ln Φ(∆Xs)− t∫R

(Φ(x)− 1)F1(dx)

)= exp

((λ+

1 − λ+2 )X+

t −Ψ+(λ+1 − λ

+2 )t)

× exp(

(λ−1 − λ−2 )X−t −Ψ−(λ−1 − λ

−2 )t), t ≥ 0

which finishes the proof.

8.3. Remark. With the notation of Section 12, the measure Q is the bilateral

Esscher transform Q = P(λ+1 −λ

+2 ,λ−2 −λ

−1 ), see Definition 12.1 below.

9. The p-variation index of a tempered stable process

In this section, we compute the p-variation index, which is a measure of thesmoothness of the sample paths, for tempered stable processes. The p-variation hasbeen investigated in various different contexts, see, e.g., [5, 17, 18, 37, 1, 9, 21, 22,20].

For t ≥ 0 let Z[0, t] be set of all decompositions

Π = 0 = t0 < t1 < . . . < tn = tof the interval [0, t]. For a function f : R+ → R we define the p-variation Vp(f) :

R+ → R+ as

Vp(f)t := supΠ∈Z[0,t]

∑ti∈Πti 6=t

|fti+1 − fti |p, t ≥ 0.

Note that for any t ≥ 0 the relation Vp(f)t < ∞ implies that Vq(f)t < ∞ for allq > p.

9.1. Remark. There exist functions f : R+ → R with Vp(f)t =∞ for all t > 0 andall p > 0. Indeed, let f := 1Q+

and fix an arbitrary t > 0. For n ∈ N we set ti := in t,

i = 0, . . . , n, for each i = 1, . . . , n we choose an irrational number si ∈ R \ Q withti−1 < si < ti and we define the partition

Πn := 0 = t0 < s1 < t1 < . . . < sn < tn = t.

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30 UWE KUCHLER AND STEFAN TAPPE

Then, for each p > 0 we have∑ui∈Πnui 6=t

|fui+1 − fui |p = 2n→∞ for n→∞,

showing that Vp(f)t =∞.

For a function f : R+ → R we define the p-variation index

γ(f) := infp > 0 : Vp(f)t <∞ for all t ≥ 0,

where we agree to set inf ∅ :=∞. The p-variation index is a measure of the smooth-ness of a function f in the sense that smaller p-variation indices mean smootherbehaviour of f . Here are some examples:

• For every absolutely continuous function f we have γ(f) ≤ 1;• For a Brownian motion W we have γ(W (ω)) = 2 for almost all ω ∈ Ω;• For the function f := 1Q+

we have γ(f) =∞, see Remark 9.1.

In fact, for a Levy process X the p-variation index γ(X(ω)) does not depend onω ∈ Ω. In order to determine γ(X), we introduce the Blumenthal-Getoor indexβ(X) (which goes back to [5]) as

β(X) := inf

p > 0 :

∫ 1

−1

|x|pF (dx) <∞,

where F denotes the Levy measure of X. Note that β(X) ≤ 2. According to [5,Thm. 4.1, 4.2] and [37, Thm. 2] we have γ(X) = β(X) almost surely. Moreover, by[5, Thm. 3.1, 3.3] we have almost surely

limt→0

t−1/pXt = 0, p > β(X)

lim supt→0

t−1/p|Xt| =∞, p < β(X).

Now, let X be a tempered stable process

X ∼ Γ(α+, β+, λ+;α−, β−, λ−).

9.2. Proposition. We have γ(X) = β(X) = maxβ+, β−.

Proof. For any p > 0 we have∫ 1

−1

|x|pF (dx) = α+

∫ 1

0

e−λ+x

x1+β+−p dx+ α−∫ 0

−1

e−λ−|x|

|x|1+β−−p dx

= α+

∫ 1

0

e−λ+x

x1+β+−p dx+ α−∫ 1

0

e−λ−x

x1+β−−p dx,

which is finite if and only if p > maxβ+, β−. This shows β(X) = maxβ+, β−.Since γ(X) = β(X), the proof is complete.

We obtain the following characterization of bilateral Gamma processes withinthe class of tempered stable processes:

9.3. Corollary. We have γ(X) = 0 if and only if X is a bilateral Gamma process.

Consequently, we may regard bilateral Gamma processes as the smoothest classof processes among all tempered stable processes.

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TEMPERED STABLE DISTRIBUTIONS 31

10. Stock price models driven by tempered stable processes

Now, we turn to our applications in finance and consider stock models drivenby tempered stable processes. Let (Ω,F , (Ft)t≥0,P) be a filtered probability spacesatisfying the usual conditions. A tempered stable stock model is an exponentialLevy model of the type

St = S0eXt

Bt = ert(10.1)

where X denotes a tempered stable process and S is a dividend paying stock withdeterministic initial value S0 > 0 and dividend rate q ≥ 0. Furthermore, B is thebank account with interest rate r ≥ 0. In what follows, we assume that r ≥ q ≥ 0,that is, the dividend rate q of the stock cannot exceed the interest rate r of thebank account and none of them is negative.

An equivalent probability measure Q ∼ P is a local martingale measure if thediscounted stock price process

St := e−(r−q)tSt = S0eXt−(r−q)t, t ≥ 0(10.2)

is a local Q-martingale. In fact, an equivalent measure Q ∼ P is a local martingalemeasure if and only if it is a martingale measure, that is, the discounted stock price(10.2) is a martingale, see [32, Lemma 2.6].

The existence of a martingale measure Q ∼ P ensures that the stock market isfree of arbitrage, and the price of an European option Φ(ST ), where T > 0 is thetime of maturity and Φ : R→ R the payoff profile, is given by

π = e−rTEQ[Φ(ST )].

10.1. Lemma. Let X be a tempered stable distribution

X ∼ Γ(α+, β+, λ+;α−, β−, λ−)

under P with β+, β− ∈ (0, 1). Then the following statements are valid:

(1) If λ+ ≥ 1, then P is a martingale measure for S if and only if

α+Γ(−β+)[(λ+ − 1)β+

− (λ+)β+

] + α−Γ(−β−)[(λ− + 1)β−− (λ−)β

−] = r − q.

(10.3)

(2) If λ+ < 1, then P is never a martingale measure for S.

Proof. If λ+ ≥ 1, we have E[eX1 ] < ∞. By [32, Lemma 2.6] the discounted price

process S in (10.2) is a local martingale if and only if E[eX1−(r−q)] = 1, which, bytaking into account (2.9), is the case if and only if (10.3) holds.

In the case λ+ < 1 we have E[eX1 ] =∞. [32, Lemma 2.6] implies that S cannotbe a local martingale.

11. Existence of Esscher martingale measures in tempered stablestock models

One method to find an equivalent martingale measure is to use the so-calledEsscher transform, which was pioneered in [14]. We shall study the existence ofEsscher martingale measures in this section.

11.1. Definition. Let X be a tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

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32 UWE KUCHLER AND STEFAN TAPPE

under P and let Θ ∈ (−λ−, λ+) be arbitrary. The Esscher transform PΘ is definedas the locally equivalent probability measure with likelihood process

Λt(PΘ,P) :=dPΘ

dP

∣∣∣∣Ft

= eΘXt−Ψ(Θ)t, t ≥ 0(11.1)

where Ψ denotes the cumulant generating function given by (2.9) resp. (2.10).

11.2. Lemma. Let X be a tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

under P and let Θ ∈ (−λ−, λ+) be arbitrary. Then we have

X ∼ TS(α+, β+, λ+ −Θ;α−, β−, λ− + Θ)

under PΘ.

Proof. This follows from Proposition 2.1.3 and Example 2.1.4 in [29].

In the sequel, we assume that β+, β− ∈ (0, 1). For a bilateral Gamma process(i.e. β+ = β− = 0) the existence of Esscher martingale measures was treated in[32]. We define the function f : [−λ−, λ+ − 1]→ R as

f(Θ) := f+(Θ) + f−(Θ),

where we have set

f+(Θ) := α+Γ(−β+)[(λ+ −Θ− 1)β

+

− (λ+ −Θ)β+],

f−(Θ) := α−Γ(−β−)[(λ− + Θ + 1)β

−− (λ− + Θ)β

−].

11.3. Theorem. Let X be a tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

under P with β+, β− ∈ (0, 1). Then, there exists Θ ∈ (−λ−, λ+) such that PΘ is amartingale measure if and only if

λ+ + λ− > 1(11.2)

and r − q ∈ (f(−λ−), f(λ+ − 1)],(11.3)

and condition (11.3) is equivalent to

α+Γ(−β+)[(λ+ + λ− − 1)β

+

− (λ+ + λ−)β+]

+ α−Γ(−β−)

< r − q ≤ −α+Γ(−β+) + α−Γ(−β−)[(λ+ + λ−)β

−− (λ+ + λ− − 1)β

−].

If (11.2) and (11.3) are satisfied, Θ is unique, belongs to the interval (−λ−, λ+−1],and it is the unique solution of the equation

f(Θ) = r − q.(11.4)

Moreover, we have

X ∼ Γ(α+, β+, λ+ −Θ;α−, β−, λ− + Θ)

under PΘ.

Proof. Let Θ ∈ (−λ−, λ+) be arbitrary. In view of Lemma 11.2 and Lemma 10.1,the probability measure PΘ is a martingale measure if and only if λ+ −Θ ≥ 1, i.e.Θ ∈ (−λ−, λ+ − 1], and (11.4) is fulfilled. Note that (−λ−, λ+ − 1] 6= ∅ if and onlyif (11.2) is satisfied. For f+ and f− we obtain the derivatives

(f+)′(Θ) = −α+β+Γ(−β+)[(λ+ −Θ− 1)β

+−1 − (λ+ −Θ)β+−1

],

(f−)′(Θ) = −α−β−Γ(−β−)[(λ− + Θ)β

−−1 − (λ− + Θ + 1)β−−1

]

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TEMPERED STABLE DISTRIBUTIONS 33

for Θ ∈ (−λ−, λ+ − 1). Noting that β+, β− ∈ (0, 1) we see that (f+)′, (f−)′ > 0 onthe interval (−λ−, λ+ − 1]. Hence, f is strictly increasing on (−λ−, λ+ − 1], whichcompletes the proof.

11.4. Remark. In contrast to the present situation, for bilateral Gamma stockmodels (β+ = β− = 0) condition (11.2) alone is already sufficient for the existenceof an Esscher martingale measure, cf. [32, Remark 4.4].

12. Existence of minimal entropy measures preserving the class oftempered stable processes

In the literature, one often performs option pricing by finding an equivalentmartingale measure Q ∼ P which minimizes

E[f(Λ1(Q,P))]

for a strictly convex function f : (0,∞) → R. Here are some popular choices forthe functional f :

• For f(x) = x lnx we call Q the minimal entropy martingale measure.• For f(x) = xp with p > 1 we call Q the p-optimal martingale measure.• For p = 2 we call Q the variance-optimal martingale measure.

We refer to [32, Sec. 5] for further remarks and related literature. While p-optimalequivalent martingale measures does not exist in tempered stable stock models(which follows from [2, Ex. 2.7]), we obtain a result similar to [32, Thm. 5.3] con-cerning the existence of minimal entropy martingale measures, see [32, Remark 5.4].In this section, we minimize the relative entropy

H(Q |P) := E[Λ1(Q,P) ln Λ1(Q,P)] = EQ[ln Λ1(Q,P)]

within the class of tempered stable processes by performing bilateral Esscher trans-forms.

In the sequel, we assume that β+, β− ∈ (0, 1). For a bilateral Gamma process(i.e. β+ = β− = 0) the existence of minimal entropy measures was treated in [32].We decompose the tempered stable process X = X+−X− as the difference of twoindependent subordinators and the respective cumulant generating functions are,according to (2.16), given by

Ψ+(z) = α+Γ(−β+)[(λ+ − z)β

+

− (λ+)β+], z ∈ (−∞, λ+](12.1)

Ψ−(z) = α−Γ(−β−)[(λ− − z)β

−− (λ−)β

−], z ∈ (−∞, λ−].(12.2)

Note that Ψ(z) = Ψ+(z) + Ψ−(−z) for z ∈ [−λ−, λ+].

12.1. Definition. Let X be a tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

under P and let θ+ ∈ (−∞, λ+) and θ− ∈ (−∞, λ−) be arbitrary. The bilateral

Esscher transform P(θ+,θ−) is defined as the locally equivalent probability measurewith likelihood process

Λt(P(θ+,θ−),P) :=dP(θ+,θ−)

dP

∣∣∣∣Ft

= eθ+X+

t −Ψ+(θ+)t · eθ−X−t −Ψ−(θ−)t, t ≥ 0.

Note that the Esscher transforms PΘ from Section 11 are special cases of the

just introduced bilateral Esscher transforms P(θ+,θ−). Indeed, it holds

PΘ = P(Θ,−Θ), Θ ∈ (−λ−, λ+).(12.3)

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34 UWE KUCHLER AND STEFAN TAPPE

12.2. Lemma. Let X be a tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−)

under P and let θ+ ∈ (−∞, λ+) and θ− ∈ (−∞, λ−) be arbitrary. Then we have

X ∼ TS(α+, β+, λ+ − θ+;α−, β−, λ− − θ−)

under P(θ+,θ−).

Proof. This follows from Proposition 2.1.3 and Example 2.1.4 in [29].

12.3. Proposition. The following statements are valid:

(1) If we have

−α+Γ(−β+) ≤ r − q,(12.4)

then no pair (θ+, θ−) ∈ (−∞, λ+) × (−∞, λ−) with P(θ+,θ−) being a mar-tingale measure exists.

(2) If we have

−α+Γ(−β+) > r − q,(12.5)

then there exist −∞ ≤ θ+1 < θ+

2 ≤ λ+ − 1 and a continuous, strictlyincreasing, bijective function Φ : (θ+

1 , θ+2 )→ (−∞, λ−) such that:

• For all θ+ ∈ (θ+1 , θ

+2 ) there exists a unique θ− ∈ (−∞, λ−) with

P(θ+,θ−) being a martingale measure, and it is given by θ− = Φ(θ+).

• For all θ+ ∈ (−∞, λ+) \ (θ+1 , θ

+2 ) no θ− ∈ (−∞, λ−) with P(θ+,θ−)

being a martingale measure exists.

Proof. We introduce the functions f+ : (−∞, λ+−1]→ R and f− : (−∞, λ−]→ Ras

f+(θ+) := α+Γ(−β+)[(λ+ − θ+ − 1)β

+

− (λ+ − θ+)β+],

f−(θ−) := α−Γ(−β−)[(λ− − θ− + 1)β

−− (λ− − θ−)β

−].

By Lemmas 10.1, 12.2 the measure P(θ+,θ−) is a martingale measure if and only ifθ+ ∈ (−∞, λ+ − 1] and

f+(θ+) + f−(θ−) = r − q.(12.6)

The function f+ is continuous and strictly increasing on (−∞, λ+ − 1] with

limθ+→−∞

f+(θ+) = 0 and f+(λ+ − 1) = −α+Γ(−β+) > 0.

The function f− is continuous and strictly decreasing on (−∞, λ−] with

limθ−→−∞

f−(θ−) = 0 and f−(λ−) = α−Γ(−β−) < 0.

Therefore, if we have (12.4), then for no pair (θ+, θ−) ∈ (−∞, λ+ − 1]× (−∞, λ−)equation (12.6) is satisfied. If we have (12.5), then let −∞ ≤ θ+

1 < θ+2 ≤ λ+ − 1 be

the unique solutions of the equations

f+(θ+1 ) = r − q,

f+(θ+2 ) = r − q − α−Γ(−β−),

with the conventions

θ+1 = −∞ if r − q = 0,

θ+2 = λ+ − 1 if r − q − α−Γ(−β−) > −α+Γ(−β+),

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TEMPERED STABLE DISTRIBUTIONS 35

and define

Φ(θ+) := (f−)−1(r − q − f+(θ+)), θ+ ∈ (θ+1 , θ

+2 ).

Then Φ is continuous and strictly increasing with Φ((θ+1 , θ

+2 )) = (−∞, λ−), which

finishes the proof.

12.4. Remark. The proof of Proposition 12.3 shows that the situation θ+1 = −∞

occurs if and only if r = q and that the situation θ+2 = λ+ − 1 occurs if and only if

r − q ≤ −α+Γ(−β+) + α−Γ(−β−).

By Proposition 8.1, all equivalent measure transformations preserving the class oftempered stable processes are bilateral Esscher transforms. Hence, we introduce theset of parameters

MP := (θ+, θ−) ∈ (−∞, λ+)× (−∞, λ−) |P(θ+,θ−) is a martingale measuresuch that the bilateral Esscher transform is a martingale measure. The previousProposition 12.3 tells us that for (12.4) we have MP = ∅ and for (12.5) we have

MP = (θ,Φ(θ)) ∈ R2 | θ ∈ (θ+1 , θ

+2 ).(12.7)

Moreover, we remark that condition (12.5) is always fulfilled if r = q.

12.5. Lemma. For all (θ+, θ−) ∈ (−∞, λ+)× (−∞, λ−) we have

H(P(θ+,θ−) |P)

= −α+Γ(−β+)(λ+β+(λ+ − θ+)β

+−1 + (1− β+)(λ+ − θ+)β+

− (λ+)β+)

− α−Γ(−β−)(λ−β−(λ− − θ−)β

−−1 + (1− β−)(λ− − θ−)β−− (λ−)β

−).

Proof. The relative entropy of the bilateral Esscher transform is given by

H(P(θ+,θ−) |P) = EP(θ+,θ−) [ln Λ1(P(θ+,θ−),P)]

= EP(θ+,θ−) [θ+X+

1 −Ψ+(θ+)] + EP(θ+,θ−) [θ−X−1 −Ψ−(θ−)].

Using Lemma 12.2 we obtain

EP(θ+,θ−) [θ+X+

1 −Ψ+(θ+)]

= θ+Γ(1− β+)α+

(λ+ − θ+)1−β+ − α+Γ(−β+)[(λ+ − θ+)β

+

− (λ+)β+]

= −α+Γ(−β+)

(θ+β+

(λ+ − θ+)1−β+ + (λ+ − θ+)β+

− (λ+)β+

)= −α+Γ(−β+)

((β+θ+ + λ+ − θ+)(λ+ − θ+)β

+−1 − (λ+)β+)

= −α+Γ(−β+)((λ+β+ + (1− β+)(λ+ − θ+))

)(λ+ − θ+)β

+−1 − (λ+)β+)

= −α+Γ(−β+)(λ+β+(λ+ − θ+)β

+−1 + (1− β+)(λ+ − θ+)β+

− (λ+)β+).

An analogous calculation for EP(θ+,θ−) [θ−X−1 −Ψ−(θ−)] finishes the proof.

12.6. Theorem. Let X be a tempered stable process

X ∼ TS(α+, λ+, β+;α−, λ−, β−)

under P. The following statements are valid:

(1) If (12.4), then we have MP = ∅.(2) If (12.5), then there exist θ+ ∈ (−∞, λ+) and θ− ∈ (−∞, λ−) such that

H(P(θ+,θ−) |P) = min(ϑ+,ϑ−)∈MP

H(P(ϑ+,ϑ−) |P).(12.8)

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36 UWE KUCHLER AND STEFAN TAPPE

Proof. If (12.4), then we have MP = ∅ by Proposition 12.3. Now suppose (12.5)and let Φ : (θ+

1 , θ+2 ) → (−∞, λ−) be the function from Proposition 12.3. Let f :

(θ+1 , θ

+2 )→ R be the function

(12.9)

f(θ) := −α+Γ(−β+)(λ+β+(λ+ − θ)β

+−1 + (1− β+)(λ+ − θ)β+

− (λ+)β+)

− α−Γ(−β−)(λ−β−(λ− − Φ(θ))β

−−1 + (1− β−)(λ− − Φ(θ))β−− (λ−)β

−).

By Proposition 12.3 and Lemma 12.5, for each θ ∈ (θ+1 , θ

+2 ) the measure P(θ,Φ(θ)) is

a martingale measure and we have H(P(θ,Φ(θ)) |P) = f(θ). The function Φ is strictlyincreasing with

limθ↓θ+

1

Φ(θ) = −∞ and limθ↑θ+

2

Φ(θ) = λ−,

which gives us

limθ↓θ+

1

f(θ) =∞ and limθ↑θ+

2

f(θ) =∞.

Since f is continuous, it attains a minimum and the assertion follows.

12.7. Remark. In contrast to bilateral Gamma stock models, it may happen thatMP = ∅ (i.e., there is no equivalent martingale measure under which X is a tem-pered stable process), and the function Φ from Proposition 12.3 is not available inclosed form, cf. [32, Remark 6.7].

As mentioned at the beginning of this section, for tempered stable stock modelsthe p-optimal martingale measure does not exist. However, we can, as providedfor the minimal entropy martingale measure, determine the p-optimal martingalemeasure within the class of tempered stable processes.

The p-distance of a bilateral Esscher transform is easy to compute. Indeed, since

X+ and X− are independent, for p > 1 and θ+ ∈ (−∞, λ+

p ), θ− ∈ (−∞, λ−

p ) the

p-distance is given by(12.10)

E

[(dP(θ+,θ−)

dP

)p]= e−p(Ψ

+(θ+)+Ψ−(θ−))E[epθ

+X+1]E[epθ−X−1

]= exp

(− p(Ψ+(θ+) + Ψ−(θ−)) + Ψ+(pθ+) + Ψ−(pθ−)

)= exp

(− α+Γ(−β+)

[p[(λ+ − θ+)β

+

− (λ+)β+]−[(λ+ − pθ+)β

+

− (λ+)β+]]

− α−Γ(−β−)[p[(λ− − θ−)β

−− (λ−)β

−]−[(λ− − pθ−)β

−− (λ−)β

−]]).

A similar argumentation as in Theorem 12.6 shows that, provided condition (12.5)holds true, there exists a pair (θ+, θ−) minimizing the p-distance (12.10), and inthis case we also have θ− = Φ(θ+), where θ+ minimizes the function(12.11)

fp(θ) = −α+Γ(−β+)[p[(λ+ − θ)β

+

− (λ+)β+]−[(λ+ − pθ)β

+

− (λ+)β+]]

− α−Γ(−β−)[p[(λ− − Φ(θ))β

−− (λ−)β

−]−[(λ− − pΦ(θ))β

−− (λ−)β

−]].

Numerical computations for concrete examples show that θp → θ for p ↓ 1, wherefor each p > 1 the parameter θp minimizes (12.11) and θ minimizes (12.9). This isnot surprising, since it is known that, under suitable technical conditions, the p-optimal martingale measure converges to the minimal entropy martingale measure

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TEMPERED STABLE DISTRIBUTIONS 37

for p ↓ 1, see, e.g., [15, 16, 41, 23, 2, 26]. Here, we shall only give an intuitiveargument. For α, λ > 0 and β ∈ (0, 1) consider the functions

h(θ) = λβ(λ− θ)β−1 + (1− β)(λ− θ)β − λβ ,

hp(θ) = p[(λ− θ)β − λβ

]−[(λ− pθ)β − λβ

].

Note that h and hp have a global minimum at θ = 0. Moreover, if p > 1 is closeto 1, then the functions (p− 1)h(θ) and hp(θ) are very close to each other. Indeed,Taylor’s theorem shows that, for appropriate ξ1, ξ2 ∈ R which are between 0 and θ,we have

h(θ) = λβ(λ− θ)β−1 + (1− β)

[λβ − βλβ−1θ +

β(β − 1)(λ− ξ1)β−2

2θ2

]− λβ

= λβ(λ− θ)β−1 − βλβ + β(β − 1)λβ−1θ − β(β − 1)2(λ− ξ1)β−2

2θ2

=λβ(β − 1)(β − 2)(λ− ξ2)β−3

2θ2 − β(β − 1)2(λ− ξ1)β−2

2θ2

=β(β − 1)

2

[λ(β − 2)(λ− ξ2)β−3 − (β − 1)(λ− ξ1)β−2

]θ2,

and for appropriate ζ, ζp ∈ R, which are between 0 and θ, we have

hp(θ) = p

[− βλβ−1θ +

β(β − 1)(λ− ζ)β−2

2θ2

]−[− pβλβ−1θ +

q2β(β − 1)(λ− pζp)β−2

2θ2

]=pβ(β − 1)

2

[(λ− ζ)β−2 − p(λ− pζp)β−2

]θ2.

Noting that ζp → ζ for p ↓ 1 we see that

(p− 1)h(θ) = ϑ1(p, θ)θ2 and hp(θ) = ϑ2(p, θ)θ2

with ϑi(p, θ)→ 0 as p ↓ 1 for i = 1, 2. Therefore, the distance |(p− 1)h(θ)− hp(θ)|becomes small for p > 1 being close to 1. Intuitively, this explains the convergenceθp → θ for p ↓ 1, which implies the convergence of the p-optimal martingale measureto the minimal entropy martingale measure.

13. Option pricing in tempered stable stock models

We are now interested in computing the price π of an European call option withstrike price K > 0 and maturity date T > 0.

After performing a (bilateral) Esscher transform as in the previous Sections 11,12, the process X is again a tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−).

Hence, we shall assume in the sequel that the probability measure P is already amartingale measure. Then we have

π = e−rTE[(ST −K)+].

If λ+ > 1, then the price of the call option is given by

π = −e−rTK

∫ iν+∞

iν−∞

(K

S0

)izϕXT (−z)z(z − i)

dz,

where ν ∈ (1, λ+) is arbitrary. This follows from [33, Thm. 3.2]. In fact, we get anoption pricing formula in closed form. In the sequel,

Fα+,β+,λ+;α−,β−,λ−

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38 UWE KUCHLER AND STEFAN TAPPE

denotes the TS(α+, β+, λ+;α−, β−, λ−)-distribution function and

Fα+,β+,λ+;α−,β−,λ− = 1− Fα+,β+,λ+;α−,β−,λ− .

13.1. Proposition. Suppose that λ+ > 1. Then, the price of the call option is givenby

(13.1)π = S0e

(Ψ(1)−r)TFα+T,β+,λ+−1;α−T,β−,λ−+1(ln(K/S0))

−KFα+T,β+,λ+;α−,β−T,λ−(ln(K/S0)).

Proof. We follow an idea from [24, Sec. 8.1]. By the definition of the density (11.1)we obtain

π = e−rTE[(ST −K)+] = e−rTE[(S0eXT −K)+]

= S0e−rTE[eXT 1XT≥ln(K/S0)]−KP(XT ≥ ln(K/S0))

= S0e−rTEP1

[eXT 1XT≥ln(K/S0)

dPdP1

∣∣∣∣FT

]−KP(XT ≥ ln(K/S0))

= S0e−rT eΨ(1)TP1(XT ≥ ln(K/S0))−KP(XT ≥ ln(K/S0)),

which, in view of Lemma 11.2, provides the formula (13.1).

13.2. Remark. We remark that the concrete values of Ψ(1) in the option pricingformula (13.1) are given by

Ψ(1) = α+Γ(−β+)[(λ+ − 1)β

+

− (λ+)β+]

+ α−Γ(−β−)[(λ− + 1)β

−− (λ−)β

−]in the tempered stable case β+, β− ∈ (0, 1), and by

Ψ(1) = α+ ln

(λ+

λ+ − 1

)+ α− ln

(λ−

λ− + 1

),

in the bilateral Gamma case β+ = β− = 0. This follows from (2.9) and (2.10).

14. Existence of minimal martingale measures in tempered stablestock models

In this section, we deal with the existence of the minimal martingale measurein tempered stable stock models. The minimal martingale measure was introducedin [13] with the motivation of constructing optimal hedging strategies. Throughoutthis section, we fix a finite time horizon T > 0 and assume that λ+ ≥ 2. Then theconstant

c = c(α+, α−, λ+, λ−, r, q) =Ψ(1)− (r − q)Ψ(2)− 2Ψ(1)

,(14.1)

is well-defined. For technical reasons, we shall also assume that the filtration (Ft)t≥0

is generated by the tempered stable process

X ∼ TS(α+, β+, λ+;α−, β−, λ−).

As in [32, Lemma 7.1] we show that the discounted stock price process S is a special

semimartingale. Let S = S0 +M +A be its canonical decomposition and let Z bethe stochastic exponential

Zt = E(−∫ •

0

c

Ss−dMs

)t

, t ∈ [0, T ](14.2)

Then the (possibly signed) measure

dPdP

:= ZT(14.3)

is the so-called minimal martingale measure for S.

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TEMPERED STABLE DISTRIBUTIONS 39

14.1. Theorem. Suppose that β+, β− ∈ (0, 1). The following statements are equiv-alent:

(1) Z is a strict martingale density for S.

(2) Z is a strictly positive P-martingale.(3) We have

−1 ≤ c ≤ 0.(14.4)

(4) We have

α+Γ(−β+)[(λ+ − 1)β+

− (λ+)β+

](14.5)

+ α−Γ(−β)[(λ− + 1)β−− (λ−)β

−] ≤ r − q

and α+Γ(−β+)[(λ+ − 1)β+

− (λ+ − 2)β+

](14.6)

+ α−Γ(−β)[(λ− + 1)β−− (λ− + 2)β

−] ≤ −(r − q).

If the previous conditions are satisfied, then under the minimal martingale measureP we have

(14.7)X ∼ TS((c+ 1)α+, β+, λ+; (c+ 1)α−, β−, λ−)

∗ TS(−cα+, β+, λ+ − 1;−cα−, β−, λ− + 1).

14.2. Remark. Relation (14.7) means that under P the driving process X is the sumof two independent tempered stable processes. There are the following two boundaryvalues:

• In the case c = 0 we have

X ∼ TS(α+, β+, λ+, α−, β−, λ−) under P,

i.e., the minimal martingale measure P coincides with the physical measureP. Indeed, the definition (14.1) of c and Lemma 10.1 show that P already

is a martingale measure for S.• In the case c = −1 we have

X ∼ TS(α+, β+, λ+ − 1, α−, β−, λ− + 1) under P,

i.e., the minimal martingale measure P coincides with the Esscher trans-form P1, see Theorem 11.3. Indeed, the definition (14.1) of c shows thatequation (11.4) is satisfied with Θ = 1.

Proof. We only have to show the equivalence (3) ⇔ (4), as the rest follows byarguing as in the proof of [32, Thm. 7.3]. We observe that (14.4) is equivalent tothe two conditions

Ψ(1) ≤ r − q and Ψ(1)−Ψ(2) ≤ −(r − q),

and, in view of the cumulant generating function given by (2.9), these two conditionsare fulfilled if and only if we have (14.5) and (14.6).

As outlined at the end of [32, Sec. 7], under the minimal martingale measure Pwe can construct a trading strategy ξ which minimizes the quadratic hedging error.The arguments transfer to our present situation with a driving tempered stableprocess.

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40 UWE KUCHLER AND STEFAN TAPPE

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Humboldt Universitat zu Berlin, Institut fur Mathematik, Unter den Linden 6, D-10099 Berlin, Germany

E-mail address: [email protected]

Leibniz Universitat Hannover, Institut fur Mathematische Stochastik, Welfengarten1, D-30167 Hannover, Germany

E-mail address: [email protected]