Lévy Processes

Ernst Eberlein; Jan Kallsen

doi:10.1007/978-3-030-26106-1_2

Mathematical Finance pp 97-169 | Cite as

Lévy Processes

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Ernst Eberlein
Jan Kallsen

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First Online: 04 December 2019

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Abstract

The continuous-time analogue of a random walk is called a Lévy process. One may also view Lévy processes as the stochastic counterpart of linear functions. Both viewpoints illustrate that these processes play a fundamental role.

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Appendix 1: Problems

The exercises correspond to the section with the same number.

2.1

Show that an adapted càdlàg process X is a martingale if and only if X(τ) is integrable and E(X(τ)) = E(X(0)) for any bounded stopping time τ.

2.2

Let X be a Lévy process and B a Borel set. Let $\tau :=\inf \{t\geq 0:\Delta X(t)\in B\}$, i.e. τ denotes the first time where X has a jump of size in B. Show that τ has exponential distribution unless it is almost surely 0 or ∞.

Hint: recall that the exponential distribution is characterised by having no memory, i.e. P(τ > t + s|τ > t) = P(τ > s) for any s, t ≥ 0.

2.3

Show that if a counting process N is a Lévy process with E(N(1)) = λ, then

$$\displaystyle \begin{aligned}E(e^{iuN(1)})=\exp((e^{iu}-1)\lambda),\quad u\in\mathbb R,\end{aligned}$$

i.e. N is a Poisson process with parameter λ.

Hint: consider $N_n:=|\{i\in \{1,\dots ,2^n\}:N_{i/2^n}-N_{(i-1)/2^n}\geq 1\}|$ and show N_n → N(1) in law as n →∞.

2.4

Let N be a Poisson process with parameter λ and $(Y_n)_{n\in \mathbb N}$ a sequence of random variables with law Q such that N and the Y_n are all independent. Define the process X by

$$\displaystyle \begin{aligned}X(t):=\sum_{n=1}^{N(t)}Y_n,\quad t\in\mathbb R_+.\end{aligned}$$

Show that X is a Lévy process relative to its own filtration and has characteristic function

$$\displaystyle \begin{aligned}E(e^{iuX(1)})=\exp\left(\int(e^{iux}-1)\lambda Q(dx)\right),\quad u\in\mathbb R.\end{aligned}$$

Hint: condition on the value of N.

2.5

For an adapted càdlàg process X define its random measure of jumps μ^X by

$$\displaystyle \begin{aligned}\mu^X([0,t]\times B):=\big|\{(s,x)\in[0,t]\times B:\Delta X(s)=x\}\big|\end{aligned}$$

for t ≥ 0, $B\in \mathcal {B}$. Show that μ^X is an integer-valued random measure on $\mathbb R_+\times \mathbb R$ in the sense that

1.
μ^X(ω, ⋅) is a measure on $\mathbb R_+\times \mathbb R$ for fixed ω ∈ Ω,
2.
$\mu ^X(\omega ,\{0\}\times \mathbb R)=0$ for any ω ∈ Ω,
3.
μ^X(ω, ⋅) has values in $\mathbb N\cup \{\infty \}$ for any ω ∈ Ω,
4.
$\mu ^X(\omega ,\{t\}\times \mathbb R)\le 1$ for any ω ∈ Ω, t ≥ 0,
5.
μ^X is adapted in the sense that μ^X(⋅, [0, t] × B) is $\mathcal F_t$-measurable for fixed t ≥ 0 and any $B\in \mathcal {B}$,
6.
μ^X is predictably σ-finite in the sense that there exists some strictly positive $\mathcal P\otimes \mathcal {B}$-measurable function η on $\Omega \times \mathbb R_+\times \mathbb R$ such that
$$\displaystyle \begin{aligned}\iint_{\mathbb R_+\times\mathbb R}\eta(\omega,t,x)\mu^X(\omega, d(t,x))P(d\omega)<\infty.\end{aligned}$$

Moreover, show that the random measure of jumps μ^X of a Lévy process satisfies in addition:

7.
the random variable μ^X(⋅, (s, t] × B) is independent of $\mathcal F_s$ for s ≤ t, $B\in \mathcal {B}$,
8.
E(μ^X(⋅, [0, t] × B)) = tK(B), t ≥ 0, $B\in \mathcal {B}$ for some (non-random) intensity measure K on $\mathbb R$. (One can show that K is in fact the Lévy measure of X.)

Hint: You may use without proof that $\tau :=\inf \{t\geq 0:\Delta X(t)\in B\}$ is a stopping time for any adapted càdlàg process X and any Borel set B. Strictly speaking, this can be guaranteed only if the filtration is complete, i.e. if $\mathcal F_0$ contains all $\mathcal F$-null sets.

Appendix 2: Notes and Comments

For Sect. 2.1, see for example [154]. General references on Lévy processes include [2, 238, 259]. References [23, 202] treat the fluctuation theory which is not discussed here, whereas [38, 60, 263] consider Lévy processes from the point of view of applications in finance.

For Theorem 2.24 we refer to [259, Theorem 30.1]. Proposition 2.25 can be viewed as a special case of [259, Proposition 11.10], cf. also [60, Example 4.1]. Proposition 2.26 can be obtained as an application of [182, Corollary 2.21] and [154, Corollary II.4.19]. A law of large numbers related to Theorem 2.27 is stated in [259, Theorem 36.5].

Most of the Lévy processes in Sect. 2.4 and their properties are discussed in [60, Chapter 4] and [263, Section 5.3]. For specific processes see also [9, 82, 88, 90, 91, 195, 210, 220]. The excellent fit of hyperbolic distributions to empirical return distributions of German stock price data led to the introduction of hyperbolic Lévy motions in finance, cf. [82]. The bilateral gamma process is introduced in [200], and for multivariate generalised hyperbolic processes we refer to [288, 289]. Stable Lévy motions are discussed in detail in [256, 259], cf. also [60, Section 3.7]. For the results in Sect. 2.4.11 we refer in particular to [256, Sections 1.1, 2.3, 3.1, 7.5]. The Lévy–Itô decomposition of Theorem 2.33 can be found in [259, Theorem 19.2], even stated with almost sure convergence.

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Ernst Eberlein
- 1
Jan Kallsen
- 2

1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
2.Department of MathematicsKiel UniversityKielGermany

Lévy Processes

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