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Lévy copulas: review of recent results

Peter Tankov

Abstract We review and extend the now considerable literature on Lévy copulas. First, we focus on Monte Carlo methods and present a new robust algorithm for the simulation of multidimensional Ĺevy processes with dependence given by a Lévy copula. Next, we review statistical estimation techniques in a parametric and a non-parametric setting. Finally, we discuss the interplaybetween Ĺevy copulas and multivariate regular variation and briefly review the applications of Ĺevy copulas in risk management. In particular, we provide a new easy-to-use sufficient condition for multivariate regular variation of Ĺevy measures in terms of their Lévy copulas.

Key Words: Lévy processes, Ĺevy copulas, Monte Carlo simulation, statistical es- timation, risk management, regular variation

1 Introduction

Introduced in [13, 31, 42], the concept of Lévy copula allows to characterize in a time-independent fashion the dependence structure of the pure jump part of a Ĺevy process. During the past ten years, several authors have proposed extensions of Lévy copulas, developed simulation and estimation techniques for these and related ob- jects, and studied the applications of these tools to financial risk management. In this paper we review the early developments and the subsequent literature on Ĺevy cop- ulas, present new simulation algorithms for Lévy processes with dependence given by a Lévy copula, discuss the link between Lévy copulas and multivariate regular variation and mention some risk management applications. The aim is to provide a summary of available tools and an entry point to the now considerable literature on

Peter Tankov Laboratoire de Probabilités et Mod̀eles Aĺeatoires, Université Paris Diderot, Paris, France and In- ternational Laboratory of Quantitative Finance, NationalResearch University Higher School of Economics, Moscow, Russia., e-mail: tankov@math.univ-paris-diderot.fr

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2 Peter Tankov

Lévy copulas and more generally dependence models for multidimensional Ĺevy processes. We focus on practical aspects such as statistical estimation and Monte Carlo simulation rather than theoretical properties of Lévy copulas.

This chapter is structured as follows. In Section 2 we recallthe main definitions and results from the theory of Lévy copulas and review some alternative construc- tions and dependence models proposed in the literature. Section 3 presents new algorithms for simulating Ĺevy processes with a given Lévy copula, via a series rep- resentation. Section 4 reviews the statistical proceduresproposed in the literature for estimating Ĺevy copulas in the parametric or non-parametric setting. InSection 5 we discuss the interplay between these objects and multivariate regular variation. In particular, we present a new easy-to-use sufficient condition for multivariate regular variation of Ĺevy measures in terms of their Lévy copulas. In Section 6 we review the applications of Ĺevy copulas in risk management. Section 7 concludes the paper and discusses some directions for further research.

Remarks on notation In this chapter, the components of a vector are denoted by the same letter with superscripts:X = (X1, . . . ,Xn). The scalar product of two vec- tors is written with angle brackets:〈X,Y〉 = ∑ni=1XiYi , and the Euclidean norm of the vectorX is denoted by|X|. The extended real line is denoted byR̄ := (−∞,∞].

2 A primer on L évy copulas

This section contains a brief review of the theory of Lévy copulas as exposed in [13, 31, 42]. We invite the readers to consult these references for additional details.

Recall that a Ĺevy process is a stochastic process with stationary and independent increments, which is continuous in probability. The law of aLévy process(Xt)t≥0 is completely determined by the law ofXt at any given timet > 0. The characteristic function of this law is given explicitly by the Ĺevy-Khintchine formula:

E[ei〈u,Xt 〉] = etψ(u), u∈ Rn,

ψ(u) =−〈Au,u〉 2

+ i〈γ ,u〉+ ∫

Rn (ei〈u,x〉−1− i〈u,x〉1|x|≤1)ν(dx),

whereγ ∈ Rn, A is a positive semi-definiten×n matrix andν is a positive measure onRn with ν({0})= 0 such that∫

Rn(|x|2∧1)ν(dx)1 xJ(ds×dx), (1)

whereB is a Brownian motion (centered Gaussian process with independent incre- ments) with covariance matrixA at timet = 1, J is a Poisson random measure with

Lévy copulas: review of recent results 3

intensity measuredt×ν(dx) andJ̃ is the compensated version ofJ. (Bt)t≥0 is thus the continuous martingale part of the processX, and the remaining terms

γt + ∫ t

0

∫

|x|≤1 xJ̃(ds×dx)+

∫ t

0

∫

|x|>1 xJ(ds×dx)

may be called the pure jump part ofX. Sinceγ corresponds to a deterministic shift of every component, the law of the pure jump part of a Lévy process is determined essentially by the Ĺevy measureν .

Lévy copulas provide a representation of the Lévy measure of a multidimensional Lévy process, which allows to specify separately the Lévy measures of the compo- nents and the information about the dependence between the components1. Simi- larly to copulas for probability measures, this gives a flexible approach for building multidimensional dynamic models based on Lévy processes.

The main ideas of Ĺevy copulas are simpler to explain in the context of Lévy measures on[0,∞)n, which correspond to Ĺevy processes with only positive jumps in every component. Formally, the definitions of Lévy copula, tail integrals etc. are different for Lévy measures on[0,∞)n and on the full space, and we shall speak of Lévy copulas on[0,∞]n and of Ĺevy copulas on(−∞,∞]n, respectively. However, when there is no ambiguity, the explicit mention of the domain will be dropped. Moreover, by comparing the two definitions below it is easy tosee that from a Ĺevy copula on[0,∞]n one can always construct a Lévy copula on(−∞,∞]n by setting it to zero outside its original domain.

Lévy copulas on[0,∞]n Similarly to probability measures, which can be repre- sented through their distribution functions, Lévy measures can be represented by tail integrals.

Definition 1 (Tail integral). Let ν be a Ĺevy measure on[0,∞)n. The tail integral U of ν is a function[0,∞)n → [0,∞] such that 1. U(0, . . . ,0) = ∞. 2. For(x1, . . . ,xn) ∈ [0,∞)n\{0},

U(x1, . . . ,xn) = ν([x1,∞)×·· ·× [xn,∞)).

The i-th one-dimensional marginal tail integralUi of aRn-valued Ĺevy process X = (X1, . . . ,Xn) is the tail integral of the processXi and can be computed as

Ui(z) = (U(x1, . . . ,xn)|xi = z;x j = 0 for j 6= i), z≥ 0.

We recall that a functionF : DomF ⊆ R̄n → R̄ is calledn-increasing if for all a∈ DomF andb∈ DomF with ai ≤ bi for all i we have

VF((a,b]) := ∑ c∈DomF :ci=ai or bi ,i=1...n

sgn(c)F(c)≥ 0,

1 By “dependence” we mean the information on the law of a random vector which remains to be determined once the marginal laws of its components have been specified.

4 Peter Tankov

sgn(c) =

{ 1, if ck = ak for an even number of indices,

−1, if ck = ak for an odd number of indices.

Definition 2 (Lévy copula). A function F : [0,∞]n → [0,∞] is a Lévy copulaon [0,∞]n if

1. F(u1, . . . ,un)< ∞ for (u1, . . . ,un) 6= (∞, . . . ,∞), 2. F(u1, . . . ,un) = 0 wheneverui = 0 for at least onei ∈ {1, . . . ,n}, 3. F is n-increasing, 4. Fi(u) = u for any i ∈ {1, . . . ,n}, u∈ [0,∞], where

Fi(u) = (F(v1, . . . ,vn)|vi = u;v j = 0 for j 6= i).

The following theorem gives a representation of the tail integral of a Ĺevy mea- sure (and thus of the Ĺevy measure itself) in terms of its marginal tail integrals and a Lévy copula. It may be calledSklar’s theoremfor Lévy copulas on[0,∞]n. Theorem 1.Let ν be a Ĺevy measure on[0,∞)n with tail integral U and marginal Lévy measuresν1, . . . ,νn. Then there exists a Lévy copula F on[0,∞]n such that

U(x1, . . . ,xn) = F(U1(x1), . . . ,Un(xn)), (x1, . . . ,xn) ∈ [0,∞)n, (2)

where U1, . . . ,Un are tail integrals ofν1, . . . ,νn. This Ĺevy copula is unique on ∏ni=1RanUi .

Conversely, if F is a Ĺevy copula on[0,∞]n andν1, . . . ,νn are Lévy measures on [0,∞) with tail integrals U1, . . . ,Un then (2) defines a tail integral of a Lévy measure on [0,∞)n with marginal Ĺevy measuresν1, . . . ,νn.

A basic example of a one-parameter family of Lévy copulas on[0,∞]n is the Clayton family, given by

Fθ (u1, . . . ,un) = (u −θ 1 + · · ·+u−θn )−1/θ , θ > 0. (3)

This family has as limiting cases the independence Lévy copula (whenθ → 0)

F⊥(u1, . . . ,un) = n

∑ i=1

ui ∏ j 6=i

1{∞}(u j)

and the complete dependence Lévy copula (whenθ → ∞)

F‖(u1, . . . ,un) = min(u1, . . . ,un).

Since Ĺevy copulas are closely related to distribution copulas, many of the clas- sical copula constructions can be modified to build Lévy copulas. This allows to defineArchimedean Ĺevy copulas(see Propositions 5.6 and 5.7 in [13] for the case of Lévy copulas on[0,∞]n). Another example is the vine construction of Lévy cop- ulas [26], where a Ĺevy copula on[0,∞]n is constructed fromn(n−1)/2 bivariate dependence functions (n−1 Lévy copulas and(n−2)(n−1)/2 distributional cop- ulas).

Lévy copulas: review of recent re