4.2.1 Definition and Fundamental Examples
A L´evy process is a probabilistic model for an unpredictable measurement Xt
that evolves in time t, in such a way that the change of the measurement in disjoint time intervals of equal duration, say XsC Xs and XtC Xt with sC t, are independent from one another but with identical distribution. For instance, ifStrepresents the time-t price of an asset andXt is the log return during Œ0; t, defined by
Xt Dlog.St=S0/ ;
then the previous property will imply that daily or weekly log returns will be independent from one another with common distribution. Formally, a L´evy process is defined as follows:
Definition 1. A L´evy processX D fXtgt1 is aRd-valued stochastic process (collection of random vectors inRdindexed by time) defined on a probability space .;F;P/such that:
(i) X0D0.
(ii) X has independent increments:Xt1Xt0; : : : ; XtnXtn1 are independent for any0t0< : : : < tn.
(iii) Xhas stationary increments: the distribution ofXtCXtis the same asX, for allt; 0.
(iv) its paths are right-continuous with left-limits (rcll).
(v) it has no fixed jump-times; that is,P.Xt ¤0/D0, for any timet.
The last property can be replaced by asking thatX is continuous in probability, namely,Xs
!P Xt, ass!t, for anyt. Also, ifXsatisfies all the other properties except (iv), then there exists a rcll version of the process (see e.g. Sato1999).
There are three fundamental examples of L´evy processes that deserve some attention: Brownian motion, Poisson process, and compound Poisson process.
Definition 2. A (standard) Brownian motionW is a real-valued process such that (i)W0D0, (ii) it has independent increments, (iii)WtWshas normal distribution with mean0and variancets, for anys < t, and (iv) it has continuous paths.
It turns out that the only real L´evy processes with continuous paths are of the form Xt DWtCbt;for constants > 0andb.
A Poisson process is another fundamental type of L´evy process that is often used as building blocks of other processes.
Definition 3. A Poisson process N is an integer-valued process such that (i)N0D0, (ii) it has independent increments, (iii)NtNshas Poisson distribution with parameter .ts/, for anys < t, and (iv) its paths are rcll. The parameter is called the intensity of the process.
The Poisson process is frequently used as a model to count events of certain type (say, car accidents) occurring randomly through time. Concretely, suppose thatT1<
T2 < : : :represent random occurrence times of a certain event and let Nt be the number of events occurring by timet:
NtD X1 iD1
1fTitg: (4.6)
Then, if the events occur independently from one another, homogeneously in time, and with an intensity of events per unit time, fNtgt0 given by (4.6) will be approximately a Poisson process with intensity . This fact is a consequence of the Binomial approximation to the Poisson distribution (see, e.g., Feller1968for this heuristic construction of a Poisson process). It turns out that any Poisson process can be written in the form (4.6) withfTigi1(called arrival times) such that the waiting times
i WDTiTi1;
are independent exponential r.v.’s with common mean1= (so, the bigger the , the smaller the expected waiting time between arrivals and the higher the intensity of arrivals).
To introduce the last fundamental example, the compound Poisson process, we recall the concept of probability distribution. Given a random vectorJ inRddefined on some probability space.;P/, the distribution ofJ is the mappingdefined on setsARdas follows:
.A/WDP.J 2A/:
Thus,.A/measures the probability that the random vectorJ belongs to the set A. A compound Poisson process with jump distributionand jump intensity is a process of the form
Zt WD
Nt
X
iD1
Ji;
wherefJigi1 are independent with common distribution and N is a Poisson process with intensity that is independent offJigi. Whend D1, one can say that the compound Poisson processZ WD fZtgt0is like a Poisson process with random jump sizes independent from one another. A compound Poisson process is the only L´evy process that has piece-wise constant paths with finitely-many jumps in any time intervalŒ0; T . Note that the distribution of the compound Poisson processZ
is characterized by the finite measure:
.A/WD .A/; ARd;
called the L´evy measure of Z. Furthermore, for any finite measure , one can associate a compound Poisson processZ with L´evy measure(namely, the com- pound Poisson process with intensity of jumps WD .Rd/and jump distribution .dx/WD.dx/=.Rd/).
For future reference, it is useful to note that the characteristic function ofZt is given by
Eeihu;ZtiDexp
t Z
Rd
eihu;xi1
.dx/
(4.7) Also, if EjJij D R
jxj.dx/ < 1, thenEZt D tR
x.dx/ and the so-called compensated compound Poisson processZNt WDZtEZthas characteristic function
Eeihu;ZNti Dexp
t Z
Rd
eihu;xi1ihu; xi .dx/
: (4.8)
One of the most fundamental results establishes that any L´evy process can be approximated arbitrarily close by the superposition of a Brownian motion with drift, Wt Cbt, and an independent compound Poisson processZ. The reminderRt WD Xt .WtCbtCZt/is a pure-jump L´evy process with jump sizes smaller than say an" > 0, which can be taken arbitrarily small. The previous fundamental fact is a consequence of the L´evy-Itˆo decomposition that we review in Sect.4.3.2.
4.2.2 Infinitely Divisible Distributions and the L´evy–Khintchine Formula
The marginal distributions of a L´evy processX are infinitely-divisible. A random variableis said to be infinitely divisible if for eachn2, one can constructni.i.d.
r.v’s .n;1; : : : ; n;nsuch that
DDn;1C: : :Cn;n: ThatXt is infinitely divisible is clear since
Xt D Xn1 kD0
X.kC1/t=nXkt=n
;
andfX.kC1/t=nXkt=ngn1kD0are i.i.d. The class of infinitely divisible distributions is closely related to limits in distribution of an array of row-wise i.i.d. r.v.’s:
Theorem 1 (Kallenberg1997). is infinitely divisible iff for eachnthere exists i.i.d. random variablesfn;kgkkD1n such that
kn
X
kD1
n;k
!D ; as n! 1:
In term of the characteristic function'.u/WDEeihu;i; is infinitely divisible if and only if '.u/¤0, for all u, and its distinguished nth-root '.u/1=n is the characteristic function of some other variable for eachn(see Lemma 7.6 in Sato 1999). This property of the characteristic function turns out to be sufficient to determine its form in terms of three “parameters”.A; b; /, called the L´evy triplet of, as defined below.
Theorem 2 (L´evy-Khintchine formula). is infinitely divisible iff Eeihu;iDexp
ihb;ui 1
2hu; Aui CZ
eihu;xi1ihu; xi1jxj1
.dx/
; (4.9) for some symmetric nonnegative-definite matrixA, a vectorb2Rd, and a measure (called the L´evy measure) onRd0 WDRdnf0gsuch that
Z
Rd0.jxj2^1/.dx/ <1: (4.10) Moreover, all triplets.A; b; /with the stated properties may occur.
The following remarks are important:
Remark 1. The previous result implies that the time-t marginal distribution of a L´evy processfXtgt0is identified with a L´evy triplet.At; bt; t/. Given thatXhas stationary and independent increments, it follows thatEeihu;Xti D˚
Eeihu;X1it
;for any rationalt and by the right-continuity ofX, for any realt. Thus, if.A; b; /is the L´evy triplet ofX1, then.At; bt; t/Dt.A; b; /and
'Xt.u/WDEeihu;XtiDet .u/; where (4.11) .u/WDihb;ui 1
2hu; Aui CZ
eihu;xi1ihu; xi1jxj1
.dx/: (4.12) The triple.A; b; /is called the L´evy or characteristic triplet of the L´evy processX.
Remark 2. The exponent (4.12) is called the L´evy exponent of the L´evy process fXtgt0. We can see that its first term is the L´evy exponent of the L´evy process bt. The second term is the L´evy exponent of the L´evy process †Wt, where W D.W1; : : : ; Wd/T aredindependent Wiener processes and†is add lower triangular matrix in the Cholesky decompositionA D ††T. The last term in the
L´evy exponent can be decomposed into two terms:
cp.u/D Z
jxj>1
eihu;xi1
.dx/;
lccp.u/D Z
jxj1
eihu;xi1ihu; xi .dx/:
The first term above is the L´evy exponent of a compound Poisson process Xcp with L´evy measure1.dx/ WD 1jxj>1.dx/ (see (4.7)). The exponent lccp cor- responds to the limit in distribution of compensated compound Poisson processes.
Concretely, suppose thatX."/ is a compound Poisson process with L´evy measure ".dx/WD 1"<jxj1.dx/, then the processXt."/EXt."/converges in distribution to a process with characteristic function expft lccpg(see (4.8)). L´evy-Khintchine formula implies that, in distribution,X is the superposition of four independent L´evy processes as follows:
Xt DD „ƒ‚…bt
Drift
C „ƒ‚…† Wt Brownian part
C „ƒ‚…Xtcp
Cmpnd. Poisson
C lim
"&0
Xt."/EXt."/
„ ƒ‚ …
Limit of cmpstd cmpnd Poisson
; (4.13)
where equality is in the sense of finite-dimensional distributions. The condition (4.10) on guarantees that theXcp is indeed well defined and the compensated compound Poisson converges in distribution.
In the rest of this section, we go over some other fundamental distributional properties of the L´evy process and their applications.
4.2.3 Short-Term Distributional Behavior
The characteristic function (4.11) of X determines uniquely the L´evy triple .A; b; /. For instance, the uniqueness of the matrix A is a consequence of the following result:
h!0limhlog'Xt
h1=2u D t
2hu; Aui I (4.14) see pp. 40 inSato(1999). In term of the processX, (4.14) implies that
p1 hXht
t0
! f† WD tgt0; ash!0: (4.15)
whereW D.W1; : : : ; Wd/T aredindependent Wiener processes and†is a lower triangular matrix such thatAD††T.
From a statistical point of view, (4.15) means that, when†¤0, the short-term incrementsfX.kC1/hXkhgnkD1, properly scaled, behave like the increments of a Wiener process. In the context of the exponential L´evy model (4.34), the result (4.15) will imply that the log returns of the stock, properly scaled, are normally distributed when the Brownian component of the L´evy processXis non-zero. This property is not consistent with the empirical heavy tails of high-frequency financial returns. Recently,Rosi´nski(2007) proposes a pure-jump class of L´evy processes, called tempered stable (TS) L´evy processes, such that
1 h1=˛Xht
t0
! fZD tgt0; ash!0; (4.16)
whereZis a stable process with index˛ < 2.
4.2.4 Moments and Short-Term Moment Asymptotics
LetgWRd!RCbe a nonnegative locally bounded function andX be a L´evy pro- cess with L´evy triplet.A; b; /. The expected valueEg./is called theg-moment of a random variable . Let us now consider submultiplicative or subadditive moment functions g. Recall that a nonnegative locally bounded function g is submultiplicative (resp. subadditive) if there exists a constantK > 0such that g.x C y/ Kg.x/g.y/ (resp. g.x C y/ K.g.x/C g.y//), for all x; y. Examples of this kind of functions areg.x1; : : : ; xd/ D jxjjp, for p 1, and g.x1; : : : ; xd/ D expfjxjjˇg, forˇ 2 .0; 1. In the case of a compound Poisson process, it is easy to check that
Eg.Xt/ <1, for anyt > 0if and only ifR
jxj>1g.x/.dx/ <1.
The previous fact holds for general L´evy processes (see Kruglov1970and (Sato, 1999, Theorem 25.3)). In particular, X.t/WD.X1.t/; : : : ; Xd.t//WDXt has finite mean if and only ifR
fjxj>1gjxj.dx/ < 1. In that case, by differentiation of the characteristic function, it follows that
EXj.t/Dt Z
fjxj>1gxj.dx/Cbj
; Similarly,EjX.t/j2<1if and only ifR
fjxj>1gjxj2.dx/ <1, in which case, Cov
Xj.t/; Xk.t/
Dt Aj kC Z
xjxk.dx/
:
The two above equations show the connection between the the L´evy triplet .A; b; /, and the mean and covariance of the process. Note that the variance rate
Var.Xj.t//=t remains constant over time. It can also be shown that the kurtosis is inversely proportional to timet. In the risk-neutral world, these properties are not empirically supported under the exponential L´evy model (4.2), which rather support a model where both measurements increase with timet(see e.g. Carr et al.2003and references therein).
The L´evy measurecontrols the short-term ergodic behavior ofX. Namely, for any bounded continuous function' WRd !Rvanishing on a neighborhood of the origin, it holds that
t!0lim 1
t E'.Xt/D Z
'.x/.dx/I (4.17)
cf. (Sato, 1999, Corollary 8.9). For a real L´evy processes X with L´evy triplet .2; b; /, (4.17) can be extended to incorporate unbounded functions and different behaviors at the origin. Suppose that'WR!Riscontinuous such thatj'j g for a subadditive or submultiplicative functiongWR ! RC. Furthermore, fixing IWDfr 0WR
.jxjr ^1/ .dx/ <1g;assume that' exhibits any of the following behaviors asx!0:
(a) i. '.x/Do.jxj2/.
ii. '.x/DO.jxjr/, for somer 2I \.1; 2/and D0.
iii. '.x/Do.jxj/,12I and D0.
iv. '.x/R D sO.jxjr/, for some r 2 I \.0; 1/, D 0, and bN WD b
jxj1x.dx/D0.
(b) '.x/x2.
(c) '.x/ jxjand D0.
Building on results inWoerner (2003) andJacod (2007),Figueroa-L´opez(2008) proves that
limt!0
1
t E'.Xt/WD 8ˆ ˆˆ
<
ˆˆ ˆ:
R'.x/.dx/; if (a) holds; 2CR
'.x/.dx/;if (b) holds; j Nbj CR
'.x/.dx/;if (c) holds:
(4.18)
Woerner (2003) and also Figueroa-L´opez (2004) used the previous short-term ergodic property to show the consistency of the statistics
ˇO.'/WD 1 tn
Xn kD1
'
Xtk Xtk1
; (4.19)
towards the integral parameter ˇ.'/ WD R
'.x/.dx/; when tn ! 1 and maxftktk1g !0, for test functions'as in (a). When.dx/Ds.x/dx,Figueroa- L´opez(2004) applied the estimators (4.19) to analyze the asymptotic properties of nonparametric sieve-type estimatorssOfors. The problem of model selection was analyzed further in Figueroa-L´opez and Houdr´e(2006);Figueroa-L´opez (2009),
where it was proved that sieve estimatorsesT can match the rate of convergence of the minimax risk of estimatorssO. Concretely, it turns out that
lim sup
T!1
EksesTk2
infsOsups2Eks Osk2 <1;
whereŒ0; T is the time horizon over which we observe the processX,is certain class of smooth functions, and the infimum in the denominator is over all estimators sOwhich are based on whole trajectoryfXtgtT. The optimal rate of the estimatoresT is attained by choosing appropriately the dimension of the sieve and the sampling frequency in function ofT and the smoothness of the class of functions.
4.2.5 Extraction of the L´evy Measure
The L´evy measure can be inferred from the characteristic function'Xt.u/ of the L´evy process (see, e.g.,Sato 1999, pp. 40–41). Concretely, by first recovering hu; Auifrom (4.14), one can obtain
‰.u/WDlog'X1.u/C 1
2hu; Aui: Then, it turns out that
Z
Œ1;1d.‰.u/‰.uCw//dwD Z
Rdeihz;xi.Q dx/; (4.20) whereQis the finite measure
Q
.dx/WD2d 0
@1 Yd jD1
sinxj
xj
1 A.dx/:
Hence,can be recovered from the inverse Fourier transform of the left-hand side of (4.20).
The above method can be applied to devise non-parametric estimation of the L´evy measure by replacing the Fourier transform'X1by its empirical version:
'OX1.u/WD 1 n
Xn kD1
expfihu; XkXk1ig:
given discrete observationsX1; : : : ; Xnof the process. Recently, similar nonpara- metric methods have been proposed in the literature to estimate the L´evy density s.x/D .dx/=dxof a real L´evy processX (c.f. Neumann and Reiss2007; Comte
and Genon-Catalot2008; Gugushvili2008). For instance, based on the increments X1X0; : : : ; XnX.n1/,Neumann and Reiss(2007) consider a nonparametric estimator forsthat minimizes the distance between the “population” characteristic function'X1.Is/and the empirical characteristic function'OX1./. By appropriately defining the distance metric, Neumann and Reiss (2008) showed the consistency of the proposed estimators. Another approach, followed for instance byWatteel and Kulperger(2003) and Comte and Genon-Catalot (2008), relies on an “explicit”
formula for the L´evy density s in terms of the derivatives of the characteristic function'X1. For instance, under certain regularity conditions,
F.xs.x// ./D i'X10 ./
'X1./; whereF.f /.u/ D R
eiuxf .x/dx denotes the Fourier transform of a functionf. Hence, an estimator fors can be built by replacing by a smooth version of the empirical estimate'OX1 and applying inverse Fourier transformF1.