DENSITY IN SMALL TIME FOR LEVY PROCESSES
JEAN PICARD
Abstract. The density of real-valued Levy processes is studied in small time under the assumption that the process has many small jumps. We prove that the real line can be divided into three subsets on which the density is smaller and smaller: the set of points that the process can reach with a nite number of jumps (-accessible points) the set of points that the process can reach with an innite number of jumps (asymptotically -accessible points) and the set of points that the process cannot reach by jumping (-inaccessible points).
1. Introduction
The problem of the absolute continuity of innitely divisible laws has been studied for a long time in the literature, especially in dimension 1, see Tucker (1965) (in larger dimension, more geometry is involved, see Yamazato (1994) for an example). Variables corresponding to these laws can be viewed as the values at a xed time of Levy processes Xt (processes with independent and stationary increments such that X0 = 0), or as linear functionals of Poisson random measures. The problem of the absolute continuity and of the smoothness of the density can also be extended to some non linear functionals of Poisson measures, especially to Markov processes Xt with jumps (see for instance Bismut (1983), Bichteler et al. (1987), Picard (1996)).
In order to study these processes, one has to consider the measure
(ydx) =PXt+dt2(y+dx)Xt=ydt
which describes how the process can jump fromy toy+x in the case of Levy processes, (ydx) = (dx) does not depend on y and is called the Levy measure of the process. The sucient conditions which are known for the existence of a smooth density involve two types of conditions, namely the mass of near 0 (the process must have many small jumps) and the smoothness of . Actually, one type of condition can be weakened if the other one is strengthened for instance, a Levy process has an absolutely continuous law if its Levy measure is absolutely continuous and if it has innitely many jumps (Tucker (1962)), but if the Levy measure is singu- lar, one has to impose a stronger condition on the number of small jumps moreover a critical behaviour is possible, where the law is singular for small times and absolutely continuous for large times (Tucker (1965)). If now we assume that the law ofXt is absolutely continuous for any t, a natural
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Received by the journal November 30, 1995. Revised January 7, 1997. Accepted for publication September 18, 1997.
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question is to study the behaviour of the densityp(tx) as t!0. This has been considered in Leandre (1987), Ishikawa (1993 and 1994) in the case of absolutely continuous Levy measures, and for a class of points x which the process can reach with a nite number of jumps (these points will be called -accessible in this paper). Our aim here is to consider the case of processes with possibly singular Levy measures and with a large enough number of jumps, so that aC1 density p(tx) exists for anyt however, in order to avoid many technicalities, we consider neither the general case of Markov processes (which requires the use of Malliavin's calculus), nor the multidimensional case (which involves more geometry) we limit ourselves to real-valued Levy processes and study the logarithmic behaviour ofp(tx) ast! 0 it appears that this simple case already involves some interesting geometrical properties.
We now state our results without making precise all the conditions. We need two main assumptions on the Levy processXt. Loosely speaking, (a) the rst assumption says that X has approximately the same number of small jumps than stable processes with some index 0< 2 the case
= 2 means that X contains a non trivial Brownian part, and if < 2, the precise statement of the assumption says that the tail at 0 of the Levy measure ofX satises an approximate scaling property
(b)the second assumption requires that the process goes suciently up and down this assumption is always satised if > 1, and otherwise, it says that the Levy measure has enough mass on bothR?+and R?;.
Under these two assumptions, it appears that the points x 2 Rcan be divided into three classes for which the behaviour ofp(tx) ast!0 is quite dierent.
(i)The rst class is the setA of -accessible points xthat the process can reach with a nite number of jumps (in particular x = 0 that the process can reach without any jump) more precisely, A is the set of points of the formPn1 xiwherexi is in the support of the Levy measure for these points (and under additional regularity conditions),
logp(tx) = ;(x)logt+o(log(1=t))
where the rate function ;(x) depends on the jumps xi which drive the pro- cess from 0 to x in particular, ;(0) = ;1=. Notice that this set A may be countable. A large deviation principle is easily proved for the law ofXt, but the rate function M(x), which is the minimal number of jumps which are necessary to reachx, is dierent from ;(x) in the singular case.
(ii)The second class is the setAnA of asymptotically -accessible points that the process can reach with an innite number of jumps for these points,
log(1=t)log(1=p(tx))C(logt)2: Actually, we will describe an example for which
logp(tx); (x)(logt)2
for some pointsxand a function (x) depending on the sequences of jumps driving the process from 0 tox.
(iii)Finally the third class is the setRnA of -inaccessible points that the process cannot reach with only jumps (this does not mean that these points are inaccessible for the process) under our two assumptions (a) and (b), this set can be non empty only if= 2 (Xhas a non trivial Brownian part), or if 1<<2 and all the jumps ofXhave the same sign, for instance positive (the process is said to be completely asymmetric, or spectrally positive) in this caseRnAis R?; and log(1=p(tx)) is of order t;1=(;1).
We also consider the case of non-decreasing Levy processes (or subordi- nators). In this case, the second assumption (b) is not satised however, similar properties can also be proved the main dierence concerns the be- haviour at -accessible points (notice for instance thatp(t0) = 0 under the rst assumption (a)).
Let us now set the notations which are used throughout this paper. Con- sider rst an innitely divisible law Q on R(or equivalently an innitely divisible variableXwith lawQ) its Fourier transform is given by the Levy- Khintchine formula
(w) =
Z
e iw x
dQ(x) = expiw ;22 +w2
Z
;
e iw x
;1;iw x101](jxj)d(x) (1:1) with2R,0 (diusion coecient) and a measure onR?satisfying
Z
;
x 2
^1d(x)<1
and which is the Levy measure ofQ. More generally, if = 1+i2 is a complex number such thate1x is -integrable on ;11]c, then
Z
e x
dQ(x) = exp+22 +2
Z
;
e x
;1;x101](jxj)d(x): (1:2) We also introduce a nite measure onRwhich is deduced fromand by
(dx) = (x2^1)(dx) +20(dx):
Then Q is characterized by the parameters (), or equivalently by () which are called its -parameter and -measure. It follows easily from (1.1) that the sum of two independent innitely divisible variables with parameters (11) and (22) is innitely divisible with parameters (1 +21 +2) thus, if we decompose the measure into 1 +2, we deduce a decomposition of the variableX into the sum of two independent variables. It is also an easy consequence of (1.1) that if the restriction of to ;11]chas a rst moment or a second moment, then a variable X with lawQsatises the same property and
EX =+
Z
jxj>1
xd(x) varX=2+
Z
x 2
d(x): (1:3) Now consider a Levy process Xt the law of each variable Xt is innitely divisible, and the parameters, , , of the process (Xt) are dened to
be the corresponding parameters of the variableX1 then the parameters of the variableXtare given bytandt thus the characteristic functiontof
X
tis deduced from the Levy-Khintchine formula (1.1), and a decomposition of also leads to a decomposition of Xt into the sum of independent Levy processes. In the particular case where= 0 and
Z
;
jxj^1d(x)<1 (1:4) thenXthas nite variation and can be written as
X
t=X
st
Xs+t (1:5)
where the-parameter is dened as
=;
Z
;11]
xd(x): (1:6) In this case, the law of X can be characterized by (). In particular,
= 0 means thatXt is a pure jump process similarly, we will say that an innitely divisible law satisfying (1.4) and = 0 is of pure jump type if its
-parameter dened by (1.6) is 0. If0 and is supported byR?+, then
X
t is non decreasing (it is a subordinator).
The paper is organized as follows. Inx2, we prove some preliminary results concerning innitely divisible variables. Inx3, assuming that the tail of at 0 satises an approximate scaling property, we estimate supxp(tx). In x4, we studyp(t0) and in the three subsequent sections, we studyp(tx) when
x is -accessible (x5), asymptotically -accessible (x6), or -inaccessible (x7) in x5, we also give the large deviation principle. The various constant numbers will be denoted byc or C and may vary from an equation to the other.
2. Preliminary results
Lemma 2.1. Let Qi be a family of innitely divisible laws with parameters
i and i. Suppose that the total mass i(R) is bounded, and that each Qi has a continuous densitypi. Then there exists a family xi such that xi;i is bounded andpi(xi) is bounded below by a positive number. In particular, supxpi(x) is bounded below.
Proof. LetXi be a variable with lawQi, and write Xi=i+Yi+Zi where
Y
i andZi are independent innitely divisible variables with-parameter 0, and the-measures of which are respectively the restriction ofi to ;11]
and its complement notice thatZi has-parameter 0 (see (1.6)). Then
EY 2
i =i(;11]) PZi= 0]exp;i(;11]c) = exp;i(;11]c) where the second relation is obtained by consideringZias the value at time 1 of a pure jump Levy process, and by noticing that the right-hand side is
the probability for the process to contain no jump. Thus
PjXi;ijh]PjYijh]PZi= 0]
;1;i(;11])=h2exp;i(;11]c)
;1;i(R)=h2exp;i(R)
if we choose h large enough this expression can be bounded below by a positive constant, so the supremum of pi on i;hi+h] can also be
bounded below. tu
Lemma 2.2. Consider a family Qi of innitely divisible laws of pure jump type, with Levy measurei satisfying
lim
"!0
sup
i Z
;""]
jxjd
i(x) = 0 sup
i
i(R)<1:
Suppose that eachQihas a continuous densitypi. Then for any xedh>0, there exists a family xi such that jxijh and pi(xi) is bounded below by a positive number.
Proof. Let 0<"1 be a number which will be chosen later. We decompose a variable with lawQi intoXi=Yi+Zi, where Yi and Ziare of pure jump type, and their Levy measures are the restriction of i to ;""] and its complement. Then
EY 2
i =
Z
;""]
xd
i(x)2+
Z
;""]
x 2
d
i(x)
Z
;""]
xd
i(x)2+"
Z
;""]
jxjd
i(x)
converges uniformly to 0 as"!0, so we can choose"such thatEYi2 h2=2.
On the other hand,
PZi= 0]exp;i(;""]c)exp;i(R)="2 so by proceeding as in Lemma 2.1,
PjXijh](1;EYi2=h2)exp;i(R)="2 ;exp;i(R)="22
is bounded below and we can conclude. tu
Lemma 2.3. Consider a family of innitely divisible lawsQiwith parameters (ii) and suppose that
i(;""])c"2; (2:1) for any 0 " 1 and for some c > 0 and 0 < 2. Then Qi has a smooth density pi, and pi(x), as well as all the derivatives p(k )i (x), are bounded uniformly in (ix).
Remark 2.4. The proof relies on integrability properties of the character- istic function, and is classical, see x4f of Bismut (1983) for related results
actually, it can be extended to some Markov processes with jumps, see Pi- card (1996).
Proof. We deduce from the Levy-Khintchine formula (1.1) that the charac- teristic function ofQj satises
j
j(w)j= exp;
Z
(1;cos(w x))dj(x);j2w2=2
exp;;c0w2j(;1=w 1=w])
forjw j1 and somec0 >0, where we have used the inequality 1;cos(w x)c0jw xj2
on the setfx jw xj1g. Thus, from our assumption (2.1),
j
j(w)jexp;cjw j
for jw j 1. In particular, j is integrable with respect to the Lebesgue measure, soQj has a density pj which is given by the inversion formula
p
j(x) = 12
Z
e
;iw x
j(w)dw 1 2
Z
j
j(w)jdw :
Thuspj is uniformly bounded. Similarly, by dierentiating this formula, one proves that
sup
x
p (k )
j (x) 1 2
Z
jw j k
j
j(w)jdw
is bounded. tu
The following lemma is a simple result concerning the closed support of innitely divisible laws. In particular, it does not say anything about much more complicated problems (such as the Hausdor dimension of the law, see Rubin (1967)) which have been studied in previous literature it is however sucient for our purpose, since our laws will be easily proved to have a smooth density from previous lemma.
Lemma 2.5. Consider an innitely divisible law Q without Gaussian part ( = 0) and the Levy measure of which satises(;""])>0 for any">0.
Suppose either that
(];10)(]0+1)>0 (2:2)
or that Z
(jxj^1)d(x) = +1: (2:3) Then the support of Q is R. If these two conditions are not satised, then the support is];1] or +1.
Remark 2.6. It is evident that the support is Rif Q has a non trivial Gaussian part.
Proof. For any">0, if " is the restriction of to ;""]c, one can check thatQ" is absolutely continuous with respect toQ. One deduces that
suppQ+ suppsuppQ: (2:4)
From our assumption, the support of contains numbers with arbitrarily small absolute value suppose for instance that it contains arbitrarily small positive numbers then we deduce from (2.4) that
x2suppQ=) x+1suppQ:
It is thus sucient to prove that under each of our two assumptions (2.2) and (2.3), the support ofQcontains arbitrarily large negative numbers, and that if neither (2.2) nor (2.3) hold, the smallest number in the support is. Under (2.2), the support of contains some negative number ;y, so (2.4) implies
x2suppQ=) (x;y)2suppQ
and we conclude. Let us now suppose that (2.2) does not hold, so that is supported by positive numbers. If we view Q as the law of X1 for some Levy processXt, then for any", the law of
X
"
1 =X1;X
t1
Xt1f Xt>"g
is absolutely continous with respect toQ. If (2.3) holds, thenX1" #;1 as
"#0, so its support contains somex(") which diverges to;1, and therefore suppQ also contains x(") if (2.3) does not hold, then X1" # from (1.5), and we can deduce that is in the support of Q moreover in this case,
X
t
;t is non-decreasing, so X1 and the support cannot contain a
smaller number. tu
Lemma 2.7. LetX be an innitely divisible variable with parameters() such that is supported by a bounded interval; ]. There exists a C>0 depending only on upper bounds onjj, (R) and such that for any h,
PjXjh]Ce;h:
Remark 2.8. Equivalently, one can say from (1.3) thatC depends only on upper bounds on ,jEXj and varX.
Proof. It is sucient to estimate the expectation of expX and exp;X. But from (1.2),
EexpX= exp+ 2 +2
Z
(ex;1;x1fjxj1g)d(x):
Since the support ofis bounded, the function in the integral is dominated by x2^1, so the term in the exponential is dominated by jj+(R). The expectation of exp;X is dealt with similarly. tu Lemma 2.9. Consider innitely divisible variablesX, the Levy measures of which are supported by some bounded interval ; ], and which have zero mean. Then there existsCn=Cn( ) such that
PjXjn ]Cn(varX)n:
Remark 2.10. One can say equivalently that
PjXjn ]Cn0(R)n:
Proof. Put "= varX and r = r(") = "1=(2(n+1)) one can suppose " 1.
ConsiderXas the value X1 of a Levy processXt at time 1, and decompose it intoXt=Xt1+Xt2 where
X 1
t =X
st
1fj Xsj>r gXs:
Let us consider the event fjX11j (n;1=2) g the process Xt1 is a pure jump process, and its jumps are at most (in absolute value), so on this event, there is at leastn jumps moreover, one more jump is needed if one of the jumps is less than =2. Thus, in order for jX11j to be greater than (n;1=2) , there must be at least n jumps satisfying =2 jXj , or at leastn+ 1 jumps satisfyingrjXj . These two numbers of jumps are Poisson variables with mean values
fjxj =2g4"= 2 fjxjr g"=r2 ="n=(n+1) so
PjX11j(n;1=2) ]fjxj =2gn+fjxjr gn+1 =O("n): (2:5) The Levy measure ofX2=ris supported by ;11],
E
X 2
1
r
=
E
X 1
1
r
=
Z
jxj>r x
r d(x)
"=r
2
1 varX12
r
"=r 2
1 so from Lemma 2.7,
PjX12j =2]Ce; =(2r ) =o("k) (2:6) for anyk. We conclude by adding (2.5) and (2.6). tu
3. Estimation of the supremum of the density
We have seen inx2 that the supremum of the density of an innitely divisible variable can be estimated from the behaviour of its -measure. We now translate these results in the case of a Levy process in small time.
Theorem 3.1. Let Xt be a Levy process with parameters ().
(a) Suppose that
liminf
"!0
"
;2
(;""])>0 (3:1) for some 0< 2. ThenXt has a C1 density satisfying
sup
x
p(tx) =O(t;1=)
ast!0, and more generally sup
x
p
(k )(tx)=O(t;(k +1)=):
(b) Suppose that Xt has a continuous density p(t:) and that limsup
"!0
"
;2
(;""])<1 (3:2) for some 0< 2. Then, for t small enough,
sup
x
p(tx)ct;1=: (3:3) Proof. Consider Yt =Xt=t1=. Of course, Yt is not a Levy process but is for each t an innitely divisible variable. Its Levy measure and diusion coecient are given by
t(A) =t(t1=A) t=t1=2;1=
so its-measure satises
t(R) =t
Z
x 2
t 2=
^1d(x) +t 2
t
2= =t
Z
d(x) (x2^1)_t2=
if t1, and
t(;""]) =t1;2=(;"t1="t1=]) (3:4) for"1. In case (a), we deduce that
t(;""])c"2;
so from Lemma 2.3, the variablesYt have uniformly bounded densitiesq(t:) with bounded derivatives thusXt has a density given by
p(tx) =t;1=q(tt;1=x) (3:5) and we can conclude. In case (b), from an integration by parts,
t(R) =t(R)+ 2t
Z
1
t 1=
(;xx])
x 3
dx (3:6)
which is bounded from our assumption. Thus we can apply Lemma 2.1 and deduce that the supremum of the density ofYt is bounded below. We again conclude from the formula (3.5) relating the densities ofXt and Yt. tu Remark 3.2. The assumption (3.2) of case (b) is always satised with
= 2, so the Wiener process provides a lower bound for the concentration of Levy processes in small time.
Remark 3.3. IfXtis a-stable process, thensatises the scaling property
(;""]) ="2;(;11])
and both conditions (a) and (b) hold. These conditions can actually be viewed as an approximate scaling property on the tail of at 0.
Remark 3.4. If we assume condition (3.2) with <1, then
Z
;11]
jxjd(x) =
Z
;11]
d(x)=jxj=(;11])+
Z
1
0
(;xx])=x2dx<1 soXt has nite variation. If we assume condition (3.1) with 1, then
Z
;""]
jxjd(x) =
Z
;""]
d(x)=jxj(;""])="c"1; c (3:7)
so Z
;11]
jxjd(x) =1 andXt has innite variation.
Remark 3.5. By joining the two parts of Theorem 3.1, we deduce that
c"
2;
(;""])C"2; =)c0t;1= sup
x
p(tx)C0t;1=
and that
lim
"!0
;log(;""])log"= 2; (3:8) implies
lim
t!0
;sup
x
logp(tx)logt=;1=:
4. Estimation near 0
We now want to nd a condition ensuring thatp(t0) also satises a lower bound similar to (3.3), so that it has the order of magnitude of supxp(tx).
Consider rst the symmetric case= 0 and symmetric in this case, the function x 7!p(tx) has its maximum at x= 0 (because the characteristic function is positive), and therefore the behaviour of p(t0) follows from Theorem 3.1. Let us now consider the general non-symmetric case.
Lemma 4.1. Let Qi be a family of innitely divisible laws with parameters (ii), and let 0< 2. Suppose thatiandi(R) are uniformly bounded and that one of the two following conditions is satised
(a) 1 and i satises the lower bound condition (2.1) (b) <1 and both restrictions of i to R+and R; satisfy (2.1).
Then for any > 0, the density pi of Qi (which exists from Lemma 2.3) satises
inf
i
inf
jxj p
i(x)>0:
Proof. Let i1 be the restriction of i=2 to ;11], let i2 = i;i1, and decompose a variable Xi with law Qi into independent variables Xi1+Xi2 with respective coecients (i10) and (i2i). Letp2i be the density ofXi2
the mass i2(R) and the coecient i are bounded, so, from Lemma 2.1, there exists a bounded xi such that p2i(xi) is bounded below by a positive number moreover, the derivative of p2i is uniformly bounded from Lemma 2.3, so there exists a positive csuch that
8x2R jx;x
i
jc=)p2i(x)c:
If we notice thatpi(x) is the expectation ofp2i(x;Xi1), we obtain inf
jxj p
i(x)c inf
jxj
Pjx;X 1
i
;x
i jc
c inf
jxj +C PjX
1
i
;xjc
:
Thus it is sucient to check that for anyC and c, inf
i
inf
jxjC
PjXi1;xj<c]>0 or that
inf
X2X
inf
jxjC
PjX;xj<c]>0
where X denotes the set of laws of Xi1. The measures i1 are relatively compact for the topology of convergence on bounded functions (they are bounded and supported by ;11]), and for any converging subsequence, the corresponding Xi1 also converges in law (this is an easy application of the Levy-Khintchine formula (1.1)) thusX is relatively compact. Since the map (xX)7!PjX;xj<c] is lower semicontinuous, it is sucient to prove that
PjX;xj<c]>0
for anyx,c and anyX in the closure X of X. This means that one has to prove that the closed support of any X 2 X is R the -measure 0 of X satises an estimate of type
0(;""])c"2;
because these estimates hold uniformly fori1 if = 2,X has a non trivial Gaussian part, so the result is immediate if 1 < 2 and if 0 is the Levy measure ofX, the function jxj^1 is not 0-integrable from (3.7), so the result follows from Lemma 2.5 in the case <1, both 0(;"0) and
0(]0"]) satisfy the above estimate, so one can also apply Lemma 2.5. tu Lemma 4.2. LetQi be a family of innitely divisible laws of pure jump type we suppose that the-measures i are uniformly bounded, supported by R?+, and that
c"
2;
i(0"])C"2;
for some0<<1. Then for any 0< 1 < 2, the densitiespi ofQisatisfy inf
i
inf
1 x
2 p
i(x)>0:
Proof. We follow the argument of the proof of Lemma 4.1. We decompose
X
i into Xi1+Xi2 of pure jump type, with-measures i1 and i2 dened as in Lemma 4.1. Notice that
Z
0"]
d 2
i(y)
y
= i2(0"])
"
+
Z
"
0
2
i(0y])
y 2
dyC"
1;
so Lemma 2.2 and the fact thatXi2 0 show that there exists 0<xi< 1 such thatp2i(xi)c. From the boundedness of the derivative ofp2i, we also deduce
jx;x
i
jc=)p2i(x)c:
Thus inf
1 x
2
p(x)c inf
1 x
2
Pjx;X 1
i
;x
i jc
c inf
0x
2 PjX
1
i
;xjc
:
Therefore one only has to prove that inf
X2X
inf
0x 2
PjX;xj<c]>0
or that the support of anyX 2X containsR+ but this follows from Lemma
2.5, becauseX is of pure jump type. tu
Theorem 4.3. Assume that
c"
2;
(;""])C"2; (4:1) for any0"1 and some index 0< 2. Suppose also that one of the four following conditions is satised
(a) >1 (b) = 1 and
limsup
"!0
Z
f"jxj1g
xd(x)<1
(c) < 1, both restrictions of to R?; and R?+ satisfy (4.1), and the - parameter is 0 (so thatXt is a pure jump process)
(d) <1, is supported by R?+ and the -parameter is 0 (so that Xt is a pure jump non decreasing process).
Then, in cases (abc), for any positive , one has
jxj t
1==)ct;1= p(tx)Ct;1=
fortsmall enough, so in particularp(t0) is of ordert;1=. In case (d), for any positive 1 and 2 and for smallt,
1 t
1=
x
2 t
1= =)ct;1=p(tx)Ct;1=:
Remark 4.4. The conclusion of case (d) is weaker. This is not surprising sincep(t0) = 0. One can say in this case thatXslips on the right, and one can refer to the cases (abc) as the non slipping case.
