On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise
The local properties of distributions of solutions of SDE's with jumps are studied. Using the method based on the “time-wise” differentiation on the space of functionals of Poisson point measure, we give a full analog of Hormander condition, sufficient for the solution to have a regular distrib...
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irk-123456789-1658332020-02-17T01:27:51Z On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise Kulik, A.M. Статті The local properties of distributions of solutions of SDE's with jumps are studied. Using the method based on the “time-wise” differentiation on the space of functionals of Poisson point measure, we give a full analog of Hormander condition, sufficient for the solution to have a regular distribution. This condition is formulated only in terms of coefficients of the equation and does not require any regularity properties of the Levy measure of the noise. Вивчаються локальні властивості розв'язків СДР зі стрибками. При застосуванні методу, який базується на „диференціюванні за часом" на просторі функціоналів від пуассонової точкової міри, наведено умову, яка аналогічна умові Хьормандера та достатня для того, щоб розв'язок мав регулярний розподіл. Ця умова формулюється тільки у термінах коефіцієнтів рівняння та не вимагає від міри Леві виконання будь-яких властивостей регулярності. 2005 Article On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise / A.M. Kulik // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1261–1283. — Бібліогр.: 13 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165833 519.21 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Kulik, A.M. On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise Український математичний журнал |
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The local properties of distributions of solutions of SDE's with jumps are studied. Using the method based on the “time-wise” differentiation on the space of functionals of Poisson point measure, we give a full analog of Hormander condition, sufficient for the solution to have a regular distribution. This condition is formulated only in terms of coefficients of the equation and does not require any regularity properties of the Levy measure of the noise. |
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Article |
| author |
Kulik, A.M. |
| author_facet |
Kulik, A.M. |
| author_sort |
Kulik, A.M. |
| title |
On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise |
| title_short |
On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise |
| title_full |
On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise |
| title_fullStr |
On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise |
| title_full_unstemmed |
On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise |
| title_sort |
on the regularity of distribution for a solution of sde of a jump type with arbitrary levy measure of the noise |
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Інститут математики НАН України |
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2005 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165833 |
| citation_txt |
On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise / A.M. Kulik // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1261–1283. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT kulikam ontheregularityofdistributionforasolutionofsdeofajumptypewitharbitrarylevymeasureofthenoise |
| first_indexed |
2025-07-14T20:05:38Z |
| last_indexed |
2025-07-14T20:05:38Z |
| _version_ |
1837654122547904512 |
| fulltext |
UDC 519.21
A. M. Kulik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ON A REGULARITY OF DISTRIBUTION
FOR SOLUTION OF SDE OF A JUMP TYPE
WITH ARBITRARY LEVY MEASURE OF THE NOISE∗
PRO REHULQRNIST\ ROZPODILU ROZV’QZKU SDR
ZI STRYBKAMY Z DOVIL\NOG MIROG LEVI
In the paper the local properties of distributions of solutions of SDE’s with jumps are studied. Using the method,
based on the ”time-wise” differentiation on the space of functionals from Poisson point measure, we give a full
analogue of Hörmander condition, sufficient for the solution to have a regular distribution. This condition is
formulated only in terms of coefficients of the equation and does not require any regularity properties of the
Levy measure of the noise.
Vyvçagt\sq lokal\ni vlastyvosti rozv’qzkiv SDR zi strybkamy. Pry zastosuvanni metodu, qkyj bazu-
[t\sq na „dyferencigvanni za çasom” na prostori funkcionaliv vid puassonovo] toçkovo] miry, nave-
deno umovu, qka analohiçna umovi X\ormandera ta dostatnq dlq toho, wob rozv’qzok mav rehulqrnyj
rozpodil. Cq umova formulg[t\sq til\ky u terminax koefici[ntiv rivnqnnq ta ne vymaha[ vid miry
Levi vykonannq bud\-qkyx vlastyvostej rehulqrnosti.
Introduction. In this article we deal with the following general problem. Let X(x, t, r)
be the solution of the SDE
X(x, t, r) = x+
t∫
r
a(s,X(x, s, r)) ds+
+
t∫
r
∫
Rd
c(s,X(x, s−, r), u)ν̃(ds, du), t ∈ [r,+∞), (0.1)
where ν is a random Poisson point measure on Rd×R+ with the Levy measure Π, ν̃ is
corresponding compensated measure (we are not going into details with introducing this
standard objects, referring the reader, if necessary, to [1]), and coefficients a, c satisfy
standard conditions, sufficient for equation (0.1) to have unique strong solution. Denote
by P (x, t, r, dy) the distribution of this solution, P (x, t, r, dy) ≡ P (X(x, t, r) ∈ dy).
It is natural both from probabilistic and analytical points of view to consider the follow-
ing family of questions: does measure P (x, t, r, dy) have a density p(x, t, r, y) w.r.t.
Lebesgue measure? Does this density, considered either as a function from y under fixed
t, x, or as a function from (x, t, r, y), possess some regularity property, for instance be-
longs to some Lp,loc, is locally bounded, belongs to classes Ck or C∞, etc? These
questions were studied by numerous authors, let us emphasize two big groups of results
in this direction, which are based on different ideas and impose essentially different con-
ditions on the Levy measure Π of the noise.
The first group is based on the approach proposed by J. Bismut [2], in which some
Malliavin-type calculus on a space of trajectories of Levy processes is introduced via
transformations of trajectories, which change values of its jumps (see [2 – 5] and refer-
ences there). In this approach Levy measure is supposed to have some (regular) density
∗ Research supported in part by the Ministry of Education and Science of Ukraine (Grant of the President
of Ukraine), project No. GP/F8/0086.
c© A. M. KULIK, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9 1261
1262 A. M. KULIK
w.r.t. Lebesgue measure, which is a natural condition, sufficient for such transformations
to be admissible.
The second group is based on the method by J. Picard [6], in which some version
of stochastic calculus of variations for Poisson point measure is proposed. This method
uses perturbations of the point measure by adding point into it and requires method some
limitations on the asymptotic behavior of the Levy measure at the origin.
It is natural to try to give some sufficient conditions for the regular density to ex-
ist, which would not involve any specific conditions on the Levy measure. As a first
possible answer on this question, let us mention recent results by V. N. Kolokol’tsov
and A. D. Tyukov [7], who developed an analytical approach for SDE’s of some special
form, which, we believe, is not crucial and is caused by the framework of characteristics
method for stochastic heat equation with a jump noise. This approach allows to prove
regularity results for small time part of the initial distribution, this means that instead of
P (x, t, r, dy) the measures E1IX(x,t,r)∈dy · 1It≤τ , where τ is some specific stopping
time, are considered.
Another point of view on this problem was given in the recent work by the author
[8]. It was motivated by a natural idea, that without any conditions on the Levy measure
there always exist admissible transformations of the Poisson point measure ν, which
change the moments of jumps, and one can construct some kind of stochastic calculus of
variations based on these transformations. This idea is not very new, it was mentioned
in the introduction to [6]. However, the rigorous development of this idea is nontrivial, it
appears that the corresponding calculus have some new properties, which does not exist
in Malliavin calculus for diffusions or Bismut calculus for jump processes with regular
Levy measures (see discussions in [8] and Example 1.4 below).
In the work [8] the following two problems remained unsolved. First, sufficient condi-
tion for P (x, t, r, dy) to have a density was given in the following form: some combina-
tion of differential and difference operators, defined by the coefficients of initial equation,
has to be nondegenerated. This can be interpreted as a partial analogue of Hörmander
condition, as soon as Hörmander condition is formulated in the terms not of one, but of
a sequence of vector fields. Thus, it is natural to try to give a regularity result under a
full analogue of Hörmander condition. Another problem is regularity properties of the
density. It was shown in [8] (see also Example 1.4 below), that the density, considered
as a function of y, can be extremely nonregular, for instance, there exist situations in
which it does not belong to L1+ε,loc for ε > 0. At the same time, the properties of the
density as a function of (t, x) were not studied. In this paper we solve the first problem
and prove the regularity of the density under a full analogue of Hörmander condition.
1. Main result. We suppose that coefficients a : R+×Rm → Rm, c : R+×Rm×
× Rd → Rm of equation (0.1) are measurable functions which are infinitely differen-
tiable in (s, x) and locally bounded together with their derivatives. We also assume that
∃K ∀x, y ∈ Rm, s ∈ R+ :∫
Rd
‖c(s, x, u)− c(s, y, u)‖2Π(du) ≤ K‖x− y‖2,
so that (0.1) has a unique strong solution which is a process with cádlág trajectories. Also
we suppose the following condition to hold true,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1263
∫
Rd
sup
s∈[0,T ],‖x‖≤R
[
‖c(s, x, u)‖Rm + ‖∇xc(s, x, u)‖Rm×Rm
]
Π(du) < +∞ (1.1)
for any positive T, R.
Under this condition (0.1) can be rewritten in the equivalent form
X(x, t, r) = x+
t∫
r
ã(s,X(x, s, r)) ds+
t∫
r
∫
Rd
c(s,X(x, s−, r), u)ν(ds, du), t ∈ R+,
(1.2)
with ã(s, x) = a(s, x)−
∫
Rd
c(s, x, u)Π(du).
Let us introduce some notations. For every function Υ(s, x, u), s ∈ R+, x ∈ Rm,
u ∈ Rd, which takes values in Rm and is smooth w.r.t. (s, x), we define ΛΥ ≡ ΛãΥ
by
(ΛΥ)(s, x, u) = ∇xΥ(s, x, u)ã(s, x) + Υ′
s(s, x, u)−∇xã(s, x)Υ(s, x, u),
here ∇x denotes vector derivative w.r.t. variable x. We also define ΞuΥ ≡ Ξc,uΥ by
(ΞuΥ)(s, x, u) = [IRm +∇xc(s, x, u))]−1Υ(s, x+ c(x, u), u).
Note that the function ΞuΥ is well defined only for s, x, u satisfying assumption
−1
∈ σ(∇xc(s, x, u)), (1.3)
we denote the set of such (s, x, u) by Θ and put Θs,x = {u|(s, x, u) ∈ Θ}.
For (s, x, u) ∈ Θ we put
∆(s, x, u) = [IRm +∇xc(s, x, u))]−1×
×
[
{ã(s, x+ c(s, x, u))− ã(s, x)} − ∇xc(s, x, u)ã(s, x)− c′s(s, x, u)
]
,
and introduce the family of Rm-valued functions {∆i0,...,ik
k , k ≥ 0, ir ∈ Z+, r =
= 0, . . . , k} by
∆i0,...,ik
k (s, x, u0, . . . , uk) = ΛikΞuk
Λik−1 . . .Λi1Ξu1Λ
i0∆(s, x, u0),
s ∈ R+, x ∈ Rm, u0, . . . , uk ∈ Θs,x.
Next, we denote by Lk(s, x, u0, . . . , uk), k ≥ 0 the linear span (in Rm ) of the vectors{
∆i0,...,ij
j (s, x, u0+r, . . . , uj+r), i0, . . . , ij ≥ 0, r = 0, . . . , k− j, j = 0, . . . , k
}
.
One can see that the family {Lk} is monotonous in a sense that Lk(s, x, u0, . . . , uk) ⊂
⊂ Lk+1(s, x, u0, . . . , uk+1).
At last, let us denote
Π∗
k+1(A) = sup
n≥1
Π⊗(k+1)
(
A ∩
{
(u0, . . . , uk) ∈ (Rd)k+1
∣∣∣∣ ‖ui‖ > 1
n
, i = 0, . . . , k
})
{
Π
({
u ∈ Rd| ‖u‖ > 1
n
})}k ,
A ∈ B
((
Rd
)k+1
)
.
For every k function Π∗
k+1 posses the following properties:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1264 A. M. KULIK
1) Π∗
k+1(A) ≤ Π∗
k+1(B) for A ⊂ B;
2) Π∗
k+1(A) = limn→∞ Π∗
k+1(An) for An ↑ A, n→ +∞;
3) Π∗
k+1(A) ≤
∑∞
n=1
Π∗
k+1(An) for A ⊂
⋃∞
n=1An.
Informally one can interpret the space (Rd)k+1 as the space of (k+ 1)-point config-
urations and Π∗
k+1 as the outer measure, generated on this space by initial Levy measure.
Now we can formulate our main result.
Theorem 1.1. A. Suppose that
∀x ∈ Rm, s ∈ R+, l̄ ∈ Rm\{0} :
Π
{
u ∈ Θs,x : l̄ is not orthogonal to L0(s, x, u)
}
= +∞. (1.4)
Then for every x ∈ Rm, 0 ≤ r < t
P ◦ [X(x, t, r)]−1 � λm.
B. Suppose that there exists k > 0 such that for every x ∈ Rm, s ∈ R+,
l̄ ∈ Rm\{0}
Π∗
k+1
{
(u0, . . . , uk) ∈ [Θs,x]k+1 :
l̄ is not orthogonal to Lk(s, x, u0, . . . , uk)
}
= +∞. (1.5)
Suppose also two following additional conditions to hold true:
A) there exists C > 0 such that functions a(·, ·) and c(·, ·, u) for Π-almost all
u ∈ Rd are analytical functions in every point (s, x) ∈ R+ × Rm with the radius of
analyticity not less than C;
B)
∫
Rd
sup
s∈[0,T ],‖x‖≤R
[
‖(∇x)jc(s, x, u)‖(Rm)×j
]
Π(du) < +∞ for any j ∈ N,
T > 0, R > 0.
Then for every x ∈ Rm, 0 ≤ r < t
P ◦ [X(x, t, r)]−1 � λm.
The proof of Theorem 1.1 will be given in Section 2, some improvements will be
given in Section 3. Here let us make some discussion.
It was shown in [8] (see also Example 1.4 below) that the density p(x, t, r, y) of
distribution of X(x, t, r), considered as a function of y, can be extremely nonregular,
for instance, there exist situations in which it does not belong to Lp,loc for every p > 1.
At the same time, the properties of this density, considered as a function of (t, x), were
not studied.
Proposition 1.1. Under conditions of Theorem 1.1 the function
Rm × {(t, r) ∈ (R+)2|t > r} � (x, t, r) �→ p(x, t, r, ·) ∈ L1(Rm)
is continuous.
The proof of this statement is a subject of a separate paper [9] and is based in the
methods, developed recently in [10].
Next, the statement of Theorem 1.1 can be rewritten in a form, which is natural
from the point of view of theory of pseudo-differential operators, let us do this in time-
homogeneous case.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1265
For an Rm-valued function Υ(x) denote by the same letter Υ differential operator
on Rm, defined by
(Υf) = (∇f,Υ)Rm , f ∈ C1
b (R
m),
then it is clear that
ΛΥ = [ã,Υ] ≡ ã ·Υ−Υ · ã.
Also denote by Cu (for every u ∈ Rd ) difference operator defined by
Cuf(x) = f(x+ c(x, u))− f(x), f ∈ Cb(Rm).
Under more strong version of condition (1.3),
sup
u∈Rd,x∈Rm
‖∇xc(x, u)‖ < 1, (1.3′)
operator I + Cu is invertible (here I is identity operator), and one can see that
ΞuΥ = (I + Cu)Υ(I + Cu)−1, ∆(·, u) = ã− Ξuã.
Statements of Theorem 1.1.A and Proposition 1.1 now can be reformulated in the follow-
ing form.
Corollary 1.1. Let us say that the family of operators {Ψu,k, k ≥ 0}, indexed by
u ∈ Rd, is nondegenerated w.r.t. measure Π if for every x ∈ Rm and every f ∈
∈ C1(Rm) such that ∇f(x)
= 0
Π
(
{u|∃k ≥ 0 : Ψu,kf(x)
= 0}
)
= +∞.
Consider the family
Ψ0,u = [ã, Cu](I+Cu)−1 = ã−(I+Cu)ã(I+Cu)−1, Ψk,u = [ã,Ψk−1,u], k > 0,
and suppose that it is nondegenerated w.r.t. measure Π. Then for the PDO L, given by
the formula
L = ã+
∫
Rd
CuΠ(du),
the fundamental solution of equation
u′t = Lu
is usual (not generalized) function, which is continuous while considered as a function
from Rm × R+ to L1(Rm).
Statement B of Theorem 1.1 also can be reformulated in the same way, we omit this
in order to shorten exposition.
At last, let us give several examples, illustrating different features of the regularity
result, given by Theorem 1.1.
The first example shows that Theorem 1.1 is a crucial improvement of Theorem 4.2
[8]. It is motivated by the classical Kolmogorov’s example of a diffusion, which hy-
poellipticity can not be provided only by condition on a diffusion part, see [11] or [12],
Chapter 5, Example 8.1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1266 A. M. KULIK
Example 1.1. Consider SDE
X1(t, x̄) = x1 +
t∫
0
X2(s, x̄) ds+ ηt,
X2(t, x̄) = x2 +
t∫
0
X1(s, x̄) ds,
(1.6)
where x̄ = (x1, x2) ∈ R2 and ηt is some compensated Levy process with infinite Levy
measure. Then this equation is of the type (0.1) with m = 2, d = 1 and
a(s, x̄) = (x2, x1)T , c(s, x̄, u) = (u, 0)T .
One has
∆(x̄, u) = (0, u)T , ∆1
0(x̄, u) = −(u, 0)T ,
which means that for every x ∈ R2, u
= 0 L0(x, u) = R2 and therefore condition of
Theorem 1.1 holds true. Note that for l̄ = (1, 0)T Π(u|(l̄,∆(x̄, u))
= 0) = 0 for every
x ∈ R2, which means that condition of Theorem 4.2 [8] fails.
It is worth to mention that equation (1.6) is not the full analogue of Example 8.1 [12].
The corresponding analogue should be written as follows:
X1(t, x̄) = x1 +
t∫
0
X1(s, x̄) ds+ ηt,
X2(t, x̄) = x2 +
t∫
0
X1(s, x̄) ds.
(1.6′)
Equation (1.6′) gives the simple counterexample, which shows that conditions of the
Theorem 1.1 are close to necessary ones. Namely, in this case ∆i0,...,ik
k (x̄, u0, . . . , uk) =
= (u0, u0)T for every k ≥ 1, i0, . . . , ik ≥ 1, x̄ ∈ R2, u0, . . . , uk ∈ R, and conditions
of the Theorem fail. On the other hand, one can choose Π in such a way that the distri-
bution of X1(t)−X2(t) = ηt is not absolutely continuous (see Example 1.4 below), and
for Π the joint distribution (X1(t), X2(t)) definitely is not absolutely continuous.
Next, let ψ ∈ C∞(R) be globally Lipschitz. Let us consider equation
X1(t, x̄) = x1 +
t∫
0
ψ(X2(s, x)) ds+ ηt,
X2(t, x̄) = x2 +
t∫
0
X1(s, x) ds.
(1.7)
One has that
∆(x̄, u) = (0, u)T , ∆r
0(x̄, u) = (−u)r(ψ(r)(x2), 0)T .
Now let us take ψ(x) which is equal in some neighborhood of 0 to xr, r ∈ N, then
equation (1.7) shows that, in general, we can not replace in statement A the family
L0(x, u) by Lr0(x, u) ≡
〈
∆i
0(x, u), i ≤ r
〉
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1267
In the second example conditions of statement B of the Theorem hold true, but condi-
tions of statement A fail.
Example 1.2. Consider SDE
X1(t, x̄) = x1 +
t∫
0
X1(s, x̄) ds+ η1
t ,
X2(t, x̄) = x2 +
t∫
0
X1(s, x̄)X2(s, x̄) ds+ η2
t .
(1.8)
We suppose that the Levy measure Π of the Levy process ηt = (η1
t , η
2
t ) is concentrated
on the set
{
(u1, u2) ∈ R2 : u1 ·u2 = 0
}
and denote by Π1, Π2 restrictions of Π on the
axis
{
(u1, u2) : u2 = 0
}
and
{
(u1, u2) : u1 = 0
}
correspondingly. Straightforward
computations give that
∆(x̄, ū) = (u1, u1x2 + u2x1)T , ∆r
0(x̄, ū) = (−1)r(u1, Pr,u1,u2(x1, x2))T ,
where Pr,u1,u2 is some polynom with the free term equal to zero. This means that for
x̄ = (0, 0)T , l̄ = (0, 1)T condition (1.4) fails.
On the other hand, consider vectors
∆(x̄, ū0), ∆0,0
1 (x̄, ū0, ū1) =
(
u0
1, u
0
1(x2 + u1
2) + u0
2(x1 + u1
1)
)T
,
one can see that these vectors generate R2 for every ū0, ū1 ∈ supp Π such that u0
1
= 0,
u1
2
= 0. For the set A of such pairs (ū0, ū1) we have that
Π∗
2(A) = sup
n
Π1
(
‖ū‖ > 1
n
)
Π2
(
‖ū‖ > 1
n
)
Π
(
‖ū‖ > 1
n
) ,
and Π∗
2(A) = +∞ under condition
Π1(R2) = +∞, Π2(R2) = +∞. (1.9)
It is easy to see that condition (1.9) is a nessesary one: if it fails, then for the solution of
(1.8) with x1 = x2 = 0 the distribution either of X1(t) or X2(t) has an atom.
The third example shows the following interesting feature. In the usual Hörmander
condition the linear subspace, generated by the corresponding family of vector fields, is
supposed to have maximal possible dimension. This example shows, that condition of
Theorem 1.1 can hold true even if dim Lk(s, x, u) < m for every s, x, u, k.
Example 1.3. Consider SDE
X1(t, x̄) = x1 +
t∫
0
X1(s, x̄) ds+ η1
t ,
X2(t, x̄) = x2 +
t∫
0
X2(s, x̄) ds+ η2
t ,
where ηt =
(
η1
t , η
2
t
)
is a compensated two-dimensional Levy process. One has that for
every k, i0, . . . , ik ∈ Z+
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1268 A. M. KULIK
∆i0,...,ik
k (x̄, u0, u1, . . . , uk) = ∆(x̄, u0) = u0, u0, . . . , uk, x̄ ∈ R2,
and therefore dim Lk(x̄, u0, . . . , uk) = 1 for u0
= 0 and dim Lk(x̄, u0, . . . , uk) = 0
otherwise. On the other hand, if for every l̄ ∈ R2
Π(ū
∈ 〈l̄〉) = +∞,
condition of Theorem 1.1 hold true.
One can say, that in the considered examples solution of equation
X(t, x) = x+
t∫
0
X(s, x) ds+ ηt (1.10)
plays the role of an analogue of one-dimensional diffusion with constant coefficients.
Indeed, in condition of Theorem 1.1, considered as an analogue of Hörmander condition,
function ∆ corresponds to the vector field generated by diffusion coefficient, and for
equation (1.10) ∆ does not depend on x. The following example (last in this section)
shows, that even in this simplest case one can hardly expect to obtain for the density, given
by Theorem 1.1, any regularity properties better than given in Proposition 1.1.
Example 1.4. Consider (1.10) with ηt =
∑∞
k=1
1
αk
ηkt , where α > 1, {ηk} are
independent Poisson processes with the same intensity λ. Let for simplicity x = 0, then
one can show (see [8], Chapter 5) that there exists β = β(α) > 0 such that
lim sup
ε→0+
ε
−λt
β P(X(t, 0) > ε) > 0. (1.11)
We have ∆(x, u) = u, i.e., conditions of Theorem 1.1 hold true and there exists a density
p(t, ·) of distribution of X(t, 0). We have P(X(t, 0) < 0) = 0, therefore from (1.11)
we obtain that
p(t, ·)
∈ Ck(R), if λt < β(k + 1), k ≥ 1,
p(t, ·)
∈ L∞,loc(R), if λt < β, (1.12)
p(t, ·)
∈ Lp,loc(R), if λt < β
(
1− 1
p
)
, p ∈ (1,+∞).
It appears, that statements (1.12) are rather precise, namely, as soon as (1.10) is a linear
equation, one can calculate the characteristic function ϕt(·) of X(t, 0) explicitly and
then obtain the estimate ϕt(z) = o
(
|z|−λt
γ
)
, |z| → ∞, with some γ = γ(α) > 0,
which means that
p(t, ·) ∈ Ck(R), λt > γ(k + 1), k ≥ 0.
This forms the phenomenon, which can be call “gradual hypoellipticity”: fundamental
solution of the corresponding equation with PDO becomes smooth not instantly, but after
some period of time, and this period depends linearly on the rate of smoothness which is
to be achieved.
The feature of “gradual hypoellipticity” is interesting, but not very common, this can
be illustrated by the following modification of the example. Using the same arguments
to those made in [8] (Chapter 5), one can construct for every given function ϕ with
ϕ(0+) = +∞ a sequence {αk} ∈ R+, such that for the solution of (1.10) with ηt =
=
∑
k
αkη
k
t the following analogue of (1.11) holds true:
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ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1269
∀t > 0 : lim sup
ε→0+
ϕ(ε)P(X(t, 0) > ε) > 0. (1.13)
This can be reformulated in the following form. Let Φ be positive convex function on
R, denote by LΦ
loc the space of the functions f on R such that∫
I
Φ(f(x)) dx < +∞
for every finite interval I. The following statement is due to (1.13) and Jensen’s in-
equality.
Proposition 1.2. For any fixed Φ with
Φ(x)
|x| → ∞, |x| → ∞, there exists an
equation of the type (1.10) such that
p(t, ·)
∈ LΦ
loc, t > 0.
It is worth to be mentioned that these interesting features do not occur in classes of
equations, which can be treated by methods of J. Bismut or J. Picard. Another new feature
is that, at least for some values of α (say, integers or so called P.V. numbers), distribu-
tion of ηt =
∑∞
k=1
1
αk
ηkt is singular for every t. The situation, when the solution
of SDE is regular while the noise, driving this equation, is not, seems not to be studied
systematically yet.
As a conclusive remark let us say that our method of proof (some modification of
stratification method) appears to be well suited for a “boundary region” of equations of
the type (0.1), in which such phenomenons, as “gradual hypoellipticity” or singularity of
initial noise, hold, and which can not be treated by other known methods. However, the
price is that this method does not allow one to obtain general results on regularity, more
strong than given in Proposition 1.1.
2. Proof. First we will prove the theorem, supposing coefficient c to satisfy addi-
tional condition
sup
s∈R+, u∈Rd, x∈Rm
‖∇xc(s, x, u)‖ < 1. (2.1)
Denote by
{
Etr
}
the m×m-matrix valued process satisfying equation
Etr = IRm +
t∫
r
∇xã(s,X(s))Esr ds+
+
∫∫
[r,t]×Rd
∇xc(s,X(s−), u)Es−r ν(ds, du).
Under condition (2.1) matrix Etr is a.s. invertible for every r, t.
The starting point in our proof is the following statement (see [8], Theorem 4.1).
Denote by p(·) the point process corresponding to random point measure ν .
Proposition 2.1. Denote by St the linear span of the set of vectors
{
[Eτ−0 ]−1∆(τ,
X(τ−), p(τ)), τ ≤ t
}
, where τ ’s are taken from the domain D of the process p(·).
Let Ωt = {ω| dimSt(ω) = m}, then
P|Ωt
◦ [X(t)]−1 � λm.
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1270 A. M. KULIK
Our aim is to show that under conditions of Theorem 1.1 the set Ωt coincides with
Ω almost surely. The method of proof is up to [8, 13] and is based on so called “time-
stretching” transformations of the jump process, let us briefly give here necessary con-
structions. Denote H = L2(R+), H0 = L∞(R+) ∩ L2(R+), Jh(·) =
∫ ·
0
h(s) ds,
h ∈ H. For a fixed h ∈ H0 define the family {T th, t ∈ R} of transformations of the
axis R+ by putting T thx, x ∈ R+ equal to the value at the point s = t of the solution
of the Cauchy problem
z′x,h(s) = Jh(zx,h(s)), s ∈ R, zx,h(0) = x.
The following properties hold:
a) T s+th = T sh ◦ T th;
b)
d
dt
T thx|t=0 = Jh(x);
c) T th = T 1
th.
We denote Th ≡ T 1
h . Denote also Πfin =
{
Γ ∈ B(Rd), Π(Γ) < +∞
}
and define
for h ∈ H0, Γ ∈ Πfin transformation TΓ
h of the random measure ν by[
TΓ
h ν
]
([0, t]×∆) =
= ν
(
[0, T−ht]× (∆ ∩ Γ)) + ν([0, t]× (∆\Γ)
)
, t ∈ R+, ∆ ∈ Πfin.
Transformation TΓ
h is admissible for the distribution of ν in a sense that there exists
function pΓ
h (which can be given explicitly), such that for every {t1, . . . , tn} ⊂ R+,
{∆1, . . . ,∆n} ⊂ Πfin and Borel function ϕ : Rn → R
Eϕ
(
[TΓ
h ν]([0, t1]×∆1), . . . , [TΓ
h ν]([0, tn]×∆n)
)
=
= EpΓ
hϕ
(
ν([0, t1]×∆1), . . . , ν([0, tn]×∆n)
)
.
This fact, under additional condition that σ -algebra of all random events is generated by
ν, imply that TΓ
h generates the corresponding transformation of random variables, we
denote it also by TΓ
h .
For a given h ∈ H0, Γ ∈ Πfin and random variable f denote
∂Γ
hf = lim
ε→0
TΓ
εhf − f
ε
, (2.2)
the variable in the left hand side of (2.2) is defined on the set of such ω ∈ Ω, that the
limit in the right-hand side exists. The key point in our considerations is the following
simple statement.
Lemma 2.1. Let f be a random variable, h1, . . . , hk ∈ H0, Γ1, . . . ,Γk ∈ Πfin
and
A =
{
(∂Γ1
h1
) . . . (∂Γk
hk
)f is defined and
= 0
}
,
then
P|A ◦ f−1 � λ1.
Sketch of the proof. One can verify that for a fixed h, Γ the transformation TΓ
h gen-
erates measurable stratification of the initial space Ω, and therefore using stratification
method (see [5], Chapter 2.5) one can show that
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ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1271
P|{∂Γ
hf �=0} ◦ f−1 � λ1, (2.3)
i.e., the needed statement holds true for k = 1. Statement (2.3) implies that P
({
∂Γ
hf
=
= 0
}
∩ {f = 0}
)
= 0, which gives an opportunity to prove statement of the lemma by
induction.
Let us prove first the statement A of Theorem 1.1, which is more simple. Let Υ(s, x, u)
be a fixed vector-valued function, S ⊂ Rm be some subspace.
Lemma 2.2. For every s < t{
∃τ ∈ D ∩ (s, t) : [Eτ−0 ]−1(ΛΥ)(τ,X(τ−), p(τ))
∈ S
}
⊂
⊂
{
∃τ ∈ D ∩ (s, t) : [Eτ−0 ]−1Υ(τ,X(τ−), p(τ))
∈ S
}
almost surely.
Proof. Denote by l1, . . . , lk some basis in S⊥, and put
Ωs,t,j,n ≡
{
∃τ ∈ Dn ∩ (s, t) :
(
[Eτ−0 ]−1(ΛΥ)(τ,X(τ−), p(τ)), lj
)
Rm
= 0
}
, (2.4)
where Dn ≡
{
τ ∈ D|‖p(τ)‖ > 1
n
}
. In order to prove the needed statement it is enough
to show that for every s < t, j ≤ k, n ≥ 1,
Ωs,t,j,n ⊂
{
∃τ ∈ Dn ∩ (s, t) : [Eτ−0 ]−1Υ(τ,X(τ−), p(τ))
∈ S
}
(2.5)
almost surely. Let s < t, j ≤ k, n ≥ 1 be fixed, we define τ̃ on the set Ωs,t,j,n as
the first point from Dn, satisfying condition in the right-hand side of (2.4), and denote
Ψ =
(
[E τ̃−0 ]−1Υ(τ̃ , X(τ̃−), p(τ̃)), lj
)
Rm
. We shall prove that
P|Ωs,t,j,n
◦Ψ−1 � λ1, (2.6)
this will provide (2.5). For N, r ∈ N denote ΩrN =
{
Dn ∩
(
r − 1
N
,
r
N
]
= {τ̃}
}
, one
can see that P(
⋃
N,r ΩrN ) = 1. Let us show that for hrN = 1I( r−1
N , r
N ], Γn =
{
u
∣∣ ‖u‖ >
>
1
n
}
almost surely on the set ΩrN there exist
lim
ε→0
TΓn
εhr
N
Ψ−Ψ
ε
= −[JhrN ](τ̃)
([
E τ̃−0
]−1
(ΛΥ)(τ̃ , X(τ̃−), p(τ̃)), lj
)
Rm
. (2.7)
This, together with Lemma 2.1, will give the needed statement, as soon as [JhrN ](τ̃)
= 0
on ΩrN . By the construction,
d
dε
∣∣∣∣
ε=0
[
TΓn
εhr
N
τ̃
]
= −[JhrN ](τ̃). (2.8)
In order to find
d
dε
∣∣∣
ε=0
[
TΓn
εhr
N
X(τ̃−)
]
, let us note that on the set ΩrN for ε small enough
TΓn
εhr
N
X(τ̃−) = X̃
(
TΓn
εhr
N
τ̃
)
, where X̃ is a solution of equation
X̃(v) = X(s) +
v∫
s
ã(z, X̃(z)) dz +
v∫
s
∫
‖u‖≤ 1
n
c(z, X̃(z−), u)ν(ds, du).
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1272 A. M. KULIK
Under condition (1.1) almost every trajectory of the process X̃ is differentiable by v for
almost all v w.r.t. Lebesgue measure on [s,+∞), and the corresponding derivative is
equal to X̃ ′(v) = ã(v, X̃(v)). Distribution of τ̃ is absolutely continuous, τ̃ and X̃ are
independent. Therefore
d
dε
∣∣∣∣
ε=0
[
TΓn
εhr
N
X̃(τ̃)
]
= −[JhrN ](τ̃)ã(τ̃ , X(τ̃−)) (2.9)
almost surely on ΩrN . The same considerations give that almost surely on ΩrN
d
dε
∣∣∣∣
ε=0
[
TΓn
εhr
N
[
E τ̃−0
]−1
]
= [JhrN ](τ̃)[E τ̃−0 ]−1∇xã(τ̃ , X(τ̃−)). (2.10)
Equalities (2.8) – (2.10) together with the chain rule give (2.7).
The lemma is proved.
The end of the proof of statement A repeats the proof of Theorem 4.2 [8], let us give
it here briefly. First let us give the following useful statement, which is a generalization
of Lemma 4.3 [8].
Lemma 2.3. Suppose that the following objects are chosen.
1. A measurable space (U,U) with a measure µ on it and a compact metric space Z.
2. A sequence of functions {fn : Z × U → R, n ∈ N}, such that every fn is
measurable w.r.t. second coordinate when the first one is fixed and is continuous w.r.t.
first coordinate when the second one is fixed.
3. A sequence {αr} ⊂ R+, a sequence of open sets {On,k ⊂ R, n, k ∈ N},
monotonously increasing by k for every fixed n, and a monotonously increasing se-
quence of measurable sets {Ur ⊂ U} with µ(Ur) < +∞ and ∪rUr = U.
Denote On = ∪kOn,k and suppose that for every z ∈ Z
sup
r
[
αrµ{u ∈ Ur| ∃n ∈ N : fn(z, u) ∈ On}
]
= +∞,
then
lim
n,k,r→∞
inf
z∈Z
sup
q≤r
[
αqµ{u ∈ Uq| ∃i ≤ n : fi(z, u) ∈ Oi,k}
]
= +∞.
Proof. Let us consider functions
ϕn,k,r(z) = sup
q≤r
[
αqµ{u ∈ Uq| ∃i ≤ n : fi(z, u) ∈ Oi,k}
]
,
due to conditions of the lemma for every z ∈ Z ϕn,k,r(z) tends to +∞ and is
monotonous w.r.t. every index n, k, r while others are fixed. Moreover, every function
ϕn,k,r is lower semicontinuous, i.e., for every sequence zj → z we have the inequality
ϕn,k,r(z) ≤ lim infj ϕn,k,r(zj). Therefore the needed statement holds true due to the
correspondent version of the Dini theorem.
The lemma is proved.
Corollary 2.1. Let K be some compact subset in Rm, take Z = [0, T ] × K ×
× {l̄ ∈ Rm : ‖l̄‖ = 1}, µ = Π, Ur =
{
u : ‖u‖ >
1
r
}
, αr ≡ 1, fn(s, x, l̄, u) =
=
(
∆n
0 (s, x, u), l̄
)
Rm , On,k ≡ R\{0}. Then due to lemma under condition (1.4) for
every T < +∞ and compact set K ⊂ Rm
lim
n,k→+∞
inf
s≤T,x∈K,l̄ �=0̄
Π
{
u|‖u‖ ≥ 1
n
, ∃j ≤ k :
l̄ is not orthogonal to ∆j
0(s, x, u)
}
= +∞. (2.11)
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ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1273
Another corollary will be given below, in the proof of statement B (see (2.18)).
In order to shorten notations we suppose further that for some compact K ⊂ Rm
X(s) ∈ K, s ≤ t a.s., the standard way to give rigorous basis for this supposition is
the following one. Take the Markov moment ζK of the exit of X(·) from the set K
and consider the new process XK(·) = X(· ∧ ζK). For this process all estimates, given
below, hold true, and for every given t the probability of the set {X|[0,t]
= XK |[0,t]} ⊂
⊂ {ζK > t} can be made arbitrary small by an appropriate choice of K.
Let n, k be fixed, denote by τni the i-th point from Dn, Snt =
〈[
Eτ−0
]−1∆(τ,
X(τ−), p(τ)), τ ∈ Dn, τ ≤ t
〉
. Due to Fubini theorem, for every s ≤ t
P
(
dimSnt = dimSns
∣∣∣ dimSns < m
)
=
=
∫
{dimS<m}×Rm
P(∀τ ∈ Dn ∩ (s, t)×
×Q
[
Eτ−s
]−1∆(τ,X(y, τ−, s), ρ(τ)) ∈ S
)
κs,n(dS, dy, dQ),
here we suppose that the space of all subspaces of Rm is parameterized in such a way that
it becomes a Polish space, and κs,n is the joint distribution of Sns , X(s) and [Es0 ]−1.
Due to Lemma 2.2 for every S
= Rm, y ∈ Rm one has
P
(
∀τ ∈ Dn ∩ (s, t), [Eτ−0 ]−1∆(τ,X(y, τ−, s), ρ(τ))
∈ S
)
=
= P
(
∀τ ∈ Dn ∩ (s, t), ∃j ≤ k [Eτ−0 ]−1∆j
0(τ,X(y, τ−, s), ρ(τ))
∈ S
)
≥
≥ inf
l̄ �=0̄
P
(
∀τ ∈ Dn ∩ (s, t), ∃j ≤ k [Eτ−0 ]−1∆j(τ,X(y, τ−, s), ρ(τ))
⊥ l̄
)
. (2.12)
The variable p(τni ) (the value of the i-th jump from Dn ) is independent from the values
of others jumps, from the moments of all jumps and from
[
Eτ
n
i −
0
]−1
. The distribution of
p(τni ) is equal
Π|{u:‖u‖≥ 1
n}
Π
({
u : ‖u‖ ≥ 1
n
}) . Therefore, denoting
γn,k = inf
s≤T,x∈K,l̄ �=0
Π
(
u : ‖u‖ > 1
n
, ∃j ≤ k : (∆j(x, s, u), l̄ )Rd
= 0
)
,
λn = Π
({
u : ‖u‖ > 1
n
})
,
Nn
s,t = #(Dn ∩ (s, t)) (it has the Poisson distribution with intensity λn(t − s) ), one
can estimate the last term in (2.12) by
E
(
1− γn,k
λn
)Nn
s,t
= exp{−(t− s)γn,k}.
This implies that
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1274 A. M. KULIK
P(dimSt = m) ≥ lim
n→+∞
m∏
r=1
P
(
dimSntr
m
> dimSnt(r−1)
m
∣∣∣ dimSnt(r−1)
m
< m
)
≥
≥ lim
n,k→+∞
(
1− exp
{
− t
m
γn,k
})m
= 1,
which gives the needed statement.
Now let us proceed with the proof of statement B. In order to shorten notations we will
consider only the time-homogeneous case. Also, without loss of generality, we suppose
that there are some compacts K ⊂ Rm, K̃ ⊂ Rm×m such that X(t) ∈ K,
[
Et−0
]−1 ∈
∈ K̃ a.s., t ≥ 0.
Let us introduce some notations. For a given ordered set t̄ ≡ {t0 < t2 < · · · < tk},
tj ∈ Q∩R+, k ≥ 1 denote d(t̄) = minj(tj − tj−1). For every such t̄ and every l ≥ 1
let us choose a sequence h̄t̄,l =
{
ht̄,lj ∈ H0, j = 1, . . . , k
}
such that
a) supp Jht̄,lj ⊂ (tj , tj−1);
b) Jht̄,lj = 1 on
(
tj +
d(t̄ )
3l
, tj−1 −
d(t̄ )
3l
)
.
Next, for a given t̄ and n, l ∈ N we put
Ωt̄,n =
k−1⋂
j=0
{
#
[
Dn ∩ (tj , tj+1)
]
= 1
}
,
Ωt̄,l,n = Ωt̄,n ∩
k−1⋂
j=0
{
#
[
Dn ∩
(
tj+1 +
d(t̄ )
3l
, tj −
d(t̄ )
3l
)]
= 1
}
.
Denote T j,t̄,l,nε = T
{u|‖u‖≥ 1
n}
εht̄,l
j
, j = 1, . . . , k. The following properties hold true:
1) the set Ωt̄,n is invariant w.r.t. every transformation T j,t̄,l,nε ;
2) for every ε1,2, j1,2 transformations T j1,t̄,l,nε1 and T j2,t̄,l,nε2 commute.
Denote ∂ t̄,l,nj =
d
dε
T j,t̄,l,nε
∣∣∣∣
ε=0
, derivative is taken in an a.s sense. One can see that
∂ t̄,l,ni τ t̄,l,nj = −δi,j almost surely on Ωt̄,l,n, i, j ≤ k,
here by τ t̄,nj we denote the unique point from Dn ∩ (tj , tj+1), δi,j is the Kronecker
symbol. In order to shorten notations we will further omit the subscripts t̄, l, n, t̄,n over
∂j , τj .
For a given t̄, n consider processes X̃, Ẽ
(
= X̃ t̄,n, Ẽ t̄,n
)
, defined as the solutions
of SDE’s
X̃(t) = x+
t∫
0
ã(X̃(s)) ds+
t∧t0∫
0
∫
Rd
+
t∫
t∧t0
∫
{u|‖p(u)> 1
n}
c(X̃(s−), u)ν(ds, du),
Ẽt0 = I +
t∫
0
∇xã(X̃(s))Ẽs0 ds+
t∧t0∫
0
∫
Rd
+
t∫
t∧t0
∫
{u|‖p(u)> 1
n}
×
×∇xc(X̃(s−), u) Ẽs−0 ν(ds, du).
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ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1275
Lemma 2.4. Under condition B) there exist functions V Nk ∈ C(R+), V Nk (0) = 0,
k, N ∈ N, such that for every n, l ∈ N, t̄ = {t0 < · · · < tk} and every i1, . . . , ik ∈
∈ {0, . . . , N}
i)
∥∥∥(∂k)ik(∂k−1 + ∂k)ik−1 . . . (∂1 + . . .+ ∂k)i1
(
[Eτk−
0 ]−1∆(X(τk−), p(τk))
)
−
−(∂k)ik(∂k−1 + ∂k)ik−1 . . . (∂1 +. . .+ ∂k)i1
(
[Ẽτk−
0 ]−1∆(X̃(τk−), p(τk))
)∥∥∥ ≤
≤ V Nk (ηn,(k+1)N
tk
− η
n,(k+1)N
t0 ),
ii)
∥∥∥(∂k)ik(∂k−1 + ∂k)ik−1 . . . (∂1 + . . .+ ∂k)i1
(
[Ẽτk−
0 ]−1∆(X̃(τk−), p(τk))
)
−
−[Eτ1−0 ]∆i1,...,ik
k−1 (X(τ1−), p(τ1), . . . , p(τk))
∥∥∥
Rm
≤ V Nk (tk − t0)
almost surely on Ωt̄,l,n, where
ηn,rt =
t∫
0
∫
{‖u‖≤ 1
n}
supx∈K(‖c(x, u)‖Rm + . . .+ ‖(∇x)rc(x, u)‖(Rm)×r )ν(ds, du).
Proof. By the definition X̃ = X and Ẽ = E on [0, τ1). Due to (2.7)
(∂1)i1
(
[Eτ1−0 ]−1∆(τ1, X(τ1−), p(τ1))
)
= [Eτ1−0 ]−1∆i1
0 (X(τ1−), p(τ1))
almost surely on Ωt̄,l,n, which means that the case k = 1 is already proved.
To proceed with the case k > 1 we need two auxiliary technical results. Supposing
n to be fixed, denote by Ψr,t(x) ≡ Ψ0
r,t(x) solution of SDE
X(t) = x+
t∫
r
ã(X(s)) ds+
t∫
r
∫
{u|‖p(u)‖≤ 1
n}
c(X(s−), u)ν(ds, du), t ≥ r.
It follows from the general results about differentiability of the solution of differential
equation w.r.t. initial value that functions Ψj
r,t ≡ (∇x)jΨr,t are well defined almost
surely. We denote by Φr,t(x) ≡ Φ0
r,t(x) solution of ODE
X(t) = x+
t∫
r
ã(X(s)) ds, t ≥ r,
and put Φjr,t ≡ (∇x)jΦr,t.
Proposition 2.2. For every N ∈ N there exists function WN ∈ C(R+) with
WN (0) = 0 such that for every j ≤ N,x ∈ Rm, t > r
‖Ψj
r,t(x)− Φjr,t(x)‖(Rm)×(j+1) ≤WN (ηn,Ntk − ηn,Nt0 ).
almost surely.
Sketch of the proof. One can write down iteratively differential equations both on
Ψj
r,t and Φjr,t (stochastic for Ψj
r,t and ordinary for Ψj
r,t). These equations are linear
nonhomogeneous equations with free terms constructed (in a same regular manner) from
functions a, c with their derivatives up to the order j and functions {Ψi
r,t, i < j} or
{Φir,t, i < j} correspondingly. Now the needed statement can be obtained by induction
using condition B) and Gronwall lemma.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1276 A. M. KULIK
The same considerations together with the fact that the process ηn,Nt in every point
t almost surely has derivative w.r.t. t, equal to 0, provide the following statement.
Proposition 2.3. The function WN in previous proposition can be chosen in such a
way that for every j1, j2, j3 ≤ N, x ∈ Rm, t > r∥∥∥∥ ∂j1
∂rj1
∂j2
∂tj2
Ψj3
r,t(x)− ∂j1
∂rj1
∂j2
∂tj2
Φj3r,t(x)
∥∥∥∥
(Rm)×(j3+1)
≤WN
(
ηn,Ntk − ηn,Nt0
)
almost surely.
Now let us return to the proof of the lemma. In order to shorten notations we will
consider only the case k = 2, the arguments for k > 2 will be the same.
Using (2.7), we obtain that
(∂2)i2
(
[Eτ2−0 ]−1∆(X(τ2−), p(τ2))
)
= [Eτ2−0 ]−1(Λi2∆)(X(τ2−), p(τ2)).
Let us estimate (∂1)i1
(
[Eτ2−0 ]−1Υ(X(τ2−), p(τ2))
)
for a vector-valued function Υ.
One can write down
[Eτ2−0 ]−1Υ(X(τ2−), p(τ2)) =
= [Eτ1−0 ]−1
[
I +∇xc(X(τ1−), p(τ1))
]−1[
Ψ1
τ1,τ2(X(τ1−) + c(X(τ1−), p(τ1)))
]−1
×
×Υ
(
Ψτ1,τ2(X(τ1−) + c(X(τ1−), p(τ1)), p(τ2))
)
, (2.13)
[Ẽτ2−0 ]−1Υ(X̃(τ2−), p(τ2)) =
= [Ẽτ1−0 ]−1
[
I +∇xc(X̃(τ1−), p(τ1))
]−1[
Φ1
τ1,τ2(X̃(τ1−) + c(X̃(τ1−), p(τ1))
]−1
×
×Υ
(
Φτ1,τ2(X̃(τ1−) + c(X̃(τ1−), p(τ1))), p(τ2)
)
. (2.14)
We know that almost surely on the set Ωt̄,l,n
∂1τ1 = −1, ∂1[Eτ1−0 ]−1 = [Eτ1−0 ]−1∇xã(X(τ1−)),
∂1X(τ1−) = −ã(X(τ1−)), ∂1[Ẽτ1−0 ]−1 = [Ẽτ1−0 ]−1∇xã(X̃(τ1−)),
∂1X̃(τ1−) = −ã(X̃(τ1−)).
(2.15)
Taking iteratively ∂1 from the right-hand sides of equalities (2.13), (2.14) and using
(2.15) and Propositions 2.2, 2.3 we obtain statement i) of the lemma.
Now let us estimate the value (∂1 + ∂2)i1(∂2)i2
(
[Ẽτ2−0 ]−1∆(X̃(τ2−), p(τ2))
)
. As
soon as function Φ is defined by a homogeneous equation, one has that for every j ≥ 0(
∂
∂r
+
∂
∂t
)
Φjr,t(x) = 0. This together with (2.15) means that for ϕ ∈ C1
(∂1 + ∂2)
[
Φjτ1,τ2(ϕ(X̃(τ1−)))
]
= −
(
Φj+1
τ1,τ2(ϕ(X̃(τ1−))), [∇ϕ∇xã](X(τ1−))
)
Rm
.
(2.16)
Note that ‖Φjτ1,τ2(x)‖(Rm)×(j+1) = O(t2 − t0) on Ωt̄,l,n for all j ≥ 2, thus iterating
(2.16) we obtain that∥∥∥(∂1 +∂2)i
[
Φ1
τ1,τ2(X̃(τ1−)+c(X̃(τ1−), p(τ1)))
]−1∥∥∥
Rm×m
= O(t2− t0) on Ωt̄,l,n.
The same considerations give that
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ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1277
∥∥∥(∂1 + ∂2)iΥ(Φτ1,τ2(X̃(τ1−) + c(X̃(τ1−), p(τ1))), p(τ2))−
−(∂1 + ∂2)iΥ(X̃(τ1−) + c(X̃(τ1−), p(τ1)), p(τ2))
∥∥∥
Rm
= O(t2 − t0) on Ωt̄,l,n.
Therefore, as soon as ∂2X̃(τ1−) = 0 and ∂2Ẽτ1−0 = 0, we have that, up to some
O(t2 − t0) term, (∂1 + ∂2)i
([
Ẽτ2−0
]−1Υ(X̃(τ2−), p(τ2))
)
is equal to
(∂1)i
{[
Ẽτ1−0
]−1
[
I +∇xc(X̃(τ1−), p(τ1))
]−1
×
× Υ((X̃(τ1−) + c(X̃(τ1−), p(τ1)), p(τ2))
}
=
= [Ẽτ1−0 ]−1
[
ΛiΞp(τ1)Υ(·, p(τ2))
]
(X̃(τ1−)),
which gives the needed statement.
The lemma is proved.
For a given s < t and l̄
= 0 let us consider the event
As,t,l̄ =
{
∃l, n ∈ N, t̄ = {t0, . . . , tk} ⊂ (s, t) ∩Q, i0, . . . ik ≥ 0 :
(∂ t̄,l,n1 )i0 . . . (∂ t̄,l,n1 + . . .+ ∂ t̄,l,nk )ik×
×
([(
Ẽ t̄,n̄0
)τ t̄,n
k −]
∆
(
X̃ t̄,n
(
τ t̄,nk −
)
, p
(
τ t̄,nk
))
, l̄
)
Rm
= 0
}
.
Due to representation (2.14) and condition A), for every n, t̄ = {t0 < . . .
. . . < tk} there exists a function
ϕt̄,l,n : (Rm)× (Rm×m)× (Rd)k × {(v1, . . . , vk) : t0 ≤ v1 ≤ . . . ≤ vk} → Rm,
which is analytical in every point w.r.t. coordinates v1, . . . , vk with the radius of analyt-
icity not less than C, and such that
[(
Ẽ t̄,n̄0
)τ t̄,n
k −]
∆
(
X̃ t̄,n(τ t̄,nk −), p(τ t̄,nk )
)
=
= ϕt̄,l,n
(
X(t0), Et00 , p(τ t̄,n1 ), . . . , p(τ t̄,nk ), τ t̄,n1 , . . . , τ t̄,nk
)
.
The following fact is well known: if some function is analytical on some subset of Rk
and is not equal to 0 in some point, then it is not equal to 0 in almost every point w.r.t.
λk. Variables τ t̄,n1 , . . . , τ t̄,nk are independent from X(t0), Et00 , p(τ t̄,n1 ), . . . , p(τ t̄,nk ) and
their joint distribution is absolutely continuous w.r.t. λk. This together with statement ii)
of Lemma 2.4 implies that for a given n, l, t̄ and l̄
= 0 almost surely
Ωt̄,l,n ∩
{
(∂ t̄,l,n1 )i0 . . . (∂ t̄,l,n1 + . . .+ ∂ t̄,l,nk )ik×
×
([
(Ẽ t̄,n̄0 )τ
t̄,n
k −
]
∆(X̃ t̄,n(τ t̄,nk −), p(τ t̄,nk )), l̄
)
Rm
= 0
}
⊃
⊃ Ωt̄,l,n ∩
{([
Eτ
t̄,n
1 −
0
]−1
∆i0,...,ik
k−1 (X(τ t̄,n1 −), p(τ t̄,n1 ), . . . , p(τ t̄,nk )), l̄
)
Rm
= 0
}
.
This gives that almost surely
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1278 A. M. KULIK
As,t,l̄ ⊃ Bs,t,l̄ ≡
⋃
n,k≥1
{
∃j ≥ 1 : τnj , . . . , τ
n
j+k−1 ∈ (s, t)
and
([
Eτ
n
j −
0
]−1)∗
l̄ is not orthogonal to
Lk(X(τnj −), p(τnj ), . . . , p(τnj+k−1))
}
.
Let us show that
P(Bs,t,l̄) = 1. (2.17)
Denote by LMk (s, x, u0, . . . , uk), k ≥ 0 the linear span of the vectors{
∆i0,...,ij
j (s, x, u0+r, . . . , uj+r), i0, . . . , ij ≤M, r = 0, . . . , k − j, j = 0, . . . , k
}
.
Let condition (1.5) to hold true with some given k > 0. Then due to Lemma 2.3 for every
R there exist n,M such that for every x ∈ K, b̄
= 0 there exists N = NM,R(x, b̄) ≤ n
satisfying condition
δNM (x, b̄) ≡ Π⊗(k+1)
({
(u0, . . . , uk) ∈
{
‖u‖ > 1
N
}k+1
:
b̄ is not orthogonal to LMk (x, u0, . . . , uk)
})
×
×
({
Π
({
u ∈ Rd| ‖u‖ > 1
N
})}k)−1
≥ R. (2.18)
Note that λN ≡ Π
({
‖u‖ > 1
N
})
is not less than R and therefore infM,x,b̄NM,R(x,
b̄ ) → +∞, R→ +∞. Let us denote
Bn,M
s,t,l̄
≡
⋃
N≤n
{
∃j ≥ 1 : τNj , . . . , τ
N
j+k−1 ∈ (s, t) and
([
Eτ
N
j −
0
]−1)∗
l̄ is not orthogonal to LMk
(
X(τNj −), p(τNj ), . . . , p(τNj+k−1)
)}
and estimate probability of Bn,M
s,t,l̄
. First we take constant C = C(k) such that e−C <
<
1
3
and
∑
i≥k
e−C
Ci
i!
>
1
2
. Then we construct inductively a random covering of the
interval (s,+∞) in the following way. Let us take the interval
(
s, s+
C
λn
)
and consider
the set Dn ∩
(
s, s+
C
λn
)
. If this set is empty, we put I1 =
(
s, s+
C
λn
]
, otherwise we
take the first point θ from this set, define Ñ = NM,R
(
X(θ−),
([
Eθ−0
]−1
)∗
l̄
)
and put
I1 =
(
s, θ +
C
λÑ
]
. Then we take I1 as the first set in the covering which we are going
to construct, replace (s,+∞) by (s,+∞)\I1 and repeat the preceding procedure. We
obtain a countable covering of the interval (s,+∞) by a segments
{
Ir = (vr−1, vr],
r ≥ 1
}
, which can be separated in two groups:
1) some segments of the length
C
λn
, we denote this group by G1;
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1279
2) some segments of the length >
C
λn
, we denote this group by G∈.
Note that the length of every segment is not greater than
2C
R
, we suppose that R
is taken sufficiently large and
2C
R
≤ t− s
3
. Next, by the construction every vr is a
stopping time and vr+1 is independent from Fvr−, random event
{
Ir = (vr−1, vr] ∈
∈ G1
}
is independent from Fvr− and its probability is equal to e−C . Denote by Zs,t
the total length of all segments Ir in the first group such that vr−1 < t, then
EZs,t =
C
λn
∑
r
P(Ir ∈ G1, vr−1 < t) =
=
Ce−C
λn
∑
r
P(vr−1 < t) ≤
{[λn(t− s)
C
]
+ 1
}
Ce−C
λn
,
here we used the obvious fact that P(vr−1 < t) = 0, r >
[
λn(t− s)
C
]
+1. Analogously
one can verify that
DZs,t ≤
{[λn(t− s)
C
]
+ 1
}
C2(e−C − e−2C)
(λn)2
,
which means that
Zs,t − EZs,t
P→ 0 and P− lim sup
λn→+∞
Zs,t ≤ e−C(t− s) <
t− s
3
.
Therefore for every fixed p ∈ (0, 1) one can choose initial number R (and, conse-
quently, number n ) large enough to provide estimate
P
(
Zs,t ≤
t− s
3
)
≥ p. (2.19)
Next, let us monotonously enumerate the second group, G2 = {Jj}. For a given j let
θj be the first point from Dn∩Ir, Nj = NM,R
(
X(θj−), θj ,
([
Eθj−
0
]−1
)∗
l̄
)
. Denote
by Dj the event
{
the segment
(
θj , θj+
C
λNj
]
contains at least k points from DNj
}
,
P (Dj) =
∑
i≥k
e−C
Ci
i!
>
1
2
. Denote the first k points from DNj ∩ (θj ,+∞) by
θ1
j , . . . , θ
k
j . Due to the choice of Nj probability of the event
Cj =
{([
Eθ
k
j −
0
]−1
)∗
l̄ is not orthogonal to
LMk
(
X(θkj−), p(θj), p(θ1
j , . . . , p(θ
k
j )
)}
s not less than
R
λNj
. Events Cj , Dj are independent both from Fθj− and from each
other, therefore
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1280 A. M. KULIK
P
(
Bn,M
s,t,l̄
)
≥ 1− P
⋂
j:θj<
s+2t
3
[
Ω\(Cj ∩Dj)
] ≥
≥ 1− E
∏
j:Jj⊂(s,t)
(
1− R
2λNj
)
≥ 1− E exp
− ∑
j:Jj⊂(s,t)
R
2λNj
.
The variable Ws,t =
∑
j:Jj⊂(s,t)
C
λNj
is just the total length of the intervals from the
second group, which are contained in (s, t). One have that Ws,t ≥
2(t− s)
3
−Zs,t, and
under (2.19) we have that Ws,t ≥
t− s
3
with probability ≥ p, which gives that
P
(
Bn,M
s,t,l̄
)
≥ p− exp
[
− R
2C
(t− s)
]
. (2.20)
Now we proceed in a following way: for a given p ∈ (0, 1) we take Rp such that
(2.19) holds for every R ≥ Rp, then take R → ∞ in (2.20) and therefore obtain that
P(Bs,t,l̄) ≥ p. At last, we take p ↑ 1 and obtain (2.17).
Denote
AM,j,N
s,t,l̄
=
{
∃l, n ≥ N, t̄ = {t0, . . . , tk} ⊂ (s, t) ∩Q, k, i0, . . . ik ≤M :
∣∣∣(∂ t̄,l,n1 )i0 . . . (∂ t̄,l,n1 + . . .+ ∂ t̄,l,nk )ik×
×
([(
Ẽ t̄,n̄0
)τ t̄,n
k −]
∆
(
X̃ t̄,n
(
τ t̄,nk −
)
, p
(
τ t̄,nk
))
, l̄
)
Rm
∣∣∣ >
1
j
}
,
by the construction AM,j,N
s,t,l̄
⊂ AM̃,j̃,Ñ
s,t,l̄
, N ≤ Ñ , M ≤ M̃, j ≤ j̃, and due to (2.17)
P
(
AM,j
s,t,l̄
)
→ 1 as M, j → +∞ for every l̄
= 0, s < t. For a given ε ∈ (0, 1) let
us take N∗, j∗, M∗ such that P
(
AM∗,j∗,N
s,t,l̄
)
≥ p for every N ≥ N∗. Next, we take
N∗ such that for every n ≥ N∗ P
(
VMM
(
η
n,M(M+1)
t − η
n,M(M+1)
s
)
>
1
j∗
)
≤ ε.
Now we can apply the statement ii) of Lemma 2.4 for n ≥ N∗ ∨N∗ and obtain that the
probability of the event
Cs,t,l̄ ≡
{
∃l, n ∈ N, t̄ = {t0, . . . , tk} ⊂ (s, t) ∩Q, i0, . . . ik ≥ 0 :
(∂ t̄,l,n1 )i0 . . . (∂ t̄,l,n1 + . . .+ ∂ t̄,l,nk )ik×
×
([(
E t̄,n̄0
)τ t̄,n
k −]
∆
(
X t̄,n
(
τ t̄,nk −
)
, p
(
τ t̄,nk
))
, l̄
)
Rm
= 0
}
is not less than 1 − 2ε and therefore P(Cs,t,l̄) = 1. Using this fact and Lemma 2.1,
we obtain analogously to the proof of statement A that P(dimSt = dimSs|dimSs <
< m) = 0 for every s < t, which gives the needed statement.
The last thing, which we need to do, is to remove condition (2.1). Denote by ζ
the moment of the first jump such that
∥∥∇xc(ζ,X(ζ−), p(ζ))
∥∥ ≥ 1. Considerations,
analogous to those made before, imply that
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ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1281
P|{ζ>t} ◦ [X(t)]−1 ≤ λm.
Next, let ΓR,t =
{
u
∣∣∣ sups≤t,‖x‖≤R ‖∇xc(s, x, u)‖ ≥ 1
}
, ζR,t−δ = inf
{
r ≥ t− δ|r ∈
∈ D, p(r) ∈ ΓR,t
}
. The same considerations, together with evolutionary property of the
family X(x, t, s), gives that for every R, δ
P|{ζR,t−δ>t}∩{sups≤t ‖X(s)‖≤R} ◦ [X(t)]−1 ≤ λm.
This means that the total mass of the singular part of the distribution of X(t) can be
estimated by
P
(
{ζR,t−δ < t} ∪
{
sup
s≤t
‖X(s)‖ > R
})
,
which can be made arbitrarily small by taking first R large enough and then δ small
enough.
The theorem is proved.
3. Appendix: some improvements and unsolved problems. One can see from the
proof of the Theorem 1.1 that conditions A), B) are a technical ones, which are used to
calculate and estimate compositions of derivatives w.r.t. the first k − 1 jumps in a given
set of k jumps
(
derivatives ∂ t̄,l,n1 , . . . , ∂ t̄,l,nk−1 , see notations before Lemma 2.4). This
remark immediately gives the following version of statement B.
Proposition 3.1. Denote by L̃k(x, u0, . . . , uk) the span of the vectors{
∆i,0,...,0
j (x, u0+r, . . . , uj+r), j = 0, . . . , k, r = 0, . . . , k − j, i ≥ 0
}
.
Suppose that for some k > 0 for every x ∈ Rm, s ∈ R+, l̄ ∈ Rm\{0}
Π∗
k+1
{
(u0, . . . , uk) ∈ [Θs,x]k+1 :
l̄ is not orthogonal to L̃k(s, x, u0, . . . , uk)
}
= +∞. (3.1)
Then for every x ∈ Rm, 0 ≤ r < t
P ◦ [X(x, t, r)]−1 � λm.
The proof is analogous to the proof of statement B and is omitted. Proposition 3.1
allows, in particular, to consider SDE’s such that their drift coefficients have a rot of
zeros.
Example 3.1. a) Consider one-dimensional SDE
X(t, x) = x+
t∫
0
a(X(s, x)) ds+ ηt, (3.2)
where ηt is the Levy process with the Levy measure Π =
∑
k≥1
αkδ3−k , where∑
k
αk = +∞. Suppose that a ∈ C∞(R) is such that in every point of the Cantor
set K ⊂ [0, 1] function a together with all its derivatives is equal to zero, and a
= 0
outside K. Theorem 1.1 can not be applied here. Indeed, as soon as supp Π ⊂ K, one
has that ∆i
0(0, u) = 0 for every u ∈ supp Π, i ≥ 0, which means that condition (1.4)
fails and statement A. is not applicable. Statement B we can not apply because function
a is not analytical.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1282 A. M. KULIK
On the other hand, for every point x ∈ R and every i
= j one has that at least one
of the numbers x + 3−i, x + 3−j , x + 3−i + 3−j does not belong to K. This means
that for every x
Π∗
2
{
(u0, u1) : L̃2(x, u0, u1)
= {0}
}
≥ sup
n
∑
i<j≤n
αiαj∑
j≤n
αj
= +∞,
and (3.1) holds true with k = 1. Therefore solution of (3.2) has absolutely continuous
distribution.
It is worth to be mentioned that regularity properties of the solution of SDE of the type
(3.2) can essentially depend on the mutual properties of the set of zeros of the function a
and Levy measure of the process ηt.
Example 3.1. b) Let a be equal to zero on the set K1,0,1,1
4 of the points y ∈ [0, 1]
such that in their representations
y =
∞∑
j=1
yj
4j
, yj ∈ {0, 1, 2, 3}, j ≥ 1, (3.3)
every digit yj is not equal to 1 (note that the classical Cantor set from the previous
example can be written in these notations as K1,0,1
3 ). Let us consider SDE’s of the
type (3.2) with two different processes ηt in the right-hand side, having Levy measures
equal Πρ =
∑
k≥1
kρδ4−k , ρ > 1, and Π−1 =
∑
k≥1
1
k
δ4−k correspondingly. The
first case can be treated analogously to the previous example. Namely, for every x ∈ R
and every given i > j there exist numbers ε1 ∈ {0, 1, 2, 3}, ε2 ∈ {0, 1}, not equal
simultaneously to 0, such that x + ε14−i + ε24−j
∈ K1,0,1,1
4 . This means that if in
every point x
∈ K1,0,1,1
4 some derivative of a is not equal to zero, then
Π∗
4
{
(u0, . . . , u4) : L̃4(x, u0, . . . , u4)
= {0}
}
≥ sup
n
∑
j<i≤n
i3ρjρ
4! · [
∑
i≤n
iρ]3
= +∞,
and solution of (3.2) has absolutely continuous distribution.
On the other hand, process η· with Levy measure Π−1 on the interval (0, 1) does
not have multiply jumps (i.e., all its jumps have different values) with probability
p∗ =
∞∏
n=1
e−
1
n
(
1 +
1
n
)
= e−γ
∗
> 0,
here γ∗ = 0,577215 . . . is the Euler’s constant. This means that with probability p∗ the
value ηs in every point s ≤ 1 has in its representation (3.3) all digits equal to either 0
or 1. Let us take by starting point x =
2
3
, all its digits in (3.3) are equal 2, and therefore
with probability p∗ all digits of
2
3
+ ηs for every s ≤ 1 are equal 2 or 3, which
means that
2
3
+ ηs ∈ K1,0,1,1
4 . If a = 0 on K1,0,1,1
4 , then with the same probability
X
(
1,
2
3
)
=
2
3
+ η1 ∈ K1,0,1,1
4 . Remind that λ1(K1,0,1,1
4 ) = 0, and this together with
the preceding arguments gives that the distribution of X
(
1,
2
3
)
has a nontrivial singular
component.
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ON A REGULARITY OF DISTRIBUTION FOR SOLUTION OF SDE OF A JUMP TYPE . . . 1283
At this time we can not give any general condition, say, in the terms of the entropy of
the set of zeroes of the drift coefficient in (3.2), sufficient for solution to have an absolutely
continuous distribution.
At the end, let us give another improvement of statement B. One can see that the
constant C = C(k) in the proof can be chosen in the form C(k) = C∗k. Repeating the
rest of the proof, we obtain that statement B holds true with the condition (1.5) replaced
by the weaker condition
1
k
inf
x,s,l̄ �=0
Π∗
k+1
{
(u0, . . . , uk) ∈ [Θs,x]k+1 :
l̄ is not orthogonal to Lk(s, x, u0, . . . , uk)
}
→ +∞, k → +∞. (3.4)
The question whether the term
1
k
in the left hand side of (3.4) is sharp or it can be
replaced by some term, increasing more slowly (or maybe removed at all), is still open.
Acknowledgements. The author wants to thank Professor T. Komatsu for the kind
invitation to visit Osaka City University and fruitful discussions on Malliavin calculus
for SDE’s with jumps during the visit. The question, asked by Professor T. Komatsu,
motivated the main theorem of this work, moreover, he gave an idea, which author did not
understood immediately, but which contained a key for the proof of this result.
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Received 17.06.2005
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