the Density of the Law of a Poisson Process

doi:10.1155/2011/803508

Research Article

A Criterium for the Strict Positivity of

the Density of the Law of a Poisson Process

R ´emi L ´eandre

Institut de Math´ematiques, Universit´e de Bourgogne, 21000 Dijon, France

Correspondence should be addressed to R´emi L´eandre,[email protected] Received 26 August 2010; Revised 23 December 2010; Accepted 9 January 2011 Academic Editor: Dumitru Baleanu

Copyrightq2011 R´emi L´eandre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We translate in semigroup theory our resultL´eandre, 1990giving a necessary condition so that the law of a Markov process with jumps could have a strictly positive density. This result express, that we have to jump in a finite number of jumps in a “submersive” way from the starting point xto the end pointyif the density of the jump processp1x, zis strictly positive inx, y. We use the Malliavin Calculus of Bismut type ofL´eandre,2008;2010translated in semi-group theory as a tool, and the interpretation in semi-group theory of some classical results of the stochastic analy- sis for Poisson process as, for instance, the formula giving the law of a compound Poisson process.

1. Introduction

We are interested in this paper in the following problem.

Problem*. Let X be a random variable given by the solution of a stochastic differential equation, with lawpdy. For whaty pdyis bounded below byqydy, whereq·is strictly positive continuous neary?

This problem was solved by using the Malliavin Calculus. See the survey paper of L´eandre1on that. For various applications of the Malliavin Calculus on heat kernels, we refer to the review of Kusuoka2, L´eandre3, and Watanabe4.

Let us explain the state of the art in the case of a diffusion. We considerm1 smooth vectors fields with bounded derivatives at each orderXion ^d and the diffusion generator L 1/2

i>0X_i²X0. It generates a linear semigroupPt acting on differentiable bounded functionsfon ^d:

∂

∂tPtfx LPtfx. 1.1

It is a semigroup in probability measures. It has a probabilistic representation5. Letwⁱ_tbe a ^m valued Brownian motion. Let us use the notation of formal path integrals of physics.

The law of the Brownian motion is given formally as the Gaussian measure

dμ 1 Zexp

−

i>0

₁

0

d/dsw_sⁱ² 2ds

dDw., 1.2

wheredDw.is a kind of formal Lebesgue measure. We introduce the stochastic differential equation in the Stratonovitch case issued fromx:

dxtx

i>0

Xixtxdwⁱ_tX0xtxdt. 1.3

Then,

P_tfx E

fxtx . 1.4

Ifw_. → x₁x is a submersion in some sense inw, then we can apply in some sense the implicit function theorem in order to get a lower bound of the law ofx1xby a measure having a strictly positive density in the values ofx1x inw with respect of the Lebesgue measure on ^d. The problem is that the solution of the stochastic differential equation1.3 is only almost surely defined. So the use of the implicit theorem leds to some difficulties which were overcome by Bismut in6. The use of Bismut’s procedure allows to7to solve Problem^∗. See8for a translation of the proof of7in semigroup theory.

Plenty of the standard tools of stochastic analysis were translated recently by L´eandre in semigroup theory. See the review9,10on that. Problem^∗was solved for a diffusion by using the Malliavin Calculus of Bismut type in semigroup theory in8.

We are interested in solving Problem^∗ in the case of a jump process. Let us consider a generator of L´evy type. Iffis a differentiable function

Lfx

Êd

fxz−fx

μdz, 1.5

it generates a linear semigroupPtsatisfying the parabolic equation

∂

∂tPtfx LPtfx. 1.6

It is a semigroup in probability measures11–13. It is represented by a jump process with independent incrementszt:

Ptfx E

fxzt . 1.7

Tortrat14studied the supportSof the law ofzt: ify1 ∈Sandy2 ∈S, theny1y2 belong toS. Ifμhas a finite massλ, then the processz.has the law of a compound Poisson process:

z_tis sum of his jumps. There is only a finite number of jumps. The jumps are all independents with lawμdz/λand the times where the jumps occur follow the law of a standard Poisson process with parameterλ. We will give a proof, uniquely based upon algebraic computations on semi groups, of this fact in the paper.

Problem^∗was solved in1by using the Malliavin Calculus for jump processessee 15–17for related works.μis called the L´evy measure. For that we need some regularity on the L´evy measureμdz. Under regularity assumption onμdz,1used another time the implicit function theorem, when we can jumps in a finite number of jumps in a “submersive”

way between the starting point and the end point. Recently we have translated in semigroup theory plenty of tools of the stochastic analysis for Poisson processes18–23. Our goal is to translate in semigroup theory the result of1.

For material on stochastic differential equations driven by jump processes, we refer to the books13,24,25. For the analytic side of the theory of Markov processes with jumps, we refer to the books11–13.

This paper enters in a general program which would like that stochastic analysis tools become available for partial differential equation different of the parabolic equations whose generators satisfy the maximum principle26,27.

2. Statement of the Main Theorems

The goal of this paper is to give the proof of the two next theorems originally proved by1 by using stochastic analysis and the Malliavin Calculus of Bismut type for jump processes of 28.

Let us considerm functions gjz positive with compact support on continuous except in 0 equal to|z|^−1−αjnear 0 withαj∈0,1.

Let us introducemfunctionsγjzwith bounded derivatives at each order, equal to 0 in 0 with values in ^d.

We consider the Markov generator

Lfx

Ê

f

xγjz

−fx

gjzdz. 2.1

We do the following hypothesis.

Hypothesis 2.1. There exists an r such that the family of vectors {

j,k≤rd^k/dz^kγj0}

generates ^d.

L generates a convolution linear semigroup Pt in probability measures acting on differentiable bounded functionsf.P_tsatisfies the parabolic equation

∂

∂tP_tfx LP_tfx. 2.2

Under Hypothesis2.1,20,29,30proved thatP1has a smooth heat kernelp1x, y:

P1fx

Êd

f y

p1

x, y

dy. 2.3

We denote

F_k,j.z.

γ_jlzl, 2.4

wherej_.{j1, . . . , j_k}.

Theorem 2.2. Ifp1x, y>0, then there existsk, jl,z⁰_l/0 such thatgjlz⁰_l>0 such that, iF_k,j.z⁰_. y−x,

ii z. → Fk,j.z.is a submersion inz⁰_i.

Remark 2.3. Let us explain heuristically the theorem. Letz^j_tbe the process with independent increments associated to the generator

L_jf y

Ê

f yz

−f y

g_jzdz, 2.5

wherey∈ . The processesz^j_. are independents, and the time of their jumps are disjoints. We put

xtx x

s≤t,j

γj

Δz^j_s

. 2.6

Then,

Ptfx E

fxtx . 2.7

The theorem explains that we have to jump in a finite numbers of jumps in a submersive way fromxtoyif we wantp1x, y>0. Let us give some explanations what we mean about this fact, because the jump process has in fact an infinite number of jumps because the L´evy measure is of infinite mass. We take

z^j,_t

s≤t

−,^c

Δz^js

Δz^js. 2.8

z^j,_t has generator

L_jf y

|z|>

f yz

−f y

gjzdz. 2.9

The jump process

x_tx x

s≤t,j

γj

Δz^j,s

2.10

has only a finite number of jumps because its L´evy measure is of finite mass and its law gives a good approximation of the law ofx_txifis small enough!

We consider some vectorse_jand a smooth vector fieldsX₀with bounded derivatives at each order. We consider the generator

Lfx

Ê

f

xejz

−fx

gjzdz

dfx, X0x

. 2.11

It generates a Markov semigroupP_t,

∂

∂tPtfx LPtfx 2.12

if is bounded differentiable. Ifgjz |z|^1−αj, theLis classically related to fractional powers of the Laplacian31.

We do the following Hypothesis.

Hypothesis 2.4. Consider inf_x∈Êd,|ξ|1

j|ξ, ej||ξ,∂/∂xX0xej|>0.

In such a case,19,29,30has proven that there exists a smooth heat kernelp1x, y:

P₁fx

Ê

d

f y

p₁ x, y

dy. 2.13

We considert1 < t2 <· · ·< tk <1 and we denote byt. {t1, . . . , tk}. We introduce the differential impulsive equation starting fromx:

dxs

j., t., z.

x X0

xs

j., t., z.

x

ds, Δxtl

j., t., z.

x ejlzl. 2.14

We denote

Fk,j.,t.z. x1

j., t., z.

x. 2.15

Theorem 2.5. The conditionp₁x, y > 0 implies that there exists j_.,k,t_., andz⁰_l/0,g_jlz⁰_l > 0 such that:

iFk,j.,t.z⁰_. y,

iiz → Fk,j.,t.z.is a submersion inz⁰_..

Remark 2.6. Let us explain heuristically this theorem. We consider the processes with inde- pendent incrementsz^j_t. We consider the stochastic differential equation

xtx x z^j_tej

_t

0

X0xsxds. 2.16

Then,

P_tfx E

fxtx . 2.17

It has since

Ê

gjz ∞an infinite number of jumps. We take z^j,_t

s≤t

−,^c

Δz^j_s

Δz^j_s. 2.18

z^j,_t has a finite number of jumps and has generator L_jf

y

|z|>

f yz

−f y

gjzdz. 2.19

We consider the stochastic differential equation

x_tx x z^j,_t ej

_t

0

X0x_sxds. 2.20

The law ofx₁xis a good approximation of the law ofx₁xifis small enough. This express the fact that by a finite number of jumps,x_sxhas to pass fromxtoyin a submersive way ifp1x, y>0.

3. Two Results on Jump Processes Translated in Semigroup Theory

We consider ^d,x∈ ^d,z∈ ^d, andμa positive measure on ^dsuch thatλ

μdz <∞.

We introduce the expression

Gⁿfx λ⁻ⁿ

^Êdn

f

x

i

z_i

dμ z_i. 3.1

We consider the generator L

f x

Êhatd

fxz −fx

dμ z. 3.2

It is a bounded operator on the space of continuous bounded functions endowed with the uniform norm. It generates therefore a semigroupPt.

Theorem 3.1compound Poisson process. We have the formula Ptfx exp

−λt

n

λt_n

n! Gⁿfx. 3.3

Proof. In order to simplify the exposition, we supposeλ1.

We have the recursion formula

Gⁿfx −Gⁿ⁻¹fx LGⁿ⁻¹fx 3.4

such that

Gⁿ IL_n

3.5

ButL is a bounded operator on the set of continuous functions endowed with the uniform norm. Therefore, the semigroupPtsatisfies to

Ptfx

n≥0

tⁿ/n!

L_n

fx. 3.6

We write

L_n

LI−I_n

3.7

such that

L_n

p

−1^pC^p_nG^n−p. 3.8

Therefore,

P_tfx

n≥0,p≤n

tⁿ/n!−1^pC^p_nG^n−pfx, V

tⁿ/n!G^n,fx exp−t. 3.9

Let us consider now a generator L

f x

Ê

d

fxz −fx

dμxz

dfx, X0x

. 3.10

We suppose that the total mass ofμxis finite and is equal to the constant quantityλand that

μxdepends continuously ofxfor the strong topology.L generates a semigroup on the space of continuous functions endowed with the uniform norm.

LetL¹be the generator L¹

f x

dfx,X0x

. 3.11

It generates a semigroupP_t¹. We suppose thatP_t¹1 1. It is the same to suppose that the solution of the ordinary differential equation

dx¹_sX0

x_s¹

ds 3.12

does not blow up

L² f

x

Ê

d

fxz −fx

dμxz. 3.13

It is a bounded operator on the set of uniformly bounded functions endowed with the uniform topology. Therefore it generates a semigroup on the set of bounded continuous functions. We get the following translation of2.20in semigroup theory.

Theorem 3.2. We have iffis bounded continuous Pt

f

x fx exp

−λt

×

0<s1<s2<···<sr<t

P_s¹₁G²· · ·G²P_t−s¹ _rfxds1· · ·dsr. 3.14

Proof. We suppose to simplify thatλ1. By the classical Volterra expansion, we get P_t

f

x fx

0<s1<s2<···<sr<t

P_s¹₁L²· · ·L²P_t−s¹ _rfxds1· · ·ds_r. 3.15

We write

L²G²−I. 3.16

The previous Volterra expansion can be written as Pt

f

x fx

0<s1<s2<···<sr<t

P_s¹₁ G²−I

· · · G²−I

P_t−s¹ _rfxds 1· · ·dsr. 3.17

We distribute in the last expression, and we use the two formulas

P_s¹₁P_s¹₂P_s¹_1s2, 3.18

t1<s2<···sr<t2

ds₁· · ·ds_r t2−t1^r

r! . 3.19

We recognize

PtI

r

n1,...,nr

−1ⁿⁱ×

0<s1<···<sr<t

sⁿ₁¹

n1!P_s¹₁G²s2−s1ⁿ²

n2! · · ·G²P₁^t−srt−sr^nr

nr! ds1· · ·dsr

I

r

0<s1<···<sr<t

×exp−s1G²P_s¹₁exp−s2−s₁· · ·

×G²P¹t−s_rexp−t−s_rds1· · ·ds_r.

3.20

The result follows from the fact that

exp−t exp−s1exp−s2−s1· · ·exp−t−sr. 3.21

4. Proof of Theorem 2.2

LetL the Malliavin generator acting on smooth function on ^d ×d, where_d is the space of symmetric matrices on ^d:

Lfx, V

Ê

f

xγ_jz, V νz

·, γ_jz₂

−fx, V

g_jzdz. 4.1

νzis a smooth positive function with compact support equal toz⁴on a neighborhood of 0.

V is called the Malliavin matrix.L generates a semigroupPtcalled the Malliavin semigroup.

Under Hypothesis2.1, we have for allp20 P₁

V^−p x,0<∞. 4.2

Letgbe a smooth positive function equals to 1 if|V|<1 and equal to 0 if|V|>2. We consider the measureμK

f−→P1

g

V⁻¹ K

f

x,0. 4.3

Proposition 4.1. The measureμKhas a smooth densityp^Kx, y, and, whenK → ∞,p^Kx, y → p1x, yuniformly.

Proof. We follow the argument of20. We put^{l d}×d× ^d2×· · ·× ^drx^l x1, v, x2, . . . , xr. We consider a bounded mapg,g0 0, from ^m into^l. Its values in ^d isγz

γjzj and its values ind is

νzj·, γ_jzj².dμz

gjzjdzj. We consider the generator

L^lf

x^l

Êm

f^l

x^lgz

−f^l

x^l

dμz. 4.4

This generates a semigroupP_t^l. We putg_KV gV/K. By using the integration by parts formulas of20,

P1

1−gK

V⁻¹

D^αf

x,0 P₁^l

fθ x,0, . . ., 4.5

whereP_t^lis a semi group of the previous type,θa polynomial in the components,V⁻¹, and of valuation 1 in1−g_KV⁻¹and the derivatives ofg_KV⁻¹.αis a multi-index. By Theorem 3 of20, we deduce that

P1

1−gK

V⁻¹

D^αf

x,0≤CKf

∞ 4.6

whenCK → 0 whenK → ∞. Therefore the result is obtained.

Let >0. Let Lfx, V

|z|>

f

xγjz, Vνz

·, γ_jz2

−fx, V

gjzdz. 4.7

By the same procedure, we define analog generatorsL^l. We deduce several semigroupsP_t andP_t^,l. We consider the measureμ_K

f−→P₁ g_K

V⁻¹ f

x,0. 4.8

Proposition 4.2. μ_K has a densityp^,K₁ x, y, and, when → 0, p^,K₁ x, y tends uniformly to p^K₁x, y.

Proof. Letαbe a multi-index. We have μK

D^αf −μ_K

D^αf P₁^l

fθ x,0, . . .−P₁^l,

fθ x,0, . . ., 4.9 whereθis a polynomial inu^l,V⁻¹and of valuation 1 ingKV⁻¹and its derivatives. The result will come from the next lemma.

Lemma 4.3. Letθbe a polynomial inu^l,V⁻¹and of valuation 1 ing_KV⁻¹and its derivatives. Then when → 0

P₁^l,

fθ x,0, . . .−→P₁^l

fθ x,0, . . .. 4.10

Proof. Ifθis smooth bounded, we have by Duhamel formula

P₁^l

fθ x,0 P₁^l,

fθ x,0, . . . ₁

0

P_s^l,

L^l−L^l P_1−s

fθ x,0, . . ., 4.11

and the result goes. It remains to remark that under the previous condition L^l − L^lP_1−sfθx,0, . . . has a polynomial behaviour whose component tends to zero and to apply Theorem 3 of20. This comes from the fact thatP_t−sfθis a polynomial inx₁, . . . x_d and is differentiable bounded invbecause we keep only bounded values ofV⁻¹ due to the apparition ofgV⁻¹/K.

Proof ofTheorem 2.2. Ifp1x, y>0, there exists aK,such thatp^,Kx, y>0.

Let us introduce >0. We put Lfx, V

|z|>

f

x, V γz

−fx, V

dμz. 4.12

To simplify the exposition, we suppose that

|z|>dμz 1.

We put

G^n,fx, V

|zⁱ|>f F_n

z¹, . . . , zⁿ;x, V dμ zⁱ

, 4.13

whereFis defined as in2.4but withγ. ByTheorem 3.1, P_tfx, V

tⁿ/n!G^n,fx, V exp−t. 4.14

Since p^,Kx, y > 0, the measure f → G^n,f·gV⁻¹/K has a strictly positive density inyfor somen. This measure is equal to the measure

f −→

j1,...,jn

· · ·

|zl|>0f Fn,j.

g

⎛

⎝V_n,j⁻¹

.

K

⎞

⎠

gjlzldzl. 4.15

One of the measure in the above sum has a stricly positive density iny, and, therefore, nearby y. So there exists foryclose fromyn, jl,|zl|> ,gjlzl>0 such that

iFn,j.z. y−x, iiThe matrixVn,j.z.

νzl·, γ_jlzl²has an inverse bounded byK.

It remains to remark that the Gram matrix associated to Fn,j.z. is equal to ·, γ_j

lzl²is larger toCVn,.z.and to apply the implicit function theorem.

5. Proof of Theorem 2.5

Let us consider the Malliavin generator Lfx, U, V

Ê

f

xzej, U, V νz

·, U⁻¹ej

2

−fx, U, V

gjzdz

dxfx, U, V, X0x

!

dUfx, U, V , ∂

∂xX0xU

"

.

5.1

Ubelong tod, the space of invertible matrices, andVbelong tod.Vis called the Malliavin matrix.

As in the previous part, we approximateL by a generator whose L´evy measure is of finite mass. We get for >0,

Lfx, U, V

|z|>

f

xzej, U, V νz

·, U⁻¹ej

₂

−fx, U, V

gjzdz

d_xfx, U, V , X0x

!

d_Ufx, U, V, ∂

∂xX₀xU

"

.

5.2

LandLgenerate Markov semigroupPtandP_t.

We repeat with some algebraic modifications due to 19 the considerations of the previous part. LetK >0. We consider the measureμ_K

f−→P₁

g V⁻¹

K

f

x, I,0. 5.3

It has a density p^,Kx, y. WhenK → ∞the densityp^0,Kx, z of μ⁰_K tends uniformly to p₁x, zinz. When → ∞, the densityp^,Kx, ztends uniformly inztop^0,Kx, z. Therefore, ifp1x, y>0, we can findandKsuch thatp^,Kx, y>0.

LetP_sbe the semi group generated byL:

Lfx, U, V

d_xfx, U, V , X0x

!

d_Ufx, U, V , ∂

∂xX₀xU

"

. 5.4

LetLdefined by:

L¹fx, U, V

|z|>

f

xzej, U, V νz

·, U⁻¹ej

₂

−fx, U, V

gjzdz. 5.5

We suppose to simplify the exposition that _|z|>gjzdz1.

We put

Gfx, U, V

|z|>

f

xzej, U, V νz

·, U⁻¹ej

₂

gjzdz. 5.6

Let us useTheorem 3.2. Ifp^,Kx, y>0, then there exists aksuch that the mesure

f −→

0<s1<···<sk<1

Ps1G· · ·GP_1−sk

g V⁻¹

K

f

x, I,0ds1· · ·dsk 5.7

has a densityp^,K_k x, y>0. Therefore, there exist 0< t1<· · ·tk<1 such that the measure

f −→Pt1G· · ·GPtk

g

V⁻¹ K

f

x, I,0 5.8

has a strictly positive density neary. We consider the system of impulsive equation issued fromx, I,0:

dxs

j., t., z.

x X0

xs

j., t., z.

x

ds; Δxtl

j., t., z.

x ejlzl,

dU_s j_., t_., z_.

∂

∂xX₀ x_s

j_., t_., z_. x

U_s j_., t_., z_.

ds, ΔVtl

j., t., z.

νzl

·, U⁻¹_tl ejl

₂ .

5.9

We remark that

f−→P_t1G· · ·GP_tk

g V⁻¹

K

f

x, I,0

j1,...,jk

· · ·

|z_jl|>f

x1

j., t., z.

g V1

j., t., z.

K

gjlzldzl.

5.10

Therefore, the density of one of the measure

f −→

· · ·

|z_jl|>f

x₁ j_., t_., z_.

g V1

j., t., z.

K

g_jlzldzl 5.11

is strictly positive iny!

From5.11, we see that there existsj. t.such that, for some|zj| > ,gjlzk > 0 we have foryclose fromy

ix1j., t., z.x y,

iiV1j., t., z.⁻¹is bounded byK.

But the Gram matrix associated tox1j., t., z.x is equal to

·, U1U⁻¹_t

l ejl². It has therefore an inverse bounded byCK. The result arises by the implicit function theorem.

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