Density regularity for multidimensional jump diffusions with position dependent jump rate

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Preprint submitted on 10 May 2016

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Density regularity for multidimensional jump diffusions with position dependent jump rate

Victor Rabiet

To cite this version:

Victor Rabiet. Density regularity for multidimensional jump diffusions with position dependent jump rate. 2016. �hal-01313405�

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Density regularity for multidimensional jump diffusions with position dependent jump rate

Victor Rabiet May 10, 2016

Résumé

On considère une diffusionX = (Xt)t, avec des sauts, correspondant au générateur infinitésimal suivant :

Lψ(x) =1 2

X

1≤i,j≤d

aij(x)∂²ψ(x)

∂xi∂xj

+g(x)∇ψ(x) + Z

R^d

(ψ(x+c(z, x))−ψ(x))γ(z, x)µ(dz)

oùµest de masse totale infinie. En notant(Xt(x))un tel processus partant du pointx, et en utilisant une approche basé sur un Calcul de Malliavin fini-dimensionnel, nous étudions la régularité jointe de celui-ci dans le sens suivant : on fixeb≥1etp >1,Kun ensemble compact deR^d, et nous donnons des conditions suffisantes pour avoirP(Xt(x)∈dy) =pt(x, y)dyavec(x, y)7→pt(x, y)appartenant à W^bp(K×R^d).

Abstract

We consider a jump type diffusionX= (Xt)t with infinitesimal generator given by

Lψ(x) =1 2

X

1≤i,j≤d

aij(x)∂²ψ(x)

∂xi∂xj

+g(x)∇ψ(x) + Z

R^d

(ψ(x+c(z, x))−ψ(x))γ(z, x)µ(dz)

where µis of infinite total mass. Denoting(Xt(x))such process starting from pointx, and using an finite-dimensional Malliavin Calculus based approach, we study the joint regularity in the following sense : letb≥1,p >1and Kbe a compact set inR^d, then we give sufficient conditions in order to haveP(Xt(x)∈dy) =pt(x, y)dywith(x, y)7→pt(x, y)inW^bp(K×R^d).

Key words: Diffusions with jumps, Malliavin Calculus, Regularity, Density, Finite dimensional Malli- avin Calculus.

MSC 2000 : 60 J 55, 60 J 35, 60 F 10, 62 M 05

1 Differential calculus and Integration by part

1.1 Introduction

The purpose of this work is to study the regularity of the density of the following stochastic equation Xt=x+

m

X

l=1

Z t 0

σl(Xs) dW_s^l+ Z t

0

b(Xs) ds +

Z t 0

Z

E×R+

c(Xs−, z)1{u≤γ(Xs−,z}N(ds,dz,du)

where the existence and uniqueness is proved for example in [10] or Graham [6] (1992).

The way to do so used here is based on a two step strategy. First, we construct an approximation(FM) of the process Xt (basically given a non-decreasing sequence of subsets BM

M∈N^∗, with µ(BM)<∞, recoveringE, the approximationFM will be constructed (for eachM) from a restriction of the processes Xtbased on the restriction of the random measureN on the subsetBM) verifying an integration by part formula :

E [ϕ⁰(FM)] = E [ϕ(FM)HM].

This integration by part is obtained within a general framework developed in [3] by V. Bally and E.

Clément, whose main results used in the sequel are presented in this section.

The second step consists in proving the density regularity itself. The idea is to use a certain balance between the error E [|FM −Xt|] (which tends to0) and the weight E [|HM|] (which tends to ∞). This was the strategy used in [3] as well. But here the estimates of E [|HM|]will appear to be more delicate than the corresponding one in [3] because of the additional Brownian partσdW. Moreover, the balance used in [3] was based on a Fourier transform method while here we use the new method developed by V.

Bally and L. Caramellino in [2].

This new method allowed us also to extend the result to the regularity of the density considering additionally the variation of the starting point of the process, which was fixed in [3] ; the part of [2] used for our purpose is presented, in this section, subsection 1.6.

1.2 Notations, tools of differential calculus

1.2.1 Notations and differentials operators

We consider a sequence(Vi)_i∈N^∗ of random variables on a probability space(Ω,F,P), aσ-algebra G ⊂ F and aG-measurable random variableJ, with values inN. We assume that the variables(Vi)andJ satisfy the following integrability condition :

∀p≥1, E [J^p] + E

X^J

i=1V_i² p

<∞.

Following Bally and Clément, we will define a differential calculus based on the variables(Vi), conditionally onG.

First we will define the following set

Definition 1.1 LetM be the class of functionsf : Ω×R^N

∗→Rsuch that :

• f can be written as

f(ω, v) =

∞

X

j=1

f^j(ω, v₁, . . . , v_j)1{J(ω)=j}

where f^j : Ω×R^j →RareG × B(R^j)-measurable functions ;

• there exists a random variable C∈T

q≥1L^q(Ω,F,P)andp∈N^∗ such that

|f(ω, v)| ≤C(ω) 1 +

X^J(ω)

i=1 v_i² p

(in other words, conditionally on G, the functions of Mhave polynomial growth with respect to the variables(v_i)).

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We will define also

• Gi theσ-algebra generated byG ∪σ(Vj,1≤j ≤J, j6=i),

• (ai(ω))and(bi(ω))two sequences ofGi-measurable random variables satisfying

−∞ ≤a_i(ω)< b_i(ω)≤+∞, ∀i∈N^∗

• O_i the open set of R^N

∗ defined by O_i = P_i⁻¹(]a_i, b_i[), where P_i is the coordinate map R^N

∗ (ie.

P_i(v) =v_i).

We localize the differential calculus on the sets(Oi)by introducing some weights(πi), satisfying the following hypothesis.

Hypothesis 1.1 For alln∈N^∗,πi∈ M and

{πi>0} ⊂Oi.

Moreover for allj≥1,π^j_i is infinitely differentiable with bounded derivatives with respect to the variables (v1, . . . , vj).

At last, we associate to these weightsπi, the spacesC_π^k⊂ M,k∈N^∗, defined recursively as follows.

• For k= 1, C_π¹ denotes the space of functions f ∈ Msuch that for eachi∈N^∗, f admits a partial derivative with respect to the variablev_i on the open setO_i. We then define

∂^π_if(ω, v) ^def= π(ω, v) ∂

∂v_if(ω, v) and we assume that∂_i^πf ∈ M.

• Suppose now that C_π^k is already defined. For a multi-index α = (α₁, . . . , α_k) ∈ N^∗k, we define recursively ∂_α^π =∂_α^π

1· · ·∂_α^π

k and C_π^k+1 is the space of functions f ∈C_π^k such that for every multi- index α= (α1, . . . , αk)∈N^∗^k we have ∂_α^πf ∈C_π¹. Notice that if∂_α^πf ∈ Mfor each αwith|α| ≤k.

• Finally we define

C_π^∞ ^def= \

k∈N^∗

C_π^k.

Definition 1.2 (Simple functionals) A random variable F is called a simple functional if there exist f ∈C_π^∞ such that F =f(ω, V), where V = (Vi). We denote byS the space of the simple functionals (it is an algebra) ; moreover, it is worth to notice that, conditionally on G,F=f^J(V1, . . . , VJ).

Definition 1.3 (Simple processes.) A simple process is a sequence of random variables U = (Ui)i∈N^∗

such that for each i∈N^∗,U_i ∈ S. Consequently, conditionally onG, we have U_i =u^J_i(V₁, . . . , V_J). We denote byP the space of the simple processes and we define the scalar product

hU, ViJ=

J

X

i=1

UiVi (∈ S).

We can now define the derivative operator and state the integration by parts formula.

Definition 1.4 (The derivative operator.) We define D :S → P by DF ^def= (D_iF)∈ P where D_iF ^def= ∂^π_if(ω, v).

Notice thatDiF = 0, for i > J.

Definition 1.5 (Malliavin covariance matrix ) For F = (F¹, . . . , F^d) ∈ S^d, the Malliavin covariance matrix is defined by

σ^k,k⁰(F) =hDF^k,DF^k⁰iJ=

J

X

i=1

DiF^kDiF^k⁰

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We denote

Λ(F) ={detσ(F)6= 0} and γ(F)(ω) =σ⁻¹(F)(ω), ω∈Λ(F)

In order to derive an integration by parts formula, we need some additional assumptions on the random variables(V_i). The main hypothesis is that conditionally onG, the law of the vector(V₁, . . . , V_J), admits a locally smooth density with respect to the Lebesgue measure onR^J.

Hypothesis 1.2 1. Conditionally on G, the vector (V1, . . . , VJ) is absolutely continuous with respect to the Lebesgue measure on R^J and we denote bypJ the conditional density.

2. The set{pJ >0} is open inR^J and on{pJ >0},lnpJ∈C_π^∞. 3. For allq >1, there exists a constant C_q such that

(1 +|v|^q)pJ≤Cq

where |v| stands for the euclidean norm of the vector (v1, . . . , vJ).

Assumption 3) implies in particular that conditionally on G, the functions ofM are integrable with respect top_J and that for f ∈ M:

E_G[f(ω, V)] = Z

R^J

f^J×pJ(ω, v1, . . . , vJ) dv1, . . . ,dvJ.

Definition 1.6 (The divergence operator) Let U = (Ui)_i∈_N^∗∈ P with U ∈ S. We defineδ:P → S by

δi(U) ^def= −(∂vi(πiUi) +Ui1{pJ>0}∂_i^πlnpJ) (1.1) δ(U) =

J

X

i=1

δ_i(U) (1.2)

ForF ∈ S, we then define

L(F) ^def= δ(DF) (1.3)

1.3 Duality and integration by parts formulae 1.4 IPP

The duality betweenδandD is given by the following proposition.

Proposition 1.1 Assuming the two preceding hypothesis, then for all F ∈ S and for allU ∈ P we have E_G[hDF, UiJ] = E_G[F δ(U)].

Lemma 1.2 Let φ:R^d→Rbe a smooth function andF= (F¹, . . . , F^d)∈ S^d. Then φ(F)∈ S and Dφ(F) =

d

X

r=1

∂rφ(F) DF^r. (1.4)

If F ∈ S andU ∈ P, then

δ(F U) =F δ(U)− hDF, Ui_J. Moreover, forF = (F¹, . . . , F^d)∈ S^d, we have

Lφ(F) =

d

X

r=1

∂rφ(F)LF^r−

d

X

r,r⁰=1

∂r,r⁰φ(F)hDF^r,DF^r⁰iJ. We can now state the main results of this subsection.

Theorem 1.3 Assuming the two preceding hypothesis, letF = (F¹, . . . , F^d)∈ S^d,G∈ S andφ:R^d→R be a smooth bounded function with bounded derivatives. Let Λ∈ G,Λ⊂Λ(F)such that

E

|detγ(F)|^p1^Λ

<∞, ∀p≥1.

Then,

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1. for everyr= 1, . . . , d,

E_G[∂rφ(F)G]1^Λ= E_G[φ(F)Hr(F, G)]1^Λ with

Hr(F, G) =

d

X

r⁰=1

δ(Gγ^r⁰^,r(F) DF^r⁰) =

d

X

r⁰=1

Gδ(γ^r⁰^,r(F) DF^r⁰)−γ^r⁰^,rhDF^r⁰,DGiJ

; (1.5) 2. for every multi-indexβ = (β1, . . . , βq)∈ {1, . . . , d}^q

E_G[∂_βφ(F)G]1Λ= E_G[φ(F)H_β^q(F, G)]1Λ (1.6) where the weights H^q are defined recursively by (1.5)and

H_β^q(F, G) =Hβ₁

F, H_(β^q−1

2,...,βq)(F, G)

. (1.7)

1.5 Estimations of H

^q

In order to estimate the weights H^q appearing in the integration by parts formulae of the previous subsection, we first need to define iterations of the derivative operator. Letα= (α1, . . . , αk)be a multi- index, withαi∈ {1, . . . , J}, fori= 1, . . . , k and|α|=k.

ForF ∈ S we define recursively D^k_(α₁_,...,α

k)F ^def= Dα_k D^k−1_(α

1,...,αk−1)F and D^kF ^def=

D^k_(α₁_,...,α

k)F

αi∈{1,...,J}. Notice that D^kF ∈R^J⊗k, and consequently we define the norm ofD^kF as

D^kF

def= v u u t

J

X

α₁,...,α_k=1

D^k_(α₁_,...,α_k₎F

2. (1.8)

Moreover we introduce the following norms, forF ∈ S :

|F|1,l

def=

l

X

k=1

D^kF

and |F|l

def= |F|+|F|1,l =

l

X

k=0

D^kF

. (1.9)

ForF = (F₁, . . . , F_d)∈ S^d :

|F|1,l

def=

d

X

r=1

F^r

_1,l and |F|l

def=

d

X

r=1

F^r _l, and, similarly, forF = F^r,r⁰

r,r⁰=1,...,d

|F|1,l

def=

d

X

r,r⁰=1

F^r,r⁰

_1,l and |F|l

def=

d

X

r,r⁰=1

F^r,r⁰ _l.

Notation 1.4 • In the sequel, we will generally denote simply D^k_α by Dα (where α is a multi-index of length k).

• We will also use the following generalisation for F ∈ S^d andG∈ S^d×k : we will simply set

DαF ^def= DαFi

1≤i≤d and D^kG ^def= D^kGi,j

1≤i≤d 1≤j≤k

.

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1.5.1 Differentiability lemmas

In this subsection we will use directly the notations from Section 2 defined in 2.9 and 2.10, where we will apply the previous general differential framework.

In order to express the form of the different multi-derivatives we will use in the next section, let us set the following notations :

• ifF ∈ S^d, we will denote then-th derivative D(k_n,r_n)

D(k_n−1,r_n−1)

· · ·

D(k₁,r₁)(F) by

Dα(F)

withα= (α₁, . . . , α_n)and, for alli∈ {1, . . . , n},α_i ^def= (k_i, r_i);

• for1≤l≤n, we denote by Mn(l) =n

M = (M1, . . . , Ml), [

i∈J1,lK

Mi={1, . . . , n}andMi∩Mj=∅, fori6=jo

; the set of the partitions of lengthl of{1, . . . , n}.

Remark 1.5 The multi-derivatives defined above are not commutative : in general D_(k,r)

D_(m,n)(F)

6= D(m,n)

D_(k,r)(F) . We can now state :

Lemma 1.6 LetA, B∈ S,φ:R^d→Randc:R^d×R^d→Rbe smooth functions andF = (F¹, . . . , F^d), G= (G¹, . . . , G^d)∈ S^d. Then

1. for every(k, r)∈J1, JK×J1, dK,

D_k,r(AB) = D_k,r(A)B+AD_k,r(B) ; (1.10) and for every α= (α₁, . . . , α_n), withα_i ^def= (k_i, r_i)∈J1, JK×J1, dK,

D_α(AB) = X

αi⊕αj=α α_i,α_j ordered

D_α_iAD_α_jB; (1.11)

(by “ordered” we mean that if α_i= (α_i₁, . . . , α_i_k), theni₁<· · ·< i_k) 2. for everyα= (α1, . . . , αn), withαi

def= (ki, ri)∈J1, JK×J1, dK, Dαφ(F) =

n

X

l=1

X

β=(β₁,...,β_l) β_i∈J1,dK

X

M∈Mn(l)

∂βφ(F) DM₁(α)Fβ₁· · ·DM_l(α)Fβ_l (1.12)

=Tα φ

(F) +∇φ(F) DαF, (1.13)

where

• forM = (M1, . . . , Ml)∈ Mn(l), ifMj= (i1, . . . , ir)⊆ {1, . . . , n}, Mj(α) ^def= (αi₁, . . . , αi_r),

• and

Tα φ (F) ^def=

n

X

l=2

X

β=(β₁,...,β_l) β_i∈J1,dK

X

M∈Mn(l)

∂βφ(F) D_M₁_(α)Fβ₁· · ·D_M_l_(α)Fβ_l (1.14)

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3. for everyα= (α₁, . . . , α_n), withα_i ^def= (k_i, r_i)∈J1, JK×J1, dK, and using the same notations, Dαc(F, G) =

n

X

l=1

X

β=(β₁,...,β_r,β_r+1⁰ ,...,β⁰_l) βi∈J1,dK, β_j⁰∈Jd+1,2dK

X

M∈Mn(l)

∂βc(F, G) D_M₁_(α)Fβ₁· · ·D_M_r_(α)Fβ_r

×D_M_r+1_(α)G_β⁰

r+1· · ·D_M_l_(α)G_β⁰

l

=Uα(c)(F, G) +∇fc(F, G) DαF+∇gc(F, G) DαG.

Remark 1.7 The non-symmetric form (1.13) is used in the sequel in recurrence’s purpose : all the elements M_i(α) from T_α are such that |Mi(α)| < α so the degree of derivation of D_M_i_(α) is strictly inferior to the one ofD_αitself.

With the same notations :

Lemma 1.8 Let φ : R^d → R a smooth function and F = (F¹, . . . , F^d) ∈ S^d, α a multi-index and n ^def= |α|. Then there exists Cn,p,d>0 such that

|Dαφ(F)|^2p≤Cn,p,d |φ|n(F)^2p

n

X

l=0

X

M∈Mn(l)

|D_M₁_(α)F|^2pd+· · ·+|D_M_l_(α)F|^2pd

(1.15)

with|φ|n(F) ^def= sup_|β|≤n|∂βφ(F)|.

Proof :

From Lemma 1.6 we have D_αφ(F) =

n

X

l=1

X

β=(β₁,...,β_l) β_i∈J1,dK

X

M∈M_n(l)

∂_βφ(F) D_M₁_(α)F_β₁· · ·D_M_l_(α)F_β_l.

It follows

|Dαφ(F)| ≤Cn|φ|n(F)

n

X

l=1

X

β=(β1,...,βl) β_i∈J1,dK

X

M∈Mn(l)

|D_M₁_(α)Fβ₁| · · · |D_M_d_(α)Fβ_l|

|Dαφ(F)|^2p≤Cn,p |φ|n(F)^2p

n

X

l=1

X

β=(β1,...,βl) βi∈J1,dK

X

M∈Mn(l)

|D_M₁_(α)Fβ₁|^2p· · · |D_M_d_(α)Fβ_l|^2p

Now¹

|DM₁(α)F|^2p· · · |DM_l(α)F|^2p≤ 1 d

|DM₁(α)F|^2pd+· · ·+|DM_d(α)F|^2pd , so

|Dαφ(F)|^2p≤Cn,p,d |φ|n(F)2p n

X

l=1

X

M∈Mn(l)

|D_M₁_(α)F|^2pd+· · ·+|D_M_d_(α)F|^2pd .

• We will also need an extended version of the first item of Lemma 1.6 :

1Since, ifa1, . . . ,an∈R^∗+, it is well-known that pQⁿ n

i=1ai≤¹_nPn i=1ai,

n

Y

i=1

ai≤ 1 nⁿ

Xⁿ

i=1

ai

n

≤ 1 nⁿnⁿ⁻¹

n

X

i=1

aⁿ_i,

so n

Y

i=1

ai≤ 1 n

n

X

i=1

aⁿ_i.

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Lemma 1.9 Let A, B∈ S^d×d. Then for every (k, r)∈J1, JK×J1, dK,

Dk,r(AB) = Dk,r(A)B+ADk,r(B) ; (1.16) and for everyα= (α1, . . . , αn), withαi

def= (ki, ri)∈J1, JK×J1, dK, Dα(AB) = X

αi⊕αj=α αi,αj ordered

DαiADαjB; (1.17)

(by “ordered” we mean that if αi= (αi1, . . . , αi_k), then i1<· · ·< ik).

Proof : LetA= (A_i,j)_1≤i,j≤d andB = (B_i,j)_1≤i,j≤d withA_i,j,B_i,j∈ S. Then, Dk,r(AB) = Dk,r

X^d

m=1

Ai,mBm,j

1≤i,j≤d

= X^d

m=1

Dk,r Ai,mBm,j

1≤i,j≤d

= X^d

m=1

D_k,r A_i,m

B_m,j+A_i,mD_k,r B_m,j

1≤i,j≤d

(using (1.10))

= Dk,r(A)B+ADk,r(B).

But the proof of (1.17) only requires an induction over the formal relation (1.16) (and does not need

any commutativity in the product ofAbyB). •

Corollary 1.10 Let A = (Ai,j)_1≤i,j≤d, B = (Bi,j)_1≤i,j≤d with Ai,j, Bi,j ∈ S and l ∈ N^∗. Then there existsCl>0 such that

|AB|l≤Cl|A|l|B|l. (1.18)

We also have the following result proven in [3] (Lemma 8) :

Lemma 1.11 Let φ:R^d→Rbe a C^∞ function and F ∈ S^d then for alll≥1we have

|φ(F)|1,l≤ |∇φ(F)||F|1,l+Cl sup

2≤β≤l

|∂βφ(F)||F|^l_1,l−1. Result that we will essentially use in this work through this corollary :

Corollary 1.12 Let φ : R^d →R be a C^∞ bounded function with bounded derivatives of any order and F ∈ S^d then for all l≥1there exists C_φ,l>0 such that

|φ(F)|_l≤C_φ,l 1 +|F|_l+|F|^l_l−1

. (1.19)

1.5.2 Some bounds onH^q

The further theorem, proven in [3], gives some estimates for the weightsH^q in terms of the derivatives of G,F,LF andγ(F).

Theorem 1.13 ForF ∈ S^d, G∈ S and for all q∈N^∗ there exists a universal constant C_q,d such that for every multi-indexβ= (β₁, . . . , β_q)

H_β^q(F, G)

≤C_q,d|G|_q(1 +|F|_q+1)^(6d+1)q

|detσ(F)|^3q−1 1 +|LF|^q_q−1 .

Remark 1.14 In the sequel, we will simply denoteH_β^q(F,1) by H_β^q(F).

1.6 Interpolation method : notations and theoretical result

All this subsection is directly taken from the article of Bally and Caramelino [2].

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1.6.1 Notations and definitions Let us define forξandγ some multi-indexes

x^γ ^def=

d

Y

i=1

x^γ_iⁱ (1.20)

fξ,γ(x) ^def= x^γ∂ξf(x) (1.21)

kfkk,l,p= X

0≤|γ|≤l

X

0≤|ξ|≤k

kfξ,γkp (1.22)

For allγ such that|γ| ≤l,

|x^γ| ^def=

d

Y

i=1

|xi|^γⁱ ≤

d

Y

i=1

|x|^γⁱ ≤ |x|^P^γⁱ≤(1 +|x|)^l so

kfkk,l,p≤C_d X

0≤|ξ|≤k

k(1 +|x|)^l∂_ξf(x)kp. (1.23) Since in the sequel we will have to bound the quantitykfMk2m+q,2m,p, let us notice that we have directly

kfMk2m+q,2m,p≤Cd

X

0≤|ξ|≤2m+q

Z

R^d×R^d

(1 +|(x, y)|^2m)∂ξ fM(x, y)p

dxdy_p¹

. (1.24) Moreover, we will define a distance between two measuresµandν in the following way :

dk(µ, ν) ^def= supn Z

φdµ− Z

φdν

:φ∈ C^∞(R^d), X

0≤|ξ|≤k

k∂ξφk_∞≤1 o

. (1.25)

Remark 1.15 Here, we will only use the case k= 1, which is called the bounded variation distance (or, also, the Fortet-Mourier distance).

1.6.2 Main result

Theorem 1.16 Let q, k∈N,m∈N^∗,p >1 and set η >q+k+d/p^∗

2m . (1.26)

We consider a non negative finite measure µand a family of finite non negative measures µδ(dx) =fδ(x) dx, δ >0.

We assume that there existC,r >0such that λq,m(δ) ^def= sup

δ≤δ⁰≤1

kfδ⁰k2m+q,2m,p ≤Cδ^−r

and moreover, with η given in (1.26),

λq,m(δ)^ηdk(µ, µδ)≤C. (1.27)

Thenµ(dx) =f(x) dxwith f ∈W^q,p.

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2 Regularity of the Density

2.1 Introduction

As we briefly mentioned in the introduction of the last section, the main purpose of this second part is to study the regularity of the law of the random variableXtsolution of the following stochastic equation with jumps :

X_t=x+ Z t

0

σ(X_s) dW_s+ Z t+

0

Z

E×R+

c(z, X_s−)1{u≤γ(z,X_s−)}N(ds,dz,du) + Z t

0

g(X_s) ds (2.28) (again, for the existence and the uniqueness of such stochastic equation see [6]).

Our global aim is to give sufficient conditions in order to prove that the law of X_t is absolutely continuous with respect to the Lebesgue measure and has a smooth density. That was the point of the study made in [3] as well, with an equation of this type but without the Brownian part.

But, here, we will not only consider the existence and the regularity of the densityy7→pX_t(y)(defined byPX_t(dy) =pX_t(y) dy) with a given starting point x∈R^d : we will consider instead the behaviour of (x, y)7→pX_t^x(y)(withPX_t^x(dy) =pX_t^x(y) dy where X_t^x stands for the solution of (2.28) starting atx).

This joint density regularity property, in addition to being obviously a stronger result, will allow us (and it was, at first, one of the motivations to extend the result obtained for the regularity ofy7→pX_t^x(y)) to obtain an interesting application concerning the Harris-recurrence of the process (section 5).

2.2 Hypothesis and notations

Let us recall that the associated intensity measure of the counting measureN is given by Nˆ(dt,dz,du) = dt×µ(dz)×1{0,∞}(u) du

where(z, u)∈X =R^d×R+ andµ(dz) =h(z) dz.

In this subsection we make the following hypothesis on the functionsγ,g,handc.

Hypothesis 2.1 We assume thatγ,g,hand c are infinitely differentiable functions in both variables z andx. Moreover we assume that

• g and its derivatives are bounded ;

• lnhhas bounded derivatives ;

• both γ andlnγ have bounded derivatives.

Hypothesis 2.2 We assume that there exist two functionsγ, γ:R^d →R+ such that C≥γ(z)≥γ(z, x)≥γ(z)≥0, ∀x∈R^d

whereC is a constant.

Hypothesis 2.3 1. We assume that there exists a non negative and bounded function c : R^d → R+

such that R

R^dc(z)µ(dz)<∞and

|c(z, x)|+|∂_z^β∂_x^αc(z, x)| ≤c(z), ∀z, x∈R^d. We need this hypothesis in order to estimate the Sobolev norms.

2. There exists a measurable functioncˆ:R^d→R+ such that R

R^dˆc(z)µ(dz)<∞and

||∇xc×(Id +∇xc)⁻¹(z, x)|| ≤c(z),ˆ ∀z, x∈R^d. In order to simplify the notations we assume that ˆc(z) =c(z).

3. There exists a non negative functionc:R^d→R+ such that, for all z∈R^d,

d

X

r=1

h∂z^rc(z, x), ξi²≥c²(z)|ξ|², ∀ξ∈R^d and we assume that there exists θ∈R

∗

+ such that lim inf

a→∞

1 lna

Z

{c²≥_a¹}

γ(z)µ(dz) =θ. (2.29)

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Remark: assumptions 2) and 3) give sufficient conditions to prove the non degeneracy of the Malliavin covariance matrix as defined in the previous section.

2.3 Main result

We are now able to state the density property of X_t^x and the joint regularity (in x and y) of it : we fix q ≥ 1 and p > 1, K a compact set of R^d, and we will give sufficient conditions in order to have PX_t^x(dy) =pX_t^xdy with(x, y)7→pX_t^x ∈W^q,p(K×R^d).

Theorem 2.1 Let q,p≥1. We assume that hypotheses 2.1, 2.2 and 2.3 hold.

Let B_M

M∈N^∗ such thatS

M∈N^∗B_M =E and, for all i∈N^∗

B_i⊂B_i+1 and µ(B_i)<+∞.

Let K a compact set ofR^d and (with p^∗ such that ¹_p+_p¹∗ = 1) η > q+ 1 +d/p^∗

2 . (2.30)

If there exists C,r >0such that

µ(BM)^6(d+q+3)³ ≤CM^r (2.31)

and if

lim sup

M

µ(BM)^6(d+q+3)³^ηZ

B^c_M

c(z)γ(z) dµ(z) + sZ

B_M+1^c

c²(z)γ(z) dµ(z)

!

<+∞, (2.32) then, fort > ^4d(3q_θ⁰⁻¹⁾, with q⁰ =d+q+ 2, and for everyx∈K, the law of X_t^x is absolutely continuous with respect to the Lebesgue measure, ie. PX_t^x(dy) =pX^x_t(y) dy, and the function(x, y)7→pX_t^x(y)belongs toW^q,p(K×R^d).

Remark 2.2 The quantity η and the related condition 2.30, come directly from the main theorem of the interpolation method (Theorem 1.16), in the particular casek= 1, as we stated in Remark 1.15, and with m= 1 (this last choice is discussed in Remark 2.12).

Remark 2.3 If θ =∞, then, for all t > 0, the law of X_t^x is absolutely continuous with respect to the Lebesgue measure and the density pX_t^x belongs to W^q,p(K×R^d).

Remark 2.4 Recalling that (cf. Brézis [5], p.168, Corollaire IX.13, for example), with k ^def= q− ^d_p

, we have (with O ∈ Rⁿ an open ball), W^q,p(O) ⊂ C^k(O), in the sense that each element of W^q,p(O) has a C^k representative, this theorem can also be used to characterize the C^k behaviour of the function (x, y)7→pX_t^x(y)(as we will briefly see in the examples at 2.3.1).

Notation 2.5 In the sequel, since x will belong to a fixed compact set, we will often write simply Y_t instead of Y_t^x for any process Y starting at x, if this precision is not strictly needed (that is why the starting point will never explicitly appear within the Section 3 but will be always used in Section 4).

Before starting the proof itself, we will first try to give a sketch of the strategy that we will use. The global idea is articulated in two steps :

1. to obtain an integration by part formula on an appropriate approximation of the processX_t; 2. to use this last result to prove the regularity of the density.

The terms from the condition (2.32) are a direct consequence of this pattern.

For the first step, and first of all, given a non-decreasing sequence of subsets B_M

M∈N^∗, withµ(B_M)<

∞, recovering E, we construct (for each M) an approximation X_t^M of the process Xt based on the restrictionNM of the random measureN on the subset BM.

Using a similar result as the Lemma ??, given in the first part of this work, we can then say that the L¹-distance between these two processes is bounded as follows :

∀t≤T, E

X_t−X_t^M

≤C_T Z

B^c_M

c(z)γ(z)µ(dz),

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which explains the presence of the termR

B^c_Mc(z)γ(z) dµ(z)in the condition (2.32).

Since µ(B_M)< +∞, the random measure N_M may be represented as a compound Poisson process (where the jump times will be denoted by T_k^M, k ∈ N) and the Poisson part of process X_t^M could be expressed as a sum ; nevertheless, because of the indicator function from the original equation, the coefficients of the equation verified byX_t^M are still (for the Poisson part) discontinuous and therefore, we cannot use directly the differential calculus presented earlier. Instead we prove that X_t^M has the same law as the processX^M_t which verifies an equation with smooth coefficients.

At this point, one would like to obtain an integration by part formula forX^M_t , but there remains one last difficulty : it is clear that, fort < T₁^M (the first jump ofNM), the random measureNM produces no noise, and consequently there is no chance to use it for an integration by part (the Malliavin covariance matrix being, of course, degenerated).

That is why one last process will be introduced : FM

def= X^M_t +p

UM(t)×∆, where∆ Gaussian and whereUM(t)is defined byUM(t) =tR

B_M+1^c c²(z)γ(z) dµ(z).

TheL¹-distance betweenF_M andX^M_t is then bounded, fort≤T, by KT

sZ

B_M+1^c

c²(z)γ(z) dµ(z),

which gives a natural interpretation for the last term of the condition (2.32).

We are now able to obtain an integration by part formula for the processFM :

E [ϕ⁰(FM)] = E [ϕ(FM)HM]. (IM) The second step consists in proving the density regularity. The idea is to use a certain balance between the error E [|F_M −X_t|] (which tends to0) and the weight E [|H_M|] (which tends to ∞). This was the strategy used in [3] as well. But here the estimates ofE [|H_M|]have been more delicate then the corresponding one in [3] because of the additional brownian partσdW. Moreover, the balance used in [3]

was based on a Fourier transform method while here we use the new method developed in [2].

This new method allows us also to extend the result to the regularity of the density considering additionally the variation of the starting point of the process, which was fixed in [3] ; finally, we give an application of this improvement since we can then consider a regenerative scheme to obtain an interesting result concerning the Harris-recurrence of the process (section 5).

2.3.1 Examples

In this example we assume thath= 1soµ(dz) = dzandγ(z)is equal to a constantγ >0. We then have µ(B_M) =r_dM^d

where rd is the volume of the unit ball in R^d. We will also assume that x is in some compact set K ^def= B(0, R),R >0.

We will consider two types of behaviour for c.

i) Exponential decay : we assume that c(z) = e^−a|z|^c for some constants 0 < b ≤a and c > 0. We

then have Z

{c²>¹_u}

γ(z) dµ(dz) =γ rd

(2a)^d^c ×(lnu)^d^c. we then deduce for the constantθ(definied in (2.29))

θ= 0 if c > d,

θ=∞ if 0< c < d, (2.33)

θ=γr_d

2a if c=d.

Ifc > d, hypothesis 2.3.3 fails, which is coherent with the result of Bichteler, Gravereaux and Jacod in [4]. Now observe that

Z

B_M^c

B^c_M+1

c²(z)γ(z) dµ(z)≤Ke^−ξM^c

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for some ξ >0, so the condition (2.32) is always well verified.

When0< c < d, sinceθ=∞, for everyt >0,(x, y)7→pX_t^x(y)belongs toW^∞,p(K×R^d)(∀p≥1), which implies, according to the Remark 2.4, that (x, y)7→pX^x_t(y)can be considered as an element ofC^∞(K×R^d).

If c = d, then appears a more particular behaviour and it is interesting to compare the result obtained here with the example from [3] (recalling that, in this last case, Xt does not possess a Brownian part, though), assuming here, for that sake, that xis fixed,k∈Nandp >1 :

[3] Present work

Domain t > ^8da_γr

d(3d+ 2) t > ^8da_γr

d

3d

1 +¹_p

+ 3k+ 8

Regularity ofpX_t C^k with C^k with

k≤ ¹₃

1 +^γr_8da^dt

−d k≤ ¹₃

1 + ^γr_8da^dt

−d 1 +¹_p

−2 Remark 2.6 In fact, in this work,(x, y)7→pX_t^x(y)belongs to W^q,p(K×R^d)(∀p≥1) with

q≤ 1 3

1 + γr_d 8dat

−d−2, so, using again the Remark 2.4, (x, y)7→p_X^x

t(y) can be considered as an element of C^k(K×R^d), with

k ^def= h q−d

p

i≤q−d p ≤ 1

3

1 + γrd

8dat

−d 1 +1

p −2.

In particular it requires, at least, q≥^d_p to obtain some regularity.

ii) Polynomial decay : We assume now that c(z) = _1+|z|^b v and c(z) = _1+|z|^a v for some constants 0 <

a≤b andv > d. We have here Z

{c²>¹_u}

γ(z) dµ(dz) =γr_d×(a√

u−1)^d^v, so θ = lim sup_u→∞_ln¹_uγr_d(a√

u−1)^d^v =∞ and then, in this case, the regularity result stands for every t >0.

A simple computation gives us the following bounds : Z

B^c_M

c(z)γ(z) dµ(z)≤ C

M^v−d and Z

B^c_M+1

c²(z)γ(z) dµ(z)≤ C M^2v−d. So withC andr >0 such thatµ(BM)^6(d+q+3)³ ≤CM^r(condition (2.31)), we have

µ(B_M)^6(d+q+3)³^ηZ

B_M^c

B^c_M+1

c²(z)γ(z) dµ(z)

≤C⁰M^rη 1

2M^v−d + 1 2M^2v−d

≤C⁰M^rη−v+d. Hence, the condition (2.32) is true if

η≤v−d r

and, since here µ(BM) = rdM^d, (2.31) givesr= 6d(d+q+ 3)³ and, with (2.30), we have the following condition :

q+ 1 +d/p^∗

2 ≤v−d

r . Finally, withv such that

v >6d(d+q+ 3)³q+ 1 +d

2 , (2.34)

(x, y)7→pX_t^x(y)belongs toW^q,p(K×R^d), for allp≥1.

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2.4 Approximation of X

_t

In order to prove that the processX_t, solution of (2.28), has a smooth density, we will apply the differential calculus and the integration by parts formula from Section 1. But since the random variableX_tcannot be viewed as a simple functional, the first step consists in approximating it. We describe in this subsection our approximation procedure. We consider a non-negative and smooth functionϕsuch thatϕ(z) = 0for

|z|>1andR

R^dϕ(z) dz= 1. And for M ∈N, we denote ΦM(z) =ϕ∗1^BM

with BM ={z ∈R^d : |z| < M}. ThenΦM ∈C_b^∞ and we have 1^BM−1 <ΦM <1^BM+1. We denote by X_t^M the solution of the equation

X_t^M =x+ Z t

0

σ(X_s^M) dW_s+ Z t

0

Z

E

c_M(z, X_s−^M)1{u≤γ(z,X_s−^M)}N(ds,dz,du) + Z t

0

g(X_s^M) ds. (2.35) where

cM(z, x) ^def= c(z, x)ΦM(z).

If we set

NM(ds,dz,du) ^def= 1^BM+1(z)×1[0,2C](u)N(ds,dz,du),

since{u < γ(z, X_s−^M)} ⊂ {u <2 ¯C}andΦM(z) = 0 if|z|> M+ 1,we may replaceN byNM in the above equation and consequentlyX_t^M is solution of the equation

X_t^M =x+ Z t

0

σ(X_s^M) dWs+ Z t

0

Z

E

cM(z, X_s−^M)1{u≤γ(z,X_s−^M)}dNM(s, z, u) + Z t

0

g(X_s^M) ds.

Since the intensity measure NˆM is finite, we may represent the random measureNM by a compound Poisson process. Let

λM

def= 2 ¯C×µ(BM+1) =t⁻¹E [NM(t, E)]

and let J_t^M a Poisson process of parameter λM. We denote by T_k^M, k ∈N, the jump times ofJ_t^M. We also consider two sequences of independent random variables (Z_k^M)_k∈N and (Uk)_k∈N, respectively in R^d andR+, which are independent ofJ_t^M and such that

Z_k^M ∼ 1

µ(BM+1)1^BM+1(z)µ(dz), and Uk∼ 1

2 ¯C1[0,2 ¯C](u) du.

Then, the last equation may be written as

X_t^M =x+ Z t

0

σ(X_s^M) dW_s+

J_t^M

X

k=1

c_M(Z_k^M, X_T^MM

k −)1(U_k,∞)(γ(Z_k^M, X_T^MM k −)) +

Z t 0

g(X_s^M) ds. (2.36) The random variable X_t^M solution of (2.36) is a function of (Z1, . . . , Z_JM

t ) but it is not a simple functional, as defined in Section 1, because the coefficient cM(z, x)1{u≤γ(z,x)} is not differentiable with respect toz. In order to avoid this difficulty we use the following alternative representation. Letz_M^∗ ∈R^d such that|z_M^∗ |=M+ 3. We define

q_M(z, x) ^def= ϕ(z−z_M^∗ )θ_M,γ(x) + 1

2Cµ(BM+1)1BM+1(z)γ(z, x)h(z) (2.37) θ_M,γ(x) ^def= 1

µ(BM+1) Z

{|z|≤M+1}

1− 1

2Cγ(z, x)

µ(dz). (2.38)

We recall that ϕ is the function defined at the beginning of this subsubsection : a non-negative and smooth function with R

ϕ = 1 and which is null outside the unit ball. Moreover from Hypothesis 2.2, 0≤γ(z, x)≤C and then

1≥θ_M,γ(x)≥1

2. (2.39)

From this last inequality it is easy to deduce the following result :

Lemma 2.7 Let qM defined as in (2.37). ThenlnqM has bounded derivatives of any order.

Density regularity for multidimensional jump diffusions with position dependent jump rate

HAL Id: hal-01313405

https://hal.archives-ouvertes.fr/hal-01313405

Density regularity for multidimensional jump diffusions with position dependent jump rate

Victor Rabiet

To cite this version:

Density regularity for multidimensional jump diffusions with position dependent jump rate

Victor Rabiet May 10, 2016

Contents

1 Differential calculus and Integration by part

1.1 Introduction

1.2 Notations, tools of differential calculus

1.3 Duality and integration by parts formulae 1.4 IPP

1.5 Estimations of H

1.6 Interpolation method : notations and theoretical result

2 Regularity of the Density

2.1 Introduction

2.2 Hypothesis and notations

2.3 Main result

2.4 Approximation of X