Asymptotic behavior for multi-scale PDMP's

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Preprint submitted on 21 Apr 2015

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Asymptotic behavior for multi-scale PDMP’s

Vlad Bally, Victor Rabiet

To cite this version:

Vlad Bally, Victor Rabiet. Asymptotic behavior for multi-scale PDMP’s. 2015. �hal-01144107�

Asymptotic behavior for multi-scale PDMP’s

Vlad Bally

^∗

and Victor Rabiet

^†

University Paris Est Marne la Vall´ ee.

April 15, 2015

Key words : Stochastic Differential Equations with jumps, Multi-scale Piecewise Deterministic Markov Processes, Averaging Phenomenon, Regularity of the Markov Semigroup

MSC 2000 : 60 J 75, 60 G 57, 47 D 07

1 Abstract

We study the asymptotic behaviour of a sequence of Piecewise Constant Markov Processes (in short PDMP) in which three different scales are at work: a rapid, a medium and a slow one. At the limit the rapid scale gives rise to a diffusion part (this is a CLT type regime), the medium scale produces a drift part (this is the law of large numbers type regime) and the slaw rate gives a finite variation jump process. So at the limit we obtain a stochastic differential equation which is similar to the P DM P evolution but now, in-between two jumps the equation evolutes as a general diffusion process including a Brownian part and moreover, an infinity of jumps occur in each finite time interval. This type of equations seems to be new in the literature and our first goal is to prove existence and uniqueness of the solution for them. Afterwords we study the regularity of the semigroup and we use it in order to prove the convergence result mentioned in the beginning.

2 Introduction

In this paper we introduce the following class of jump type stochastic equations:

X

t

= x +

m

X

l=1

Z

t

0

σ

l

(X

s

)dW

_s^l

+ Z

t

0

b(X

s

)ds (1)

+ Z

t

0

Z

E

Z

(0,∞)

c(z, X

_s−

)1

_{{u≤γ(z,Xs−)}}

N

µ

(ds, dz, du).

Here E is a measurable space, N

µ

(ds, dz, du) is a homogeneous Poisson point measure on E × (0, ∞) with intensity measure µ(dz) × 1

_(0,∞)

(u)du and the coefficients are σ

l

, b : R

^d

→ R

^d

and c : R

^d

× E → R

^d

, γ : R

^d

× E → [0, ∞). Suppose for a moment that µ is a finite measure and σ

_l

= 0, l = 1, ..., m.

Then the solution of the above equation is a Piecewise Deterministic Markov Process (P DM P in short) and existence and uniqueness of the solution are well known. But, if µ is an infinite measure and we have a non null diffusion component, this type of equations have not been considered in the literature. So our first aim is to prove that under reasonable hypothesis equation (1) has a unique solution - this is done Theorem 3. The proof is based on some non trivial L

¹

estimates (we thank to Nicolas Fournier who gave us an important hint in this direction).

∗LAMA (UMR CNRS, UPEMLV, UPEC), INRIA, F-77454 Marne-la-Vall´ee, France. Email: [email protected]

†LAMA (UMR CNRS, UPEMLV, UPEC), F-77454 Marne-la-Vall´ee, France. Email: [email protected]

The equation (1) naturally appears as the limit of sequences of P DM P

⁰

s with three different regimes, that we describe now. We consider a sequence of processes X

_tⁿ

, n ∈ N, which solve

X

_tⁿ

= x +

3

X

i=1

Z

t

0

Z

E

Z

(0,∞)

c

_(i),n

(z, X

_s−ⁿ

)1

_{{u≤γ(i),n(z,Xs−)}}

N

µ_(i),n

(ds, dz, du) (2) where N

_µ(i),n

(ds, dz, du), i = 1, 2, 3 are three independent Poisson point measures of intensities

µ

(i),n

(dz) × 1

_(0,∞)

(u)du.

Each of them represents a different regime. For i = 1 we consider a CLT type regime, described by the following hypothesis:

lim

n

Z

E

c

_(1),n

c

^∗_(1),n

(z, x) × γ

_(1),n

(z, x)dµ

_(1),n

(z) = σσ

^∗

(x) (3) where σ is the matrix with columns σ

l

, l = 1, ..., m and σ

^∗

designs the transposed matrix. Moreover we assume that

lim

n

Z

E

c

(1),n

(z, x)

3

× γ

(1),n

(z, x)dµ

(1),n

(z) = 0. (4) For i = 2 we have a Law of Large Numbers type regime: we assume that

lim

n

Z

E

c

_(2),n

(z, x) × γ

_(2),n

(z, x)dµ

_(2),n

(z) = b(x) (5) and

lim

n

Z

E

c

(2),n

(z, x)

2

× γ

(2),n

(z, x)dµ

(2),n

(z) = 0. (6) Finally, for i = 3 we have a ”finite variation” type regime: we assume that µ

_(3),n

(dz) = 1

_En

(z)dµ(z) where µ is the intensity measure which appears in the limit equation (1) and E

_n

↑ E is a sequence of measurable sets such that µ(E

_n

) < ∞ and

lim

n

Z

E\E_n

|c(z, x)|γ(z, x)dµ(z) = 0. (7) And we assume that

lim

n

Z

E

c

_(3),n

(z, x)γ

_(3),n

(z, x)dµ

_(3),n

(z) = Z

E

c(z, x)γ(z, x)dµ(z) (8) We stress that the convergence in (3),(5) and (8) has to be given in a more precise and quantitative way (see (69)) - here we just give the general direction. Some other technical hypothesis are in force.

Then we are able to prove that, for f ∈ C

_b³

(R

^d

), one has lim

n

E(f (X

_tⁿ

)) = E(f (X

t

)) and to control the speed of convergence.

The weak convergence of Markov chains to diffusion processes has been widely discussed in the literature (see e.g. [Kur71], [Kur78], [Kus84], [JS03])) but in our framework we have the following specific difficulty. In the case of standard jump type equations (that is: 1

_{{u≤γ(z,Xs−)}}

does not appear in the equation (1)) the flow x → X

t

(x) is differentiable and consequently, if f ∈ C

_b³

(R

^d

), then x → E(f (X

t

(x))) is three times differentiable as well. Using this, one proves in a straightforward way the convergence of the semigroups and moreover, obtains an estimate of the error. But, because of the indicator function, this is not true here - and so a key point in our approach is to study the regularity of x → E(f(X

t

(x))). This is done in Theorem 7 and 12.

We also mention that P DM P

⁰

s with several regimes have recently been considered in the literature

for modelling and numerically solving problems in gene networks (see [CDMR12] and [ACT

⁺

04],

[ACT

⁺

05]) and chemical networks (see [BKPR06]). However we do not enter in our paper in the specific framework on such physical phenomenons.

The paper is organized as follows: In Section 3 we prove existence and uniqueness for the solution of equation (1) and in Section 4 we prove the regularity of x → E(f (X

_t

(x))). In Section 5 we prove the convergence result and in the Appendix we give some moment inequalities used in the paper.

Acknowledgements: We are grateful to Nicolas Fournier and to Eva L¨ ocherbach for their useful suggestions and commentaries.

3 Existence and uniqueness

3.1 Notation and main result

We consider a measurable space (E, E ) and, for a σ finite measure µ on E we denote by N

_µ

the Poisson point measure on E × [0, ∞) of compensator N b

µ

(dt, dz, du) = dt × µ(dz) × du (we refer to Ikeda Watanabe [IW89] for definitions and notation concerning Poisson point measures). Moreover we consider a m dimensional Brownian motion W = (W

¹

, ..., W

^m

) which is independent of the Poisson measure N

_µ

and we look to the d dimensional stochastic equation

X

t

= x +

m

X

l=1

Z

t

0

σ

l

(X

s

)dW

_s^l

+ Z

t

0

b(X

s

)ds (9)

+ Z

t+

0

Z

E×[0,∞)

c(z, X

_s−

)1

_{{u≤γ(z,Xs−)}}

N

_µ

(ds, dz, du).

with σ

l

, b : R

^d

→ R

^d

and c : E × R

^d

→ R

^d

, γ : E × R

^d

→ [0, ∞).

Definition 1 A process (X

t

)

t≥0

is called a L

¹

solution of the equation (9) if it is adapted, c` adl` ag and, for every T > 0

sup

t≤T

E(|X

t

|) < ∞. (10)

Remark 2 We precise that X

_t

, t ≥ 0 is a c` adl` ag process if it is right continuous and has finite left hand limits almost surely. In particular X

_t

may not blow up in finite time: if τ

_R

= inf{t : |X

_t

| ≥ R}

then sup

_R

τ

_R

= ∞ (indeed, if sup

_R

τ

_R

= τ

_∞

≤ T, then X

_T∧τ∞−

= ∞).

We give now the hypothesis which are needed in order to obtain existence and uniqueness for a L

¹

solution of the above equation. We assume that there exist a constant L ∈ R

+

and some functions l

c

, l

γ

: E → R

+

such that, for every x, y ∈ R

^d

|b(x) − b(y)| +

m

X

l=1

|σ

_l

(x) − σ

_l

(y)| ≤ L |x − y| (11)

and for every x, y ∈ R

^d

and z ∈ E

|c(z, x) − c(z, y)| ≤ l

c

(z) |x − y| , |γ(z, x) − γ(z, y)| ≤ l

γ

(z) |x − y| . (12)

Moreover we assume that

C

_µ

(γ, c) := sup

x∈R^d

Z

E

(l

_γ

(z) |c(z, x)| + l

_c

(z)γ(z, x))dµ(z) < ∞. (13) For a measurable set G ⊂ E and Γ ≥ 1 we denote

λ(G) = sup

x∈R^d

Z

G

|c(z, x)| γ(z, x)dµ(z), (14) β(Γ) = sup

x∈R^d

Z

E

|c(z, x)| γ(z, x)1

_{{Γ≤γ(z,x)}}

dµ(z) (15) and we assume that

λ(E) < ∞ and lim

Γ→∞

β(Γ) = 0. (16)

Our main result is the following:

Theorem 3 Suppose that (11),(12), (13) and (16) hold. Then the equation (9) has a unique L

¹

solution.

3.2 The basic estimate

In this section we give the main estimate which allows to prove Theorem 3. We will work with some truncated versions of the equation (9) that we construct now. We consider a family of smooth functions ψ

Γ

: R

+

→ [0, Γ] such that

ψ

_Γ

(x) = x if x ≤ Γ − 1, (17)

= Γ if x ≥ Γ

and such that the derivatives of any order of ψ

Γ

are bounded, unifromely with respect to Γ. Then we construct

γ

Γ

(z, x) = ψ

Γ

(γ(z, x)). (18)

This is a smooth version of Γ ∧ γ(z, x).

For a measurable set G ⊂ E and a constant Γ > 1 we denote by X

^G,Γ

the L

¹

solution (if such a solution exists) of the equation

X

_t^G,Γ

= x +

m

X

l=1

Z

t

0

σ

_l

(X

_s^G,Γ

)dW

_s^l

+ Z

t

0

b(X

_s^G,Γ

)ds (19)

+ Z

t+

0

Z

E×[0,∞)

1

G

(z)c(z, X

_s−^G,Γ

)1

_{u≤γ

Γ(z,X_s−^G,Γ)}

N

µ

(ds, dz, du).

Remark 4 Notice that we accept the case G = E and Γ = ∞ and then X

_t^G,Γ

= X

t

the solution of the equation (9).

Remark 5 If µ(G) < ∞ and Γ < ∞ then it is easy to prove that the equation (19) has a unique L

¹

solution: indeed if T

k

, k ∈ N are the jump times of the Poisson process t → N

µ

(t, G) then, for t ∈ [T

_k−1

, T

_k

) one solves the standard diffusion equation dX

_s^G,Γ

= P

m

l=1

σ

_l

(X

_s^G,Γ

)dW

_s^l

+ b(X

_s^G,Γ

)ds and then defines X

_T^G,Γ

k

= X

_T^G

k−

+ c(Z

k

, X

_s−^G,Γ

)1

_{{Uk≤γΓ(z,X}G

s−)}

where Z

k

∼

_µ(G)¹

1

G

(z)µ(dz) and U

k

∼

1

Γ

1

_[0,Γ]

(u)du.

Lemma 6 Suppose that (11),(12),(13) and (16) hold. Let G

1

⊂ G

2

⊂ E be two measurable sets and 1 < Γ

₁

≤ Γ

₂

(the case G

₁

= G

₂

= E and Γ

₁

= Γ

₂

= ∞ is included) and let X

_t¹

:= X

_t^G1,Γ1

and X

_t²

:= X

_t^G2,Γ2

be two L

¹

solutions of the equation (19) (corresponding to G

1

, Γ

1

respectively to G

2

, Γ

2

). There exists an universal constant C such that for every T ≥ 0 one has

sup

t≤T

E(

X

_t¹

− X

_t²

) ≤ CT exp(CT (L + C

µ

(γ, c))) × (β(Γ

1

) + λ(G

2

8 G

1

)) (20) with L, C

_µ

(γ, c), β(Γ

₁

) and λ(G

₂

8 G

₁

) defined in (11), (13),(14) and (15). Moreover for every ρ > 0

P (sup

t≤T

X

_t¹

− X

_t²

≥ ρ) ≤ CT

ρ exp(CT (L + C

µ

(γ, c))) × (β (Γ

1

) + λ(G

2

8 G

1

)). (21) Proof. Step 1. We will use a cut-off procedure inspired from [BF11]. Let us introduce some notation. Let ϕ(x) = α1

_(−1,1)

(x) exp(−

_1−x¹2

) with α such that R

ϕ(x)dx = 1, and, for ε > 0 let ϕ

_ε

(x) =

¹_ε

ϕ(

^x_ε

). We also denote h

_ε

(x) = 2ε ∨ |x| and we define

φ

ε

(x) = h

ε

∗ ϕ

ε

(x), and f

ε

(z) = φ

ε

(|z|).

The basic property of f

ε

is the following: there exists an universal constant C such that for every ε > 0

∂f

ε

∂z

_i

(z)

≤ C and (22)

∂

²

f

_ε

∂z

i

∂z

j

(z)

≤ C

|z| . (23)

Proof. We have

∂f

ε

∂z

i

(z) = φ

⁰_ε

(|z|) z

i

|z| and

∂

²

f

ε

∂z

_i

∂z

_j

(z) =

φ

⁰⁰_ε

(|z|) − φ

⁰_ε

(|z|)

|z|

z

i

z

j

|z|

²

+ δ

i,j

φ

⁰_ε

(|z|)

|z| .

Since φ

⁰_ε

is bounded, (22) follows. Let us check (23). If |z| ≤ ε then φ

⁰_ε

(|z|) = φ

⁰⁰_ε

(|z|) = 0 so f

ε

(z) = 0.

If ε ≤ |z| ≤ 3ε then φ

⁰_ε

(|z|) ≤ C and φ

⁰⁰_ε

(|z|) ≤

¹_ε

so that

∂

²

f

ε

∂z

i

∂z

j

(z)

≤ C( 1 ε + 1

|z| ) ≤ C

|z| .

Finally, if x ≥ 3ε then φ

⁰_ε

(x) = 1 and φ

⁰⁰_ε

(x) = 0 so we obtain (23) for |z| ≥ 3ε as well . Step 2. We denote

∆

j

σ

t

= σ

j

(X

_t¹

) − σ

j

(X

_t²

), ∆b

t

= b(X

_t¹

) − b(X

_t²

) and (24) H

_t^Γ1,Γ2

(z, u) = 1

_G1

(z)c(z, X

_t−¹

)1

_{u≤γ

Γ1(z,X_t−¹ )}

− 1

_G2

(z)c(z, X

_t−²

)1

_{u≤Γγ

Γ2(z,X²_t−)}

. Then Z

_t

:= X

_t¹

− X

_t²

verifies the equation

Z

t

=

m

X

l=1

Z

t

0

∆

l

σ

s

dW

_s^l

+ Z

t

0

∆b

s

ds + Z

t+

0

Z

E×[0,1]

H

_s−^Γ1,Γ2

(z, u)N

µ

(ds, dz, du).

For R > 0 we define the stopping time

τ

R

= inf{t : X

_t¹

∨ X

_t²

> R}

and we notice that lim

_R→∞

τ

_R

= ∞ (see Remark 2). We denote

Z

_t^R

= Z

t∧τ_R

.

Using Itˆ o’s formula we write

f

ε

(Z

_t^R

) − f

ε

(Z

₀^R

) = M

ε

(t ∧ τ

R

) +

3

X

i=1

I

_εⁱ

(t ∧ τ

R

) (25) with

M

ε

(t) =

d

X

i=1 m

X

l=1

Z

t

0

∂f

ε

∂z

i

(Z

_s^R

)∆

ⁱ_l

σ

s

dW

_s^l

and

I

_ε¹

(t) = 1 2

d

X

i,j=1 m

X

l=1

Z

t

0

∂

²

f

ε

∂z

_i

∂z

_j

(Z

_s^R

)∆

ⁱ_l

σ

s

∆

^j_l

σ

s

ds, I

_ε²

(t) =

d

X

i=1

Z

t

0

∂f

_ε

∂z

i

(Z

_s^R

)∆

ⁱ

b

s

ds, I

_ε³

(t) =

Z

t+

0

Z

E×(0,1)

(f

ε

(Z

s−

+ H

_s−^Γ1,Γ2

(z, u)) − f

ε

(Z

s−

))dN (s, z, u).

Since

∆

ⁱ_l

σ

s

≤ L |Z

_s∧τR

| ≤ 2LR and ∂

i

f

ε

is bounded, the process M

ε

is a martingale and this gives E(M

ε

(t ∧ τ

R

)) = 0. Then (we have f

ε

(Z

₀^R

) = f

ε

(0) = ε)

E(f

ε

(Z

_t^R

)) ≤ ε +

3

X

i=1

E(I

_εⁱ

(t ∧ τ

R

)) . We will prove that

3

X

i=1

E(

I

_εⁱ

(t ∧ τ

_R

)

) ≤ t(β(Γ

₁

) + λ(G

₂

8 G

₁

)) + C(L + C

_µ

(γ, c)) Z

t

0

E(

Z

_s^R

)ds. (26) We estimate the terms in the RHS of the above inequality. Since σ

_l

is Lipschitz continuous, for s ≤ t ∧ τ

_R

we have

∆

ⁱ_l

σ

_s

≤ L

Z

_s^R

. Then, using (23) we obtain I

_ε¹

(t ∧ τ

R

)

≤ CL Z

t

0

Z

_s^R

ds and using (22) we get a similar upper bound for

I

_ε²

(t ∧ τ

R

)

. So (26) is verified for i = 1, 2.

We estimate now I

_ε³

. Since N(ds, dz, du) is a positive measure and f

ε

is Lipschitz continuous I

_ε³

(t ∧ τ

_R

)

≤ C

Z

(t∧τR)+

0

Z

E×(0,∞)

H

_s^Γ1,Γ2

(z, u)

dN(s, z, u) and then, using the isometry property

E(

I

_ε³

(t ∧ τ

R

) ) ≤ E(

Z

t∧τR

0

Z

E×(0,∞)

H

_s^Γ1,Γ2

(z, u)

dµ(z)duds) ≤ J

1

+ J

2

with

J

₁

= E(

Z

t∧τR

0

Z

E×(0,1)

H

_s^Γ1,Γ2

(z, u) − H

_s^∞,∞

(z, u)

dµ(z)duds) J

2

= E(

Z

t∧τR

0

Z

E×(0,1)

|H

_s^∞,∞

(z, u)| dµ(z)duds).

Then

J

2

≤ E(

Z

t∧τR

0

Z

G28G1

c(z, X

_s²

)

γ(z, X

_s²

)dµ(z)ds) +E(

Z

t∧τR

0

Z

G₁

(l

_γ

(z)

c(z, X

_s¹

)

+ l

_c

(z)γ(z, X

_s²

))

X

_s²

− X

_s¹

dµ(z)ds)

≤ tλ(G

₂

8 G

₁

) + C

_µ

(γ, c)E(

Z

t∧τ_R

0

Z

_s^R

ds).

And

J

1

≤ K

1

+ K

2

with

K

_i

≤ E(

Z

t∧τR

0

Z

G_i

c(z, X

_sⁱ

)

(γ(z, X

_sⁱ

) − γ

_Γi

(z, X

_sⁱ

))dµ(z)ds)

≤ E(

Z

t∧τR

0

Z

G_i

c(z, X

_sⁱ

)

γ(z, X

_sⁱ

)1

_{{Γi≤γ(z,X}i

s)}

dµ(z)ds) ≤ β(Γ

_i

)t.

So (26) is proved. In particular we obtain E(f

_ε

(Z

_t^R

))

≤ ε + t(β(Γ

₁

) + λ(G

₂

8 G

₁

)) + C(L + C

_µ

(γ, c)) Z

t

0

E(

Z

_s^R

)ds.

We have lim

_ε→0

f

ε

(z) = |z| , so, using Fatou’s lemma E(

Z

_t^R

) ≤ t(β(Γ

1

) + λ(G

2

8 G

1

)) + C(L + C

µ

(γ, c)) Z

t

0

E(

Z

_s^R

)ds.

Then, by Gronwall’s lemma E(

Z

_t^R

) ≤ Ct(β(Γ

₁

) + λ(G

₂

8 G

₁

)) exp(Ct(L + C

_µ

(γ, c))). (27) We recall that lim

_R→∞

τ

_R

= ∞ so, using again Fatou’s lemma, we pass to the limit with R → ∞ and we obtain

E(|Z

t

|) = E(lim

_R→∞

Z

_t^R

) ≤ Ct(β(Γ

1

) + λ(G

2

8 G

1

)) exp(Ct(L + C

µ

(γ, c))) so (20) is proved.

Step 3. Let us prove (21). Using (25) and the fact that f

ε

is Lipschitz continuous, we obtain E(|M

ε

(t ∧ τ

R

)|) ≤ E(

f

ε

(Z

_t^R

) − f

ε

(Z

₀^R

) ) +

3

X

i=1

E(

I

_εⁱ

(t ∧ τ

R

) )

≤ CE(

Z

_t^R

) +

3

X

i=1

E(

I

_εⁱ

(t ∧ τ

R

) )

≤ Ct(β(Γ

₁

) + λ(G

₂

8 G

₁

)) exp(Ct(L + C

_µ

(γ, c))) the last inequality being a consequence of (26) and (27).

We take now ρ > 0 and we use Doob’s inequality and Chebyshev’s inequality in order to get P (sup

t≤T

f

_ε

(Z

_t^R

)

≥ ρ) ≤ P(sup

t≤T

|M

ε

(t ∧ τ

_R

)| ≥ ρ 4 ) +

3

X

i=1

P(sup

t≤T

I

_εⁱ

(t ∧ τ

_R

) ≥ ρ

4 )

≤ C

ρ (E(|M

ε

(t ∧ τ

_R

)|) +

3

X

i=1

E(sup

t≤T

I

_εⁱ

(t ∧ τ

_R

) )

≤ Ct

ρ (β(Γ

1

) + λ(G

2

8 G

1

)) exp(Ct(L + C

µ

(γ, c)))).

Using Fatou’s lemma we pass to the limit with ε → 0 and with R → ∞ and we obtain(21).

3.3 Proof of Theorem 3

Uniqueness of the solution immediately follows from (20) with G

₁

= G

₂

= E and Γ

₁

= Γ

₂

= ∞. Let us prove existence. We take a sequence of subsets E

n

↑ E such that µ(E

n

) < ∞ and Γ

n

= n. By (16)

n,m→∞

lim (β(Γ

n

) + λ(E

n

8 E

m

)) = 0.

Since µ(E

_n

) < ∞, Γ

_n

< ∞ we may construct a solution X

_tⁿ

:= X

_t^En,Γn

and then, by (21),

n,m→∞

lim sup

t≤T

|X

_tⁿ

− X

_t^m

| = 0

in probability. Passing to a subsequence, the above convergence holds almost surely, so we may construct a process X

t

such that

n→∞

lim sup

t≤T

|X

t

− X

_tⁿ

| = 0 almost surely.

Since X

_tⁿ

are adapted and c` adl` ag processes, so is X

t

. Using (20) we conclude that for every n

sup

t≤T

E(|X

_tⁿ

|) ≤ Cλ(E

n

) (28)

and using (20) again we get

sup

t≤T

E(|X

t

|) ≤ Cλ(E). (29)

It remains to check that X

_t

verifies the equation (9) which reads

X

t

= x + M (t) + I

¹

(t) + I

²

(t) (30)

where

M (t) =

m

X

l=1

Z

t

0

σ

l

(X

s

)dW

_s^l

, I

¹

(t) = Z

t

0

b(X

s

)ds, I

²

(t) =

Z

t+

0

Z

E×[0,∞)

c(z, X

_s−

)1

_{{u≤γ(z,Xs−)}}

N

_µ

(ds, dz, du).

In a similar way we write

X

_tⁿ

= x + M

n

(t) + I

_n¹

(t) + I

_n²

(t) (31) where

M

_n

(t) =

m

X

l=1

Z

t

0

σ

_l

(X

_sⁿ

)dW

_s^l

, I

_n¹

(t) = Z

t

0

b(X

_sⁿ

)ds, I

_n²

(t) =

Z

t+

0

Z

E_n×[0,∞)

c(z, X

_s−ⁿ

)1

_{u≤γΓ

n(z,X_s−ⁿ )}

N

µ

(ds, dz, du).

Since (31) holds true and X

_tⁿ

→ X

_t

almost surely, it remains to prove that the terms in the right side of (31) converge in probability also. We have

E(

I

²

(t) − I

_n²

(t) ) ≤ E(

Z

t

0

Z

E×(0,∞)

H

_s^∞,Γn

(z, u)

dµ(z)duds)

with H

_s^∞,Γn

(z, u) defined as in (24) with G

1

= E, Γ

1

= ∞ and G

2

= E

n

, Γ

n

= n. Using (26) E(

I

²

(t) − I

_n²

(t)

) ≤ Ct(β(Γ

_n

) + λ(E

_n^c

) + Z

t

0

E(|X

s

− X

_sⁿ

|)ds).

since lim

n

E(|X

s

− X

_sⁿ

|) = 0 for every s, we use Lebesgue’s theorem (recall (28) and (29)) and we conclude that I

_n²

(t) → I

²

(t) in L

¹

. The same is true for I

_n¹

(t).

Let us now treat M

_n

(t). We denote A

_n

= {sup

_s≤t

|X

s

− X

_sⁿ

| ≤ 1} and, for ρ > 0, we write P (|M

n

(t) − M (t)| ≥ ρ) ≤ P(A

^c_n

) + P (A

_n

∩ {|M

n

(t) − M (t)| ≥ ρ}).

On A

_n

we have

M (t) − M

n

(t) =

m

X

l=1

Z

t

0

(σ

l

(X

s

) − σ

l

(X

_sⁿ

))1

_{|Xs−Xn

s|≤1}

dW

_s^l

so that

P(A

n

∩ {|M

n

(t) − M (t)| ≥ ρ}) ≤ 1 ρ

m

X

l=1

E(

Z

t

0

(σ

l

(X

s

) − σ

l

(X

_sⁿ

))1

_{

|

^Xs−Xs^En

|

≤1}

dW

_s^l

)

≤ C ρ (E(

Z

t

0

|X

s

− X

_sⁿ

|

²

1

_{|Xs−Xsⁿ_|≤1}

ds))

^1/2

≤ C ρ (E(

Z

t

0

|X

_s

− X

_sⁿ

| ds))

^1/2

→ 0.

We also have lim

n→∞

P (A

^c_n

) = 0 so that lim

n→∞

P (|M

n

(t) − M (t)| ≥ ρ) = 0.

4 Regularity of the semigroup

Our aim is to study the regularity of the semigroup P

t

f (x) := E(f (X

t

(x))) where X

t

(x) is the solution of the equation (9) which starts from X

0

= x. We have to introduce some more notation. For a function f : R

^d

→ R which is k time differentiable we denote

kf k

_k,∞

= sup

x∈R^d

X

|α|≤k

|∂

_x^α

f (x)| . (32)

For a function f : E × R

^d

→ R,which is q times differentiable with respect to a, for a set G ⊂ E and for p ≥ 1 we denote

f

_q,p

(G) = sup

x∈R^d

X

1≤|α|≤q

Z

G

|∂

_x^α

f (z, x)|

^p

γ(z, x)dµ(z) (33) Moreover, for q ∈ N and p > 1, we define (with σ, b and c the coefficients in the equation (9))

θ

q,p

(G) = 1 + kσk

^2p_q,∞

+ kbk

^2p_q,∞

+ c

q,1

(G) + c

q,2p

(G). (34) We also denote

γ(G) = inf{γ(z, x) : x ∈ R

^d

, z ∈ G}, (35)

γ(G) = sup

x∈R^d

Z

G

γ(z, x)dµ(x), (36)

e γ

q

(G) = X

1≤|α|≤q

1 µ(G) sup

x∈R^d

Z

G

|∂

_x^α

γ(z, x)| dµ(x) (37)

and

α

_q,p

(t, G) = C(t ∨ 1)( γ

_q,2q

(G)

γ

^q

(G) + θ

_q,p

(G)e

^tCθq,p(G)

(1 + γ(G) γ

^q

(G) +

q

X

j=1

e γ

_j^q−j+1

(G))) (38)

Here C is an universal constant.

In the following we will repeatedly use the following inequalities:

sup

x∈R^d

X

1≤|α|≤q

Z

G

|∂

_x^α

f (z, x)|

^p

γ

_Γ

(z, x)dµ(z) ≤ f

_q,p

(G) (39)

and

X

1≤|α|≤q

1 µ(G) sup

x∈R^d

Z

G

|∂

_x^α

γ

Γ

(z, x)| dµ(x) ≤ C e γ

q

(G). (40) The first one is a consequence of γ

Γ

≤ γ and the second one follows from the definition γ

Γ

= ψ

Γ

(γ) and the fact that ψ

Γ

has derivatives which are bounded uniformly with respect to Γ.

Theorem 7 Let q ∈ N and G ⊂ E with µ(G) < ∞. We assume that (11), (12), (13) and (16) hold, that σ ∈ C

_b^q+1

(R

^d

), b ∈ C

_b^q

(R

^d

), c(◦, z) ∈ C

_b^q

(R

^d

), γ(◦, z) ∈ C

_b^q

(R

^d

) and γ(G) > 0. Let Γ > 1 be such that

m

_G

(Γ) := 1 µ(G)

Z

G

1 ∧ γ(z, x)

Γ dµ(z) < 1 (41)

and let P

_t^G,Γ

f (x) = E(f (X

_t^G,Γ

(x)) where X

_t^G,Γ

(x) is the solution of equation (19) which starts from x. There exits some constants C

q

and l

q

, depending on q only, such that for every f ∈ C

_b^q

(R

^d

) one has

P

_t^G,Γ

f

q,∞

≤ C

q

1 − m

G

(Γ) α

q,ql_q

(t, G) kf k

_q,∞

(42) For q = 1, 2, 3 we have l

q

= q.

Remark 8 Notice that α

q,ql_q

(t, G) appears as the constant which controls the regularity of x → P

_t^G,Γ

f (x). Roughly speaking we expect that α

q,ql_q

(t, G) < ∞ if µ(G) < ∞. But in the following we will consider a sequence of sets E

n

↑ E such that µ(E

n

) < ∞ and α

q,qlq

(t, E

n

) < ∞ but α

q,qlq

(t, E

n

) ↑ ∞.

So we have regularity for the semigroup of the truncated equations but we loose control when passing with n → ∞. This is the delicate point in our approach. The rate of the blow up α

_q,qlq

(t, E

_n

) ↑ ∞ becomes critical ; see also Remark 13.

In order to prove Theorem 7 we need some preparation. Since µ(G) < ∞ we have an alternative representation of the solution X

_t^G,Γ

of the equation (19) by means of a compound Poisson process:

we consider a Poisson process J

t

with parameter µ(G)Γ < ∞ and we denote by T

k

, k ∈ N the jump times of J

t

(since G and Γ are fixed, we do not mention them in the notation). Moreover we take a sequence Z

_k

, U

_k

, k ∈ N of independent random variables (which are independent of W and of J ) with laws

P (Z

k

∈ dz) = 1

µ(G) 1

G

(z)µ(dz), P (U

k

∈ du) = 1

Γ 1

(0,Γ)

(u)du.

Then the equation (19) may be represented as X

_t^G,Γ

(x) = x +

m

X

l=1

Z

t

0

σ

_l

(X

_s^G,Γ

(x))dW

_s^l

+ Z

t

0

b(X

_s^G,Γ

(x))ds (43)

+ X

k≤Jt

c(Z

_k

, X

_T^G,Γ

k−

(x))1

_{U

k≤γΓ(Z_k,X^G,Γ

Tk−(x))}

.

We give now a second representation which does no more contain the indicator function 1

_{U

k≤γΓ(Zk,X_Tk^G,Γ₋(x))}

and so it is suitable when discussing the regularity with respect to x. We denote by Φ

_t,s

(x), 0 ≤ t ≤ s the solution of the standard diffusion equation

Φ

_t,s

(x) = x +

m

X

l=1

Z

s

t

σ

_l

(Φ

_t,r

(x))dW

_r^l

+ Z

s

t

b(Φ

_t,r

(x))dr. (44)

Notice that, since σ ∈ C

_b^q+1

(R

^d

) and b ∈ C

_b^q

(R

^d

), we may choose a version of Φ which is q times differentiable with respect to x (see [IW89]). Moreover we consider a sequence (z) := (z

k

)

k∈N

with z

_k

∈ E, we denote

z

^k

= (z

₁

, ..., z

_k

)

and we construct a process x

t

(x, (z)) in the following way: we put x

0

(x) = x and, if x

T_k−

(x, z

^k−1

) is given, we define

x

T_k

(x, z

^k

) = x

T_k−

(x, z

^k−1

) + c(x

T_k−

(x, z

^k−1

), z

k

)1

G

(z

k

) (45) x

t

(x, z

^k

) = Φ

T_k,t

(x

T_k

(x, z

^k

)) T

k

≤ t < T

k+1

.

Since Φ and c(◦, z) are q times differentiable with respect to x, so is x → x

t

(x, z

^k

).

We take now function ψ : E → R

+

such that ψ(z) = 0 for z ∈ G and R

ψdµ = 1 and we construct the probability density

q

_G,Γ

(z, x) = θ

_G

(x)ψ(z) + 1

µ(G)Γ 1

_G

(z)γ

_Γ

(z, x) with (46) θ

G,Γ

(x) = 1 − 1

µ(G)Γ Z

G

γ

Γ

(z, x))µ(dz).

By the very definition of θ

G,Γ

(x) we have R

E

q

G,Γ

(z, x)dµ(z) = 1. And since m

G

(Γ) < 1 we have θ

G,Γ

(x) ≥ 1 − m

G

(Γ) > 0.

We construct a sequence of random variables Z

k

in the following way. Z

1

has conditional law P (Z

1

∈ dz | x

T₁−

(x) = y) = q

G,Γ

(z, y)µ(dz).

Then, if Z

_i

, i ≤ k − 1 are given, we construct Z

_k

to be a random variable with conditional law P (Z

k

∈ dz | x

T_k−

(x, Z

^k−1

) = y) = q

G,Γ

(z, y)µ(dz) (47) where Z

^k

= (Z

1

, ..., Z

k

). Notice that the density of the law of Z

ⁿ

with respect to µ(dz

1

)...µ(dz

n

) is given by

p

n

(x, z

1

, ..., z

n

) =

n

Y

k=1

q

G

(x

T_k−

(x, z

1

, ..., z

_k−1

), z

k

). (48) Finally we define

X

^G,Γ_t

(x) = x

t

(x, Z

^k−1

), T

_k−1

≤ t < T

k

(49) and we notice that, according to (45)

X

^G,Γ_Tk

(x) = X

^G,Γ_Tk−

(x) + c(Z

k

, X

^G,Γ_Tk−

(x))1

G

(Z

k

) (50) X

^G,Γ_t

(x, z

^k

) = Φ

T_k,t

(X

^G,Γ_Tk

(x)) T

k

≤ t < T

k+1

.

Remark 9 In mathematical physics the above equation are known as ”transport equations” and the equation (43) is called the ”fictive chock” representation and the recurrence relation (50) is the ”real chock” representation: see [LPS98] pg 49. The above book gives a complete view of the numerical methods used in the Monte Carlo approach to such equations as well as several possible applications.

Lemma 10 The law of X

_t^G,Γ

(x) coincides with the law of X

^G,Γ_t

(x). Moreover, for any non negative and measurable function Ψ the law of S

t

= P

Jt

k=1

Ψ(Z

k

)1

_{U

k≤γΓ(Z_k,X_k^G,Γ)}

coincides with the law of S

t

= P

Jt

k=1

Ψ(Z

k

).

Proof. We have E(f (X

_T^G,Γ

j

) | X

_T^G,Γ

j−

= x)

= E(f (x + c(Z

_j

, x)1

_G

(Z

_j

))1

_{{Uj≤γΓ(Zj,x)}}

) + E(f (x)1

_{{Uj>γΓ(Zj,x)}}

)

= : I + J.

A simple computation shows that P (U

j

> γ(Z

j

, x)) = θ

G,Γ

(x) and moreover

I = 1

Γ Z

E

Z

Γ

0

f (x + c(z, x)1

G

(z))1

_{{u≤γΓ(z,x)}}

1

µ(G) duµ(dz)

= Z

E

f(x + c(z, x)1

_G

(z))γ

_Γ

(z, x) 1

Γµ(G) µ(dz) so that

E(f (X

_T^G,Γ

j

) | X

_T^G,Γ

j−

= x) = Z

E

f (x + c(z, x)1

G

(z))γ

Γ

(z, x) 1

Γµ(G) µ(dz) +θ

_G,Γ

(x)f (x) =

Z

E

f (x + c(z, x)1

_G

(z))q

_G,Γ

(z, x)µ(dz) = E(f (X

^G,Γ_T

j

) | X

^G,Γ_T

j−

= x).

We conclude that the laws of X

_t^G,Γ

coincides with the law of X

^G,Γ_t

. In order to check that the law of S

t

and of S

t

are the same, we just use the previous result for the couple (X

_t^G,Γ

, S

t

) and (X

^G,Γ_t

, S

t

).

The process X

^G,Γ_t

(x) satisfy the equation:

X

^G,Γ_t

(x) = x +

m

X

l=1

Z

t

0

σ

l

(X

^G,Γ_s

(x))dW

_s^l

+ Z

t

0

b(X

^G,Γ_s

(x))ds (51)

+

Jt

X

k=1

c(Z

k

, X

^G,Γ_Tk−

(x))1

G

(Z

k

).

Since x → x

t

(x, z

^k

)is differentiable, so is x → X

^G,Γ_t

(x). Our first aim is to estimate the derivatives of this process.

Proposition 11 A. For every q, p ∈ N there exists some constants C (depending on q and p), and l

q

(depending on q) such that, for every multi-index α with |α| = q

E(

∂

_x^α

X

^G,Γ_t

(x)

p

) ≤ Cθ

q,plq

(G)e

^tCθq,plq(G)

(52)

with θ

q,p

(G) defined in (34). One has l

q

≤ 2

^q

and, for q = 1, 2, 3, one has l

q

= q.

B. Moreover, with γ(G) defined in (36), E(

Jt

X

k=1

1

_G

(Z

_k

)

∂

_x^α

X

^G,Γ_t

(x)

p

) ≤ Ctγ(G) × θ

_q,plq

(G)e

^tCθq,plq(G)

. (53)

Proof. We treat the first derivatives. We have (with e

i

= (0, ..., 0, 1, 0, ..., 0) with 1 on the i

⁰

th position)

∂

_xi

X

^G,Γ_t

(x) = e

_i

+

m

X

l=1

Z

t

0

D ∇σ

l

(X

^G,Γ_s

(x)), ∂

_xi

X

^G,Γ_s

(x) E

dW

_s^l

(54)

+ Z

t

0

D ∇b(X

^G,Γ_s

(x)), ∂

_xi

X

^G,Γ_s

(x) E ds +

J_t

X

k=1

D ∇

x

c(X

^G,Γ_Tk−

(x), Z

k

), ∂

_xi

X

^G,Γ_Tk−

(x) E

1

G

(Z

k

).

Using the identity of laws given in Lemma 10 for the system (X

^G,Γ_t

(x), ∇

x

X

^G,Γ_t

(x))

t≥0

we conclude

that the law of this process coincides with the law of the process (X

_t^G,Γ

(x), V

(1),t

(x))

t≥0

where X

_t^G,Γ

(x)

is the solution of the equation (19) and V

_(1),tⁱ

∈ R

^d

, i = 1, ..., d solves the equation V

(1),t

(x) = e

i

+

m

X

l=1

Z

t

0

∇σ

l

(X

_s^G,Γ

(x)), V

(1),s

(x)

dW

_s^l

(55)

+ Z

t

0

∇b(X

_s^G,Γ

(x)), V

_(1),t

(x) ds +

J_t

X

k=1

D ∇

x

c(Z

k

, X

_T^G,Γ

k−

(x)), V

_(1),Tk−

(x) E

1

G

(Z

k

)1

_{U

k≤γΓ(Z_k,X^G,Γ

Tk−(x))}

.

We will use Proposition 21 from the Appendix (with E replaced by G) in order to estimate the moments of V

(1),t

(x). In order to identify notations we mention that the index set is now Λ = {1, ..., d}

and α = i. Moreover V

_(1),0ⁱ

= e

_i

and H

ⁱ

= h

ⁱ

= Q

ⁱ

= 0 so, in particular, q = 0 and R

ⁱ

= 0. Let us now identify b c

(1)

(p) which is defined in (87):

b c

₍₁₎

(p) = sup

x∈R^d

Z

G

|∇

x

c(z, x)|)(1 + |∇

x

c(z, x)|)

^2p−1

γ

Γ

(z, x)dµ(z) ≤ θ

1,p

(G).

Here the lower index in b c

₍₁₎

(p) indicates that we are dealing with the solutions of (55) which concern the first order derivatives. And we have used the inequality |∂

_x^α

γ

Γ

(z, x)| ≤ C |∂

_x^α

γ(z, x)| .

Then, using the identity of law and (88) we obtain E(

∂

xⁱ

X

^G,Γ_t

(x)

2p

) = E(

V

_(1),tⁱ

(x)

2p

) ≤ exp(tC

p

θ

1,p

(G)) so (52) is proved. And

E(

Jt

X

k=1

1

_G

(Z

_k

)

∂

_xi

X

^G,Γ_T

k−

(x)

p

) = E(

Jt

X

k=1

1

_G

(Z

_k

) V

_(1),Tⁱ

k−

(x)

p

1

_{U

k≤γΓ(Z_k,X^G,Γ

Tk−(x))}

)

= E(

Z

t

0

Z

G

V

_(1),Tⁱ

k−

(x)

p

γ

Γ

(z, X

_T^G,Γ

k−

(x))µ(dz)

≤ sup

x∈R^d

Z

G

γ(z, x)dµ(z) Z

t

0

E(

V

_(1),sⁱ

(x)

2p

)ds so (53) is also proved (with l

₁

= 1).

We estimate now the second order derivatives. We take derivatives in (54) and we obtain

∂

_xj

∂

_xi

X

^G,Γ_t

(x) =

m

X

l=1

Z

t

0

H

^i,j_l

(s)dW

_s^l

+ Z

t

0

h

^i,j

(s)ds +

Jt

X

k=1

Q

^i,j

(T

_k

−, Z

_k

)1

_G

(Z

_k

) (56)

+

m

X

l=1

Z

t

0

D ∇σ

l

(X

^G,Γ_s

(x)), ∂

_xj

∂

_xi

X

^G,Γ_s

(x) E dW

_s^l

+ Z

t

0

D ∇b(X

^G,Γ_s

(x)), ∂

_xj

∂

_xi

X

^G,Γ_s

(x) E ds

+

J_t^G

X

k=1

D ∇

x

c(Z

k

, X

^G,Γ_Tk−

(x)), ∂

_xj

∂

_xi

X

^G,Γ_Tk−

(x) E

1

G

(Z

k

).

with

H

^i,j_l

(s) =

d

X

r,r⁰=1

∂

r

∂

r⁰

σ

l

(X

^G_s

(x))∂

_xi

X

^G,r_s

(x)∂

_xj

X

^G,r

0

s

(x), h

^i,j

(s) =

d

X

r,r⁰=1

∂

r

∂

r⁰

b(X

^G_s

(x))∂

_xi

X

^G,r_s

(x)∂

_xj

X

^G,r

0

s

(x),

and

Q

^i,j

(s, Z

k

) =

d

X

r,r⁰=1

∂

x^r

∂

_xr0

c(Z

k

, X

^G,Γ_s

(x))∂

xⁱ

X

^G,Γ,r_s

(x)∂

x^j

X

^G,Γ,r

0

s

(x).

Using the identity of laws given in Lemma 10 for the system (X

^G,Γ_t

(x), ∇

x

X

^G,Γ_t

(x), ∇

²_x

X

^G,Γ_t

(x))

_t≥0

we conclude that the law of this process coincides with the law of the process (X

_t^G,Γ

(x), V

_(1),t

(x), V

_(2),t

(x))

_t≥0

where X

_t^G,Γ

(x) is the solution of the equation (19), V

_(1),tⁱ

∈ R

^d

, i = 1, ..., d solves the equation (55) and V

_(2),t^i,j

(x) ∈ R

^d

, i, j = 1, ..., d solves the following equation:

V

_(2),t^i,j

(x) =

m

X

l=1

Z

t

0

H

_l^i,j

(s)dW

_s^l

+ Z

t

0

h

^i,j

(s)ds (57)

+

J_t^G

X

k=1

Q

^i,j

(T

k

−, Z

k

)1

G

(Z

k

)1

_{{Uk≤γΓ(Zk,X}G Tk−(x),)}

+

m

X

l=1

Z

t

0

D ∇σ

l

(X

_s^G,Γ

(x)), V

_(2),s^i,j

E dW

_s^l

+

Z

t

0

D ∇b(X

_s^G,Γ

(x)), V

_(2),s^i,j

E ds

+

J_t^G

X

k=1

D ∇

x

c(Z

k

, X

_T^G,Γ

k−

(x)), V

_(2),T^i,j

k−

E

1

G

(Z

k

)1

_{U

k≤γΓ(Z_k,X_Tk^G,Γ₋(x))}

with

H

_l^i,j

(s) =

d

X

r,r⁰=1

∂

r

∂

r⁰

σ

l

(X

_s^G,Γ

(x))(V

_(1),sⁱ

)

^r

(x)(V

_(1),s^j

)

^r0

(x),

h

^i,j

(s) =

d

X

r,r⁰=1

∂

_r

∂

_r⁰

b(X

_s^G,Γ

(x))(V

_(1),sⁱ

)

^r

(x)(V

_(1),s^j

)

^r0

(x), and

Q

^i,j

(s, a

k

) =

d

X

r,r⁰=1

∂

x^r

∂

_xr0

c(Z

k

, X

_s^G,Γ

(x))(V

_(1),sⁱ

)

^r

(x)(V

_(1),s^j

)

^r0

(x).

We will again use Proposition 21 (with E = G) in order to estimate the moments of V

_(2),t

(x). Now the index set is Λ = {(i, j); i, j = 1, ..., d} and α = (i, j). Moreover V

_(2),0^i,j

= 0 and H

^i,j

, h

^i,j

, Q

^i,j

are given above. In particular we have

Q

^i,j

(s, Z

_k

)

≤ q(Z

_k

, X

_s^G

)R

^i,j

(s) with q(z, x) = P

|α|=2

|∂

_x^α

c(z, x)|

and R

_s^i,j

= V

(1),s

2

. So

b c

₍₂₎

(p) = sup

x∈R^d

Z

G

( X

1≤|α|≤2

|∂

_x^α

c(z, x)|)(1 + X

1≤|α|≤2

|∂

_x^α

c(z, x)|)

^2p−1

γ

Γ

(z, x)dµ(z)

≤ θ

2,p

(G)

Moreover, using the estimates for V

_(1),t

we obtain Z

t

0

(E(

m

X

l=1

|H

_l^α

(s)|

^2p

+ |h

^α_l

(s)|

^2p

+ b c

₍₂₎

(p) R

^α_s−

2p

)ds) (58)

≤ Z

t

0

C

p

(kσk

^2p_2,∞

+ kbk

^2p_2,∞

+ θ

2,p

(G))E(

V

(1),s

4p

)ds

≤ tC

_p

(kσk

^2p_2,∞

+ kbk

^2p_2,∞

+ θ

_2,p

(G)) exp(tC

_p

θ

_1,2p

(G)).

Then

E(

∂

_xj

∂

_xi

X

^G,Γ_t

(x)

2p

) = E(

V

_t^i,j

2p

) ≤ tC

_p

θ

_2,p

(G) exp(tC

_p

θ

_2,2p

(G)).

So the proof of (52) is finished and then (53) follows as above. Notice that l

2

= 2 here.

For the third order derivatives the proof is similar: now the set of multi-indexes is Λ = {α = (i, j, k) : 1 ≤ i, j, k ≤ d} and H

^α

, h

^α

, Q

^α

are defined in a similar way. Moreover one has |H

^α

| + |h

^α

| ≤ C(kσk

_3,∞

+ kbk

_3,∞

)(

V

₍₁₎

3

+ V

₍₂₎

V

₍₁₎

) and |Q

^α

| ≤ Cc

_[3]

( V

₍₁₎

3

+ V

₍₂₎

V

₍₁₎

). Using Proposition 21, H¨ older’s inequality and the recurrence hypothesis one obtains (52) with l

₃

= 3. For higher order derivatives the proof is the same, but it is more difficult to give a precise expression for l

_q

- this is why we keep the bound l

_q

≤ 2

^q

which is clearly sufficient.

We are now ready to give:

Proof of Theorem 7. Recall (45) and (48). Recall also the notation z

^k

= (z

1

, ..., z

k

), and recall that J

t

represents the number of jumps up to t. Then we write

E(f (X

^G,Γ_t

(x))) = E(

Z

f (x

t

(x, z

^Jt

))p

J_t

(x, z

^Jt

)µ(dz

1

), ..., µ(dz

J_t

)) where

p

_Jt

(x, z

^Jt

) =

Jt

Y

k=1

q

_G,Γ

(x

_Tk−

(x, z

^k−1

), z

_k

).

It follows that

∂

x_i

E(f (X

^G,Γ_t

(x))) = A + B with

A =

d

X

l=1

E(

Z

∂

_l

f (x

_t

(x, z

^Jt

))∂

_xi

x

^l_t

(x, z

^Jt

)p

_Jt

(x, z

^Jt

)µ(dz

₁

), ..., µ(dz

_Jt

))

=

d

X

l=1

E(∂

_l

f (x

_t

(x, Z

^Jt

))∂

_xi

x

^l_t

(x, Z

^Jt

)) and

B = E(

Z

f (x

_t

(x, z

^Jt

))∂

_xi

p

_Jt

(x, z

^Jt

)µ(dz

₁

), ..., µ(dz

_Jt

))

= E(

Z

f (x

t

(x, z

^Jt

))∂

xi

ln p

Jt

(x, z

^Jt

) × p

Jt

(x, z

^Jt

)µ(dz

1

), ..., µ(dz

Jt

))

= E(f (x

t

(x, Z

^Jt

))∂

x_i

ln p

J_t

(x, Z

^Jt

)).

Let us estimate A. Using (52) (with q = 1, l

q

= 1 and p = 1)

|A| ≤ kf k

_1,∞

d

X

l=1

E(

∂

x_i

x

^l_t

(x, Z

^Jt

)

) = kf k

_1,∞

E(

∇

x

X

^G,Γ_t

(x) )

≤ C kf k

_1,∞

θ

1,1

(G)e

^tCθ1,1(G)

. Let us estimate B. We have

∂

_xi

ln p

_Jt

(x, z

^Jt

) =

Jt

X

k=1

ψ(z

_k

)∂

_xi

ln θ

_G,Γ

(x

_Tk−

(x, z

^k−1

))

+

Jt

X

k=1

1

_G

(z

_k

)∂

_xi

ln γ

_Γ

(z

_k

, x

_Tk−

(x, z

^k−1

)

= : S

1

(x, z

^Jt

) + S

2

(x, z

^Jt

).

Then

|B| ≤ kf k

_∞

(E(

S

1

(x, Z

^Jt

) ) + E(

S

2

(x, Z

^Jt

)

)