Approximation of Markov semigroups in total variation distance

HAL Id: hal-01110015

https://hal-upec-upem.archives-ouvertes.fr/hal-01110015

Submitted on 27 Jan 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Approximation of Markov semigroups in total variation distance

Vlad Bally, Clément Rey

To cite this version:

Vlad Bally, Clément Rey. Approximation of Markov semigroups in total variation distance. Electronic

Journal of Probability, Institute of Mathematical Statistics (IMS), 2016, 21 (12). �hal-01110015�

Approximation of Markov semigroups in total variation distance

Vlad Bally ¹ Clément Rey ²

Abstract

The first goal of this paper is to prove that, regularization properties of a Markov semigroup enable to prove convergence in total variation distance for approximation schemes for the semigroup. Moreover, using an interpolation argument we obtain estimates for the error in distribution sense (at the level of the densities of the semigroup with respect to the Lebesgue measure). In a second step, we build an abstract Malliavin calculus based on a splitting procedure, which turns out to be the suited instrument in order to prove the above mentioned regularization properties. Finally, we use these results in order to estimate the error in total variation distance for the Ninomiya Victoir scheme (which is an approximation scheme, of order 2, for diffusion processes).

1 Introduction

In this paper we study the total variation distance between two discrete time Markov semi- groups and we give applications for the speed of convergence of approximation schemes. In order to do it we use an abstract Malliavin type calculus based on a splitting procedure which enables us to prove regularization properties of the semigroup - and it turns out that such regularization properties are crucial in order to be able to deal with measurable test functions.

Moreover, we take a step further and we give estimates for the distance between the density function of the Markov semigroup and the density function of the approximation scheme. At this level we have to use an interpolation argument which has been recently obtained in [8].

Let us be more specific and describe the different steps of our approach. We consider the d dimensional Markov chain

X

_k+1ⁿ

= ψ

_k

(X

_kⁿ

, Z

_k+1

√ n ), k ∈ N , (1)

where ψ

_k

: R

^d

× R

^N

→ R

^d

is a smooth function such that ψ

_k

(x, 0) = x and Z

_k

∈ R

^N

, k ∈ N is a sequence of independent random variables. The semigroup of the Markov chain X

_kⁿ

is denoted

1

[email protected], Université de Marne-la-Vallée Cité Descartes 5, boulevard Descartes Champs sur Marne 77454 Marne la Vallée Cedex 2 France - France.

Projet MathRisk ENPC-INRIA-UMLV.

2

[email protected], CERMICS-École des Ponts Paris Tech, 6 et 8 avenue Blaise Pascal, Cité Descartes - Champs sur Marne, 77455 Marne la Vallée Cedex 2 - France.

Projet MathRisk ENPC-INRIA-UMLV, This research benefited from the support of the “Chaire Risques Fi-

nanciers”, Fondation du Risque.

1 INTRODUCTION 2 by P

_kⁿ

and the transition probabilities are µ

ⁿ_k

(x, dy) = P (X

_k+1ⁿ

(x) ∈ dy|X

_kⁿ

= x). Moreover we consider a Markov process in continuous time (X

t

)

_t>0

with semigroup P

t

and we denote ν

_kⁿ

(x, dy) = P (X

_tk

∈ dy|X

_tk

= x) where t

_k

= kδ =

_n^k

with δ =

¹_n

. A first standard result is the following: let us assume that there exists h > 0, p ∈ N such that for every f ∈ C

^p

( R

^d

), k ∈ N and x ∈ R

^d

,

µ

ⁿ_k

f(x) − ν

_kⁿ

f (x) =

R f (y)µ

ⁿ_k

(x, dy) − R

f (y)ν

_kⁿ

(x, dy)

6 Ckf k

p,∞

δ

^1+h

(2) where kfk

_p,∞

denotes the supremum norm of f and of its derivatives up to order p. Then, for every T > 0,

sup

tk6T

kP

_tk

f − P

_kⁿ

f k

∞

6 Ckf k

p,∞

δ

^h

. (3) It means that (X

_kⁿ

)

k∈N

is an approximation scheme of weak order h for the Markov process (X

t

)

_t>0

. In the case of the Euler scheme for diffusion processes, this result, with h = 1, has initially been proved in the seminal papers of Milstein [26] and of Talay and Tubaro [32] (see also [17]). Similar results were obtained in various situations: diffusion processes with jumps (see [31], [15]) or diffusion processes with boundary conditions (see [12], [11], [13]). See [16]

for an overview of this subject. More recently, approximation schemes of higher orders (e.g., h = 2), based on a cubature method, have been introduced and studied by Kusuoka [21], Lyons [25], Ninomiya, Victoir [27], Alfonsi [1], Kohatsu-Higa and Tankov [18].

Another result concerns convergence in total variation distance: we want to obtain (3) with kfk

p,∞

replaced by kf k

∞

when f is a measurable function. In the case of the Euler scheme for diffusion processes, a first result of this type has been obtained by Bally and Talay [6], [7] using the Malliavin calculus (see also Guyon [14]). Afterwards Konakov, Menozzi and Molchanov [19], [20] obtained similar results using a parametrix method. Recently Kusuoka [22] obtained estimates of the error in total variation distance for the Victoir Ninomiya scheme (which corresponds to the case h = 2). We will obtain a similar result using our approach. Moreover, we give estimates of the rate of convergence of the density function and its derivatives.

Regularization properties. We first remark that the crucial property which is used in order to replace kf k

p,∞

by kf k

∞

in (3), is the regularization property of the semigroup. Let us be more precise: let η > 0, p ∈ N be fixed. Given the time grid t

_k

= kδ, we say that a semigroup (P

_k

)

k∈N

satisfies R

_p,η

, if

R

_p,η

kP

_k

fk

p,∞

6 C

t

^ηp_k

kf k

∞

. (4)

We also introduce a dual regularization property: we consider the dual semigroup P

_k^∗

(i.e.

hP

_k^∗

g, f i = hg, P

_k

f i with the scalar product in L

²

( R

^d

)) and we assume that R

^∗_p,η

kP

_k^∗

f k

_p,1

6 C

t

^ηp_k

kf k

₁

, (5)

where kf k

_p,1

denotes the L

¹

norm of f and of its derivatives up to order p. Finally, we consider the following stronger regularization property: for every multi-index α, β with |α| + |β| = p,

R

p,η

k∂

^α

P

k

∂

^β

fk

∞

6 C

t

^ηp_k

kf k

∞

. (6)

One notices that R

_p,η

implies both R

_p,η

and R

^∗_p,η

and that a semigroup satisfying R

_p,η

is absolutely continuous with respect to the Lebesgue measure.

In addition to (2), we will also suppose that the following dual estimate of the error in short time holds:

| hg, (µ

ⁿ_k

− ν

_kⁿ

)f i | 6 Ckgk

p,1

kfk

∞

δ

^1+h

. (7) Keeping these properties in mind, we can state the following result.

Theorem 1.1. We fix T, h > 0, p ∈ N and we assume that the short time estimates (2) and (7) hold (with this p and h). Moreover, we assume that (4) holds for (P

_tk

)

_k∈N

and that (5) holds for (P

_kⁿ

)

k∈N

. Then,

∀0 < S 6 T, sup

S6tk6T

kP

t_k

f − P

_kⁿ

f k

_∞

6 C

S

^ηp

kfk

∞

δ

^h

. (8) Integration by parts formulae. Once we have this abstract result, the following step is to give sufficient conditions in order to obtain R

_p,η

, R

^∗_p,η

and R

_p,η

. We will use Malliavin type integration by parts formulae based on the noise Z

_k

∈ R

^N

. In order to do it, we assume that the law of each Z

_k

is locally lower bounded by the Lebesgue measure: there exists some z

∗,k

∈ R

^N

and r

∗

, ε

∗

> 0 such that for every measurable set A ⊂ B

_r∗

(z

∗,k

) one has

P (Z

_k

∈ A) > ε

_∗

λ(A) (9) where λ is the Lebesgue measure. If this property holds then a "splitting method" can be used in order to represent Z

k

as

Z

_k

√ n = χ

_k

U

_k

+ (1 − χ

_k

)V

_k

,

where χ

_k

, U

_k

, V

_k

are independent random variables, χ

_k

is a Bernoulli random variable and

√ nU

_k

∼ ϕ

_r∗

(u)du with ϕ

_r∗

∈ C

^∞

( R

^N

). Then we use the abstract Malliavin calculus based on U

_k

, developed in [5] and [3], in order to obtain integration by parts formulae. The crucial point is that the density ϕ

_r∗

of √

nU

_k

is smooth and we control its logarithmic derivatives. Using this, we construct integration by parts formulae and obtain relevant estimates for the weights which appear in these formulae. It is worth mentioning that, a variant of the Malliavin calculus based on a similar splitting method has already been used by Nourdin and Poly [29] (see also [28] and [23]). They use the so called Γ calculus introduced by Backry, Gentil and Ledoux [2]. Roughly speaking the difference between the approach in our paper and the one in [2] is the following: our construction is similar to the "simple functionals" approach in Malliavin calculus and has the derivative operator as basic object. In contrast, in the Γ calculus, the basic object is the Ornstein Uhlenbeck operator.

In order to state the main result of our paper, we introduce some additional assumptions:

1 INTRODUCTION 4

∀p ∈ N , sup

k∈N

E [|Z

_k

|

^p

] < ∞, (10)

∀r ∈ N

^∗

, sup

k∈N

X

16|β|6r

X

06|α|6r−|β|

k∂

_x^α

∂

_z^β

ψ

_k

k

∞

< ∞, (11)

∃λ

∗

> 0, ∀k ∈ N , inf

x∈R^d

inf

|ζ|=1 N

X

i=1

h∂

_zi

ψ

_k

(x, 0), ζi

²

> λ

∗

. (12) Moreover, we introduce the following regularized version of the approximation scheme X

_kⁿ

:

X

_k^n,θ

(x) = 1

n

^θ

G + X

_kⁿ

(x), (13)

with G a standard normal random variable independent from X

_kⁿ

and θ > h + 1. Here X

_kⁿ

(x) is the Markov chain which starts from x: X

₀ⁿ

(x) = x. We denote

P

^n,θ

(x, dy) = P (X

_k^n,θ

(x) ∈ dy) = p

^n,θ_k

(x, y)dy. (14) Theorem 1.2. Consider a Markov semigroup (P

_t

)

_t>0

and the approximation Markov chain (P

_kⁿ

)

_k∈N

defined in (1). We fix T, h > 0, p ∈ N and we assume that the short time estimates (2) and (7) hold (with this p and h). Moreover, we assume (9), (10), (11) and (12).

A. For every 0 < S 6 T , we have sup

S6tk6T

kP

t_k

f − P

_kⁿ

f k

∞

6 C

S

^p

kf k

∞

δ

^h

. (15) B. For every t > 0, P

_t

(x, dy) = p

_t

(x, y)dy with (x, y) → p

_t

(x, y) belonging to C

^∞

( R

^d

× R

^d

).

C. For every R, ε > 0 and every multi-index α, β, we have sup

S6tk6T

sup

|x|+|y|6R

|∂

_x^α

∂

_y^β

p

_tk

(x, y) − ∂

_x^α

∂

_y^β

p

^n,θ_k

(x, y)| 6 C

_ε

δ

^h(1−ε)

. (16) We notice that (15) gives the total variation distance between the semigroups (P

_t

)

_t>0

and (P

_kⁿ

)

k∈N

. Once the appropriate regularization properties are obtained (using the abstract Malliavin calculus), the proof of (15) is rather elementary. In contrast, the estimate (16) is based on a non trivial interpolation result recently obtained in [8]. Notice, however, that the estimate (16) is sub-optimal (because of ε > 0). We will illustrate (15) by taking X

_kⁿ

to be the Ninomiya Victoir scheme of a diffusion process. This is a variant of the result already ob- tained by Kusuoka [22] in the case where Z

_k

has a Gaussian distribution (and so the standard Malliavin calculus is available). Since in our paper Z

k

has an arbitrary distribution (except the property (9)), our result may be seen as an invariance principle as well.

The paper is organized as follows. In Section 2, we prove Theorem 1.1. In Section 3, we settle the abstract Malliavin calculus based on the splitting method and we use it in order to prove the regularization properties for the approximation scheme X

_kⁿ

(in fact for the regularization X

_k^n,θ

) and we prove Theorem 1.2. Finally, in Section 4, we use the previous results in order to give estimates of the total variation distance for the Ninomiya Victoir approximation scheme.

In the Appendix, we prove some technical estimates concerning the Sobolev norms of X

_kⁿ

.

2 The distance between two Markov semigroups

Throughout this section the following notations will prevail. We fix T > 0 the horizon of the underlying processes and we denote n ∈ N

^∗

, the number of time step between 0 and T . Then, we set δ := δ

_n

=

_n¹

and introduce the time homogeneous time grid t

_k

= kT δ = kT /n. Notice that, all the results from this paper remains true with non homogeneous time step but, for sake of simplicity, we will not consider this case. First, we state some results for smooth test functions.

2.1 Regular test functions

We consider a sequence of finite transition measures µ

_k

(x, dy), k ∈ N from R

^d

to R

^d

. This means that for each fixed x and k, µ

_k

(x, dy) is a finite measure on R

^d

with the borelian σ field and for each bounded measurable function f : R

^d

→ R , the application

x 7→ µ

_k

f (x) :=

Z

R^d

f(y)µ

_k

(x, dy) (17)

is measurable. We also denote

|µ

_k

| := sup

x∈R^d

sup

kfk∞61

Z

R^d

f(y)µ

_k

(x, dy)

, (18) and, we assume that all the sequences of measures we consider in this paper satisfies:

sup

k∈N

|µ

_k

| < ∞. (19)

Although the main application concerns the case where µ

_k

(x, dy) is a probability measure, we do not assume this here: we allow µ

_k

(x, dy) to be a signed measure of finite (but arbitrary) total mass. This is because one may use the results from this section not only in order to estimate the distance between two semigroups but also in order to obtain a development of the error. To the sequence µ

_k

, k ∈ N we associate the discrete semigroup

P

₀

f(x) = f(x), P

_k+1

f (x) = µ

_k+1

P

_k

f (x) = Z

R^d

P

_k

f (y)µ

_k+1

(x, dy).

More generally, for r > k we define P

_k,r

f by

P

_k,k

f(x) = f(x), P

_k,r+1

f(x) = µ

_r+1

P

_k,r

f(x).

For f ∈ C

^∞

( R

^d

) and for a multi-index α = (α

₁

, · · · , α

_d

) ∈ N

^d

we denote |α| = α

₁

+ ... + α

_d

and ∂

_α

f = ∂

_x^α₁¹

...∂

_x^αd

d

f (x). We include the multi-index α = (0, ..., 0) and in this case ∂

_α

f = f.

We use the norms

kf k

p,∞

= sup

x∈R^d

X

06|α|6p

|∂

_α

f (x)|, kfk

_p,1

= X

06|α|6p

Z

R^d

|∂

_α

f (x)|dx.

In particular kfk

0,∞

= kf k

∞

is the usual supremum norm.

2 THE DISTANCE BETWEEN TWO MARKOV SEMIGROUPS 6 We will consider the following hypothesis: let p ∈ N and 0 6 k 6 r. If f ∈ C

^p

( R

^d

) then P

k,r

f ∈ C

^p

( R

^d

) and

sup

06tk6tr

kP

_k,r

fk

p,∞

6 Ckf k

p,∞

. (20) We consider now a second sequence of finite transition measures ν

_k

(x, dy), k ∈ N and the corresponding semigroup Q

k

defined as above. Our aim is to estimate the distance between P

_k

f and Q

_k

f in terms of the distance between the transition measures µ

_k

(x, dy) and ν

_k

(x, dy), so we denote

∆

k

= µ

k

− ν

k

.

P

_k

can be seen as a semigroup in continuous time considered on the time grid t

_k

, k ∈ N , while Q

k

would be its approximation discrete semigroup. Let p ∈ N , h > 0 be fixed. We introduce a short time error approximation assumption: there exists a constant C > 0 (depending on p only) such that for every k ∈ N , we have

E

_n

(h, p) k∆

_k

f k

∞

6 Ckf k

p,∞

δ

^h+1

. (21) Proposition 2.1. Let p ∈ N be fixed. Suppose that µ

k

and ν

k

satisfy (20) and (21) with p = 0. Then for every f ∈ C

^p

( R

^d

),

sup

tm6T

kP

m

f − Q

m

f k

∞

6 Ckfk

p,∞

δ

^h

. (22) Proof. We have

kP

_m

f − Q

_m

f k

∞

6

m−1

X

k=0

kP

_k+1,m

P

_k,k+1

Q

_k

f − P

_k+1,m

Q

_k,k+1

Q

_k

f k

∞

(23)

=

m−1

X

k=0

kP

k+1,m

∆

k+1

Q

k

f k

∞

. Using (19) and (21), we obtain

kP

_k+1,m

∆

_k+1

Q

_k

f k

_∞

6 Ck∆

_k+1

Q

_k

fk

_∞

6 Cδ

^1+h

kQ

_k

fk

_p,∞

6 Cδ

^1+h

kf k

_p,∞

. Summing over k = 0, ..., m − 1, we conclude.

2.2 Measurable test functions (convergence in total variation dis- tance)

The estimate (22) requires a lot of regularity for the test function f. Our aim is to show that, if the semigroups at work have a regularization property, then we may obtain estimates of the error for measurable test functions. In order to state this result we have to give some hypothesis on the adjoint semigroup. Let p ∈ N . We assume that there exists a constant C > 1 such that for every measurable function f and any g ∈ C

^p

( R

^d

)

E

_n^∗

(h, p) | hg, ∆

_k

fi | 6 Ckgk

_p,1

kfk

∞

δ

^1+h

. (24)

where hg, f i = R

g(x)f (x)dx is the scalar product in L

²

( R

^d

).

Our regularization hypothesis is the following. Let p ∈ N , S > 0 and η > 0 be given. We assume that there exists a constant C > 1 such that

R

p,η

(S) kP

k,r

f k

p,∞

6 C

S

^ηp

kfk

∞

for S 6 t

r

− t

k

. (25) We also consider the "adjoint regularization hypothesis". We assume that there exists an adjoint semigroup P

_k,r^∗

, that is

P

_k,r^∗

g, f

= hg, P

_k,r

f i

for every bounded measurable function f and every function g ∈ C

_c^∞

( R

^d

). We assume that P

_k,r^∗

satisfies (20) and moreover

R

^∗_p,η

(S) kP

_k,r^∗

fk

_p,1

6 C

S

^ηp

kfk

₁

for S 6 t

_r

− t

_k

. (26) Notice that a sufficient condition in order that R

^∗_p,η

(S) holds is the following: for every multi index α with |α| 6 p

kP

_k,r

∂

_α

fk

∞

6 C

S

^ηp

kf k

∞

for S 6 t

_r

− t

_k

. (27) Indeed:

k∂

_α

P

_k,r^∗

f k

₁

6 sup

kgk∞61

|

∂

_α

P

_k,r^∗

f, g

| = sup

kgk∞61

| hf, P

_k,r

(∂

_α

g)i | 6 kfk

₁

sup

kgk∞61

kP

_k,r

(∂

_α

g)k

∞

6 C S

^ηp

kf k

₁

.

Proposition 2.2. Let p ∈ N , η > 0, h > 0 and 0 < S 6 T /2 be fixed. We suppose that (20), (21) and (24) hold for P

_m

and Q

_m

. We also suppose that P satisfies R

_p,η

(S) (see (25)) and Q satisfies R

^∗_p,η

(S) (see (26)). Then,

sup

2S6tm6T

kP

_m

f − Q

_m

f k

∞

6 C

S

^ηp

kfk

∞

δ

^h

. (28) Proof. Using a density argument we may assume that f ∈ C

^p

( R

^d

). Moreover, by (23), it is sufficient to prove that

kQ

k+1,m

∆

k+1

P

k

f k

∞

6 C

S

^ηp

kf k

∞

δ

^1+h

. (29) Since t

_m

> 2S we have t

_k

> S or t

_m

− t

_k+1

> S. Suppose first that t

_k

> S. Using (19) for Q, (21) and (25) for P ,

kQ

_k+1,m

∆

_k+1

P

_k

fk

∞

6 Ck∆

_k+1

P

_k

f k

∞

6 CkP

_k

fk

p,∞

δ

^1+h

6 CS

^−ηp

kfk

∞

δ

^1+h

.

2 THE DISTANCE BETWEEN TWO MARKOV SEMIGROUPS 8 Suppose now that t

_m

− t

_k+1

> S. We take φ

_ε

(x) = ε

^−d

φ(ε

⁻¹

x) with φ ∈ C

_c

( R

^d

), φ > 0.

Then, for a fixed x

0

, we define φ

ε,x0

(x) = φ

ε

(x − x

0

). By (20) Q

k+1,m

∆

k

P

k

f ∈ C

^p

( R

^d

) so it is continuous. Then

|Q

k+1,m

∆

k

P

k

f (x

0

)| = lim

ε→0

| hφ

ε,x0

, Q

k+1,m

∆

k

P

k

fi |.

Using (24), (26) and then (20), we obtain

| hφ

_ε,x0

, Q

_k+1,m

∆

_k+1

P

_k

f i | = |

Q

^∗_k+1,m

φ

_ε,x0

, ∆

_k+1

P

_k

f

| 6 CkQ

^∗_k+1,m

φ

_ε,x0

k

_p,1

kP

_k

fk

∞

δ

^1+h

6 CS

^−ηp

kφ

_ε,x0

k

₁

kf k

∞

δ

^1+h

and since kφ

_ε,x0

k

₁

= kφk

₁

6 C, the proof is completed.

In concrete applications the following slightly more general variant of the above proposition will be useful.

Proposition 2.3. Let p ∈ N , η > 0, h > 0 and 0 < S 6 T /2 be fixed. We assume that (20), (21) and (24) hold for P and Q with these p, η, h and S. Moreover, we assume that there exists some kernels P

_k,r

which satisfies R

_p,η

(S) (see(25)) and Q

_k,r

which satisfies R

^∗_p,η

(S) (see (26)). We also assume that for every 0 6 k 6 r, with t

_r

− t

_k

> S

kQ

k,r

f − Q

_k,r

fk

∞

+ kP

k,r

f − P

k,r

f k

∞

6 CS

^−ηp

δ

^h+1

kf k

∞

. (30) Then,

sup

2S6tm6T

kP

_m

f − Q

_m

fk

∞

6 C sup

k6n

(|µ

_k

| + |ν

_k

|)S

^−ηp

δ

^h

kfk

∞

. (31) Remark 2.1. Notice that P

_k,r

and Q

_k,r

are not supposed to satisfy the semigroup property and are not directly related to µ

_k

and ν

_k

.

Proof. The proof follows the same line as the one of the previous proposition. Suppose first that t

_k

> S. Then, (19) implies

kQ

_k,m

∆

_k+1

P

_k

f k

∞

6 kQ

_k,m

∆

_k+1

P

_k

f k

∞

+ kQ

_k,m

∆

_k+1

(P

_k

− P

_k

)fk

∞

6 k∆

k+1

P

k

fk

∞

+ k∆

k+1

(P

k

− P

k

)f k

∞

. Since P

_k

verifies R

_p,η

(S), we deduce from (21) that

k∆

_k+1

P

_k

f k

∞

6 Cδ

^1+h

kP

_k

fk

p,∞

6 CS

^−pη

kf k

∞

δ

^1+h

. Using (30), it follows

|∆

_k+1

(P

_k

− P

_k

)f (x)| 6 | Z

(P

_k

− P

_k

)f (y)ν

_k+1

(x, dy)| + | Z

(P

_k

− P

_k

)f(y)µ

_k+1

(x, dy)|

6 (|ν

k+1

| + |µ

k+1

|)k(P

k

− P

k

)fk

∞

6 C(|ν

_k+1

| + |µ

_k+1

|)S

^−ηp

kf k

∞

δ

^h+1

. Suppose now that t

_m

− t

_k+1

> S. We write

kQ

_k+1,m

∆

_k+1

P

_k

f k

_∞

6 kQ

_k+1,m

∆

_k+1

P

_k

f k

_∞

+ k(Q

_k+1,m

− Q

_k+1,m

)∆

_k+1

P

_k

f k

_∞

.

In order to bound kQ

_k+1,m

∆

_k+1

P

_k

f k

∞

we use the same reasoning as in the proof of the previous

proposition. And the second term is bounded using (30).

2.3 Convergence of the density functions

In this section we will consider a Markov semigroup (P

_t

)

_t>0

and we will give an approximation result and a regularity criterion for it. The regularization property that we assume for the approximation processes is stronger than the one considered in the previous section and, instead of Proposition 2.2 we will use a general approximation result based on an interpolation inequality, proved in [8]. We recall that we have fixed T > 0 and that, δ = δ

ⁿ

= 1/n and we denote t

ⁿ_k

= t

k

=

^kT_n

. For k ∈ N , we consider µ

ⁿ_k

(x, dy) = µ

ⁿ

(x, dy) = P

δ

(x, dy), for all k ∈ N , the homogeneous sequence of finite transition measures which satisfy (20).

Moreover we introduce a sequence of transition probability measures ν

_kⁿ

(x, dy), k ∈ N , and the corresponding discrete semigroups P

ⁿ

(x, dy) defined by P

_k,kⁿ

= Id and P

_k,r+1ⁿ

= ν

_r+1ⁿ

P

_k,rⁿ

. We recall that P

_kⁿ

f = P

_0,kⁿ

f. We assume that for f ∈ C

^p

( R

^d

), we have P

_kⁿ

f ∈ C

^p

( R

^d

) and it verifies (20) :

sup

06tⁿ_k6tⁿr

kP

_k,rⁿ

fk

p,∞

6 Ckf k

p,∞

. (32) For h > 0 and p ∈ N , we assume that for all n ∈ N ,

E

_n

(h, p) k(µ

ⁿ

− ν

_kⁿ

)fk

∞

6 Cδ

^1+h

kf k

p,∞

. (33) and,

E

_n^∗

(h, p) | hg, (µ

ⁿ

− ν

_kⁿ

)fi | 6 Ckgk

_p,1

kf k

∞

δ

^1+h

. (34) We introduce now (P

ⁿ_k

)

k∈N

, a modification of (P

_kⁿ

)

k∈N

in the sense that for every measurable and bounded function f : R

^d

→ R , we have

sup

S6tⁿr−tⁿ_k

kP

_k,rⁿ

f − P

ⁿ_k,r

fk

∞

6 C

S

^ηp

δ

^h+1

kfk

∞

. (35) We assume that (P

ⁿ_k

)

k∈N

satisfies the following strong regularization property. We fix q ∈ N S, η > 0, and we assume that for every multi-index α, β with |α| + |β| 6 q and f ∈ C

^q

( R

^d

) one has

R

_p,η

(S) k∂

^α

P

ⁿ_k,r

∂

^β

f k

∞

6 CS

^−ηp

kf k

∞

, t

ⁿ_r

− t

ⁿ_k

> S. (36) Notice that if R

_q+d,η

(S) holds, then there exists p

ⁿ_k

∈ C

^q

( R

^d

× R

^d

) such that P

ⁿ_k

(x, dy) = p

ⁿ_k

(x, y)dy and, for every R > 1, t

ⁿ_k

> S and |α| + |β| 6 q, then

sup

|x|+|y|6R

|∂

_x^α

∂

_y^β

p

ⁿ_k

(x, y)| 6 CS

^−ηq

. (37) Moreover, the regularization properties R

_p,η

(S) and R

^∗_p,η

(S) hold when R

_p,η

(S) is satisfied.

Theorem 2.1. A. We fix p ∈ N , p and h, η > 0, 0 6 S 6 T /2, and we assume that (20) holds for P and that (32), (33), (34), (35) and (36) hold for this p, h, η and S, and for every n ∈ N . Then

sup

2S6tⁿ_k6T

kP

_tⁿ

k

f − P

_kⁿ

fk

∞

6 CS

^−ηp

n

^h

kfk

∞

. (38)

2 THE DISTANCE BETWEEN TWO MARKOV SEMIGROUPS 10 B. Suppose that (36) holds for every p ∈ N . Then, for every t > 0, P

_t

(x, dy) = p

_t

(x, y )dy with (x, y ) → p

t

(x, y) belonging to C

^∞

( R

^d

× R

^d

).

C. For every R, ε > 0 and every multi-index α, β we have sup

2S6tⁿ_k6T

sup

|x|+|y|6R

|∂

_x^α

∂

_y^β

p

tⁿ_k

(x, y) − ∂

_x^α

∂

_y^β

p

ⁿ_k

(x, y)| 6 C

n

^h(1−ε)

(39)

with a constant C which depends on R, S, T, ε and on |α| + |β| (and may go to infinity as ε ↓ 0).

Remark 2.2. The inequality (38) is essentially a consequence of Proposition 2.3. However, we may not use directly this result, because we do not assume that the semigroup (P

_t

)

_t>0

has the regularization property (25). This is pleasant, because we have to check the regularization property on the approximation scheme (P

_kⁿ

)

_k∈N

only.

Remark 2.3. The estimate (39) is sub-optimal because of ε > 0. One may wonder if opti- mal estimates (with n

^h

instead of n

^h(1−ε)

) may be obtained - as it was the case in the paper of Bally and Talay [6] concerning the Euler scheme. Notice that, in the above paper, spe- cific properties related to the dynamics of the diffusion process which gives the semigoup are used, and in particular properties of the tangent flow. For example, if X

_t

(x) denotes the diffusion process starting from x then we have E [f

⁰

(X

_t

(x))] = ∂

_x

E [f (X

_t

(x))(∂

_x

X

_t

(x))

⁻¹

] − E [f (X

_t

(x))∂

_x

(∂

_x

X

_t

(x))

⁻¹

)]. Such properties are crucial in the above paper - but are difficult to express in terms of general semigroups.

Proof. Let m = ζn, ζ ∈ N

^∗

. Using (33), (34), we obtain k(P

_k,k+1ⁿ

− P

_ζk,ζ(k+1)^m

)fk

_∞

6 Ckf k

p,∞

n

^−h−1

and |hg, (P

_k,k+1ⁿ

− P

_ζk,ζ(k+1)^m

)fi| 6 Ckgk

_p,1

kf k

∞

n

^−h−1

. Then Proposition 2.3 implies that: ∀2S 6 t

ⁿ_k

6 T, kP

_k^m

f − P

_kⁿ

fk

∞

6 CS

^−ηp

n

^−h

kf k

∞

. So the sequence (P

ⁿ

)

n∈N

is Cauchy and then converges with rate CS

^−ηp

n

^−h

kfk

∞

. It remains to identify the limit. By Proposition 2.1, this limit is (P

_t

f)

_t>0

for f ∈ C

_b^p

, so we conclude.

Let us prove B. We are going to use a result from [8]. First, we introduce some notations. Let d

_p

be the distance defined by

d

p

(µ, ν) = sup

| R

f dµ − R

f dν k : kf k

p,∞

6 1 . For q, l ∈ N , r > 1 and f ∈ C

^q

( R

^d

× R

^d

), we denote

kf k

q,l,r

= X

06|α|6q

R R (1 + |x|

^l

+ |y|

^l

)|∂

α

f (x, y)|

^r

dxdy

1/r

.

We have the following result which is Theorem 2.11 from [8].

Theorem 2.2. Let q, p, l, m ∈ N and r > 1 be given and let r

^∗

be the conjugate of r. Consider some measures µ(dx, dy) and µ

_n

(dx, dy) = g

_n

(x, y)dxdy with g

_n

∈ C

^q+2m

( R

^d

× R

^d

). Suppose that for some α > (q + p + d/r

^∗

)/m, we have

d

p

(µ, µ

n

)kg

n

k

^α_q+2m,2m,r

6 C ∀n ∈ N . (40)

Then µ(dx, dy) = g(x, y)dxdy with g ∈ W

^q,r

( R

^d

) and

kg − g

_n

k

_Wq,r(R^d)

6 C(d

^1/α_p

(µ, µ

_n

) + d

^1−(q+p+d/r_p ^∗)/αm

(µ, µ

_n

)). (41) In particular, if (40) is satisfied for all m ∈ N

^∗

and any α > (q + p + d/r

^∗

)/m, then for every ε > 0 we have

kg − g

n

k

_W^q,r_(R^d₎

6 Cd

^1−ε₀

(µ, µ

n

). (42) with a constant C which depends on ε and may go to infinity as ε ↓ 0.

We come back to our framework. We fix R > 0, 2S 6 t 6 T and choose k(n, t) ∈ N such that t

ⁿ_k(n,t)

= t. Let us consider a function Φ

_R

∈ C

_b^∞

( R

^d

× R

^d

) such that 1

BR×B_R

(x, y) 6 Φ

_R

(x, y) 6 1

BR+1×BR+1

(x, y) and denote

g

_k(n,t)^n,R

(x, y) = Φ

_R

(x, y)p

ⁿ_k(n,t)

(x, y).

We use the result above for the sequence g

_n

:= g

_k(n,t)^n,R

, n ∈ N and µ(dx, dy) := P

_t

(x, dy)dx.

In our specific case (35) and (38) give d

₀

(g, g

_n

) 6 Cn

^−h

and the hypothesis (36) ensures that sup

_n

kg

_n

k

_q+2m,2m,r

< ∞ so (40) holds for every α ∈ R

+

and r > 1. Using Sobolev’s embedding theorem, for u 6 q − d/r we have

kg − g

_n

k

_u,∞

6 Ckg − g

_n

k

_Wq,r(R^d)

6 Cn

^−h(1−ε)

and we conclude.

3 A class of Markov chains

3.1 Integration by parts using a splitting method

In this section we consider a sequence of independent random variables Z

_k

= (Z

_k¹

, · · · , Z

_k^N

) ∈ R

^N

, k ∈ {1, · · · , n} and we denote Z = (Z

₁

, ..., Z

_n

). The number n is fixed throughout this section (so there is no asymptotic procedure going on; but morally n is large because we are interested in estimating the error as n → ∞). Our aim is to settle an integration by parts formula based on the law of Z. The basic assumption is the following: there exists z

∗,k

∈ R

^N

and ε

∗

, r

∗

> 0 such that for every Borel set A ⊂ R

^N

and every k ∈ {1, · · · , n}

L

_z∗

(ε

∗

, r

∗

) P (Z

_k

∈ A) > ε

∗

λ(A ∩ B

_r∗

(z

∗,k

)) (43) where λ is the Lebesgue measure on R

^N

. We also define

M

_p

(Z ) := 1 ∨ sup

k6n

E [|Z

_k

|

^p

] (44)

and assume that M

p

(Z ) < ∞ for every p > 1.

It is easy to check that (43) holds if and only if there exists some non negative measures µ

_k

with total mass µ

_k

( R

^N

) < 1 and a lower semi-continuous function ϕ > 0 such that P (Z

_k

∈

3 A CLASS OF MARKOV CHAINS 12 dz) = µ

_k

(dz) + ϕ(z − z

∗,k

)dz. Notice that the random variables Z

₁

, · · · , Z

_n

are not assumed to be identically distributed. However, the fact that r

∗

> 0 and ε

∗

> 0 are the same for all k represents a mild substitute of this property. In order to construct ϕ we have to introduce the following function: For a > 0, set ϕ

_a

: R

^N

→ R defined by

ϕ

_a

(z) = 1

^|z|6a

+ exp

1 − a

²

a

²

− (|z| − a)

²

1

a<|z|<2a

. (45) Then ϕ

_a

∈ C

_b^∞

( R

^N

), 0 6 ϕ

_a

6 1 and we have the following crucial property: for every p, k ∈ N there exists a universal constant C

_q,p

such that for every x ∈ R

^N

, q ∈ N and i

₁

, · · · , i

_q

∈ {1, · · · , N}, we have

ϕ

_a

(z)| ∂

^q

∂z

ⁱ¹

· ∂z

^iq

(ln ϕ

_a

)(z)|

^p

6 C

_q,p

a

^pq

, (46)

with the convention ln ϕ

a

(z) = 0 for |z| > 2a. As an immediate consequence of (43), for every non negative function f : R

^N

→ R

+

E [f (Z

_k

)] > ε

_∗

Z

R^N

ϕ

_r∗/2

( z − z

_∗,k

)f (z)dz. (47) By a change of variable

E [f( 1

√ n Z

_k

)] > ε

∗

Z

R^N

n

^N/2

ϕ

_r∗/2

√

n(z − z

∗,k

√ n )

f(z)dz. (48)

We denote

m

∗

= ε

∗

Z

R^N

ϕ

r∗/2

(z)dz = ε

∗

Z

R^N

ϕ

r∗/2

(z − z

∗,k

)dz (49) and

φ

_n

(z) = n

^N/2

ϕ

_r∗/2

( √

nz) (50)

and we notice that R

φ

_n

(z)dz = m

∗

ε

⁻¹_∗

.

We consider a sequence of independent random variables χ

_k

∈ {0, 1}, U

_k

, V

_k

∈ R

^N

, k ∈ {1, · · · , n} with laws given by

P (χ

_k

= 1) = m

∗

, P (χ

_k

= 0) = 1 − m

∗

, (51) P (U

_k

∈ dz) = ε

∗

m

∗

φ

_n

(z − z

∗,k

√ n )dz, P (V

_k

∈ dz) = 1

1 − m

∗

( P ( 1

√ n Z

_k

∈ dz) − ε

∗

φ

_n

(z − z

∗,k

√ n )dz).

Notice that (48) guarantees that P (V

_k

∈ dz) > 0. Then a direct computation shows that P (χ

_k

U

_k

+ (1 − χ

_k

)V

_k

∈ dz) = P ( 1

√ n Z

_k

∈ dz). (52)

This is the splitting procedure for

^√1_n

Z

_k

. Now on we will work with this representation of the law of

^√1_n

Z

k

. So, we always take

√ 1

n Z

_k

= χ

_k

U

_k

+ (1 − χ

_k

)V

_k

.

Remark 3.1. The above splitting procedure has already been widely used in the litterature: in [30] and [24], it is used in order to prove convergence to equilibrium of Markov processes. In [9], [10] and [33], it is used to study the Central Limit Theorem. Last but not least, in [29], the above splitting method (with 1

Br∗(z∗,k)

instead of φ

_n

(z −

^z√∗,k_n

)) is used in a framework which is similar to the one in this paper.

In the following we denote χ = (χ

₁

, · · · , χ

_n

), U = (U

₁

, · · · , U

_n

) and V = (V

₁

, · · · , V

_n

) and we consider the following class of random variables:

S = {F = f (χ, U, V ) : f is measurable and u → f (χ, u, v) ∈ C

_b^∞

( R

ⁿ

× R

^N

), ∀χ, v}. (53) For a multi index α = (α

₁

, · · · , α

_q

) with α

_j

= (k

_j

, i

_j

), k

_j

∈ {1, · · · , n}, i

_j

∈ {1, · · · , N }, we denote |α| = q the length of α and

∂

_u^α

f(χ, u, v) = ∂

^q

∂u

ⁱ_k¹

1

· · · ∂u

ⁱ_k^q

q

f (χ, u, v).

We construct now a differential calculus based on the laws of the random variables U

_k

, k = 1, · · · , n which mimics the Malliavin calculus, following the ideas from [5], [3] and [4]. In order to be self contained we shortly present the results that we need. For F = f(χ, U, V ) ∈ S we define the Malliavin derivatives

D

_(k,i)

F = χ

_k

1

√ n

∂F

∂U

_kⁱ

= χ

_k

1

√ n

∂f

∂u

ⁱ_k

(χ, U, V ), k = 1, · · · , n, i = 1, · · · , N. (54) We denote by h◦, ◦i the usual scalar product on R

^N

× R

ⁿ

. The Malliavin covariance matrix for a multi dimensional functional F = (F

¹

, · · · , F

^d

) is defined as

σ

_F^i,j

=

DF

ⁱ

, DF

^j

=

n

X

k=1 N

X

r=1

D

_(k,r)

F

ⁱ

× D

_(k,r)

F

^j

, i, j = 1, · · · , d. (55) The higher order derivatives are defined by iterating D:

D

_α

F = D

_α1

· · · D

_αm

F. (56)

Now we define the Ornstein Uhlenbeck operator L : S → S. We denote Γ

_k

= ln φ

_n

(U

_k

− z

∗,k

√ n ) ∈ S (57)

and we notice that

D

_(k,i)

Γ

_k

= 1

√ n χ

_k

∂

_uⁱ

k

ln φ

_n

(U

_k

− z

∗,k

√ n )

= χ

_k

∂

_zⁱ

(ln ϕ

_r∗/2

) √

n(U

_k

− z

∗,k

√ n ) .

Finally, we define

−LF =

n

X

k=1 N

X

i=1

D

_(k,i)

D

_(k,i)

F +

n

X

k=1 N

X

i=1

D

_(k,i)

F × D

_(k,i)

Γ

_k

. (58)

3 A CLASS OF MARKOV CHAINS 14 Remark 3.2. The basic random variables in our calculus are Z

_k

, k = 1, · · · , n so we precise the way in which the differential operators act on them. Since χ

k

Z

k

= √

nχ

k

U

k

it follows that

D

_(m,j)

Z

_kⁱ

= χ

_k

δ

_m,k

δ

_i,j

, (59)

LZ

_kⁱ

= −χ

_k

∂

_zⁱ

(ln ϕ

_r∗/2

) √

n(U

_k

− z

∗,k

√ n )

. (60)

In our framework, the duality formula in Malliavin calculus reads as follows: for each F, G ∈ S E [F LG] = E [hDF, DGi] = E [GLF ]. (61) This follows immediately using the independence structure and standard integration by parts on R

^N

: indeed, if f, g ∈ C

_b¹

( R

^N

) and k ∈ {1, · · · , n}, then

N

X

i=1

E [∂

_uⁱ

k

f(U

k

)∂

_uⁱ

k

g(U

k

)]

= ε

_∗

m

∗

N

X

i=1

Z

R^N

∂

_uⁱ

k

f(u)∂

_uⁱ

k

g(u)φ

n

(u − z

_∗,k

√ n )du

= − ε

∗

m

∗ N

X

i=1

Z

R^N

f(u)(∂

_u²i

k

g(u) + ∂

_uⁱ

k

g(u) ∂

_uⁱ

k

φ

_n

(u −

^z√∗,k_n

)

φ

_n

(u −

^z√∗,k_n

) )φ

_n

(u − z

∗,k

√ n )du

= − E h

f (U

_k

)

N

X

i=1

∂

_i²

g(U

_k

) + ∂

_uⁱ

k

g(U

_k

)∂

_uⁱ

k

ln φ

_n

(U

_k

− z

∗,k

√ n ) i .

It follows that

n

X

k=1 N

X

i=1

E [D

_(k,i)

F × D

_(k,i)

G]

= 1 n

n

X

k=1 N

X

i=1

E [χ

_k

∂

_uⁱ

k

f (χ, U, V ) × ∂

_uⁱ

k

g(χ, U, V )]

= − E h

f(χ, U, V )

n

X

k=1

χ

_k

N

X

i=1

1 n ∂

_u²i

k

g(χ, U, V ) + 1

√ n ∂

_uⁱ

k

g(χ, U, V ) 1

√ n ∂

_uⁱ

k

ln φ

_n

(U

_k

− z

∗,k

√ n ) i

= − E h

f(χ, U, V )

n

X

k=1

χ

_k

N

X

i=1

D

_(k,i)

D

_(k,i)

G + D

_(k,i)

GD

_(k,i)

Γ

_k

i

= E [F LG],

which is exactly (61). We have the following standard chain rule: for φ ∈ C

¹

( R

^d

) and F ∈ S

^d

Dφ(F ) =

d

X

j=1

∂

_j

φ(F )DF

^j

. (62)

Moreover, one may prove, using (62) and the duality relation (or direct computation), that Lφ(F ) =

d

X

j=1

∂

j

φ(F )LF

^j

+

d

X

i,j=1

∂

i

∂

j

φ(F )

DF

ⁱ

, DF

^j

. (63)

In particular for F, G ∈ S,

L(F G) = F LG + GLF + 2 hDF, DGi . (64) We are now able to give the Malliavin integration by parts formula:

Theorem 3.1. Let F ∈ S

^d

and G ∈ S be such that E [(det σ

_F

)

^−p

] < ∞ for every p > 1. We denote γ

F

= σ

_F⁻¹

. Then for every φ ∈ C

_c^∞

( R

^d

) and every i = 1, · · · , d

E [∂

_i

φ(F )G] = E [φ(F )H

_i

(F, G)] (65) with

−H(F, G) = Gγ

_F

LF + hD(Gγ

_F

), DF i (66) and

H

_i

(F, G) = −

d

X

j=1

Gγ

_F^i,j

LF

^j

+ D(Gγ

_F^i,j

)DF

^j

. (67) Moreover, for every multi index α = (α

₁

, · · · , α

_m

) ∈ {1, · · · , d}

^m

E [∂

_α

φ(F )G] = E [φ(F )H

_α

(F, G)] (68) with H

α

(F, G) defined by the recurrence relation H

_(α1,···,αm)

(F, G) = H

αm

(F, H

_(α1,···,αm−1)

(F, G)).

Proof. Using the chain rule Dφ(F ) = ∇φ(F )DF we have

hDφ(F ), DF i = ∇φ(F ) hDF, DF i = ∇φ(F )σ

_F

.

It follows that ∇φ(F ) = γ

_F

hDφ(F ), DF i . Then, using (64) and the duality formula (61), E [G∇φ(F )] = E [Gγ

_F

hDφ(F ), DF i] = 1

2 E [Gγ

_F

(L(φ(F )F ) − φ(F )LF − F Lφ(F ))]

= 1

2 E [φ(F )(F L(Gγ

_F

) − Gγ

_F

LF − L(Gγ

_F

F ))].

We use once again (64) in order to obtain H(F, G) in (66).

We give now estimates of the weights H

_α

(F, G) which appear in the above integration by parts formulas. We will work with the norms:

|F |

²_1,m

= X

16|α|6m

|D

_α

F |

²

, |F |

²_m

= |F |

²

+ |F |

²_1,m

, (69) and

kF k

_1,m,p

=

|F |

_1,m

p

= E [|F |

^p_1,m

]

^1/p

(70) kF k

m,p

= kF k

p

+

|F |

1,m

p

.

3 A CLASS OF MARKOV CHAINS 16 Proposition 3.1. For each m, q ∈ N , there exists a universal constant C > 1 (depending on d, m, q only) such that for every multi index α with |α| 6 q and every F ∈ S

^d

and G ∈ S on has

|H

_α

(F, G)|

_m

6 C(1 ∨ (det σ

_F

)

⁻¹

)

^q(q+m+1)

(1 + |F |

^2dq(q+m+2)_1,m+q+1

+ |LF |

^2q_m+q−1

)|G|

_m+q

. (71) The proof is long but straightforward so we skip it. The reader may find the detailed proof in [5] and in [3], Proposition 3.3.

We finish this section with an estimate of kLZ

_kⁱ

k

_q,p

:

Lemma 3.1. A. For every k = 1, · · · , n and i = 1, · · · , N, we have

E [LZ

_kⁱ

] = 0. (72)

B. For every q ∈ N and p > 2 there exists a constant C depending on q, p only kLZ

_kⁱ

k

_q,p

6 Cm

^1/p∗

r

∗

(1 + r

_∗^−q

) (73)

Proof. A. Using the duality relation we have E [1 × LZ

_kⁱ

] = E [hD1, DZ

_kⁱ

i] = 0. In order to prove B we recall (see (60)) that

LZ

_kⁱ

= −χ

_k

∂

_zⁱ

(ln ϕ

_r∗/2

) √

n(U

_k

− z

∗,k

√ n ) .

Let Λ

k,q

be the set of the multi-index α = (α

1

, · · · , α

q

) such that α

j

= (k, i

j

). Notice that for a multi-index α of length q, such that α / ∈ Λ

_k,q

, we have D

_α

LZ

_kⁱ

= 0. Suppose now that α ∈ Λ

_k,q

and let α = (i

₁

, · · · , i

_q

). It follows

D

_α

LZ

_kⁱ

= −χ

_k

∂

_z^α

∂

_zⁱ

(ln ϕ

_r∗/2

) √

n(U

_k

− z

∗,k

√ n ) .

Using (46), we obtain kD

_α

LZ

_kⁱ

k

^p_p

= ε

∗

kχ

_k

k

^p_p

m

∗

Z

|v|6r∗

n

^N/2

∂

_z^α

∂

_zⁱ

(ln ϕ

_r∗/2

) √

n(u − z

∗,k

√ n )

p

ϕ

_r∗/2

√

n(u − z

∗,k

√ n ) du

= ε

∗

kχ

k

k

^p_p

m

∗

Z

r∗/26|v|6r∗

∂

_z^α

∂

_zⁱ

(ln ϕ

_r∗/2

)(v )

p

ϕ

_r∗/2

(v)dv 6 C

_q+1,p

m

∗

r

∗^p(q+1)

.

and then

kLZ

_kⁱ

k

_q,p

6 C sup

l6q

sup

α∈Λ_k,l

kD

_α

LZ

_kⁱ

k

_p

6 Cm

^1/p∗

r

∗

(1 + r

_∗^−q

).

3.1.1 Localizaton

In the following, we will not work under P , but under a localized probability measure defined as follows. We fix M 6 n and we consider the set

Λ

_M

= { 1 M

M

X

k=1

χ

_k

> m

_∗

2 }. (74)

Using Hoeffding’s inequality and the fact that E [χ

_k

] = m

∗

, it can be checked that

P (Λ

^c_M

) 6 C exp(−CM m

²_∗

) (75)

We consider also the localization function ϕ

_n^1/4_/2

, defined in (45), and we construct the random variable

Θ = Θ

_M,n

= 1

ΛM

×

n

Y

k=1

ϕ

_n1/4/2

(Z

_k

). (76) Since Z

_k

has finite moments of any order, the following inequality can be shown: for every q ∈ N there exists C such that

P (Θ

_M,n

= 0) 6 P (Λ

^c_M

) +

n

X

k=1

P (|Z

_k

| > n

^1/4

) 6 C exp(−CM m

²_∗

) + M

_4(q+1)

(Z)

n

^q

. (77) We define the probability measure

d P

Θ

= 1

E [Θ] Θd P . (78)

Corollary 3.1. Let F ∈ S

^d

and G ∈ S be such that E

Θ

[(det σ

_F

)

^−p

] < ∞ for every p > 1. We denote γ

_F

= σ

_F⁻¹

. Then, for every φ ∈ C

_c^∞

( R

^d

) and every i = 1, · · · , d

E

Θ

[∂

_i

φ(F )G] = E

Θ

[φ(F )H

_i^Θ

(F, G)] (79) with

−H

^Θ

(F, G) = Gγ

F

LF + hD(Gγ

F

, DF i + Gγ

F

hD ln Θ, DF i (80) and

H

_i^Θ

(F, G) = −

d

X

j=1

Gγ

_F^i,j

LF

^j

+ D(Gγ

_F^i,j

)DF

^j

+ Gγ

_F^i,j

D ln Θ, DF

^j

. (81)

And for every multi index α = (α

₁

, · · · , α

_m

) ∈ {1, · · · , d}

^m

,

E

Θ

[∂

_α

φ(F )G] = E

Θ

[φ(F )H

_α^Θ

(F, G)], (82) with H

_α^Θ

(F, G) defined by the recurrence relation H

_(α^Θ

1,···,αm)

(F, G) = H

_α^Θ_m

(F, H

_(α^Θ

1,···,αm−1)

(F, G)).

Moreover there exists an universal constant C such that for every multi index α with |α| = q E

Θ

[|H

_α^Θ

(F, G)|

^p_m

] 6 C × C

_q,Θ

(F, G) (83) with

C

q,Θ

(F, G) = E

^Θ

[(1 ∨ (det σ

F

)

⁻¹

)

^2pq(q+m+1)

]

^1/2

× (1 + E [|F |

8pqd(q+m+2)

1,m+q+1

]

^1/4

+ E [|LF |

^8pq_m+q−1

]

^1/4

) E [|G|

^4p_m+q

]

^1/4

. (84)

3 A CLASS OF MARKOV CHAINS 18 Proof. Using (66) with G replaced by GΘ we obtain E [∂

_i

φ(F )GΘ] = E [φ(F )H] with

H = −ΘGγ

_F

LF − hD(ΘGγ

_F

), DF i = ΘH

_i

(F, G) − Gγ

_F

hDΘ, DF i . It follows that

E

Θ

[∂

_i

φ(F )G] = 1

E [Θ] E [∂

_i

φ(F )GΘ] = 1

E [Θ] E [φ(F )(ΘH

_i

(F, G) − Gγ

_F

hDΘ, DF i]

= E

Θ

[φ(F )(H

_i

(F, G) − Gγ

_F

hD ln Θ, DF i)].

So (79) is proved and (82) follows by recurrence. Notice that by (46) we have E

Θ

[|Gγ

_F

hD ln Θ, DF i |

^p

] 6 C E

Θ

[|Gγ

_F

DF |

^p

].

Then (83) follows from (71).

3.2 Markov chains

Throughout this section, n ∈ N will still be fixed and will be the number of time step between 0 and T and also the number of increments that we consider in our abstract Malliavin calculus.

We consider two sequences of independent random variables Z

_k

∈ R

^N

, κ

_k

∈ R , k ∈ N and we assume that Z

_k

verifies (43). We also assume that Z

_k

has finite moments of any order and we recall that

M

_p

(Z ) = 1 ∨ sup

k6n

E [|Z

_k

|

^p

].

We construct the R

^d

valued Markov chain

X

_k+1ⁿ

= ψ(κ

_k

, X

_kⁿ

, Z

_k+1

√ n ), k ∈ N (85)

where

ψ ∈ C

^∞

( R × R

^d

× R

^N

; R

^d

) and ψ(κ, x, 0) = x. (86) We denote

kψk

1,r,∞

= X

06|α|6r

X

16|β|6r−|α|

k∂

_x^α

∂

_z^β

ψk

∞

(87)

and, for C, r > 1 we denote

N

_ψ

(C, r) = (1 + kψk

1,r,∞

) exp(Ckψk

²_1,3,∞

). (88) Since |β| > 1 in the above definition, we have at least one derivative with respect to z. All our estimates will be done in terms of kψk

1,r,∞

so we may assume without loss of generality that

E [Z

_kⁱ

] = 0, k ∈ N , i = 1, · · · , N. (89)

Indeed, if this is not true, we denote by m

_k

= ( E [Z

_k¹

], · · · , E [Z

_k^N

]) and we work with Z e

_k

=

Z

_k

− m

_k

instead of Z

_k

and with ψ e (x, z) = ψ(κ, x, z + m

_k

) instead of ψ. Since ∇

_z

ψ = ∇

_z

ψ e all

the results remain true.