A Malliavin calculus method to study densities of additive functionals of SDE's with irregular drifts

www.imstat.org/aihp 2012, Vol. 48, No. 3, 871–883

DOI:10.1214/11-AIHP418

© Association des Publications de l’Institut Henri Poincaré, 2012

A Malliavin Calculus method to study densities of additive functionals of SDE’s with irregular drifts ¹

Arturo Kohatsu-Higa

and Akihiro Tanaka

aRitsumeikan University and Japan Science and Technology Agency, Department of Mathematical Sciences, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan. E-mail:arturokohatsu@gmail.com

bSettsu-shi Higashibehu 4-18019-302, Osaka 566-0042, Japan. E-mail:tnkaki2000@gmail.com

Received 28 September 2010; revised 11 January 2011; accepted 7 February 2011 Dedicated to the memory of Prof. Paul Malliavin

Abstract. We present a general method which allows to use Malliavin Calculus for additive functionals of stochastic equations with irregular drift. This method uses the Girsanov theorem combined with Itô–Taylor expansion in order to obtain regularity properties for this density. We apply the methodology to the case of the Lebesgue integral of a diffusion with bounded and measurable drift.

Résumé. On introduit une méthode générale qui permet l’utilisation du Calcul de Malliavin pour des fonctionnelles additives gé- nérées par des équations stochastiques avec une dérive irrégulière. Cette méthode utilise le théorème de Girsanov avec l’expansion d’Itô–Taylor pour obtenir la régularité de la densité. On applique cette méthodologie pour au cas de l’intégrale en temps d’une diffusion avec derive mesurable bornée.

MSC:Primary 60H07; secondary 60H10

Keywords:Malliavin Calculus; Non-smooth drift; Density function

1. Introduction

Paul Malliavin in the eighties developed a stochastic method to study the regularity of probability laws arising on the Wiener space. In particular, using this stochastic method he obtained the existence and smoothness of densities of diffusions under the hypoelliptic condition. This celebrated theory has been applied successfully to a variety of stochastic equations providing useful properties of densities of random variables associated to these equations. It is based on the integration by parts formula which requires “stochastic” smoothness of the random variable in question.

For this reason, one usually requires smoothness of the coefficients of the stochastic equation. On the other hand, many results from partial differential equations provide the same results for the density of uniformly elliptic diffusions under weaker conditions on (some of) the coefficients, such as Hölder continuity or even bounded and measurable.

Still, many stochastic equations (e.g. stochastic partial differential equations) do not have a partial differential equation counterpart and therefore the current results on existence and smoothness of probability laws are limited to the case of smooth coefficients.

1This research was supported by grants from the Japan Ministry of Education and Science and the Japan Science and Technology Agency. The authors would like to thank all the people that gave us information about related results and the anonymous referee who suggested a simplification of the proof.

2Previous results related with this article appeared in the author’s MSc thesis (in Japanese) at Osaka University.

Here we consider ad₁-dimensional diffusion X^x_t =x+

_t

0

b(X^x_s)ds+ _t

0

σ (X_s^x)◦dWs, (1)

whereσ:R^d1→R^d1×m,b:R^d1→R^d1,x∈R^d1. We assume thatσ is smooth and uniformly elliptic whilebis mea- surable and bounded. HereW denotes am-dimensional Brownian motion defined on the probability space(Ω,F, P ) and the above stochastic integral is the Fisk-Stratonovich integral. Under these conditions, we obtain the existence and smoothness of the density ofYt whereYt=_t

0ψ (X_s^x)ds whereψ:R^d1→R^d2 is a smooth function with bounded derivatives. Note that this random vector is non-elliptic in the sense that there is no Wiener component in the equation forY and that the system(X, Y )is of dimensiond=d₁+d₂.

Now, we introduce our main condition. Define the vector fields V0=

d2

i=1

ψi

∂

∂x_i+d1 −1 2

d1

i,j=1

m l=1

σj l∂jσil

∂

∂x_i,

V_i=

d₁

j=1

σ_{j i} ∂

∂xj

, 1≤i≤m.

Then, define

A0= {V1, . . . , Vm}, An=

[Vi, Z];i=0, . . . , m, Z∈An−1

, n≥1.

We assume that there existsk∈Nandc0>0 such that for allξ∈R^dandx∈R^d1

V∈∪^kl=1Al

V (x), ξ2

≥c₀ξ². (2)

For example, in the one dimensional case (d1=d₂=m=1), if we assume that|ψ(x)| ≥c₀andσ is uniformly elliptic then the above hypothesis (2) is satisfied.

Our goal is then to prove

Theorem 1. Consider the random vectorYt=t

0ψ (X^x_s)dswhereXsatisfies(1).Assume that the coefficientsσ and ψare smooth with bounded derivatives,bis a bounded and measurable function,σ is uniformly elliptic and that the system(X, Y )satisfies(2).Then the vectorY_t has a smooth density.

The regularity of the joint density of time averages of diffusions is a problem that appears in various applied fields ranging from average quantities in economics, finance³and filtering problems⁴between others. One possible example is to consider one dimensional diffusions with drifts that have a discontinuity of the first kind (for some examples, see e.g. [27] and the references therein. Also see the references on existence and uniqueness of solutions for SDE’s with irregular drift).

From the mathematical point of view, another related attempt to deal with irregularities in the coefficients of the diffusion using Malliavin Calculus can be found in [26]. Note that if we are only considering the density of the random vectorX_t^xthen the problem could be studied by using techniques from partial differential equations as can be seen in [15] or [24] but only limited regularity is obtained. The goal of the present research is to try to establish an extension of the Malliavin Calculus technique to deal with situations where the drift is irregular.

3Notably the Asian option considers as its main random variable the integral of the stock price.

4The signal process is an example that belongs to an extended version of the above theorem to be considered in Theorem7.

Our method of proof can be succinctly explained as follows. First, we use Girsanov’s theorem in order to remove the irregular drift from the stochastic equation. This is why we require uniform ellipticity. This reduces our study to the case where the stochastic equation is regular driven by a new Brownian motion, sayB. Still, one has to deal with the non-smooth change of measure as this will appear in the expression for the characteristic function.

The change of measure is then expanded using Itô–Taylor expansion up to a high order. We prove that the residue is small. The problem is to be able to carry out the integration by parts formula without requiring derivatives of the multiple integrals as they contain the non-smooth drift.

Next, we divide the integration region of the multiple stochastic integral so that a fixed interval is not included in the region of integration. We call this interval,[α1, α2]. Using this interval we take advantage that the noiseB within this interval regularizes the irregular drift by use of conditional expectations.

Then we perform a conditional integration by parts (ibp) in this interval with respect toB. If we perform a usual ibp we will not avoid considering derivatives of the drift.

The final idea consists in taking expectations with respect to the Wiener process,B, in the interval[^α1+₂^α2, α₂]in order to regularize the drift appearing in the multiple integrals, and then applying the ibp in the interval[α₁,^α1+₂^α2] (where no drift appears) as many times as necessary, in order to obtain the smoothness of the density. Here is where the additivity ofY plays an important role.

The method just described is rather general and can be applied to many situations. In fact, we will consider a slight generalization of the problem proposed here (see also the conclusions section at the end of the article).

This article has the following sections. In Section2, we give the main set up of the problem. In Section2.1, we give notation from Malliavin Calculus that will be used in this article. Then in Section3we give our general set-up for the problem and the generalization of Theorem1which we prove in Section4.

Throughout the article, we assume that all stochastic differential equations have unique strong solutions. In recent years, various applications of the system described here have appeared (in particular the 3D-Navier Stokes equation between others) and has led to the study of existence and uniqueness of stochastic differential equations with irregular drifts in weak and pathwise form (the interested reader may look at [2–6,10,11,16–18,22,25,28] between others). In order to avoid this discussion, we decided to use a set-up where the Girsanov theorem has already been applied (see Section3).

For a multi-indexα=(α₁, . . . , α_k)∈N^k, we denote its length by|α| =k. Without further mention we use some- times the convention of summation over double indices (Einstein convention). When the exact values of the constants are not important we may repeat the same symbol for constants that may change from one line to the next. These constants will be denoted by an over bar symbol. For a matrixA,A^Tdenotes the transpose ofA.

Vectors will be always considered as column vectors except for vectors denoting time which are considered as line vectors. For a vector of time variables s=(s₁, . . . , s_n), we use the following notations: ds=dsn· · ·ds1,s^j = (s₁, . . . , s_j),s¯^n−j+1=(s_j, . . . , s_n). Similarly we denote byπⁱ(s)andπ¯ⁱ(s)the projection operator for the firsti components and the lasticomponents ofsrespectively. Sometimes, we may usesto denote a real time (instead of a vector one) integration variable. The difference should be clear from the context. Partial derivatives will be denoted by

∂if (x),x∈Rⁿwhere this means theith variable appearing in the functionf. For a multi-indexα=(α1, . . . , αk)∈ {1, . . . , d}^k we denote(iθ )^−α=i^{−k k}_i=1θ_α⁻¹

i ,|θ|^−α= ^k_i=1|θ_α⁻¹

i |and∂_α= ^k_i=1∂_αi. The inner product between two vectors u andv is denoted byu·v. Similar notation is used for products of vectors and matrices or between matrices. Stochastic processes may be denoted interchangeably byX(t )orX_t. As usual, constants may change from one line to the next even if the same notation is used. Finally,jwill denote the imaginary unit and although there may be a slight abuse of notation (occasionallyj may denote an index in a summation) the context will determine clearly the intended meaning.

2. Preliminaries

Let(Ω,F, P , (Ft)_t≥0)be a filtered standardm-dimensional Wiener space with filtration given byFt(see [8], Chap- ter 1, Section 2.25). We will use the following classical result which characterizes the existence and smoothness of densities.

Lemma 2. LetF be a random vector and denote byh_F(θ )=E[exp(j θ·F )]its characteristic function.If fork∈N, we have that

θ^kh_F(θ )dθ <∞

thenF has a density and it belongs to the classC_b^k.

Note that in order to obtain the smoothness of the density ofF it is enough to prove that for anyk∈Nthere exists a positive constantCksuch that for allθ ≥1, we have thatθ^k|hF(θ )| ≤Ck.

2.1. Malliavin calculus

We give a brief introduction to Malliavin calculus. For details, we refer the reader to [7,19,20] or [21]. From now on we fixT >0. For any measurable functionh∈L²([0, T];R^m)=:H0,T we denote its stochastic integral byW (h)= T

0 h(s)·dWs and bySthe class of smooth functionals S=

F: F =f

W (h₁), . . . , W (h_n)

;h₁, . . . , h_n∈H0,T;f ∈C_p^∞(Rⁿ;R);n∈N .

HereC_p^∞ denotes the class ofC^∞-functions with at most polynomial growth at infinity. For F ∈S we define the Malliavin derivative as

D^j_tF = n i=1

∂_if

W (h₁), . . . , W (h_n) h^j_i(t).

Fork∈Z₊andp≥1 we define the following semi-norms Fk,p=

E

|F|^p +

k i=1

EDⁱF^p

H^⊗0,Tⁱ

^1/p , DⁱF

H^⊗0,Tⁱ = _T

0

· · · _T

0

|Dⁱ_s

1,...,siF|²ds1· · ·dsi

1/2

.

As usual,F0,p≡ Fp. If we complete these semi-norms appropriately, one defines the spaceD^k,p. In such a case forF ∈D^k,p, we can define thekth order derivative asD_s1,...,siF =D_s1· · ·D_siF. As with multiple derivatives we use the symbolD_s^αF for a multiple derivative with respect to the noises indexed inαat the times indicated by the time vectors. Furthermore we define the space of smooth random variablesD^∞=

k,pD^k,p. Similarly, for a Hilbert spaceV and aV-valued random variable one can defineD^k,p(V )andD^∞(V ).In particular, for aR-valued random processus, s≤T, we define the following semi-norm

uk,p=

E

u^p_H0,T +

k i=1

EDⁱu^p

H^⊗0,Tⁱ⁺¹

1/p

.

We define the Skorokhod integralδas the dual operator of the closable operatorD. Whenδis restricted toFt-adapted L²stochastic processes{u_s, s≤T}then the Skorokhod integral coincides with the Itô integralδ(u).For this reason we sometimes writeδ(u)=_T

0 u_t·dWt orδ(u)=_T

0 u_t·δW_t if we want to be more explicit. Furthermore ifuis an element in the domain ofδ(this is the case if e.g.u∈D^1,2(L²[0, T])) andF ∈D^1,2and it satisfiesE[F²_T

0 u²_tdt]<

∞then

δ(F u_t)=F δ(u)− _T

0

(D_tF )·u_tdt

is satisfied provided the right-hand side is square integrable and furthermore we have the following duality formula E

F δ(u)

=E _T

0

D_tF·u_tdt

. (3)

Definition 3. For a random vectorF =(F¹, . . . , F^d)∈(D^1,2)^d,we define thed×dMalliavin covariance matrixM_F

M_F^ij= DFⁱ, DF^j_H0,T. (4)

If we have that for anyp≥1, E

(detM_F)^−p

<+∞

then we say that the random vectorF is non-degenerate.

In order to simplify notation we sometimes consider thed×mmatrixDF=(D^jFⁱ)_ij forF=(F¹, . . . , F^d). We also have the following integration by parts formula.

Proposition 4. Let F ∈ (D^∞)^d be nondegenerate and let G∈ D^∞, φ∈ C_p^∞. Then for any multi-index α∈ {1, . . . , d}^|α|,we have

E

∂αφ(F )G

=E

φ(F )Hα(F, G)

for a random variableHα(F, G)∈D^∞which has the following explicit expression H(α₁,...,α_k)(F, G)=H(α_k)

F, H(α₁,...,α_k−1)(F, G) , H_(α1)(F, G)=δ

G

M_F⁻¹α₁j

DF^j .

Furthermore there exists positive integersβ,γ,q,n₁andn₂such that H_α(F, G)

p≤C(detM_F)⁻¹ⁿ¹

β DFⁿ_|α²_|,γG_|α|,q. (5)

We will also use conditional Malliavin Calculus in various part of the article. For this we introduce the notation E_t[G] =E[G|Ft]

Ft,t+h,k,p=

Et

|F|^p +

k i=1

EtDⁱF^p

H^⊗it,t+h

1/p

, F ∈D^k,p, k≥1, p≥2, h >0.

We also use the following notation for the Skorohod integralδ_t,t+h(u)=_t+h

t u_s·dWs. We remark as before that ifuis in the domain ofδandE[F²_T

0 u²_tdt]<∞andF ∈D^1,2then we have the duality formula E_t

F δ_t,t+h(u)

=E_t t+h

t

D_sF ·u_sds

(6) if both terms inside the expectations are conditionally square integrable. In many situations we will have to change probability spaces in order to carry the Malliavin Calculus in the standard Wiener space, given that the quantity to be computed is determined only by the law ofW. We will do this without any further mention.

Note that usually the spaceHused in the definition of the Malliavin covariance matrix (4) isH=L²[0, T]. Some other times we may useH=L²[t, t+h]. In such a case we may use the notationM_F^[t,t+h],Hα^[t,t+h]etc.

In the next result we give an extension of various classical theorems that can be found in [9,12–14,20]. The proof is quite similar so we leave it for the reader. We will apply the following results in various situations.

3. Problem set-up and main result

The problem that we will treat is the integration by parts formula for the following expression h(θ )=E

exp

j θ·Y_t^0,z ρ(t )

, ρ(t )=exp

_t

0

b¯ X^0,x_s

·dWs−1 2

_t

0

b¯^Tb¯ X_s^0,x

ds

, whereW is am-dimensional Wiener process and

X^s,x_t =x+ _t

s

b X^s,x_u

du+ _t

s

σ X_u^s,x

◦dWu, (7)

Y_t^s,z=y+ _t

s

ψ (X^s,x_u )du+ _t

s

ϕ(X^s,x_u )◦dWu (8)

forz=(x, y). The following hypotheses will be used in our main theorem:

Hypothesis (A1). b¯:R^d1→R^mis a bounded and measurable function.

Hypothesis (A2). All the coefficients b:R^d1 →R^d1, σ:R^d1 →R^d1×m,ϕ:R^d1 →R^d2×m and ψ:R^d1 →R^d2 are smooth with bounded derivatives.

Hypothesis (A3). σ^Tσ (x)is an invertible matrix for allx∈R^m.

Remark 5. 1. The process Y^s,(x,0) will appear frequently in our calculations so we may simplify the notation to Y^s,x≡Y^s,(x,0).Similarly,we will also useYt=Y_t^0,xandXt≡X_t^0,x.Note that due to the fact that the initial value of Y is zero the flow properties are slightly modified.In fact,the classical flow property is written as

exp

j θ·Y_t^0,z

=exp

j θ·Y_s^0,z exp

j θ·Y^s,(X

0,x s ,Y_s^0,z) t

. (9)

2.We also remark that under Hypothesis(A2),we have that there exists a pathwise unique solution for(X, Y )and it has smooth flows satisfying the Markov property.We define its associated semigroup by

Ptf (z)=E f

Z^0,z_t

forZ^s,z_t :=(X^s,x_t , Y_t^s,z).Note that the following basic estimate is satisfied P_tf (z)≤Cf_∞.

3.Furthermore under Hypothesis(A3)we remark that (σ^Tσ )⁻¹σ (X_t)^Tb(X˜ _t)dt−dXt

=dWt

and therefore

σ (X_s, s≤t )=σ (W_s, s≤t ) is satisfied.Here

˜

b_i=bi=1 2

d1

j=1

m l=1

∂jσilσj l.

Therefore we also have that X satisfies the Markov property(see e.g. [23],SectionV.6,Theorem32).For example,we have

E

f b

a

g₁(X_s)dWs+ _b

a

g₂(X_s)dsFa

=E

f b

a

g₁(X_s)dWs+ _b

a

g₂(X_s)dsX_a . This property can also be generalized to multiple stochastic integrals.

We define the vector fields associated toZ V₀=

d₁

i=1

b_i ∂

∂x_i +

d₂

i=1

ψ_i ∂

∂x_i+d1 −1 2

d₁

i,j=1

m l=1

σ_{j l}∂_jσ_il ∂

∂x_i −1 2

d₂

i=1 d₁

j=1

m l=1

σ_{j l}∂_jϕ_il ∂

∂x_i+d1,

V_i=

d₁

j=1

σ_{j i} ∂

∂x_j +

d₂

j=1

ϕ_{j i} ∂

∂x_j+d1, 1≤i≤m.

We assume that there existsk∈Nandc₀>0 satisfying that for allξ ∈R^dandx∈R^d1

V∈∪^kl=1Al

V (x), ξ2

≥c₀ξ². (10)

This condition is a uniform version of the Hörmander condition. Furthermore we also have the following classical result.

Theorem 6. Assume Hypothesis(A2)and (10).ThenZ_t^s,z:=(X^s,x_t , Y_t^s,z)∈D^∞and in factZ_t^s,zs,t,k,p≤C(1+ z)^μfor anys < t and k, p∈Nand some positive constantsC and μ.The Malliavin covariance matrix has all inverse moments.That is,there exists positive constantsCandμsuch that

E

detM^[a,t]

Z_t^s,z

₋p

≤C(t−a)^−μ

1+ zμ

for alla∈ [s, t)and there exists positive constantsC≡C(α)andμ≡μ(α)such that for any multi-indexα H_α^[a,t]

Z^s,z_t ,1

p≤C

1+ zμ

(t−a)^−μ. (11)

Therefore the semigroup associated toZ^s,z_t satisfies

∂_αP_tf (z)≤C

1+ zμ

t^−μf_∞

and in particular,the characteristic function ofZ_t^s,zsatisfies E

e^{j θ·Zts,z}≤C

1+ zμ

(t−s)^−μθ^−k

for anyθ∈R^d+1− {0},anyk∈Nand some constantsCandμ.ThereforeZ_t^s,zhas a smooth densitypt−s(z,·).

We will prove the following result which is more general than Theorem1.

Theorem 7. Assume Hypotheses(A1),(A2),(A3)and that the system(7)–(8)is uniformly hypoelliptic in the sense of(10).Then there exists constantsC_ksuch that for allk∈N,we have for allθ∈R^d2

θ^kh(θ )= θ^kE exp

j θ·Y_t^0,xρ(t )≤C_k.

The conclusions on the densities ofY will follow from Lemma2.

4. Proof of Theorem7

The goal is to prove thatθ^k|h(θ )|is a bounded function ofθfor anyk >0. We start with a lemma which expands h(θ ). The proof uses the Itô–Taylor expansion.

Lemma 8. For anyθ∈Randt≥0,then h(θ )=E

exp(j θ·Yt) +E

IN(t, θ )+RN(t, θ )

, (12)

where

I_N(t, θ )=exp(j θ·Y_t) N n=1

I˜_n(t), (13)

I˜_n(t)= _t

0

_s1

0 · · · _sn−1

0

b(X¯ _sn)·dWsn· · · ¯b(X_s1)·dWs₁,

RN(t, θ )=ξt(0, x) t

0

_s1

0

· · · s_N

0

ρs_N+1b(X¯ s_N+1)·dWs_N+1· · · ¯b(Xs₁)·dWs₁. Furthermore

ER_N(t, θ )≤exp

2⁻¹t ¯b²_∞(C(d)t ¯b_∞)^(N+1)/2

√(N+1)! . (14)

HereC(d)is a universal constant that only depends on the dimensiond. Proof. First we just need to apply the Itô–Taylor expansion ofρas follows

ρ_t= N n=1

I˜_n(t)+ _t

0

_s1

0

· · · _sN

0

ρ_sN+1b(X¯ _sN+1)·dWsN+1· · · ¯b(X_s1)·dWs1. In order to obtain the estimate onR_N(t, θ )we just perform anL²(Ω)estimate as follows

ER_N(t, θ )≤E t

0

s₁ 0

· · · s_N

0

ρ_sN+1b(X¯ _sN+1)·dWsN+1· · · ¯b(X_s1)·dWs1

²1/2

≤exp

2⁻¹t ¯b²_∞(C(d)t ¯b_∞)^(N+1)/2

√(N+1)! . Note that here we have used that

E ρ_t²

=E

exp

2 _t

0

b(X¯ _s)·dWs−2 _t

0

b¯^Tb(X¯ _s)ds

exp _t

0

b¯^Tb(X¯ _s)ds

≤exp t ¯b²_∞

.

Considering the result in Lemma8, we have to find upper bounds for each of the terms in (12). We will start with each of the terms in the sum (13). The important idea is to separate the domain of integration in each integralI˜_n in a clever way so that we can apply the integration by parts formula. In order to do this, we define for eachn∈N, i=1, . . . , nthe following points:a_i=₂^ti andb_i=₂^3ti+1. and the following sets

Aⁿ=

(s₁, . . . , s_n)∈ [0, t]ⁿ;s_n≤ · · · ≤s₁≤t , Aⁿ_i =

(s₁, . . . , s_n)∈ [0, t]ⁿ;s₁> a₁, . . . , s_i−1> a_i−1, s_i≤a_i

∩Aⁿ, A^∗_n=

(s1, . . . , sn)∈ [0, t]ⁿ;s1> a1, . . . , sn−1> an−1, sn> an

∩Aⁿ.

What will be important in what follows is that on Aⁿ_i, we have that ass_i ≤a_i ands_i−1> a_i−1 then s_i−1−s_i >

a_i−1−a_i. Then we also define the sets forn∈N ⁿ_i =πⁱ⁻¹

Aⁿ_i

, i=2, . . . , n,

¯

ⁿ_i = ¯πⁿ⁻ⁱ⁺¹ Aⁿ_i

, i=1, . . . , n.

Then we have the following important decompositions. The proof is straightforward Lemma 9. Forn∈Nandi=1, . . . , nwe have

Aⁿ= n i=1

Aⁿ_i +A^∗_n. (15)

FurthermoreAⁿ₁= ¯ⁿ₁and forn∈Nandi=2, . . . , nwe have

Aⁿ_i =ⁿ_i × ¯ⁿ_i. (16)

From this lemma one obtains the following decomposition for the integrals in (13).

Lemma 10. Forn∈N,we have I˜_n(t)= ¯U₁ⁿ+

n i=2

U¯_iⁿU_iⁿ+U_nⁿ₊⁺₁¹, (17)

whereU_iⁿandU¯_iⁿ∈Fai are given by U_iⁿ=

[0,t]ⁱ⁻¹1ⁿ

i

sⁱ⁻¹b(X¯ _si−1)·dWs_i−1· · · ¯b(X_s1)·dWs₁, i=2, . . . , n, (18)

U¯_iⁿ=

[0,t]ⁿ⁻ⁱ⁺¹1_¯n i

sⁿ⁻ⁱ⁺¹b(X¯ _sn)·dWsn· · · ¯b(X_si)·dWs_i, i=1, . . . , n. (19)

The following L²(Ω) estimates are also satisfied: ¯U_iⁿ2 ≤ ¯bⁿ_∞⁻ⁱ⁺¹_(n^a₋ⁱⁿ⁻_i+ⁱ⁺₁₎¹_! and U_iⁿa_i−1,t,0,2≤ ¯bⁱ_∞⁻¹(t − a_i−1)ⁱ⁻¹.

Proof. The proof starts from the previous lemma as I˜_n(t)=

[0,t]ⁿ1Aⁿb(X¯ _sn)·dWsn· · · ¯b(X_s1)·dWs1.

Therefore by (15), we first decompose the above stochastic integral inn+1 stochastic integrals. The first and the last are already the integrals announced in (17). The last term in (17) follows becauseⁿ_n⁺₊¹₁=A^∗_n.

For the other stochastic integrals one uses (16) and finally note thatU¯_iⁿ∈Faiand therefore the first innern−i+1 stochastic integrals can be taken outside of the externali−1 integrals as their region of integration is always taken for values of time strictly greater thana_i. TheL²(Ω)estimates forU_iⁿandU¯_iⁿ follows straightforwardly from the

Itô-isometry.

Proof of Theorem7. Our goal is to prove thatθ^k|E[e^iθ·Yt]| ≤C_kforθbig enough. Throughout the proof various constants may depend onkbut this is not important askis fixed for the rest of the proof (although we may try to be explicit at some points).

First, we note that due to (12) we will only consider its middle term (the other terms which are simpler will be analyzed at the end of the proof):

E

I_N(t, θ )

= N n=1

E

exp(j θ·Y_t)I˜_n(t)

. (20)

Furthermore considering (13) and (17) our goal can be decomposed as follows: Prove that for anyk∈N, anyθ∈R^d2 withθbig enough, there exists a constantC_n,kso that fori=1, . . . , n,

θ^kE

exp(j θ·Y_t)U¯₁ⁿ≤C_n,k, (21)

θ^kE

exp(j θ·Y_t)U¯_iⁿU_iⁿ≤C_n,k, i=2, . . . , n, (22)

θ^kE

exp(j θ·Y_t)U_nⁿ₊⁺₁¹≤C_n,k. (23)

The proof of the first and third statement follow by a simplification of the argument for the second inequality, so we just prove the second. In fact, using the flow property (see Remark5.1 (9)), we have

E

exp(j θ·Yt)U¯_iⁿU_iⁿ

=E

exp(j θ·Ya_i−1)U¯_iⁿE exp

j θ·Y_t^{ai−1,Xai−1}

U_iⁿ|Fa_i−1

=E

exp(j θ·Y_bi)U¯_iⁿE exp

j θ·Y_a^b_iⁱ₋^,X₁^bi

F_iⁿ(X_ai−1)|Fbi

.

Here we have obviously defined F_iⁿ(x)≡F_iⁿ(x;θ ):=E

exp

j θ·Y_t^{ai−1,Xai−1}

U_iⁿ|X_ai−1=x

. (24)

The important issue in the above decomposition and the calculations to follow is that asb¯is not smooth thenU_iⁿ andU¯_iⁿare not smooth random variables in the Malliavin sense and therefore a direct application of the integration by parts formula in (22) or (24) using Proposition4would not work. Instead, we will use the partition of time intervals given in Lemma9so as to allow for the application of partial Malliavin Calculus or conditional integration by parts in the appropriately selected time interval[a_i, b_i].

Now, letφ_iⁿ(z)≡φ_iⁿ(z, θ )=exp(j θ·y)F_iⁿ(x)wherez=(x, y). Therefore using the definition of the semigroup P and the flow property as explained in Remark5.1, we have that

E[exp(j θ·Yt)U¯_iⁿU_iⁿ] =E

exp(j θ·Yb_i)U¯_iⁿPa_i−1−b_iφ_iⁿ(Xb_i,0)

=E

exp(j θ·Y_ai)U¯_iⁿE exp

j θ·Y_b^ai,Xai

i

P_ai−1−biφ_iⁿ(X_bi,0)|Fai

. (25)

Now, we perform integration by parts in the inner conditional expectation with respect to the multi-indexβ ∈ {d₁+1, . . . , d2}^|β|on the interval[a_i, b_i]and the functiong_iⁿ(z)=exp(j θ·y)P_ai−1−biφⁿ_i(x,0). This gives (here we abuse the notation, lettingZ_b^ai,Xai

i ≡Z_b^ai,(Xai,0)

i )

E exp

j θ·Y_b^ai,Xai

i

P_ai−1−biφⁿ_i(X_bi,0)|Fa_i

=(j θ )^{− ˜β}E

∂_βg_iⁿ Z_b^ai,Xai

i

|Fa_i

=(j θ )^{− ˜β}E gⁿ_i

Z^a_b^i,Xai

i

H_β^[ai,bi] Z^a_b^i,Xai

i ,1

|Fa_i

=(j θ )^{− ˜β}E exp

j θ·Y_b^ai,Xai

i

P_ai−1−biφⁿ_i(X_bi,0)H_β^[ai,bi] Z_b^ai,Xai

i ,1

|Fa_i

, (26)

whereβ˜=(β1−d1, . . . , β_|β|−d1).