Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions

www.imstat.org/aihp 2014, Vol. 50, No. 1, 111–135

DOI:10.1214/12-AIHP522

© Association des Publications de l’Institut Henri Poincaré, 2014

Upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motions

Fabrice Baudoin

, Cheng Ouyang

and Samy Tindel

aDepartment of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907-2067, USA. E-mail:fbaudoin@math.purdue.edu bDepartment of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA.

E-mail:couyang@math.uic.edu

cInstitut Élie Cartan Nancy, Université de Lorraine, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France. E-mail:samy.tindel@univ-lorraine.fr Received 22 November 2011; revised 27 August 2012; accepted 29 August 2012

Abstract. In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameterH >1/3. We show that under some geometric conditions, in the regular caseH >1/2, the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough caseH >1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

Résumé. Dans ce papier nous étudions des bornes supérieures pour la densité d’une solution déquation différentielle conduite par un mouvement brownien fractionnaire d’indice de HurstH >1/3. Nous montrons, que sous certaines conditions géomètriques, dans le cas régulierH >1/2, la densité de la solution satisfait l’inégalité de log-Sobolev, l’inégalité de concentration gaussienne et admet une borne supérieure gaullienne. Dans le casH >1/3 et sous la même condition géomètrique, nous montrons que la densité est infiniment différentiable et admet une borne supérieure sous-gaussienne.

1. Introduction

Let B=(B¹, . . . , B^d)be a d dimensional fractional Brownian motion (fBm in the sequel) defined on a complete probability space (Ω,F,P), with Hurst parameter H ∈(0,1). Recall that it means that B is a centered Gaussian process indexed byR₊, whose coordinates are independent and satisfy

E

B_t^j−B_s^j2

= |t−s|^2H fors, t∈R₊. (1)

In particular, by considering the family{B^H;H∈(0,1)}, one obtains some Gaussian processes with any prescribed Hölder regularity, while fulfilling some intuitive scaling properties. This converts fBm into the most natural general- ization of Brownian motion to this day.

We are concerned here with the following class of equations driven byB:

X^x_t =x+ t

0

V0

X^x_s ds+

d i=1

t 0

V_i X_s^x

dB_sⁱ, (2)

1Supported in part by NSF Grant DMS-09-07326.

2Supported in part by the (French) ANR grant ECRU.

wherex is a generic initial condition and {V_i;0≤i≤d}is a collection of smooth vector fields ofR^d. Owing to the fact that fBm is a natural generalization of Brownian motion, this kind of model is often used by practitioners in different contexts, among which we would like to highlight recent sophisticated models in Biophysics [24,33,34].

As far as mathematical results are concerned, equation (2) is now a fairly well understood object: existence and uniqueness results are obtained forH >¹₂ thanks to Young integral type tools [31,36], while rough paths methods [16,26] are required for ¹₄< H <¹₂. Numerical schemes can be implemented for this kind of systems [14,16], and a notion of ergodicity is also available [19,20]. Finally, the law ofX^x_t has been analyzed by means of semi-group type methods [1,28] and its density has also been investigated in [2,8,23,32].

In spite of these advances, concentrations results and Gaussian bounds for the solution to (2) are scarce: we are only aware of the large deviation results [27] in this line of investigation. The current article is thus an attempt to make a step in this direction, by analyzing a special but nontrivial situation.

Indeed, we consider here equation (2) driven by a fBm with Hurst parameterH∈(¹₃,1), and we suppose that our vector fieldsV0, . . . , Vdfulfill either of the following non-degeneracy and antisymmetric hypothesis:

Hypothesis 1.1. The vector fieldsV0, . . . , VdareC^∞-bounded,andV1, . . . , Vd satisfy (i) For everyx∈R^d,the vectorsV₁(x), . . . , V_d(x)form a basis ofR^d.

(ii) There exist smooth and bounded functionsω^k_ij such that:

[V_i, V_j] = d k=1

ω^k_ijV_k and ω_ij^k = −ω^j_ik. (3)

The second assumption (ii) is of geometric nature and actually means that the Levi–Civita connection associated with the Riemannian structure given by the vector fieldsV_i’s is

∇XY=1 2[X, Y].

In a Lie group structure, this is equivalent to the fact that the Lie algebra is of compact type, or in other words that the adjoint representation is unitary. Such geometric assumption already appeared in the work [3] where it was used to prove a small-time asymptotics of the density.

Hypothesis 1.2. The Hypothesis1.1is satisfied and moreover,the vector fieldsV₁, . . . , V_d form moreover a uniform elliptic system.That is

v^TV V^Tv≥λ|v|² for allv∈R^d.

HereV =(V_jⁱ)_i,j=1,...,dandλis a positive constant.

WhenH >¹₂, under Hypothesis1.1our main result can be loosely summarized as follows (see Theorem3.13for a precise statement):

Theorem 1.3. FixH >¹₂.LetX^x be the solution to equation(2),and suppose Hypothesis1.1is satisfied.Then for anyt∈R^∗₊,the random variableX^x_t admits a smooth densityp_X(t,·).Furthermore,there exist3positive constants c_t⁽¹⁾, c_t⁽²⁾, c⁽³⁾_t,xsuch that

p_X(t, y)≤c⁽¹⁾_t exp

−c⁽³⁾_t

|y| −c⁽²⁾_t,x2 ,

for anyy∈R^d.

We do not claim any optimality in the quantitiesc⁽¹⁾_t , c⁽²⁾_t andc_t,x⁽³⁾above (whose exact definitions are postponed to Section3.3). Nevertheless, this is (to the best of our knowledge) the first Gaussian type bound available for solutions

of differential equations driven by fBm. Let us observe that such global Gaussian bounds of course does not hold in general and crucially relies on some geometric assumptions. Typically in the Brownian motion case, a non negative lower bound for the Ricci curvature is required. The assumption (ii) here is stronger and it is easily seen that it implies that the Ricci curvature of the Levi–Civita connection associated with the Riemannian structure given by the vector fieldsV_i’s is non negative.

Let us say a few words about the strategy we have followed in order to prove Theorem1.3. It is mostly based on stochastic analysis tools, and particularly on a general integration by parts formula giving an exact expression for the densityp_X(t,·)in terms of Malliavin derivatives in the non-degenerate case we are dealing with. In this context, it is crucial to bound the first Malliavin derivative of X_t^x (calledDX_t^x in the sequel) efficiently. This is where our asymmetry hypothesis on the vector fieldsV₁, . . . , V_nenter into the picture, and we shall see (at Theorem3.1) how asymmetry properties yield an easy deterministic bound onDX^x_t. This result enables to get concentration results for the law ofX_t^x, and is the key to our density bounds as well.

As another interesting consequences of the deterministic bound on DX_t^x, we also obtain logarithmic Sobolev inequality and Poincaré inequality for the law ofX_t.

Once the picture for the smooth case (when H > ¹₂) becomes clear, we are able to extend some of our results described above to the irregular case when ¹₃< H <¹₂. In particular, we are able to prove

Theorem 1.4. FixH∈(¹₃,¹₂).Assume Hypothesis1.2.LetX^xbe the solution to equation(2)andγ_Xt the Malliavin matrix ofX_t^x,t >0.We have|detγ_Xt|⁻¹∈L^∞(P).The random variableX_t^xadmits a smooth densityp_X(t,·)and for anyδ < Hthere exist2positive constantsc_t⁽¹⁾such that

p_X(t, y)≤c⁽¹⁾_t exp

−|y−x|^δ

, (4)

for ally∈R^d.

The existence of a density for solutions to stochastic differential equations of the form (2) under Hörmander’s condi- tion has been obtained by Cass and Friz [7] for any ¹₄< H <¹₂. While finishing the current article, an important step towards the study of regular densities in the rough case ¹₄< H <¹₂ has been accomplished in [9], where integrability estimates for the Jacobian of equation (2) are established. Nevertheless, as of today, besides the result of P. Driscoll [15], when H < ¹₂, to the best of our knowledge, Hypothesis1.2is a first wide class of examples where we have an affirmative answer for the smoothness of the density. Our bound on the inverse of the Malliavin matrix together with polynomial bounds on the Hölder norm of the Malliavin derivative allows then to obtain the sub-Gaussian upper bound (4).

Notations. Throughout this paper,unless otherwise specified we use| · |for Euclidean norms and · L^p for theL^p norm with respect to the underlying probability measureP.

Consider a finite-dimensional vector space V.The space of V-valued Hölder continuous functions defined on [0,1],with Hölder continuity exponentγ∈(0,1),will be denoted byC^γ(V ),or justC^γ when this does not yield any ambiguity.For a functiong∈C^γ(V )and0≤s < t≤1,we shall consider the semi-norms

gs,t,γ= sup

s≤u<v≤t

|g_v−g_u|V

|v−u|^γ . (5)

The semi-normg0,1,γ will simply be denoted bygγ. 2. Stochastic calculus for fractional Brownian motion

For some fixed H∈(¹₃,1), we consider (Ω,F,P)the canonical probability space associated with the fractional Brownian motion (in short fBm) with Hurst parameterH. That is,Ω=C0([0,1])is the Banach space of continuous functions vanishing at 0 equipped with the supremum norm,Fis the Borel sigma-algebra andPis the unique prob- ability measure onΩ such that the canonical processB= {Bt =(B_t¹, . . . , B_t^d), t∈ [0,1]} is a fractional Brownian

motion with Hurst parameterH. In this context, let us recall thatB is ad-dimensional centered Gaussian process, whose covariance structure is induced by equation (1). This can be equivalently stated as

R(t, s):=E B_s^jB_t^j

=1 2

s^2H+t^2H− |t−s|^2H

, fors, t∈ [0,1]andj=1, . . . , d.

In particular it can be shown, by a standard application of Kolmogorov’s criterion, thatBadmits a continuous version whose paths areγ-Hölder continuous for anyγ < H.

This section is devoted to give the basic elements of stochastic calculus with respect toBwhich allow to understand the remainder of the paper.

2.1. Malliavin calculus tools

Gaussian techniques are obviously essential in the analysis of fBm, and we proceed here to introduce some of them (see [30] for further details): letEbe the space ofR^d-valued step functions on[0,1], andHthe closure ofEfor the scalar product:

(1_[0,t₁], . . . ,1_[0,t_d]), (1_[0,s₁], . . . ,1_[0,s_d])

H= d i=1

R(ti, si).

We denote byK_H^∗ the isometry betweenHandL²([0,1]). WhenH > ¹₂it can be shown thatL^1/H([0,1],R^d)⊂H, and when¹₃< H <¹₂one has

C^γ⊂H⊂L² [0,1]

for allγ >¹₂−H.

Some isometry arguments allow to define the Wiener integralB(h)=₁

0h_s,dBsfor any elementh∈H, with the additional propertyE[B(h₁)B(h₂)] = h₁, h_2Hfor anyh₁, h₂∈H. AF-measurable real valued random variableF is then said to be cylindrical if it can be written, for a givenn≥1, as

F=f B

h¹ , . . . , B

hⁿ

=f ₁

0

h¹_s,dBs

, . . . , ₁

0

hⁿ_s,dBs

,

wherehⁱ ∈Handf :Rⁿ→Ris aC^∞bounded function with bounded derivatives. The set of cylindrical random variables is denotedS.

The Malliavin derivative is defined as follows: forF ∈S, the derivative ofF is theR^dvalued stochastic process (DtF )0≤t≤1given by

DtF = n i=1

hⁱ(t)∂f

∂xi

B h¹

, . . . , B hⁿ

.

More generally, we can introduce iterated derivatives. IfF ∈S, we set D^k_t1,...,tkF =Dt1· · ·DtkF.

For anyp≥1, it can be checked that the operatorD^k is closable fromS intoL^p(Ω;H^⊗k). We denote byD^k,p the closure of the class of cylindrical random variables with respect to the norm

Fk,p=

E F^p

+ k j=1

ED^jF^p

H^⊗j

1/p

,

and

D^∞=

p≥1

k≥1

D^k,p.

2.2. Differential equations driven by fBm

Recall that we consider the following kind of equation:

X^x_t =x+ _t

0

V₀ X^x_s

ds+ d i=1

_t

0

V_i X_s^x

dB_sⁱ, (6)

where the vector fieldsV₀, . . . , V_nareC^∞-bounded.

When equation (6) is driven by a fBm with Hurst parameter H > ¹₂ it can be solved, thanks to a fixed point argument, with the stochastic integral interpreted in the (pathwise) Young sense (see e.g. [17]). Let us recall that Young’s integral can be defined in the following way:

Proposition 2.1. Letf∈C^γ,g∈C^κwithγ +κ >1,and0≤s≤t≤1.Then the integral_t

s gξdfξ is well-defined as limit of Riemann sums along partitions of[s, t].Moreover,the following estimation is fulfilled:

_t

s

g_ξdfξ

≤Cfγgκ|t−s|^γ, (7)

where the constantConly depends onγandκ.A sharper estimate is also available:

_t

s

g_ξdfξ

≤ |g_s|fγ|t−s|^γ+c_{γ ,κ}fγgκ|t−s|^γ+κ. (8) With this definition in mind, we can solve our differential system of interest, and the following moments bounds are proven in [23]:

Proposition 2.2. (Hu-Nualart)Consider equation(6)driven by a fBm B with Hurst parameterH > ¹₂.Let us call X^xits uniqueβ-Hölder continuous solution,for anyβ < H.Then

(1) When vector fieldsV areC^∞-bounded,we have sup

t∈[0,T]

X_t^x≤ |x| +cV ,TB^1/β_{0,T ,β}.

(2) If we only assume that vector fieldsV have linear growth,with∇V ,∇²V,bounded,the following estimate holds true:

sup

t∈[0,T]

X_t^x≤ 1+ |x|

exp

cV ,TB^1/β_{0,T ,β}

. (9)

Remark 2.3. The framework of fractional integrals is used in[23]in order to define integrals with respect toB.It is however easily seen to be equivalent to the Young setting we have chosen to work with.

When the Hurst parameter¹₃< H <¹₂, equation (6) can be solved, again by fixed point argument, with the stochas- tic integral interpreted in the (pathwise) rough path theory (see e.g. [17] and [26]). In this case, we obtain

Proposition 2.4. (Besalú-Nualart[5])Consider equation(6)driven by a fBmBwith Hurst parameter ¹₃< H <¹₂. Denote byX^xits uniqueβ-Hölder continuous solution,for anyβ < H.If the vector fieldsV areC^∞-bounded,then for anyλ >0andδ < H

E expλ

sup

0≤t≤T

|X_t−x|^δ

<∞.

Once equation (6) is solved, the vectorX^x_t is a typical example of random variable which can be differentiated in the Malliavin sense. In fact, fixH∈(¹₃,1), one gets the following results (see [7] and [32] for further details):

Proposition 2.5. LetX^xbe the solution to equation(6)and supposeV_i’s areC^∞-bounded vector fields onR^d.Then for everyi=1, . . . , d,t >0,andx∈R^d,we haveX_t^x,i∈D^∞and

D^jsX^x_t =J0→tJ⁻_0→¹_sVj(Xs), j=1, . . . , d,0≤s≤t, whereD^jsX^x,i_t is thejth component ofDsX_t^x,i,andJ0→t=^∂X_∂x^xt .

Finally the following approximation result, which can be found for instance in [16], will also be used in the sequel:

Proposition 2.6. Form≥1andT >0,letB^m= {B_t^m;t∈ [0, T]}be the sequence of linear interpolations ofBalong the dyadic subdivision of[0, T]of meshm;that is ift_i^m=i2^−mT fori=0, . . . ,2^m;then fort∈(t_i^m, t_i^m₊₁],

B_t^m=B_t^m

i + t−t_i^m t_i^m₊₁−t_i^m(B_t^m

i+1−B_t^m

i ).

ConsiderX^mthe solution to equation(6)restricted to[0, T],whereB has been replaced byB^m.Set alsoD^jsX^m_t = J0→tJ⁻_0→¹_sV_j(X^m_s ),forj =1, . . . , dand0≤s≤t.Then almost surely,for anyγ < H the following holds true:

mlim→∞X^x−X^m

0,T ,γ+D^jX^x_t −D^jX^m_t

0,T ,γ

=0. (10)

3. Estimates for solutions of SDEs driven by fBm: The smooth case

Throughout this section, we fixH∈(¹₂,1). Recall thatX^xdesignates the solution to (6). This section is devoted to getting some further bounds forX_t^xand its Malliavin derivatives, under Hypothesis1.1.

We are now ready to prove the main result of this section, which is an almost sure deterministic bound for the Malliavin derivative ofX_t^x:

Theorem 3.1. Under Hypothesis1.1,the Malliavin derivative of the solutionX^x_T to equation(6)can be bounded as follows for anyT ∈ [0,1](in the almost sure sense):

DX^x_T

∞≤Mexp(CT ), withM= sup

x∈Rⁿ sup

λ≤1

d i=1

λ_iV_i(x)

2

, (11)

and where the constantClinearly depends onV₀.In particular,one also hasDX_T^x_H≤Mexp(CT ).

Proof. Let us focus on the proof of (11). Indeed, sincef_His dominated by the supremum norm whenH > ¹₂, this will be sufficient in order to prove the second claim of our theorem. We now split our proof in two steps.

Step 1: Matricial expression for the derivative.Let us first restate Proposition2.5in the following form:DX^x_T is solution to

D^jsX^x_T =J0→T

Φ_s^∗V_j

(x), 0≤s≤T ,

whereΦ_s^∗Vj denotes the pullback action of the diffeomorphismΦs =X^·_s:R^d→R^d on the vector fieldVj. Now, a simple application of the change of variable formula for Young type integrals yields

d Φ_s^∗Vj

(x)=

Φ_s^∗[V0, Vj] (x)ds+

d i=1

Φ_s^∗[Vi, Vj] (x)dB_sⁱ.

Moreover, recall that the Lie brackets[V_i, V_j]can be decomposed, according to Assumption1.1, into [V0, V_j] =

d k=1

ω^k_0jV_k, and [V_i, V_j] = d k=1

ω_ij^kV_k,

withω^k_ij= −ω^j_ikfori, j, k≥1. Hence,

d Φ_s^∗V_j

(x)= d k=1

ω_0j^k X_s^x

Φ_s^∗V_k (x)ds+

d k=1

d i=1

ω^k_ij X_s^x

Φ_s^∗V_k (x)dB_sⁱ.

By denotingMsthed×dmatrix with columns M^js =D^jsX^x_T =J0→T

Φ_s^∗V_j (x),

we therefore obtain the equation

dMs=Ms

ω0

X_s^x ds+

d i=1

ω_i X^x_s

dB_sⁱ

, MT =V X_T^x

, (12)

whereV (X^x_T)is the matrix with columnsV_j(X^x_T), 1≤j ≤d and whereω_i(X^x_s)is the skew symmetric matrix with entriesω^k_ij(X^x_s).

Step 2: Bound onM.We now prove that the processMis uniformly bounded. We have d

MsM^Ts

=(dMs)M^Ts +Ms

dM^Ts

=Ms

ω₀

X^x_s ds+

d i=1

ω_i X_s^x

dB_sⁱ

M^T_s +Ms

ω₀^T

X^x_s ds+

d i=1

ω_i^T X_s^x

dB_sⁱ

M^T_s

=Ms

ω₀ X_s^x

+ω^T₀ X^x_s

M^Ts ds.

Taking into account the terminal conditionMT =V (X^x_T)gives then the expected result as a straightforward conse-

quence of Gronwall’s inequality.

Once the bound (11) onDX_T^x_∞is obtained, one can also retrieve some information on the Hölder norms ofDX^x_T improving the general estimate (9). This is the content of the following proposition:

Proposition 3.2. Consider ¹₂< γ < H and setc^T₀ =Mexp(CT ),which is the constant appearing in relation(11).

Under Hypothesis1.1,the Malliavin derivative of the solutionX_T^x to equation(6)can be bounded as follows for any T ∈ [0,1]:

DX_T^x

γ ≤c_{T ,V ,d}

1+ |x| + B^1/γγ

B⁽¹γ^{−γ )/γ}, (13)

for a strictly positive constantc_{T ,V ,d}.

Proof. We have shown at Theorem3.1thatDX^x_T is governed by equation (12), and thatM_∞≤c^T₀. We will now separate our proof into a local and a global estimate, and notice that the constants appearing in the computations below might change from line to line.

Step 1: Local estimate.Consider 0≤s < t≤T and setε=t−s. Letu < vbe two generic elements of[s, t].

Applying relation (8) to the expression ofMv−Mugiven by equation (12), we obtain

|Mv−Mu| ≤c₀^Tc_V|v−u| +

d i=1

|Mu|ω_i

X^x_uBγ|v−u|^γ+c_γMω_i X^x

s,t,γBγ|v−u|^2γ .

We thus obtain, for a constantcV depending on the vector fieldsV, Ms,t,γ ≤c₀^Tc_V|t−s|^1−γ+dc^T₀c_VBγ+c_γ

c^T₀c_VX^x

γ+c_VMs,t,γ

Bγ|t−s|^γ

≤c₀^TcV

T^1−γ +dBγ

+dc^T₀cVX^x

γBγ|t−s|^γ+dcVBγ|t−s|^γMs,t,γ. (14) Take nowt−s=εsuch that dcVBγε^γ =1/2, namelyε= [2dcVBγ]^−1/γ. Recall also thatX^xγ ≤c(1+ B^1/γγ )according to [16]. It is then easily seen that relation (14) yields

Ms,t,γ≤c₀^Tc_{T ,V ,d}

1+ |x| + B^1/γ_γ

:=a_{T ,V ,d,B}, for a strictly positive constantc_{T ,V ,d}.

Step 2: Global estimate.We consider nows, t∈ [0, T]such thatiε≤s < (i+1)ε≤j ε≤t < (j+1)ε, whereε has been defined at Step 1. Set alsot_i=s,t_k=kεfori+1≤k≤j, andt_j+1=t. Then

|Mt−Ms| =

j k=i

Mt_k+1−Mt_k

≤a_{T ,V ,d,B} j

k=i

(t_k+1−t_k)^γ

≤a_{T ,V ,d,B}(j−i+1)^1−γ(t−s)^γ, (15)

where we have used the fact thatr→r^λis a concave function. Note that the indicesi, j above satisfy(j−i+1)≤ 2T /ε. Plugging this into the last series of inequalities, we end up with our claim (13).

3.1. Log-Sobolev inequality

In this section, we present some interesting functional inequalities which are usually studied in a Markov setting;

namely, the logarithmic Sobolev inequality and Poincaré inequality. As we will see, these inequalities become avail- able in our non-Markov case when we have uniform boundedness for the Malliavin derivative of X_T (see Theo- rem3.1).

We start with the following version of logarithmic Sobolev inequality for the law ofXT. Theorem 3.3. LetCandMbe as in Theorem3.1.We have for allf∈C¹andT ∈ [0,1],

Ef (X_T)²lnf (X_T)²−

Ef (X_T)² ln

Ef (X_T)²

≤2M²e^2CTT^2HE∇f (X_T)² provided the right hand side in the above is finite.

Proof. The proof is standard by applying Clark–Ocone formula (see e.g. [6]). First recall the representation of frac- tional Brownian motion

B_t= t

0

K_H(t, s)dWs.

HereW is ad-dimensional Wiener process. Denote byD^W the Malliavin derivative with respect to the Wiener pro- cessW. We have

K_H^∗D=D^W, (16)

whereK_H^∗ is the isometry fromH, the reproducing kernel space ofB, toL². By Clark–Ocone formula we have g(X_T)−Eg(X_T)=

_T

0

E

D^W_s g(X_T)|Fs

dWs= _T

0

E K_H^∗

Dg(XT)

s|Fs

dWs.

Hence, if we denoteM_s=E(g(X_T)|Fs),0≤s≤T, we have dMs=E

K_H^∗

Dg(XT)

s|Fs

dWs.

For simplicity we may assume thatg≥εfor someε >0, which can be removed afterwards by lettingεtend to 0.

Applying Itô’s formula toMslnMs, we get Eg(XT)ln

g(XT)

−Eg(XT)ln

Eg(XT)

=E(MTlnMT)−E(M0lnM0)

=1 2E

T 0

1 Ms

E K_H^∗

Dg(XT)

s|Fs²ds. (17)

Replace nowgbyf²in the above. By the Cauchy–Schwarz inequality, E

K_H^∗

Df (XT)²

s|Fs²=4E

f (XT)∇f (XT), K_H^∗(DXT)s

|Fs²

≤4E

f (X_T)²|Fs

E ∇f (X_T), K_H^∗(DX_T)_s2

|Fs

.

Substituting the above to (17), together with Theorem3.1, we obtain the desired result.

As a corollary of the logarithmic Sobolev inequality obtained above, we have the following Poincaré inequality (see e.g. [22, Theorem 8.6.8]).

Theorem 3.4. LetCandMbe in Theorem3.1.We have for allf ∈C¹, Ef (X_T)²−

Ef (X_T)2

≤M²e^2CTT^2HE∇f (X_T)², provided the right hand side in the above is finite.

Remark 3.5. In the above,assume further that the vector fieldsV₁, . . . , V_d form an uniform elliptic system,we ob- tain the following natural expression of logarithmic Sobolev inequality and Poincaré inequality when working on a Riemannian manifold

Ef (X_T)²lnf (XT)²−

Ef (X_T)² ln

Ef (X_T)²

≤2aM²e^2CTT^2H d i=1

E|V_if|², and

Ef (X_T)²−

Ef (X_T)2

≤aM²e^2CTT^2H d i=1

E|V_if|².

3.2. Concentration inequality

It is classical that the boundedness of the Malliavin derivative inHimplies the Gaussian concentration inequality, more precisely (see [35, Theorem 9.1.1] or [25]):

Lemma 3.6. LetF ∈D^1,2such that almost surelyDF_H≤C,whereCis a non random constant.Then,for every θ≥0,

E e^{θ F}

≤e^{E(F )θ+(1/2)C2θ2}

and therefore for everyx≥0, P

F−E(F )≥x

≤e^−x2/(2C2). (18)

As a corollary of this lemma, we deduce

Proposition 3.7. Assume that the Hypothesis1.1is satisfied,then there existCandMsuch that for everyT ≥0and λ≥0,

P sup

0≤t≤T

X_t^x−E sup

0≤t≤T

X^x_t≥λ ≤exp

− λ² 2M²e^2CTT^2H

.

Proof. Let F= sup

0≤t≤T

X^x_t.

By Theorem3.1, it is not hard to see that (see e.g. [30]) DF_H≤Me^CTT^H,

whereMandCare the same as in Theorem3.1. Now an easy application of Lemma3.6completes our proof.

Remark 3.8. Notice that this relation can only be obtained forH >¹₂.Indeed,denote byC⁰the space of continuous functions endowed with the supremum norm.Then the norm involved in[35,Theorem9.1.1]isDXtL^∞(H).This norm is dominated byDXtL^∞(C⁰)only ifH >¹₂.

3.3. Gaussian upper bound

One natural way to estimate the density ofX_t is to apply the results in [30, Chapter 2]. More precisely, we first have the following integration by parts formula for non-degenerate random vectors.

Proposition 3.9. Let F =(F¹, . . . , F^d) be a non-degenerate random vector whose components are in D^∞, and γ_F the Malliavin matrix of F.LetG∈D^∞ and ϕ be a function in the spaceC_p^∞(R^d). Then for any multi-index α∈ {1,2, . . . , d}^k, k≥1,there exists an elementH_α∈D^∞such that

E

∂_αϕ(F )G

=E

ϕ(F )H_α .

Moreover,the elementsH_α are recursively given by H_(i)=

d j=1

δ G

γ_F⁻¹ij

DF^j ,

H_α=H_(αk)(H_{(α1,...,αk−1)}), and for1≤p < q <∞we have HαL^p≤Cp,qγ_F⁻¹DF^k

k,2^k−1rGk,q, where_p¹ =_q¹+¹_r.

Remark 3.10. By the estimates forH_α above,one can conclude that there exist constantsβ, γ >1and integersm, n such that

H_αL^p≤C_p,qdetγ_F^−1m

L^βDFⁿ_k,γGk,q. (19)

Remark 3.11. In what follows,we useH_α(F, G)to emphasize its dependence onF andG.

As a consequence of the above proposition, one has

Proposition 3.12. LetF =(F¹, . . . , F^d)be a non-degenerate random vector whose components are inD^∞.Then the densityp_F(x)ofF belongs to the Schwartz space,and

p_F(x)=E

I_{F >x}H(1,2,...,d)(F,1) .

In the above,F > xis used to denote the eventFⁱ> xⁱ, i=1,2, . . . , d.

Now we state and prove a global Gaussian upper bound for the density function ofX^x_t.

Theorem 3.13. Denote byp_X(t, y) the density function ofX^x_t.There exist positive constants c⁽¹⁾_t , c_t,x⁽²⁾, c₃, andc₄ such that for allt∈ [0,∞),

pX(t, y)≤c⁽¹⁾_t exp

−(|y| −c⁽²⁾_t,x)² c₃e^c4tt^2H

. (20)

Herec_t⁽¹⁾is of the form

c⁽¹⁾_t =O 1

t^α

ast↓0,

for some positive numberα;andc_t,x⁽²⁾converges to a constant ast↓0.

Remark 3.14. In the above theorem,we do not prove that the value ofαis indeeddH,as one should expect.This will be studied in a subsequent paper[4]that discusses the strict positivity and Gaussian bounds of the density,which are important ingredients toward the analysis of some hitting probabilities ofXt (cf. [12]).Since obtainingα=dH requires assumptions and involves detailed computation for orders ofD^kX_t intfor eachkas well as a more careful application of Proposition3.9,for sake of conciseness we omit the details here and refer the reader to the forthcoming paper[4].

Proof. Fixβ∈(¹₂,1). By (19) and Proposition3.12, we have p_X(t, y)≤C

P{X_t> y}1/qdetγ_X⁻¹

t ^m

L^αDXtⁿ_k,γ, (21)

for some constantsα, γ >1 and integersm, n. Without loss of generality, we may assumey_i≥0 for 1≤i≤d. Since otherwise, for exampley_j<0, we can consider the alternative expression for the density

p_X(t, y)=E

i=j

I_{yi<Xⁱ_t}I_{

y^j>X^j_t}H(1,2,...,d)(X_t,1)

,

and deduce similar estimate. In the following, we provide estimate forDXtⁿ_k,γ,detγ_X⁻¹

t ^m_LαandP{X_t > y}respec- tively.

To this end, first note that we have the linear stochastic differential equation satisfied by the matrix valued process η_t=γ_X⁻¹

t (see e.g. [10, Proposition 3.8] or [23]), that is, ηt=α₀⁻¹−

d =1

_t

0

ηuα(Xu)+α^∗(Xu)ηu

dB_u− _t

0

ηuβ(Xu)+β^∗(Xu)ηu

du,