Probability density for a hyperbolic SPDE with time dependent coefficients

August 2007, Vol. 11, p. 365–380 www.edpsciences.org/ps

DOI: 10.1051/ps:2007024

PROBABILITY DENSITY FOR A HYPERBOLIC SPDE WITH TIME DEPENDENT COEFFICIENTS

Marta Sanz-Sol´ e

and Iv´ an Torrecilla-Tarantino

Abstract. We prove the existence and smoothness of density for the solution of a hyperbolic SPDE with free term coefficients depending on time, under hypoelliptic non degeneracy conditions. The result extends those proved in Cattiaux and Mesnager, PTRF 123(2002) 453-483 [2] to an infinite dimensional setting.

Mathematics Subject Classification. 60H07, 60H15, 60G60.

Received July 10, 2006. Accepted December 26, 2006.

Introduction

The initial developments of Malliavin Calculus provided a probabilistic proof of H¨ormander’s theorem for hypoelliptic operators in square form. As an application, the existence and smoothness of the density for the solution of diffusion processes with coefficients depending only on the spatial variable were obtained (see [5]).

There have been several attempts to extend this result to diffusions with coefficients depending on two variables, time and space. The first results in this direction, [3, 11], apply to pretty smooth coefficients (see the cases termed in Section 2 assmooth,factorableandregular H¨older). More recently, Cattiaux and Mesnager [2] solved the problem for hypoelliptic coefficients under less restrictive smoothness conditions on the coefficients.

The classical application of Malliavin Calculus mentioned before has been extended in [8] to the two-parameter Itˆo equation – a wave equation in reduced form. Rules of two-parameter Itˆo calculus differ from those of the classical one. As a consequence, the analogue of H¨ormander’s condition is formulated in terms of thecovariant derivativeinstead of the Lie brackets. In view of the results of [2], a natural question is whether one could also extend the results from [8] to coefficients of the equation depending on the two-dimensional time parameter.

This article is devoted to study this problem. Combining the techniques of [8] with those of [2], we prove in Theorem 1.1 such an extension.

When studying the inverse of the Malliavin matrix corresponding to homogeneous diffusion processes, one needs estimates of the type

P _S

0 Yt²dt≤aε^δ, _S

0 α²tdt≥bε^η

≤ε^p,

Keywords and phrases. Malliavin calculus. Stochastic partial differential equations. Two-parameter processes.

∗Supported by the grant BMF 2003-01345 from the Direcci´on General de Investigaci´on, Ministerio de Ciencia y Tecnolog´ıa, Spain.

1 Facultat de Matem`atiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain;

[email protected]; [email protected]

c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.esaim-ps.org or http://dx.doi.org/10.1051/ps:2007024

for small ε and bounded stopping time S. Here Yt = Y0+Mt+Vt, t ≥ 0, is a continuous semimartingale with martingale and bounded variation components,MtandVt, respectively; it is assumed that the quadratic variation ofMtis of the form_t

0α²_sds(see for instance [6, 10] for different proofs of this result).

When dealing with non-homogeneous diffusions with somehow rough coefficients, such result does not suffice.

Actually, the crucial steps of the proofs of Theorem 2.6 and Proposition 3.2 in [2] consist of establishing alternate suitable extensions using a new approach an novel ideas. Setting these ideas in an abstract framework, we prove in Section 2 more sophisticated versions of Stroock-Norris type estimates which are shown to be useful for two-parameter processes.

The paper is organized as follows. In Section 1, we introduce the notation, the different hypothesis we are going to consider along the paper, and we state the main result. Section 2 is devoted to the extension of Stroock-Norris estimates. With these tools, we prove in Section 3 the main result.

1. Notations, assumptions and the main result

Consider the stochastic differential equation onR^m

Xz=x0+

Rz

_d

l=1

Al(r, Xr) dWr^l+A0(r, Xr) dr

, (1)

where z = (s, t)∈ [0, S]×[0, T], 0 ≤S, T < ∞, x0 ∈ R^m, Rz = [0, s]×[0, t] and Wz =

W_z¹, . . . , W_z^d is a d-dimensional Brownian sheet (see [1]).

SetE={(s, t)∈RS,T : st= 0}. We assume that the coefficients of (1) satisfy the conditions

(h1) Al :RS,T ×R^m−→R^m, 0≤l≤d, areγ-H¨older continuous in t, for some γ∈]0,1[, measurable with respect to s, and infinitely differentiable with respect to the spatial variable. Moreover,

Kγ:= sup

y∈R^m sup

0≤θ≤S max

0≤l≤dAl(θ,·, y)_γ<∞, (2)

where the notation · γ refers to the usual H¨older norm.

(h2) for any multi-indexα= (α1, . . . , αm),|α| ≥0, and 0≤l≤d, the partial derivatives∂_α^xAl with respect tox∈R^mexist and

K:= sup

0≤θ≤S sup

0≤τ≤T max

0≤|α|≤N+2 max

0≤j≤d∂_α^xAj(θ, τ,·)_∞<∞, (3) (h3) for a fixedz = (s, t)∈ E, the vector fieldsAl’s, 1≤l≤dsatisfy the restricted H¨ormander’s condition

stated as follows:

The vector space spanned by the vector fields A1, . . . , Ad, A^∇i Aj, 1 ≤ i, j ≤ d, A^∇i

A^∇jAk , 1 ≤ i, j, k≤d,. . .,A^∇_i1

. . .

A^∇_in−1Ain

. . .

, 1≤i1, . . . , in ≤d,. . ., at the point (0, t, x0) has full rank.

Here, the notationA^∇_i Aj denotes the covariant derivative of the vector fieldAj alongAi.

Assumption (h2) implies that the coefficients of the equation are Lipschitz functions and have linear growth in the spatial variable, uniformly in the time variables. By the usual method of Picard’s iterates, one can prove existence and uniqueness of solution for the equation (1) and moreover, the solution has almost surely continuous paths (see Lem. 3.1 in [8]).

Condition (h3) clearly implies the following. There exist N := N(x0, t) ∈ N, and positive real numbers cN :=cN(x0, t),s0:=s0(t), R:=R(t) such that for anyv∈S^m−1,

N k=0

V∈Σk

v, V(θ, t, y) ²≥cN, (4)

for any (θ, y)∈[0, s0]×B(x0, R), where Σ₀={Al, 1≤l≤d}, Σk={A^∇_l V, 1≤l≤d, V ∈Σ_k−1},k≥1.

We consider different combinations of regularity of the coefficients and non degeneracy conditions of the underlying differential operator, as follows.

(1) Elliptic case: (h1) and (h2) holds. Moreover, the vector space spanned by the vector fieldsA1, . . . , Ad

at the point (0, t, x0) has full rank.

(2) Smooth case: (h1) to (h3) hold. In addition, for each 0≤l≤d, the functions∂_α^xAl’s areC_b¹ ins, for all multi-index α,

K1:= sup

0≤θ≤S sup

0≤τ≤T max

0≤|α|≤N max

1≤j≤d∂^θ∂_α^xAj(θ, τ,·)

∞<∞. (5)

(3) Factorable case: Al(θ, τ, x) =fl(θ)Al(τ, x), 0≤l ≤d, withfl measurable, _c¹ ≥ |f_l| ≥c >0. The functions Al’s, 0 ≤ l ≤ d, satisfy the analogue of hypothesis (h1)-(h3) for coefficients which do not depend onθ.

(4) Regular H¨older case: (h1) to (h3) hold. In addition, for any multi-index α, the functions ∂_α^xAj, 0≤j≤d, areβ(α)-H¨older continuous in the arguments, for someβ(α)∈₁

2,1

. Moreover, K_β(α):= sup

y∈R^m sup

0≤τ≤T max

0≤|α|≤N+2 max

0≤l≤d∂_α^xAl(·, τ, y)_β(α)<∞. (6) (5) Irregular H¨older case: The same assumptions as in the preceding case, except that hereβ(α)∈

0,¹₂ . The main result of this paper is the next theorem, stating the existence and smoothness of density for the probability law of the solution of (1) at any fixed pointz∈ E.

Theorem 1.1. Let X ={Xz, z∈RS,T} be the solution of (1). Each one of the set of assumptions termed before as elliptic, smooth, factorable, regular H¨older and irregular H¨older, imply that the random vector Xz, for fixed z∈ E, has an infinitely differentiable density with respect to the Lebesgue measure.

Remark 1.2. In the formulation of assumptions (h1)–(h3) and of the different scenaries, the roles of the time componentss andt might be exchanged.

2. Stroock-Norris type lemmas for continuous semimartingales depending on a parameter

In this section, we prove two extensions of the Stroock-Norris estimates. The difference between them stands on the order of H¨older continuity of the involved processes. We notice that Lemma 2.1 could provide as a by-product an alternate proof of Norris Lemma [6].

Lemma 2.1. Let (Ys(λ), s∈[0, S])be a real continuous semimartingale depending on a parameter λ≥0, with decomposition

Ys(λ) =Y0(λ) +Ms(λ) +Vs(λ), (7) whereMs(λ),Vs(λ)denote a local martingale and a bounded variation process, respectively, satisfying

Ms(λ) = m j=1

_s

0 Ψ^j_η(λ) dMη^j,

M^j,M^k

s= _s

0 Θ^j,k_η dη, Vs(λ) =

_s

0 Φ_η(λ) dη.

Assume that:

(i) For each λ ≥ 0, Ψ^j_η(λ), Φ_η(λ) and Θ^j,k_η , 1 ≤ j, k ≤ m, are adapted continuous processes, indexed by η∈[0, S], bounded by some constant K, uniformly in η, λ.

(ii) For eachη∈[0, S],1≤j≤m,Yη(λ),Y0(λ),Ψ^j_η(λ),Φ_η(λ)as functions ofλ, areβ-H¨older continuous, with ¹₂ < β <1, uniformly inη.

Set M(λ)_s=_s

0 Υ_η(λ) dη,where

Υ_η(λ) = m j,k=1

Ψ^j_η(λ) Ψ^k_η(λ) Θ^j,k_η . (8)

Fix ν > _2β−1³ . Then, for anyρ >3 + 2ν, positive constantsα1,α2, p≥2, andε small enough, there exists a constant C such that

P _s

0 Y_u²(u) du≤α1ε^ρ, _s

0

Υ_u(u) du≥α2ε

≤Cε^p.

Proof. Fixn≥1 and setsi=^is_n,i= 0, . . . , n. For eachi= 0, . . . , n−2, we define Di=

_s

0 Y_u²(u) du≤α1ε^ρ, _si+1

si

Υ_u(u) du≥ α2ε 2 (n−1)

.

We also define

Dn−1= s

0 Y_u²(u) du≤α1ε^ρ, _s

(1−_n¹)s

Υ_u(u) du≥α2ε 2

.

Then _s

0 Y_u²(u) du≤α1ε^ρ, _s

0 Υ_u(u) du≥α2ε

⊂

n−1

i=0

Di. (9)

Chebyshev’s inequality and the boundedness of Ψ^j, Θ^j,k, 1≤j, k≤m, yield for allp≥1,

P{Dn−1} ≤P _s

(1−_n¹)s

Υ_u(u) du≥α2ε 2

≤

2m²K³s α2

p

ε^−p n^p · Takingn= [ε^−ν] (where [·] denotes the integer part) andν >1, we have ^ε−p

n^p ≤2^pε^(ν−1)p, for anyε < ε0= 2^−1ν. Therefore,

P{Dn−1} ≤Cε^p, (10)

for allp= (ν−1)p≥2.

We now study the terms P(Di), for alli= 0, . . . , n−2. Setting Fi=

_si+2

si

Y_u²(u) du≤α1ε^ρ, _si+1

si

Υ_u(u) du≥ α2ε 2n

, we clearly haveDi⊂Fi.

By Itˆo’s formula, foru∈]si, si+2], it holds that Y_u²(λ) =Y_s²_i(λ) + 2

m j=1

_u

si

Yη(λ) Ψ^j_η(λ) dM_η^j+ 2 _u

si

Yη(λ) Φ_η(λ) dη+ _u

si

Υ_η(λ) dη. (11) Setλ=uin (11). By integrating with respect to theuvariable and applying Fubini’s theorem and a stochastic Fubini theorem, we obtain _si+2

si

Y_u²(u) du≥ Csi+2+Asi+2+Msi+2, (12) where, for anys≥si,

Cs = _s

si

_s

η

Υ_η(u) du

dη, As = 2

_s

si

_s

η Yη(u) Φ_η(u) du

dη, Ms = 2

m j=1

_s

si

_s

η

Yη(u) Ψ^j_η(u) du

dM_η^j. We devote the remaining of the proof to show the inclusion

Fi⊂

sup

si≤u≤si+2

|Mu| ≥ α2s

16 ε^1+2ν, M _si+2 <α²₂s² 256ε^3+4ν

(13) for someν >0. Then, applying the martingale exponential inequality,

P

sup

si≤u≤si+2

|Mu| ≥ α2s

16 ε^1+2ν, M _si+2 <α²₂s² 256ε^3+4ν

≤2 exp

−1 2ε⁻¹

, and this will finish the proof.

The above inclusion is obtained by proving a lower bound for Csi+2, and upper bounds for Asi+2 and M_{· si+2}.

Lower bound for Csi+2. Clearly, on Fi, the triangular inequality and the H¨older continuity of Υ_η(·) (with constantKβ), imply

Csi+2 ≥ _si+2

si

_si+2

η

Υ_η(η) du

dη− _si+2

si

_si+2

η

|Υη(u)−Υ_η(η)|du

dη

≥ _si+1

si

(si+2−η) Υ_η(η) dη− _si+2

si

_si+2

η

Kβ|u−η|^βdu

dη

≥(si+2−si+1) _si+2

si

Υ_η(η) dη− Kβ(si+2−si)^β+2

≥α2sε 2n² − Kβ

(2s)^β+2 n^β+2 ·

Takingn= [ε^−ν] andν >0, yields α2sε

2n² − Kβ

(2s)^β+2 n^β+2 ≥ α2s

2 ε^1+2ν− Kβ(4sε^ν)^β+2,

for anyε < ε0. Chooseν >0 such thatνβ >1, so thatν(β+ 2)>1+2ν. Thus, takingε1=

α2

Kβ4^β+3s^β+1

_νβ−1¹ , we have that for allε < ε1∧ε0, onFi,

Csi+2≥ α2s

4 ε^1+2ν. (14)

Upper bound for Asi+2. Jensen’s inequality implies Asi+2≤2K

_si+2

si

_si+2

η

|Yη(u)|du

dη.

Sinceλ→Yu(λ) isβ-H¨older continuous (H¨older constantK_β), we obtain _si+2

si

_si+2

η

|Yη(u)|du

dη≤ _si+2

si

_si+2

η

|Yη(η)|du

dη+ _si+2

si

_si+2

η

|Yη(u)−Yη(η)|du

dη

≤ _si+2

si

(si+2−η)|Yη(η)|dη+K_{β si+2}

si

_si+2

η

|u−η|^βdu

dη

≤ _si+2

si

(si+2−η)|Yη(η)|dη+K_β(si+2−si)^β+2. But,

_si+2

si

(si+2−η)|Yη(η)|dη≤

_si+2

si

(si+2−η)²dη

¹_{2 si+2}

si

Yη²(η) dη ¹₂

≤(si+2−si)³²

_si+2

si

Y_η²(η) dη ¹₂

·

Thus, onFi, we have

A_si+2≤2K 2s

n ³₂

α1¹²ε^ρ2 +K_β 2s

n

_β+2 .

As before, takingn= [ε^−ν],ν >0, one obtains 2s

n ³₂

α1¹²ε^ρ2 +K_β 2s

n _β+2

≤(4s)³²α1¹²ε^3ν2+ρ2 +K_β(4s)^β+2ε^ν(β+2)

≤

(4s)³²α1¹² +K_β(4s)^β+2

ε^{(3ν2+ρ2)∧(ν(β+2))}, for anyε < ε0. Chooseν such thatνβ >1 andρ >2 +ν. Thenm(ρ, ν, β) =_ρ

2+^3ν₂

∧ {ν(2 +β)}>1 + 2ν.

Thus, taking

ε₁=

⎛

⎝ α2

32K

4³²s¹²α1¹² +K_β4^β+2s^β+1

⎞

⎠

m(ρ,ν,β)−1−2ν1

,

onFi, we obtain for allε < ε₁∧ε0,

Asi+2≤ α2s

16 ε^1+2ν. (15)

Upper bound for M _si+2. Clearly,

M _si+2 = 4 m j,k=1

_si+2

si

_si+2

η

Yη(u) Ψ^j_η(u) du

si+2

η

Yη(u) Ψ^k_η(u) du

Θ^j,k_η dη.

Jensen’s inequality implies _si+2

η Yη(u) Ψ^j_η(u) du

≤ K(si+2−η)|Yη(η)|+KK_β(si+2−η)^β+1.

Thus,

M _si+2 ≤C1(si+2−si)² _si+2

si

Yη²(η) dη+C2(si+2−si)^2β+3, and onFi,

M _si+2 ≤C1

2s n

₂

α1ε^ρ+C2

2s n

_2β+3 .

Proceeding as before, we may chooseν >0 such thatν(2β−1)>3, thenρ >3 + 2ν, and findε₁ such that on Fi, for allε < ε₁,

M _si+2 <α²₂s²

256ε^3+4ν. (16)

The set Fi∩ sup_{si≤u≤si+2}|Mu|< ^α₁₆^2sε^1+2ν

!

is empty. In fact, on this set, by (12), (14), (15) and (16), we obtain

α1ε^ρ≥ _si+2

si

Y_u²(u) du≥ Csi+2−Asi+2−Msi+2≥ α2s

8 ε^1+2ν. (17)

Hence, takingβ > ¹₂,ν > _2β−1³ ,ρ >(3 + 2ν),ε <ε˜0, with

ε˜0:= min α2s

8α1

_ρ−1−2ν¹

, ε0, ε1, ε₁, ε₁

,

(17) cannot be satisfied.

Consequently, for anyε <ε˜0,

Fi=Fi∩

sup

si≤u≤si+2

|Mu| ≥ α2s 16 ε^1+2ν

⊂

sup

si≤u≤si+2

|Mu| ≥ α2s

16 ε^1+2ν, M _si+2 <α²₂s² 256ε^3+4ν

.

Thus, we obtain (13), and this ends the proof of the lemma.

The next lemma applies to the more irregular situation whereβ ∈]0,¹₂].

Lemma 2.2. Consider a continuous semimartingale (Ys(λ),0 ≤s≤S), with decomposition given in (7) and satisfying the assumption (i) of Lemma 2.1. Fix β∈]0,¹₂]and assume that:

(ii’) For each η∈[0, S],1 ≤j≤m,Y0(λ),Ψ^j_η(λ),Φ_η(λ), as functions ofλ, are β-H¨older continuous in λ, uniformly in η.

(iii’) For all β < β, there exists versions ofΨ^j_η(η)andΘ^j,k_η ,1≤j, k≤m, respectively, which areβ-H¨older continuous on [0, s].

Furthermore, for allp≥2

EΨ^j_·(·)^p

β+Θ^j,k_· (·)^p

β

≤Cp<∞.

Then, for any ρ >

11

2 +_β⁴ 1 +_β¹

, positive constantsα1,α2,p≥2and εsufficiently small, there exists a constant C such that

P _s

0 Yu²(u) du≤α1ε^ρ, _s

0 Υ_u(u) du≥α2ε

≤Cε^p. Proof. We follow the arguments of Proposition 3.2 in [2] in a more abstract presentation.

Fact 1. Fixβ < β,ε < s∧1,a >2

1 + _β¹

,p≥2. Condition (ii’) implies the existence of a positive constant cp such that

P _s

0 Y_u²(u) du≤α1ε^a, sup

0≤u≤s|Y_u(u)|>(1 +√ α1)ε

≤cpε^p. This can be checked following the proof of Lemma 3.4 on [2] withξ,Φ^∗−1_u Y (x) :=Yu(u).

As a consequence, takingρ=ra,ε^r< s∧1, withr >0 andaas before, we reduce the problem to estimate the probability of the set

B=

0≤u≤ssup |Yu(u)| ≤(1 +√ α1)ε^r,

_s

0 Υ_u(u) du≥α2ε

.

As in the previous Lemma 2.1, we consider the discretization of the interval [0, s] given bysi= ^is_n,i= 0, . . . , n, withn=

"

ε^−β4

# .

Fori= 0, . . . , n−1, set Ci :=

∃u∈[si, si+1] : Υ_u(u)< α2ε²³, _s

0

Υ_u(u) du≥α2ε

, Di :=

sup

si≤u≤si+1

|Yu(u)| ≤(1 +√

α1)ε^r, Υ_u(u)≥α2ε³², ∀u∈[si, si+1]

.

Clearly,B∩D^c_i ⊂Ci.

Fact 2. Condition (iii’) implies that for anyp≥2, andε¹² <_2s¹,

P _n−1

$

i=0

Ci

≤cpε^p.

For the proof of this fact, we can follow the arguments of Lemma 3.7 in [2] with%_m

j=1ξ,Φ^∗−1_u [Xj, Z] ²(x) :=

Υ_u(u).

LetD=&_n−1

i=0 Di, According to Fact 2, we haveP(B∩D^c)≤cpε^p for anyε¹² <_2s¹. Therefore, we have only to estimateP(B∩D). Moreover, since

P(B∩D)≤P(D)≤nmax

i {P(Di)} ≤ε^−β4max

i {P(Di)},

it suffices to show that P(Di)≤Cpε^p,for anyp≥1, with some constantCp, not depending oni. Recall the definition of Υ given by (8). Owning to assumptions (i) and (ii’),

sup

(u,λ)∈[si,si+1]²|Υu(u)−Υ_u(λ)| ≤C|s_i+1−si|^β =C s

n _β

≤Cs^βε^4ββ, (18)

for allε < ε0= 2^−β4.

For anyi= 0, . . . , n−1, set

Fi=

sup

si≤u≤si+1

|Yu(u)| ≤(1 +√

α1)ε^r, inf

(u,λ)∈[si,si+1]²Υ_u(λ)≥ α2

2 ε³²

.

Using (18), we prove that Di⊂Fi , for allε < ε1∧ε0, for someε1>0.

Indeed, the triangular inequality and the estimate (18) implies that, for anyλ∈[si, si+1], onDi

Υ_u(λ)≥Υ_u(u)−Cs^βε^4ββ ≥α2ε³² −Cs^βε^4ββ.

Since β < β, we have ^4β_β −³₂ >0. Choosing ε1= _α

2Cs2^β 4β1 β −3

2, we obtain Υ_u(λ)≥ ^α₂²ε³², for allε < ε1∧ε0

and for any (u, λ)∈[si, si+1]².

Thus, we have now reduced the proof to estimateP{Fi}.

We shall only consider the casei= 0. In fact, it will become clear from the proof that the arguments depend only on the length of the interval [si, si+1].

We shall prove the existence of the arg sup associated to Y·(λ). This is done following the same arguments as in [2], Proposition 3.2, that is, using Girsanov’s theorem. Our setting needs a more general version of this theorem than the one used in [2]. For instance, we can apply Theorem 35 in [9], p. 132. With this, on a new probability space, the semimartingaleY·(λ) is transformed into a local martingale and then, by a standard time change, into a Brownian motion, for which the arg sup does exist. We only give a sketch of this procedure, since it is very similar as in [2].

Applying a Girsanov transformation needs to work on the whole probability space Ω, and not only on F0. For this reason, we have to modify the processY·(λ), as follows. Define

φ(z) =

z if |z| ≤ K,

±2K if |z|>3K, otherwise,|φ|is bounded by 2K and with derivative bounded by 1;

ψε(z) =

⎧⎨

⎩

1 if |z| ≥ ^α₂²ε³², 0 if |z|< ^α₄²ε³²,

ψεis even and non-decreasing on [0,∞). Set

Φ_·(∗) =φ(Φ_·(∗)), (19)

Ψ^j_·(∗) =φ

Ψ^j_·(∗) ψε(Υ_·(∗)), 1≤j≤m, (20) Ψ^m+1_· (∗) =

*α2

2 ε³²[1−ψε(Υ_·(∗))]. (21)

Consider a Brownian motionM^m+1, independent of

M¹, . . . ,M^m

. Let

Yu(λ) =Y0(λ) +

m+1

j=1

_u

0 Ψ^j_η(λ) dM_η^j+ _u

0 Φ_η(λ) dη. (22)

This is the analogue of (3.12) in [2].

Observe that if ω∈F0 foru≤s1, thenYu(λ, ω) =Yu(λ, ω). Hence,Y·(λ) is a modification ofY·(λ).

We define the modification ofMu(λ) byMu(λ) :=%_m+1

j=1

_u

0 Ψ^j_η(λ) dMη^j, with quadratic variation given by +M·(λ),

u=_u

0 Υ_η(λ) dη, where Υ_η(λ) =

m j,k=1

Ψ^j_η(λ) Ψ^k_η(λ) Θ^j,k_η +

Ψ^m+1_{η 2}

(λ). (23)

One can check that for allη.

Υ_η(λ)≥ α2

8 ε³². (24)

This is the crucial fact ensuring the existence of a probability measureP, equivalent to- P, such that on eachF_s, dP-

dP = exp

− _s

0

Φ_η(λ)

Υ_η(λ)dMη(λ)−1 2

_s

0

Φ²_η(λ) Υ²_η(λ)

d+

M·(λ),

η

,

and such that

M_u(λ) =Mu(λ) + _u

0 Φ_η(λ) dη,

is a P- local martingale, (see [9]). Thus, P-a.s.,- Yu(λ) =Y0(λ) +M_u(λ), is a local martingale.

Set

A_u(λ) =+ Y·(λ),

u= _u

0 Υ_η(λ) dη, T_u(λ) = inf{η≥0, A_η(λ)≥u}.

Then, there exists aP-Brownian motion- B such that for allu,Yu(λ) =Y0(λ) +B_Au(λ), that is,Y_Tu(λ)(λ) = Y0(λ) +BuP-a.s., (see [4], p. 174, for details).-

LetS1=^α2ε

32

8 s1≥^α₈²sε^23+β4. Using (24), we can prove uα2

8 ε³² ≤A_u(λ)≤2m²K³u. (25)

Consequently, _2m^u_2K3 ≤T_u(λ)≤ ^8u

α2ε³2. Hence, α2sε^32+β4

16m²K³ ≤ S1

2m²K³ ≤T_S1(λ)≤ 8S1

α2ε³² =s1. (26)

Set Π₁(λ) :=T_S1(λ). The supremum of the absolute value of a linear Brownian motion on a deterministic time interval is a.s. attained at a single time. Thus, for allλ∈[0, s1],P-a.s., there exists an unique (random) time- such that

η1(ω, λ) = arg sup

0≤η≤Π1(λ)

Yη(λ). (27)

SincePandP- are equivalent and Π₁(λ)≤s1, (27) holdsP-a.s.

The final step consists of proving some control on the modified process Y·(λ), because we only have the existence of the arg sup for this process.

Letθ1=^α2sε

32 +4 β

16m²K³ . Owing to (26),

0≤η≤θsup1

Yη(λ)≤ sup

0≤η≤Π1(λ)

Yη(λ). (28)

The term sup_0≤η≤θ1Yη(λ) can be estimated in a similar manner as in [6]. We consider the Itˆo formula forY_θ²₁(λ),

Y²_θ1(λ) =Y²₀(λ) + 2 _θ1

0 Yη(λ)dMη(λ) + 2 _θ1

0 Yη(λ) Φ_η(λ) dη+ _θ1

0 Υ_η(λ) dη. (29) Fixr> _β² +¹¹₄. We have the following facts concerningY·(λ):

Fact 3. There existsε₀>0 such that for allε < ε₀

P

0≤η≤θsup1

Yη(λ)≤(2 +√ α1)ε^r

≤2 exp

−1 2ε⁻¹

.

Fact 4. There existsε₀ >0 such that for allε < ε₀

P

∃λ∈[0, s1], sup

0≤η≤θ1

Yη(λ)≤(1 +√ α1)ε^r

≤cpε^p,

for allp≥2.

For their proofs, we can follow Lemmas 3.21 and 3.22 of [2], respectively.

Let us return to Y·(λ). The processes Yu(λ), A_u(λ) and T_u(λ), have jointly continuous sample paths in (u, λ), a.s. Thus, forω∈F0, for all (u, λ)∈[0, s1]²,Yu(λ, ω) =Yu(λ, ω).

Consider the set

G0=F0∩

∀λ∈[0, s1], sup

0≤η≤Π1(λ)|Yη(λ)|>(1 +√ α1)ε^r

.

As in [2], one checks that G0 = ∅. Consequently, with the result stated in Fact 4, we end the proof of the

lemma.

3. Proof of Theorem 1.1

Proof. To simplify the notation, we omit the summation sign on repeated indices. Using the classical approach going back to Malliavin –and also by Bismut, Ikeda and Watanabe, Stroock, Bouleau and Hirsch, etc.– (see for instance [7]), the proof of the theorem consists of checking that

(i) X_zⁱ ∈D^∞, for all 1≤i≤m;

(ii) detC_z⁻¹∈L^p, for allp≥2, whereCz is them×mmatrix whose entries are C_z^ij =

Rz

(D_r^lX_zⁱ)(D_r^lX_z^j)dr.

Proving (i) is straightforward. Indeed, due to the condition (h2), one can proceed as in [8], Proposition 3.3.

To prove (ii), it suffices to show that for anyp≥2 there existsε0(p) such that for every ε≤ε0(p) sup

|v|=1

P

v^TCzv≤ε

≤ε^p (30)

(seee.g. [7]).

We next give some preparations for the proof of (30), valid in each one of the set of assumptions of Theo- rem 1.1. Following similar arguments as in [8], pp. 585–586, we write

v^TCv= d

l=1

Rz

v, ξ(r, z)Al(r, Xr) ²dr,

wherev∈R^m, and forr∈RS,T,ξ_jⁱ(r, z), rz, 1≤i, j≤m, is the solution to ξ_jⁱ(r, z) =δ_jⁱ+

[r,z]∂_k^xAⁱ_l(u, Xu)ξ_j^k(r, u)dW_u^l+

[r,z]∂_k^xAⁱ₀(u, Xu)ξ_j^k(r, u)du. (31) Then, for any v∈S^m−1, 0< ε <1, 0< µ <1, we have

P

v^TCzv≤ε

≤P(E0) +P(B), where

E0= _d

l=1

_s

0 v, Al(η, t, Xη,t) ²dη≤4ε^µ

,

B= _d

l=1

_s

0

_t

t−ε^1−µv, Al(η, t, Xη,t)−ξ(η, τ, s, t)Al(η, τ, Xη,τ) ²dηdτ > ε

.