On the positive correlations in Wiener space via fractional calculus

On the positive correlations in Wiener space via fractional calculus

Toufik Guendouzi

Laboratory of Mathematics, Djillali Liabes University

PO. Box 89, 22000 Sidi Bel Abbes, Algeria email: [email protected]

Abstract. In this paper we study the correlation inequality in the Wiener space using the Malliavin and the fractional calculus. Under positivity and monotonicity conditions, we give a proof of the positive correlation between two random functionalsFandG which are assumed smooth enough. The main argument is the Itˆo-Clark representation for- mula for the functionals of a fractional Brownian motion.

1 Introduction

It is well-known that the correlations inequalities are one of the most power- ful tools of the stochastic analysis due to its vast range of applications. So, The theoretical study of these inequalities has matured tremendously since the seminal work of Fortuin, Kasteleyn and Ginibre [5]. In general, several au- thors have been interested in finding applications of these inequalities in some areas including statistical mechanics (see, for instance, Bakry and Michel [1], Preston [15]).

Recently, Mayer-Wolf, ¨Ust¨unel and Zakai obtained general covariance in- equalities in an abstract Wiener space. They consider such inequalities for functionals satisfying either monotonicity or convexity properties [13]. Hence Houdr´e and Perez-Abreu in [9] used Malliavin calculus techniques to obtain

2010 Mathematics Subject Classification: 60E15, 60F15, 60G05, 60G10, 60H07, 60J60 Key words and phrases: Wiener space, positive correlations, Malliavin calculus, Clark formula, FKG inequality, fractional Brownian motion

206

covariance identities and inequalities for functionals of the Wiener and the Poisson processes.

The purpose of this paper is to use the Malliavin calculus techniques to study the positive correlations between two functionals on the Wiener space via fractional calculus. Our proofs rely in general on the Itˆo-Clark represen- tation formula for the functionals of a fractional Brownian motion and the monotonicity condition forFand Gon the Wiener space. Here, the fractional Brownian motion of index H ∈ (0, 1) is the centred Gaussian process whose covariance kernel is given by

RH(s, t) =E_H[W_s^HW_t^H], and for fgiven in [a, b], each of the expressions

(D^α_a+f)(x) = d dx

[α]+1

I^1−{α_a+ ^}f(x), (D^α_b−f)(x) =

− d dx

[α]+1

I^1−{α_b− ^}f(x), are respectively called right and left fractional derivative where [α] denotes the integer part of α, {α} = α− [α] and (I^α_a+f(x))(x), (I^α_b−f(x))(x) are right and left fractional integral of the order α > 0 (see [3]). Hence for H∈(0, 1) the integral transformK_Hfis defined as

K_Hf = I^2H₀+x^1/2−HI^1/2−H₀+ x^H−1/2f, H≤1/2 K_Hf = I¹₀+x^H−1/2I^H−1/2₀+ x^1/2−Hf, H≥1/2,

K_H is an isomorphism from L²([0, 1]) onto I^H+1/2₀+ (L²([0, 1])). If H ≥ 1/2, r→KH(t, r) is continuous on(0, t].

The organization of this paper is as follows: in Section 2, we shall give some preparation and state main results. We begin by recalling the basic notions of Malliavin calculus, the gradient operator and Sobolev-type space D2,1, the Ornstein-Uhlenbeck semigroup, the Itˆo-Clark representation formula for func- tional of Brownian motion. In Section 3, we shall study the positive correlation between two functionals of the Wiener space satisfying monotonicity property.

2 Preliminaries

This section gives some basic notions of analysis on the Wiener space(W,F^H, P_H). The reader can consult [14] for a complete survey on this topic. Let W represented as C₀([0, 1],R) of continuous function ω : [0, 1] −→ R with

w(0) = 0, equipped with the ||.||_∞-norm i.e W is also a (separable) Banach- space, W^∗ is its topological dual and (W_t)_t∈[0,1] be a canonical Brownian motion generating the filtration(F_t^H)_t∈[0,1]. Random-variables onWare called Wiener functionals and the coordinate process ω(t) is a Brownian motion underP_H. So we writeω(t) =W(t, ω) =W(t). Recall thatP_His the unique probability measure on W such that the canonical process (W(t))_t∈R is a centered Gaussian process with the covariance Kernel RH:

E_H[W(t)W(s)] =R_H(t, s).

The Cameron-Martinspace HHis an subspace ofW defined as HH={K_Hh;˙ h˙ ∈L²([0, 1], dt)},

i.e, any h ∈ HH can be represented as h(t) = KHh(t) =˙ Z1

0

KH(s, t)h(s)ds,˙ h˙ belongs to L²([0, 1]). The scalar product on the space HH is given by (h, g)_HH = (K_Hh, K˙ _Hg)˙ _HH = (h,˙ g)˙ _L2([0,1]).

We note that for anyH∈(0, 1),R_H(t, s) can be written as R_H(t, s) =

Z1 0

K_H(t, r)K_H(s, r)dr,

andR_H=K_HK^∗_H, where K_His the Hilbert-Schmidt operator introduced in the first section. R_His also the injection fromW^∗ into the spaceHHand it can be decomposed as R_Hη =K_H(K^∗_Hη), for anyη in W^∗ (see, [18]). The restriction of K^∗_HtoW^∗ is the injection fromW^∗ into L²([0, 1]).

Ifyis anHH-valued random variable, we denote by ˙ytheL²([0, 1],R)-valued random variable such that y(ω, t) =

Zt 0

K_H(t, s)y(ω, s)ds. Here, for˙ F∈ S(χ) theH-Gross-Sobolev derivative ofF, denoted by ∇Fand is theHH⊗χ-valued mapping defined by

∇F(ω) = Xn

i=1

∂f

∂x_i(hl1, ωi, . . . ,hln, ωi)RH(l_i)⊗x, (1) whereχis a separable Hilbert space,S(χ) is the set ofχ-valued smooth cylin- dric functionals, and for each 1 ≤ i ≤ n, l_i is in W^∗ and x_i belongs to χ.

Hence, for anyRHη∈ HHwe have by the Cameron-Martin theorem E_H[F(ω+R_Hη)] =

Z

F(ω)exp

hη, ωi−||R_Hη||²_HH/2

dP_H(ω). (2)

The Ornstein-Uhlenbeck semigroup {T_t^H, t ≥ 0} of bounded operators which acts on L^p(P_H, χ) for anyp≥1 can be described by the Mehler formula:

(T_t^HF)(ω) = Z

W

F

e^−tω+p

1−e^−2tω^′

P_H(dω^′). (3) The directional derivative ofF∈ S(χ) in the the direction R_Hη∈ HHis given by

(∇F, RHη)H_H = d

dtF(ω+t.R_Hη)

_t=0, (4) and from (2) we have∇Fdepends only on the equivalence classes with respect toP_Hand E_H((∇F, RHη)_HH) =E_H(Fhω, ηi).

For any p ≥ 1 we define Sobolev space D^H_p,k(χ), k ∈ Z, as the completion of S(χ)with respect to the norm

||F||_p,k,H=||F||_L^p

H +||∇^kF||_L^p₍P_H;χ),

hence the operator ∇ can be extended as continuous linear operator from D^H_p,k(χ) to D_p,k−1^H (HH ⊗χ) for any p > 1 and k ∈ Z (see [18]). Thus

∇:D^H_p,k(χ)→D_p,k−1^H (HH⊗χ); its formal adjoint with respect toP_His the op- eratorδ_Hin the sense that∀F∈ S,∀y∈ S(HH),E_H[Fδ_Hy] =E_Hh

(∇F, y)H_H

i, and since∇has continuous extensions,δ_Hhas also a continuous linear exten- sion fromD_p,k^H (HH) toD^H_p,k−1 for anyp > 1 and k∈N.

Recall the following, unique, Wiener-Itˆo chaos expansion for all P_H-square integrable functional Ffrom W toR

F=EF+ X∞

1

J^H_nF, (5)

whereJ^H_nis then-fold iterated Itˆo integral ofF. Ify∈ HHandϑ^y₁ =exp(δHy−

1/2||y||²_H

H), then we have

J^H_nϑ^y₁ = 1

n!δ⁽ⁿ⁾_H y^⊗n. (6)

More precisely, if F∈ ∪k∈ZD_2,k^H , J^H_nF= 1

n!δ⁽ⁿ⁾_H

E_H∇⁽ⁿ⁾F .

ForH∈(0, 1), let{π^H_t;t∈[0, 1]}be the family of orthogonal projection inHH

defined by

π^H_t(K_Hy) =KH(y1_[0,1]), y∈L²([0, 1]). (7) The operator Υ(π^H_t) is the second quantization ofπ^H_t from L²(P_H) into itself defined by

F=X

n≥0

δ⁽ⁿ⁾_H f_n7→Υπ^H_t(F) = X

n≥0

δ⁽ⁿ_H⁾

(π^H_t)^⊗nf_n . Thus we have, for y∈ HH,

Υ(π^H_t) ϑ^y₁

=exp(δH(π^H_ty) −1/2||π^H_ty||²_HH) =ϑ^y₁, (8) hence the bijectivity of the operator KHhas the following consequence

F_t^H=σ{δ_H(π^H_ty), y∈ HH} ∨NH, whereNHis the set of theP_H-negligible events.

We also note that for any F∈L²(P_H),

Υ(π^H_t)F=E_H[F|F_t^H], and in particular

E_H[W_t|F_t^H] = Zt

0

K_H(t, s)1_[0,1](s)δ_HW_s,

E_H[exp(δ_Hy−1/2||y||²_HH)|F_t^H] =exp(δ_H(π^H_ty) −1/2||π^H_ty||²_HH), for anyy∈ HH.

We shall recall the following results

Theorem 1 ([3]) Let F be D_2,1^H . Then F belongs toF_t^H iff∇F=π^H_t∇F.

Theorem 2 (Itˆo-Clark representation formula) For any F∈ D_2,1^H , F−E_H[F] =

Z1 0

E_H[K⁻¹_H(∇F)(s)|F_s^H]δ_HW_s

= δ_H

K_H(E_H[K⁻¹_H(∇F)(.)|F.]) .

3 Monotonicity and positive correlations

Our method relies on the Itˆo-Clark formula which plays a crucial role to es- tablish positive correlation between two random functionals under some hy- potheses. Thus we recall here the following correlation identity, in the first lemma, which is based on the Clark formula and the Itˆo isometry. We refer to [3] and [9] for tutorial references on this identity.

Lemma 1 For any F, G∈L²(P_H) we have Cov(F, G) =E_HhZ1

0

E_H[K⁻¹_H(∇F)(s)|F_s^H]E_H[K⁻¹_H(∇G)(s)|F_s^H]dsi

. (9) Proof. We have

Cov(F, G) = E_Hh

(F−E_H[F])(G−E_H[G])i

= E_H hZ1

0

E_H[K⁻¹_H (∇F)(s)|F_s^H]δ_HW_s Z1

0

E_H[K⁻¹_H(∇G)(s)|F_s^H]δ_HW_si

= E_H hZ1

0

E_H[K⁻¹_H (∇F)(s)|F_s^H]E_H[K⁻¹_H (∇G)(s)|F_s^H]dsi .

Proposition 1 Let Gbe aF_t^H-measurable element ofD^H_2,1. Then the identity (9) can be written as

Cov(F, G) = E_HhZ1 0

E_H[K⁻¹_H (∇F)(s)|F_s^H]K⁻¹_H (∇G)(s)dsi

= E_H hZ1

0

E_H[K⁻¹_H (∇F)(s)|F_s^H]K⁻¹_H (π^H_t∇G)(s)dsi . The next result is an immediate consequence of (9).

Lemma 2 Let F, G∈L²(P_H) such that

E_H[K⁻¹_H (∇F)(s)|F_s^H]E_H[K⁻¹_H (∇G)(s)|F_s^H]≥0, ds×dP−a.s.

Then F and Gare positively correlated and we have Cov(F, G)≥0.

The main results of this section are the following:

Corollary 1 If F, G∈ D_2,1^H satisfy K⁻¹_H [∇F](t)≥0, K⁻¹_H[∇G](t)≥0a.s., then F and Gare positively correlated.

Corollary 2 If G ∈ D^H_2,1, and if E_H[K⁻¹_H(∇F)(s)|F_s^H] ≥ 0, K⁻¹_H [∇G](t) ≥ 0a.s., then F and Gare positively correlated.

The next theorem studies the positivity of K⁻¹_H [∇F](t), for any functional F∈ D^H_2,1under the monotonicity assumption.

Theorem 3 For any increasing functional F∈ D_2,1^H we have K⁻¹_H[∇F](t)≥0, dt×dP_H−a.s.

Proof. LetF be increasing functional i.e. F(.+y)≥F(.) a.s., for all y∈ HH

and {u^t_n, n ≥ 0} be an orthonormal basis of L²([0, 1]), for H ∈ (0, 1), V_n^t be the σ field generated by {δ_HK_Hu^t_i, i ≤ n}. Since ∨_nV_n^t = F_t^H, the sequence Fn= E_Hh

F/V_n^ti

converge to Fin D_2,1^H, and from π^H_tK_Hu^t_n=K_Hu^t_n, forFnwe have∇Fn=π^H_t∇Fnand∇F=π^H_t∇Ffollows. Hence, by the Cameron-Martin formula (2) we have for any V_n^t-measurable and square-integrable random variableϑ^t_n,

E_H

Fn(ω+y^t_n)

=E_Hh

exp(δHy^t_n−1/2||y^t_n||²_H

H)F_n(ω)i

=E_H

ϑ^t_nFn(ω)

=E_H

ϑ^t_nE_H[F/V_n^t](ω)

=E_H

E_H[ϑ^t_nF/V_n^t](ω)

=E_H[ϑ^t_nF(ω)]

=E_H[F(ω+y^t_n)].

On the other hand, for any square-integrable functionf on [0, 1]ⁿwe have Fn(ω+y) = Fn(ω+y^t)

= f

δ_HK_Hu^t₀+ (K_Hu^t₀, K⁻¹_H yt)_L2([0,1]), . . . , . . . , δ_HK_Hu^t_n+ (K_Hu^t_n, K⁻¹_H y_t)_L2([0,1])

= f

δHKHu^t₀+ (K_Hu^t₀, π^H_tK⁻¹_H y^t_n)_L2([0,1]), . . . , . . . , δ_HK_Hu^t_n+ (K_Hu^t_n, π^H_tK⁻¹_H y^t_n)_L2([0,1])

= F_n(ω+y^t_n)

= E_H[F(ω+y^t_n)/V_n^t]

≥ E_H[F/V_n^t](ω)

= F_n(ω) −a.s.

Thus, we conclude that the smooth function F_n(ω+τy) is increasing in τ, for any τ ∈ R where π^H_tK⁻¹_H y^t_n = K⁻¹_H y^t_n is positive, hence we have from (4) that (∇Fn, K⁻¹_H y)_L2([0,1])is positive. Since∇Fnis positive and∇Fn→∇F then∇F is also positive.

To complete the proof, it suffices to use the fact that δ_H(π^H_t∇F) =

Zt 0

K⁻¹_H [∇F](s)δHW_s

and because∇F positive we get K⁻¹_H[∇F](s)≥0, a.s.

Theorem 4 For any increasing functional F∈L²(P_H) we have E_H[K⁻¹_H (∇F)(s)|F_s^H]≥0, dt×dP_H−a.s.

Proof. Let {T_t^H, t ≥0} be a semigroup defined as in (3), and assume that F is increasing functional in L²(P_H). Taking t = 1/n, ∀n ≥ 1, we have T_1/n^H F is also increasing from (3) and element ofD^H_2,1. Hence from lemma 3, ∇T_1/n^H F is positive and also K⁻¹_H [∇T_1/n^H F](s)≥0, a.s., thenE_H[K⁻¹_H(∇T_1/n^H F)(t)|F_t^H] follows. Finally, using the fact thatT_1/n^H F→Fasngoes to infinity we get the

result.

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Received: February 25, 2010