Chaotic extensions and the lent particle method for Brownian motion

In previous works, we have developed a new Malliavin calculus on the Poisson space based on the lent particle formula. The aim of this work is to prove that, on the Wiener space for the standard Ornstein-Uhlenbeck structure, we also have such a formula which permits to calculate easily and intuitively the Malliavin derivative of a functional. Our approach uses chaos extensions associated to stationary processes of rotations of normal martingales.


Introduction
When a measurable space with a σ-finite measure ν is equipped on L 2 (ν) with a local Dirichlet form with carré du champ γ, the associated Poisson space, i.e. the probability space of a random Poisson measure with intensity ν, may itself be endowed with a local Dirichlet structure with carré du champ Γ (cf. [16], [17]). If a gradient ♭ has been chosen associated with the operator γ, a gradient ♯ associated with Γ may be constructed on the Poisson space (cf [1], [10], [14], [2]) and we have shown [2], [3], that such a gradient is provided by the lent particle formula which amounts to add a point to the configuration, to derive with respect to this point, and then to take back the point before integrating with respect to a random Poisson measure variant of the initial one.
On the example of a Lévy process, in order to find the gradient of the functional V = t 0 ϕ(Y u− )dY u , this method consists in adding a jump to the process Y at time s and then deriving with respect to the size of this jump.
If we think the Brownian motion as a Lévy process, this addresses naturally the question of knowing whether to obtain the Malliavin derivative of a Wiener functional we could add a jump to the Brownian path and derive with respect to the size of the jump, in other words whether we have, denoting D s F the Malliavin derivative of F Formula (1) is satisfied in the case F = Φ( 1 0 h 1 dB, . . . , 1 0 h n dB) with Φ regular and h i continuous. But this formula has no sense in general, since the mapping t → 1 {t s} does not belong to the Cameron-Martin space.
We tackle this question by means of the chaotic extension of a Wiener functional to a normal martingale weighted combination of a Brownian motion and a Poisson process, and we show that the gradient and its domain are characterized in terms of derivative of a second order stationary process.
We show that a formula similar to (1) is valid and yields the gradient if F belongs to the domain of the Ornstein-Uhlenbeck Dirichlet form, but whose meaning and justification involve chaotic decompositions. This gives rise to a concrete calculus allowing C 1 changes of variables. Let us also mention the works of B. Dupire ([8]), R. Cont and D.A. Fournié ( [5]) which use an idea somewhat similar but in a completely different mathematical approach and context.

The second order stationary process of rotations of normal martingales.
Let B be a standard one-dimensional Brownian motion defined on the Wiener space Ω 1 under the Wiener measure P 1 . In this section, we considerÑ a standard compensated Poisson process independent of B. We denote by P 2 the law of the Poisson process N and P = P 1 × P 2 . Let us point out that in the next sections, we shall replaceÑ by any normal martingale.

The chaotic extension
For real θ, let us consider the normal martingale If f n is a symmetric function of L 2 (R n , λ n ), we denote I n (f n ) the Brownian multiple stochastic integral and I θ n (f n ) the multiple stochastic integral with respect to X θ . We have classically cf [6] I n (f n ) 2 It follows that if F ∈ L 2 (P 1 ) has the expansion on the Wiener chaos the same sequence f n defines a chaotic extension of F : F θ = ∞ n=0 I θ n (f n ).

Remark 1.
Let us emphasize that the chaotic extension F → F θ is not compatible with composition of applications : Φ • Ψ θ = (Φ • Ψ) θ except obvious cases as seen by taking Φ(x) = x 2 , Ψ = I 1 (f ) and θ = π/2. Thus it is important that the sequence (f n ) n appears in the notation : we will use the "short notation" of [6].
We denote P (resp. P(t)) the set of finite subsets of ]0, ∞[ (resp. ]0, t]). We write A = {s 1 < · · · < s n } for the current element of P and dA for the measure whose restriction to each simplex is the Lebesgue measure, cf [6] p201 et seq. Thus In all the paper we confond the stochastic integrals H s− dX θ s and H s dX θ s thanks to the fact that X θ is normal. Proposition 1. Let be f and g ∈ L 2 (P), and h = f + ig ∈ L 2 C (P). The random variable H θ = P h(A)dX θ A defines a second order stationary process continuous in L 2 C (P). Proof. Let us denote similarly F θ = P f (A)dX θ A and G θ = P g(A)dX θ A . It is enough to show that F θ and G θ are measurable, second order stationary and stationary correlated. Using the chaos expansions F θ+ϕ = n I θ+ϕ n (f n ) and G θ = n I θ n (g n ) that comes from the fact that the bracket of the martingales X θ+ϕ and X θ is So E[I θ+ϕ n (f n )I θ n (g n )] = n! f n , g n L 2 (λn) cos n ϕ and E[I θ+ϕ m (f m )I θ n (g n )] = 0 if m is different of n.
It follows that the stationary process H θ possesses a spectral representation where the c n are real 0 and the ξ n are orthonormal in L 2 C (P). The norm H θ 2 which doesn't depend on θ is the total mass of the spectral measure c 2 n .
The c n are linked with the norms of the components of H on the chaos by formulas involving Bessel functions. In the case where H is an exponential vector, which is the Fourier transform of the spectral measure hence equal to n c 2 n e inϕ . By the relation defining the Bessel functions J n (formula of Schlömilch) it comes c 2 n = i n J n (−i h 2 ) and for n 0 (cf [15]) The variables c n ξ n may be also expressed in terms of Bessel functions using the expression of exponential vectors for X θ , cf formula (5) below.

Chaotic structure of L 2 (P).
This part is independent of the rest of the paper. It is devoted to the study of chaotic representations for X θ . Let us first remark that the above considerations dont use the chaotic representation property for X θ which is false if sin θ cos θ = 0, as it is well known cf for instance [7]. Let us denote L 2 (P X θ ) the vector space of σ(X θ )-measurable random variables belonging to L 2 (P). That means that, if sin θ cos θ = 0, the vector space C(X θ ) = {F θ : F ∈ L 2 (P 1 )} which is closed in L 2 (P X θ ) has a non empty complement.
If we consider the simplest example of the square of a functional of the first Brownian chaos F = hdB with h ∈ L 1 ∩ L ∞ , we have F θ = hdX θ and by Ito formula sinceÑ = sin θ X θ + cos θ X θ+π/2 , we see that with U ∈ C(X θ ) and h 2 dX θ+π/2 orthogonal to C(X θ ). It follows that for k ∈ L 2 (R + ), kdX θ+π/2 ∈ L 2 (P X θ ) and this implies Proposition 2. Let us suppose sin θ cos θ = 0. The σ-fields generated by X θ and X θ+π/2 are the same. The spaces L 2 (P X θ ) do not depend on θ and are equal to L 2 (P).
The intuitive meaning of this proposition is that on a sample path of X θ it is possible to measurably detect the underlying Brownian and Poisson paths.
The multiple stochastic integrals w.r. to X θ are not enough to fill L 2 (P X θ ). In view of the previous example, we may think to add the stochastic integrals w.r. to X θ+π/2 , i.e. to add C(X θ+π/2 ), which is orthogonal to C(X θ ) and included in L 2 (P X θ ). But this is not sufficient, we must add also the crossed chaos in the following manner: Let us consider the vector martingale X θ = (X θ , X θ+π/2 ). For h = (h 1 , h 2 ) ∈ L 2 (R + , R 2 ) we may consider the stochastic integral h.dX θ , and more generally (notation of [4] Chap These stochastic integals define orthogonal sub-spaces of L 2 (P X θ ) = L 2 (P). Now considering the exponential vector , we see that the following SDE is satisfied . We obtain Proposition 3. For any θ, the stochastic integrals (4) define a complete orthogonal decomposition of L 2 (P).
Proof. a) Let us suppose first sin θ cos θ = 0. Starting with (5) an easy computation yields that If we take a step function ξ ∈ L 2 (R + ) and choose h 1 and h 2 such that we obtain that exp[ ξdX θ ] belongs to the space generated by the chaos, and the result follows.
b) Now if θ = 0, X θ = (B,Ñ ). The above argument is still valid and with h 2 = e u 2 − 1. That gives easily the result and the same in the other cases where sin θ cos θ = 0.
In other words L 2 (P) is isomorphic to the symmetric Fock space F ock(L 2 (R + , R 2 )). This implies the predictable representation property with respect to X θ .

Derivative in θ and gradient of Malliavin.
We come back to the setting of subsection 2.1 with stochastic integrals with respect to the real process X θ . We want to study the behavior near θ = 0 using the fact that X 0 = B.
But since we deal no more with chaotic representation we may replaceÑ by any normal martingale M independent of B (for instance a centered normalized Lévy process) define under a probability that we still denote P 2 and as in subsection 2.1, where from now on, I θ n denotes the multiple stochastic integral with respect to Y θ .
To see the connection with the Brownian chaos expansion, let us remark that -as in the preceding part -for any θ ∈ R, the pair this allows to prove the following property n!n f n 2 L 2 Proof. Our notation is F θ = n k=0 I θ k (f k ) and F 0 = F = n k=0 I k (f k ). Let us consider first I θ n (f n ) in the case of an elementary tensor f n = g 1 ⊗ · · · ⊗ g n . We can write this multiple integral (with the notation of Bouleau-Hirsch [4] p79) what gives, running the induction down, This extends to general tensors and similarly we can show that if k = ℓ what yields the proposition.
We denote by D, the domain of the Ornstein-Uhlenbeck form. We recall that an element Let us take now an F ∈ D, the random variables n k=0 I θ k (f k ) converge in L 2 (P) to F θ uniformly in θ. Their derivatives -because F ∈ D -form a Cauchy sequence and converge also uniformly in θ. This implies that F θ is differentiable and that the derivatives of the where E is the Ornstein-Uhlenbeck form and Γ the associated carré du champ operator.
We also have the converse property: Proposition 6. Let F ∈ L 2 (P 1 ). If the map θ → F θ is differentiable in L 2 (P) at a certain point θ 0 ∈ R then F belongs to D.
Proof. We write F = n I n (f n ) and consider a sequence (θ k ) k 1 which converges to θ 0 and such that θ k = θ 0 , for all k 1. As exists in L 2 (P) we deduce that there exists a constant C > 0 such that By the Fatou's Lemma and the previous Proposition, we get which yields the result.
This provides the following result : the righthand term is a gradient for the Ornstein-Uhlenbeck form that we may denote F ♯ , so we have in the sense of L 2 (P) = L 2 (P 1 × P 2 ) Proof. Let be F ∈ D. We assess the distance between 1 θ (F θ − F ) and D s F dM s by steps : distance between 1 θ (F θ − F ) and 1 θ (F θ n − F n ) ; between 1 θ (F θ n − F n ) and D s F n dM s ; then between D s F n dM s and D s F dM s .
By the preceding propositions We may choose n so that the first one and the third one be both small independently of θ. And n being fixed we do θ → 0 in the second one and apply the argument of the proof of Proposition 4.
The classical integration by part formula, i.e. the property that the divergence operator, dual of D, can be expressed by a stochasitic integral on predictable processes, is a consequence of propositions 5 and 7 by derivation in θ.
Indeed let us denote A the closed sub-vector space of L 2 (P, L 2 (R + , dt)) generated by the processes of the form ∆(t) f n d (n) B with f n ∈ L 2 (R n ).
Proof. By relation (6) the property is true if F has a finite expansion on the chaos hence also if F ∈ L 2 .
Let us denote now D A the closed vector space generated by the processes ∆(t) f n d (n) B with f n ∈ L 2 (R n ) for the norm of D with values in L 2 (R + ).
so that taking θ = 0 Proof. We differentiate F θ G θ t dY θ+π/2 t taking in account the lemma and the fact that

Remark 2.
Taking anewÑ for M , we may apply the previous reasoning at the point θ = π/2. Denoting D (N ) the operator of Nualart-Vivès [12] which acts on the Poisson chaos as D acts on the Brownian ones, Proposition 7 says that for (f n ) such that n!n f n 2 < ∞ the Poisson functional F = I π/2 n (f n ) is such that d dθ F θ | θ=π/2 = D (N ) F dB. And by Proposition 9 we obtain that the finite difference operator D (N ) of the Ornstein-Uhlenbeck structure on the Poisson space satisfies an integration by part formula (cf Øksendal and al [9] Thm 12.10) despite its non local character.
Remark 3. In the case of another standard Brownian motionB for M , Proposition 7 gives exactly the derivation operator in the sense of Feyel-La Pradelle cf [4] Chap. II §2.
In that case the situation is quite different from the one we had in Section 2. Indeed Y θ = B cos θ +B sin θ does satisfy the chaotic representation property, so that the space {F θ : F ∈ L 2 (P 1 )} is L 2 (P Y θ ). It is not possible to measurably detect the paths of B andB on those of Y θ . But the concept of chaotic extension becomes simpler because it is compatible with the composition of the functions. It is valid to write in this case Indeed, it is correct for F = Φ( h 1 dB, . . . , h k dB) with Φ a polynomial by Ito formula and induction (what was false in the case of the Poisson process), and then for general F in L 2 by approximation. As a consequence, Proposition 7 gives a formula of Mehler type without integration for the gradient and with integration for the carré du champ whereÊ acts onB as usual. By the change of variable cos θ = e −t/2 this may be also written in a form similar to Mehler formula: what gives denoting P t the Ornstein-Uhlenbeck semi-group well known formula when F and F 2 are in the domain of the generator and that we obtain here for F ∈ D.

Functional calculus of class C 1 ∩ Lip.
Proposition 10. Let us suppose that the process H s be in D A (cf proposition 9) then Proof. The functional F = H s dB s is in D and has a chaotic expansion F = P f (A)dB A . Following the short notation of [6] (p203) if we put for E ∈ Ṗ what proves the proposition.
where in a natural way Lip(R k , R) denotes the set of uniformly Lipschitz real-valued functions defined on R k . It comes from Proposition 7 and from the functional calculus in local Dirichlet structures the following result Proposition 11. When θ → 0, we have in L 2 (P) Proof. We have indeed in the sense of L 2 , the function Φ being Lipschitz and C 1 It follows that we may replace (Φ(F 1 , . . . , F k )) θ by Φ(F θ 1 , . . . , F θ k ) in applying the method. Let us define an equivalence relation denoted ∼ = in the set of functionals in L 2 (P) depending on θ and differentiable in L 2 at θ = 0 by Let us also define a weaker equivalence relation denoted ≃ for functionals in L 0 (P) depending on θ and differentiable in probability at θ = 0 by the limits in the derivations being in probability.
Proof. The equality of the value at zero of the two terms is evident, and differentiating the lefthand term in zero gives which is equal to the derivative of the righthand term.
Let us consider a stochastic differential equation (SDE) with C 1 ∩ Lip coefficients with respect to the argument x Let us recall that we are considering chaotic extensions F → F θ with respect to Y θ = B cos θ + M sin θ.
The following proposition shows that we can calculate the Malliavin gradient of the diffusion by perturbing the Brownian trajectories using an independent normal martingale such as a compensated Poisson process.
Proposition 13. The chaotic extension X θ t of the solution of (11) is equivalent (relation ∼ =) to the solution Z t (θ) of the SDE Proof. Let us denote (X n ) n∈N (resp. (Z n ) n∈N ) the approximating sequence in the Picard iteration applied to equation satisfied by X (resp. Z). We have first X 0 t = Z 0 t = x and By Propositions 10 and 11 Then, what gives by Propositions 12 and 11: By induction, we get easily that for all n ∈ N, (X n t ) θ ∼ = Z n t (θ). But we know that X n t converges to X t as n tend to +∞ not only in L 2 but in D since the coefficients of the SDE are Lipschitz (cf [4] Chap IV), and this implies that (X n t ) θ converges to X θ t and that d dθ (X n t ) θ converges to d dθ X θ t . Now, Z n t (θ) converges to Z t (θ). and its derivative converges to a solution of equation which has a unique explicit solution as linear equation in Z ′ (θ) which is the derivative of Z(θ). That proves the proposition.

The unit jump on the interval [0, 1].
In order to express the preceding results on [0, 1] with a single jump, we propose two different approaches.

First approach
We come back to the case M =Ñ and to express the preceding results on the interval [0, 1], we are conditioning by {N 1 = 1}. This amounts to reasoning on Ω 1 × {N 1 = 1} under the measure P = P 1 × P 2 which gives it the mass e −1 . Then the unique jump is uniformly distributed on [0, 1]. For a functional F ∈ L 2 with expansion F = n I n (f n ), the expansion of the chaotic extension F θ = n I θ n (f n ) considered on the event {N 1 = 1} is the same sum of stochastic integrals but with integrant the semi-martingale V t (θ) = B t cos θ + (1 {t U } − t) sin θ, in other words In the sequel, we use the fact that the notation P f (A)dS A makes sense for any semi-martingale S which admits the decomposition where M is a local martingale whose skew bracket is absolutely continuous w.r.t. the Lebesgue measure and L an absolutely continuous process. For example if U is uniform on [0, 1], then By absolutely continuous change of probability measure we may remove the term in −t sin θ: For the proof let us state the b) If F ∈ L 0 then F (B + tξ) converges to F in probability as t tends to 0.
Proof. a) We develop the square. The first term is as F ∈ L p p > 2 it is uniformly integrable and it converges to EF 2 . For the rectangle term, it is easily seen by change of probability measure that converges to E[F G] for G bounded and continuous. And we can reduce to this case by the above argument. b) We truncate F . If A n = {B : |F | n} by uniform intégrability we can find n such that the probability of A n (B + tξ) be ε for all t. The result comes now from part a).
proof of proposition 14 : Puting C = {N 1 = 1}, we are working under the probability measure Q = e×P 1 ×(P 2 | C ). The conditioning explained above yields the following relation in probability whose second member is we use that the identity map j is a Cameron-Martin function and that 1 0 D s F ds = DF, dj ds L 2 (ds) . If we change of measure and take exp(−B 1 sin θ − sin 2 θ 2 ).Q relation (13) writes saying that, for all ε > 0, tends to zero, where we denote 1) Let us observe that converges in probability to 1 0 D s F ds. The other two terms are processed by the lemma.
We obtain indeed that under Q, 1 In other words we obtain the following result which might have been easily directly verified Proposition 16. The gradient of Malliavin D u X x t of the solution of the SDE may be computed by considering the solution of the equation and taking the derivative in θ at θ = 0.
Let us remark that since u is defined du-almost surely, we may in the equation defining X x t (θ) put either σ(s, X x s (θ)) or σ(s, X x s− (θ)). Let us perform the calculation suggested in the proposition. That gives for u < t Hence, denoting we see by comparing (14) and (15) that so, derivating (14) with respect to θ and (15) with respect to y, and then derivating the second relation of (16) with respect to x This is a fast way of obtaining this classical result (transfert principle by the flow of Malliavin cf [11] Chap VIII). Proposition 16 is the lent particle formula for the Brownian motion. We see that the method of proof allows to obtain this formula without sinus nor cosinus for general F in D provided that we be able to find a functional regular in θ equivalent to the chaotic extension of F . In particular the example of the introduction generalises in the following way : if F = Φ s 1 <···<s k 1 f k 1 dB s 1 · · · dB s k 1 , . . . , s 1 <···<s kn f kn dB s 1 · · · dB s kn with f k i ∈ L 2 (λ k i ) and Φ of class C 1 ∩ Lip, we have The limit is in probability as in Proposition 14.

Second approach
Instead of performing a change of probability measure to remove the term −t sin θ as in the previous approach, we consider M a Lévy process with Lévy measure 1 2 (δ −1 (dx) + δ 1 (dx)) in place ofÑ . Let us remark that we might have considered any Lévy process whose Lévy measure has mean 0 and variance 1. M can be expressed as where N is a Poisson process with intensity 1 and (J n ) n a sequence of i.i.d. variables, independent of N such that P 2 (J 1 = 1) = P 2 (J 1 = −1) = 1 2 .
Proposition 17. Let U be uniform on [0, 1] independent of B. We put R t (θ) = B t cos θ + 1 {t U } sin θ. Let be F ∈ D, F = P f (A)dB A . we have Proof. We denote by U 1 the time of the first jump and by P 2 the conditional law e1 {N 1 =1} P 2 . We consider R 1 t (θ) = B t cos θ + 1 {t U 1 } sin θ and R 1 t (θ) = B t cos θ + J 1 1 {t U 1 } sin θ. The chaotic extension related to Y t (θ) = B t cos θ + M t sin θ satisfies As a consequence of Proposition 7, we have in the sense of L 2 (P 1 × P 2 ): Then, we remark that and obtain lim θ→0 1 θ ( P f (A)dR 1 A (θ) − F ) = D U 1 F in L 2 (P 1 × P 2 ).

Another example
Remark 5. When we enlarge the field of validity of the calculus, by using the equivalence relation ≃ instead of ∼ = to functionals in L 0 (P) depending on θ and differentiable in probabibility at θ = 0, the authorized functional calculus becomes C 1 instead of C 1 ∩ Lip.