Time reversal for drifted fractional Brownian motion with Hurst index H>1/2

Let X be a drifted fractional Brownian motion with Hurst index H > 1 / 2. We prove that there exists a fractional backward representation of X , i


Introduction
Time reversal for diffusion processes driven by a Brownian motion (Bm in short) has already been studied by several authors in the Markovian case (9; 17; 10) and by Föllmer (7) in the Markovian and non Markovian case.The question can be summed up as to know whether the time reversed process is again a diffusion and how to compute its reversed drift and its reversed diffusion coefficient.Different approaches have been proposed.Haussmann and Pardoux (9) tackle this problem by means of weak solutions of backward and forward Kolmogorov equations; Pardoux (17) bases its approach on the enlargement of a filtration.In both cases, the reversed drift, the reversed diffusion coefficient and the Brownian motion driving the reversed diffusion are explicitely identified.Millet, Nualart and Sanz (10) use the integration by parts from Malliavin Calculus and obtain under mild assumptions the expressions of the reversed drift and diffusion coefficient.
These approaches find their roots in Föllmer's work (7) for a class of drifted Brownian motions.He gives, under a finite entropy condition, a formula for the reversed drift of a non Markovian diffusion with a constant diffusion coefficient.He deeply uses the relation between drifts and the forward and backward derivatives introduced by Nelson (12) in his dynamical theory of Brownian diffusions.From a dynamical point of view, Nelson's operators are fundamental tools as regards Brownian diffusions.Based on these operators, it is possible to define an operator which extends classical differentiation from smooth deterministic functions to classical diffusion processes and which allows to give stochastic analogue to standard differential operators (see (3)).Unfortunately these operators fail to exist for a simple fractional Brownian motion (fBm in short) when H = 1/2 (cf Proposition 10 of this paper).
The question one may then address is to know if we can obtain a drifted fBm for the time reversed process of a drifted fBm, which extends the one obtained in the Brownian case.Despite the non existence of Nelson's derivatives for a fBm, we prove that the answer to this question is positive by using the transfer principle and Föllmer's formula in the non Markovian Brownian diffusion case.
Let us explain more precisely our result in the case of a fractional diffusion.This example is further described in Section 5. Let 1/2 H < 1 and (Y H t ) t∈[0,T ] be the solution of the stochastic differential equation dY H t = u(Y H t )dt + dB H t , where the function u is bounded and has bounded first derivative and (B H ) H is a family of fBm transfered from a unique Bm B 1/2 .When H = 1/2, we know (see (17)) that the time reversed process Y 1/2 defined by Y Why this result may be interesting?In the Brownian case, the drift and the time reversed drift are respectively the forward Nelson derivative and minus the backward Nelson derivative which are actually notions of mean velocities.Although these objects do not exist in the fractional case, the drift of a fractional diffusion can be always thought as a forward velocity with respect to the driving process.From this point of view, our structure theorem for the time reversed fractional diffusion gives a backward velocity which is explicit from the initial drift and coherent in the following sense: this quantity is a "continuous" extension in H of the well known notion of the Wiener case.This prevents from non relevant decompositions.Our method of construction is natural but non trivial.We finally mention that an other notion of velocity based on stochastic derivatives with respect to some "differentiating" σ-fields, can be found in (4).
Our paper is organized as follows.In Section 2 we present some preliminary definitions and results about fractional Brownian motion.We recall in Section 3 Föllmer's strategy to tackle the time reversal problem for Brownian diffusion both in the Markovian and the non Markovian case.
In Section 4, we state our main result about the existence of a reversed drift for a drifted fBm with 1/2 < H < 1 which "continuously" extends the one obtained in the Wiener case.Moreover, we prove that Nelson's derivatives are inappropriate tools for a fBm.Section 5 is devoted to the application of our result to fractional diffusions.In Section 6, we discuss the way to construct operators extending Nelson's operators in the fractional case.For the sake of completeness, we finally include in an appendix the proofs of some crucial results from Föllmer (7).

Notations and preliminary results
We briefly recall some basic facts about stochastic integration with respect to a fBm.One refers to (14; 1; 2) for further details.Let B H = (B H t ) t∈[0,T ] be a fractional Brownian motion (fBm in short) with Hurst parameter H > 1/2 defined on a complete filtered probability space (Ω, F, (F t ) t∈[0,T ] , P).We mean that B H is a centered Gaussian process with the covariance function If H = 1/2, then B H is clearly a Brownian motion (Bm in short).From (1), one can easily see that where G is a centered Gaussian variable with variance 1.So the process B H has α−Hölder continuous paths for all α ∈ (0, H).

Spaces of deterministic integrands
We denote by E the set of step R−valued functions on [0,T ].Let H be the Hilbert space defined as the closure of E with respect to the scalar product Then the scalar product between two elements ϕ and ψ of E is given by When H > 1/2, the space H contains L 1 H (0, T ; R) but its elements may be distributions.However, Formula (2) holds for ϕ, ψ ∈ L The covariance kernel R H (t, s) introduced in (1) can be written as where K H (t, s) is the square integrable kernel defined by where 1/2 , for s < t (β denotes the Beta function).We set K H (t, s) = 0 if s t.

We introduce the operator K *
H : E → L 2 (0, T ; R) defined by: It holds that (see ( 14)) for any ϕ, ψ ∈ E and then K * H provides an isometry between the Hilbert space H and L 2 (0, T ; R).We finally denote by K H the operator defined by The space K H L 2 (0, T ; R) is the fractional version of the Cameron-Martin space.In the case of a classical Brownian motion K H (t, s) = 1 [0,t] (s).

Transfer principle
In this work, we shall often use the link between the stochastic integration of deterministic integrand with respect to the fBm and with respect to a Wiener process which is naturally associated with B H .This correspondence is usually called the transfer principle.
The process W = (W t ) t∈[0,T ] defined by is a Wiener process, and the process B H has an integral representation of the form For any ϕ ∈ H, it holds that

Fractional Calculus
In order to describe more precisely some spaces related to the integration of deterministic elements of H, we need further notations.
where Γ denotes the Euler function.This integral extends the classical integral of f when α = 1.
Let I α a+ (L p ) the image of L p (a, b) by the operator I α a+ .If f ∈ I α a+ (L p ) and α ∈ (0, 1), then for almost all x ∈ (a, b), the left-sided Riemann-Liouville derivative of f of order α is defined by its Weyl representation and I α a+ D α a+ f = f .In this framework, the operator K H has the following properties.First, the square integrable kernel K H is given by (see (6)): where F is the Gauss hypergeometric function.The operator K H is an isomorphism from L 2 (0, T ) and it can be expressed as follows when H > 1/2: where h ∈ L 2 (0, T ).The inverse operator K −1 H is given by 1185 for all ϕ ∈ I A fundamental remark for the sequel is that the expression (8) shows that K H h is an absolutely continuous function when H 1/2.In this case, we then set We will need in the sequel the following technical lemma related to the fractional operator O H .
Lemma 1. Set H 0 > 1/2.Let f ∈ L 1 (0, T ) and assume that f satisfies the condition and lim Proof.Fix t ∈ (0, T ).We use (3) and Fubini theorem to write when H > 1/2: where the constant is the one given in the definition of But for all H ∈ (1/2, H 0 ), then the inequality (11).Moreover, using u H−1/2 s H−1/2 t H−1/2 , the following limit holds: lim The hypothesis on the function f allows us to apply the dominated convergence theorem, which yields the convergence (12).

Sample path properties
We finally need the following Lemma about path regularity of processes parameterized by H.
We essentially do the same computations as in the proof of Lemma 7.4 in ( 16) but we give precisions about the dependence on the parameter H of the quantities involved in (13), especially the fact that the random variable ξ H has moments of order q = 1, ..., 5 independent of the parameter H.This will play an important role in our application to stochastic differential equations driven by a fBm.
Of course, this result applies to a fBm since (E|B Proof.With ψ(u) = u p and p(u) = u H in Lemma 1.1 of ( 8), the Garsia-Rodemich-Rumsey inequality reads as follows: where the random variable B is We denote by ξ p,H = B 1/p and we have and since p 5 and H ∈ [1/2, 1), we have We set C p,T = 80 × 4 1/p T 2 c p and the result is proved.

Reminder of time reversal on the canonical probability space
In this section, we recall fundamental results on time reversal on the Wiener space.We essentially use the tools and the results stated by Föllmer in (7).
Let us denote by (X t ) t∈[0,T ] the coordinate process defined on the canonical probability space (Ω * , (F t ), W * ) where Ω * = C([0, T ]) is the space of real valued continuous functions on [0, T ] endowed with the supremum norm, (F t ) t∈[0,T ] is the canonical filtration (generated by the coordinate process X) and W * the Wiener measure.Let W be an equivalent measure to W * .By the Girsanov Theorem, there exists an adapted process such that the process defined by is a Bm under W.
We say that W has finite entropy with respect to W * if According to Proposition 2.11 p.122 in (7), this condition is equivalent to the following finite energy condition (with respect to W): When no confusion is possible, we will omit the measure.
We denote by The following result of Föllmer (cf Lemma 3.1 in ( 7)) ensures the existence of this reversed drift: Lemma 4. If W has finite entropy with respect to W * , then there exists an adapted process ( b t ) t∈[0,T ] with finite energy on [0, τ ], for any τ < T , such that Notice that the reversed drift has only finite energy on [0, τ ] for any τ < T and not on the entire time interval [0, T ].
Föllmer starts from a finite entropy measure and thus produces a finite energy drift.He then works with several measures and their "reversal".Nevertheless for our main result stated in the next section, it is important to work with a unique probability measure.
We stress on the obvious fact that the reversed process X t • R = X T −t under W has the same law than the process X t under W.So by considering the reversed processes and the filtration R(F t ) = σ{X s , T − t s T } we can rewrite Lemma 4 in terms of a unique probability measure.
We start from a drifted Bm defined on a probability space and we have to impose a condition on its drift to obtain the corresponding measure (the one from which Föllmer starts).We choose the Novikov condition, namely The finite energy of the drift is then a straightforward consequence of (16).
We now state a fundamental result of Föllmer on which our main result (Theorem 9) is based on.
The original Föllmer's result only mentions that the time reversed drift has finite energy on [0, τ ], 0 < τ < T .But it turns out to be also in L p (Ω × (0, T )) for any p ∈ (1, 2).This fact induces significant simplifications in the proof of Lemma 11.We give the proof of this theorem in Appendix A.1.
Under the finite entropy condition (15), one can express the drift process (b t ) t∈[0,T ] appearing in (14) in terms of Nelson derivative of the process X. Definition 6.Let X be a F t −adapted process and (G t ) t∈[0,T ] be a decreasing filtration with respect to which X is adapted.The forward and backward Nelson derivative of X are respectively defined for almost all t ∈ (0, T ) as for some p 1, when these limits exist.
The above expressions turn out to be the key point for the explicit computation of the reversed drift of the diffusion X both in Markovian and non Markovian case.
We henceforth work with G t := F T −t = σ(X s ; t s T ).We might refer to the filtrations (F t ) t∈[0,T ] and (G t ) t∈[0,T ] as respectively the past of X and the future of X.
The drift process b of X as well as the drift b of X have the following expression in terms of Nelson derivatives.Proposition 7. Let X be of the form dX t = b t dt + dW t where the process (b t ) t∈[0,T ] is F t −adapted and has finite energy on [0, T ].We denote by b the drift of X (its existence is ensured by Theorem 5).Then for all t ∈ (0, T ), Proof.We refer to Proposition 2.5 p.121 in (7) for a detail proof of (18).
and using that b has finite energy on [0, τ ] for all τ ∈ (0, T ), we deduce that and ( 19) is then proved.
We now recall Föllmer's formula of the reversed drift b.This result will be useful in the last part of the paper when we apply our main result to a fractional diffusion process.
To this end, we notice that since the drift satisfies the Novikov condition ( 16), the Girsanov Theorem insures us that (X t ) t∈[0,T ] is a (F t , Q)-Bm under the probability measure Q defined by dQ/dP = G where We use the classical notations of Malliavin Calculus with respect to the Bm X on (Ω, F, (F t ) t∈[0,T ] , Q).More precisely, we denote D the Malliavin derivative operator, D 1,2 its domain and L 1,2 the Hilbert space which is isomorphic to L 2 ([0, T ]; D 1,2 ) as it is defined in Definition 1.3.2 in (13).
Theorem 8. Let (Ω, F, (F t ) t∈[0,T ] , P) be a complete filtered probability space and let X be a drifted Bm which writes: where (W t ) t∈[0,T ] is a (F t )-Bm and the drift process (b t ) t∈[0,T ] is F t -adapted and satisfies the Novikov condition ( 16) and the following conditions: for almost all t, the process (D t b s ) s∈[0,T ] is Skorohod integrable,

there exists a version of the process
Then the reversed drift reads For the sake of completeness, we also give the proof of Theorem 8 in Appendix A.2.
4 Existence of a continuously extended drift for the time reversed drifted fBm for H > 1/2 In this section, we consider a family of fBm (B H ) H∈[1/2,1) defined on a complete filtered probability space (Ω, F, (F t ) t∈[0,T ] , P) transfered from a unique Bm W : for all H > 1/2

Main result
We are interested in drifted processes of the form where y ∈ R and (u H t ) t∈[0,T ] is an F t −adapted process.A natural question is to know if the time reversed drifted fBm Y H is again a drifted fBm, which extends the one obtained in the Brownian case.We mean that if the formula is parameterized by H, we have to recover the results stated in Theorem 8 for the drifted Brownian motion defined by ( 22) when H = 1 2 : We show in the next theorem that the reversed process of the drifted fBm Y H can be driven by a fBm B H which is related to the Wiener process B 1/2 driving the reversed process Y in L 1 (Ω).We will also give a relation between the drifts of Y H and the one of Y 1/2 .
We will need the following conditions: Remark that the condition (i) is also given for H = 1/2.This implies that the process (u t ) t∈[0,T ] also satisfies the Novikov condition.Moreover, this condition implies that b H ∈ L 2 (Ω × [0, T ]).
Applying the operator K H we deduce that the drift has the special form Besides, this fact will be used in Theorem 9 via Lemma 11 below.
We can now state the main result of our work.
Theorem 9. Given a family of processes (u H t ) t∈[0,T ] which satisfies conditions (i) and (ii), let (Y H t ) t∈[0,t] be a family of processes such that Then there exists a family of continuous processes u H H 1/2 and a family of fBm ( B H ) H 1/2 such that the time reversed process (Y with for all t ∈ (0, T ) lim where W and u are respectively the F Y 1/2 -adapted Bm (reversed drift) produced by the time reversal of the process Y 1/2 defined by Remark that the assumptions (i) insure us that the results on time reversal for the drifted Bm Actually, this assumption is sufficient to construct the reversed drift and the reversed fBm for our drifted fBm Y H .The assumptions of (ii) are used in order to prove that the drift we construct satisfies a kind of robustness with respect to the parameter H as it is explained in the following subsection.

Continuous extension as a structure constraint
The property ( 25) is important if we want to formalize the idea that the reversed formula (24) has to extend the reversed one in the classical Wiener case.In that sense, we might say that our formula is a continuously extended formula of the Wiener case.One might think about this extension as the "commutativity" of the following informal diagram: where R is the time reversal procedure based on the transfer principle.When H = 1/2, there is no transfer to do.
This notion of continuously extension plays its hole part if we consider the naive and trivial decomposition The process T is a fBm (it is centered Gaussian process and has R H as covariance function), but (27) is not a formula which extends in our sense the one obtained in the Wiener case since lim Actually, the decomposition Y Although the decomposition of Theorem 9 is an extension of the classical Wiener formula, we have lost the structure of adaptation with respect to F Y : in the example of the next section we can show for instance that the fBm B H produced by our theorem is not adapted with respect to F B H by showing that the drift is not adapted.

Non existence of Nelson derivatives
Moreover, one may wonder if we can hope to obtain a drifted reversed fBm using Nelson's derivatives.Unfortunately, Nelson's derivatives are inappropriate as an operator acting on drifted fBm thanks to the following proposition.
Proposition 10.Set H = 1/2.The limit exists neither as an element in L p (Ω) for any p ∈ [1, ∞) nor as an almost sure limit.
Proof.Let p ∈ [1, ∞).The process defined by Then we immediately deduce that We fix t ∈ (0, T ).We note that (Z h ) h>0 is a centered Gaussian process.The variance of Z h is given by: If Z h converges in L p (Ω) or almost surely to a random variable Z when h tends to 0, then Z h converges in law to Z, and we know that Z is centered Gaussian variable with variance σ 2 = lim h↓0 σ 2 h .But we shall prove that σ 2 h does not converge when h tends to 0. Indeed, since t → K H (t, s) is differentiable with we have: Therefore we deduce from Fatou Lemma that lim inf So we conclude that when h tends to 0, Z h converges neither in L p (Ω) nor almost surely.
Some related results are extended and studied in more details in (4).

The case H < 1/2
The techniques we have developed may provide a analogous theorem in the case H < 1/2, where moreover the formulas and the study of the operators K −1 H are more tractable.However, we lost the structure of a "drifted process" for the time reversed representation.As we will see in the proof of Theorem 9, we will construct in the case H > 1/2 a drift u H • such that where g is a process.Actually in the case H < 1/2, the operator K H does not map L 2 into a space of absolutely continuous functions (e.g.see (15) for the expressions of K H when H < 1/2).So, although we can still write for Y H a continuous extension formula from the Wiener case: the process U H is not in general of the form

Proof of Theorem 9
In the sequel, we will use the letter X H for a semi-martingale driven by the Bm W , and b H to design its drift.The notation X H means that the semi-martingale depends on H.We will have We need the following lemma: Lemma 11.Let X be a drifted Bm with drift (b t ) t∈[0,T ] satisfying the assumptions of Theorem 8: and let X its time reversed process: Then for any 0 t T , we have the following formula: Proof.We first prove the following equality where 0 < γ < δ < T − t.Remind that The integration by part formula w.r.t. the semimartingale X leads to With the definition of X and the change of variable u = T − t, we then write: We deduce (29) by the integration by part formula w.r.t. the semimartingale X.
Proof.We divide the proof in two steps.
First step.Using the transfer principle and the isometry K H , it holds that where Thanks to the condition (i), we can apply Theorem 5 to the drifted Bm X H with finite energy drift process (b H t ) t∈[0,T ] defined by If ( F t ) t∈[0,T ] is the filtration generated by the reversed process X H t := X H T −t , then there exists a ( F t )-adapted processes ( b H t ) t∈[0,T ] ∈ L p (Ω × [0, T ]) for any p ∈ (1, 2), and a ( F t )-Bm ( W H t ) t∈[0,T ] such that We deduce from Lemma 11 that We then write: where We compute the covariance of the centered Gaussian process ( B H t ) t∈[0,T ] : Since we deduce that Hence the process ( B H t ) t∈[0,T ] is a fBm.Remark moreover that writing the process u H shows that it is continuous (see (10)).
Second step.Let us show that for all t ∈ (0, T ), First of all, we study the first term of the r.h.s. of (30).Lemma 1 implies that We have thanks to Proposition 7: So, by Jensen inequality and Fubini's theorem and then We have where Using and Let 1 < p < 1 H 0 < 2, we use Hölder inequality .
Thanks to the hypothesis (ii), {s → T , thus this family is uniformly integrable.By (34), s 1/2−H 0 E|b H s − u s | → 0 , ds T a.s.when H tends to 1/2, so this convergence also holds in L 1 [0, T ], dt T .Reporting this convergence result in (31), when H tends to 1/2.We now study the second term of the r.h.s. of (30).We write: By Theorem 5, u ∈ L p (0, T ) a.s.for any p ∈ (1, 2), and consequently u satisfies a.s. the hypothesis of Lemma 1 which then yields the following estimation and convergence: Since for any p ∈ (1, 2), u ∈ L p (Ω × (0, T )), we have and we can apply the dominated convergence theorem and write lim So we conclude that for all t ∈ (0, T ), when H tends to 1/2.Using Lemma 3.2 in (5), we have which concludes the proof of the

Application to stochastic differential equations driven by a fBm
First of all, we apply our result to the reversal of a fBm.We yet consider that B H is a fBm having the integral representation B H t = t 0 K H (t, s)dW s .It is well known (see (17)) that the reversed process W solves: where the Brownian motion W t is given by Therefore, thanks to Theorem 9, we deduce that the reversed fBm reads: where the fBm ( B H t ) t∈[0,T ] is given by This situation can be extended in the case of stochastic differential equations driven by B H .

SDE driven by a single fBm
Using successively Theorem 9, Theorem 8 and the results in (17), we can state the following proposition: Proposition 12. (a) Let Y H be the process defined as the unique solution of where the function u is bounded with bounded first derivative.
for all t ∈ (0, T ), where (t, y) → p t (y) is the density of the law of Y (b) The process Y H is not a "fractional diffusion", i.e. of the form (12).
Proof.It is proved in (15) that there exists a unique strong solution of stochastic differential equation (35).
First step.In order to prove (a), we have to verify that all the assumptions of Theorem 9 are fulfilled.We recall that b The process (b H t ) t∈[0,t] satisfies the Novikov condition (16) as noticed in the section 3.3 p.110−111 in (15) and the condition (i) holds true.
We check the assumptions of (ii).
The trajectories of the process (Y t ) t∈[0,T ] are Hölder continuous of order H − ǫ for all ǫ > 0 (see (16; 15)).Since the function u has a bounded first derivative, the process (u(Y t )) t∈[0,T ] has Second step.We now prove (b).We have to use Theorem 8 to obtain the explicit form of the drift.In order to verify the conditions 1, 2 and 3 of Theorem 8, we compute the Malliavin derivatives with respect to the process (X t ) t∈[0,T ] defined by which is a Bm under the probability measure Q defined by dQ/dP = G where G is given by (20).Let Y t = t 0 K H (t, s)dX s where we omit the index H for Y for simplicity.In view of the form of Y , (Y t ) t∈[0,T ] is a fBm with respect to the new probability measure Q and we have the following relations between the Malliavin derivative with respect to X (denoted by D) and the Malliavin derivative with respect to Y (denoted D Y ): for any random variable and one remarks that for any r t The following computations are quite the same one that those carried out in the proof of Lemma 14 of (11) in a different framework.For sake of completeness, we include them. where for r t and the functions f i , i = 1, ..., 4 vanish when r > t.Remind that (see (3)) that we have and it follows that The above expression implies that the process (D r b t ) t∈[0,T ] is adapted with respect to the filtration generated by the Bm X because it is the same one that the filtration generated by the process Y .Therefore if we have the assumptions 1, 2 and 3 of Theorem 8 will be checked.
Using the fact that (Y t ) t∈[0,T ] is a fBm under the probability Q and Lemma 2, we get that for any ǫ > 0, there exists a square integrable random variable ζ H,ǫ such that Since the function u is Lipschitz, we get for 0 < ǫ < 1/2 Reporting in the expression (39) the fact that θ α r α for θ r yields We conclude that From the inequalities (42) and |t −α − s −α | α(t − s)t −α+1 for t s r, we get and the change of variable s = (t − r)ξ + r yields We finally get We use another time the inequality θ α r α for θ r and the change of variable θ = (t − r)ξ + r in order to have and consequently The expression (40) and the inequalities (43), ( 44) and (45) imply that (41) is satisfied.Consequently, Theorem 8 asserts that there exists a reversed drift b of the form (21) for the time reversal of X.Since the drift u , we deduce that Y H cannot be a "fractional diffusion".

A remark on fractional SDE with a non linear diffusion coefficient
We are now interested in fractional SDE with a non linear diffusion coefficient.Let X H be the solution of where the stochastic integral is understood in the Young sense.
Let us assume the conditions given in (10) to ensure that the time reversed process of the diffusion X 1/2 is again a diffusion: 1. b : R → R and σ : R → R are Borel measurable functions satisfying the hypothesis: there exists a constant K > 0 such that for every x, y ∈ R we have 2. For any t ∈ (0, T ), X , so: If we assume that σ is bounded, the derivative of h −1 will also be bounded, hence ) and σ Lipschitz.As a consequence, the stochastic integral s is also continuous in L 1 (Ω) when H ↓ 1/2.But X T −t − X T −s is independent of G T −s and for all r ∈ [s, t], X T −r = X T −r − X 0 is also independent of G T −s .Therefore: T − r dr.
Since (X t ) t∈[0,T ] is a Q-Bm and E Q [(X T −u − α(u)X T −s )X T −s ] = 0, we can write and W (1) is a ( F t , Q)-martingale.The fact that the quadratic variation of the continuous ( F t , Q)martingale W (1) is equal to t together with Lévy theorem conclude the proof of the lemma.
Since P ∼ Q, Girsanov theorem insures the existence of a ( F t )-adapted process (a t ) t∈[0,T ] such that W t := W  Proof.We follow the ideas of Föllmer.
We denote G t = F T −t = σ{X u ; t u ≤ T } = F [t,T ] ∨σ{X t }, where F [t,T ] denotes the sigma-field generated by the increments of the Bm X between t and T .
Remind that the drift (b t ) t∈[0,T ] can be expressed in term of forward Nelson derivative.The reversed drift is expressed thanks to (backward) Nelson derivative: We introduce the following subset of L 2 (Ω): It is easy to check that T t 0 is a total subset of L 2 (Ω, G t 0 , Q).
Then, in order to compute the conditional expectation (48), we have to compute for any random variable F in T t 0 .It is straightforward that for all t t 0 , D t 0 F = D t F .
We now write for all square integrable deterministic function h t :  Taking h t = 1 [t 0 −h,t 0 ] (t), we thus obtain: Since F ∈ T t 0 , we have 1209 With we deduce that From (49), we write: Let a, b ∈ R, a < b.For any p 1, we denote by L p (a, b) the usual Lebesgue spaces of functions on [a, b] and |.| L p (a,b) the associated norm.Let f ∈ L 1 (a, b) and α > 0. The left fractional Riemann-Liouville integral of f of order α is defined for almost all x ∈ (a, b) by is a ( F t , P)-Bm.The process (a t ) t∈[0,T ] has finite energy with respect to P which has finite entropy with respect to Q (see Lemma 3.1 in (7)).We then writeW t = X t − X 0 − t 0 b s ds where b s = a s + X s T − s (47)is a ( F s )-adapted process with a priori only local finite energy, namely finite energy on any time interval [0, τ ] for τ < T .However, one can prove that b ∈ L p (Ω × [0, T ]), 1 < p < 2. The expression (47) indeed gives:| b T −s | |a T −s |

T 0 D 0 b s dX s − 1/ 2 T 0 b 2 s
t (G −1 ).h t dt .using G −1 = exp T ds and the commutativity relationship between the Malliavin derivative and the stochastic integral (see(13), p.38) yieldG.D t (G −1 ) = b t + T t D t b s dX s − T 0 b s .D t b s ds = b t + T t D t b s dW s .
(•)is the density of the law of the process Y 1/2 at time t and B 1/2 is a Brownian motion with respect to the filtration generated by Y 1/2 .
Our main result extends this formula in the following way: we are able to find both a drift process u H and a fBm B H such thatdY H t = u H t dt + d B H tand such that the following convergences hold in L 1 (Ω):s ) ds.
H t ) t∈[0,T ] has Hölder continuous trajectories of order H 0 −1/2, and there exists η > H − 1 2 such that E|u H t − u H s | c|t − s| η b) There exists H 0 > 1/2 such that H• 0 u H s ds satisfies the Novikov condition (16),(ii) There exists H 0 > 1/2 such that a) the process (u Then there exists a family of processes u H H 1/2 and a family of fBm ( B H ) H 1/2 such that the time reversed process (Y . Using the change of variables formula, we obtain that Y verifies bounded with bounded first derivative, we can apply our previous theorem and obtain a time reversed representation for Y 0 − D t b s dW s dt .Using (49), we conclude that:− E ( b T −t 0 + b t 0 )F = E F t b s dW s dt , and the formula for the reversed drift (21) is proved.