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Such objects were introduced by Nelson in his dynamical theory of Brownian diffusion [9]. For a fixed time t, he calculates a forward (resp., backward) derivative with respect to a given σ-field Pt which can be seen as the past of the process up to time t (resp., Ft, the future of the process after time t). The main class with which this can be done turns out to be that of Wiener diffusions. The purpose of this paper is, on one hand, to introduce notions which can be used to study the aforementioned quantities for general processes and, on the other hand, to treat some examples. We mainly study these notions on solutions of stochastic differential equations driven by a fractional Brownian 1 motion with Hurst index H ≥ 2 . In particular, we recall results on Wiener 1 diffusions (case H = 2 ) in our framework. We prove that for a suitable σ- algebra, the stochastic derivatives of a solution of the fractional stochastic differential equation exist and we are able to give explicit formulas. Our paper is organized as follows. In Section 2, we recall some now clas- sical facts on stochastic for fractional . In Section 3, we introduce the fundamental notions related to the so-called stochastic derivatives. In Section 4, we study stochastic derivatives of Nelson’s type for fractional diffusions. We show in Section 5 that stochastic derivatives with respect to the present turn out to be adequate tools for treating fractional 1 Brownian motion with H > 2 . We also treat the more difficult case of a fractional diffusion.

2. Basic notions for fractional Brownian motion. We briefly recall some basic facts concerning stochastic with respect to a fractional Brown- ian motion; refer to [12] for further details. Let B = (Bt)t∈[0,T ] be a fractional Brownian motion with Hurst parameter H ∈ (0, 1) defined on a probability space (Ω, F, P). This means that B is a centered with the covariance function E(BsBt)= RH (s,t), where

1 2H 2H 2H (1) RH (s,t)= 2 (t + s − |t − s| ). If H = 1/2, then B is a Brownian motion. From (1), one can easily see 2 2H that E|Bt − Bs| = |t − s| , so B has α-H¨older continuous paths for any α ∈ (0,H). STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 3

2.1. Space 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 1 1 h [0,t], [0,s]iH = RH (t,s). 1 We denote by | · |H the associated norm. The mapping [0,t] 7→ Bt can be ex- tended to an isometry between H and the Gaussian space H1(B) associated with B. We denote this isometry by ϕ 7→ B(ϕ). 1 When H ∈ ( 2 , 1), it follows from [14] that the elements of H may not be functions but distributions of negative order. It will be more convenient to work with a subspace of H which contains only functions. Such a space is the set |H| of all measurable functions f on [0, T ] such that T T 2 2H−2 |f||H| := H(2H − 1) |f(u)||f(v)||u − v| du dv < ∞. Z0 Z0

We know that (|H|, | · ||H|) is a Banach space, but that (|H|, h·, ·iH) is not complete (see, e.g., [14]). Moreover, we have the inclusions (2) L2([0, T ]) ⊂ L1/H ([0, T ]) ⊂|H|⊂H.

2.2. Fractional operators. The covariance kernel RH (t,s) introduced in (1) can be written as s∧t RH (t,s)= KH (s, u)KH (t, u) du, Z0 where KH (t,s) is the square integrable kernel defined by t 1/2−H H−3/2 H−1/2 (3) KH (t,s)= cH s (u − s) u du, 0

1 2 When H> 2 , we introduce the operator OH on L ([0, T ]) defined by d (4) (O ϕ)(s) := K (ϕ)(s)= sH−1/2IH−1/2s1/2−H ϕ(s). H dt H  0+ Let f : [a, b] → R be α-H¨older continuous and g : [a, b] → R be β-H¨older continuous with α + β> 1. Then for any s,t ∈ [a, b], the Young integral [18] t s f dg exists and we can express it in terms of fractional derivatives (see R[19]): for any γ ∈ (1 − β, α), we have t t γ γ 1−γ (5) f dg = (−1) Ds+f(x)Dt− gt−(x) dx, Zs Zs where gt−(x)= g(x) − g(t). In particular, we deduce that t α+β (6) ∀s

2.3. . Let S be the set of all smooth cylindrical random variables, that is, which can be expressed as F = f(B(φ1),...,B(φn)) where n n ≥ 1, f : R → R is a smooth function with compact support and φi ∈H. The Malliavin derivative of F with respect to B is the element of L2(Ω, H) defined by n B ∂f Ds F = (B(φ1),...,B(φn))φi(s), s ∈ [0, T ]. Xi=1 ∂xi B 1 1,2 In particular, Ds Bt = [0,t](s). As usual, D denotes the closure of the set of smooth random variables with respect to the norm 2 2 B 2 kF k1,2 = E[F ]+E[|D· F |H]. B n 1 The Malliavin derivative D verifies the : if ϕ : R → R is Cb and 1,2 1,2 (Fi)i=1,...,n is a sequence of elements of D , then ϕ(F1,...,Fn) ∈ D and we have for any s ∈ [0, T ], n ∂ϕ DB ϕ(F ,...,F )= (F ,...,F )DBF . s 1 n ∂x 1 n s i Xi=1 i The operator δB is the adjoint of the derivative operator DB. If a u ∈ L2(Ω, H) belongs to the domain of the divergence operator, that is, if it verifies B |EhD F, uiH|≤ cukF kL2 for any F ∈ S, 6 S. DARSES AND I. NOURDIN then δB (u) is defined by the duality relationship

B B E(F δ (u))=EhD F, uiH for every F ∈ D1,2.

2.4. Pathwise integration with respect to B. If X = (Xt)t∈[0,T ] and Z = (Zt)t∈[0,T ] are two continuous processes, then we define the forward integral of Z with respect to X, in the sense of Russo–Vallois, by · · −1 (7) Zs dXs = lim ucp ε Zs(Xs+ε − Xs) ds, t ∈ [0, T ], Z0 ε→0 Z0 provided the limit exists. Here “ucp” means “uniform convergence in prob- ability.” If X (resp., Z) has a.s. H¨older continuous paths of order α (resp., · β) with α + β> 1, then 0 Zs dXs exists and coincides with the usual Young integral: see [15], PropositionR 2.12.

2.5. Stochastic differential equation driven by B. Here, we assume that k H> 1/2. We denote by Cb the set of all functions whose derivatives from 2 1 order 1 to order k are bounded. If σ ∈ Cb and b ∈ Cb , then the equation t t (8) Xt = x0 + σ(Xs) dBs + b(Xs) ds, t ∈ [0, T ], Z0 Z0 admits a unique solution X in the set of processes whose paths are H¨older continuous of order α> 1 − H. Here, the integral with respect to B is in the sense of Russo–Vallois; see (7). Moreover, we have a Doss–Sussmann-type [5, 16] representation,

Xt = φ(At,Bt), t ∈ [0, T ], where φ and A are given, respectively, by ∂φ (x1,x2)= σ(φ(x1,x2)), φ(x1, 0) = x1, x1,x2 ∈ R ∂x2 and

Bt ′ ′ At = exp − σ (φ(At,s)) ds b(φ(At,Bt)), A0 = x0, t ∈ [0, T ].  Z0  Using this representation, we can show that X belongs to D1,2 and that t t B ′ ′ 1 Ds Xt = σ(Xs)exp b (Xu) du + σ (Xu) dBu [0,t](s), s,t ∈ [0, T ]; Zs Zs  see [10], proof of Theorem B. STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 7

3. Notions related to stochastic derivatives. Let (Zt)t∈[0,T ] be a stochas- tic process defined on (Ω, F, P). In the sequel, we always assume that for 2 any t ∈ [0, T ], Zt ∈ L (Ω, F, P). For all t ∈ (0, T ) and h 6= 0 such that t + h ∈ (0, T ), we set Z − Z ∆ Z = t+h t . h t h

3.1. Stochastic derivatives in a strong sense.

Definition 1. Set t ∈ (0, T ). We say that At (resp., Bt) is a forward differentiating σ-field (resp., backward differentiating σ-field) for Z at t if t t + E[∆hZt|A ] (resp., E[∆−hZt|B ]) converges in probability when h ↓ 0 . In these cases, we define the so-called forward and backward derivatives,

At t (9) D+ Zt = lim E[∆hZt|A ], h↓0+

Bt t (10) D− Zt = lim E[∆−hZt|B ]. h↓0+

The set of all forward (resp., backward) differentiating σ-fields for Z at +(t) −(t) time t is denoted by MZ (resp., MZ ). The intuition we can have is that the larger M±(t) is, the more regular Z is at time t. For instance, one +(t) −(t) obviously has that {∅, Ω} ∈ MZ (resp., ∈ MZ ) if and only if s 7→ E(Zs) is right differentiable (resp., left differentiable) at time t. At the opposite +(t) −(t) extreme, one has that F ∈ MZ (resp., ∈ MZ ) if and only if s 7→ Zs is a.s. right differentiable (resp., left differentiable) at time t.

t t Definition 2. We say that (A , B )t∈(0,T ) is a differentiating collection of σ-fields for Z if for any t ∈ (0, T ), At (resp., Bt) is a forward (resp., backward) differentiating σ-field for Z at t. If At = Bt for any t ∈ (0, T ), we t t t write, for simplicity, (A )t∈(0,T ) instead of (A , B )t∈(0,T ).

On one hand, we may introduce the following definition.

Definition 3. Set t ∈ (0, T ). We say that At (resp. Bt) is a nondegen- erate forward σ-field (resp., nondegenerate backward σ-field) for Z at t if it is forward (resp., backward) differentiating at t and if

At Bt (11) for any c ∈ R P(D+ Zt = c) < 1 [resp., P(D− Zt = c) < 1].

For instance, if Z is a process such that s 7→ E(Zs) is differentiable at t ∈ (0, T ), then {∅, Ω} is a forward and backward differentiating σ-field at t 8 S. DARSES AND I. NOURDIN but is degenerate. Let us also note that condition (11) is obviously equiva- At Bt At 2 lent to Var(D+ Zt) 6= 0 [resp., Var(D− Zt) 6=0] when D+ Zt ∈ L (Ω) [resp., Bt 2 D− Zt ∈ L (Ω)]. On the other hand, one could hope that such stochastic derivatives con- serve the property which holds for ordinary derivatives on functions: “it can discriminate the constants among the other processes.” So we introduce the following.

t t Definition 4. We say that (A , B )t∈(0,T ) is a discriminating collection t t of σ-fields for Z if (A , B )t∈(0,T ) is a differentiating collection of σ-fields for Z and if it satisfies the following property:

At Bt (∀t ∈ (0, T ), D+ Zt = D− Zt = 0) ⇒ Z is a.s. a constant process on [0, T]. t t t As in Definition 2, we write, for simplicity, (A )t∈(0,T ) instead of (A , B )t∈(0,T ) when At = Bt for any t ∈ (0, T ).

An obvious example of a discriminating collection of σ-fields for a process with differentiable paths is {At = F,t ∈ (0, T )}. If Z is a process such that t s 7→ E(Zs) is differentiable on (0, T ), then the collection {A = {∅, Ω},t ∈ [0, T ]} is differentiating, but, in general, not discriminating. Let us now consider a more advanced example. Let B = (Bt)t∈[0,T ] be a fractional Brownian motion with Hurst index H ∈ (1/2, 1). Let us denote g by Pt the σ-field generated by Bs for 0 ≤ s ≤ t and, if g : R → R, by Tt the σ-field generated by g(Bt).

Proposition 5. 1. For any t ∈ (0, T ), Pt is not a forward differentiating σ-field for B at t. g 2. For any even function g : R → R and for any t ∈ (0, T ), Tt is a forward and backward differentiating (but generate) σ-field for B at t. id 3. For any t ∈ (0, T ), Tt is a forward and backward differentiating and nondegenerate σ-field for B at t.

Proof. 1. We refer to Proposition 10 of [4] for a proof. This result is extended to the case of Volterra processes in the current paper; see Propo- sition 13. 2. Since B and −B have the same law, it follows that E[∆hBt|g(Bt)] = 0 for any t ∈ (0, T ) and h 6= 0 such that t + h ∈ (0, T ). The conclusion follows easily. 3. Using a linear Gaussian regression we can write 2H 2H (1 + h/t) − 1 − (h/t) Bt E[∆hBt|Bt]= Bt −→ H in probability. 2 h→0 t STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 9

−1 id  Since Var(Ht Bt) > 0, Tt is nondegenerate.

Thus, for fractional Brownian motion, stochastic derivatives with respect id to the present (i.e., with respect to Tt ) turn out to be adequate tools (see Section 5 below, for a more precise study).

3.2. Stochastic derivatives in a weak sense. A way to weaken Definition 1 is to consider stochastic derivatives as follows.

Definition 6. Set t ∈ (0, T ) and let A be a sub-σ-field of F. We say that Z is weak forward differentiable with respect to A at t if limh↓0+ E[V ∆hZt] exists, for all random variables V belonging to a dense subspace of the closed subspace L2(Ω, A, P) ⊂ L2(Ω, F, P). We similarly define the notion of weak backward differentiation with re- spect to A at t by considering ∆−hZt instead of ∆hZt.

If the process Z is weak forward differentiable at t and such that the 2 sequence (∆hZt)h is bounded in L (Ω), then we can associate with it a weak forward stochastic derivative with respect to A at t. Indeed, in that case, let us denote by Θ the dense subspace involved. The linear form ψ : V 7→ 2 limh↓0+ E[V ∆hZt] defined on Θ ⊂ L (Ω, A, P) is continuous and so can be extended in a unique continuous linear form on L2(Ω, A, P), still denoted by ′ 2 ′ ψ. Thus, there exists a unique Zt ∈ L (Ω, A, P) such that ψ(V )=E[ZtV ]. ′ ′ One can easily show that Zt does not depend on Θ. We will say that Zt is the weak forward stochastic derivative of Z with respect to A at t.

2 Remark 7. The boundedness of (∆hZt)h in L (Ω) may appear to be quite a restrictive condition (e.g., it is not satisfied for a fractional Brownian motion). But it allows us to relate our notion to the usual notion of weak limit.

If At (resp., Bt) is a forward (resp., backward) differentiating σ-field for Z at t and if, moreover, the convergence (9) [resp., (10)] also holds in L2, then Z is weak forward (resp., backward) differentiable with respect to At (resp., Bt) at t. But the converse is not true in general. Let Υ be the set of the so-called fractional diffusions X = (Xt)t∈[0,T ] de- fined by t t (12) Xt = x0 + σs dBs + bs ds, t ∈ [0, T ]. Z0 Z0 Here, σ and b are processes which are adapted with respect to the natural filtration associated with B and X and satisfying the following conditions: σ ∈ Cα a.s. with α> 1 − H and b ∈ L1([0, T ]) a.s. 10 S. DARSES AND I. NOURDIN

Lemma 8. The decomposition ( 12) is unique. That is, if t t t t (13) x0 + σs dBs + bs ds =x ˜0 + σ˜s dBs + ˜bs ds, t ∈ [0, T ], Z0 Z0 Z0 Z0 then x0 =x ˜0, σ =σ ˜ and b = ˜b.

Proof. The equality x0 =x ˜0 is obvious and (13) is then equivalent to t t (σs − σ˜s) dBs = (˜bs − bs) ds, t ∈ [0, T ], Z0 Z0 kT which implies, by setting tk = n , that 1/H (|σtk − σ˜tk ||Btk+1 − Btk |)

tk tk tk 1/H +1 ˜ +1 +1 = (bs − bs) ds + (σs − σtk ) dBs + (˜σs − σ˜tk ) dBs Z Z Z tk tk tk

tk 1/H tk 1/H +1 ˜ +1 ≤ C (bs − bs) ds + (σs − σtk ) dBs  Z Z tk tk

tk+1 1/H + (˜σs − σ˜tk ) dBs . Z  tk

We easily deduce, using (6), that n−1 1/H 1/H lim |σtk − σ˜tk | |Btk − Btk | = 0 in probability. n→∞ +1 kX=0 But, on the other hand, it is easy to obtain (see, e.g., Theorem 4.4 in [7]) that n−1 T 1/H 1/H 1/H lim |σtk −σ˜tk | |Btk −Btk | = |σs −σ˜s| ds in probability. n→∞ +1 Z kX=0 0 We deduce that σ =σ ˜ and so b = ˜b. 

In Section 4, we will see that the past of X ∈ Υ before time t is not, in general, a forward differentiating σ-field at time t. We will see in Section 5 that the present of X ∈ Υ is, in general, a differentiating collection of σ-fields. However, X is weak differentiable for a large class of σ-fields. We intro- b duce the set S of all r.v.’s ϕ(B(φ1),...,B(φn)) ∈ S such that φ1,...,φn are bounded functions. Let ℘ be the set of all sub-σ-fields A ⊂ F such that L2(Ω, A, P) ∩ Sb is dense in L2(Ω, A, P). For instance, any σ-field can be expressed as A[r,s] = ς(Bv, r ≤ v ≤ s) belongs to ℘ (see, e.g., [12], page 24). STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 11

Proposition 9. Let A∈ ℘ and t ∈ (0, T ). Let X ∈ Υ be given by ( 12), satisfying the following conditions: 1 (i) the map s 7→ bs is continuous from (0, T ) into L (Ω); 1,2 B (ii) for all s ∈ [0, T ], σs ∈ D and sups∈[0,T ] E|Ds σt| < +∞; p (iii) E(|σ|α) < +∞ for some p> 1 and α> 1 − H. Then X is weak forward and backward differentiable at t with respect to A.

Proof. For simplicity, we only prove the forward case, the backward case being similar. Let t ∈ (0, T ). We write t+h t+h (14) Xt+h − Xt = σt(Bt+h − Bt)+ bs ds + (σs − σt) dBs. Zt Zt First, we treat the second term of the right-hand side of (14). Let V ∈ 2 b L (Ω, A, P) ∩ S . Since V is bounded and the map s 7→ bs is continuous from 1 (0, T ) into L (Ω), the function s 7→ E[V bs] is continuous. We then deduce, by means of Fubini’s theorem, that 1 t+h (15) lim E V bs ds ds = E[V bt]. h↓0 h  Zt  p Then using inequality (6) and the hypothesis E(|σ|α) < +∞, the following limit holds: 1 t+h (16) lim E V (σs − σt) dBs = 0. h→0 h  Zt  Finally, we show that the limit

lim E[σtV ∆hBt] h↓0 1,2 exists. Since σtV ∈ D (see Exercise 1.2.13 in [11]), we have B 1 E[σt(Bt+h − Bt)V ]=E[δ ( [t,t+h])σtV ] 1 B 1 B = E[σth [t,t+h], D V iH]+E[V h [t,t+h], D σtiH]. Condition (ii) and the fact that V ∈ Sb allow us, in particular, to write

(17) E[σt(Bt+h − Bt)V ]= H(2H − 1)(Ψt,h(σt, V )+Ψt,h(V,σt)), where T t+h B 2H−2 Ψt,h(X,Y )=E X Ds Y |v − s| dv ds .  Z0 Zt 

When X or Y denotes σt or V , Fubini’s theorem yields t+h Ψt,h(X,Y )= f(v,X,Y ) dv Zt 12 S. DARSES AND I. NOURDIN with T B 2H−2 f(v,X,Y )= E[XDs Y ]|v − s| ds. Z0 We have, due to condition (ii) and the fact that V ∈ Sb, that T |f(v,X,Y ) − f(w,X,Y )|≤ C(X,Y ) ||v − s|2H−2 − |w − s|2H−2| ds, Z0 where C(X,Y ) is a constant depending only on X and Y . The previous integral tends to 0 when w tends to v. The continuity of the function v 7→ f(v,X,Y ) follows. Therefore, the limit −1 lim h E[σt(Bt+h − Bt)V ] h→0 exists and equals T T B 2H−2 B 2H−2 H(2H − 1)E σt Ds V |t − s| ds + V Ds σt|t − s| ds .   Z0 Z0  4. Stochastic derivatives of Nelson’s type. Let Z be a defined on (Ω, F, P). We define the past of Z before time t, Z Pt := ς(Zs, 0 ≤ s ≤ t) and the future of Z after time t, Z Ft := ς(Zs,t ≤ s ≤ T ). Z Z If Pt and Ft are, respectively, forward and backward differentiating σ- Z Z Pt Ft fields for Z at t, we call D+ Zt and D− Zt, respectively, the forward and backward stochastic derivatives of Nelson’s type in reference to Nelson’s P F work [9]. In the sequel, we denote them by D+Zt and D− Zt, for simplicity.

4.1. The case of Wiener diffusions. We denote by Λ the space of diffu- sion processes X satisfying the following conditions. 1. X solves the stochastic differential equation

(18) dXt = b(t, Xt) dt + σ(t, Xt) dWt, X0 = x0, d d d d d d where x0 ∈ R , b : [0, T ] × R → R and σ : [0, T ] × R → R ⊗ R are Borel measurable functions satisfying the following hypothesis: there exists a constant K> 0 such that for every x,y ∈ Rd, we have sup(|σ(t,x) − σ(t,y)| + |b(t,x) − b(t,y)|) ≤ K|x − y|, t sup(|σ(t,x)| + |b(t,x)|) ≤ K(1 + |x|). t STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 13

2. For any t ∈ (0, T ), Xt has a density pt. ∗ 3. Setting aij = (σσ )ij, for any i, j ∈ {1,...,n} any t0 ∈ (0, T ) and any bounded open set O ⊂ Rd, T |∂j(aij(t,x)pt(x))| dx dt < +∞. Zt0 ZO 1 4. The functions b and (t,x) 7→ pt(x) ∂j(aij(t,x)pt(x)) are bounded, belong to C1,2([0, T ] × Rd) and have all first- and second-order derivatives bounded 1 [we use the usual convention that the term involving pt(x) is 0 if pt(x) = 0]. These conditions are introduced in [8] and ensure the existence of a drift and a diffusion coefficient for the time-reversed process Xt := XT −t. F¨ollmer focuses in [6], Proposition 2.5, on the important relation between drifts and derivatives of Nelson’s type. It allows the computation the drift of the time reversal of a Brownian diffusion with constant diffusion coefficient, both in the Markov and non-Markov case (see Theorem 3.10 and 4.7 in [6]). For a Markov diffusion with a rather general diffusion coefficient, we have the following result.

Theorem 10. Let X ∈ Λ be given by ( 18). Then X is a Markov diffu- sion with respect to PX and F X . Moreover, PX and F X are differentiating and, in general, nondegenerate: P D+Xt = b(t, Xt),

F 1 (D− Xt)i = bi(t, Xt) − ∂j(aij(t, Xt)pt(Xt)), pt(Xt) Xj 1 where we use the convention that the term involving pt(x) is 0 if pt(x) = 0.

We refer to [3] for a proof; it is based on the proof of Proposition 4.1 in [17] and Theorem 2.3 in [8].

4.2. The case of fractional Brownian motion and Volterra processes. Let K be an L2-kernel, that is, a function K : [0, T ] × [0, T ] → R satisfying 2 ∂+K [0,T ]2 K(t,s) dt ds < +∞. We denote by ∂t the right derivative of K with respectR to t (with the convention that it equals +∞ if it does not exist). We assume, moreover, that K is Volterra, that is, that it vanishes on {(t,s) ∈ [0, T ]2 : s>t} and is nondegenerate. In other words, the family {K(t, ·),t ∈ [0, T ]} is free and spans a vector space dense in L2([0, T ]). With such a kernel K, we associate the so-called Volterra process t (19) Gt = K(t,s) dWs, 0 ≤ t ≤ T, Z0 14 S. DARSES AND I. NOURDIN where W denotes a standard Brownian motion. The assumptions made on K imply, in particular, that the natural filtrations associated with W and G are the same (see, e.g., [1], Remark 3).

Proposition 11. Let t ∈ (0, T ) and G be a Volterra process associated with a nondegenerate Volterra kernel K satisfying the condition K(t + h, ·) − K(t, ·) ∂+K (20) −→ in L2([0,t]). h h↓0 ∂t

P t ∂+K 2 The forward Nelson derivative D+Gt at t exists if and only if 0 ∂t (t,s) ds < P t ∂+K RG +∞. In this case, we have D+Gt = 0 ∂t (t,s) dWs and Pt is nondegen- t ∂+K 2 R erate at t if and only if 0 ∂t (t,s) ds > 0. R Proof. We adapt the proof of [4], Proposition 10. Using the represen- tation (19), we deduce that G W E[∆hGt|Pt ]=E[∆hGt|Pt ] 1 t = [K(t + h, s) − K(t,s)] dWs =: Zh. h Z0

Note that Z = (Zh)h>0 is a centered Gaussian process. First, assume that t ∂+K 2 0 ∂t (t,s) ds =+∞. It is a classical result that if Zh converges in proba- Rbility as h ↓ 0, then Var(Zh) converges as h ↓ 0. But, from Fatou’s lemma, we deduce that t + ∂ K 2 lim inf Var(Zh) ≥ (t,s) ds =+∞. h↓0 Z0 ∂t

Thus, Zh does not converge in probability as h ↓ 0. Conversely, assume that t ∂+K 2 0 ∂t (t,s) ds < +∞. In this case, assumption (20) implies that Zh → R t ∂+K P 0 ∂t (t,s) dWs in probability as h ↓ 0. In other words, D+Gt exists and R t ∂+K G equals 0 ∂t (t,s) dWs. We easily deduce that Pt is nondegenerate at t if R t ∂+K 2  and only if 0 ∂t (t,s) ds > 0. R The result of Proposition 10 in [4] is then a particular case: if B denotes a fractional Brownian motion with Hurst index H ∈ (0, 1/2) ∪ (1/2, 1) and if P t t ∈ (0, T ), then D+Bt does not exist. Indeed, we have Bt = 0 KH (t,s) dWs, where KH is the nondegenerate Volterra kernel given by (3R) and verifying ∂K t H−1/2 H (t,s)= c (t − s)H−3/2. ∂t H s STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 15

Remark 12. For a stochastic process Z, let us define P (21) ξ(Z)=Leb{t ∈ [0, T ], D+Zt exists}. For instance, if B is a fractional Brownian motion with Hurst index H ∈ (0, 1), then ξ(B)= T if H = 1/2 and ξ(B) = 0 otherwise. A real c ∈ [0, T ] being fixed, it is, in fact, not difficult, using Proposition 11, to construct a continuous process Z such that ξ(Z)= c. For instance, we can consider the Volterra process associated with the Volterra kernel H(t) K(t,s)= (t − s) , if s ≤ t,  0, otherwise, 0, if t ≤ c, with H(t)=  (t − c) ∧ 1/4, if t > c.

The study of backward derivatives seems to be more difficult. Among the difficulties, we note that it is not easy to obtain backward representation of fractional diffusions (see [4]). However, for a fractional Brownian motion, we are able to prove the following proposition.

Proposition 13. Set H> 1/2. The limit

Bt − Bt−h B lim E Ft h↓0  h  exists neither as an element of Lp(Ω) for any p ∈ [1, ∞) nor as an almost sure limit.

Proof. Fix t ∈ (0, T ) and set

Bt − Bt−h B Bt − Bt−h Gh := E Ft and Zh := E Bt,Bt+h .  h   h 

Since (Gh)h>0 is a family of Gaussian random variables, it suffices to prove that Var(Gh) diverges when h goes to 0. 2 2 We have Zh = E[Gh|Bt,Bt+h]. So, by Jensen’s inequality, Zh ≤ E[Gh|Bt, Bt+h] and Var(Zh) ≤ Var(Gh). Let us show that limh↓0 Var(Zh)=+∞. The covariance matrix of the Gaussian vector (Bt−h − Bt,Bt,Bt+h) is a v ,  v∗ M  where a = Var(Bt−h − Bt), v = (R(t − h, t) − R(t,t); R(t − h, t + h) − R(t,t + h)) and R(t,t) R(t,t + h) M = .  R(t,t + h) R(t + h, t + h)  16 S. DARSES AND I. NOURDIN

2 Since dh := R(t,t)R(t + h, t + h) − R(t + h, t) 6= 0, M is invertible. There- −1 ∗ ∗ fore, hZh = vM Q , where Q = (Bt,Bt+h). Since M = E[Q Q], we deduce that −1 ∗ −1 ∗ ∗ −1 ∗ Var(hZh)=E[vM Q (vM Q ) ]= vM v . Hence,

1 2 2 Var(hZh)= (R(t + h, t + h)v1 − 2R(t + h, t)v1v2 + R(t,t)v2). dh This expression is homogeneous in t2H , so we henceforth work with t = 1. 2H Tedious computation gives dh ∼ h as h ↓ 0. Moreover, we note that v2 = 2H v1 + ch , where c is a constant depending only on H. Thus, 2H 2H 2H 2H 2 2 4h dh Var(hZh)= v1ch (1 − (1 + h) + h )+ h v1 + c h . 2H v1 Since 2H > 1 and the function x 7→ x is derivable, the quantities h and H H H 1−(1+h)2 +h2 h4 2H−2 h converge as h ↓ 0. But 2H< 2 and h2h2H = h → +∞ as h ↓ 0. Thus,

lim Var(Zh)=+∞, h↓0 which concludes the proof. 

4.3. The case of fractional diffusions.

Proposition 14. Let X ∈ Υ be given by ( 12) and satisfy the following T p conditions: E( 0 |bs| ds) < +∞ and E(|σ|α) < +∞ for some p> 1 and α> X 1 − H. If for anyR t ∈ (0, T ), σt 6= 0 a.s., then for almost all t ∈ (0, T ), Pt is not a forward differentiating σ-field for X at t.

Proof. Remember we assumed that σ and b are adapted with respect to the natural filtration associated with B and X, see (12). In particular, X B we deduce from (12) that Pt ⊂ Pt . Since we can also write t t 1 bs Bt = dXs − ds, Z0 σs Z0 σs X B we finally have Pt = Pt . X B Thus, we deduce that E[∆hBt|Pt ] = E[∆hBt|Pt ] does not converge in probability as h ↓ 0, as a consequence of Proposition 10 in [4] or Proposi- tion 13 of the current paper. T Consider expression (14). The hypothesis E 0 |bs| ds < +∞ allows us to use the techniques of the proof of PropositionR 2.5 in [6] to show that 1 t+h X h E[ t bs ds|Pt ] converges in probability for almost all t. Using inequality p X (6) andR the hypothesis E(|σ|α) < +∞, we can finally conclude that Pt is not a forward differentiating σ-field for X at almost all times t.  STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 17

4.4. The case of fractional differential equations with analytic volatility.

Proposition 15. Let X ∈ Ξ be given by ( 8) and let t ∈ (0, T ). We X assume, moreover, that σ is a real analytic function. Then Pt is a for- ward differentiating σ-field for X at t if and only if σ ≡ 0. In this case, X X X P = {Pt ,t ∈ (0, T )} is a discriminating collection of σ-fields and Pt is degenerate at any t ∈ (0, T ).

Proof. If σ ≡ 0, then X is deterministic and differentiable in t. Conse- X quently, Pt is a forward differentiating σ-field, but is degenerate. Assume, now, that σ 6≡ 0. According to the Bouleau–Hirsch optimal criterion for frac- tional differential equations (see [10], Theorem B), we have that the law of Xt is absolutely continuous with respect to the Lebesgue measure for any −1 t [indeed, we have int σ ({0})= ∅]. We deduce that P(σ(Xt)=0)=0 for any t, since Leb(σ−1({0})) = 0 (σ has only isolated zeros). Proposition 14 X  allows us to conclude that Pt is not a forward differentiating σ-field.

Remark 16. The case where σ is not assumed to be analytic seems X more difficult to handle. We conjecture, however, that in this case, Pt is a forward differentiating σ-field for X if and only if t

5. Stochastic derivatives with respect to the present.

X 5.1. Definition. A consequence of Proposition 11 is that the σ-field Pt generated by Xs, 0 ≤ s ≤ t (the past of X), cannot be used for differentiating when we work with fractional Brownian motion. Moreover, we stress the following important fact: the Markov property of a Wiener diffusion X ∈ Λd X implies that taking expectations with respect to Pt produces the same effect as taking expectations only with respect to Xt. The following definition is then natural.

Definition 17. Let Z = (Zt)t∈[0,T ] be a stochastic process defined on Z a complete probability space (Ω, F, P) and for any t ∈ (0, T ), let Tt be the σ-field generated by Zt. We say that Z admits a forward (resp., backward) Z stochastic derivative with respect to the present t ∈ (0, T ) if Tt is a for- ward (resp., backward) differentiating σ-field for Z at t. In this case, we set Z Z T Tt T Tt D+Zt := D+ Zt (resp., D−Zt := D− Zt). 18 S. DARSES AND I. NOURDIN

Example 18. Let B be a fractional Brownian motion with Hurst index H ∈ (0, 1) and t ∈ (0, T ). Then −1 Ht Bt, if H> 1/2, T D+Bt =  0, if H = 1/2,  does not exist, if H< 1/2, and  −1 Ht Bt, if H> 1/2, T −1 D−Zt =  t Bt, if H = 1/2,  does not exist, if H< 1/2, (see also Proposition 5). In particular, we would say that the fractional Brownian motion with Hurst index H> 1/2 is more regular than Brownian motion (H = 1/2) because of the equality between the forward and backward derivatives in the case H > 1/2, contrary to the case H = 1/2. We can identify the cause of these different regularities: the covariance function RH is differentiable along the diagonal (t,t) in the case H> 1/2, while it is not when H = 1/2.

5.2. Case of fractional differential equations. We denote by Ξ the set of fractional differential equations, that is, the subset of Υ whose elements are 2 1 processes X = (Xt)t∈[0,T ] solving (8) with σ ∈Cb and b ∈Cb . T In the sequel, we compute D±Xt for X ∈ Ξ and t ∈ (0, T ). Let us begin with a simple case.

Proposition 19. Let X ∈ Ξ be given by ( 8) and let t ∈ (0, T ). Assume, moreover, that σ and b are proportional. Then X admits a forward and a backward stochastic derivative with respect to the present t, given by T T −1 (22) D+Xt = D−Xt = Ht σ(Xt)Bt + b(Xt). X In particular, the present Tt is nondegenerate at t if and only if σ(x0) 6= 0 X X and the collection of σ-fields T = {Tt ,t ∈ (0, T )} is discriminating for X.

T Proof. We will only provide the proof for D+Xt, the computation for T D−Xt being similar. Assume that b(x)= rσ(x) with r ∈ R. Then Xt = ′ f(Bt + rt), where f : R → R is defined by f(0) = x0 and f = σ(f). If σ(x0)= T −1 0, then Xt ≡ x0 and D+Xt =0= σ(Xt)Ht Bt + b(Xt). If σ(x0) 6= 0, then it −1 is classical that f is strictly monotone. We can then write Bt = f (Xt)−rt. In particular, the random variables which are measurable with respect to Xt are measurable with respect to Bt and vice versa. On the other hand, by T −1 using a linear Gaussian regression, it is easy to show that D+Bt = Ht Bt (see also Proposition 5). Finally, the convergences (15) and (16) and the equality (14) allow us to conclude that we have (22). STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 19

Now, let us prove that the present is nondegenerate for X at t if and only if σ(x0) 6=0. When σ(x0) = 0, it is clear that the present is degenerate at t (see the first part of this proof). On the other hand, if the present is degenerated at t, then there exists c ∈ R such that −1 Ht σ ◦ f(Bt + rt)Bt + rσ ◦ f(Bt + rt)= c. By rearranging, we obtain that σ ◦f(X)(X +α)= β for some α, β ∈ R, where X = Bt + rt. By using the fact that X has a strictly positive density on R, we deduce that σ ◦ f(x)(x + α)= β for any x ∈ R. Necessarily, β = 0 (with x = −α) and then f ′ = σ ◦ f = 0. We deduce that f is constant and then that f ≡ x0, that is, σ(x0)=0. −1 −1 Finally, if Ht σ(Xt)Bt + b(Xt)= σ(Xt)(Ht Bt + r) = 0 a.s. for any t, then σ(Xt)=0= b(Xt) a.s. for any t and Xt ≡ x0 a.s. for any t; see (8). In X X other words, the collection of σ-fields T = {Tt ,t ∈ (0, T )} is discriminat- ing. 

Let us now describe a more general case.

Theorem 20. Let X ∈ Ξ be given by ( 8) and let t ∈ (0, T ). Assume, 2 2 moreover, that b ∈Cb and that σ ∈Cb is elliptic, that is, satisfies infx∈R |σ(x)| > 0. Then X admits a forward and a backward stochastic derivative with re- spect to the present t, given by T T D+Xt = D−Xt Xt σ(Xt) dy = b(Xt)+ H t Z0 σ(y) (23) t t t b H − E (Xs) ds + βr (s) δBr ds Z0 σ Z0 Z0 t H − t βr (t)δBr Xt , Z0  where r ′ ′ H b σ − bσ 1 βr (t)= OH (Xs) s≥· ds (t).  Z0 σ 

Recall that OH is defined by ( 4).

H Proof. First, note that βr (t) belongs to the domain of the divergence operator δB , due to the additional hypothesis on b and σ. We provide only T T the proof for D+Xt, the computation for D−Xt being similar. F¨olmer [6], Section 4, tackles the problem of the computation of the time reversed drift of a non-Markovian diffusion by means of a Girsanov transformation and the 20 S. DARSES AND I. NOURDIN

Malliavin calculus. Our proof uses such a strategy, coupled with the transfer principle. First step. Assume that σ ≡ 1. Using the transfer principle and the isom- etry KH , it holds that t Xt = KH (t,s) dYs, Z0 where t Yt = Wt + ar dr. Z0 Here, we set · −1 ar = KH b(Xs) ds (r).  Z0 

We know (see [13], Theorem 2) that the process X = (Xt)t∈[0,T ] is a fractional Brownian motion under the new probability measure Q = G · P, where T T 1 2 G = exp − as dWs − 2 as ds .  Z0 Z0  Using the , of Malliavin calculus, we can write, for g : R → 1 R ∈ Cb , −1 X 1 E[(Xt+h − Xt)g(Xt)]=EQ[G g(Xt)δ ( [t,t+h])] −1 1 X =EQ[G h [t,t+h], D g(Xt)iH] 1 X −1 +EQ[g(Xt)h [t,t+h], D G iH] ′ 1 1 = E[g (Xt)]h [t,t+h], [0,t]iH ∗ 1 ∗ X −1 + E[Gg(Xt)hKH [t,t+h], KH D G iL2 ]. ∗ X −1 Y −1 But KH D G = D G (transfer principle). Since T T −1 1 2 G = exp as dYs − 2 as ds , Z0 Z0  we have T T Y −1 Y Y G × Dt (G )= at + Dt as dYs − asDt as ds Z0 Z0 T Y = at + Dt as dWs. Z0 Moreover, T T T Y ∗ X X Ds ar dWr = (KH Ds a)(r) dWr = Ds ar δBr := Φ(s) Z0 Z0 Z0 STOCHASTIC DERIVATIVES FOR FRACTIONAL DIFFUSIONS 21 and ∗ 1 1 (KH [0,t])(s)= KH (t,s) [0,t](s). Therefore, ∗ 1 ∗ X −1 hKH [t,t+h], GKH D G iL2

= (KH a)(t + h) − (KH a)(t) + (KH Φ)(t + h) − (KH Φ)(t) t+h = b(Xu) du + (KH Φ)(t + h) − (KH Φ)(t). Zt T X By the stochastic Fubini theorem, we have (OH Φ)(t)= 0 (OH D· ar)(t) δBr. We set R r H X ′ 1 βr (t) = (OH D· ar)(t)= OH b (Xs) s≥· ds (t).  Z0  We then deduce that

E[(Xt+h − Xt)g(Xt)] ′ 1 1 (24) = E[g (Xt)]h [t,t+h], [0,t]iH t+h t+h T H +E g(Xt) b(Xs) ds + βr (s) δBr ds .  Zt Zt Z0 

By developing E[Xt g(Xt)] as in (24), we obtain t t T 2H ′ H t E[g (Xt)]=E g(Xt) Xt − b(Xs) ds − βr (s) δBr ds .   Z0 Z0 Z0  Then

E[∆hXt | Xt] t t T −1 1 1 H = h h [t,t+h], [0,t]iH Xt − E b(Xs) ds + βr (s) δBr ds Xt  Z0 Z0 Z0 

t+h t+h T −1 H + h E b(Xs) ds + βr (s) δBr ds Xt . Zt Zt Z0 

We deduce that E[∆hXt | Xt] converges in probability, as h ↓ 0, to t t T T H H H H b(Xt)+ Xt − E b(Xs) ds + βr (s) δBr ds − βr (t) δBr Xt . t t Z0 Z0 Z0 Z0 

Since limh↓0 E[∆hXt | Xt] does not depend on T , we finally obtain (23 ) in the particular case where σ ≡ 1 by letting T ↓ t. Second step. Assume that σ does not vanish. Set Yt = h(Xt), where x dy h(x)= 0 σ(y) . Using the formula, we obtain that Y satisfiesR t b −1 Yt = y0 + Bt + ◦ h (Ys) ds, t ∈ [0, T ]. Z0 σ 22 S. DARSES AND I. NOURDIN

Since, on the one hand, the σ-fields generated by Xt and Yt are the same and, on the other hand, X has α-H¨older continuous paths with α> 1/2, we have T T D+Xt = σ(Xt)D+Yt. Expression (23) is then a consequence of the first step of the proof. 

Remark 21. When σ does not vanish and b ≡ rσ with r ∈ R, we can T apply either Proposition 19 or Theorem 20 to compute D±Xt. Of course, the conclusions are the same. Indeed, since we have, in this case, b′σ − bσ′ ≡ 0 Xt dy ′ and 0 σ(y) = Bt + rt [since Xt = f(Bt + rt) with f satisfying f = σ ◦ f], formulaR (23) can be simplified to (22).

Compared to the case where σ and b are proportional, here, it is more difficult to decide if the present (i.e., the collection of σ-fields generated by Xt) is discriminating or not. In the framework of the stochastic embedding of dynamical systems in- troduced in [2], the set of processes, called Nelson differentiable processes, which satisfy the equality between stochastic forward and stochastic back- ward derivatives plays a fundamental role (see [3], Chapters 3 and 7). We stress on the fact that solutions of stochastic differential equations driven by a fractional Brownian motion with H> 1/2 provide examples of Nelson differentiable processes which are not absolutely continuous.

Acknowledgments. This work was carried out while the first author ben- efited from the hospitality of Universit´eParis 6. This institution is gratefully acknowledged. We also wish to thank the anonymous referee for a careful and thorough reading of this work and for his constructive remarks. We finally acknowledge C. Stricker for pointing out an error in a previous version.

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