On the absolute continuity of one-dimensional SDE s driven by a fractional Brownian motion

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1 On the abolute continuity of one-dimenional SDE driven by a fractional Brownian motion Ivan Nourdin Univerité Henri Poincaré, Intitut de Mathématique Élie Cartan, B.P Vandœuvre-lè-Nancy Cédex, France Ivan.Nourdin@iecn.u-nancy.fr Thoma Simon Univerité d Évry-Val d Eonne, Équipe d Analye et Probabilité Boulevard Françoi Mitterand, 9125 Évry Cédex, France Thoma.Simon@math.univ-evry.fr Abtract The problem of abolute continuity for a cla of SDE driven by a real fractional Brownian motion of any Hurt index i adreed. Firt, we give an elementary proof of the fact that the olution to the SDE ha a poitive denity for all t > when the diffuion coefficient doe not vanih, echoing in the fractional Brownian framework the main reult we had previouly obtained for Marcu equation driven by Lévy procee [9. Second, we extend in our etting the claical entrance-time criterion of Bouleau-Hirch[2. Keyword: Abolute continuity - Do-Sumann tranformation - Fractional Brownian motion - Newton-Côte SDE. MSC 2: 6G18, 6H1. 1 Introduction In thi note we tudy the abolute continuity of the olution at any time t > to SDE of the type: X t = x + b(x ) d + σ(x ) db H, (1) where b, σ are real function and B H i a linear fractional Brownian motion (fbm) with Hurt index H (, 1). In (1), mean a particular type of linear non-emimartingale integrator, the o-called Newton-Côte integrator, which wa recently introduced by one of u et al. [7 [8. Roughly peaking, i an operator defined through a limiting procedure involving the uual Newton-Côte linear approximator (whoe order depend on the roughne of the path B H ), and a forward-backward decompoition à la Ruo-Valloi [12. Thi give a reaonable cla of olution to (1) a oon a σ i regular enough. We refer to [7 and [8 for more detail on thi topic. The main interet of i that it yield a firt order Itô formula: if f : R 2 R i regular enough and Y : Ω R + R i a bounded variation proce, then for every t f(b H t, Y t ) = f(, Y ) + f x(b H, Y ) db H + 1 f y(b H, Y )dy, (2)

2 ee [8 for detail. Thi formula allow to olve (1) through Do [5 and Sumann [13 claical computation. More preciely, our olution X i given by X t = ϕ(b H t, Y t ) for every t > where (x, y) ϕ(x, y) i the flow aociated to σ: ϕ x(x, y) = σ(ϕ(x, y)), ϕ(, y) = y for every (x, y) R 2, (3) and Y i the olution to the random ODE with the notation a(x, y) = b(ϕ(x, y)) ϕ y(x, y) Y t = x + a(b H, Y ) d, { = b(ϕ(x, y)) exp x } σ (ϕ(u, y)) du (4) for every (x, y) R 2. In the equel, we will only refer to X a given by the above Do- Sumann tranformation and we will tudy the abolute continuity with repect to the Lebegue meaure of X t for any t >. Our firt reult, which i given in Section 2, tate that X t ha a poitive denity for every t > a oon a σ doe not vanih. Notice that in the much more difficult framework where the driving proce of (1) i a non Gauian Lévy proce with infinitely many jump, the ame criterion wa obtained in [9. Here, the imple proof relie on a uitable Giranov tranformation [11 which reduce to the eay cae when b, i.e. when X t = ϕ(bt H, x ) for every t >. Thi poitivity reult i related to Propoition 6 in [1, where in a multidimenional etting but without drift, a ufficient condition (which become σ(x ) in dimenion one) under which X t ha a denity for every t > wa given, a well a an equivalent of the denity f t at x when t. We remark that in dimenion one, a cloed formula - ee (5) below - can be readily obtained. Of coure, thi non-vanihing condition on σ i not optimal. For intance, thinking of the equation dx t = X t dbt H whoe olution i X t = X exp Bt H, we ee that the poitivity aumption on σ i not neceary. Moreover, in the Brownian cae H = 1/2, it i well-known that thi criterion can be relaxed either into a condition of Hörmander type when σ i regular enough - ee e.g. [1 p. 111, or into an optimal criterion involving the entrance time into {σ(x) } when σ ha little regularity - ee Theorem 6.3. in [2. We did not try to go in the Hörmander direction, ince the computation involving Newton-Côte integral become quite mey. Neverthele, we were able to obtain a literal extenion of Bouleau-Hirch criterion for any H (, 1). Thi extenion may eem a little urpriing, ince Bouleau-Hirch criterion bear a Markovian flavour, wherea the olution to our SDE i not Markovian in general. The proof, which i given in Section 3, conit in computing the Malliavin derivative of X t via the Do-Sumann tranformation, and then uing a general non-degeneracy criterion of Nualart-Zakai. Notice finally that the computation of thi Malliavin derivative relie mainly on the exitence of a Stratonovich change of variable formula. Hence, our Theorem B below could probably be extended to other type of rough equation driven by fbm, ee e.g. [4 and [6. In thee two paper there are retriction from below on the Hurt parameter of the driving fbm, but on the other hand thi latter i allowed to be multidimenional. Since Bouleau- Hirch criterion alo work in a multidimenional framework (with a more complicated formulation for the entrance-time), one may ak for a general fractional extenion of thi reult. The preent note can be viewed a a firt attempt in thi direction. 2

3 2 A non-vanihing criterion on the diffuion coefficient The following theorem, whoe proof i elementary, yield a firt imple criterion on σ according to which X t ha a poitive denity on R for every t >. Theorem A If σ doe not vanih, then X t ha a poitive denity on R for every t >. Proof. Conidering B H intead of B H if neceary, we may uppoe that σ >. Recalling that ϕ : R 2 R i the flow aociated with σ, we firt notice that for every fixed y R, the function x ϕ(x, y) i a bijection onto R. Indeed, ϕ(, y) i clearly increaing and l = lim x + ϕ(x, y) exit in R {+ }. If l +, then lim x ϕ x(x, y) = σ(l) > and lim x + ϕ(x, y) = +. Similarly, we can how that lim x ϕ(x, y) =, which yield the deired property. We will denote by ψ : R 2 R the invere of ϕ, i.e. ψ(x, y) i the unique olution to ϕ(ψ(x, y), y) = x. (i) When b, we have X t = ϕ(bt H, x ) for every t > and we can write, for every A B(R), P(X t A) = P(Bt H 1 ψ(a, x )) = e u2 2t 2H du 2πt 2H ψ(a,x ) = 1 2πt 2H A Hence, X t ha an explicit poitive denity given by e ϕ(v,x ) 2 2t 2H σ(ϕ(v, x )) dv. f Xt (v) = 1 ϕ(v,x ) 2 2πt 2H e 2t 2H σ(ϕ(v, x )). (5) (ii) When b, we can firt uppoe that b ha compact upport, by an immediate approximation argument. Beide, for every t >, we have X t = ϕ(b H t, Y t ) = ϕ(b H t, ϕ(ψ(y t, x ), x )) = ϕ(b H t + ψ(y t, x ), x ), the lat equality coming from the flow property of ϕ. Since b ha compact upport, it i eay to ee from (4) and the bijection property of ϕ that Y t i a bounded random variable for every t >. Hence ψ(y t, x ) i alo bounded for every t > and we can appeal to Giranov theorem for fbm (ee Theorem 3.1 in [11), which yield X t = ϕ( B H t, x ), where M i a fbm under a probability Q equivalent to P. Hence we are reduced to the cae b and we can conclude from above that, under Q, X t ha a poitive denity over R. Since P and Q are equivalent, the ame hold under P. Remark Theorem A entail in particular that Supp X t = R for every t >. Actually, thi upport property can be extended on the functional level: when σ doe not vanih, it follow eaily from Do argument [5 that Supp X = C x, where X = {X t, t } i viewed a a random variable valued in C x, the et of continuou function from R + to R tarting from x endowed with the local upremum norm. 3

4 3 Extenion of a reult of Bouleau-Hirch In thi ection we extend Theorem A quite coniderably, giving a neceary and ufficient condition on σ in the pirit of Bouleau-Hirch [2 criterion. However our argument are omewhat more elaborate, and we firt need to recall a few fact about the Gauian analyi related to fractional Brownian motion. In order to implify the preentation and without lo of generality, we will fix an horizon T > to (1), hence we will define fbm on [, T only. 3.1 Some recall about fractional Brownian Motion Let u give a few fact about the Gauian tructure of fbm and it Malliavin derivative proce, following Sect. 3.1 in [11 and Chap. 1.2 in [1. Set R H (t, ) := 1 2 (2H + t 2H t 2H ),, t [, T. Let E be the et of tep-function on [, T. Conider the Hilbert pace H defined a the cloure of E wih repect to the calar product ( ) 1[,t, 1 [, = R H(t, ). More preciely, if we et H K H (t, ) = Γ (H + 1/2) 1 (t ) H 1/2 F (H 1/2, 1/2 H; H + 1/2, 1 t/), where F tand for the tandard hypergeometric function, and define the linear operator KH from E to L2 ([, T ) by (K Hϕ)() = K H (T, )ϕ() + T then H i iometric to L 2 ([, T ) thank to the equality (ϕ, ρ) H = T (ϕ(r) ϕ()) K H (r, ) dr, r (K Hϕ)()(K Hρ)() d. (6) B H i a centred Gauian proce with covariance function R H (t, ), hence it aociated Gauian pace i iometric to H through the mapping 1 [,t B H t. Let f : R n R be a mooth function with compact upport and conider the random variable F = f(bt H 1,..., Bt H n ) (we then ay that F i a mooth random variable). The derivative proce of F i the element of L 2 (Ω, H) defined by D F = n i=1 f x i (B H t 1,..., B H t n )1 [,ti (). In particular D Bt H = 1 [,t (). A uual, D 1,1 i the cloure of mooth random variable with repect to the norm F 1,1 = E [ F + E [ D.F H and D 1,1 loc i it aociated local domain, that i the et of random variable F uch that there exit a equence {(Ω n, F n ), n 1} F D 1,1 uch that Ω n Ω a.. and F = F n a.. on Ω n (ee [1 p. 45 for more detail). We finally recall the following criterion which i due to Nualart-Zakai (ee Theorem in [1) : Theorem 1 (Nualart-Zakai) If F D 1,1 loc and a.. D.F H >, then F ha a denity with repect to Lebegue meaure on R. 4

5 3.2 Statement and proof of the main reult Let J = σ 1 ({}) and int J be the interior of J. Conider the determinitic equation and the determinitic time x t = x + b (x ) d (7) t x = up{t : x t int J}. When H = 1/2, it wa proved by Bouleau-Hirch (ee e.g. Theorem 6.3. in [2) that X t ha a denity with repect to Lebegue meaure if and only if t > t x. In particular X t ha a denity for all t a oon a σ(x ), which alo follow from Hörmander condition. Notice that Bouleau-Hirch reult hold in a more general multidimenional context (but then t x i the entrance time of X into the et where σ ha maximal rank, and t x i no more determinitic). In dimenion 1, we aim to extend thi reult to fbm of any Hurt index : Theorem B Let {x t, t }, {X t, t } and t x be defined a above. Then X t ha a denity with repect to Lebegue meaure if and only if t > t x. We will need a lemma which extend Prop in [3, Chap. IV, to fbm. Lemma 2 With the above notation, t x = inf{t > : X t int J} a.. Proof. According to (3), it i obviou that ϕ(x, y) = y for all x R et y J and then ϕ y(x, y) = 1 for all x R and y int J. Set τ = inf{t > : X t int J} and T = inf{t > : Y t int J}. We have a.. t < T t: Y int J t: X = ϕ(b H, Y ) = Y t: X int J t τ, which yield T τ. t < τ t: X int J t: ϕ(b H, X ) = X = ϕ(b H, Y ) t: X = Y t: Y int J t T. Hence τ T. t < t x t: x int J t: x = b(x ) = b ϕ(bh,x ) ϕ y (BH,x) t: Y int J t T, o that t x T. t: x = Y t < T t: Y int J t: Y = b ϕ(bh,y ) ϕ y (BH,Y) t: x int J t t x, whence T t x. = b(y ) t: Y = x Finally, thi prove that a.. t x = T = τ, and complete the proof of the Lemma. Proof of Theorem B. Suppoe firt that t > t x. Recall that X t = ϕ(bt H, Y t ), where ϕ i given by (3) and Y i the unique olution to where we et L u Y = x + = ϕ y(b H u, Y u ) = exp 5 L 1 u b(x u ) du, [ B H u σ (ϕ(z, Y u )) dz

6 for every u - the econd equality being an obviou conequence of (3). Notice that L u > a.. for every u. We will alo ue the notation BH M u = ϕ yy(bu H u, Y u ) = L u σ (ϕ(z, Y u ))ϕ y(z, Y u ) dz, the econd equality coming readily from (3) a well. We now differentiate the random variable X u, u t. Fixing [, t once and for all, the Chain Rule (ee Prop in [1) yield D X u = (σ(x u ) + L u D Y u ) 1 [,u (). In particular, etting N u = L 1 u D X u for every u t, we get N t = L 1 t σ(x t ) + D Y t. Itô formula (2) entail and L t = 1 + L 1 t σ(x t ) = L 1 σ(x ) + L u σ (X u ) db H u + M u dy u ( σ (X u ) L 2 u M u σ(x u ) ) dy u. On the other hand, differentiating Y t yield D Y t = N u b (X u ) du ( σ (X u ) L 2 u M u σ(x u ) + L 1 ) u M u N u dyu. Putting everything together, we get [ N t = L 1 ( σ(x ) exp b (X u ) L 2 u M u b(x u ) ) du. Hence, [ ( [ ) D X t = σ(x ) exp b Lt (X u ) du exp L 1 u M u dy u. L Notice that by Itô formula [ u L u = exp σ (X v ) dbv H + u L 1 v M v dy v, o that [ D X t = σ(x ) exp b (X u ) du + σ (X u ) db H u. 6

7 Now ince t > t x, it follow from Lemma 2 and the a.. continuity of σ(x ) that the function D X t doe not vanih on a ubet of [, t with poitive Lebegue meaure. It i then not difficult to ee that the ame hold for the function (K H D.X t)(). Uing (6), we obtain D.X t 2 H = (D.X t, D.X t ) H = T (K HD.X t ) 2 () d > a.. Thank to Theorem 1, we can conclude that X t ha a denity with repect to Lebegue meaure. Suppoe finally that t t x. Then it follow by uniquene that X t = x t a.. where x t i determinitic, o that X t cannot have a denity. Thi complete the proof of Theorem B. Reference [1 F. Baudoin and L. Coutin. Etude en temp petit de olution d EDS conduite par de mouvement brownien fractionnaire. To appear in C. R. Acad. Sci. Pari. [2 N. Bouleau and F. Hirch. Forme de Dirichlet générale et denité de variable aléatoire réelle ur l epace de Wiener. J. Funct. Analyi 69 (2), pp , [3 N. Bouleau and F. Hirch. Dirichlet Form and Analyi on Wiener Space. De Gruyter, Berlin, [4 L. Coutin and Z. Qian. Stochatic analyi, rough path analyi and fractional Brownian motion. Probab. Theory Related Field 122 (1), pp , 22. [5 H. Do. Lien entre équation différentielle tochatique et ordinaire. Ann. Int. H. Poincaré Probab. Statit. XIII (1), pp , [6 D. Feyel and A. de la Pradelle. Curvilinear integral along rough path. Preprint Evry, 24. [7 M. Gradinaru, I. Nourdin, F. Ruo and P. Valloi. m-order integral and Itô formula for nonemimartingale procee; the cae of a fractional Brownian motion with any Hurt index. To appear in Ann. Int. H. Poincaré Probab. Statit. [8 I. Nourdin. PhD Thei, Nancy, 24. [9 I. Nourdin and T. Simon. On the abolute continuity of Lévy procee with drift. To appear in Annal of Probability. [1 D. Nualart. The Malliavin Calculu and Related Topic. Springer-Verlag, [11 D. Nualart and Y. Ouknine. Stochatic differential equation with additive fractional noie and locally unbounded drift. Preprint Barcelona, 23. [12 F. Ruo and P. Valloi. Forward, backward and ymmetric tochatic integration. Probab. Theory Rel. Field 97 (4), pp , [13 H. J. Sumann. On the gap between determinitic and tochatic ordinary differential equation. Ann. Prob. 6, pp ,