Stochastic differential equations driven by generalized grey noise
aa r X i v : . [ m a t h . P R ] D ec Stochastic differential equations driven bygeneralized grey noise
José Luís da Silva ,CCM, University of Madeira, Campus da Penteada,9020-105 Funchal, Portugal.Email: [email protected]
Mohamed Erraoui
Université Cadi Ayyad, Faculté des Sciences Semlalia,Département de Mathématiques, BP 2390, Marrakech, MarocEmail: [email protected] 25, 2018
Abstract
In this paper we establish a substitution formula for stochasticdifferential equation driven by generalized grey noise. We then applythis formula to investigate the absolute continuity of the solution withrespect to the Lebesgue measure and the positivity of the density.Finally, we derive an upper bound and show the smoothness of thedensity.
Keywords : Generalized grey Brownian motion, fractional Brownianmotion, stochastic differential equations, absolute continuity.
Contents Substitution theorem 74 Applications 10
A number of stochastic models for explaining anomalous diffusion have beenintroduced in the literature, among them we would like to quote the frac-tional Brownian motion (fBm), see e.g. [MvN68], [Taq03], the Lévy flights[DSU08], the grey Brownian motion (gBm) [Sch90], [Sch92], the generalizedgrey Brownian motion (ggBm) denoted by B α,β [Mur08], [MM09], [MP08]and references therein. The latter is a family of self-similar with stationaryincrements processes ( α -sssi) where the two real parameters α ∈ (0 , and β ∈ (0 , . It includes fBm when α ∈ (0 , and β = 1 , and time-fractionaldiffusion stochastic processes when α = β ∈ (0 , . The gBm corresponds tothe choice α = β , with < β < . Finally, the standard Brownian motion(Bm) is recovered by setting α = β = 1 . We observe that only in the particu-lar case of Bm the corresponding process is Markovian. Moreover the process B α,β has ( α − ε ) -Hölder continuous trajectories for all ε > and it can berepresented (in law) as a scale mixture ( p Y β B H ) where B H is a standardfBm with Hurst parameter H = α/ and Y β is an independent non-negativerandom variable, for the details see Section 2.We will consider the following stochastic differential equation (SDE) on R n X t = x + d X j =1 ˆ t V j ( X s ) dB jα,β ( s ) + ˆ t V ( X s ) ds, t ∈ [0 , T ] , (1.1)where x ∈ R n , T > is a fixed time, B α,β = ( B α,β , . . . , B dα,β ) is a d -dimensional ggBm, α ∈ (1 , , β ∈ (0 , and { V j ; 0 ≤ j ≤ d } is a collectionof vector fields of R n .The stochastic integral appearing in (1.1) is a pathwise Riemann-Stieltjesintegral, see [You36]. It is well known that, under suitable assumptions on V = ( V , . . . , V d ) , the equation (1.1) has a unique solution which is ( α − ε ) -Hölder continuous for all ε > . This result was obtained in [Lyo94] using2he notion of p -variation. The theory of rough paths, introduced by Lyonsin [Lyo94], was used by Coutin and Qian in order to prove an existence anduniqueness result for the equation (1.1) driven by fBm, see [CQ02]. Nualartand Răşcanu [NR02] have established the existence of a unique solution fora class of general differential equations that includes (1.1) using the frac-tional integration by parts formula obtained by Zähle for Young integral, see[Zäh98].The representation in law of B α,β , see (2.8) below, allows us to consider,instead of the equation (1.1), the following equation X Ht = x + d X j =1 ˆ t V j ( X Hs ) d (cid:0)p Y β B jH (cid:1) ( s )+ ˆ t V ( X Hs ) ds, t ∈ [0 , T ] . (1.2)This is due to the fact that the solutions of the SDEs (1.1) and (1.2) inducesthe same distribution on the space of continuous functions C ([0 , T ] ; R n ) .Furthermore, since the stochastic integral in (1.2) is a pathwise Riemann-Stieltjes integral, then the SDE (1.2) can be written as X Ht = x + p Y β d X j =1 ˆ t V j ( X Hs ) dB jH ( s ) + ˆ t V ( X Hs ) ds, t ∈ [0 , T ] . (1.3)The main purpose of this paper is to establish a substitution formula (SF)for equation (1.3). Let us now describe our approach. For each y > , weconsider the following equation X Ht ( y ) = x + √ y d X j =1 ˆ t V j ( X Hs ( y )) dB jH ( s ) + ˆ t V ( X Hs ( y )) ds. (1.4)It is well known that, under suitable assumptions, see e.g. Nualart and Răş-canu [HN07], that if − H < λ < the SDE (1.4) has a strong (1 − λ ) -Höldercontinuous solution X H · ( y ) . To establish a SF, the natural idea is to replace y in (1.4) by the random variable Y β and prove that X H · ( Y β ) satisfies theSDE (1.3). For more details on the SF we refer to [Nua06]. To handle thisproblem, the key is to prove, for each t ∈ [0 , T ] , the following equalities ˆ t V j ( X Hs ( y )) dB jH ( s ) (cid:12)(cid:12)(cid:12)(cid:12) y = Y β = ˆ t V j ( X Hs ( Y β )) dB jH ( s ) , j = 1 , . . . , d, (1.5)3nd ˆ t V ( X Hs ( y )) ds (cid:12)(cid:12)(cid:12)(cid:12) y = Y β = ˆ t V ( X Hs ( Y β )) ds. (1.6)To this end we need to study the regularity of the solution X Ht ( y ) of the SDE(1.4) with respect to y . Once this is accomplished, we use the SF to showthe absolute continuity of the law of the solution X H ( Y β ) and the positivityof its density p X Ht ( Y β ) . Subsequently to give a Gaussian mixture type upperbound and to study the smoothness of p X Ht ( Y β ) . We emphasize the fact thatthese results are essentially due to those established for the density p X Ht ( y ) of the law of X Ht ( y ) , see [BOT14, BNOT14, NS09], and the dependencewith respect to y of p X Ht ( y ) . Indeed, using the SF and the independence of { X Ht ( y ) , ≤ t ≤ T, y > } and Y β , the density p X H ( Y β ) is given by p X Ht ( Y β ) ( z ) = ˆ + ∞ p X Ht ( y ) ( z ) p Y β ( y ) dy, z ∈ R n , where p Y β is the density of the law of Y β . Hence, the density p X Ht ( Y β ) is givenin terms of a parameter dependent integral, implying that all the propertiesof p X Ht ( Y β ) will be deducted from those of p X Ht ( y ) . This persuade us to borrowthe hypotheses of the cited works to realize the above results. According to Mura and Pagnini [MP08], the ggBm B α,β is a stochastic pro-cess defined on a probability space (Ω , F , P ) such that for any collection ≤ t < t < . . . < t n < ∞ the joint probability density function of ( B α,β ( t ) , . . . , B α,β ( t n )) is given by f α,β ( x, t , . . . , t n ) = (2 π ) − n p det(Σ α ) ˆ ∞ τ n/ exp (cid:18) − x ⊤ Σ − α x τ (cid:19) M β ( τ ) dτ, (2.1)where n ∈ N , x ∈ R n , Σ α = ( a i,j ) ni,j =1 is the matrix given by a i,j = t αi + t αj − | t i − t j | α , and M β is the so-called M -Wright probability density function (a naturalgeneralization of the Gaussian density) which is related to the Mittag-Leffler4unction through the following Laplace transform ˆ ∞ e − sτ M β ( τ ) dτ = E β ( − s ) . (2.2)Here E β is the Mittag-Leffler function of order β , defined by E β ( x ) = ∞ X n =0 x n Γ( βn + 1) , x ∈ R . It follows from (2.1) that for a given u = ( u , . . . , u n ) ∈ R n , n ∈ N and anycollection { B α,β, ( t ) , . . . , B α,β, ( t n ) } with ≤ t < t < . . . < t n < ∞ we have E exp i n X k =1 u k B α,β ( t k ) !! = E β (cid:18) − u ⊤ Σ α u (cid:19) . (2.3)Equation (2.3) shows that ggBm, which is not Gaussian in general, is astochastic process defined only through its first and second moments whichis a property of Gaussian processes.The following properties can be easily derived from (2.3).1. B α,β (0) = 0 almost surely. In addition, for each t ≥ , the moments ofany order are given by ( E ( B n +1 α,β ( t )) = 0 , E ( B nα,β ( t )) = (2 n )!2 n Γ( βn +1) t nα .
2. For each t, s ≥ , the characteristic function of the increments is E (cid:0) e iu ( B α,β ( t ) − B α,β ( s )) (cid:1) = E β (cid:18) − u | t − s | α (cid:19) , u ∈ R . (2.4)3. The covariance function has the form E ( B α,β ( t ) B α,β ( s )) = 12Γ( β + 1) ( t α + s α − | t − s | α ) , t, s ≥ . (2.5)It was shown in [MP08] that the ggBm B α,β admits the following represen-tation (cid:8) B α,β ( t ) , t ≥ (cid:9) d = (cid:8)p Y β B H ( t ) , t ≥ (cid:9) , (2.6)5here d = denotes the equality of the finite dimensional distribution and B H isa standard fBm with Hurst parameter H = α/ . Y β is an independent non-negative random variable with probability density function M β . A processwith the representation given as in (2.6) is known to be variance mixture ofnormal distributions. A consequence of the representation (2.6) is the Höldercontinuity of the trajectories of ggBm which reduces to the Hölder continuityof the fBm. Thus we have E ( | B α,β ( t ) − B α,β ( s ) | p ) = c p | t − s | pα/ . (2.7)We conclude that the process B α,β has ( α − ε ) -Hölder continuous trajectoriesfor all ε > . So, we can use the integral introduced by Young [You36] withrespect to B α,β . That is, for any Hölder continuous function f of order γ such that γ + ( α/ > and every subdivision ( t ni ) i =0 ,...,T of [0 , T ] , whosemesh tends to , as n goes to ∞ , the Riemann sums n − X i =0 f ( t ni ) (cid:0) B α,β ( t ni +1 ) − B α,β ( t ni ) (cid:1) converge to a limit which is independent of the subdivision ( t ni ) i =0 ,...,T . Wedenote this limit by ˆ T f ( t ) dB α,β ( t ) . Till now we have recalled the ggBm in -dimension, but from now on we usea d -dimensional ggBm B β,α = ( B α,β , . . . , B dα,β ) ( < β ≤ , < α ≤ ) withcharacteristic function E (cid:0) e i ( x , B α,β ( t )) R d (cid:1) = E β (cid:18) −
12 ( x , x ) R d t α (cid:19) and the representation in law B α,β ( t ) = p Y β B H ( t ) , t ≥ , (2.8)where Y β is independent of B H ( t ) , B H is a d -dimensional fBm with Hurstparameter H = α/ . Notations : Throughout this paper, unless otherwise specified we will makeuse of the following notations: 6or < λ < we denote by C λ (cid:0) , T, R d (cid:1) the space of all λ -Höldercontinuous functions f : [0 , T ] −→ R d , equipped with the norm k f k λ = k f k ,T ∞ + k f k ,T,λ where k f k ,T, ∞ = sup ≤ t ≤ T | f ( t ) | , k f k ,T,λ = sup ≤ s
Let
T > and / < δ < H < be given. Under Hypothesis( H.1 ) there exist a positive constant C n depending on T, δ, H, k V k C b and k V k C b such that k X H ( y ) − X H ( e y ) k δ ≤ C n (cid:12)(cid:12)(cid:12) √ y − pe y (cid:12)(cid:12)(cid:12) k V k C b k B H k ,T,δ × (1 + k B H k ,T,δ ) /δ exp (cid:16) C n k B H k /δ ,T,δ (cid:17) for all | y | , | e y | ≤ n . Now we are ready to state the regularity of the solution X Ht ( y ) of theSDE (1.4) with respect to y . Proposition 4.
Let
T > and / < δ < H < be given. Under Hypothesis( H.1 ) there exist a positive e C n > depending on T , δ , H , k V k C b and k V k C b such that E (cid:18) sup s ≤ t (cid:12)(cid:12) X Hs ( y ) − X Hs ( e y ) (cid:12)(cid:12) (cid:19) ≤ e C n | y − e y | , t ∈ [0 , T ] for all | y | , | e y | ≤ n .Proof. Let t ∈ [0 , T ] and | y | , | e y | ≤ n be fixed. Using the estimate in Propo-sition 3 we obtain E (cid:18) sup s ≤ t (cid:12)(cid:12) X Ht ( y ) − X Ht ( e y ) (cid:12)(cid:12) (cid:19) ≤ C n (cid:12)(cid:12)(cid:12) √ y − pe y (cid:12)(cid:12)(cid:12) k V k C b × E (cid:18) k B H k ,T,δ (cid:16) k B H k ,T,δ (cid:17) /β exp (cid:16) C n k B H k /δ ,T,δ (cid:17)(cid:19) .
8t follows from the assertions (i) and (ii) of Lemma 1 and the following Younginequality C n k B H k /δ ,T,δ ≤ δ − δ (cid:18) C n ε (cid:19) δ/ (2 δ − + ε δ k B H k ,T,δ that, for small enough ε , there exist a constant e C n > depending on T, δ, H, k V k C b and k V k C b such that E (cid:18) sup s ≤ t (cid:12)(cid:12) X Hs ( y ) − X Hs ( e y ) (cid:12)(cid:12) (cid:19) ≤ e C n | y − e y | . The following proposition provides the substitution formulas (1.5) and(1.6).
Proposition 5.
Under Hypothesis (
H.1 ) the equalities (1.5) and (1.6) aresatisfied.Proof.
First let’s recall that, for j = 1 , . . . , d and any y > , the Youngintegrals ˆ T V j ( X Hs ( y )) dB jH ( s ) (3.2)and ˆ T V j ( X Hs ( Y β )) dB jH ( s ) (3.3)exist. Indeed, if − H < λ < , then for each y > , the SDE (1.4)has a strong (1 − λ ) -Hölder continuous solution X H · ( y ) . Therefore, the pro-cess X H. ( Y β ) has (1 − λ ) -Hölder continuous paths. Then the existence ofthe preceding integrals follows from the Lipschitz condition of V j and theHölder continuity of the paths of B jH . As a consequence, for any subdivision ( t nk ) k =0 ,...,n − of [0 , T ] , whose mesh tends to as n goes to ∞ , and each y ≥ ,the Riemann sums S jn ( y ) = n − X k =0 V j ( X Ht nk ( y )) (cid:0) B jH ( t nk +1 ) − B jH ( t nk ) (cid:1) R jn = n − X k =0 V j ( X Ht nk ( Y β )) (cid:0) B jH ( t nk +1 ) − B jH ( t nk ) (cid:1) converge to (3.2) and (3.3), respectively. Now to prove (1.5), it suffices toshow that S jn ( Y β ) = R jn , converge, as n goes to ∞ , to ˆ T V j ( X Hs ( y )) dB jH ( s ) (cid:12)(cid:12)(cid:12)(cid:12) y = Y β . Taking into account that the fBm with Hurst parameter H has locally bounded p -variation for p > /H and the regularity of the solution X Ht ( y ) with respectto y , cf. Proposition 4, then the above mentioned convergence follows fromLemma 3.2.2 in Nualart and the following estimate, E (cid:12)(cid:12) S jn ( y ) − S jn ( e y ) (cid:12)(cid:12) = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 (cid:16) V j ( X Ht nk ( y )) − V j ( X Ht nk ( e y )) (cid:17) (cid:0) B jH ( t nk +1 ) − B jH ( t nk ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | y − e y | for all | y | , | e y | ≤ n . The equality (1.6) is easy to prove.The main result of this section is the following theorem. Theorem 6.
The process (cid:8) X Ht ( Y β ) , t ∈ [0 , T ] (cid:9) satisfies the SDE (1.2).Proof. It follows from the classical Kolmogorov criterion that, for each t ∈ [0 , T ] , there exists a modification of the process (cid:8) X Ht ( y ) , y ≥ (cid:9) that is acontinuous process whose paths are γ -Hölder for every γ ∈ [0 , ) . Now usingthe equalities (1.5) and (1.6) we obtain that the process (cid:8) X Ht ( Y β ) , t ∈ [0 , T ] (cid:9) satisfies the SDE (1.2) by substituting y = Y β ( w ) in the SDE (1.4). Thiscompletes the proof. As an application of the SF obtained in the previous section we first deduce,under suitable non degeneracy condition on the vector field V , the absolutecontinuity (with respect to the Lebesgue measure on R n ) of the law of thesolution X Ht ( Y β ) at any time t > . Secondly we give sufficient conditionsfor the strict positivity of the density, cf. Subsection 4.1. Finally we de-rive a Gaussian mixture upper bound for the density and its smoothness inSubsections 4.2 and 4.3. 10 .1 Absolute continuity In order to investigate the absolute continuity of the law of X Ht ( Y β ) on R n and the strict positivity of the density we assume: (H.2) The vector fields V , . . . , V d are C ∞ b . (H.3) For every x ∈ R n and every non vanishing λ ∈ R d , the vector spacespanned by { V j ( x ) , [ V j , Z ] , ≤ j ≤ d } is R n , where Z is given by Z = P dj =1 λ j V j . Proposition 7.
Assume that Hypotheses ( H.2 ) and ( H.3 ) hold. Then forany t ∈ (0 , T ] , we have:1. The law of the solution X Ht ( Y β ) of the SDE (1.2) has a density p X Ht ( Y β ) with respect to the Lebesgue measure on R n .2. The density p X Ht ( Y β ) is strictly positive, that is p X Ht ( Y β ) ( z ) > for all z ∈ R n .Proof.
1. It follows from Theorem . in Baudoin and Hairer [BH07] that,for any y > and t ∈ (0 , T ] , the law of the the solution X Ht ( y ) of the SDE(1.4) has a smooth density p X Ht ( y ) with respect to the Lebesgue measure on R n . Since { B H ( t ) , ≤ t ≤ T } and Y β are independent, then { X Ht ( y ) , ≤ t ≤ T, y > } and Y β are also independent. Now it is easy to see that thedensity function of X Ht ( Y β ) is given by p X Ht ( Y β ) ( z ) = ˆ ∞ p X Ht ( y ) ( z ) M β ( y ) dy, z ∈ R n . (4.1)2. Let t ∈ (0 , T ] be given. It is follows from Baudoin et al. [BNOT14] that,under Hypotheses ( H.2 ) and (
H.3 ), for any y > the density p X Ht ( y ) of thethe solution X Ht ( y ) of the SDE (1.4) fulfills p X Ht ( y ) ( z ) > , for all z ∈ R n .Then for any z ∈ R n we have p X Ht ( y ) ( z ) > for all y > . It follows that thedensity (4.1) p X Ht ( Y β ) ( z ) > for all z ∈ R n . Remark . The absolute continuity of the law of X Ht ( Y β ) may be obtainedusing Theorem 8 in Nualart and Saussereau [NS09] under weaker regularityconditions on V j , ≤ j ≤ d . Namely, V j ∈ C b , ≤ j ≤ d and the followingnon degeneracy hypothesis: (H.4) For every x ∈ R n , the vector space spanned by V ( x ) , . . . , V d ( x ) is R n .11 .2 Upper bound of the density First of all, we recall the result of Baudoin [BOT14] on the global Gaussianupper bound for the density function p X Ht ( y ) of the solution X Ht ( y ) , for y > .Moreover, we highlight the dependence of p X Ht ( y ) with respect to y . For thiswe need to assume the same assumptions and also keep the same notation asin the original work [BOT14]. We suppose that our vector fields V , . . . , V d fulfill the following antisymmetric hypothesis: (H.5) There exist smooth and bounded functions ω ki,j such that: [ V i , V j ] = d X k =1 ω ki,j V k and ω ki,j = − ω ji,k , i, j = 1 , . . . , d. The following theorem is an adaptation of Theorem 1.3 in [BOT14] forthe equation (1.4).
Theorem 9.
Assume that Hypotheses (
H.2 ), ( H.4 ) and ( H.5 ) are satisfied.Then, for t ∈ (0 , T ] , the random variable X Ht ( y ) admits a smooth density p X Ht ( y ) . Furthermore, there exist positive constants c (1) t ( y ) , c (2) t ( y ) , c (3) t ( y ) such that p X Ht ( y ) ( z ) ≤ c (1) t ( y ) exp (cid:18) − c (3) t ( y ) (cid:16) | z | − c (2) t ( y ) (cid:17) (cid:19) for any z ∈ R n . We would like to emphasize the dependence of the constants c (1) t ( y ) , c (2) t ( y ) , c (3) t ( y ) with respect to y . For that we need a careful reading ofthe the proof of Theorem 1.3 in [BOT14] taking into account the dependencewith respect to y. In a first step we look for the constants c (2) t ( y ) , c (3) t ( y ) . Itshould be noted that these two constants come from the tail estimate of thesolution P [ X Ht ( y ) > z ] . It follows from Proposition 2.2 in [BOT14] (see alsoHu and Nualart [HN07]) that there exist a constant C > depending on V , V , k and x , such that sup ≤ t ≤ T (cid:12)(cid:12) X Ht ( y ) (cid:12)(cid:12) ≤ | x | + y / (2 δ ) CT k B H k /δ ,T,δ (4.2) sup ≤ t ≤ T (cid:13)(cid:13)(cid:13) γ − X Ht ( y ) (cid:13)(cid:13)(cid:13) ≤ CT Hd h (cid:16) y / (2 δ ) CT k B H k /δ ,T,δ (cid:17)i (4.3)12 up ≤ t,r i ≤ T (cid:12)(cid:12) D j k r k . . . D j r X Ht ( y ) (cid:12)(cid:12) ≤ C exp (cid:16) y / (2 δ ) CT k B H k /δ ,T,δ (cid:17) (4.4)where D and γ X Ht denote the Malliavin derivative and the Malliavin matrixof X Ht ( y ) , respectively.On the other hand, we obtain from Theorem 3.1 in [BOT14] the followingdeterministic bound of the Malliavin derivative of the solution X Ht ( y ) , almostsurely k D X Ht ( y ) k ∞ ≤ M y exp( θt ) , y > , where the constant θ linearly depend on V and M = sup x ∈ R n sup k λ k≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X j =1 λ j V j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Now using the concentration property and the inequality (4.2) we obtain P [ X t ( y ) > z ] ≤ exp (cid:16) − c (3) t ( y ) (cid:0) | z | − c (2) t ( y ) (cid:1) (cid:17) (4.5)where c (2) t ( y ) = √ d E (cid:18) max i =1 ,...,n (cid:16)(cid:12)(cid:12)(cid:12) X H,it ( y ) (cid:12)(cid:12)(cid:12)(cid:17)(cid:19) and c (3) t ( y ) = 12 dM e θt t H y . For the constant c (1) t ( y ) , it derived from the norms of the Malliavin derivativeand the Malliavin matrix of X Ht ( y ) . Indeed, using the inequalities (4.2)-(4.4)and the tail estimate (3.1), Theorem 3.14 in [BOT14] gives us the followingGaussian upper bound of the density p X Ht ( y ) , for y > , p X Ht ( y ) ( z ) ≤ c (1) t ( y ) exp (cid:18) − c (3) t ( y ) (cid:16) | z | − c (2) t ( y ) (cid:17) (cid:19) , (4.6)where the constant c (1) t ( y ) is given by c (1) t ( y ) = (cid:13)(cid:13)(cid:13) det γ − X t ( y ) (cid:13)(cid:13)(cid:13) mL p k D X t ( y ) k m ′ k,p ′ p, p ′ > and integers m, m ′ . Let us note that for τ < / (128(2 T ) H − δ ) ) , c (1) t ( y ) satisfy c (1) t ( y ) ≤ C (cid:18) t m/p F ( t, δ − , δ , Cy / δ ) m/p (cid:19) × (cid:18) t nH ( k +1)+ m ′ p ′ F ( t, δ − , δ , Cy / δ ) m ′ /p ′ (cid:19) (4.7)where F (cid:18) t, δ − , δ , Cy / (2 δ ) (cid:19) := ˆ + ∞ u (1 /δ ) − exp (cid:0) − τ u (cid:1) exp (cid:0) Cty / (2 δ ) u /δ (cid:1) du. Now we are ready to give the upper bound of the density p X Ht ( Y β ) . Proposition 10.
Assume that Hypotheses (
H.2 ), (
H.4 ) and (
H.5 ) are sat-isfied. Then for t ∈ (0 , T ] , the density p X Ht ( Y β ) satisfies the following Gaussianmixture type upper bound, for all z ∈ R n p X Ht ( Y β ) ( z ) ≤ ˆ ∞ ρ H ( z, y ) M β ( y ) dy, (4.8) where ρ H ( z, y ) := c (1) t ( y ) exp (cid:18) − c (3) t ( y ) (cid:16) | z | − c (2) t ( y ) (cid:17) (cid:19) . (4.9) Proof.
First we point out the asymptotic behavior of the function M β ( y ) when y goes to ∞ , see Eq. (4.5) in [MMP10]: M β ( y/β ) ∼ p π (1 − β ) y ( β − / / (1 − β ) exp (cid:18) − − ββ y / (1 − β ) (cid:19) . (4.10)With this and the fact that δ < < − β we see that, for any p > , theintegral ˆ + ∞ (cid:18) F (cid:18) t, δ − , δ , Cy / (2 δ ) (cid:19)(cid:19) p M β ( y ) dy is finite. This allows us to conclude that the integral ´ ∞ ρ H ( z, y ) M β ( y ) dy is well defined and as a consequence the density function p X Ht ( Y β ) satisfy theGaussian mixture type upper bound (4.8).14 .3 Smoothness of the density To show the smoothness of the density p X Ht ( Y β ) we use the differentiationunder the integral sign in representation (4.1). Since p X Ht ( y ) is smooth forany y > , then it is sufficient to obtain an upper bound of | ∂ κ p X Ht ( y ) | ≤ h κ ( y ) ,for any multi-index κ , such that ˆ ∞ h κ ( y ) M β ( y ) dy < ∞ . It follows from the proof of Proposition 2.1.5 in [Nua06] that | ∂ κ p X Ht ( y ) ( z ) | ≤ c (1) t,κ ( y ) exp (cid:18) − c (3) t ( y ) (cid:16) | z | − c (2) t ( y ) (cid:17) (cid:19) , where c (1) t,κ ( y ) = (cid:13)(cid:13)(cid:13) det γ − X t ( y ) (cid:13)(cid:13)(cid:13) lL q k D X t ( y ) k l ′ k ′ ,q ′ for some integer l, l ′ , k ′ and constants q, q ′ > . The function c (1) t,κ may beestimated as in (4.7) which implies that ˆ ∞ c (1) t,κ ( y ) M β ( y ) dy < ∞ . This is sufficient to guarantee the smoothness of the density p X Ht ( Y β ) . Westate this result in the following proposition. Proposition 11.
Assume that Hypotheses (
H.2 ), (
H.4 ) and (
H.5 ) are sat-isfied. Then, for t ∈ (0 , T ] , the density p X Ht ( Y β ) is a smooth ( C ∞ ) function. Acknowledgments
We would like to thank Professor David Nualart for reading the first versionof that paper and the suggestion for studying a more general case presentedhere. Financial support of the project CCM - PEst-OE/MAT/UI0219/2014and Laboratory LIBMA form the University Cadi Ayyad Marrakech aregratefully acknowledged. 15 eferences [BH07] F. Baudoin and M. Hairer. A version of Hörmander’s theorem forthe fractional Brownian motion.
Probab. Theory Related Fields ,139(3-4):373–395, 2007.[BNOT14] F. Baudoin, E. Nualart, C. Ouyang, and S. Tindel. On probabilitylaws of solutions to differential systems driven by a fractionalBrownian motion. arXiv.org , January 2014.[BOT14] F. Baudoin, C. Ouyang, and S. Tindel. Upper bounds for thedensity of solutions to stochastic differential equations drivenby fractional Brownian motions.
Ann. Inst. Henri PoincaréProbab. Stat. , 50(1):111–135, 2014.[CQ02] L. Coutin and Z. Qian. Stochastic analysis, rough path analysisand fractional Brownian motions.
Probab. Theory Related Fields ,122(1):108–140, 2002.[DSU08] A. A. Dubkov, B. Spagnolo, and V. V. Uchaikin. Lévy flightsuperdiffusion: an introduction.
Internat. J. Bifur. ChaosAppl. Sci. Engrg. , 18(9):2649–2672, 2008.[HN07] Y. Hu and D. Nualart. Differential equations driven by Höldercontinuous functions of order greater than 1/2. In
Stochasticanalysis and applications , pages 399–413. Springer, 2007.[Lyo94] T. Lyons. Differential equations driven by rough signals. I. An ex-tension of an inequality of L. C. Young.
Math. Res. Lett , 1(4):451–464, 1994.[MM09] A. Mura and F. Mainardi. A class of self-similar stochastic pro-cesses with stationary increments to model anomalous diffusionin physics.
Integr. Transf. Spec. F. , 20(3-4):185–198, 2009.[MMP10] F. Mainardi, A. Mura, and G. Pagnini. The M -Wright function intime-fractional diffusion processes: A tutorial survey. Int. J. Dif-ferential Equ. , 2010:Art. ID 104505, 29, 2010.[MP08] A. Mura and G. Pagnini. Characterizations and simulationsof a class of stochastic processes to model anomalous diffusion.
J. Phys. A , 41(28):285003, 22, 2008.16Mur08] A. Mura.
Non-Markovian Stochastic Processes and their Applica-tions: From Anomalous Diffusions to Time Series Analysis . PhDthesis, Bologna, 2008.[MvN68] B. B. Mandelbrot and J. W. van Ness. Fractional Brownian mo-tions, fractional noises and applications.
SIAM Review , 10:422–437, 1968.[NR02] D. Nualart and A. Răşcanu. Differential equations driven byfractional Brownian motion.
Collect. Math. , 53(1):55–81, 2002.[NS09] D. Nualart and B. Saussereau. Malliavin calculus for stochasticdifferential equations driven by a fractional Brownian motion.
Stochastic Process. Appl. , 119(2):391–409, 2009.[Nua06] D. Nualart.
The Malliavin calculus and related topics . Springer,2006.[Sau12] B. Saussereau. A stability result for stochastic differential equa-tions driven by fractional Brownian motions.
Int. J. Stoch. Anal. ,2012:1–13, 2012.[Sch90] W. R. Schneider. Grey noise. In S. Albeverio et al., editor,
Stochastic processes, physics and geometry , pages 676–681. WorldSci. Publ., Teaneck, NJ, 1990.[Sch92] W. R. Schneider. Grey noise. In S. Albeverio, J. .E. Fenstad,H. Holden, and T. Lindstrøm, editors,
Ideas and methods inmathematical analysis, stochastics, and applications (Oslo, 1988) ,pages 261–282. Cambridge Univ. Press, Cambridge, 1992.[Taq03] M. S. Taqqu. Fractional Brownian motion and long-range de-pendence. In
Theory and applications of long-range dependence ,pages 5–38. Birkhäuser Boston, Boston, MA, 2003.[You36] L. C. Young. An inequality of the Hölder type, connected withStieltjes integration.
Acta Math. , 67(1):251–282, 1936.[Zäh98] M. Zähle. Integration with respect to fractal functions andstochastic calculus. I.