SMOOTHNESS OF THE LAW OF THE SUPREMUM OF THE FRACTIONAL BROWNIAN MOTION

This note is devoted to prove that the supremum of a fractional Brownian motion with Hurst parameter H 2 (0 ; 1) has an inﬂnitely diﬁerentiable density on (0 ; 1 ). The proof of this result is based on the techniques of the Malliavin calculus.


Introduction
A fractional Brownian motion (fBm for short) of Hurst parameter H ∈ (0, 1) is a centered Gaussian process B = {B t , t ∈ [0, 1]} with the covariance function Notice that if H = 1 2 , the process B is a standard Brownian motion. From (1) it follows that and, as consequence, B has α-Hölder continuous paths for any α < H. The Malliavin calculus is a suitable tool for the study of the regularity of the densities of functionals of a Gaussian process. We refer to [7] and [8] for a detailed presentation of this theory. This approach is particularly useful when analytical methods are not available. In [5] the Malliavin calculus has been applied to derive the smoothness of the law of the supremum of the Brownian sheet. In order to obtain this result, the authors establish a general criterion for the smoothness of the density, assuming that the random variable is locally in D ∞ . The aim of this paper is to study the smoothness of the law of the supremum of a fBm using the general criterion obtained in [5]. The organization of this note is as follows. In Section 2 we present some preliminaries on the fBm and we review the basic facts on the Malliavin calculus and on the fractional calculus that will be used in the sequel. In Section 3 we state the general criterion for the smoothness of densities and we apply it to the supremum of the fBm.

Fractional Brownian motion
Fix H ∈ (0, 1) and let B = {B t , t ∈ [0, 1]} be a fBm with Hurst parameter H. That is, B is a zero mean Gaussian process with covariance function given by (1). Let {F t , t ∈ [0, 1]} be the family of sub-σ-fields of F generated by B and the P -null sets of F. We denote by E ⊂ H the class of step functions on [0, 1]. Let H be the Hilbert space defined as the closure of E with respect to the scalar product The mapping 1 [0,t] −→ B t can be extended to an isometry between H and the Gaussian space The covariance kernel R H (t, s) can be written as where K H is a square integrable kernel given by (see [4]): F (a, b, c, z) being the Gauss hypergeometric function. Consider the linear operator K * H from E to L 2 ([0, 1]) defined by For any pair of step functions ϕ and ψ in E we have (see [3]) As a consequence, the operator K * H provides an isometry between the Hilbert spaces H and L 2 ([0, 1]). Hence, 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 because K * H 1 [0,t] (s) = K H (t, s).

Fractional calculus
We refer to [9] for a complete survey of the fractional calculus. Let us introduce here the main definitions. If f ∈ L 1 ([0, 1]) and α > 0, the right and left-sided fractional Riemann-Liouville integrals of f of order α on [0, 1] are given almost surely for all t ∈ [0, 1] by and respectively, where Γ denotes the Gamma function.
Fractional differentiation can be introduced as an inverse operation. For any p > 1 and 1]) and is given by where the convergence of the integrals at the singularity t = s holds in the L p -sense. When αp > 1 any function in I α a + (L p ) is α − 1 p -Hölder continuous. On the other hand, any Hölder continuous function of order β > α has fractional derivative of order α. That is, The operator K * H can be expressed in terms of fractional integrals or derivatives. In fact, if H > 1 2 , we have where where d H = c H Γ(H + 1 2 ).

Malliavin calculus
We briefly recall some basic elements of the stochastic calculus of variations with respect to the fBm B. For more complete presentation on the subject, see [7] and [8].
The process B = {B t , t ∈ [0, 1]} is Gaussian and, hence, we can develop a stochastic calculus of variations (or Malliavin calculus) with respect to it. Let C ∞ b (R) be the class of infinitely differentiable functions f : R n → R such that f and all its partial derivatives are bounded. We denote by S the class of smooth cylindrical random variables F of the form where n ≥ 1, f ∈ C ∞ b (R n ) and h 1 , ..., h n ∈ H. The derivative operator D of a smooth and cylindrical random variable F of the form (12) is defined as the H-valued random variable In this way the derivative DF is an element of L 2 (Ω; H). The iterated derivative operator of D is denoted by D k . It is a closable unbounded operator from L p (Ω) into L p Ω; H ⊗k ) for each k ≥ 1, and each p ≥ 1. We denote by D k,p the closure of S with respect to the norm defined by We set D ∞ = ∩ k,p D k,p . For any given Hilbert space V , the corresponding Sobolev space of V -valued random variables can also be introduced. More precisely, let S V denote the family of V -valued smooth random variables of the form We define Then D k is a closable operator from S V ⊂ L p (Ω; V ) into L p (Ω; H ⊗k ⊗ V ) for any p ≥ 1. For any integer k ≥ 1 and for any real number p ≥ 1, a norm is defined on S V by We denote by D k,p (V ) the completion of S V with respect to the norm . k,p,V . We set Our main result will be based on the application of the following general criterion for smoothness of densities for one-dimensional random variable established in [5].
Theorem 1 Let F be a random variable in D 1,2 . Let A be an open subset of R. Suppose that there exist an H-valued random variable u A and a random variable G A such that Then the random variable F possesses an infinitely differentiable density on the set A.

Supremum of the fractional Brownian motion
The process B has a version with continuous paths as result of being α-Hölder continuous for any α < H. Set From results of [10] we know that M possesses an absolutely continuous density on (0, ∞). In order to apply Theorem 1, we will first recall some results on this supremum .

Lemma 2
The process B attains its maximum on a unique random point T.
Proof. The proof of this lemma would follow by the same arguments as the proof of Lemma 3.1 of [5], applying the criterion for absolute continuity of the supremum of a Gaussian process established in [10].
The following lemma will ensure the weak differentiability of the supremum of the fBm and give the value of its derivative. Proof. Similar to the proof of Lemma 3.2. in [5].
With the above results in hands, we are in position to prove our main result.
Proof. Fix a > 0 and set A = (a, ∞). Define the following random variable Recall that T a is a stopping time with respect to the filtration {F t , t ∈ [0, 1]} and notice that T a ≤ T on the set {M > a} . Hence, by Lemma 3, it holds that For any (p, γ) ∈ ∆, we define the process Y on [0, 1] by setting, for any t ∈ [0, 1] |s − r| 2pγ+1 dsdr.
We will need the following property: There exists a constant R depending on a, γ and p such that To prove this fact we use the Garsia, Rodemich and Rumsey Lemma in [6]. This lemma applied to the function s ∈ [0, t] → B s , with the hypothesis that Y t < R, implies This implies that sup 0≤s≤t |B s | ≤ C p,γ R 1 2p . It suffices to choose R in such a way that C p,γ R 1 2p < a. Let ψ : R + → [0, 1] be an infinitely differentiable function such that Consider the H-valued random variable given by where K * H is the operator defined in (2) and K * ,adj H denotes its adjoint in L 2 ([0, 1]). We claim that the random element u A introduced in (15) and the random variable G A = 1 0 ψ (Y t ) dt satisfy the conditions of Theorem 1. Let us first show that u A belongs to D ∞ (H). Fix an integer j ≥ 0. It suffices to show that for any q ≥ 1, The j-th order derivative D j of the function ψ (Y t ) is evaluated with the help of the Faà di Bruno formula, see formula [24. 1.2] in [1], as follows Hence, in order to show (16) it suffices to check that for all 1 ≤ n ≤ j, By (3) From (10), if H > 1 2 , we obtain After some computations we get where On the other hand, if H < 1 2 , from (11) we obtain where e H = c H Γ(H + 1 2 ) −1 .
In the sequel C H will denote a generic constant depending on H. If H > 1 2 , (19) yields and for H < 1 2 , (20) yields We have Taking into account that and this implies that sup for any q ≥ 1.
On the other hand, from From and this implies that sup for any q ≥ 1. On the other hand, on the set {M > a}, taking into account (13) and (14), it holds that ψ (Y t ) > 0 =⇒ t ≤ T, and, as a consequence, T 0 ψ (Y t ) dt = G A . Finally, it remains to show condition (ii), that is, G −1 A ∈ L q (Ω) for any q ≥ 2. We have because Y is non-decreasing and is continuous. For any ε > 0 we get This completes the proof of the proposition.