Existence and smoothness of the density of the solution to fractional stochastic integral Volterra equations

We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter . We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and smoothness of the density under suitable nondegeneracy conditions. This extends the results in Hu and Nualart [Differential equations driven by Hölder continuous functions of order greater than 1/2, Stoch. Anal. Appl. Abel Symp. 2 (2007), pp. 399–413] and Nualart and Saussereau [Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stoch. Process. Appl. 119 (2009), pp. 391–409] where stochastic differential equations driven by fractional Brownian motion are considered. The proof uses a priori estimates for deterministic differential equations driven by a function in a suitable Sobolev space.


Introduction
We consider the stochastic integral Volterra equation on R d . . , m} are independent fractional Brownian motions (fBm) with Hurst parameter H > 1 2 defined in a complete probability space ( , F, P), and X 0 is a d-dimensional random variable.
As H > 1 2 , the integral with respect to W H can be defined as a pathwise Riemann-Stieltjes integral using the results by Young [19]. Moreover, Zähle [20] introduced a generalized Stieltjes integral using the techniques of fractional calculus. In particular, she obtained a formula for the Riemann-Stieltjes integral using fractional derivatives (see (3)). Using this formula, Nualart and Rascanu [14] proved a general result on existence, uniqueness and finite moments of the solution to a class of general differential equations included CONTACT Mireia Besalú mbesalu@ub.edu in (1). These results were extended by Besalú and Rovira [5] for the Volterra equation (1). The proof of these results uses a priori estimates for a deterministic differential equation driven by a function in a suitable Sobolev space. The first aim of this paper it is to obtain supremum norm estimates of the solution to (1). We first consider the case where σ is bounded since, in this case, the estimates are of polynomial type, while in the general case are of exponential type. In the case where σ is bounded, we also obtain estimates for the Malliavin derivative of the solution and show existence and smoothness of the density. To obtain these results, we first derive a priori estimates for some deterministic equations. Finally, in the case where σ is not necessarily bounded, we also show existence of the density by first showing the Fréchet differentiability of the solution to the corresponding deterministic equation.
These results provide extensions of the works by Hu and Nualart [11] and Nualart and Saussereau [15], where stochastic differential equations driven by fBm are considered. In particular, we provide a corrected proof of [11,Theorem 7], as there is a problem in their argument. The techniques used to obtain the a priori estimates in the present paper are much more involved than those in [11] and [15] due to the time-dependence of the coefficients. As in those papers, our nondegeneracy assumption is an ellipticity-type condition, see Baudoin and Hairer [2] for the existence and smoothness of the density under Hörmander's condition for stochastic differential equations driven by a fBm with Hurst parameter H > 1 2 . Volterra equations driven by general Itô processes or semimartingales are widely studied, see for instance [1,3,4,16]. Concerning Volterra equations driven by fBm, the main references are the papers of Deya and Tindel [7,8], where existence and uniqueness is studied separately for the case H > 1 3 and H > 1 2 , using an algebraic integration setting and the Young integral, respectively. For the case H > 1/2 and using the Young integral, existence and uniqueness of the solution to equation (1) with an extra term driven by an independent Wiener process is proved in [18]. See also [9,17] for the existence and uniqueness of fBm driven Volterra equations in a Hilbert space. In [21], a class of fractional stochastic Volterra equations of convolution type driven by infinite dimensional fBm with Hurst index H ∈ (0, 1) is considered, and existence and regularity results of the stochastic convolution process are established. Last but not least, existence of the density of the solution to equation (1) in the one dimensional case is obtained in [10] as a consequence of a Bismut type formula. However, supremum norm estimates and existence and smoothness of the density in the multidimensional case do not seem to be studied yet in the literature for this kind of equations.
The structure of this paper is as follows: in the next section we introduce all the spaces, norms and operators used through the paper. In Section 3, we obtain a priori estimates for the solution of some systems of equations in a deterministic framework and study the Fréchet differentiability of one of them. Section 4 is devoted to apply the results obtained in Section 3 to the Volterra equation (1) and derive the existence and smoothness of the density.
Notation: For any integer k ≥ 1, we denote by C k b the class of real-valued functions on R d which are k times continuously differentiable with bounded partial derivatives up to the kth order. We denote by C ∞ b the the class of real-valued functions on R d which are infinitely differentiable and bounded together with all their derivatives. Throughout all the paper, C α , C α,β , c α,T , etc. will denote generic constants that may change from line to line.
and K H (t, s) is the square integrable kernel defined by where can be extended to an isometry between H and the Gaussian space H 1 associated to W H . We denote this isometry by ϕ → W H (ϕ).
Consider the operator K * H from E to L 2 (0, T; R m ) defined by From (4), we get Notice that For any ϕ, ψ ∈ E, and K * H provides an isometry between the Hilbert space H and a closed subspace of L 2 (0, T; R m ).
Following [15], we consider the fractional version of the Cameron-Martin space H H := K H (L 2 (0, T; R m )), where for h ∈ L 2 (0, T; R m ), We finally denote by We remark that for any ϕ ∈ H, R H ϕ is Hölder continuous of order H. Therefore, for any 1 − H < α < 1/2, As a consequence, ( , H, P) is an abstract Wiener space in the sense of Gross.

Deterministic differential equations
Fix 0 < α < 1 2 . Consider the deterministic differential equation on R d where g ∈ W 1−α 2 (0, T; R m ), x 0 ∈ R d , and b and σ are as in (1). Consider the following hypotheses on b and σ : Moreover, there exist some constants 0 < β, μ, δ ≤ 1 and for every N ≥ 0 there exists K N > 0 such that the following properties hold: Remark 3.1: Actually, we can consider σ and b defined only in the set The following existence and uniqueness result holds.

Theorem 3.2 ([5, Theorem 4.1]): Assume that σ and b satisfy hypotheses (H1) and (H2)
The first aim of this section is to obtain estimates for the supremum norm of the solution to (5). We first consider the case where σ is bounded and the bound on b does not depend on x.
Assume that σ is bounded. Then, there exists a constant C α,β > 0 such that where K (1)

Remark 3.4:
The techniques used in the proof do not seem to extend to the case 0 < μ < 1, thus it is left open for future work. More specifically, if μ < 1, the first term in equation (18) is of order i μ+1−α . Then, when dividing by (t − s) 1−α we obtain a term of order i μ which cannot be bounded by T.

Proof:
We divide the interval [0, T] into n = [T/ ] + 1 subintervals, where [a] denotes the largest integer strictly bounded by a and ≤ 1 will be chosen below.
Step 1. We start studying x 0, ,1−α . For s, t ∈ [0, ], s < t, Using the Hypothesis (H2) (2), the term A is easy to bound For the second term we use (6) to obtain For the next term, we use [5, Lemma A.2] to get Putting together the previous estimate, equation (3) and the estimate in [11, (3.5)] we conclude that For term D, we obtain, proceeding similarly as for term C, Therefore, Next, introducing (9), (10), (11) and (12) into (8), we obtain Thus, we obtain that Therefore, if is such that Step 2. We next study The terms A, B, and D can be bounded exactly as in Step 1. Thus, it suffices to bound the terms C i 1 and C i 2 . We start with C i 1 . We write Then, by the estimate in [11, (3.5)], we obtain Similarly, for the term C i 2 we obtain Hence, from (9), (10), (12), (18) and (19), and using the fact that Choosing such that we obtain that where Step 3. We now use an induction argument in order to show that for all i ≥ 0, For i = 0 it is proved in Step 1. Assuming that it is true up to i−1 and using (21), we get that Finally, it suffices to choose such that to conclude the desired claim. Therefore, we have that Applying this inequality recursively, we conclude that sup 0≤t≤T |x t | ≤ sup 0≤t≤(n−1) |x t | + 1 ≤ · · · ≤ |x 0 | + n, and the desired bound follows choosing such that where C α,β is such that (13), (16), (20) and (22) hold.
The next result is an exponential bound for the supremum norm of the solution to (5) under more general hypotheses than the previous theorem.

Proof:
The proof follows similarly as the proof of Theorem 3.3. We divide the interval [0, T] into n = [T/ ] + 1 subintervals, where ≤ 1 will be chosen below.
Step 1. We start bounding x 0, ,1−α . We can use the same bound for |x t − x s | obtained in (8). Then, terms A and C can be bounded as in (9) and (11) respectively. For term B, using (H2)(3), we get that For term D, we obtain Thus, we get that Hence, as ≤ 1, Thus, Therefore, using the fact that where Step 2. We next study x i ,(i+1) ,β , for i ≥ 0. For s, t ∈ [i , (i + 1) ], s < t, |x t − x s | can be bounded as in (17). Then using (9), (18), (19), (24), and (25), we get that Therefore, we obtain that Thus, where C 4 = 1 − C −1 0 C 2 1−α . We next show by induction that for all i ≥ 0, For i = 0 it is proved in (26) that Then, it suffices to choose such that B 2 ≤ 1 2 and to conclude the claim for i = 0. Assuming that it is true up to i−1 and using (28), we get that Finally, it suffices to choose such that C 0 ≥ 2 and to conclude the desired claim.
By (29), we conclude that Step 3. Using (30), we get that where Iterating, we obtain that We next choose such that C 2 1−α ≤ 1 3 and C −1 0 ≤ 3 2 . Then, C −1 4 ≤ 2. Moreover, we choose such that TC 3 C −1 In order to bound K 2 , it suffices to choose such that C 1 1−α ≤ 1 3 . Then, we easily obtain that K 2 ≤ 1. We finally bound sup 0≤t≤ |x t | using (27). Again we choose such that (1 − B 2 ) −1 ≤ 3 2 and B 1 1−α ≤ 1 3 so that B −1 3 ≤ 2. We also choose such that 1−α B 0 ≤ 1 4 so that Finally, we conclude that The next result provides a supremum norm estimate of the solution z t of the following system of equations where g belongs to W 1−α We will use the following hypotheses on h, f and w: h is Lipschitz continuous with respect to t and bounded. f is bounded and satisfies (H1). w is Lipschitz continuous and bounded.

Proof:
The existence and uniqueness of the solution follows similarly as [14,Theorem 5.1]. We next prove the estimate of the supremum norm of the solution. We divide the interval [0, T] into n = [T/˜ ] + 1 subintervals, where˜ ≤ 1 will be chosen below.
Step 1. We first estimate z 0,˜ ,∞ . Let t, t ∈ [0,˜ ] with t < t . We write The first three terms are easily bounded as We next bound H and I. Using (3) and the estimate in [11, (3.5)], we get and Therefore, we obtain Similarly, Hence, we conclude that Thus, Moreover, Choosing˜ satisfying (16), we obtain by (15) that x 0,˜ ,1−α˜ 1−α ≤ 1 2 ≤ 1. We next choose˜ such that˜ 1−α K ≤ 1,˜ 1−α D 1 ≤ 1, and D 2 ≤ 1 2 . Then, we obtain that z 0,˜ ,∞ ≤ 2 w ∞ + 1. (32) Step 2. We next estimate z i˜ ,(i+1)˜ ,∞ for i = 1, . . . , n. Fix t, t ∈ [i˜ , (i + 1)˜ ] with t < t . Similar bounds can be obtained for the corresponding terms E, F, G and H as in Step 1. Thus, we just need to bound the term I i := I, that is, Following the same computations as for I, we get Therefore, the term I i is bounded by Hence, we obtain that where Choosing˜ such that E 1 ≤ 1 2 , we obtain that Choosing˜ satisfying (22), we obtain by the Step 3 in Theorem 3.3 that for all = 1, . . . , i, x ˜ ,( +1)˜ ,1−α˜ 1−α ≤ 1. Thus, Applying expression (33) recurrently we obtain that This implies that Step 3. Using the result of Step 2 yields that where We finally bound L i and K i . We choose˜ such that 2E 2˜ 1−α ≤ 1 2 , so that E −1 5 ≤ 2. We also choose˜ such that 2E 4 ≤˜ so that Hence, choosing˜ such that 4E 1 e T˜ 1−α ≤ 1 we conclude that K i ≤ 1. Moreover, as i˜ ≤ T, we have that We finally choose˜ such that that˜ 1−α C α,β g 1−α e T T ≤ 1 8 and 2˜ 1−α E 2 e T T ≤ 1 8 , so that L i ≤ e.
Iterating (34) and using (32), we conclude that sup 0≤t≤T |z t | ≤ e sup 0≤t≤(n−1)˜ |z t | + 1 ≤ · · · ≤ e n−1 sup which implies the desired result.   Assume that b(t, s, ·), σ (t, s, ·) belong to C 3 b for all s, t ∈ [0, T] and that the partial derivatives of b and σ satisfy (H2) and (H1), respectively. Then the mapping is Fréchet differentiable. Moreover, for any (h, , and i = 1, . . . , d, the Fréchet derivatives with respect to h and x are given respectively by Proof: Using [5, Proposition 2.2(2)], we get that Therefore, F is continuous in both variables (h, x). We next show the Fréchet differentiability. Let v, w ∈ W α 1 (0, T; R d ). By [5, Proposition 2.2(2) and 3.2(2)], we have that Thus, D 2 F(h, x) is a bounded linear operator. Moreover, By the mean value theorem and [5, Proposition 2.2(2)], Similarly, using [5, Proposition 3.2(2)], we obtain This shows that D 2 F is the Fréchet derivative with respect to x of F(h, x). Similarly, we show that it is Fréchet differentiable with respect to h and the derivative is given by (36).

Proposition 3.8:
Assume the hypotheses of Lemma 3.7. Then the mapping is Fréchet differentiable and for any h ∈ W 1−α 2 (0, T; R m ) the derivative in the direction h is given by for some positive constants c (1) α,T and c (2) α,T .
Proof: Existence and uniqueness follows from [5] which is bounded by Proposition 3.9. Therefore, t (s) is Hölder continuous of order 1 − α in t, uniformly in s. On the other hand, appealing again to Proposition 3.9, for s ≤ s ≤ t, we have α,T w · (s, s ) α,1 exp c (2) α,T g We next bound the · α,1 -norm of w · (s, s ). For the first term, by the definition of the · α,1 -norm, we have where we have used [5,Lemma A.2] in the last inequality.
For the second term, as ∂ x σ is bounded, we obtain Therefore, we conclude that which implies that t (s) is Hölder continuous of order β ∧ (1 − α) in s uniformly in t.

Stochastic Volterra equations driven by fBm
In this section we apply the results obtained in Section 3 to the Volterra equation ( Fix α ∈ (1 − H, 1 2 ). As the trajectories of W H are (1 − α + )-Hölder continuous for all < H + α − 1, by the first inclusion in (2), we can apply the framework of Section 3.
In particular, under the assumptions of Theorem 3.5, there exists a unique solution to equation (1) satisfying Moreover, under the further assumptions of Theorem 3.3, we have the estimate As a consequence of these estimates we can establish the following integrability properties of the solution to (1).
We next proceed with the study of the existence and smoothness of the density of the solution to (1). From now on we assume that the initial condition is constant, that is, X 0 = x 0 ∈ R d . We start by extending the results in [15] in order to show the existence of the density of the solution to the Volterra equation (1) when σ is not necessarily bounded. We first derive the (local) Malliavin differentiability of the solution.
if s 1 ∨ · · · ∨ s n ≤ t and 0 otherwise. The notationĎ j q s q means that the factor D j q s q is omitted in the sum. When n = 1 this equation coincides with (40).
Proof: By Theorem 4.2, for any t > 0 X i t belongs to D 1,2 loc and the Malliavin derivative satisfies (40). Applying Theorem 3.6 to the system formed by equations (1) and (40)   This and [12,Lemma 4.1.2] show that the random variable X i t belongs to the Sobolev space D 1,p for all p ≥ 2. Similarly, it can be proved that X i t belongs to the Sobolev space D k,p for all p, k ≥ 2. For the sake of conciseness, we only sketch the main steps. First, by induction, following exactly along the same lines as in the proofs of [15, Proposition 5 and Lemma 10] and Proposition 3.8, it can be shown that the deterministic mapping x defined in Section 3 is infinitely differentiable. Second, by a similar argument as in the proof of Theorem 4.2, we have that for all t > 0, X i t is almost surely infinitely differentiable in the directions of the Cameron-Martin space and it belongs to the space D k,p loc for all p, k ≥ 2. Finally, using equation (41), the estimate for linear equations obtained in Theorem 3.6 and an induction argument, we obtain that for all k, p ≥ 2, where D (k) denotes the kth iterated derivative. This concludes the desired claim.
The next theorem extends and corrects the proof of [11,Theorem 7] as there is a mistake in the last step of the proof.