Smoothness and Gaussian Density Estimates for Stochastic Functional Differential Equations with Fractional Noise

In this paper, we study the density of the solution to a class of stochastic functional differential equations driven by fractional Brownian motion. Based on the techniques of Malliavin calculus, we prove the smoothness and establish upper and lower Gaussian estimates for the density.


Introduction
In the last decade, Gaussian density estimates for the solutions of various stochastic equations have been intensively studied. Particularly, the class of stochastic equations with fractional noise has been discussed by several authors, see [1,2,4,8] and references therein.
We recall that fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) is a centered Gaussian process B H = (B H t ) t∈R+ with covariance function For H > 1 2 , B H t admits the so-called Volterra representation where (B t ) t∈+ is a standard Brownian motion, the kernel K H is defined by

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In this paper, we consider stochastic functional differential equations of the form where r > 0 is delay time, the kernel ρ and initial condition η are deterministic functions on [−r, 0]. The stochastic integral is interpreted as a pathwise Riemann-Stieltjes integral, which has been frequently used in the studies related to fBm. We refer the reader to [12] for a detailed presentation of this integral. The density of solutions to the equation (2) has been discussed in some special cases. When H = 1 2 , B H reduces to standard Brownian motion and in this case, the existence and smoothness of the probability density of solutions were proved by Takeuchi in [11]. When H > 1 2 , Gaussian density estimates were obtained by Dung et al. in [6] for the equation (2) with g = 0. However, the case of g ̸ = 0 has not investigated yet. Thus, in the present paper, our aim is to establish analogue results for the equation (2) with g ̸ = 0 and H > 1 2 . More specifically, we obtain the following properties: (i) the existence and Gaussian estimates for the density of solutions, (ii) the smoothness of the density of solutions.
It should be noted that the information about the density will be very useful in practical studies, see e.g. [7]. In a spirit close to [6,11], the main tools of this paper are the techniques of Malliavin calculus. However, we would like to emphasize that the complexity of stochastic integrals with respect to fBm and the appearance of delayed integral term in (2) require a fine analysis for proving the properties (i) and (ii). The rest of this article is organized as follows. In Section 2, we recall some fundamental concepts of Malliavin calculus and a general Gaussian estimate for the density of Malliavin differentiable random variables. The main results of the paper are stated and proved in Section 3. The conclusion is given in Section 4.

Preliminaries
Let us recall some elements of Malliavin calculus with respect to Brownian motion B, where B is used to present B H t as in (1) (for more details see [9]). We suppose that (B t ) t∈[0,T ] is defined on a complete probability space (Ω, F, F, P ), where F = (F t ) t∈[0,T ] is a natural filtration generated by the Brownian motion B. For h ∈ L 2 [0, T ], we denote by B(h) the Wiener integral Let S denote the dense subset of L 2 (Ω, F, P ) consisting of smooth random variables of the form If F has the form (3), we define its Malliavin derivative as the process DF : More generally, we can define the kth order derivative D k F by iterating the derivative operator k times, i.e. D k t1,...,t k F = D t k ...D t1 F . For any integer k and any p ≥ 1, we denote by D k,p the closure of S with respect to the norm

SMOOTHNESS & GAUSSIAN DENSITY ESTIMATES
A random variable F is said to be Malliavin differentiable if it belongs to D 1,2 .
In order to obtain Gaussian density estimates for solutions to the equation (2), we will use a general criterion established recently by Nourdin and Viens in [10]. We recall here [6, Theorem 2.4] for a convenient version which can be of interest for the readers who are not used to working with the Ornstein-Uhlenbeck operator.

Proposition 1
Let F be in D 1,2 with mean zero. If there exist positive constants c, C such that, for all x ∈, almost surely then the density ρ F of F exists and satisfies, for almost all x ∈

The main results
In the whole this section, we consider the equation (2) with the following fundamental assumptions. Note that the conditions on a and σ are similar to that required in Section 5 of [6].
Let us first give a short discussion about the existence and uniqueness of solutions. We denote by C 1,1 b ([0, T ]×) the space of bounded functions f : [0, T ] × R → R with bounded partial derivatives of the first order and we write We define the function We consider stochastic functional differential equation with additive noise where y 0 := F (0, x 0 ), and A(y, s) = F ′ 1 (s, G(s, y)) + a(s,G(s,y) σ(s,G(s,y)) . It was already pointed out in [6] that A(y, s) is Lipschitz and has linear growth. On the other hand, under Assumptions (A 1 ) and (A 2 ), we can check that the functions 1 σ(s,G(s,y)) and g(s, G(s, y)) are also Lipschitz and have linear growth. Hence, by repeating the computations presented in the proof of Proposition 3.1 in [3], we can infer that the equation (7) admits a unique strong solution Based on the properties of (Y t ) t∈[0,T ] , we have the following propositions.

Proposition 2
Let Assumptions (A 1 ) and (A 2 ) hold. Then, the equation (2) has a unique strong solution given by This solution is an F t -adapted process and, for all ε ∈ (0, H), whose trajectories are Hölder

Proof
The proof is similar to that of Lemma 5.1 in [6]. So we omit it.

Proposition 3 Under the Assumptions
be the unique strong solution to (7). By using the same argument as in the proof of Lemma 5.3 in [6], we have Y t ∈ D 1,2 and its Malliavin derivative is given by where G(s, Y s )) .

SMOOTHNESS & GAUSSIAN DENSITY ESTIMATES
From the relation X t = G(t, Y t ) and the chain rule of Malliavin derivatives (see Proposition 1.2.3 in [9]), we have X t ∈ D 1,2 , and This, combined with (9), gives us (8). So the proof of Proposition is complete.
From now on, we will use the symbol C to denote a generic constant, whose value may change from one line to another.

Proposition 4
Let (X t ) t∈[−r,T ] be the solution to the equation (2). Assume that (A 1 ), (A 2 ) hold. Then there exists a finite constant C > 0 such that:

Proof
From (9), (A 1 ), (A 2 ) and the boundedness of An application of Gronwall's inequality now gives that From (10), (12) and the boundedness of σ(t, x), we have

Proposition 5
Let (X t ) t∈[−r,T ] be the solution to the equation (2). Assume that (A 1 ), (A 2 ) hold. Then there exists a finite constant c > 0 such that

Proof
It follows from (9) that (D θ Y t ) t∈[θ,T ] solves the following ordinary differential equation

By the Comparison Theorem and
From the equation (9), we have where Thus we can rewrite h(t) as follows From (12) and ∂ 1 K H (t, θ) ≥ 0, On the other hand, for all 0 ≤ s ≤ t, we have Using the fact that

SMOOTHNESS & GAUSSIAN DENSITY ESTIMATES
From (14) and (15), we can get the following estimates From the definition of K H (t, r), for all 0 < r ≤ t, we deduce Now we choose ε ∈ (0, 1] such that Then, we get So we can finish the proof of Proposition because We now are ready to formulate and prove the main results of this paper.

Theorem 1
Asume that (A 1 ) and (A 2 ) hold and let (X t ) t∈[−r,T ] be the unique strong solution to the equation (2). Then, for each t ∈ (0, T ], the density ρ Xt exists and satisfies the bounds for all where c, C are finite positive constants.

Proof
For each t ∈ (0, T ], we consider the random variable F := X t − EX t . Clearly, F has mean zero and is Malliavin differentiable with D θ F = D θ X t . Hence, by Propositions 4 and 5, we can get where c, C are some finite positive constants. In view of Proposition 1, we can conclude that the density ρ F of the random variable F exits and satisfies which gives us (17) because ρ Xt (x) = ρ F (x − EX t ).

Theorem 2
Suppose the Assumptions (A 1 ) and (A 2 ). Let (X t ) t∈[−r,T ] be the solution to the equation (2). In addition, we assume that a, g and σ are infinitely differentiable functions in x with bounded derivatives of all orders. Then, for each t ∈ (0, T ], the random variable X t has an infinitely differentiable density with respect to Lebesgue measure on R.

Proof
Fix t ∈ (0, T ], thanks to Theorem 2.1.4 in [9], we have to check the following properties

SMOOTHNESS & GAUSSIAN DENSITY ESTIMATES
It is easy to verify that the coefficients of the equation (7) are infinitely differentiable in y with bounded derivatives of all orders. Hence, we can infer that Y t ∈ D ∞ . So X t does. Let us now check the property (b). By Proposition 3, we have Hence, For each y ≥ y 0 : By Markov's inequality, we have Under the assumptions (A 1 ), (A 2 ) and the inequality (|a| + |b|) p/2 ≤ 2 p/2−1 (|a| p/2 + |b| p/2 ) , we can get where C is some positive constant. By using Hölder's inequality we obtain So it holds that From (18) and (19), we deduce Now for any α ≥ 1 and p > 4Hα, we have the following estimates So the property (b) is proved. This finishes the proof of Theorem.

Conclusion
In this paper, we employed the techniques of Malliavin calculus to obtain smoothness and Gaussian density estimates for solutions to a fundamental class of stochastic functional differential equations with fractional noise. Our results develop further the studies initated in [6,11] and hence, our work partly enriches the knowledge of the theory of stochastic functional differential equations.