ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WITH MONOTONE DRIFT

Let ${B_{t}^{H},t\in \lbrack 0,T]}$ be a fractional Brownian motion with Hurst parameter $H > \frac{1}{2}$. We prove the existence of a weak solution for a stochastic differential equation of the form $X_{t}=x+B_{t}^{H}+ \int_{0}^{t}\left( b_{1}(s,X_{s})+b_{2}(s,X_{s})\right) ds$, where $ b_{1}(s,x)$ is a Holder continuous function of order strictly larger than $1-\frac{1}{2H}$ in $x$ and than $H-\frac{1}{2}$ in time and $b_{2}$ is a real bounded nondecreasing and left (or right) continuous function.


Introduction
Let B H = {B H t , t ∈ [0, T ]} be a fractional Brownian motion with Hurst parameter H ∈ (0, 1). That is, B H is a centered Gaussian process with covariance If H = 1 2 the process B H is a standard Brownian motion. Consider the following stochastic differential equation If b 2 ≡ 0 and H = 1 2 (the process B H is a standard Brownian motion), the existence of a strong solution is well-known by the results of Zvonkin [18], Veretennikov [16] and Bahlali [2]. See also the work by Nakao [11] and its generalization by Ouknine [14]. In the case of Equation (1.1) driven by the fractional Brownian motion with b 2 ≡ 0, the weak existence and uniqueness are established in [13] using a suitable version of Girsanov theorem; the existence of a strong solution could be deduced from an extension of Yamada-Watanabe's theorem or by a direct arguments. In the general case H > 1/2, to establish existence and uniqueness result, a Hölder type spacetime condition is imposed on the drift. Recently, Mishura and Nualart [9] gave an existence and uniqueness result for one discontinuous function namely the sgn function. Their approach relies on the Novikov criterion and it is valid for 1+ √ 5 4 > H > 1/2. Our aim is to establish existence and uniqueness result for general monotone function including sgn function and H > 1/2. The paper is organized as follows. In Section 2 we give some preliminaries on fractional calculus and fractional Brownian motion. In Section 3 we formulate a Girsanov theorem and show the existence of a weak solution to Equation (1.1). As a consequence we deduce the uniqueness in law and the pathwise uniqueness. Finally Section 4 discusses the existence of a strong solution.

Fractional calculus
An exhaustive survey on classical fractional calculus can be found in [15]. We recall some basic definitions and results. For f ∈ L 1 ([a, b]) and α > 0 the left fractional Riemann-Liouville integral of f of order α on (a, b) is given at almost all x by where Γ denotes the Euler function. This integral extends the usual n-order iterated integrals of f for α = n ∈ N. We have the first composition formula I α a + (I β a + f ) = I α+β a + f. The fractional derivative can be introduced as inverse operation. We assume 0 < α < 1 and p > 1. We denote by I α a + (L p ) the image of L p ([a, b]) by the operator I α a + . If f ∈ I α a + (L p ), the function φ such that f = I α a + φ is unique in L p and it agrees with the left-sided Riemann-Liouville derivative of f of order α defined by The derivative of f has the following Weil representation: where the convergence of the integrals at the singularity x = y holds in 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,

Fractional Brownian motion
Let B H = {B H t , t ∈ [0, T ]} be a fractional Brownian motion with Hurst parameter 0 < H < 1 defined on the probability space (Ω, F, P ). For each t ∈ [0, T ] we denote by F B H t the σ-field generated by the random variables B H s , s ∈ [0, t] and the sets of probability zero. We denote by E the set of step functions on [0, T ]. Let H be the Hilbert space defined as the closure of E with respect to the scalar product The mapping 1 [0,t] −→ B H t can be extended to an isometry between H and the Gaussian space H 1 (B H ) associated with B H . We will denote this isometry by ϕ −→ B H (ϕ). The covariance kernel R H (t, s) can be written as where K H is a square integrable kernel given by (see [3]): F (a, b, c, z) being the Gauss hypergeometric function. Consider the linear operator For any pair of step functions ϕ and ψ in E we have (see [1]) As a consequence, the operator K * H provides an isometry between the Hilbert spaces H and is a Wiener process, and the process B H has an integral representation of the form ) and it can be expressed in terms of fractional integrals as follows (see [3]): We will make use of the following definition of F t -fractional Brownian motion.
T ]} be a right-continuous increasing family of σ-fields on (Ω, F, P ) such that F 0 contains the sets of probability zero. A fractional Brownian motion 3 Existence of strong solution for SDE with monotone drift.
In this section we are interested by the special case b 1 ≡ 0. We will prove by approximation arguments that there is a strong solution of equation (1.1). We will discuss two cases:

1-The first case:
To treat the first situation, let us suppose that b 2 (s, .) is nondecreasing and left continuous function. We will use the following approximation lemma: Proof. First assume that b(s, .) is left continuous and let us choose for any n ≥ 1 is also a nondecreasing function for any fixed n ≥ 1. Let x, y ∈ R, we clearly have for any n ≥ 1, Obviously, we get that b n is uniformly bounded by the constant M . Let n < m, s ∈ [0, T ] and Now let x 0 ∈ R and take an increasing sequence of real numbers x n converging to x 0 . We want to show that for any s ∈ We may choose a sequence ϕ(n) ≥ n such that x n ≤ x ϕ(n) . Since (b n (s, x)) n≥1 is increasing and for any fixed n ≥ 1 the function b n (s, .) is nondecreasing, we have We deduce by (3.2) and the left continuity of b(s, .), Which ends the proof.
Let (B H ) t≥1 be a fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1). We consider the following SDE By standard Picard's iteration argument, one may show that for any n ≥ 1, the equation (3.5) has a strong solution which we denote by X n . Let n > m, we denote by ∆ t = X n t − X m t . Using the monotony argument on b n , we have We then get By Gronwall's lemma, we have for almost all w and for any t ∈ [0, T ], the sequence (X n t (w)) is a nondecreasing function of n which is bounded since b n is. Therefore it has a limit when n → ∞ and we set lim which entails in particular that X is F B H t − adapted. Applying the convergence result in Lemma 3.1 and the boundedness of b n we get by Lebesgue's dominated convergence theorem, [13]. Here, the function b 2 may have a countable set of discontinuity points. The solution constructed in Theorem 3.2 is the minimal one. For any fixed (s, x) ∈ [0, T ] × R, the sequence (b n (s, x)) n≥1 is nonincreasing and for any fixed n ≥ 1 and s ∈ [0, T ] the function b n (s, .) is nondecreasing. The same arguments as in Lemma 3.1 can be used to prove that for any sequence (x n ) n≥1 decreasing to x, we have

Remark. To show that Equation (3.4) has a weak solution, a continuity condition is imposed on the drift in
This allows us to construct the maximal solution to the equation (3.4).

2-The second case:
In this case we use the following lemma: For the proof of this lemma we refer for example to [8]. Proof. For any n ≥ 1, let b n be as in Lemma 3.3. Since b n is Lipschitz and linear growth, the result in [13] assures the existence of a strong solution X n to the equation Since (b n ) n≥1 is nonincreasing, comparison theorem entails that (X n ) n≥1 is a.s nonincreasing. By the linear growth condition on b n and Gronwall's lemma we may deduce that X n converges a.s to X, which is clearly a strong solution to the SDE (3.4). Moreover, if X 1 and X 2 are two solutions of (3.4), using the fact that b 2 (s, .) is nonincreasing, we get by applying Tanaka's formula to the continuous semi-martingale X 1 − X 2 , Then we have the pathwise uniqueness of the solution.

Girsanov transform
As in the previous section, let B H be a fractional Brownian motion with Hurst parameter 0 < H < 1 and denote by F B H t , t ∈ [0, T ] its natural filtration.
Given an adapted process with integrable trajectories u = {u t , t ∈ [0, T ]} and consider the transformation We can write ii) E(ξ T ) = 1, where Then the shifted process B H is an F B H t -fractional Brownian motion with Hurst parameter H under the new probability P defined by d P dP = ξ T .
Proof. By the standard Girsanov theorem applied to the adapted and square integrable process K −1 H · 0 u s ds we obtain that the process W defined in (4.2) is an F B H t -Brownian motion under the probability P . Hence, the result follows.
From (2.5) the inverse operator K −1 H is given by for all h ∈ I , and a sufficient condition for i) is the fact that the trajectories of u are Hölder continuous of order H − 1 2 + ε for some ε > 0.

Existence of a weak solution
Consider the stochastic differential equation: As consequence, taking into account that α < 1, we have for any λ > 1 In order to estimate the term β(s), we apply the Hölder continuity condition (1.2) and we get where we have fixed ε < H − 1 α (H − 1 2 ) and we denote By Fernique's Theorem, taking into account that α < 1, for any λ > 1 we have and we deduce condition ii) of Theorem 4.1 by means of Novikov criterion.

Uniqueness in law and pathwise uniqueness
In this subsection we will prove uniqueness in law of weak solution under the condition H 1 for b 1 , Proof. It is clear that X 2 is pathwise unique, hence the uniqueness in law holds when b 1 ≡ 0. Let X, B H be a solution of the stochastic differential equation We claim that the process u s satisfies conditions i) and ii) of Theorem 4.1. In fact, u s is an adapted process and taking into account that X t has the same regularity properties as the fBm we deduce that T 0 u 2 s ds < ∞ almost surely. Finally, we can apply again Novikov theorem in order to show that E d P dP = 1, because by Gronwall's lemma and |X t − X s | ≤ |B H t − B H s | + C 3 |t − s|(1 + X ∞ ) for some constants C i , i = 1, 2, 3. By the classical Girsanov theorem the process W t = W t + t 0 u r dr is an F t -Brownian motion under the probability P . In terms of the process W t we can write Then X satisfies the following SDE,