Consider a fractional Brownian motion (fBm), a self-similar Gaussian process with stationary increments. It was introduced by Kolmogorov [5] and studied by Mandelbrot and Van Ness [6]. The fBm with Hurst parameter H∈(0,1) is a centered Gaussian process with covariance function RH(t,s)=E(BtHBsH)=12(t2H+s2H−|t−s|2H). If H=1/2 , then the process B1/2 is a standard Brownian motion. When H≠12 , BH is neither a semimartingale nor a Markov process, so that many of the techniques employed in stochastic analysis are not available for an fBm. The self-similarity and stationarity of increments make the fBm an appropriate model for many applications in diverse fields from biology to finance. We refer to [7] for details on these notions.

Consider the following stochastic differential equation (SDE) (1) dXt=b(t,Xt)dt+dBtH,X0=x∈Rd, where b:[0,T]×Rd→Rd is a measurable function, and BH is a d-dimensional fBm with Hurst parameter H<1/2 whose components are one-dimensional independent fBms defined on a probability space (Ω,F,{Ft}t∈[0,T],P) , where the filtration {Ft}t∈[0,T] is generated by BtH , t∈[0,T] , augmented by the P-null sets. It has been proved in [2] that if b satisfies the assumption (2) b∈L∞1,∞:=L∞([0,T];L1(Rd)∩L∞(Rd)), for H<12(3d−1) , then Eq. (1) has a unique strong solution, which will be assumed throughout this paper.

Notice that if the drift coefficient is Lipschitz continuous, then Eq. (1) has a unique strong solution, which is continuous with respect to the initial condition. Moreover, the solution can be constructed using various numerical schemes.

Our purpose in this paper is to establish some stability results under the pathwise uniqueness of solutions and under weak regularity conditions on the drift coefficient b. We mention that a considerable result in this direction has been established in [1] when an fBm is replaced by a standard Brownian motion.

The paper is organized as follows. In Section 2, we introduce some properties, notation, definitions, and preliminary results. Section 3 is devoted to the study of the variation of solution with respect to the initial data. In the last section, we drop the continuity assumption on the drift and try to obtain the same result as in Section 3.

In this section, we give some properties of an fBm, definitions, and some tools used in the proofs.

For any H<1/2 , let us define the square-integrable kernel KH(t,s)=cH[(ts)H−12−(H−12)s12−H∫st(u−s)H−12uH−32du],t>s, s,\]]]> where cH=[2H(1−2H)β(1−2H,H+12))]1/2 , t>s s$]]>.

Note that ∂KH∂t(t,s)=cH(H−12)(ts)H−12(t−s)H−32. Let BH={BtH,t∈[0,T]} be an fBm defined on (Ω,F,{Ft}t∈[0,T],P) . We denote by ζ the set of step functions on [0,T] . Let H be the Hilbert space defined as the closure of ζ with respect to the scalar product ⟨1[0,t],1[0,s]⟩H=RH(t,s). The mapping 1[0,t]→BtH can be extended to an isometry between H and the Gaussian subspace of L2(Ω) associated with BH , and such an isometry is denoted by φ→BH(φ) .

Now we introduce the linear operator KH∗ from ζ to L2([0,T]) defined by (KH∗φ)(s)=KH(b,s)φ(s)+∫sb(φ(t)−φ(s))∂KH∂t(t,s)dt. The operator KH∗ is an isometry between ζ and L2([0,T]) , which can be extended to the Hilbert space H .

Define the process W={Wt,t∈[0,T]} by Wt=BH((KH∗)−11[0,t]). Then W is a Brownian motion; moreover, BH has the integral representation BtH=∫0tKH(t,s)dW(s). We need also to define an isomorphism KH from L2([0,T]) onto I0+H+12(L2) associated with the kernel KH(t,s) in terms of the fractional integrals as follows: (KHφ)(s)=I0+2Hs12−HI0+12−HsH−12φ,φ∈L2([0,T]). Note that, for φ∈L2([0,T]) , I0+α is the left fractional Riemann-Liouville integral operator of order α defined by I0+αφ(x)=1Γ(α)∫0x(x−y)α−1φ(y)dy, where Γ is the gamma function (see [3] for details).

The inverse of KH is given by (KH−1φ)(s)=s12−HD0+12−HsH−12D0+2Hφ(s),φ∈I0+H+12(L2), where for φ∈I0+H+12(L2) , D0+α is the left-sided Riemann Liouville derivative of order α defined by D0+αφ(x)=1Γ(1−α)ddx∫0xφ(y)(x−y)αdy. If φ is absolutely continuous (see [8]), then (3) (KH−1φ)(s)=sH−12I0+12−Hs12−Hφ′(s).

The main tool used in the proofs is Skorokhod’s selection theorem given by the following lemma.

We will also make use of the following result, which gives a criterion for the tightness of sequences of laws associated with continuous processes. Lemma 2.5.

([4], p. 18) Let {Xtn , t∈[0,T]} , n=1,2,… , be a sequence of d-dimensional continuous processes satisfying the following two conditions: (i)

There exist positive constants M and γ such that E[|Xn(0)|γ]≤M for every n=1,2,… ;

The purpose of this section is to ensure the continuous dependence of the solution with respect to the initial condition when the drift b is continuous and bounded. Note that, in the case of ordinary differential equation, the continuity of the coefficient is sufficient to ensure this dependence.

Next, we give a theorem that will be essential in establishing the desired result.

Before we proceed to the proof of Theorem 3.1, we state the following technical lemma.

Let us now turn to the proof of Theorem 3.1. Proof.

Suppose that the result of the theorem is false. Then there exist a constant δ>0 0$]]> and a sequence xn converging to x0 such that infnE[sup0≤t≤T|Xt(xn)−Xt(x0)|2]≥δ. Let Xn (respectively, X) be the solution of (1) corresponding to the initial condition xn (respectively, x0 ). According to Lemma 3.2, the sequence (Xn,X,BH) satisfies conditions (i) and (ii) of Lemma 2.5. Then, by Skorokhod’s selection theorem there exist a subsequence {nk,k≥1} , a probability space (Ω˜,F˜,P˜) , and stochastic processes (X˜,Y˜,B˜H) , (Xk˜,Yk˜,B˜H,k) , k≥1 , defined on (Ω˜,F˜,P˜) such that: (α)

for each k≥1 , the laws of (Xk˜,Yk˜,B˜H,k) and (Xnk,X,BH) coincide;

Thus, the processes X˜ and Y˜ satisfy the same SDE on (Ω˜,F˜,P˜) with the same driving noise B˜tH and the initial condition x0 . Then, by pathwise uniqueness, we conclude that X˜t=Y˜t for all t∈[0,T] , P˜ -a.s.

On the other hand, by uniform integrability we have that δ≤lim infnE[max0≤t≤T|Xt(xn)−Xt(x0)|2]=lim infkE˜[max0≤t≤T|X˜tk−Y˜tk|2]≤E˜[max0≤t≤T|X˜t−Y˜t|2], which is a contradiction. Then the desired result follows.  □

In this section, we drop the continuity assumption on the drift coefficient and only assume that b is bounded. The goal of this section is to generate the same result as in Theorem 3.1 without the continuity assumption.

Next, in order to use the fractional Girsanov theorem given in [8, Thm. 2], we should first check that the conditions imposed in the latter are satisfied in our context. This will be done in the following lemma.

Next, we will establish the following Krylov-type inequality that will play an essential role in the sequel.

Now we are able to state the main result of this section.