Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<12.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<\frac{1}{2}.$$\end{document} Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.


Introduction
Consider the d-dimensional stochastic differential equation (SDE) (1.1) where the driving noise B H · of this equation is a d-dimensional fractional Brownian motion, whose components are given by one-dimensional independent fractional Brownian motions with a Hurst parameter H ∈ (0, 1/2), and where α ∈ R is a constant and 1 d is the vector in R d with entries given by 1. Further, L t (X x ) is the (existing) local time at zero of X x · , which can be formally written as where δ 0 denotes the Dirac delta function in 0.
We also assume that B H · is defined on a complete probability space ( , A, P). We recall here for d = 1 and Hurst parameter H ∈ (0, 1) that B H t , 0 ≤ t ≤ T is a centered Gaussian process with covariance structure R H (t, s) given by For H = 1 2 , the fractional Brownian motion B H · coincides with the Brownian motion. Moreover, B H · has a version with (H − ε)-Hölder continuous paths for all ε ∈ (0, H ) and is the only stationary Gaussian process having the self-similarity property, that is in law for all γ > 0. Finally, we mention that for H = 1 2 the fractional Brownian motion is neither a Markov process nor a (weak) semimartingale. The latter properties, however, complicate the study of SDE's driven by B H · and in fact call for the development of new construction techniques of solutions of such equations beyond the classical Markovian framework. For further information about the fractional Brownian motion, the reader may consult, e.g., [35] and the references therein.
In this paper, we want to analyze for small Hurst parameters H ∈ (0, 1/2) strong solutions X x · to the SDE (1.1), that is solutions to (1.1), which are adapted to a Paugmented filtration F = {F t } 0≤t≤T generated by B H · . Let us mention here that solutions to (1.1) can be considered a generalization of the concept of a skew Brownian motion to the case of a fractional Brownian motion. The skew Brownian motion, which was first studied in the 1970s in [23,43] and which has applications to, e.g., astrophysics, geophysics or more recently to the simulation of diffusion processes with discontinuous coefficients (see, e.g., [18,26,48]) , is the a solution to the SDE where B · is a one-dimensional Brownian motion, L t (X x ) the local time at zero of X x · and p a parameter, which stands for the probability of positive excursions of X x · . It was shown in [22] that the SDE (1.2) has a unique strong solution if and only if p ∈ [0, 1]. The approach used by the latter authors relies on a one-to-one transformation of (1.2 ) into an SDE without drift and the symmetric Itô-Tanaka formula. Moreover, based on Skorohod's problem the authors show for 2 p − 1 = 1 or −1 that the skew Brownian motion coincides with the reflected Brownian motion-a result, which we think, does not hold true in the case of solutions to (1.1). An extension of the latter results to SDE's of the type was given in the work [25] under fairly general conditions on the coefficient σ and the measure ν, where the author also proves that strong solutions to (1.3) can be obtained through a limit of sequences of solutions to classical Itô-SDE's by using the comparison theorem. We remark here that the Walsh Brownian motion [43] also provides a natural extension of the skew Brownian motion, which is a diffusion process on rays in R 2 originating in zero and which exhibits the behavior of a Brownian motion on each of those rays. A further generalization of the latter process is the spider martingale, which has been used in the literature for the study of Brownian filtrations [47].
Other important generalizations of the skew Brownian motion to the multidimensional case in connection with weak solutions were studied in [10,40]: Using PDE techniques, Portenko in [40] gives a construction of a unique solution process associated with an infinitesimal generator with a singular drift coefficient, which is concentrated on some smooth hypersurface.
On the other hand, Bass and Chen [10] analyze (unique) weak solutions of equations of the form where B · is a d-dimensional Brownian motion and A t a process , which is obtained from limits of the form In this context, we also mention the paper [20] on SDE's with distributional drift coefficients. As for a general overview of various construction techniques with respect to the skew Brownian motion and related processes based, e.g., on the theory of Dirichlet forms or martingale problems, the reader is referred to [27]. See also the book [38].
The objective of this paper is the construction of strong solutions to the multidimensional SDE (1.1) with fractional Brownian noise initial data for small Hurst parameters H < 1 2 , where the generalized drift is given by the local time of the unknown process. Note that in contrast to [22] in the case of a skew Brownian motion, we obtain in this article the existence of strong solutions to (1.1) for all parameters α ∈ R.
Since the fractional Brownian motion is neither a Markov process nor a semimartingale, if H = 1 2 , the methods of the above-mentioned authors cannot be (directly) used for the construction of strong solutions in our setting. In fact, our construction technique considerably differs from those in the literature in the Wiener case. More specifically, we approximate the Dirac delta function in zero by means of functions ϕ ε for ε 0 given by where ϕ is, e.g., the d-dimensional standard Gaussian density. Then, we prove that the sequence of strong solutions X n t to the SDE's converges in L 2 ( ), strongly to a solution to (1.1) for n −→ ∞. In showing this, we employ a compactness criterion for sets in L 2 ( ) based on Malliavin calculus combined with a "local time variational calculus" argument. See [9] for the existence of strong solutions of SDE's driven by B H · , H < 1 2 , when, e.g., the drift coefficients b belong to L 1 (R d ) ∩ L ∞ (R d ) or see [33] in the Wiener case. We also refer to a series of other papers in the Wiener and Lévy process case and in the Hilbert space setting based on that approach: [7,8,19,32,34].
Although we can show strong uniqueness (see Proposition 5.2) with respect to (1.1) under some restrictive conditions, we remark that in contrast to, e.g., [9], our construction technique-as it is applied in this paper-does not allow for establishing this property under more general conditions. Since the fractional Brownian motion is not a semimartingale for H = 1 2 , we cannot pursue the same or similar proof strategy as, e.g., in [22] for the verification of strong uniqueness of solutions by using, e.g., the Itô-Tanaka formula. However, it is conceivable that our arguments combined with those in [4] which are based on results in [42] and a certain type of supremum concentration inequality in [44] will enable the construction of unique strong solutions to (1.1)-possibly even in the sense of Davie [15].
Here, we also want to point out a recent work of Catellier, Gubinelli [11], which came to our attention, after having finalized our article. In their striking paper, which extends the results of Davie [15] to the case of a fractional Brownian noise, the authors study the problem, which fractional Brownian paths actually regularize solutions to SDE's of the form for all H ∈ (0, 1). The (unique) solutions constructed in [11] are path by path with respect to time-dependent vector fields b in the Besov-H ölder space B α ∞,∞ , α ∈ R and in the case of distributional vector fields solutions to the SDE's, where the drift term is given by a nonlinear Young type of integral based on an averaging operator. In proving existence and uniqueness results, the authors use the Leray-Schauder-Tychonoff fixed point theorem and a comparison principle in connection with an average translation operator. Further, Lipschitz regularity of the flow (x −→ X x t ) under certain conditions is shown.
We remark that our techniques are very different from those developed by Catellier and Gubinelli [11], which seem not to work in the case of vector fields b belonging to, e.g., L 1 (R d ) ∩ L ∞ (R d ) (private communication with one of the authors in [11]). Further, their methods do not yield Malliavin differentiability of strong solutions.
Another interesting paper in the direction of path-by-path analysis of differential equations, we wish to comment on, is that of Aida [1] (see also [2]), where the author studies the existence (not uniqueness) of solutions of reflected differential equations (with a Young integral term) for certain domains by using an Euler approximation scheme and Skorohod's equation. As in the Wiener case (for d = 1 and α = 1 or −1), we believe that our constructed solutions to (1.1) do not coincide with those in [1].
Finally, we mention that the construction technique in this article may be also used for showing strong solutions of SDE's with respect to generalized drifts in the sense of (1.4) based on Kato classes. The existence of strong solutions of such equations in the Wiener case is to the best of our knowledge still an open problem. See the work of Bass, Chen [10].
Our paper is organized as follows: In Sect. 2, we introduce the framework of our paper and recall in this context some basic facts from fractional calculus and Malliavin calculus for (fractional) Brownian noise. Further, in Sect. 3 we discuss an integration by parts formula based on a local time on a simplex, which we want to employ in connection with a compactness criterion from Malliavin calculus in Sect. 5. Section 4 is devoted to the study of the local time of the fractional Brownian motion and its properties. Finally, in Sect. 5 we prove the existence of a strong solution to (1.1) by using the results of the previous sections.

Fractional Calculus
We start up here with some basic definitions and properties of fractional derivatives and integrals. For more information, see [29,41].
Let a, b ∈ R with a < b. Let f ∈ L p ([a, b]) with p ≥ 1 and α > 0. Introduce the left-and right-sided Riemann-Liouville fractional integrals by Further, for a given integer p ≥ 1, let ) and 0 < α < 1, then define the left-and right-sided Riemann-Liouville fractional derivatives by The left-and right-sided derivatives of f defined as above can be represented as follows by Finally, we see by construction that the following relations are valid ([a, b]) and similarly for I α b − and D α b − .

Shuffles
Let m and n be integers. We denote by S(m, n) the set of shuffle permutations, i.e., the set of permutations σ : {1, . . . , m+n} → {1, . . . , m+n} such that σ (1) < · · · < σ (m) and σ (m The product of two simplices then is given by the following union We hereby give a slight generalization of the above lemma, whose proof can be also found in [9]. This lemma will be used in Sect. 5. The reader may skip this lemma at first reading.
Here, A n, p is a subset of permutations of {1, . . . , n + p} such that # A n, p ≤ C n+ p for a constant C ≥ 1, and we use the definition s 0 = θ .

Proof
The proof of the result is given by induction on n. For n = 1 and k = 0, the result is trivial. For k = 1, we have where we have put w 1 = s 1 , w 2 = r 1 , . . . , w p+1 = r p .

Remark 2.2
We remark that the set A n, p in the above lemma also depends on k but we shall not make use of this fact.

Fractional Brownian motion
Denote by B H = {B H t , t ∈ [0, T ]} a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1/2). So B H is a centered Gaussian process with covariance structure We give an abridged survey on how to construct fractional Brownian motion via an isometry. We will do it in one dimension inasmuch as we will treat the multidimensional case componentwise. See [35] for further details.
Let E be the set of step functions on [0, T ], and let H be the Hilbert space given by the closure of E with respect to the inner product The mapping 1 [0,t] → B t has an extension to an isometry between H and the Gaussian subspace of L 2 ( ) associated with B H . We denote the isometry by ϕ → B H (ϕ). Let us recall the following result (see [35,Proposition 5.1.3] ) which gives an integral representation of R H (t, s) when H < 1/2: The kernel K H also has the following representation by means of fractional derivatives

Consider now the linear operator
, and then, from this fact and (2.2) one can conclude that K * H is an isometry between E and L 2 ([0, T ]) which extends to the Hilbert space H. See, e.g., [16] and [3] and the references therein.
For a given ϕ ∈ H, one proves that K * H can be represented in terms of fractional derivatives in the following ways One finds that H = I [16] and [3,Proposition 6]). Using the fact that K * H is an isometry from H into is a Wiener process and the process B H can be represented as follows see [3].
We also need to introduce the concept of fractional Brownian motion associated with a filtration. In what follows, we will denote by W a standard Wiener process on a given probability space ( , A, P) equipped with the natural filtration F = {F t } t∈[0,T ] which is generated by W and augmented by all P-null sets, we shall denote by B := B H the fractional Brownian motion with Hurst parameter H ∈ (0, 1/2) given by the representation (2.4).
In this paper, we want to make use of a version of Girsanov's theorem for fractional Brownian motion which is due to [16,Theorem 4.9]. Here, we recall the version given in [36,Theorem 2]. However, we first need the definition of an isomorphism K H It follows from this and the properties of the Riemann-Liouville fractional integrals and derivatives that the inverse of K H takes the form The latter implies that if ϕ is absolutely continuous, see [36], one has

Remark 2.6
As for the multidimensional case, define where * denotes transposition and similarly for K −1 H and K * H .
In this paper, we will also employ a crucial property of the fractional Brownian motion which was shown by [39] for general Gaussian vector fields. The latter property will be a helpful substitute for the lack of independent increments of the underlying noise.
Let m ∈ N and 0 =: t 0 < t 1 < · · · < t m < T . Then, for all ξ 1 , . . . , ξ m ∈ R d there exists a positive finite constant C > 0 (depending on m) such that (2.6) The above property is referred to the literature as local non-determinism property of the fractional Brownian motion. The reader may consult [39] or [46] for more information on this property. A stronger version of local non-determinism is also satisfied by the fractional Brownian motion. There exists a constant K > 0, depending only on H and T , such that for any t ∈ [0, T ] , 0 < r < t and for i = 1, . . . , d,

An Integration by Parts Formula
In this section, we recall an integration by parts formula, which is essentially based on the local time of the Gaussian process B H . The whole content as well as the proofs can be found in [9]. Let m be an integer, and let f : where κ j : [0, T ] → R, j = 1, . . . , m are integrable functions. Next, denote by α j a multiindex and D α j its corresponding differential operator.
In this section, we aim at deriving an integration by parts formula of the form Let us start by defining f α (θ, t, z) as above and show that it is a well-defined element of L 2 ( ).
To this end, we need the following notation: For integers k ≥ 0, let us define the expressions f k (θ, t, z) Then, defining f α (θ, t, z) as in (3.4) gives a random variable in L 2 ( ) and there exists a universal constant Proof For notational convenience, we consider θ = 0 and set where we used (2.1) in the last step.
Taking the expectation on both sides yields where Further, we see that We have that We also get that We know from Lemma [28], which is a type of Brascamp-Lieb inequality that where perm( ) is the permanent of the covariance matrix = (a i j ) of the Gaussian random vector j and where S n stands for the permutation group of size n. In addition, using an upper bound for the permanent of positive semidefinite matrices (see [5]) or direct computations we get that Further using substitution, we also have that We now want to use Lemma A.7. Then, we get that We now want to use strong local non-determinism of the form (see (2.7)): For all t ∈ [0, T ], 0 < r < t : The latter implies that for a constant C only depending on H and T . Hence, it follows from (3.9) that . Therefore, we obtain from (3.7) and (3.8) that for a constant M depending on d.
Finally, we show estimate (3.6). Using the inequality (3.5), we find that Taking the supremum over [0, T ] for each function f j , i.e., The next result is a key estimate which shows why fractional Brownian motion regularizes (1.1). It rests in fact on the earlier integration by parts formula. This estimate is given in more explicit terms when the function κ is chosen to be   .1), respectively, as in (3.2). Let θ, θ , t ∈ [0, T ], θ < θ < t and , it immediately follows that the integral in our proposition can be expressed as Taking expectation and using Theorem 3.1, we obtain We want to apply Lemma A.8. For this, we need that −H (d for all j. Hence, we have where γ (m) is defined as in Lemma A.8. The latter can be bounded above as follows ≤ C m for a large enough constant C > 0 and √ a 1 + · · · + a m ≤ √ a 1 + · · · √ a m for arbitrary nonnegative numbers a 1 , . . . , a m .
Proof The proof is similar to the previous proposition.

Local Times of a Fractional Brownian Motion and Properties
One can define, heuristically, the local time It is known that L x t B H exists and is jointly continuous in (t, x) as long as Hd < 1. See, e.g., [39] and the references therein. Moreover, by the self-similarity property of the fBm one has that L x t B H law = t 1−Hd L x/t H 1 (B H ) and, in particular The rigorous construction of L x t B H involves approximating the Dirac delta function by an approximate unity. It is convenient to consider the Gaussian approximation of unity for every x ∈ R d where ϕ is the d-dimensional standard Gaussian density. Then, we can define the smoothed local times and construct L x t (B H ) as the limit when ε tends to zero in L 2 ( ). Note that, using the Fourier transform, one can write ϕ ε (x) as follows The previous expression allows us to write Hence, by dominated convergence, we can conclude that E L x t B H , ε m converges when ε tends to zero as long as α m < ∞. If α 2 < ∞, then one can similarly show that and, therefore, if Hd < 1. Finally, we have proved the bound Remark 4. 3 We just have checked that if Hd < 1, then L x t B H exists and has moments of all orders. By checking that m≥1 α m m! < ∞, one can deduce that L x t B H has exponential moments or all orders. Furthermore, one can also show the existence of exponential moments of L x t B H 2 by doing similar computations as before. However, one may also use Theorem 4.4 to show that the exponential moments are finite.
Chen et al. [12] proved the following result on large deviations for local times of fractional Brownian motion, which we will not use in our paper but which is of independent interest:

Theorem 4.4 Let B H be a standard fractional Brownian motion with Hurst index H such that H d < 1. Then, the limit
exists and θ(H , d) satisfies the following bounds where c H is given by and

Existence of Strong Solutions
As outlined in the introduction, the object of study is a generalized SDE with additive d-dimensional fractional Brownian noise B H with Hurst parameter H ∈ (0, 1/2), i.e., where L t (X x ), t ∈ [0, T ] is a stochastic process of bounded variation which arises from taking the limit in probability, where ϕ ε are probability densities approximating δ 0 , denoting δ 0 the Dirac delta generalized function with total mass at 0. We will consider where ϕ is the d-dimensional standard Gaussian density function. Hereunder, we establish the main result of this section for H < 1 2(2+d) (see [11]).
where ϕ ε is the approximation of the Dirac delta δ 0 in (5.2) and * denotes transposition. Then, strong uniqueness holds for such solutions.
In particular, this is the case, if, e.g., uniqueness in law is satisfied.
The proof of Theorem 5.1 essentially consists of four steps: (1) In the first step, we construct a weak solution X to (5.1) by using the version of Girsanov's theorem for the fractional Brownian motion, that is we consider a probability space ( , A, P) on which a fractional Brownian motion B H and a process X x are defined such that (5.1) holds. However, a priori the solution is not a measurable functional of the driving noise, that is X x is not adapted to the filtration F = {F t } t∈[0,T ] generated by B H . (2) In the next step, we approximate the generalized drift coefficient δ 0 by the Gaussian kernels ϕ ε . Using classical Picard iteration, we know that for each smooth coefficient ϕ ε , ε > 0, there exists unique strong solution X ε · to the SDE Then, we prove that for each t ∈ [0, T ], the family {X ε t } ε>0 converges weakly as ε 0 to the conditional expectation E[X t |F t ] in the space L 2 ( ; F t ) of square integrable, F t -measurable random variables.
(3) Further, it is well known, see, e.g., [35], that for each t ∈ [0, T ] the strong solution X ε t , ε > 0, is Malliavin differentiable, and that the Malliavin derivative D s X ε t , 0 ≤ s ≤ t, with respect to W in (2.3) solves the equation where ϕ ε denotes the Jacobian of ϕ ε . Using a compactness criterion based on Malliavin calculus (see "Appendix A"), we then show that for every t ∈ [0, T ] the set of random variables {X ε t } ε>0 is relatively compact in L 2 ( ), which enables us to conclude that X ε t converges strongly as ε 0 in L 2 ( ; F t ) to E [X t |F t ]. As a consequence of the compactness criterion, we also observe that E[X t |F t ] is Malliavin differentiable. (4) Finally, we prove that E [X t |F t ] = X t , which entails that X t is F t -measurable and thus a strong solution on our specific probability space, on which we assumed our weak solution.
We assume without loss of generality that α = 1. Let us first have a look at step 1 of our program, that is we want to construct weak solutions of (5.1) by using Girsanov's theorem. Let ( , A, P) be some given probability space which carries a d-dimensional fractional Brownian motion B H with Hurst parameter H ∈ (0, 1/2) and set X x and consider the Doléans-Dade exponential formally. If we were allowed to implement Girsanov's theorem in this setting, we would arrive at the conclusion that the process is a fractional Brownian motion on ( , A, P) with Hurst parameter H ∈ (0, 1/2), where d P d P = ξ T . Hence, because of (5.5), the couple (X x , B H ) will be a weak solution of 5.1 on ( , A, P).
Therefore, in what follows we show that the requirements of Theorem 2.5 are accomplished.
Proof In order to prove Lemma 5.3, we can write Using the self-similarity of the fBm, we can write where ε(t) := εt −2H , and hence where T n (0, s) = {0 ≤ u 1 < u 2 < · · · < u n ≤ s} and Then, Moreover, Next, note that Hence, Using the last estimate, we get that where the last bound is due to Lemma A. 5

is a fractional Brownian motion with Hurst parameter H under the change of measure with respect to the Radon-Nikodym derivative ζ T .
Proof Without loss of generality, let p = 1. Then, using |e x − e y | ≤ |x − y| e x+y , Hölder's inequality, the supermartingale property of Doleans-Dade exponentials we get in connection with the previous lemma that where for constants C, μ 1 , μ 2 > 0. Now, let us have a look at the proof of the previous lemma and adopt the notation therein. In the sequel, we omit 1 d . Then, we obtain for m = 1 by using the selfsimilarity of the fBm in a similar way (but under expectation) that where ε i (t) = ε i t −2H , i = 1, 2. Using shuffling (see Sect. 2.2), we get that . . , 4m. Without loss of generality, consider the case Then, Hence, using dominated convergence in connection with Lemma A.5, we see that for ε 1 , ε 2 0. For other σ ∈ S(2m, 2m), we obtain similar limit values. In summary, we find (by also considering the case ε 1 = ε 2 ) that for ε 1 , ε 2 0. Thus, Similarly, we have that Since E = E(n, r ) is uniformly bounded with respect to n, r because of Lemma 5.3, we obtain the convergence of the Radon-Nikodym derivatives to a ζ T in L p ( ) for p = 1. The second statement of the lemma follows by using characteristic functions combined with dominated convergence.
Henceforth, we confine ourselves to the probability space ( , A, P), which carries a weak solution (X x , B H ) of (5.1) constructed from a fractional Brownian motion B H t , 0 ≤ t ≤ T with respect to a probability measure P by Girsanov's theorem. We now turn to the second step of our procedure.
Proof Without loss of generality, let x = 0. We mention that is a total subset of L 2 ( , F t , P). Denote X x,ε t by X n t for ε = 1/n and define u n s = K −1 By the classical Girsanov theorem, the process is a Wiener process under P n with Radon-Nikodym density Therefore, it follows from the definition of K −1 H that is a fractional Brownian motion with Hurst parameter H under P n . Then, using Girsanov's theorem, we find that where we used in the last equality relation (5.6), conditioning and the fact that X n t , 0 ≤ t ≤ T under P n has the same law as B H t , 0 ≤ t ≤ T under P in connection with measurable functionals (E μ expectation with respect to μ).
On the other hand, denoting by ζ T the Radon-Nikodym density associated with B H t , 0 ≤ t ≤ T in Proposition 5.4, we obtain by |e x − e y | ≤ |x − y| e x+y , Hölder's inequality, the supermartingale property of Doleans-Dade exponentials and the proof of Proposition 5.4 that where for constants C, μ 1 , μ 2 > 0. By inspecting the proof of Proposition 5.4 once again, we know that and for n −→ ∞, which completes the proof.

Remark 5.6
In fact, we can also show that Lemma 5.5 holds true for η = I d. To see this, let us adopt the notation of the proof of Lemma 5.5 and let η m : R d −→ R, m ≥ 1 be a sequence of bounded continuous functions such that Then, using Girsanov's theorem we find (without loss of generality for Hence, Hölder's inequality and the supermartingale property of Doleans-Dade exponentials yield for a constant μ > 0. Then, as in the proof of Lemma 5.5 (i.e., with respect to the upper bound for E = E(n)) we can apply (4.2), Theorem 4.4 and Lemma 5.3 and observe that Using Proposition 5.4, we can similarly show for a weak solution (X x , B H ) of (5.1) that So it follows from Lemma 5.5 that We continue with the third step of our scheme. This is the most challenging part. For notational convenience, let us from now on assume that α = 1 in (5.1) and that ϕ ε stands for the Jacobian of ϕ ε 1 d . The following result is based on a compactness criterion for subsets of L 2 ( ) which is summarized in Appendix.

Lemma 5.7
Assume H < 1 2(2+d) and let {ϕ ε } ε>0 the family of Gaussian kernels approximating Dirac's delta function δ 0 in the sense of (5.6). Fix t ∈ [0, T ] and denote by X ε t the corresponding solutions of (5.1) if we replace L t (X x ) by t 0 ϕ ε (X ε s )ds, ε > 0. Then, there exists a β ∈ (0, 1/2) such that Proof Fix t ∈ [0, T ] and take θ, θ > 0 such that 0 < θ < θ < t. Using the chain rule for the Malliavin Using Picard iteration applied to the above equation, we may write On the other hand, observe that one may again write Altogether, we can write where It follows from Lemma A.4 that for a suitably small β ∈ (0, 1/2). Let us continue with the term I n 2 (θ , θ). Then, Girsanov's theorem, Cauchy-Schwarz inequality and Lemma 5.3 imply Now, we concentrate on the expression Then, shuffling J ε 2 (θ , θ) as shown in (2.1), one can write (J ε 2 (θ , θ)) 2 as a sum of at most 2 2m summands of length 2m of the form where for each l = 1, . . . , 2m, Repeating this argument once again, we find that J ε 2 (θ , θ) 4 can be expressed as a sum of, at most, 2 8m summands of length 4m of the form where for each l = 1, . . . , 4m, It is important to note that the function K H (·, θ ) − K H (·, θ) appears only once in term (5.9) and hence only four times in term (5.11). So there are indices j 1 , . . . , j 4 ∈ {1, . . . , 4m} such that we can write (5.11) as The latter enables us to use the estimate from Proposition 3.2 with 4m [σ ( j)] = 1 for all j, |α| = 4m and Remark 3.4. Thus, we obtain that Altogether, we see that So we can find a constant C > 0 such that It is easy to see that we can choose γ ∈ (0, H ) such that there is a suitably small β ∈ (0, 1/2), 0 < β < γ < H < 1/2 so that it follows from the proof of Lemma A.4 that for every t ∈ (0, T ]. We now turn to the term I ε 3 (θ , θ). Observe that term I ε 3 (θ , θ) is the product of two terms, where the first one will simply be bounded uniformly in θ, t ∈ [0, T ] under expectation. This can be shown by following meticulously the same steps as we did for I ε 2 (θ , θ) and observing that in virtue of Proposition 3.3 with ε j = 0 for all j the singularity in θ vanishes.
Again Girsanov's theorem, Cauchy-Schwarz inequality several times and Lemma 5.3 lead to where C > 0 denotes an upper bound obtained from Lemma 5.3. Again, we have Since the exponent of |θ − θ | appearing in A 0 m (H , d, |θ − θ |) is strictly positive by assumption, we can find a small enough δ > 0 and a constant C := C H ,d,T > 0 such that . Then again, it is easy to see that we can choose β ∈ (0, 1/2) small enough so that it follows from the proof of Lemma A.4 that Corollary 5.8 Let {X ε t } ε>0 the family of approximating solutions of (5.1) in the sense of (5.6). Then, for every t ∈ [0, T ] and bounded continuous function h : Proof This is an immediate consequence of the relative compactness from Lemma 5.7 and by Lemma 5.5 in connection with Remark 5.6, we can identify the limit of X n t as being E[X t |F t ] and then the convergence holds for any bounded continuous functions as well.  Lemma 5.5). In order to obtain strong solutions with respect to arbitrary measures P, we can proceed as follows (without loss of generality for α = 1): Since X n · , n ≥ 1 (approximating sequence) and X · are strong solutions with respect to P, there exist progressively measurable functionals n (t, ·), n ≥ 1 and (t, ·) (on the space of continuous functions) such that X n t = n (t, B H · ), n ≥ 1, X t = (t, B H · ).
For a P-fractional Brownian motion B t , 0 ≤ t ≤ T , define the processes Then, we see that We also know from our construction of X · under P that  In order to formulate compactness criteria useful for our purposes, we need the following technical result which also can be found in [14]. A direct consequence of Theorem A.1 and Lemma A.2 is now the following compactness criteria. where · denotes any matrix norm. Then, the sequence X n , n = 1, 2 . . ., is relatively compact in L 2 ( ).
As a result, we have for every γ ∈ (0, H ), 0 < θ < θ < t < T , for some constant C H ,T > 0 depending only on H and T . Thus, for appropriately chosen small γ and β.
On the other hand, we have that Hence, for the symmetric group S n .
The next result corresponds to Lemma 3.19 in [13]: Lemma A.7 Let Z 1 , . . . , Z n be mean-zero Gaussian variables which are linearly independent. Then, for any measurable function g : R −→ R + we have that