Weak approximation of fractional SDES: The Donsker setting

In this note, we take up the study of weak convergence for stochastic differential equations driven by a (Liouville) fractional Brownian motion $B$ with Hurst parameter $H\in(1/3,1/2)$. In the current paper, we approximate the $d$-dimensional fBm by the convolution of a rescaled random walk with Liouville's kernel. We then show that the corresponding differential equation converges in law to a fractional SDE driven by $B$.


Introduction
The current article can be seen as a companion paper to [3], to which we refer for a further introduction.Indeed, in the latter reference, the following equation on the interval [0, 1] was considered (the generalization to [0, T ] being a matter of trivial considerations): where σ : R n → R n×d , b : R n → R n are two bounded and smooth enough functions, and B stands for a d-dimensional fBm with Hurst parameter H > 1/3.Let us be more specific about the driving process for equation (1): we consider in the sequel the so-called d-dimensional Liouville fBm B, with Hurst parameter H ∈ (1/3, 1/2).Namely, B can be written as B = (B 1 , . . ., B d ), where the B i 's are d independent centered Gaussian processes of the form for a d-dimensional Wiener process W = (W 1 , . . ., W d ).This process is very close to the usual fBm, in the sense that they only differ by a finite variation process (as pointed out in [1]), and we shall see that its simple expression (2) simplifies some of the computations in the sequel.In any case, B falls into the scope of application of the rough paths theory, which means that equation ( 1) can be solved thanks to the semi-pathwise techniques contained in [6,7,10].The natural question raised in [3] was then the following: is it possible to approximate equations like (1) in law by ordinary differential equations, thanks to a Wong-Zakai type approximation (see [9,13,14] for further references on the topic)?Some positive answer to this question had already been given in [5], where some Gaussian sequences approximations were considered in a general context.In [3], we focused on a natural and easily implementable (non Gaussian) scheme for B, based on Kac-Stroock's approximation to white noise (see [8,12]).However, another very natural way to approximate B relies on Donsker's type scheme (see [11] for the case H > 1/2 and [4] for the Brownian case), involving a rescaled random walk.We have thus decided to investigate weak approximations to (1) based on this process.
More precisely, as an approximating sequence of B, we shall choose (X ε ) ε>0 , where X ε,i is defined as follows for i = 1, . . ., d: consider a family of independent random variables {η i k ; k ≥ 1, 1 ≤ i ≤ d}, satisfying the Hypothesis 1.1.The random variables {η i k ; k ≥ 1, 1 ≤ i ≤ d} are independent and share the same law as another random variable η.Furthermore, η is assumed to satisfy E (η) = 0, E (η 2 ) = 1 and is almost surely bounded by a constant k η .
We then define X ε,i in the following way: where Notice that X ε is really a process given by the convolution of the rescaled random walk θ ε with Liouville's kernel.Let us then consider the process y ε solution to equation (1) driven by X ε , namely: Our main result is as follows: Theorem 1.2.Let (y ε ) ε>0 be the family of processes defined by (5), and let 1/3 < γ < H, where H is the Hurst parameter of B. Then, as ε → 0, y ε converges in law to the process y obtained as the solution to (1), where the convergence takes place in the Hölder space This theorem is obtained invoking many of the techniques introduced in [3].In the end, as explained at Section 2, most of the technical differences between the two articles arise in the way to evaluate the moments of quantities like 1 0 f (r)θ i,ε (r) dr for a given Hölder function f , and to compare them with the moments of Gaussian random variable.This is where we shall concentrate our efforts in the remainder of the paper, mostly at Section 3.

Reduction of the problem
We shall recall here briefly some preliminary steps contained in [3], which allow to reduce our problem to the evaluation of the moments of a specific type of Wiener integrals.
First of all, we need to recall the definition of some Hölder spaces, in which our convergences take place.We call for instance C j ([0, 1]; R d ) the space of continuous functions from [0, 1] j to R d , which will mainly be considered for j = 1 or 2 variables.The Hölder norms on those spaces are then defined in the following way: The usual Hölder spaces C µ 1 ([0, 1]; R d ) are then determined by setting g µ = δg µ for a continuous function g ∈ C 1 ([0, 1]; R d ), where δg ∈ C 2 ([0, 1]; R d ) is defined by δg st = g t − g s .
We then say that g ∈ C µ 1 ([0, 1]; R d ) iff g µ is finite.Note that • µ is only a semi-norm on C 1 ([0, 1]; R d ), but we will work in general on spaces of the type for a given a ∈ V , on which g µ is a norm.
The second crucial point one has to recall is the natural definition of a Lévy area for Liouville's fBm: Proposition 2.1.Let B be a d-dimensional Liouville fBm, and suppose that its Hurst parameter satisfies H ∈ (1/3, 1/2).Then (1) B is almost surely a γ-Hölder path for any 1/3 < γ < H.
(2) A Lévy area based on B can be defined by setting Here, the stochastic integrals are defined as Wiener-Itô integrals when i = j, while, when i = j, they are simply given by (3) The process B 2 is almost surely an element of C 2γ 2 ([0, 1]; R d×d ), and satisfies the algebraic relation These algebraic and analytic properties of the fBm path allow to invoke the rough path machinery (see [6,7,10]) in order to solve equation (1): Theorem 2.2.Let B be a Liouville fBm with Hurst parameter 1/3 < H < 1/2, and σ : R n → R n×d be a C 2 function, which is bounded together with its derivatives.Then (1) Equation ( 1) admits a unique solution y ∈ C γ 1 (R n ) for any 1/3 < γ < H, with the additional structure of weakly controlled process introduced in [7].
One of the nice aspects of rough paths theory is precisely the second point in Theorem 2.2, which allows to reduce immediately our weak convergence result for equation (1), namely Theorem 1.2, to the following result on the approximation of (B, B 2 ): Theorem 2.3.Recall that the random variables η i k satisfy Hypothesis 1.1, and let X ε be defined by (3).For any ε > 0, let X 2,ε = (X 2,ε st (i, j)) s,t≥0; i,j=1,...,d be the natural Lévy's area associated to X ε , given by where the integral is understood in the usual Lebesgue-Stieltjes sense.Then, as ε → 0, where B 2 denotes the Lévy area defined in Proposition 2.1, and where the convergence in law holds in the spaces The remainder of our work is thus devoted to the proof of Theorem 2.3.
As usual in the context of weak convergence of stochastic processes, we divide the proof into weak convergence for finite-dimensional distributions and a tightness type result.Furthermore, the tightness result in our case is easily deduced from the analogous result in [3]: The proof follows exactly the steps of [3,Proposition 4.3], the only difference being that our Lemma 3.1 has to be applied here in order to get the equivalent of inequality (28) in [3].Details are left to the reader.
With these preliminaries in hand, we can now turn to the finite dimensional distribution (f.d.d. in the sequel) convergence, which can be stated as: Proposition 2.5.Under the assumption 1.1, let (X ε , X 2,ε ) be the approximation process defined by ( 3) and (7).Then where f.d.d.−lim stands for the convergence in law of the finite dimensional distributions.Otherwise stated, for any k ≥ 1 and any family {s i , t Proof.The structure of the proof follows again closely the steps of [3, Proposition 5.1], except that other kind of estimates will be needed in order to handle the Donsker case.To be more specific, it should be observed that the first series of simplifications in the proof of [3, Proposition 5.1] can be repeated here.They allow to pass from a convergence of double iterated integrals to the convergence of some Wiener type integrals with respect to X ε .Namely, for i = 1, 2 and 0 ≤ u < t ≤ 1, set and for 0 ≤ u < t ≤ 1 and (u 1 , . . ., u 6 ) in a neighborhood of 0 in R 6 , set also Consider the analogous processes Y i,ε , Z ε defined by the same formulae, except that they are based on the approximations θ i,ε of white noise.We still need to recall a little more notation from [3]: Then it is shown in [3, Proposition 5.1] that one is reduced to prove that lim ε→0 v a ε = 0, where v a ε is given by for an arbitrary real parameter w in a neighborhood of 0. Furthermore, bounding e iw R t 0 θ ε,2 (u)du trivially by 1 and conditioning, it is easily shown that for any α ∈ (0, 1).In order to reach our aim, it is thus sufficient to check the following inequalities: sup 0 and for w < w 0 , where w 0 is a small enough constant, However, these relations can be deduced, as (39), ( 40) and (41) in [3], from Lemma 3.1 (it should be noticed however that a one-parameter version of [2, Lemma 5.1] is needed for the adaptation of the latter result to our Donsker setting).The proof is thus finished once the lemmas below are proven.

Moments estimates in the Donsker setting
In order to deal with our technical estimates, let us first introduce a new notation: set ρ 1 = (1 − 5 1/2 )/2 and ρ 2 = (1 + 5 1/2 )/2.Then the moments of any integral of a deterministic kernel f with respect to θ i,ε can be bounded as follows: Recall that the random variable η is assumed to be almost surely bounded by a constant k η .Then we have , and where φ ε f is the quantity defined at (11).
Proof.We focus first on inequality (12) and divide this proof into several steps.
Step 1: Identification of some key iterated integrals.Notice that Transforming the symmetric integral on [0, 1] 2m into an integral on the simplex, and using expression (4) for θ ε , we can write the latter expression as: + 1 and where we understand that f (x) = 0 whenever x > 1.Let us study now the quantities Separating the cases in this way for where and where the term T 2 m is defined by: with and where we have set Let us observe at this point that we have split our sum into T 1 m and T 2 m because T 1 m represents the dominant contribution to our moment estimate.This is simply due to the fact that T 1 m is obtained by assuming some pairwise equalities among the random variables η i k , while T 2 m is based on a higher number of constraints.In any case, both expressions will be analyzed through the introduction of some iterated integrals of the form Step 2: Analysis of the integrals K ν .Those iterated integrals are treated in a slightly different way according to the parity of ν.Indeed, for ν = 2n, thanks to the elementary inequality 2ab ≤ a 2 + b 2 , we obtain a bound of the form: ≤ The case ν = 2n + 1 can be treated along the same lines, except for the fact that one has to cope with some expressions of the form Combining ( 18) and ( 17) we can state the following general formula: let ν ≥ 1, and define a couple (ν * , ν) as: (i) ν * = ν/2, ν = 0 if ν is even, (ii) ν * = (ν + 1)/2, ν = 1 if ν is odd.With this notation in hand, we have: Step 3: Bound on T 1 m .It is readily checked that T 1 m can be decomposed into blocks of the form K 2 (k; w, v), for which one can apply (19).This yields Step 4: Bound on U n 1 ,...,ns .Recall that U n 1 ,...,ns is defined by ( 16).We introduce now a recursion procedure in order to control this term.Namely, integrating with respect to the last n s variables, one obtains that Plugging our bound (19) on K ns into this expression, we get We can now proceed, and integrate with respect to the variables r l for k s−1 ≤ l ≤ k s .In the end, since n s = 2m, the remaining singularity in ε is of the form ε −ns .However, each of the singularity ε −ns comes with an integral φ ε q with q = |f | 1/2 .The latter integral is easily seen to be of order ε, which compensates the singularity ε −ns (recall that ns ≤ 1).Hence, iterating the integrations with respect to the variables r, we end up with a bound of the form 1 Step 5: Bound on T 2 m .Owing to inequality (20), our bound on T 2 m can be reduced now to an estimate of the number of terms in the sum over n 1 , . . ., n s in formula (15).This boils down to the following question: given a natural number n, how can we write it as a sum of natural numbers (larger than one)?This is arguably a classical problem, and in order to recall its answer, let us take a simple example: for n = 6, the possible decompositions can be written as {6; 2 + 2 + 2; 2 + 4; 4+2; 3+3}.Furthermore, notice that the decompositions of 6 can be obtained by adding +2 to the decompositions of 4 or adding 1 to the last number of the decompositions of 5. Extrapolating to a general integer n, it is easily seen that the number of decompositions can be expressed as u n−1 , where (u n ) n≥1 stands for the Fibonacci sequence.We have thus found a number of decompositions of the form where the quantities ρ 1 , ρ 2 appear in formula (12).Moreover, the number of terms in T 2 is given by N 2m − 1, the −1 part corresponding to the term T 1 .
Putting together this expression with (20) and the result of Step 3, our claim ( 12) is now easily obtained.
Step 6: Proof of (13).The proof of (13) follows the same arguments as for (12).We briefly sketch the main difference between these two proofs, lying in the analysis of the term U n 1 ,...,ns .Indeed, since we are now dealing with an odd power 2m + 1, the equivalent of ( 20) is an upper bound of the form (21) Furthermore, applying Hölder's inequality twice, we obtain and thus we can bound (21) by 1 (m−1)!(φ ε f ) , which ends the proof.
Our next technical lemma compares the moments of a Wiener type integral with respect to θ ε and with respect to the white noise.
Then (1) We have (2) For any m > 1, the following inequality holds true, where we recall that ρ 1 , ρ 2 have been defined just before Lemma 3.1: Proof.We divide again this proof into several steps.
Step 1: Variance estimates.We prove here the first of our assertions: Notice that 1 2 On the other hand We thus get and hence this quantity can be bounded as follows: which is the first claim of our lemma.
Step 2: decomposition for higher moments: We can follow exactly the computations of Lemma 3.1, Step 1, in order to get with T j m = T j m (2m)! for j = 1, 2. Furthermore, the term T 2 m can be bounded as in Lemma 3.1, and we obtain Step 3: Study of T 1 m : We analyze T 1 m in a slightly different way as in Lemma 3.1.Namely, we first write where we have written {a, b} ≥ {c, d} for a ∧ b ≥ c ∨ d.We will now compare this quantity with another expression of the same type, called T 1 m and defined by Let us thus write T 1 m as where T 3 m represents the part of the sum taken over the indices k 1 , . . ., k m such that there exist l satisfying k l = k l+1 .However, this latter term can be bounded as in (20), yielding Step 4: Conclusion.Putting together the decompositions we have obtained so far, we end up with Invoking our estimates ( 22) and (24) on T 2 m and T 3 m , our bound on J m easily reduced to check that The latter inequality can now be obtained from the decomposition and from the estimate we have already obtained for J 1 .This finishes the proof.
Finally, the characteristic function of a Wiener type integral of the form 1 0 f (r)θ k,ε (r)dr can be compared to its expected limit 1 0 f (r)dW k r in the following way: ) for a certain α ∈ (0, 1), k ∈ {1, . . ., d} and ε > 0. For any u ∈ R, we have: Proof.Let us control first the imaginary part of the difference.Using lemma 3.1, and invoking the fact that the odd moments of a Gaussian random variable are null, we get  ≤ 4(1/5) 1/2 u 3 (φ ε f ) In order to control the real part of the difference, we will use Lemma 3.2.This yields: The latter quantity can be bounded by ), which ends the proof.