The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of c\`adl\`ag functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary It\^o integral with respect to a Brownian motion that is independent of $B$.


Introduction
The Stratonovich integral of X with respect to Y , denoted t 0 X (s) • d Y (s), can be defined as the limit in probability, if it exists, of as the mesh of the partition {t j } goes to zero. Typically, we regard (1.1) as a process in t, and require that it converges uniformly on compacts in probability (ucp).
This is closely related to the so-called symmetric integral, denoted by t 0 X (s) d • Y (s), which is the ucp limit, if it exists, of 1 t 0 X (s) + X (s + ) as → 0. The symmetric integral is an example of the regularization procedure, introduced by Russo and Vallois, and on which there is a wide body of literature. For further details on stochastic calculus via regularization, see the excellent survey article [13] and the many references therein.
A special case of interest that has received considerable attention in the literature is when Y = B H , a fractional Brownian motion with Hurst parameter H. It has been shown independently in [2] and [5] that when Y = B H and X = g(B H ) for a sufficiently differentiable function g(x), the symmetric integral exists for all H > 1/6. Moreover, in this case, the symmetric integral satisfies the classical Stratonovich change of variable formula, However, when H = 1/6, the symmetric integral does not, in general, exist. Specifically, in [2] and [5], it is shown that (1.2) does not converge in probability when Y = B 1/6 and X = (B 1/6 ) 2 . It can be similarly shown that, in this case, (1.1) also fails to converge in probability.
This brings us naturally to the notion which is the focus of this paper: the weak Stratonovich integral, which is the limit in law, if it exists, of (1.1). We focus exclusively on the case Y = B 1/6 . For simplicity, we omit the superscript and write B = B 1/6 . Our integrands shall take the form g(B(t)), for g ∈ C ∞ ( ), and we shall work only with the uniformly spaced partition, t j = j/n. In this case, (1.1) becomes where x denotes the greatest integer less than or equal to x, and ∆B j = B(t j ) − B(t j−1 ). We show that the processes I n (g, B) converge in law in D [0, ∞), the Skorohod space of càdlàg functions from [0, ∞) to . We let t 0 g(B(s)) d B(s) denote a process with this limiting law, and refer to this as the weak Stratonovich integral.
The weak Stratonovich integral with respect to B does not satisfy the classical Stratonovich change of variable formula. Rather, we show that it satisfies a change of variable formula with a correction term that is a classical Itô integral. Namely, g(B(t)) = g(B(0)) + Our precise results are actually somewhat stronger than this, in that we prove the joint convergence of the processes B, V n (B), and I n (g, B). (See Theorem 2.13.) We also discuss the joint convergence of multiple sequences of Riemann sums for different integrands. (See Theorem 2.14 and Remark 2.15.) The work in this paper is a natural follow-up to [1] and [9]. There, analogous results were proven for B 1/4 in the context of midpoint-style Riemann sums. The results in [1] and [9] were proven through different methods, and in the present work, we combine the two approaches to prove our main results.
Finally, let us stress the fact that, as a byproduct of the proof of (1.3), we show in the present paper that (See more precisely Theorem 3.7 below. Also see Theorem 3.8.) From our point of view, this result has also its own interest, and should be compared with the recent results obtained in [7,8], concerning the weighted Hermite variations of fractional Brownian motion.

Notation, preliminaries, and main result
Let B = B 1/6 be a fractional Brownian motion with Hurst parameter H = 1/6. That is, B is a centered Gaussian process, indexed by t ≥ 0, such that
Note that E|B(t) − B(s)| 2 = |t − s| 1/3 . For compactness of notation, we will sometimes write B t instead of B(t). Given a positive integer n, let ∆t = n −1 and t j = t j,n = j∆t. We shall frequently have occasion to deal with the quantity β j,n = β j = (B(t j−1 ) + B(t j ))/2. In estimating this and similar quantities, we shall adopt the notation r + = r ∨ 1, which is typically applied to nonnegative integers r. We shall also make use of the Hermite polynomials, h n (x) = (−1) n e x 2 /2 d n d x n (e −x 2 /2 ). If X is a càdlàg process, we write X (t−) = lim s↑t X (s) and ∆X (t) = X (t) − X (t−). The step function approximation to X will be denoted by X n (t) = X ( nt /n), where · is the greatest integer function. In this case, ∆X n (t j,n ) = X (t j ) − X (t j−1 ). We shall frequently use the shorthand notation ∆X j = ∆X j,n = ∆X n (t j,n ). For simplicity, positive integer powers of ∆X j shall be written without parentheses, so that ∆X k j = (∆X j ) k . The discrete p-th variation of X is defined as |∆X j | p , and the discrete signed p-th variation of X is For the discrete signed cubic variation, we shall omit the superscript, so that When we omit the index t, we mean to refer to the entire process. So, for example, V n (X ) = V n (X , ·) refers to the càdlàg process which maps t → V n (X , t). We recall the following fact which will be extensevely used in this paper.
If k is a nonnegative integer, we shall say that a function g has polynomial growth of order k if g ∈ C k ( d ) and there exist positive constants K and r such that |∂ α g(x)| ≤ K(1+|x| r ) for all x ∈ d and all |α| ≤ k. (Here, α ∈ d 0 = ( ∪{0}) d is a multi-index, and we adopt the standard multi-index notation: Given g : → and a stochastic process {X (t) : t ≥ 0}, the Stratonovich Riemann sum will be denoted by The phrase "uniformly on compacts in probability" will be abbreviated "ucp." If X n and Y n are càdlàg processes, we shall write X n ≈ Y n or X n (t) ≈ Y n (t) to mean that X n − Y n → 0 ucp. In the proofs in this paper, C shall denote a positive, finite constant that may change value from line to line.

Conditions for relative compactness
The Skorohod space of càdlàg functions It may not, however, be relatively compact in D d [0, ∞). We will therefore need the following well-known result. (For more details, see Section 2.1 of [1] and the references therein.) , and x and y have no simultaneous discontinuities, then x n + y n → x+ y in D [0, ∞). Thus, if two sequences {X n } and {Y n } are relatively compact in D [0, ∞) and every subsequential limit of {Y n } is continuous, then {X n + Y n } is relatively compact in D [0, ∞). The lemma now follows from the fact that a sequence {X (1) n , . . . , Our primary criterion for relative compactness is the following moment condition, which is a special case of Corollary 2.2 in [1].
for all n and all 0 ≤ s ≤ t ≤ T . Then {X n } is relatively compact.
Of course, a sequence {X n } converges in law in D d [0, ∞) to a process X if {X n } is relatively compact and X n → X in the sense of finite-dimensional distributions on [0, ∞). We shall also need the analogous theorem for convergence in probability, which is Lemma A2.1 in [3]. Note that if x : We will also need the following lemma, which is easily proved using the Prohorov metric.

Lemma 2.5.
Let (E, r) be a complete and separable metric space. Let X n be a sequence of Evalued random variables and suppose, for each k, there exists a sequence {X n,k } ∞ n=1 such that lim sup n→∞ E[r(X n , X n,k )] ≤ δ k , where δ k → 0 as k → ∞. Suppose also that for each k, there exists Y k such that X n,k → Y k in law as n → ∞. Then there exists X such that X n → X in law and Y k → X in law.

Elements of Malliavin calculus
In the sequel, we will need some elements of Malliavin calculus that we collect here. The reader is referred to [6] or [10] for any unexplained notion discussed in this section.
We denote by X = {X (ϕ) : ϕ ∈ H} an isonormal Gaussian process over H, a real and separable Hilbert space. By definition, X is a centered Gaussian family indexed by the elements of H and such that, for every ϕ, ψ ∈ H, We denote by H ⊗q and H q , respectively, the tensor space and the symmetric tensor space of order q ≥ 1. Let be the set of cylindrical functionals F of the form where n ≥ 1, ϕ i ∈ H and the function f ∈ C ∞ ( n ) is such that its partial derivatives have polynomial growth. The Malliavin derivative DF of a functional F of the form (2.7) is the square integrable Hvalued random variable defined as In particular, DX (ϕ) = ϕ for every ϕ ∈ H. By iteration, one can define the mth derivative D m F (which is an element of L 2 (Ω, H m )) for every m ≥ 2, giving As usual, for m ≥ 1, m,2 denotes the closure of with respect to the norm · m,2 , defined by the relation The Malliavin derivative D satisfies the following chain rule: if f : n → is in C 1 b (that is, the collection of continuously differentiable functions with a bounded derivative) and if {F i } i=1,...,n is a vector of elements of 1,2 , then f (F 1 , . . . , F n ) ∈ 1,2 and This formula can be extended to higher order derivatives as where m is the set of vectors v = (v 1 , . . . , v k ) ∈ k such that k ≥ 1, v 1 ≤ · · · ≤ v k , and v 1 + · · · + v k = m. The constants C v can be written explicitly as Remark 2.6. In (2.9), a e ⊗ b denotes the symmetrization of the tensor product a ⊗ b. Recall that, in general, the symmetrization of a function f of m variables is the function f defined by where S m denotes the set of all permutations of {1, . . . , m}.
We denote by I the adjoint of the operator D, also called the divergence operator. A random element u ∈ L 2 (Ω, H) belongs to the domain of I, noted Dom(I), if and only if it satisfies where c u is a constant depending only on u. If u ∈ Dom(I), then the random variable I(u) is defined by the duality relationship (customarily called "integration by parts formula"): which holds for every F ∈ 1,2 .
For every n ≥ 1, let n be the nth Wiener chaos of X , that is, the closed linear subspace of L 2 generated by the random variables {h n (X (ϕ)) : ϕ ∈ H, |ϕ| H = 1}, where h n is the Hermite polynomial defined by (2.1). The mapping I n (ϕ ⊗n ) = h n (X (ϕ)) (2.12) provides a linear isometry between the symmetric tensor product H n (equipped with the modified norm 1 n! · H ⊗n ) and n . We set I n ( f ) := I n ( f ) when f ∈ H ⊗n . The following duality formula holds: for any element f ∈ H n and any random variable F ∈ n,2 . We will also need the following particular case of the classical product formula between multiple integrals: if ϕ, ψ ∈ H and m, n ≥ 1, then where the sum runs over all subsets J of {t 1 , . . . , t m }, with |J| denoting the cardinality of J. Note that we may also write this as

Expansions and Gaussian estimates
A key tool of ours will be the following version of Taylor's theorem with remainder.
Theorem 2.7. Let k be a nonnegative integer. If g ∈ C k ( d ), then where h α is a continuous function with h α (a, a) = 0 for all a. Moreover, The following related expansion theorem is a slight modification of Corollary 4.
where |R| ≤ C K|η| k+1 and C depends only on K, r, ν, k, and d.
Hence, by Taylor's theorem (more specifically, by the version of Taylor's theorem which appears as Theorem 2.13 in [1]), and the fact that U and Y are independent, Hence, Since |η| 2 ≤ ν d, this completes the proof.
The following special case will be used multiple times.
has polynomial growth of order 1 with constants K and r, then where σ = (EX 2 n ) 1/2 and C depends only on r, ν, and n.
Finally, the following covariance estimates will be critical.
Lemma 2.10. Recall the notation β j = (B(t j−1 ) + B(t j ))/2 and r + = r ∨ 1. For any i, j, where C 1 , C 2 are positive, finite constants that do not depend on i or j.
Proof. (i) By symmetry, we may assume i ≤ j. First, assume j − i ≥ 2. Then where (ii) First note that by (i), This proves the lemma when either j = 1 or | j − i| + = 1. To complete the proof of (ii), suppose j > 1 and | j − i| > 1. Note that if t > 0 and s = t, then which is (ii).
(iii) This follows immediately from (ii).
(v) Without loss of generality, we may assume i < j. The upper bound follows from and the fact that E|B(t) − B(s)| 2 = |t − s| 1/3 . For the lower bound, we first assume i < j − 1 and Taking the square root in both side of this inequality we get Since ∆B i and ∆B j are negatively correlated, for some C > 0. This completes the proof when i < j − 1.

Sextic and signed cubic variations
Proof. Since V 6 n (B) is monotone, it will suffice to show that V 6 n (B, t) → 15t in L 2 for each fixed t. Indeed, the uniform convergence will then be a direct consequence of Dini's theorem. We write Applying this with ξ = ∆t −1/6 ∆B i and Y = ∆t −1/6 ∆B j , and using Lemma which completes the proof.

Main result
Given Note that, by definition, the change of variable formula (1.3) holds for all g ∈ C ∞ . We shall use the shorthand notation g(B) d B to refer to the process t → Our main result is the following.
We also have the following generalization concerning the joint convergence of multiple sequences of Riemann sums.
Remark 2.15. In less formal language, Theorem 2.14 states that the Riemann sums I n (g j , B) converge jointly, and the limiting stochastic integrals are all defined in terms of the same Brownian motion. In other words, the limiting Brownian motion remains unchanged under changes in the integrand. In this sense, the limiting Brownian motion depends only on B, despite being independent of B in the probabilistic sense.
The proofs of these two theorems are given in Section 5.

Finite-dimensional distributions
in the sense of finite-dimensional distributions on [0, ∞).
The rest of this section is devoted to the proof of Theorem 3.1.

Some technical lemmas
During the proof of Theorem 3.1, we will need technical results that are collected here. Moreover, for notational convenience, we will make use of the following shorthand notation: For future reference, let us note that by (2.10), Proof.
(ii) Observe that We deduce, for any fixed s ≤ t: where |R n | ≤ C n −1/3 , and C does not depend on s or t. The case where s > t can be obtained similarly. Taking the supremum over s ∈ [0, T ] gives us (ii).
(iii) is a direct consequence of (ii).
(v) The proof is very similar to the proof of (i v).

Lemma 3.3.
Let s ≥ 1, and suppose that φ ∈ C 6 ( s ) and g 1 , g 2 ∈ C 6 ( ) have polynomial growth of order 6, all with constants K and r. Fix a, b ∈ [0, T ]. Then is finite.
Proof. Let C denote a constant depending only on T , s, K, and r, and whose value can change from one line to another. Define f : s+3 → by .
By Lemma 3.2 (i), we have |η i | ≤ n −1/6 for any i ≤ s + 2, and |η s+3 | ≤ 1 by Lemma 2.10 (i). Moreover, we have Therefore, by taking into account these two facts, we deduce 1 Note that E[∂ α f (ξ)] < ∞ since the function ∂ α f has polynomial growth by our assumptions on φ, g 1 , and g 2 , and since ξ is a Gaussian vector.
The proof of Lemma 3.3 is done.

Lemma 3.5.
Fix an integer r ≥ 1, and some real numbers . Then, for any fixed a, b, c, d > 0, the following estimate is in order: Proof. Using the product formula (2.14), we have that As a consequence, we get (1) First, we deal with the term A (n)

.
A (n) When computing the sixth Malliavin derivative ) (using Lemma 3.4), there are three types of terms: (1a) The first type consists in terms arising when one only differentiates Φ(i 1 , i 2 , i 3 , i 4 ). By Lemma 3.2 (i), these terms are all bounded by which is less than (Here, Φ(i 1 , i 2 , i 3 , i 4 ) means a quantity having a similar form as Φ(i 1 , i 2 , i 3 , i 4 ).) Therefore, Lemma 3.3 shows that the terms of the first type in A or by the same quantity with ρ(i 4 −i 1 ) instead of ρ(i 3 −i 1 ). In order to get the previous estimate, we have used Lemma 3.2 (i) plus the fact that the sequence {ρ(r)} r∈ , introduced in (2.4), is bounded (see Remark 2.1). Moreover, by (2.13) and Lemma 3.2 (i), observe that (1c) The third and last type of terms consist of those that arise when one differentiates Φ(i 1 , i 2 , i 3 , i 4 ), I 3 (δ i 1 ) and I 3 (δ i 2 ). In this case, the corresponding terms can be bounded by expressions of the type is uniformly bounded in n on one hand, and on the other hand (by Remark 2.1), we deduce that the terms of the third type in A (n) 1 also agree with the desired conclusion (3.3).
(2) Second, we focus on the term A (n)

. We have
When computing the fourth Malliavin derivative , we have to deal with three types of terms: (2a) The first type consists in terms arising when one only differentiates Φ(i 1 , i 2 , i 3 , i 4 ). By Lemma 3.2 (i), these terms are all bounded by which is less than Hence, by Lemma 3.3, we see that the terms of the first type in A (2b) The second type consists in terms arising when one differentiates Φ(i 1 , i 2 , i 3 , i 4 ) and I 3 (δ ⊗3 is completely similar). In this case, the corresponding terms can be bounded either by or by the same quantity with ρ(i 4 − i 1 ) instead of ρ(i 3 − i 1 ). By Cauchy-Schwarz inequality, we have Since, moreover, Remark 2.1 entails that (and similarly for ρ(i 4 − i 1 ) instead of ρ(i 3 − i 1 )), we deduce that the terms of the second type in A (n) 2 also agree with the desired conclusion (3.3). (2c) The third and last type of terms consist of those that arise when one differentiates Φ(i 1 , i 2 , i 3 , i 4 ), I 3 (δ i 1 ) and I 3 (δ i 2 ). In this case, the corresponding terms can be bounded by expressions of the type is uniformly bounded in n on one hand, and on the other hand (still using Remark 2.1), we deduce that the terms of the third type in A (n) 2 also agree with the desired conclusion (3.3).
(3) Using exactly the same strategy as in point (2), we can show as well that the terms A (4) Finally, let us focus on the last term, that is A (n) 4 . We have, using successively the fact that r∈ |ρ(r)| 3 < ∞ and Lemma 3.3, Hence, the terms A  Then there exists C > 0, independent of n, such that Hence, with ρ defined by (2.4), which is (3.6). The proof of (3.7) follows the same lines, and is left to the reader.

Proof of Theorem 3.1
We are now in position to prove Theorem 3.1. For g : → , let Our main theorem which will lead us toward the proof of Theorem 3.1 is the following. Using (2.12), observe that The proof of (3.10) is divided into several steps, and follows the methodology introduced in [7].
Step 1.-We first prove that: (3.12) For g as in the statement of the theorem, we can write, for any fixed t ≥ 0: Now, let us turn to the second part of (3.12). We have By the product formula (2.14), we have Thus, for any fixed t ≥ 0, We will estimate each of these four terms using the Malliavin integration by parts formula (2.13).
For that purpose, we use Lemma 3.4 and the notation of Remark 2.6.

First, we have
Actually, in the previous double sum with respect to a and i 1 , . . . , i 6 , only the following term is non-negligible: Indeed, the other terms in A n are all of the form nt j,k=1 where x i and y i are for j or k. By Lemma 3.2 (iii), we have Hence, the quantity in (3.13) tends to zero as n → ∞. We have proved Using the integration by parts formula (2.13) as well as Lemma 3.4, we have similarly that with ρ defined by (2.4).
Using similar computations, we also have the previous convergence being obtained as in the proof of (3.27) below. Finally, we have obtained (3.14) and the proof of (3.12) is done.
Step 8.-Now, we consider the last term in (3.24), that is R (9) j,n . Since e i〈λ, n 〉 , ξ, and g p are bounded, we can write In addition we have, see (3.8), that By Lemma 3.5 we have that On the other hand, by Cauchy-Schwarz inequality, we have so that, with ρ defined by (2.4), As a consequence, combining the previous estimates with (3.6), we have shown that  and similarly for G + n , G − , and G + . By Theorems 2.12, 3.7, and 3.8, the sequence By passing to a subsequence, we may assume it converges in law in By  g, B, t)) have the same law in D 2 [0, ∞) × d . By the general fact we observed at the beginning of the proof, (G − (g, B), X ) and (G − (g, B), G − (g, B, t)) have the same law. This can be seen, for example, by letting g, B, t) , In particular, (G − (g, B, t), X ) and (G − (g, B, t), G − (g, B, t)) have the same law. But this implies G − (g, B, t) − X has the same law as the zero random variable, which gives G − (g, B, t) − X = 0 a.s.
We have thus shown that X = G − (g, B, t) a.s. Similarly, Y = G + (g, B, t) a.s. It follows that g, B), G + (g, B)), and therefore in the sense of finite-dimensional distributions on [0, ∞), which is what was to be proved.

Moment bounds
The following four moment bounds are central to our proof of relative compactness in Theorem 2.13.

Theorem 4.1. There exists a constant C such that
for all n, s, and t.
Proof. The calculations in the proof of Theorem 10 in [11] show that where g 1,∞ = g ∞ + g ∞ , and C depends only on T .
Using Lemma 2.10, we have , Hence, Substituting this into (4.1) gives which completes the proof.

Theorem 4.3. Let g ∈ C 2 ( ) have compact support. Fix T > 0 and let c and d be integers such that
where g 2,∞ = g ∞ + g ∞ + g ∞ , and C depends only on T .
Proof. Note that where 3 3 . Note that f has polynomial growth of order 2 with constants K = 1 and r = 3.
where C depends only on g and T .
Proof. Let Y j = g(β j ) − g(β c ), and note that where where σ 2 j = E|β j − β c | 2 . Note that f has polynomial growth of order 2 with constants K and r that do not depend on i or j.
It will therefore suffice to show that n (g, t) → 0 ucp.
As a corollary to this result, we find that although we do not have a bound on the second moments of I n (g, B), we can instead approximate I n (g, B), in the ucp sense, by processes whose second moments can be uniformly bounded.
Also note that that δ k → 0 as k → ∞. Since G k has compact support, we have already proven that X n,k → Y k in law. Hence, by Lemma 2.5, the sequences {X n } and {Y k } are both convergent in law, and they have the same limit. Thus, to show that X n → H(Ξ T ) in law, it will suffice to show that Y k → H(Ξ T ) in law. However, it is an immediate consequence of (2.19) that Ξ T k → Ξ T ucp, which completes the proof.
Proof of Theorem 2.14. As in the proof of Theorem 2.13, {(B, V n (B), J n )} is relatively compact. Let (B, X , Y ) be any subsequential limit. By Theorem 2.12, X = κW , where W is a standard Brownian motion, independent of B.