Approximation at First and Second Order of m-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales

: Let X be the fractional Brownian motion of any Hurst index H 2 (0 ; 1) (resp. a semimartingale) and set ﬁ = H (resp. ﬁ = 12 ). If Y is a continuous process and if m is a positive integer, we study the existence of the limit, as " ! 0, of the approximations

Abstract: Let X be the fractional Brownian motion of any Hurst index H ∈ (0, 1) (resp.a semimartingale) and set α = H (resp. α = 1  2 ).If Y is a continuous process and if m is a positive integer, we study the existence of the limit, as ε → 0, of the approximations of m-order integral of Y with respect to X.For these two choices of X, we prove that the limits are almost sure, uniformly on each compact interval, and are in terms of the m-th moment of the Gaussian standard random variable.In particular, if m is an odd integer, the limit equals to zero.In this case, the convergence in distribution, as ε → 0, of ε − 1 2 I ε (1, X) is studied.We prove that the limit is a Brownian motion when X is the fractional Brownian motion of index H ∈ (0, 1  2 ], and it is in term of a two dimensional standard Brownian motion

Introduction
In this paper we investigate the accurate convergence of some approximations of m-order integrals which appear when one performs stochastic calculus with respect to processes which are not semimartingales, for instance the fractional Brownian motion.We explain below our main motivation of this study.

Preliminaries
Recall that the fractional Brownian motion with Hurst index 0 < H < (see [16], pp.97-98).Moreover, if H > 1 2 , B H is a zero quadratic variation process.Hence, for H ≥ 1 2 , a Stratonovich type formula involving symmetric stochastic integrals holds (see, for instance [17] or [5]): (1.1) On the other hand, if 0 < H < 1 2 , some serious difficulties appear.It is a quite technical computation to prove that (1.1) is still valid when H > 1  6 (see [8] or [3]).In fact, for H > 1  6 , in [8] was proved, on the one hand, and the m-forward integral is given by (if m = 1, we write In [8] one studies, for the fractional Brownian motion, the existence of the m-order symmetric integrals.According to evenness of m, it is proved that H s exists and vanishes; • if mH = (2n − 1)H > 1  2 then integral Moreover it is also emphasized that integrals 2 ).This last statement is in fact a direct consequence of results of the present paper (as it is mentioned in the proof of Theorem 4.1 in [8]).An important consequence is that H = 1  6 is a barrier of validity for the formula (1.1) (see [1], [3], [4], [7] and [8]).

First order approximation: almost sure convergence
In the definitions (1.2) and (1.3), limits are in probability.One can ask a natural question: is it possible to have almost sure convergence?For instance, in [10], the almost sure convergence of a generalized quadratic variation of a Gaussian process is proved using a discrete observation of one sample path; in particular, their result applies to the fractional Brownian motion.Here, we prove (see Theorems 2.1 and 2.2 below for precise statements) that, as ε → 0, converge almost surely, uniformly on each compact interval, to an explicit limit when f belongs to a sufficiently large class of functions (including the case of polynomial functions, for instance f (x) = x m ), Y is any continuous process, X = B H is the fractional Brownian motion with H ∈ (0, 1) (resp.X = Z a semimartingale) and α = H (resp. α = 1 2 ).Let us remark that the case when X is a semimartingale is a non Gaussian situation unlike was the case in [10] or other papers (at our knowledge).If m = 2n is an even integer the previous result suffices to study the existence of 2n-order integrals for the fractional Brownian motion with all 0 < H < 1.Indeed, by choosing f (x) = x 2n , we can write the following equivalent, as ε ↓ 0: On the other hand, if m = 2n − 1 is an odd integer, we need to refine our analysis (especialy for Hurst index 0 < H ≤ 1 2 ) because, in this case, we do not have an almost sure non-zero equivalent.

Second order approximation: convergence in distribution
Set Y ≡ 1 and f (x) = x m in (1.4), with m ≥ 3 an odd integer.For the two same choices of X (that is X = B H with α = H or X a semimartingale with α = 1 2 ), we have, for all T > 0, After correct renormalization, is it possible to obtain the convergence in distribution of our approximation?We prove (see Theorems 2.4 and 2.5 below for precise statements) that the family of processes converges in distribution, as ε → 0, to an explicit limit: • If X = B H is the fractional Brownian motion with H ∈ (0, 1  2 ], we obtain a Brownian motion and our approach is different to those given by [6] or [19]; • If X is a semimartingale, we express the limiting process in terms of a two-dimensional standard Brownian motion.This also give an example of a non-Gaussian situation when the convergence in distribution is studied.
We can see that ds, ε > 0 by using the self-similarity of the fractional Brownian motion (that is, for all c > 0, B H ct equals in law -as a process -to c H B H t ).In [6], the authors study the convergence in distribution (but only for finite dimensional marginals) of the discrete version of our problem, that is of the sum with f a real function.On the other hand, in [19], the Hermite rank of f is used to discuss the existence of the limit in distribution of 2 (recall that, in the present paper, we assume that H is smaller than 1  2 ).
2 Statement of results

Almost sure convergence
In the following, we shall denote by N a standard Gaussian random variable independent of all processes which will appear, by B H the fractional Brownian motion with Hurst index H and by B = B 1 2 the Brownian motion.
Theorem 2.1 Assume that H ∈ (0, 1).Let f : R → R be a function satisfying for all x, y ∈ R and {Y t : t ≥ 0} be a continuous stochastic process.Then, as ε → 0, almost surely, uniformly in t on each compact interval.
The following result contains a similar statement for continuous martingales: Theorem 2.2 Let f : R → R be a polynomial function.Assume that {Y t : t ≥ 0} is a continuous process and that {J t : t ≥ 0} is an adapted locally Hölder continuous paths process.Let {Z t : t ≥ 0} be a continuous martingale given by almost surely, uniformly in t on each compact interval.Here, F t = σ(J s , s ≤ t).

Remarks:
1.For instance, if f (x) = x m then the right hand side of (2. is defined as the limit in probability, as ε → 0, of For any H ∈ (0, 1), it is a simple computation to see that the previous limit exists almost surely, on each compact interval and equals to Consequently, by using (2.4) and the part 2 of Corollary 2.3, we see that the divergence integral t 0 g(B H s )δB H s can be defined path-wise.2. In [20], it was introduced a path-wise stochastic integral with respect to B H when the integrator has γ-Hölder continuous paths with γ > 1 − H.When the integrator is of the form g(B H t ), the condition on γ implies that H > 1 2 (see p. 354).Hence, the first two parts of Corollary 2.3 could be viewed as improvements of the results in [20]. 2

Convergence in distribution
Let m be an odd integer.It is well known that the monomial x m may be expanded in terms of the Hermite polynomials: Note that the sum begin with k = 1 since m is odd (for instance x = H 1 (x), x 3 = 3H 1 (x) + H 3 (x) and so on).
Theorem 2.4 Let m ≥ 3 be an odd integer and assume that H belongs to (0, 1  2 ].Then, as ε → 0, Here {β t : t ≥ 0} denotes a one-dimensional standard Brownian motion starting from 0 and c m,H is given by where the coefficients a k,m are given by (2.5). Remark: Finally, let us state the result concerning martingales: Theorem 2.5 Let m ≥ 3 be an odd integer and assume that σ is an element of C 2 (R; R).Let {Z t : t ≥ 0} be a continuous martingale given by Here {(β t ) : t ≥ 0} denotes a two-dimensional standard Brownian motion starting from (0,0) and κ i , i = 1, 2 are some constants.

Proof of almost sure convergence
The idea to obtain almost sure convergence is firstly, to verify L 2 -type convergence and secondly, to use a Borel-Cantelli type argument and the regularity of paths (see Lemma 3.1 below).
To begin with, let us recall a classical definition: the local Hölder index γ 0 of a continuous paths process {W t : t ≥ 0} is the supremum of the exponents γ verifying, for any T > 0: We can state now the following almost sure convergence criterion which will be used in proving Theorems 2.1 and 2.2: Lemma 3.1 Let f : R → R be a function satisfying (2.1), {W t : t ≥ 0} be a locally Hölder continuous paths process with index γ 0 and {V t : t ≥ 0} be a bounded variation continuous paths process.Set and assume that for each t ≥ 0, as ε → 0, Then, for any t ≥ 0, lim ε→0 W (f ) ε (t) = V t almost surely, and if f is non-negative, for any continuous process {Y t : t ≥ 0}, as ε → 0, almost surely, uniformly in t on every compact interval.
Proof.We split the proof in several steps.
Step 3. We will show that the exceptional set of the almost sure convergence W (f ) ε (t) → V (t) can be choosed independent of t.Let Ω * the set of probability 1, such that for every and assume that {s n } and {t n } are rational sequences such that s n ↑ t and t n ↓ t.Clearly, First, letting ε goes to zero we get and then, letting n goes to infinity we deduce that for each ω ∈ Ω * and each Step 4. If f is non-negative we can apply Dini's theorem to obtain that W (f ) ε (t) converges almost surely toward V t , uniformly on every compact interval.
Step 5. Further, the reasoning is pathwise, hence we fix ω ∈ Ω, we drop the argument ω and write small letters instead capital letters.Since w  First, let us note that if f satisfies (2.1) then the positive part f + and the negative part f − also satisfy (2.1).Hence by linearity, we can assume that f is a non-negative function.We shall apply Lemma 3.1 to W = B H , the fractional Brownian motion which is a locally Hölder continuous paths process with index H (as we can see by applying the classical Kolmogorov theorem, see [15], p. 25) and to the process We need to verify (3.3).First we note that Var(B as we can see by using Taylor expansion.Hence, by classical linear regression we obtain, for uniformly with respect to u, where and M u,ε are two independent standard Gaussian random variables.We write where Using (2.1) (which implies that |f (x)| ≤ cst.(1+|x| a (1+x 2 ) b )) and Cauchy-Schwarz inequality we can prove that T 1 (ε) = O( √ ε).Using again (2.1) and (3.5) we deduce that By linearity it suffices to prove the result for f (x) = x m and by classical localization argument (see for instance [8], p. 8), it suffices to prove the result for J bounded continuous process.Recall that N denotes a standard Gaussian random variable.
Thanks to Theorem 2.1, (2.3) is true for the Brownian motion B = B j=1 δ(j).Let M be a martingale, a < b be two real numbers and we shall denote with the convention dM (1) = dM (Itô's differential) and dM (2) = d[M, M ] (Riemann-Stieltjes differential).Then, for each n ∈ N * , where c n (δ) is a contant depending only on δ and n.

Proof.
We make an induction with respect to n.
Assume that (3.9) is true for n and we verify it for n + 1: a,b,δ .
By hypothesis, paths of J are locally Hölder continuous and by using the isometry property of Itô's integral we can deduce (3.3): as ε → 0, We need now the following simple modification of Lemma 3.1: Lemma 3.3 Let us made the same assumptions on the function f and on the process W as in Lemma 3.1.Moreover, we assume that { Wt : t ≥ 0} is another locally Hölder continuous paths process with same index γ 0 and assume that W (f ) ε (t) denotes the associated process to W as in (3.2).If f is non-negative and if for each t ≥ 0, as ε → 0, then, we have almost surely, uniformly on every compact interval.
The proof is straightforward and we leave it to the reader.Using this result, we obtain that almost surely, uniformly on each compact interval.Combining (3.12) with (3.7), we get (2.3).
By a simple change of variable we can transform the left-hand side as which tends, as ε → 0, almost surely uniformly on each compact interval toward g(B H t )−g(0).The last term on the right-hand side of the previous equality converges, almost surely and uniformly on each compact interval and therefore the term which remains on the right-hand side is also forced to have a limit.The third part can be proved in a similar way.Let us turn to the second part.By setting Y s = g (3)  Consequently, it suffices to use the following Taylor formula and the dominated convergence, in order to conclude as previously.

Proofs of the convergence in distribution
Proof of Theorem 2.4.First, let us explain the main ideas in the simpliest situation of the Brownian motion (H = 1 2 ).In this case we are studying M T (t) = T −1/2 tT 0 (B s+1 − B s ) m ds (see Step 1 below) and we write it, thanks to succesive applications of Itô's formula and of the stochastic version of Fubini theorem, as tT 0 R T (s)dB s plus a remainder which tends to zero in L 2 , as T → ∞.Then we can show that lim T →∞ tT 0 R T (s) 2 ds = cst.t,hence, by Dubins-Schwarz theorem, we obtain that M T → √ cst.β, as T → ∞, in the sense of finite dimensional time marginals.Finally we prove the tightness.Let us remark that similar technics have been used in [14], precisely in the proof of Proposition 3 (see also Step 10 below).
For the fractional Brownian motion case (0 < H < 1 2 ) technical difficulties appear because the kernel K in its moving average representation (B H t = t 0 K(s, t)dB s ) is singular at the points s = 0 and s = t.Again we split the proof in several steps.
Step 1.By the self-similarity of the fractional Brownian motion, that is {B , for all c > 0, we can see that where Hence, to get (2.6) it suffices to prove that Moreover, this convergence is a consequence of the following two facts: dimensional time marginals; ii) for T ≥ 1, the family of distributions of processes M T is tight. (3.16) Step 2. Before proceeding with the proof of (3.15), let us show how the constant c m,H appears.We claim that, for each t ≥ 0, lim (3.17) . We need to estimate the expectation of the product G m 1 G m 2 .Thanks to (2.5), we have Replacing this in the expression of the second moment of M T (t), we obtain, noting also that by the change of variables x = v − u, y = u/T .Letting T goes to infinity we get on the right hand side of the previous equality c m,H t.
Step 3. We proceed now to the first technical notation which will be useful in the next step.We write M T (t) = M T (b) + M (b)  T (t), where Choose an arbitrary, but fixed > 0. Let us note that, by (3.17), we can choose b > 0 small enough such that lim Step 4. Let us recall (see, for instance, [1], p. 122) that the fractional Brownian motion can be written as where, here and elsewhere we denote by γ H the constant Γ(H + 1 2 ) −1 .We can write M (b) Since the process A has absolutely continuous trajectories and using the fact that A and B are independent as stochastic integrals on disjoint intervals, it is not difficult to prove that, for each t ≥ 0, lim Step 5.By (3.21) and (3.20) we can write We need to introduce a second technical notation.Let us denote: where the positive constant c will be fixed and specified by the statement a) below.We shall prove successively the following statements: a) for each t ≥ 0, there exists c > 0 large enough, such that lim sup b) there exists two families of stochastic processes {R T (t) : t ≥ 0} and {S T (t) : t ≥ 0} such that, for each t ≥ 0, Ň (b,c) Step 6. Suppose for a moment that a), b), c) are proved and let us finish the proof of the convergence in law in sense of finite dimensional time marginals (3.15).First, we can write, or equivalently, for each t ≥ 0, lim sup with notations a (T ) (t) := tT 0 R T (s) 2 ds and a(t) := c m,H t.Second, we fix d ∈ N * and 0 ≤ t 1 < t 2 < . . .< t d and we shall denote for any u ∈ R d and f : R + → R, u • f := d j=1 u j f (t j ).We consider the characteristic functions: By By Dubins-Schwarz theorem, we can write, for each T , , with β (T ) a one-dimensional standard Brownian motion starting from 0. Therefore, we have Combining (3.29), (3.30) and letting → 0, (3.15) follows.
Step 7. We verify (3.16), that is, the tightness of the family of distributions of processes M T .It suffices to verify the classical Kolmogorov criterion (see [15], p. 489): sup where, as in Step 2, we denoted the standard Gaussian random variables 3, 4. Let us also denote θ ij = Cov(G i , G j ), i, j = 1, . . ., 4. We need to estimate the expectation of the product G m 1 G m 2 G m 3 G m 4 .By using (2.5), we get and we need to estimate E Using the result in [18], p. 210, we can write where 1 is the sum over all indices i 1 , j 1 , . . ., i q , j q ∈ {1, 2, 3, 4} such that i 1 = j 1 , . . ., i q = j q and there are ] in terms of θ ij and so on.Since G i have variance 1, we deduce, using the conditions on the indices appearing in (3.32), that Hence (3.31) is verified so the family of distributions of processes M T is tight.The proof of Theorem 2.4 will be finished once we prove statements a)-c) in Step 5.
Step 8. We prove (3.25) and at the same time we precise the choice of the constant c.For notational convenience we will drop superscripts "(b)" or "(b,c)" during the proof.Using again (3.23) and (3.24) we can write with and where we denoted We shall prove that the second moment of each term in (3.33) can be made small enough and then (3.25) will follows.
Step 9. We can write By using Cauchy-Schwarz inequality and (3.37), we can prove that there exists c > 0 large enough, such that ).Consider now the second term in (3.35).Since m is an odd integer, the expectation equals zero for each even integer k, by independence of stochastic integrals on disjoint intervals.Hence we need to consider only odd integers k.We can write, for bT < s, s + c + 1 < s < tT , using again the independence and (3.37), where We state the following simple result: Lemma 3.5 Let m, k be odd integers and assume that (X 1 , X 2 ), (Y 1 , Y 2 ) are two independent centered Gaussian random vectors.Set θ = Cov(X 1 , X 2 ).Then If j ≥ 2, we make a similar reasoning.Hence (3.39) is verified and (3.25) follows.
Step 10.We prove now the statement b) in Step 5, that is (3.26).Again, we will drop the superscripts "(b,c)".Using Here λ denotes the Lebesgue measure and the multiple integrals I p , the tensor product f ⊗ g and the contractions f × (p) g are defined in [11] or [13].It is not difficult to prove that E[S T (t) 2 ] ≤ (cst./T ) → 0, as T → ∞, by using successively the stochastic version of Fubini theorem, (3.37) and Jensen inequality.Hence (3.26) is proved for m = 3.For m an odd integer strictly bigger than 3, (3.26) is obtained by using Lemma 3.2 and a similar reasoning as previously (using eventually the notation with multiple integrals, as in the previous remark).
Step 11.To end the proof of Theorem 2.4 we need to verify the statement c) in Step 5, that is (3.27).Assume that m = 3, the general case being similar (by using Lemma 3. X, Y continuous processes and m ≥ 1, the m-order symmetric integral is given byt 0 Y s d •(m) X s := lim ε→0

ε
simply converges toward v, the distribution function of the measure dw (f ) ε converges toward the distribution function of the measure dv, hence dw (f ) ε weakly converges toward dv.Clearly, the measure dv does not charge points and the function s → y s 1 [0,t] (s) is dv-almost everywhere continuous.Consequentely, lim ε→0 ∞ 0 y s 1 [0,t] (s) dw

y
s dv s .The proof of the almost sure convergence criterion is done.Proof of Theorem 2.1.

1 2 .Lemma 3 . 2
More precisely, as ε → 0,t 0 J m s B s+ε − B s √ ε m ds → E [N m ] t 0 J m s ds (3.7)almost surely uniformly on each compact interval.At this point we need the following technical but simple: Denote by P the set of finite sequences δ with values in {1, 2}.For δ ∈ P we denote by k = k(δ) the length of the support of the sequence δ and n(δ) := k(δ)

Proof of Corollary 2 . 3 .
To prove the first part, we set Y s = g (B H s ) and f (x) = x 2 in (2.2).Then, we obtain the existence of lim almost surely, uniformly on each compact interval.On the other hand, we can write, for a, b ∈ R, a < b, g(b) = g(a) + g (a)(b − a) + g (θ) 2 (b − a) 2 , with θ a,b ∈ (a, b).Setting a = B H s and b = B H s+ε , integrating in s on [0, t] and dividing by ε we get: (B H s ) and f (x) = x 3 in (2.2), we obtain the existence of lim almost surely, uniformly on each compact interval.On the other hand, by setting Y s ≡ 1 and f (x) = |x| 3 in (2.2) we obtain that lim , almost surely, ∀t > 0.
t}.By using (3.27) for the first term, (3.25)-(3.26)for the second term, (3.22) for the third term, (3.19) for the forth term and (3.17) for the fifth one, we obtain lim sup