Donsker-type theorems for correlated geometric fractional Brownian motions and related processes

We prove a Donsker-type theorem for vector processes of functionals of correlated Wiener integrals. This includes the case of correlated geometric fractional Brownian motions of arbitrary Hurst parameters in (0 , 1) driven by the same Brownian motion. Starting from a Donsker-type approximation of Wiener integrals of Volterra type by disturbed binary random walks, the continuous and discrete Wiener chaos representation in terms of Wick calculus is effective. The main result is the compatibility of these continuous and discrete stochastic calculi via these multivariate limit theorems.


Introduction
A fractional Brownian motion B H = (B H t ) t≥0 with Hurst parameter H ∈ (0, 1) is a continuous zero mean Gaussian process with covariance function E[B H t B H s ] = 1/2 t 2H + s 2H − |t − s| 2H , s, t ≥ 0. It is the unique Gaussian H-self-similar process with stationary increments. The process B 1/2 is a standard Brownian motion, but B H is not a semimartingale for H = 1/2. The corresponding fractional Gaussian noise (B H n+1 − B H n ) n∈N for H > 1/2 exhibits long-range dependence and is commonly used in modeling phenomena in economy, finance, physics or neuroscience (see e.g. the monographs [8] and [21] and the references therein). There is a powerful representation as a Wiener integral of Volterra type for some kernel z H (t, ·) ∈ L 2 ([0, t]), t ≥ 0 and Brownian motion B (cf. [23, 5.2]  . (1. 2) The functionals involved can be represented in terms of geometric fractional Brownian motions exp(B H t − t 2H /2) t≥0 and reformulations of the Wiener chaos expansion. This Donsker theorems extend the Fractional Donsker theorem in [28,25] and the results in [5] to multivariate functional type Donsker theorems. More generally, we consider Wiener integrals of Volterra type as in (1.1), denoted by I(f ), and the corresponding stochastic exponentials as As a primary result we obtain a Donsker theorem for vector processes of such stochastic exponentials and related functionals.
In particular, we are interested in approximating sequences which rely on disturbed random walks converging to the Wiener integrals I(f ) as in [28,25] and an appropriate discrete stochastic calculus which is justified by these convergence results, cf. [5,6]. In contrast to related multivariate invariance principles based on discrete chaos as in [2,3,4], we consider elements with an infinite chaos expansion as in (1.3).
We note that the convergence of finite-dimensional distributions is for example a consequence of a functional limit theorem in [26]. Here the main effort is assigned to the tightness of such general processes. This is handled by checking a well-known tightness criterion and some combinatorial reformulations of L p -norms of discrete counterparts of functionals as (1.3) applied on correlated Volterra-type discrete Wiener integrals.
The article is organized as follows. In Section 2 we give a brief description of the class of functionals extending the stochastic exponentials in (1.3). Section 3 is devoted to the analogue in a disturbed random walk setting. In Section 4 we state and prove the main result. The technical lemmas are postponed to Section 5.
We refer to [16,15] for further details and a reformulation in terms of multiple Wiener integrals.
The S-transform is closely related to a product imitating uncorrelated random vari- , which is implicitly contained in the Skorokhod integral and a fundamental tool in stochastic analysis. Due to the injectivity of the S-transform, the Wick product can be introduced via on a dense subset in L 2 (Ω) × L 2 (Ω). For more details on Wick product we refer to [16,15,18]. For example, for a Wiener Integral I(f ), Hermite polynomials play the role of monomials in standard calculus as Therefore the stochastic exponential is also knows as Wick exponential: The Wick exponential exp (I(f ) t ) t≥0 is the unique solution of the Doléans-Dade equation [19,Section 8.7]). Following [15,10], we denote the Wiener chaos decomposition in terms of Wick products as the Wick-analytic representation. In particular, for fixed f 1 , . . . , f K ∈ L 2 ([0, ∞)), g : R K → R and a square integrable left hand side, there exist a l1,...,l K ∈ R, l 1 , . . . , l K ≥ 0, such that This is a reformulation of the the Wiener chaos decomposition in terms of generalized Hermite polynomials, see e.g. [12,1].
We define the class of Wick-analytic functionals as in L p (Ω) for all p ∈ N (see Proposition 9 in [22]). Moreover, the analytic representation G(I(f 1 ), . . . , (see Proposition 10 in [22]). One advantage of the Wick-analytic reformulation is the characterization of Skorokhod integrands which allow exact simulation [22,Theorem 17].

Discrete stochastic calculus
As a discrete counterpart of B we consider, for every n ∈ N, a random walk approxi- As a counterpart, due to the discrete analogue of the Doléans-Dade equation the discrete Wick exponential is given by and used to introduce discrete Malliavin operators, in particular a discrete Skorokhod integral (see [7]).
The discrete Wick product is introduced via exp n (I n (f n )) n exp n (I n (g n )) = exp n (I n (f n + g n )), (3.1) where I n (f n ) and I n (g n ) are two, possibly correlated, discrete Wiener integrals. Then, (3.1) extends bilinearly to a dense subset of L 2 (Ω n , F n , P n ) × L 2 (Ω n , F n , P n ) and is equivalent to the characterization in terms of the canonical basis For example, we have the simple discrete Wiener chaos expansion in general, as illustrated by X n = I n (g n ) and Y n = I n (h n ) for g n , h n ∈ L 2 n (N). Moreover, the discrete Wick product has zero divisors, but the continuous Wick product is free of zero divisors even in more general spaces (cf. [13]).
For more information on the discrete calculus we refer to [6,11] or the monographs [27,29]. In particular, the discrete counterpart of (2.1) is true for all f n ∈ L 2 n (N) as We define the class of discrete Wick-analytic functionals as (n ∈ N fixed)

The main result
We denote a function f (t, s) t,s≥0 with f (t, ·) ∈ L 2 ([0, ∞)) for all t and f (t, s) = 0 for t ≤ s as an integrand of Volterra type. Analogously, the discrete integrand of Volterra type is given by f n (l, i) l,i∈N such that f n (l, ·) ∈ L 2 n (N) for all l ∈ N with f n (l, i) = 0 for l < i.
We specified the conditions on the continuous and discrete integrands for a Donsker theorem for Volterra type Wiener integrals in Theorem 3 of [25]. This can be reformulated for the weak convergence Then, for every Γ ∈ R m (with inner product ·, · ), in the Skorokhod space D([0, 1], R) as n tends to infinity. We notice that the normalizing term 1/ √ n is implicitly contained in the discrete Wiener integral.

Remark 4.2.
Our main result is that this weak convergence is compatible with the continuous and discrete Wick-analytic functionals. This is not trivial due to the differences of the calculi and the different correlations of discrete and continuous Wiener integrals.     are geometric fractional Brownian motion and discrete geometric fractional Brownian motion, cf. [5]. Similarly, thanks to the continuous mapping theorem, we conclude that the sequence of processes converges weakly to the process (1.2) as n tends to infinity.
The proof of Theorem 4.4 relies on the following estimate. The technical proof is postponed to Section 5. Moreover we make use of the following variant of a well-known tightness criterion Theorem 15.6 in [9]. See e.g. Remark 1 in [25] for the connection.  we obtain Kα for some constant L = L(K, C, L, Γ) > 0. Thus, for every α > 0 we find some K ∈ N with Kα > 1 and it suffices to apply the tightness criterion in Lemma 4.7.

Proof of Lemma 4.6
Firstly we note some expansion formulae and an inequality. We denote by˙ the disjoint union.
. . , n} for all i, j ∈ N. We make use of the shorthand notations for products as The following formula is clear by expansion: The generalized Cauchy-Schwarz inequality follows easily by induction: Proof. Firstly, thanks to (5.1) we observe A⊆{1,...,n} The inequality A⊆{1,...,n} is clearly true for n = 1 and then proved by induction: Thanks to (1 + x) ≤ exp(x) we conclude (5.4).
Proof of Lemma 4.6. Due to (3.2), we only have nonvanishing discrete Wick products on disjoint sets as Ξ n B1 n · · · n Ξ n B k = Ξ ṅ i=1,...,k Bi .  Analogously, for all integers K ≥ 1, the orthogonality yields where the condition D 1 D 2 · · · D 2K = ∅ gives exactly those terms in the expansion of reformulation of the sums is the content of Step 1 below. Then, the final upper bound inequality will be proved in Step 2.
Step 1 : Suppose a multiset of pairs in the first sum in (5.9), i.e. a family of nonempty sets D 1 , . . . , D 2K ⊆ {1, . . . , n} such that D 1 D 2 · · · D 2K = ∅ (and therefore n large enough, e.g. n ≥ 2K). Such a chosen family can be coded by intersection sets as follows: Let the index set Every intersection set U f is covered by an even number of sets D j which illustrates the condition D 1 D 2 · · · D 2K = ∅.
The inner sum in (5.9) is reformulated by a map: Every set of partitions is generated by a unique surjective map For shorthand, we denote by  Step 2 : Thanks to this multiplicative form in (5.11) we conclude via the generalized Cauchy-Schwarz inequality in (5.3) and the condition (4.4),