Approximating a Geometric Fractional Brownian Motion and Related

arXiv:1010.1666v1 [math.ST] 8 Oct 2010 inmto a mre se .. h oorpsb ihr rBia or Mishura by monographs the e.g., (see, emerged has motion nian dependence. hc h orsodn rcinlGusa os ( noise Gaussian fractional corresponding the which eiatnaefor semimartingale a eisrpeettosi em fWc oes e otep the Wick to the key for formula a recursion powers, Hermite Wick the of of A terms approximation equations. difference in Wick representations of series systems corresponding the in ahrcmlctdWc rdcsb hi iceecounter discrete Sott their to by due products walks Wick Donsker-type random complicated a binary rather apply disturbed we by end, motion this Brownian To motion. Brownian tional h process The HITA BENDER CHRISTIAN calculus via Wick processes discrete related and motion Brownian fractional geometric a Approximating DOI: Bernoulli 1350-7265 product Wick the semimartingales on beyond based Itô integral is the It of Itô integral. extensions the of One dr SDEs of systems linear some of motion solution the approximate We 1 Box P.O. University, E-mail: Saarland Germany. Mathematics, of Department Keywords: rcinlBona motion Brownian fractional A Introduction 1. integral enGusa rcs in process Gaussian mean hsi neetoi ern fteoiia ril publis 389–417 detail. article 2, typographic original No. the 16, of Vol. reprint 2010, electronic an is This nrcn er,alvl neeti nerto hoywt respect with theory integration in interest lively a years, recent In 10.3150/09-BEJ223 B H 16 iceeWc acls rcinlBona oin ekc weak motion; Brownian fractional calculus; Wick discrete c ihHrtparameter Hurst with B 2,21,389–417 2010, (2), 1 00ISI/BS 2010 / 2 * sasadr rwinmto,btafatoa rwinmto sno is motion Brownian fractional a but motion, Brownian standard a is [email protected] H * E = n EE PARCZEWSKI PETER and [ B 1 2 R nti ae,w etitorevst h case the to ourselves restrict we paper, this In . hsrpitdffr rmteoiia npgnto and pagination in original the from differs reprint This . t H ihsainr nrmnsadcvrac function covariance and increments stationary with B B s H H H = ] ∈ ihHrtparameter Hurst with ( 2 1 2 1 ( , | )i h ikIˆ es,icuigagoercfrac- geometric a including Wick–Itô sense, the in 1) ; t | ** 2 H [email protected] + ⋄ hc a t rgna renormalization a as origin its has which , | s | 2 H | − B e ythe by hed n H ** at cigo h iayvariables, binary the on acting part, t +1 − h ouin fteSE admit SDEs the of solutions the s H oesof powers − s 15,D601Saarbrücken, D-66041 50, 11 5 prxmto ftefractional the of approximation ofo u ue ceei an is scheme Euler our of roof | nn oevr erpaethe replace we Moreover, inen. ∈ B 2 vnb rcinlBrownian fractional a by iven H n H (0 ) . ) ISI/BS , n )i otnoszero- continuous a is 1) negne Wick–Itôonvergence; stefatoa Wick– fractional the is ∈ N B H xiislong-range exhibits ofatoa Brow- fractional to gini . in Bernoulli tal. et > H [ 4 1 , , / ,in 2, 15 ]). t 390 C. Bender and P. Parczewski operator in quantum physics. In probability theory, the Wick product with ordinary differentiation rule imitates the situation of ordinary multiplication with Itôdifferentiation rule (cf. Holden et al. [11]). Actually, this makes it a natural tool to apply for extending the Itôintegral. H We first consider the fractional Doléans–Dade SDE dSt = Std⋄Bt , S0 = 1, in terms H 1 2H of the fractional Wick–Itôintegral. The well-known solution, exp(Bt 2 t ), is the geometric fractional Brownian motion, also known as the Wick exponential− of fractional Brownian motion. Note that the Wick exponential has expectation equal to one and can therefore be interpreted as a multiplicative noise. Moreover, the ordinary exponential can be obtained from the Wick exponential by a deterministic scaling. Neither process is a 1 semimartingale for H = 2 . The name “Wick exponential” is justified by the fact that it H k exhibits a power series expansion with Wick powers (Bt )⋄ instead of ordinary powers. More generally, we consider a linear system of SDEs,

H dXt = (A1Xt + A2Yt) d⋄Bt , X0 = x0, (1) H dYt = (B1Xt + B2Yt) d⋄Bt , Y0 = y0.

One can obtain Wick power series expansions for the solution of this system, too. Our goal is to approximate these Wick analytic functionals of a fractional Brownian motion. To this end, we require an approximation of a fractional Brownian motion and an approximation of the Wick product. There are several ways to approximate a fractional Brownian motion. One of the first approximations was given by Taqqu [22] in terms of stationary Gaussian sequences. We refer to Mishura [15], Section 1.15.3, for further approaches to weak convergence to a fractional Brownian motion. Sottinen constructed a simple approximation of a fractional 1 Brownian motion on an interval for H> 2 by sums of square-integrable random variables in [21]. He used the Wiener integral representation of a fractional Brownian motion H t on an interval, Bt = 0 zH (t,s) dBs, for a suitable deterministic kernel zH (t,s), due to Molchan and Golosov, and Norros et al. [16–18]. For this purpose, he combined a pointwise approximation of the kernel zH (t,s) with Donsker’s theorem. This approach was extended by Nieminen [19] to weak convergence of perturbed martingale differences to fractional Brownian motion. We shall utilize Sottinen’s approximation with binary random variables throughout this paper. The main problem of applying the Wick product on random variables with continuous distributions is that it is not a pointwise operation. Thus, an explicit computation of the Wick–Itôintegral is only possible in rare special cases. But this is precisely the advantage of the binary random walks. In such a purely discrete setup, we apply the discrete counterpart of the Wick product as introduced in Holden et al. [10]. Starting from the binary random walk, one can build up a discrete Wiener space, and the discrete Wick product depends on this discretization. This Wiener chaos gives the analogy to the continuous Wick products. For a survey on discrete Wiener chaos, we refer to Gzyl [9]. However, we will introduce the discrete Wick product in a self-contained way in Section 3. Approximating a GFBM and related processes 391

We can now formulate a weak Euler scheme of the linear system of SDEs (1) in the Wick–Itˆosense,

n n n n H,n H,n Xl = Xl 1 + (A1Xl 1 + A2Yl 1) n (Bl/n B(l 1)/n), − − − ⋄ − − n X0 = x0, l =1, . . ., n, (2) n n n n H,n H,n Yl = Yl 1 + (B1Xl 1 + B2Yl 1) n (Bl/n B(l 1)/n), − − − ⋄ − − n Y0 = y0, l =1, . . ., n, where is the discrete Wick product and (BH,n BH,n ) are the increments of the ⋄n l/n − (l 1)/n disturbed binary random walk. As a main result, we show− that the piecewise constant interpolation of the solution of (2) converges weakly in the Skorokhod space to the solution of (1). This is the ﬁrst rigorous convergence result connecting discrete and continuous Wick calculus of which we are aware. As a special case, (2) contains the Wick diﬀerence equation

n n n H,n H,n n Xl = Xl 1 + Xl 1 n (Bl/n B(l 1)/n), X0 =1, l =1, . . ., n. (3) − − ⋄ − − As a consequence, the piecewise constant interpolation of (3) converges weakly to a geometric fractional Brownian motion, the solution of the fractional Dol´eans–Dade SDE. This was conjectured by Bender and Elliott [3] in their study of the Wick fractional Black–Scholes market. In [21], Sottinen considered the corresponding diﬀerence equation in the pathwise sense, that is, with ordinary multiplication instead of the discrete Wick product:

ˆ n ˆ n ˆ n H,n H,n ˆ n Xl = Xl 1 + Xl 1(Bl/n B(l 1)/n), X0 =1, l =1, . . ., n. (4) − − − − The solution is explicitly given by the multiplicative expression

l ˆ n H,n H,n Xl = (1 + (Bj/n B(j 1)/n)). (5) − − j=1 By the logarithmic transform of ordinary products into sums and a Taylor expansion, ˆ n one obtains an additive expression for ln(Xl ) which converges weakly to a fractional Brownian motion. In this way, Sottinen proved the convergence of Xˆ to the ordinary exponential of a fractional Brownian motion [21], Theorem 3. This approach fails for the solution of (3) since, in a product representation, analogous to (5), the discrete Wick product n appears instead of ordinary multiplication. There is, however, no straightforward way⋄ to transform discrete Wick products into sums by application of a continuous functional. However, the solution of (2) exhibits an expression which is closely related to a discrete Wick power series representation. Therefore, the convergence can be initiated explicitly for the Wick powers of a fractional Brownian motion, which fulfill the Hermite recursion 392 C. Bender and P. Parczewski formula. We obtain a discrete analog to this recursion formula for discrete Wick powers of disturbed binary random walks. Actually, the weak convergence of these discrete Wick powers is the key to the proof for our Euler scheme. The paper is organized as follows. In Section 2, we give some preliminaries on the Wick–Itôintegral with respect to a fractional Brownian motion and introduce the Wick exponential and other Wick analytic functionals. We then define the approximating sequences and state the main results in Section 3. Section 4 is devoted to some L2- estimates of the approximating sequences. We prove convergence in finite-dimensional distributions in Section 5 and tightness in Section 6.

2. Wick exponential and Wick analytic functionals

In this section, we introduce the Wick product and the Wick–Itˆointegral, and describe the Hermite recursion formula for Wick powers of a zero-mean Gaussian random variable. We then obtain the Wick power series expansions for the solutions of SDEs (1). We consider a geometric fractional Brownian motion or the so-called Wick exponential of a fractional Brownian motion exp(BH 1 t2H ). For H = 1 , this is exactly a geometric t − 2 2 Brownian motion, also known as the stochastic exponential of a standard Brownian 2H H 2 motion. For all H (0, 1) and t 0, it holds that t = E[(Bt ) ] and thus the Wick exponential generalizes∈ the stochastic≥ exponential. It is well known that exp(B 1 t) t − 2 solves the Dol´eans–Dade equation

dSt = St dBt, S0 =1, where the integral is an ordinary Itˆointegral. Actually, the Wick exponential of fractional Brownian motion solves the corresponding fractional Dol´eans–Dade equation

H dSt = St d⋄Bt , S0 =1, in terms of a fractional Wick–Itˆointegral (cf. Mishura [15], Theorem 3.3.2). We want to approximate solutions of similar SDEs. Let Φ and Ψ be two zero-mean Gaussian random variables. The Wick exponential is then deﬁned as

1 2 exp⋄(Φ) := exp(Φ E[ Φ ]). − 2 | | For a standard Brownian motion (Bt)t 0 and s

exp⋄(B B )exp⋄(B B )=exp⋄(B B ). u − t t − s u − s Forcing this renormalization property to hold for all, possibly correlated, Φ and Ψ, leads to the deﬁnition of the Wick product of two Wick exponentials: ⋄

exp⋄(Φ) exp⋄(Ψ) := exp⋄(Φ + Ψ). ⋄ Approximating a GFBM and related processes 393

The Wick product can be extended to larger classes of random variables by density arguments (cf. [2, 6, 20]). For a general introduction to the Wick product, we refer to the monographs by Kuo and Holden et al. [11, 14] and Hu and Yan [13]. Note that the Wick product is not a pointwise operation. If we suppose that Φ (0, σ), then we have, by 0 1 ∼ N deﬁnition, Φ⋄ =1, Φ⋄ =Φ and the recursion

n+1 n Φ⋄ =Φ⋄ Φ. ⋄ Observe that it holds that

d d 1 2 exp⋄(xΦ) = exp xΦ E[ xΦ ] dx dx − 2 | | x=0 x=0 1 = (Φ xσ2)exp xΦ E[ xΦ 2] =Φ. − − 2 | | x=0 Suppose we have k k d Φ⋄ = exp⋄(wΦ) dwk w=0 for all positive integers k n. Then, with z = w + x, dz = dz =1, we get ≤ dw dx n (n+1) d d Φ⋄ = exp⋄(wΦ) exp⋄(xΦ) dwn ⋄ dx w=0 x=0 dn d dn+1 = exp⋄((w + x)Φ) = exp⋄(zΦ) . dwn dx dzn+1 w=0,x=0 z=0 We now obtain, by diﬀerentiation and the Leibniz rule, the following Wick recursion formula:

n n+1 d 2 1 2 Φ⋄ = (Φ wσ )exp wΦ E[ wΦ ] dwn − − 2 | | w=0 dn 1 = (Φ wσ2) exp wΦ E[ xΦ 2] − dwn − 2 | | w=0 (6) dn 1 1 + n( σ2) − exp wΦ E[ wΦ 2] − dwn 1 − 2 | | − w=0 n 2 n 1 = ΦΦ⋄ nσ Φ⋄ − . − Deﬁne the Hermite polynomial of degree n N with parameter σ2 as ∈ 2 n 2 n 2 n x d x h 2 (x) := ( σ ) exp exp − . σ − 2σ2 dxn 2σ2 394 C. Bender and P. Parczewski

The series expansion

∞ 1 2 1 n exp x σ = h 2 (x) (7) − 2 n! σ n=0 0 1 then holds true. The ﬁrst Hermite polynomials are hσ2 (x) = 1, hσ2 (x)= x. By the Leibniz rule, we obtain the Hermite recursion formula

n+1 n 2 n 1 h 2 (x)= xh 2 (x) nσ h 2− (x). (8) σ σ − σ By the equivalent ﬁrst terms and recursions (6) and (8), we can conclude that for any Gaussian random variable Φ (0, σ) and all n N, we have ∼ N ∈ n n Φ⋄ = hσ2 (Φ). (9)

By (7), we additionally have

∞ 1 n exp⋄(Φ) = Φ⋄ . (10) n! n=0 The fractional Wick–Itôintegral, introduced by Duncan et al. [6], is an extension of the Itôintegral beyond the semimartingale framework. There are several approaches to the fractional Wick–Itôintegral. Essentially, these approaches are via white noise theory, as in Elliott and von der Hoek [7], and Hu and Øksendal [12], by Malliavin calculus in Alòs et al. [1], or by an S-transform approach in Bender [2]. In contrast to the forward integral, the fractional Wick–Itôintegral has zero mean in general. This is the crucial property for an additive noise. The Wick–Itôintegral is based on the Wick product. For a sufficiently good process (Xs)s [0,t], the fractional Wick–Itôintegral with respect to fractional Brownian H ∈ motion (Bs )[0,t] can be easily defined by Wick–Riemann sums (cf. Duncan et al. [6] or Mishura [15], Theorem 2.3.10). If we suppose that π = 0= t

H k H k 1 H H k d(Bt )⋄ = k(Bt )⋄ − d⋄Bt , (B0 )⋄ = 1 k=0 . (11) { } For the Wick exponential

H ∞ 1 H k exp⋄(B )= (B )⋄ , (12) t k! t k=0 Approximating a GFBM and related processes 395 we obtain, by summing up the identity (11), the fractional Dol´eans–Dade equation,

H dSt = St d⋄Bt , S0 =1. (13)

a ∞ k k For any analytic function F (x)= k=0 k! x , we deﬁne the Wick version as

∞ ak k F ⋄(Φ) = Φ⋄ . k! k=0 From the recursive system of SDEs (11), we obtain SDEs for other Wick analytic functionals of a fractional Brownian motion

H ∞ ak H k F ⋄(B )= (B )⋄ . t k! t k=0 Recall the linear system of SDEs (1),

H dXt = (A1Xt + A2Yt) d⋄Bt , X0 = x0, H dYt = (B1Xt + B2Yt) d⋄Bt , Y0 = y0. The coeﬃcients of the solution,

ak H k bk H k X = (B )⋄ , Y = (B )⋄ , (14) t k! t t k! t k=0 k=0 can be obtained recursively via (11) to be

a0 = x0, b0 = y0, ak = A1ak 1 + A2bk 1, bk = B1ak 1 + B2bk 1. − − − − Note that it holds that a , b Ck for a C R . This is according to the recursive | k| | k|≤ ∈ + derivation of the coeﬃcients and it ensures that the Wick analytic functionals Xt and Yt are square-integrable (cf. the proof of Proposition 6).

3. The approximation results

Here, we present the approximating sequences and discuss the main results. More precisely, we introduce the Donsker-type approximation of a fractional Brownian motion and the discrete Wick product, and obtain Wick diﬀerence equations, which correspond to the SDEs. We shall work with a fractional Brownian motion on the interval [0, 1], but all results extend to any compact interval [0,T ]. We ﬁrst consider the following kernel representation of a fractional Brownian motion on the interval [0, 1], based on works by Molchan and Golosov [16, 17],

t H Bt = zH (t,s) dBs. (15) 0 396 C. Bender and P. Parczewski

1 For H> 2 , the deterministic kernel takes the form

t 1 1/2 H H 1/2 H 3/2 zH (t,s)= 1 t s cH H s − u − (u s) − du (16) { ≥ } − 2 − s with the constant 2HΓ(3/2 H) c = − , H Γ(H +1/2)Γ(2 2H) − where Γ is the Gamma function (Norros et al. [18] or Nualart [20], Section 5.1.3). In 1 order to simplify the notation, we think of H ( 2 , 1) as ﬁxed from now on and omit the subscript H in the notation of the kernel. For∈ an introduction to some elementary properties of fractional Brownian motion, we refer to Nualart [20], Chapter 5, Mishura [15] or Biagini et al. [4]. We apply Sottinen’s approximation of a fractional Brownian motion by disturbed binary random walks. Suppose (Ω, , P ) is a probability space and, for all n N and i =1,...,n, F ∈ we have independent and identically distributed symmetric Bernoulli random variables n n n ξi :Ω 1, 1 with P (ξi =1)= P (ξi = 1). By Donsker’s theorem, the sequence of → {− } (n) 1 nt (n) − ⌊ ⌋ random walks Bt = √n i=1 ξi converges weakly to a standard Brownian motion B = (B ) [5], Theorem 16.1. The idea of Sottinen [21] is to combine these random t t [0,1] walks with∈ a pointwise approximation of the kernel in representation (15). Deﬁne the pointwise approximation of z(t,s) as

s nt z(n)(t,s) := n z ⌊ ⌋,u du. s 1/n n − The sequence of binary random walks

nt t ⌊ ⌋ i/n nt 1 BH,n := z(n)(t,s) dB(n) = n z ⌊ ⌋,s ds ξ(n) t s n √n i 0 i=1 (i 1)/n − H then converges weakly to a fractional Brownian motion (Bt )t [0,1] in the Skorokhod space D([0, 1], R)[21], Theorem 1. ∈ A major advantage of the binary random walks is that we can avoid the diﬃcult Wick product for random variables with continuous distributions. We approximate this operator on the binary random walks by discrete Wick products. N n n n For any n , let (ξ1 , ξ2 ,...,ξn ) be the n-tuple of independent and identically dis- ∈ H,n tributed symmetric Bernoulli random variables for the binary random walk Bt . The discrete Wick product is deﬁned as

ξn, if A B = , n n i ∩ ∅ ξi n ξi := i A B ⋄  ∈ ∪ i A i B 0, otherwise, ∈ ∈   Approximating a GFBM and related processes 397 where A, B 1,...,n . We denote by ⊆{ } := σ(ξn, ξn,...,ξn) Fn 1 2 n the σ-ﬁeld generated by the Bernoulli variables. Deﬁne

n n ΞA := ξi . i A ∈ n 2 Clearly, the family of functions ΞA : A 1,...,n is an orthonormal set in L (Ω, n, P ). { ⊆{ 2}} F Since its cardinality is equal to the dimension of L (Ω, n, P ), it constitutes a basis. Thus, every X L2(Ω, , P ) has a unique expansion, calledF the Walsh decomposition, ∈ Fn n n X = xAΞA, A 1,...,n ⊆{ } n R where xA . The Walsh decomposition can be regarded as a discrete version of the ∈ n n chaos expansion. By algebraic rules, one obtains, for X = A 1,...,n xAΞA and Y = n n ⊆{ } B 1,...,n yBΞB , ⊆{ } X Y = xn yn Ξn . ⋄n A B C C 1,...,n A B=C ⊆{ } A∪B= ∩ ∅ Furthermore, the L2-inner product can be computed in terms of the Walsh decomposition as n n E[XY ]= xAyA. (17) A 1,...,n ⊆{ } There exists an analogous formula for the Wick product on the white noise space via chaos expansions that justiﬁes the analogy between the discrete and ordinary Wick calculus (cf. Kuo [14]). For more information on the discrete Wick product, we refer to Holden et al. [10]. More generally, the introduction of a discrete Wiener chaos depends on the class n n n of discrete random variables (ξ1 , ξ2 ,...,ξn ). We refer to Gzyl [9] for a survey of other discrete Wiener chaos approaches. The representation

nt ⌊ ⌋ i/n nt BH,n = bn ξn with bn := √n z ⌊ ⌋,s ds t t,i i t,i n i=1 (i 1)/n − is the Walsh decomposition for the binary random walk approximating BH in L2(Ω, , P ). Note that bn = bn . Thus, we can consider BH,n = BH,n as a pro- Fn t,i nt /n,i t nt /n cess in discrete time. We can now⌊ ⌋ state our ﬁrst convergence result. ⌊ ⌋

Theorem 1. Suppose that: 398 C. Bender and P. Parczewski

1. limn an,k = ak exists for all k N; →∞ ∈ 2. there exists a C R such that a Ck for all n, k N. ∈ + | n,k|≤ ∈ n an,k H,n nk The sequence of processes k=0 k! (B )⋄ then converges weakly to the Wick power ak H k series ∞ (B )⋄ in the Skorokhod space D([0, 1], R). k=0 k! The proof is given in Sections 5 and 6. Consider now the following recursive system of Wick diﬀerence equations:

k,n k,n k 1,n H,n H,n 0,n k,n Ul = Ul 1 + kUl −1 n (Bl/n B(l 1)/n),Ul =1,U0 =0, (18) − − ⋄ − − for all l =1,...,n and k N. This is the discrete counterpart of the recursive system of ∈ 0,n H,n n0 1,n H,n n1 SDEs in (11). We observe that U =1=(B )⋄ and U = (B )⋄ , but

2,n H,n H,n H,n H,n H,n n2 U =2B B = B B = (B )⋄ . 2 1/n ⋄n 2/n 2/n ⋄n 2/n 2/n Thus, in contrast to the continuous case in (11), the discrete Wick powers are not the solutions for (18) if k 2. ≥ However, we can prove a variant of Theorem 1, based on the system of recursive Wick diﬀerence equations, whose proof will also be given in Sections 5 and 6.

k,n k,n Theorem 2. Under the assumptions of Theorem 1, deﬁne Ut := U nt as the piecewise constant interpolation of (18). ⌊ ⌋ n an,k k,n The sequence of processes k=0 k! U then converges weakly to the Wick power a ∞ k H k R series k=0 k! (B )⋄ in the Skorokhod space D([0, 1], ). Example 1 (Wick powers of a fractional Brownian motion). For an,k = l!1 k=l , { }

H,n nl d H l (B )⋄ (B )⋄ , −→ l,n d H l U (B )⋄ . −→

Example 2 (Geometric fractional Brownian motion). For an,k = ak = 1, we have

nt ⌊ ⌋ n H,n 1 H,n nk d H exp⋄ (B ) := (B )⋄ exp⋄(B ), t k! t → k=0 n n 1 k,n d H S := U exp⋄(B ). k! −→ k=0 Observe that by summing up the recursive system of Wick diﬀerence equations (18), we obtain n n n H,n H,n n Sl = Sl 1 + Sl 1 n (Bl/n B(l 1)/n), S0 =1, (19) − − ⋄ − − Approximating a GFBM and related processes 399

n n for l =1,...,n, where Sl = Sl/n. Hence, the piecewise constant interpolation of (19) converges weakly to the solution of the fractional Dol´eans–Dade equation (13). The reasoning of the previous example can be generalized as follows.

R n n Theorem 3 (Linear SDE with drift). Suppose µ,s0 , σ> 0. Then St := S nt , where Sn is the solution of the Wick diﬀerence equation ∈ ⌊ ⌋ n µ n n H,n H,n n Sl = 1+ Sl 1 + σSl 1 n (Bl/n B(l 1)/n), S0 = s0, l =1, . . ., n, (20) n − − ⋄ − − converges weakly to the solution of the linear SDE with drift

H dSt = µSt dt + σSt d⋄Bt , S0 = s0, (21) in the Skorokhod space D([0, 1], R).

Proof. First, observe that for σ σ> 0 and a = a σk , we obtain, by Theorem 2, n → n,k n,k n that n ∞ n an,k k k,n d ak H k V := σ U (σB )⋄ . k! n −→ k! k=0 k=0 With the choice an,k := 1 and σ σ := σ as n , n 1+ µ/n → → ∞

n n we observe by (18) that Vl := Vl/n satisﬁes

n n σ n H,n H,n n Vl = Vl 1 + Vl 1 n (Bl/n B(l 1)/n), V0 =1, l =1, . . ., n. − 1+ µ/n − ⋄ − − n n n Consider now the piecewise constant function (Wt )t [0,1] determined by Wt := W nt and ∈ ⌊ ⌋ n µ n n Wl = 1+ Wl 1, W0 = s0, l =1, . . ., n. n − By this well-known Euler scheme,

n (Wt )t [0,1] s0(exp(µt))t [0,1] (22) ∈ −→ ∈ in the sup-norm on [0, 1]. The product

n n µ n n σ n H,n H,n µ n Vl Wl = 1+ Vl 1Wl 1 + Vl 1 n (Bl/n B(l 1)/n) 1+ Wl 1 n − − 1+ µ/n − ⋄ − n − − 400 C. Bender and P. Parczewski

µ n n n n H,n H,n = 1+ Vl 1Wl 1 + σVl 1Wl 1 n (Bl/n B(l 1)/n), n − − − − ⋄ − − n n V0 W0 = s0, l =1, . . ., n,

n n n satisﬁes the Wick diﬀerence equation (20) for Sl = Vl Wl . The multiplication by the deterministic function s0 exp(µt) is continuous on the Skorokhod space. Thus, with (22) and Billingsley [5], Theorem 4.1, we obtain

n n n d H (St )t [0,1] = (Vt Wt )t [0,1] s0(exp(µt)exp⋄(σBt ))t [0,1] ∈ ∈ −→ ∈ R H in the Skorokhod space D([0, 1], ). As s0 exp(µt)exp⋄(σBt ) solves the SDE (21) (cf. Mishura [15], Theorem 3.3.2), the proof is complete.

Remark 1. Theorem 3 holds with additional approximations (σ ,µ ) (σ, µ). n n → Remark 2. Theorem 3 was conjectured by Bender and Elliott [3] in their study of the discrete Wick-fractional Black–Scholes market. They deduced an arbitrage in this model for suﬃciently large n. Although the arbitrage or no-arbitrage property is not preserved by weak convergence, this model showed that it is even possible to obtain arbitrage in this simple discrete Wick fractional market model. In a recent work [23], Valkeila shows that an alternative approximation to the exponential of a fractional Brownian motion by a superposition of some independent renewal reward processes leads to an arbitrage-free and complete model. We refer to Gaigalas and Kaj [8] for a general limit discussion for these superposition processes.

Theorem 4 (Linear system of SDEs). The piecewise constant interpolation

n n T n n T (Xt , Yt ) := (X nt , Y nt ) ⌊ ⌋ ⌊ ⌋ for the solution of the linear system of Wick diﬀerence equations

n n n n H,n H,n n Xl = Xl 1 + (A1Xl 1 + A2Yl 1) n (Bl/n B(l 1)/n), X0 = x0, l =1, . . ., n, − − − ⋄ − − n n n n H,n H,n n Yl = Yl 1 + (B1Xl 1 + B2Yl 1) n (Bl/n B(l 1)/n), Y0 = y0, l =1, . . ., n, − − − ⋄ − − converges weakly to the solution (X, Y )T of the corresponding linear system of SDEs (1) in the Skorokhod space D([0, 1], R)2.

Proof. Analogously to (14), we obtain, by the recursive system of Wick difference equations for U k,n in (18), the coefficients for the solution of the systems of difference equations

∞ a ∞ b Xn = k U k,n, Y n = k U k,n, l k! l l k! l k=0 k=0 Approximating a GFBM and related processes 401 recursively by

a0 = x0, b0 = y0, ak = A1ak 1 + A2bk 1, bk = B1ak 1 + B2bk 1. − − − − We deﬁne the upper bound

M := 2 max A , A , B , B . AB {| 1| | 2| | 1| | 2|} Suppose that r , r R are arbitrary. By the linear system and (18), the sequence of 1 2 ∈ processes

n r a + r b r Xn + r Y n = 1 k 2 k U k,n 1 2 k! k=0 fulﬁls the conditions in Theorem 2with

r a + r b max x , y ( r + r )M k . | 1 k 2 k|≤ {| 0| | 0|} | 1| | 2| AB Thus, we obtain the weak convergence

∞ n n d r1ak + r2bk H,n k r X + r Y (B )⋄ = r X + r Y. 1 2 −→ k! 1 2 k=0 The Cram´er–Wold device (Billingsley [5], Theorem 7.7) can now be used to complete the proof.

Remark 3. Theorem 4 can be extended to higher-dimensional linear cases. It also holds for an additional approximation of the coeﬃcients A A and B B for n . n,i → i n,i → i → ∞ Example 3 (Wick-sine and Wick-cosine). The piecewise constant interpolation of

n n n H,n H,n n Xl = Xl 1 + Yl 1 n (Bl/n B(l 1)/n), X0 =0, l =1, . . ., n, − − ⋄ − − n n n H,n H,n n Yl = Yl 1 Xl 1 n (Bl/n B(l 1)/n), Y0 =1, l =1, . . ., n, − − − ⋄ − − converges weakly to the solution of the linear system

H dXt = Yt d⋄Bt , X0 =0, H dY = X d⋄B , Y =1, t − t t 0 H H T the process (sin⋄(Bt ), cos⋄(Bt )) . By Theorem 1, it can also be approximated by the n H,n n H,n T discrete Wick version functional (sin⋄ (Bt ), cos⋄ (Bt )) . 402 C. Bender and P. Parczewski 4. Walsh decompositions and L2-estimates

In this section, we give the Walsh decompositions for the approximating sequences and obtain some L2-estimates. A key to the approximation results will be the convergence of 2 H,n 2 the L -norms of the discrete Wick powers of Bt to the corresponding L -norms of the H Wick powers of Bt . H,n nt n n Recall the Walsh decomposition Bt = ⌊i=1⌋ bt,iξi . Deﬁne

n n n n n n n bt,A := bt,i, ΞA := ξi , dl,i := bl/n,i b(l 1)/n,i − − i A i A ∈ ∈ n n n for l =1,...,n. Note that di,i = bi/n,i, dl,i =0, for i>l and that the increment has the representation

l H,n H,n n n Bl/n B(l 1)/n = dl,iξi . − − i=1 Recall the recursive system of Wick diﬀerence equations,

k,n k,n k 1,n H,n H,n 0,n k,n Ul = Ul 1 + kUl −1 n (Bl/n B(l 1)/n),Ul =1,U0 =0, (23) − − ⋄ − − for l =1,...,n and k N. ∈ Proposition 1. For all n, k N and l =0,...,n, we have the Walsh decompositions ∈ 1 U k,n = dn Ξn , (24) k! l m(p),p C C 1,...,l m:C 1,...,l p C ⊆{ } →{ } ∈ C =k injective | |

1 H,n nk n n (B )⋄ = b Ξ , (25) k! l/n l/n,C C C 1,...,l ⊆{ } C =k | |

1 H,n nk 1 k,n n n (B )⋄ U = d Ξ . (26) k! l/n − k! l m(p),p C C 1,...,l m:C 1,...,l p C ⊆{ } →{ } ∈ C =k not injective | | Proof. We use the conventions that an empty sum is zero, an empty product is one and that there exists exactly one map from the empty set to an arbitrary set. For these reasons, the formulas hold for k =0 or l = 0. We prove (24) by induction as follows. For all l =0,...,n and all k N, it is obvious that U 0,n = 1 and U k,n = 0, as in formula (24). ∈ l 0 Suppose the formula is proved for all positive integers less than or equal to a certain k and all l =0,...,n. Furthermore, for k + 1, suppose the formula is proved for all positive integers less than or equal to a certain l. For k +1 and l +1, we compute, by the diﬀerence Approximating a GFBM and related processes 403 equation (23) and the induction hypothesis,

l+1 U k+1,n U k+1,n = (k + 1)k! dn Ξn dn ξn l+1 − l m(p),p C ⋄n l+1,i i C 1,...,l m:C 1,...,l p C i=1 ⊆{ } →{ } ∈ C =k injective | | n n n = (k + 1)! dm(p),pdl+1,iΞC i (27) ∪{ } C 1,...,l m:C 1,...,l p C ⊆{ } →{ } ∈ i 1,...,l+1 injective ∈{ } C =k,i/C | | ∈ n n = (k + 1)! dm(p),pΞC′ . C′ 1,...,l+1 m′:C′ 1,...,l+1 p C′ ⊆{′ } →{ } ∈ C =k+1 injective, q:m(q)=l+1 | | ∃ Note that d = 0 for all p 1 m. Thus, by the induction hypothesis, m,p − ≥ k+1,n n n Ul = (k + 1)! dm(p),pΞC . (28) C 1,...,l+1 m:C 1,...,l+1 p C ⊆{ } →{ } ∈ C =k+1 injective, q:m(q)l. H,n We now compute the kth Wick power of (Bl/n ) as follows:

nk n ⋄ l k k n n n n bl/n,iξi = bl/n,ij ξij i=1 i1,i2,...,ik =1 j=1 j=1 pairwise distinct

n n n n = k! bl/n,i ξi = k!bl/n,CΞC . C 1,...,l i C i C C 1,...,l ⊆{ } ∈ ∈ ⊆{ } C =k C =k | | | |

H,n nk In particular, (Bl/n )⋄ = 0 for all k>l. This yields (25). The telescoping sum yields

l l l n n n n n dm(p),p = dm(p),p = (bm(p)/n,p b(m(p) 1)/n,p)= bl/n,p − − m(p)=1 m(p)=p m(p)=p and thus we get

l n n n n dm(p),p = dm(p),p = bl/n,p = bl/n,C. (29) m:C 1,...,l p C p Cm(p)=1 p C →{ } ∈ ∈ ∈ Equation (26) is, thus, implied by (24) and (25). 404 C. Bender and P. Parczewski

In the following propositions, we obtain some elementary estimates for the L2-norm of H,n discrete Wick powers of Bt .

Proposition 2. For all t [0, 1] and i 1,..., nt , ∈ ∈{ ⌊ ⌋} n (1 H) b 2c n− − . t,i ≤ H Proof. We estimate

i/n nt /n n 1/2 1 1/2 H ⌊ ⌋ H 1/2 H 3/2 bt,i = n cH H s − u − (u s) − du ds − 2 (i 1)/n s − − i/n H 1/2 H 1/2 1/2 1 1/2 H nt − 1 nt − n cH H s − ⌊ ⌋ ⌊ ⌋ s ds ≤ − 2 (i 1)/n n H 1/2 n − − − 3/2 H 3/2 H 2(H 1/2) 1 i − i 1 − nt − n1/2c − ⌊ ⌋ . ≤ H 3/2 H n − n n − 1 3/2 H 3/2 H 3/2 H Since t 1, 3/2 H 2 and x − y − x y − , the assertion follows. ≤ − ≤ | | − | | ≤ | − | Remark 4. Observe that

nt nt ⌊ ⌋ ⌊ ⌋ H,n H,n n n n n n n E[Bt Bs ]= E bt,i1 bs,i2 ξi1 ξi2 = (bt,ibs,i). (30) i1,i2=1 i=1 By Nieminen [19], we thus get, for any s,t [0, 1], the following convergence: ∈ nt ⌊ ⌋ i/n i/n H,n H,n nt ns E[Bt Bs ] = n z ⌊ ⌋,u du z ⌊ ⌋,u du (i 1)/n n (i 1)/n n i=1 − − (31) 1 z(t,u)z(s,u) du = E[BH BH ]. −→ t s 0 Moreover, we have, by the Cauchy–Schwarz inequality, the upper bound

nt 2 ⌊ ⌋ i/n nt ns E[(BH,n BH,n)2]= √n z ⌊ ⌋,u du z ⌊ ⌋,u du t − s n − n i=1 (i 1)/n − nt 2 ⌊ ⌋ i/n nt ns z ⌊ ⌋,u z ⌊ ⌋,u du (32) ≤ n − n i=1 (i 1)/n − nt ns 2H = ⌊ ⌋ ⌊ ⌋ . n − n Approximating a GFBM and related processes 405

Proposition 3. For all t s in [0, 1] and all N N such that ns N, we have ≥ ∈ ⌊ ⌋≥ 0 E[(BH,n)2]N + E[(BH,n)2]N 2E[BH,nBH,n]N ≤ t s − t s

1 H,n nN H,n nN 2 E[((B )⋄ (B )⋄ ) ] (33) − N! t − s

2 2 2H(N 1) (2 2H) 2c N t − n− − . ≤ H In particular,

H,n nN H,n nN 2 H N H N 2 lim E[((Bt )⋄ (Bs )⋄ ) ]= E[((Bt )⋄ (Bs )⋄ ) ]. n →∞ − −

Proof. As N ns , we get, making use of Proposition 1, (17) and (30) in Remark 4, ≤⌊ ⌋

1 H,n nN H,n nN E[(B )⋄ (B )⋄ ] N! t s 1 = E N! bn Ξn N! bn Ξn N! t,C C s,C C C 1,..., nt C 1,..., ns ⊆{ ⌊ ⌋} ⊆{ ⌊ ⌋} C =N C =N | | | | n n = N! bt,Cbs,C (34) C 1,..., nt ⊆{ ⌊ ⌋} C =N | | nt N nt N ⌊ ⌋ ⌊ ⌋ = (bn bn ) (bn bn ) t,ij s,ij − t,ij s,ij i1,...,i =1 j=1 i1,...,i =1 j=1 N N k,l : ik =il ∃ nt N ⌊ ⌋ = E[BH,nBH,n]N (bn bn ). t s − t,ij s,ij i1,...,i =1 j=1 N k,l : ik =il ∃ Thus, we have

E[(BH,n)2]N + E[(BH,n)2]N 2E[BH,nBH,n]N t s − t s

1 H,n nN H,n nN 2 E[((B )⋄ (B )⋄ ) ] − N! t − s (35) nt N N N ⌊ ⌋ = (bn )2 + (bn )2 2 (bn bn ) t,ij s,ij − t,ij s,ij i1,...,i =1j=1 j=1 j=1 N k,l : ik =il ∃ 406 C. Bender and P. Parczewski

nt N N 2 ⌊ ⌋ = (bn ) (bn ) 0. t,ij − s,ij ≥ i1,...,i =1j=1 j=1 N k,l : ik =il ∃ Hence, the left-hand side of the inequality in (33) follows. By Proposition 2, (30) and nt (32) in Remark 4, as well as ⌊ ⌋ t, we obtain | n |≤ nt N N 2 ⌊ ⌋ (bn ) (bn ) t,ij − s,ij i1,...,i =1j=1 j=1 N k,l : ik =il ∃ nt N 2 nt N 1 2 ⌊ ⌋ ⌊ ⌋ n N n 2 − n (bt,i ) max(bt,i) (bt,i ) (36) ≤ j ≤ 2 i j i1,...,i =1j=1 i1,...,i −1=1 j=1 N N k,l : ik =il ∃ 2 2 H,n 2 N 1 (2 2H) 2 2 2H(N 1) (2 2H) 2c N E[(B ) ] − n− − 2c N t − n− − 0 ≤ H t ≤ H → H for n . The representation of Wick powers of Bt by Hermite polynomials, as in (9), their→ orthonormality ∞ (cf. Kuo [14], page 355) and the polarization identity collectively H N H N H H N yield E[(Bt )⋄ (Bs )⋄ ]= N!E[(Bt )(Bs )] (cf. also [20], Lemma 1.1.1). Thus, we have, by (35),

H,n nN H,n nN 2 H N H N 2 E[((B )⋄ (B )⋄ ) ] E[((B )⋄ (B )⋄ ) ] t − s − t − s = N!(E[(BH,n)2]N E[(BH )2]N + E[(BH,n)2]N E[(BH )2]N t − t s − s 2E[BH,nBH,n]N +2E[BH BH ]N ) − t s t s nt N N 2 ⌊ ⌋ N! (bn ) (bn ) . − t,ij − s,ij i1,...,i =1j=1 j=1 N k,l : ik=il ∃ Applying the convergences (36) and (31) yields

H,n nN H,n nN 2 H N H N 2 E[((B )⋄ (B )⋄ ) ] E[((B )⋄ (B )⋄ ) ] 0. t − s − t − s → nt Remark 5. In particular, we obtain, by (34), (32) and ⌊ ⌋ t, | n |≤ 2 n 2 1 H,n nN 2 (b ) = E[((B )⋄ ) ] t,C N! t C 1,..., nt ⊆{ ⌊ ⌋} C =N | | (37) 1 1 nt 2HN 1 E[(BH,n)2]N ⌊ ⌋ t2HN . ≤ N! t ≤ N! n ≤ N! Approximating a GFBM and related processes 407

The next proposition estimates the diﬀerence between the approximating sequences in Theorems 1 and 2.

Proposition 4. Under the assumptions of Theorem 1, there exists a constant K > 0 such that for all t [0, 1], n 1 and k N, ∈ ≥ ∈

n n 2 an,k H,n nk an,k k,n 1 2H E (Bt )⋄ Ut Kn − (38) k! − k! ≤ k=0 k=0 for the approximating processes in Theorems 1 and 2.

n n n i/n r r 1 Proof. Recall that d = b b = √n (z( ,s) z( − ,s)) ds.By(32) r,i r/n,i − (r 1)/n,i (i 1)/n n − n in Remark 4, we obtain − − r 2H n 2 n 2 r r 1 2H (d ) (d ) − = n− . r,i ≤ r,i ≤ n − n i=1 n H Thus, we have d n− for all i,n,r 1. Hence, we obtain, as the sum in (29) tele- r,i ≤ ≥ scopes, for C 2, | |≥

n n dm(l),l = dm(l),l m:C 1,..., nt l C m:C 1,..., nt l C →{ ⌊ ⌋} ∈ →{ ⌊ ⌋} ∈ not injective u,v C : m(u)=m(v) ∃ ∈

n n = dm(l),l dm(v),u(39) u C m:C u 1,..., nt l C u v C u ∈ \{ }→{ ⌊ ⌋} ∈ \{ } ∈\{ } n n max dr,i( C 1) dm(l),l ≤ i,r | |− C′ C m:C′ 1,..., nt l C′ ′ ⊂ ∈ C = C 1 →{ ⌊ ⌋} | | | |− H n n− ( C 1) b ′ . ≤ | |− t,C C′ C ′ ⊂ C = C 1 | | | |− By (26), (39), (37) and since ( nt (k 1)) n, we obtain, for k 1, ⌊ ⌋− − ≤ ≥ 2 1 H,n nk k,n E ((B )⋄ U ) k! t − t 2 H n n− (k 1) b ′ ≤ − t,C C 1,..., nt C′ C ⊆{ ⌊ ⌋} ′ ⊂ C =k C = C 1 | | | | | |− 408 C. Bender and P. Parczewski

2H 2 n 2 n− (k 1) (k 1) (b ′ ) ≤ − − t,C C 1,..., nt C′ C ⊆{ ⌊ ⌋} ′ ⊂ C =k C = C 1 | | | | | |− 3 2H 3 n 2 (k 1) 2H(k 1) 1 2H n− (k 1) ( nt (k 1)) (b ′ ) − t − n − . ≤ − ⌊ ⌋− − t,C ≤ (k 1)! C′ 1,..., nt − ⊆{′⌊ ⌋} C =k 1 | | −

k H,n nk k,n Since a C in Theorems 1 and 2, and ((B )⋄ U ) are zero for k =0, 1 and n,k ≤ t − t orthogonal for diﬀerent k by Proposition 1 and (17), we get n n 2 an,k H,n nk an,k k,n E (Bt )⋄ Ut k! − k! k=0 k=0 nt 2 ⌊ ⌋ ∞ 3 an,k H,n nk k,n 2k (k 1) 2H(k 1) 1 2H = E ((B )⋄ U ) C − t − n − . k! t − t ≤ (k 1)! k=2 k=2 − As the series on the right-hand side converges uniformly in t [0, 1], the assertion fol- ∈ lows.

5. Convergence of the ﬁnite-dimensional distributions

We first prove that Theorems 1 and 2 hold with weak convergence replaced by convergence of the finite-dimensional distributions. To this end, we first approximate the Wick powers H of Bt by induction. We then combine these convergence results to approximate the Wick a H ∞ k H k analytic functionals F ⋄(Bt )= k=0 k! (Bt )⋄ . Finally, we conclude that convergence in finite dimensions holds in Theorem 2. H N N H We observed in Section 2 that (Bt )⋄ = h t 2H (Bt ) and that the Hermite recursion formula | | H (N+1) H H N 2H H (N 1) (B )⋄ = (B )(B )⋄ t N(B )⋄ − (40) t t t − | | t holds. For the discrete Wick powers of the discrete variables, we now obtain a discrete variant of (40).

Proposition 5 (Discrete Hermite recursion). For all N 1 and t [0, 1], ≥ ∈

H,n n(N+1) H,n H,n nN H,n 2 H,n n(N 1) H,n (B )⋄ = B (B )⋄ NE[(B ) ](B )⋄ − + R(B ,N), (41) t t t − t t t with remainder H,n n n n 2 R(Bt ,N)= N! bt,CΞC (bt,i) (42) C 1,... nt i C ⊆{⌊ ⌋} ∈ C =N 1 | | − Approximating a GFBM and related processes 409 and H,n 2 4 3 (4 4H) E[(R(B ,N)) ] 16c N!N n− − . (43) t ≤ H In particular, we will use the fact that the discrete Hermite recursion (41) converges weakly to Hermite recursion (40) for n . → ∞ Proof. By Proposition 1, we get

nt ⌊ ⌋ H,n n(N+1) H,n H,n nN n n n n (B )⋄ = B (B )⋄ = b ξ N!b Ξ t t ⋄n t t,i i ⋄n t,A A i=1 A 1,..., nt ⊆{ ⌊ ⌋} A =N | | (44) H,n H,n nN n n n n = B (B )⋄ N!b b Ξ ξ . t t − t,i t,A A i A 1,..., nt i A ⊆{ ⌊ ⌋} ∈ A =N | | For the second term in the last line in equation (44), by (30) in Remark 4 and Proposition 1, we obtain

n n n n N!bt,ibt,AΞAξi A 1,..., nt i A ⊆{ ⌊ ⌋} ∈ A =N | | n n n n 2 n n n 2 = N! bt,A i ΞA i (bt,iξi ) = N! bt,C ΞC (bt,i) \{ } \{ } A 1,..., nt i A C 1,..., nt i/C ⊆{ ⌊ ⌋} ∈ ⊆{ ⌊ ⌋} ∈ A =N C =N 1 | | | | − nt ⌊ ⌋ = N(N 1)! bn Ξn (bn )2 (bn )2 − t,C C t,i − t,i C 1,..., nt i=1 i C ⊆{ ⌊ ⌋} ∈ C =N 1 | | −

H,n n(N 1) H,n 2 n n n 2 = N(B )⋄ − E[(B ) ] N! b Ξ (b ) , t t − t,C C t,i C 1,..., nt i C ⊆{ ⌊ ⌋} ∈ C =N 1 | | − which yields (41) and (42). Thus, thanks to Proposition 2, Remark 5 and t 1, we obtain ≤ 2 H,n 2 2 n 2 n 2 E[(R(Bt ,N)) ] = (N!) (bt,C) (bt,i) C 1,..., nt i C ⊆{ ⌊ ⌋} ∈ C =N 1 | | − 2 1 2H(N 1) 2 (2 2H) 2 (N!) t − ((N 1)4c n− − ) ≤ (N 1)! − H − 4 3 (4 4H) 16c N!N n− − . ≤ H 410 C. Bender and P. Parczewski

Theorem 5. For all N N, ∈

H,n H,n nN fd H H N (1,B ,..., (B )⋄ ) (1,B ,..., (B )⋄ ). (45) −→ fd Proof. The proof proceeds by induction on N. By Sottinen’s approximation, (1,BH,n) (1,BH ). Suppose that equation (45) is proved for some N 1. Assume that k N and−→ j R ≥ ∈ ri for j =0,...,N + 1 , i =1,...,k and t1,t2,...,tk [0, 1] are chosen arbitrarily. By ∈ H,n 2 2H ∈ the pointwise convergence E[(Bt ) ] t and the generalized continuous mapping theorem (Billingsley [5], Theorem 5.5),→ the | | induction hypothesis implies that

N k k l H,n nl N+1 H,n H,n nN H,n 2 H,n n(N 1) r (B )⋄ + r (B (B )⋄ NE[(B ) ](B )⋄ − ) j tj j tj tj − tj tj j=1 j=1 l=0 N k k d l H l N+1 H H N 2H H (N 1) r (B )⋄ + r (B (B )⋄ N t (B )⋄ − ). −→ j tj j tj tj − | j | tj j=1 j=1 l=0 1 H,n 2 Since H> 2 , (43) yields R(Bt ,N) 0 in L (Ω, P ). Thus, by Slutsky’s theorem [5], Theorem 4.1, and the Hermite recursions−→ (40) and (41), we obtain

N+1 k N+1 k l H,n nl d l H l r (B )⋄ r (B )⋄ . j tj −→ j tj j=1 j=1 l=0 l=0 By the Cram´er–Wold device (Billingsley [5], Theorem 7.7), we have

H,n H,n nN+1 fd H H N+1 (1,B ,..., (B )⋄ ) (1,B ,..., (B )⋄ ) −→ and the induction is complete.

Proposition 6. In the context of Theorem 1, convergence holds in ﬁnite-dimensional distributions.

Proof. By Billingsley [5], Theorem 4.2, it suﬃces to show that the following three conditions hold:

m m an,k H,n nk fd ak H k m N, (B )⋄ (B )⋄ as n ; (46) ∀ ∈ k! t −→ k! t → ∞ k=0 k=0 n m an,k H,n nk an,k H,n nk t [0, 1], lim lim sup E (Bt )⋄ (Bt )⋄ 1 =0; (47) ∀ ∈ m n k! − k! ∧ →∞ →∞ k=0 k=0 m ak H k fd ∞ ak H k (B )⋄ (B )⋄ as m . (48) k! t −→ k! t → ∞ k=0 k=0 Approximating a GFBM and related processes 411

Condition (46) follows directly from Theorem 5 and the generalized continuous mapping theorem ([5], Theorem 5.5). For the second condition, we compute

n m 2 an,k H,n k an,k H,n nk E (Bt )⋄ (Bt )⋄ k! − k! k=0 k=0 n 2 an,k H,n nk = E (Bt )⋄ k! k=m+1 n 2 n k 2 an,k H,n nk 2 C 2Hk = E[((B )⋄ ) ] k!t , k! t ≤ k! k=m+1 k=m+1 applying the estimate of Remark 5. Here, we used the fact that discrete Wick powers of 2 n C k diﬀerent orders are orthogonal. Thus, we even obtain limm lim supn k=m+1 k! = 0 and, for all t [0, 1], a stronger result than condition (47→∞), →∞ ∈ n m 2 an,k H,n k an,k H,n nk lim lim sup E (B )⋄ (B )⋄ =0. m t t →∞ n k! − k! →∞ k=0 k=0 By the orthogonality of the diﬀerent Wick powers, we have

2 2 2k ∞ ak H k ∞ ak H k 2 ∞ C 2Hk E (Bt )⋄ = E[((Bt )⋄ ) ] t 0 k! k! ≤ k! → k=m+1 k=m+1 k=m+1 for m , which implies that condition (48) even holds in L2(Ω, P ). → ∞ In view of Proposition 4 and Slutsky’s theorem, the previous proposition also implies the following.

Proposition 7. In the context of Theorem 2, convergence holds in ﬁnite-dimensional distributions.

6. Tightness

We now show the tightness of the sequences in Theorems 1 and 2 by the following criterion, which is a variant of Theorem 15.6 in Billingsley [5].

Theorem 6. Suppose that, for the random elements Y n in the Skorokhod space ak H k R ∞ R D([0, 1], ) and k=0 k! (B )⋄ in C([0, 1], ), n fd ∞ ak H k Y (B )⋄ , −→ k! k=0 412 C. Bender and P. Parczewski and for s t in [0, 1], ≤ nt ns 2H E[(Y n Y n)2] L ⌊ ⌋ ⌊ ⌋ , t − s ≤ n − n ak n ∞ H k R where L> 0 is a constant. Then Y converges weakly to k=0 k! (B )⋄ in D([0, 1], ).

Proof. Let s

E[ Y n Y n Y n Y n ] | t − s || u − t | (E[ Y n Y n 2])1/2(E[ Y n Y n 2])1/2 ≤ | t − s | | u − t | nt ns H nu nt H nu ns 2H L ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ L ⌊ ⌋ ⌊ ⌋ . ≤ n − n n − n ≤ n − n 1 If u s , we have, since nu nu and ns ns+ 1, − ≥ n ⌊ ⌋≤ −⌊ ⌋≤− nu ns 2H ⌊ ⌋ ⌊ ⌋ (2(u s))2H n − n ≤ − and thus E[ Y n Y n Y n Y n ] L22H (u s)2H . (49) | t − s || u − t | ≤ − 1 If u s< n , then we have either ns = nt or nt = nu and so the left-hand side in (49)− is zero. Thus, the inequality⌊ (49⌋) holds⌊ ⌋ for⌊ all ⌋s

For the application of this criterion to the discrete Wick powers, we need two lemmas.

Lemma 1. Let (X, , ) be a real inner product space and x := x, x the corresponding norm on X. Then, for all x, y X and N 1, ∈ ≥ 2N 2N N N+1 2(N 1) 2 x + y 2( x, y ) 2 ( x + y ) − x y . − ≤ − Proof. It holds that

1 N 2( x, y )N =2 ( x 2 + y 2 x y 2) 2 − − N 1 − 1 2 2 N N 2 2 k N k 2(N k) = N 1 ( x + y ) + ( x + y ) ( 1) − x y − 2 − k − − k=0 1 2 2 N = N 1 ( x + y ) 2 − Approximating a GFBM and related processes 413

N 1 − 2 1 N 2 2 k N k 1 2(N k 1) x y N 1 ( x + y ) ( 1) − − x y − − . − − 2 − k − − k=0 Hence, we get

x 2N + y 2N 2( x, y )N − 2N 2N 1 2 2 N = x + y N 1 ( x + y ) (50) − 2 − N 1 − 2 1 N 2 2 k N k 1 2(N k 1) + x y N 1 ( x + y ) ( 1) − − x y − − . − 2 − k − − k=0 1 N 1 N N Since ( 2 ) − k=0 k = 2, the ﬁrst line on the right-hand side of (50) can be treated as follows: 1 x 2N + y 2N ( x 2 + y 2)N − 2N 1 − (51) N 1 2N 2N 1 − N x + y 2k 2(N k) = N 1 x y − . 2 − k 2 − k=1 As N = N , we now collect the corresponding summands in sum (51) for k = N . We k N k 2 obtain, by the− mean value theorem with

Mx,y := max (λ x + (1 λ) y )=max x , y λ [0,1] − { } ∈

2 and since k(N k) N , − ≤ 4 2N 2N 2k 2(N k) 2(N k) 2k x + y x y − x − y − − 2k 2k 2(N k) 2(N k) = ( x y )( x − y − ) − − 2k 1 2(N k) 1 2 2(N 1) 2 2kM − x y 2(N k)M − − x y N M − x y . ≤ x,y − − x,y − ≤ x,y − N Analogously, we obtain, for k = 2 ,

2N 2N x + y N N 1 N N 2 1 2(N 1) 2 2 x y = ( x y ) M − N x y . 2 − 2 − ≤ 2 x,y −

1 N 1 N 1 N 1 1 Plugging these estimates into (51) and recalling that ( ) − − = (1 N−1 ), 2 k=1 k 2 − 2 we obtain N 1 2N 2N 1 − 2 2 N 1 2 2(N 1) 2 x + y ( x + y ) 1 N M − x y . (52) − 2 ≤ − 2N 1 x,y − − 414 C. Bender and P. Parczewski

For the term in the second line on the right-hand side of (50), we observe that, by the triangle inequality,

2 2 k N k 1 2(N k 1) ( x + y ) ( 1) − − x y − − − − 2k 2(N k 1) 2(N 1) ( x + y ) ( x + y ) − − = ( x + y ) − . ≤ Applying

N 1 N 1 1 − − N 1 Mx,y x + y , =2 N 1 , ≤ 2 k − 2 − k=0 and (52) to (50), we have

x 2N + y 2N 2( x, y )N − 1 2 1 2(N 1) 2 1 N + 2 ( x + y ) − x y . ≤ − 2N 1 − 2N 1 − − − By a short calculation and induction, we obtain

1 2 1 N 17 N+1 1 N + 2 1 N=3 2 + 1 N=3 < 2 . − 2N 1 − 2N 1 ≤ { } { } 2 − − Lemma 2. For all t>s in [0, 1], we have

2H 1 H,n nN H,n nN 2 N nt ns E[((B )⋄ (B )⋄ ) ] 8 ⌊ ⌋ ⌊ ⌋ . N! t − s ≤ n − n Proof. For N = 1, the inequality is fulﬁlled by (32) in Remark 4. For N > 1, we consider H,n nN the cases N > ns and ns N separately. For nt N > ns , we have (B )⋄ = ⌊ ⌋ ⌊ ⌋≥ ⌊ ⌋≥ ⌊ ⌋ s 0. Hence, Proposition 3 and Remark 5 imply that

2HN 1 H,n nN H,n nN 2 1 H,n nN 2 nt E[((B )⋄ (B )⋄ ) ]= E[((B )⋄ ) ] ⌊ ⌋ . N! t − s N! t ≤ n nt Since N 2, 2H> 1 and ⌊ ⌋ 1, we obtain ≥ n ≤ nt 2HN nt 2 1 2 ⌊ ⌋ ⌊ ⌋ = (( nt ns )+ ns )2 n ≤ n n ⌊ ⌋−⌊ ⌋ ⌊ ⌋ 1 2 1 2 (( nt ns )+ N)2 (( nt ns )(N + 1))2 ≤ n ⌊ ⌋−⌊ ⌋ ≤ n ⌊ ⌋−⌊ ⌋ nt ns 2 = (N + 1)2 ⌊ ⌋ ⌊ ⌋ . n − n Approximating a GFBM and related processes 415

Since (N + 1)2 3N for N 2 and 2H< 2, we obtain ≤ ≥ 2H 1 H,n nN H,n nN 2 N nt ns E[((B )⋄ (B )⋄ ) ] 3 ⌊ ⌋ ⌊ ⌋ N! t − s ≤ n − n for all nt N > ns . Recall that, by Proposition 3, for nt > ns N, ⌊ ⌋≥ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋≥

1 H,n nN H,n nN 2 E[((B )⋄ (B )⋄ ) ] N! t − s E[(BH,n)2]N + E[(BH,n)2]N 2E[(BH,n)(BH,n)]N . ≤ t s − t s For any n N, we can rewrite ∈ n H,n H,n n n E[(Bt )(Bs )] = bt,ibs,i i=1 Rn n n T n n T as an ordinary inner product on of the vectors (bt,1,...,bt,n) and (bs,1,...,bs,n) . Thus, the application of Lemma 1 with t,s [0, 1] gives ∈

1 H,n nN H,n nN 2 E[((B )⋄ (B )⋄ ) ] N! t − s N+1 H,n 2 1/2 H,n 2 1/2 2(N 1) H,n H,n 2 2 (E[(B ) ] + E[(B ) ] ) − E[((B ) (B )) ] ≤ t s t − s 2H 2H N+1 2(N 1) nt ns N nt ns 2 2 − ⌊ ⌋ ⌊ ⌋ 8 ⌊ ⌋ ⌊ ⌋ . ≤ n − n ≤ n − n If N > nt , then the left-hand side of the assertion vanishes. ⌊ ⌋ Remark 6. The proofs for a fractional Brownian motion on some interval [0,T ] R ⊂ follow by a straightforward modiﬁcation. As E[(BH,n)2] T 2H for t [0,T ] and t ≤ ∈ 2HN 2H nt 2H(N 1) nt ⌊ ⌋ T − ⌊ ⌋ , n ≤ n we obtain the previous lemma for t>s in [0,T ] as

2H 1 H,n nN H,n nN 2 2H N nt ns E[((B )⋄ (B )⋄ ) ] (8T ) ⌊ ⌋ ⌊ ⌋ . N! t − s ≤ n − n We are now able to prove the weak convergence to the Wick analytic functionals of a fractional Brownian motion.

Proof of Theorem 1. We apply Theorem 6. The convergence of ﬁnite-dimensional distributions was shown in Proposition 6. Let s

H,n nk H,n nk the orthogonality of ((B )⋄ (B )⋄ ) for diﬀerent k and Lemma 2, we have t − s

n n 2 an,k H,n nk an,k H,n nk E (Bt )⋄ (Bs )⋄ k! − k! k=0 k=0 n 2 n 2k 2H an,k H,n nk H,n nk 2 C k nt ns = E[((B )⋄ (B )⋄ ) ] 8 ⌊ ⌋ ⌊ ⌋ . k! t − s ≤ k! n − n k=0 k=0 2 8kC k 2 Since 0 < ∞ = exp(8C ) =: L< , we have k=0 k! ∞

n n 2 2H an,k H,n nk an,k H,n nk nt ns E (B )⋄ (B )⋄ L ⌊ ⌋ ⌊ ⌋ . (53) k! t − k! s ≤ n − n k=0 k=0

The alternative approximation, stated in Theorem 2, follows similarly, as we shall now see.

Proof of Theorem 2. Let s 0 only if i m. Thus, by m,i ≤ Proposition 1, we can write

n nt a ⌊ ⌋ n,k U k,n = a dn Ξn . k! s n,k m(l),l C k=0 k=0 C 1,..., nt m:C 1,..., ns l C ⊆{ ⌊ ⌋} →{ ⌊ ⌋} ∈ C =k injective | | Observe that, by the telescoping sum in (29), we have

dn dn m(l),l ≤ m(l),l m:C 1,..., nt l C m:C 1,..., nt l C →{ ⌊ ⌋} ∈ →{ ⌊ ⌋} ∈ injective u : m(u)> ns ∃ ⌊ ⌋ u:m(u)> ns ∃ ⌊ ⌋ = dn dn m(l),l − m(l),l m:C 1,..., nt l C m:C 1,..., ns l C →{ ⌊ ⌋} ∈ →{ ⌊ ⌋} ∈ = bn bn . t,C − s,C Thus, due to the orthogonality of U k,n U k,n for diﬀerent values of k, Proposition 1 and t − s estimate (53), we obtain n n 2 an,k k,n an,k k,n E Ut Us k! − k! k=0 k=0 Approximating a GFBM and related processes 417

nt 2 ⌊ ⌋ 2 n = an,k dm(l),l k=0 C 1,..., nt m:C 1,..., nt l C ⊆{ ⌊ ⌋} →{ ⌊ ⌋} ∈ C =k injective | | u:m(u)> ns ∃ ⌊ ⌋ nt 2H ⌊ ⌋ nt ns a2 (bn bn )2 L ⌊ ⌋ ⌊ ⌋ ≤ n,k t,C − s,C ≤ n − n k=0 C 1,..., nt ⊂{ ⌊ ⌋} C =k | | and the result follows from Proposition 7 and Theorem 6.

Acknowledgement

The authors thank Jens-Peter Kreiß for interesting discussions.

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Received January 2009 and revised June 2009