VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA35CNF 10.15559/15-VMSTA35CNF Research Article Integral representation with respect to fractional Brownian motion under a log-Hölder assumption ShalaikoTarastosha@univ.kiev.ua https://orcid.org/0000-0003-1047-3533 ShevchenkoGeorgiyzhora@univ.kiev.ua∗ Department of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska 60, 01601 Kyiv, Ukraine Corresponding author. 2015 259201523PRESTO-2015219232 692015 1392015 1492015 © 2015 The Author(s). Published by VTeX2015 Open access article under the CC BY license.

We show that if a random variable is the final value of an adapted log-Hölder continuous process, then it can be represented as a stochastic integral with respect to a fractional Brownian motion with adapted integrand. In order to establish this representation result, we extend the definition of the fractional integral.

 Fractional Brownian motion integral representation fractional integral small deviation 60G22 60H05 26A33 Introduction

This paper can be considered as a continuation of the research started in [5, 4, 7–9]. Namely, we are interested in representations of a random variable in the form of a stochastic integral ξ=∫01ψ(s)dBH(s) with respect to a fractional Brownian motion with Hurst parameter H>1/2 1/2$]]>; the integrand ψ is assumed to be adapted to the natural filtration generated by  BH on the interval [0,1] . The motivation comes from financial mathematics, where capitals of self-financing strategies are given by stochastic integrals with respect to asset pricing processes.

The main representation results reported in [5, 4, 7–9] involve the following assumption about ξ: it is the value at 1 of an adapted Hölder-continuous process. Moreover, it was shown in [4] (see Remark 2.5) that such an assumption is unavoidable in the methods used in the cited papers.

In this paper, we generalize the existing results by showing the existence of representation (1) under a weaker assumption that ξ is the value at 1 of an adapted log-Hölder continuous process. In order to establish such a representation, we extend the definition of the fractional integral introduced in [10].

The paper is organized as follows. Section 2 gives all necessary prerequisites about the fractional Brownian motion and the definition of the extended fractional integral. In Section 3, we prove the main representation result. The Appendix contains auxiliary results concerning the extended fractional integral.

 Preliminaries General conventions

Let (Ω,F,F={Ft}t∈[0,1],P) be a standard stochastic basis. The adaptedness of processes will be understood with respect to the filtration F .

Throughout the paper, the symbol C will mean a generic constant, the value of which may change from line to line. When the dependence on some parameter(s) is important, this will be indicated by superscripts. A finite random variable of no importance will be denoted by C(ω) .

 Fractional Brownian motion

Our main object of interest in this paper is a fractional Brownian motion (fBm) with Hurst index H∈(1/2,1) on (Ω,F,P) , that is, an F -adapted centered Gaussian process BH={BH(t)}t≥0 with the covariance function RH(t,s)=E[BH(t)BH(s)]=12(t2H+s2H−|t−s|2H). By the Kolmogorov–Chentsov theorem, BH has a continuous modification, so in what follows, we assume that BH is continuous. Moreover, we will need the following statement on the uniform modulus of continuity of BH (see, e.g., [3]).

For an fBm BH , we have supt,s∈[0,1]|BH(t)−BH(s)||t−s|H|log(t−s)|1/2<∞almost surely.

Small deviations of sum of squared increments of fractional Brownian motion

First, we state a small deviation estimate for sum of squares of Gaussian random variables (see, e.g., [2]).

Let {ξi}i=1,…,n be jointly Gaussian centered random variables. Then for all x such that 0<x<∑i=1nE[ξi2] , we have P ∑i=1nξi2≤x≤exp(−(x−∑i=1nE[ξi2])2∑i,j=1n(E[ξiξj])2).

We will also need the following asymptotics of the covariance of a fractional Brownian motion. Its proof is given in the Appendix.

Let H∈(1/2,1) . Set ΔBiH=BH(i+1)−BH(i) for i=0,…,n−1 . Then the following relation holds as n→∞ : ∑i,j=0n−1(E[ΔBiHΔBjH])2∼CHn,H∈(1/2,3/4),nlogn,H=3/4,n4H−2,H∈(3/4,1).

Lemmas 1 and 2 imply the following small deviation estimate for the sum of squares of fBm increments.

Let BH={BtH}t≥0 be an fBm with Hurst index H>1/2 1/2$]]>, n≥2 , and let {ΔBkH}k=0,…,n−1 be as before. For all α∈(0,1) , we have P ∑k=0n−1(ΔBkH)2≤αn≤exp{−CH(1−α)2r(n)}, where r(n)=n,H∈(1/2,3/4),n/logn,H=3/4,n4−4H,H∈(3/4,1).

Extended fractional integral

To integrate with respect to a fractional Brownian motion, we use the fractional integral introduced in [10], but the definition is modified according to our purposes. For functions f,g:[a,b]→R and α∈(0,1) , define the fractional Riemann–Liouville derivatives (in Weyl form) (Da+αf)(x)=1Γ(1−α)(f(x)(x−a)α+α∫axf(x)−f(u)(x−u)α+1du),(Db−1−αg)(x)=e−iπαΓ(α)(g(x)(b−x)1−α+(1−α)∫xbg(x)−g(u)(u−x)2−αdu). Then the fractional integral ∫abf(x)dg(x) can be defined as ∫abf(x)dg(x)=eiπα∫ab(Da+αf)(x)(Db−1−αgb−)(x)dx, provided that the last integral is finite. However, in order to have good properties of this integral (independence of α, additivity), we need to assume more than just the finiteness of the integral; see the Appendix for details.

Now we turn to the integration with respect to a fractional Brownian motion. We will restrict our exposition to the interval [0,1] , which suffices for our purposes. Fix a number α∈(1−H,1/2) . Note that from Theorem 1 it is easy to derive the following estimate: |D1−1−αB1−H(x)|≤C(ω)|1−x|H+α−1|log(1−x)|1/2. For some μ>1/2 1/2$]]>, define the weight ρ(x)=(1−x)H+α−1|log(1−x)|μ. Let a function f:[0,1]→R be such that D0+αf∈L1([0,1],ρ) ; this will be our class of admissible integrands. Then the extended fractional integral ∫01f(x)dBH(x)=eiπα∫01(D0+αf)(x)(D1−1−αB1−H)(x)dx is well defined (see the Appendix). In particular, it is possible to take f with D0+αf∈L1[0,1] , and for such integrands, the definition agrees with the definition of the fractional integral given in [10]; see Remark on p. 340.

Furthermore, it is shown in the Appendix that if f satisfies the above assumption for a different value of α, the value of the extended fractional integral will be the same. The following estimate is obvious: |∫01f(x)dBH(x)|≤C(ω)‖D0+αf‖L1([0,1],ρ).

For each t∈(0,1) , we will define the integral ∫0tf(x)dBH(x) by a similar formula understanding it in the sense of [10] since D0+αf∈L1[0,t] , Dt−1−αBt−H∈L∞[0,t] . Under the additional assumption that Dt+αf∈L1([t,1],ρ) , we can define the integral ∫t1f(x)dBH(x) similarly to (3), and the additivity holds: ∫01f(x)dBH(x)=∫0tf(x)dBH(x)+∫t1f(x)dBH(x). Note that the additivity ∫0tf(x)dBH(x)=∫0sf(x)dBH(x)+∫stf(x)dBH(x) for s<t follows from the results of [10].

Finally, it is worth to add that for f∈Cγ[0,1] with γ>1−H 1-H$]]>, the extended fractional integral is well defined since the derivative D0+αf is bounded for any α<γ , and thus we can take α∈(1−H,γ) in the definition. The value of the integral agrees with the so-called Young integral, which is given by a limit of integral sums. An important example is f(x)=g(BH(x)) where g is Lipschitz continuous. In this case, the following change-of-variable formula holds: ∫abg(BH(x))dBH(x)=G(b)−G(a), where G(x)=∫0xg(y)dy . The formula appears to be valid (with the integral defined in the sense of [10]) even for functions h of locally bounded variation; see [1]. However, Lipschitz continuity (even continuous differentiability) will suffice for our purposes.

 Main result

In this section, for a given F1 -measurable random variable ξ, we construct an F -adapted process ψ={ψ(t)}t∈[0,1] such that (1) holds almost surely under the following “log-Hölder” assumption on ξ.

There exists an F -adapted process {Z(t)}t∈[0,1] such that Z(1)=ξ and, for some a>1 1$]]>, |Z(1)−Z(t)|≤C(ω)|log(1−t)|−a for all t∈[0,1) .

Obviously, the process Z satisfies |Z(1)−Z(t)|≤C(ω)(1−t)b for any b>0 0$]]>. So (4) is weaker than (5), which is the assumption made in [5].

In [5], the following example of a random variable not satisfying (5) was given. Assume that F={Ft=σ(BsH,s∈[0,t])}t∈[0,1] , and let ξ=∫1/21g(t)dWt , where g(t)=(1−t)−1/2|log(1−t)|−1 , and W is a Wiener process such that its natural filtration coincides with F . Using the same argument as in [5], it is possible to show that ξ does not satisfy (4) as well. However, if we take the same construction with g(t)=(1−t)−1/2|log(1−t)|−d , d>1 1$]]>, then the corresponding random variable satisfies (4), but not (5).

Next, we state a helpful lemma from [5], used in our construction of ψ.

There exists an F -adapted process ϕ on [0,1] such that for any t>0 0$]]>, D0+αϕ∈L1[0,t] , so that the integral ∫0tϕ(s)dBH(s) is defined as the fractional integral, and limt→1−∫0tϕ(s)dBH(s)=+∞ almost surely.

We can now proceed with the main result.

Let ξ satisfy Assumption 1. Then there exists an F -adapted process ψ={ψ(t)}t∈[0,1] such that (1) holds with the integral defined in the extended fractional sense.

The proof is divided into three parts.

Construction of ψ. Let κ∈(2,2a) . Put tn=1−e−κn/a , n≥1 , and let Δn=tn+1−tn . It is easy to see that (1−tn)≤CΔn. Denote for brevity ξn=Z(tn) . Then, by Assumption 1, |ξn−ξ|≤C(ω)κ−n , so that |ξn−ξ|≤2−n for all n large enough, say, for n≥N(ω) . In particular, we have |ξn−ξn−1|≤2−n+2 for all n≥N(ω)+1 .

The integrand ψ is constructed in an inductive way between the points {tn,n≥1} . Set first ψ(t)=0 , t∈[0,t1] . Assuming that ψ(t) is defined on [0,tn) , let V(t)=∫0tψ(s)dBH(s),t∈[0,tn] . The construction of the integrand [tn,tn+1) depends on whether V(tn)=ξn−1 or not.

V(tn)≠ξn−1 . In this case, thanks to Lemma 4, there exists an adapted process {ϕ(t),t∈[tn,tn+1]} such that ∫tntϕ(s)dBH(s)→+∞ as t→tn+1− . Define the stopping time τn=inf{t≥tn:∫tntϕ(s)dBH(s)≥|ξn−Vtn|} and the process ψ(t)=ϕ(t)sign(ξn−V(tn))I[tn,τn](t),t∈[tn,tn+1). It is obvious that ∫tntn+1ψ(s)dBH(s)=ξn−V(tn) and V(tn+1)=ξn .

V(tn)=ξn−1 . We consider the uniform partition sn,k=tn+kδn , k=1,…,n of [tn,tn+1] with mesh δn=Δnn−1 and auxiliary function ϕ‾(t)=an ∑k=0n−1(BH(t)−BH(sn,k))I[sn,k,sn,k+1)(t), with an=2−n+3δn−2Hn−1 . Notice that by the change-of-variable formula, ∫tntn+1ϕ‾(t)dBH(t)=an ∑k=0n−1(BH(sn,k+1)−BH(sn,k))2.

Define the stopping time σn=inf{t≥tn:∫tntϕ‾(s)dBH(s)≥|ξn−ξn−1|}∧tn+1 and set ψ(t)=sign(ξn−ξn−1)ϕ‾(t)I[tn,σn](t),t∈[tn,tn+1).

The construction approaches ξ. Our aim now is to prove that V(tn)=ξn−1 for all n large enough. By construction it suffices to show that Case II happens for all n large enough. Equivalently, we need to show that σn<tn+1 for all n large enough. In view of (8), the latter inequality holds if an ∑k=0n−1(BH(sn,k+1)−BH(sn,k))2>|ξn−ξn−1|. |\xi _{n}-\xi _{n-1}|.\]]]> Thus, in view of the Borell–Cantelli lemma and inequality (7), it suffices to verify the convergence of the series ∑n≥1Pan ∑k=1n−1(BH(sn,k+1)−BH(sn,k))2<2−n+2 or, equivalently, that ∑n≥1P ∑k=1n−1(δn−H(Bsn,k+1H−Bsn,kH))2<n/2<∞. The latter follows from Lemma 3 through the self-similarity and stationarity of increments of an fBm. Thus, for all n large enough, say, for n≥N2(ω) , V(tn)=ξn−1 .

Integrability of ψ. It is easy to see that ψ is integrable w.r.t.  BH on any interval [0,tN] . It remains to verify that the integral ∫tN1ψ(s)dBH(s) is well defined and vanishes as N→∞ (note that we did not establish the continuity of the integral as a function of the upper limit). For some μ>1/2 1/2$]]> (which will be specified later), define ρ(x)=(1−x)H+α−1|log(1−x)|μ. Clearly, it suffices to show that ‖DtN+αψ‖L1([tN,1],ρ)→0 , N→∞ . Let N≥N2(ω) . Write ∫tN1|(DtN+αψ)(s)|ρ(s)ds= ∑n=N∞∫tntn+1|(DtN+αψ)(s)|ρ(s)ds≤C ∑n=N∞ΔnH+α−1|logΔn|μ∫tntn+1|(DtN+αψ)(s)|ds, where we have used (6) and the fact that ρ is decreasing in a left neighborhood of 1.

Now we estimate ∫tntn+1|(DtN+αψ)(s)|ds≤∫tntn+1(|ψ(s)|(s−tN)α+∫tNs|ψ(s)−ψ(u)||s−u|1+αdu)ds≤C(ω)anΔn1−αδnH|log(δn)|1/2+∫tNtn+1∫tns|ψ(s)−ψ(u)||s−u|1+αduds, where we have used Theorem 1 to estimate ψ.

Consider the second term. It equals ∑k=1n∫sn,k−1sn,k(∫tNtn+∫tnsn,k−1+∫sn,k−1s)|ψ(s)−ψ(u)||s−u|1+αduds=:I1+I2+I3. Start with I1 , observing that ψ vanishes on (σn,tn+1] : I1≤∫tntn+1 ∑j=Nn∫tj−1tj|ψ(s)|+|ψ(u)||s−u|1+αduds≤C(ω)anδnH|logδn|1/2∫tntn+1(s−tn)−αds+ ∑j=Nn−1ajδjH|logδj|1/2∫tntn+1(s−tj+1)−αds≤C(ω)(anΔn1−αδnH|logδn|1/2+ ∑j=Nn−1ajδjH|logδj|1/2Δn1−α). Proceed with the second term: I2≤C(ω)anδnH|logδn|1/2 ∑k=1n∫sn,k−1sn,k∫tnsn,k−1|s−u|−1−αduds≤C(ω)anδnH|logδn|1/2 ∑k=1n∫sn,k−1sn,k(s−sn,k−1)−αds≤C(ω)annδnH+1−α|logδn|1/2=C(ω)anΔnδnH−α|logδn|1/2. Finally, assuming that σn∈[sn,l−1,sn,l) , we have I3≤C(ω) ∑k=1l−1∫sn,k−1sn,k∫sn,k−1san(s−u)H|log(s−u)|1/2(s−u)1+αduds+∫sn,l−1σn∫sn,l−1s|ψ(s)−ψ(u)||s−u|1+αduds+∫σnsn,l∫sn,l−1σn|ψ(s)−ψ(u)||s−u|1+αduds≤C(ω)an ∑k=1n∫sn,k−1sn,k(s−sn,k−1)H−α|log(s−sn,k−1)|1/2ds+C(ω)anδnH|logδn|1/2∫σnsn,l∫sn,l−1σn1|s−u|1+αduds≤C(ω)annδnH+1−α|logδn|1/2nδn1−α=C(ω)anΔnδnH−α|logδn|1/2.

Gathering all estimates, we get ∫tN1|DtN+α(ψ)(s)|ρ(s)ds≤C(ω) ∑n=N∞(anΔnHδnH|logδn|1/2|logΔn|μ+anΔnH+αδnH−α|logδn|1/2|logΔn|μ+ΔnH|logΔn|μ ∑j=Nn−1ajδjH|logδj|1/2). Consider the second sum. After changing the order of summation, we get ∑n=N∞ ∑j=Nn−1ajδjH|logδj|1/2ΔnH|logΔn|μ= ∑j=N∞ajδjH ∑n=j+1∞ΔnH|logΔn|μ≤C ∑j=N∞ajδjHΔjH|logΔj|μ ∑n=j+1∞eκj/a−κn/aκμ(n−j)/a≤C ∑j=N∞ajδjHΔjH|logΔj|μ≤C ∑j=N∞ajδjHΔjH|logΔj|μ ∑n=j+1∞e−κ(n−j)n/aκμ(n−j)/a≤C ∑j=N∞ajδjHΔjH|logΔj|μ. Consequently, noting that ΔnHδnH≤ΔnH+αδnH−α , we have ∫tN1|DtN+α(ψ)(s)|ρ(s)ds≤C(ω) ∑n=N∞anΔnH+αδnH−α|logδn|1/2|logΔn|μ≤ ∑n=N∞2−nn−1ΔnH+αδn−H−α|logδn|μ+1/2≤ ∑n=N∞2−nnH+α−1(|logΔn|μ+1/2+nμ+1/2)≤ ∑n=N∞2−nnH+α−1(C+κ(μ+1/2)n/a+nμ+1/2). Now, in order for the last series to converge, it is sufficient that (μ+1/2)/a<log2/logκ or, equivalently, μ<alog2/logκ−1/2 . Since alog2/logκ>1 1$]]> by our choice, it is possible to take some μ>1/2 1/2$]]> satisfying this requirement, thus finishing the proof.  □

 Appendix. Properties of the extended fractional integral

In this section, we establish important properties of the extended fractional integral defined by (2). We will restrict ourselves to the case [a,b]=[0,1] . Assume that the function g:[0,1]→R satisfies the following regularity requirement: there exist λ∈(0,1) and ν>0 0$]]> such that |g(x)−g(y)|≤C|x−y|λ|log|x−y||ν for all x,y∈[0,1] . We first state some properties of g that follow from this regularity.

Let g:[0,1]→R satisfy (9). Then for any β∈(0,λ) , |(D1−βg1−)(x)−(D1−βg1−)(y)|≤Cβ|x−y|λ−β|log|x−y||ν for all x,y∈[0,1] .

The proof is similar to that of Lemma 13.1 in [6] and thus omitted.

Now, for some μ>ν \nu $]]>, define ρ(x)=|1−x|λ−β|log(1−x)|μ,x∈[0,1].

Let g:[0,1]→R satisfy (9), and ψ∈C∞[−1,0]→R be a positive function with ∫−10ψ(x)dx=1 . Define ψN(x)=Nψ(Nx) , gN(x)=(g1−∗ψN)(x)=∫x(x+1/N)∧1g1−(y)ψN(x−y)dy,N≥1. Then for any μ>ν \nu $]]>, β∈(0,λ) , x∈[0,1] , and all N≥1 large enough, we have |(D1−βgN)(x)−(D1−βg1−)(x)|≤Cβ(logN)ν−μρ(x).

Note that D1−βgN=1[0,1](D1−βg1−)∗ψN . For brevity, set h(x)=(D1−βg1−)(x) , x∈[0,1] . First, suppose that x∈[0,1−1/N] . Thanks to Lemma 5, |(D1−βgN)(x)−(D1−βg1−)(x)|≤C∫xx+1/N|h(x)−h(y)|Nψ(N(x−y))dy≤C∫−10|h(x−y/N)−h(y)|ψ(y)dy≤CβNβ−λ∫−10|y|λ−β(N−log|y|)νψ(y)dy≤CNβ−λ(logN)ν≤C(logN)ν−μρ(x), where the last inequality holds for all N large enough since ρ(x) is decreasing for x close to 1.

Now consider the case x∈[1−1/N,1] . Using Lemma (5), we have |(D1−βgN)(x)−(D1−βg1−)(x)|≤|(D1−βgN)(x)|+|(D1−βg1−)(x)|≤∫x1Nψ(N(x−y))|h(1)−h(y)|dy+|h(1)−h(x)|≤C|1−x|λ−β|log(1−x)|ν≤Cρ(x)|log(1−x)|ν−μ≤Cρ(x)(logN)ν−μ since x>1−1/N 1-1/N$]]> and ν<μ . This finishes the proof.  □

Let g:[0,1]→R satisfy (9), α∈(1−λ,1) , and let ρ be given by (10) with β=1−α . Assume further that f:[0,1]→R is such that D0+αf∈L1([0,1],ρ) . Then the extended fractional integral ∫01f(x)dg(x)=eiπα∫01(D1−αf)(x)(D1−1−αg1−)(x)dx is well defined. Moreover, if f and g satisfy the above assumptions for different α, the value of the integral is preserved. Additionally, if Dt+αf∈L1([t,1],ρ) for some t∈(0,1) , then ∫01f(x)dg(x)=∫0tf(x)dg(x)+∫t1f(x)dg(x), where the first integral is understood in the sense of [10] (D0+αf∈L1[0,t],Dt−1−αgt−∈L∞[0,t]) , and the second in the above sense.

From Lemma 5 we have |(D1−1−αg)(x)|≤C(1−x)λ+α−1|log(1−x)|ν≤Cρ(x), whence the finiteness of the integral follows. Defining gN , N≥1 , as in Lemma 6, we have eiπα∫01(D1−αf)(x)(D1−1−αg1−)(x)dx=eiπαlimN→∞∫01(D1−αf)(x)(D1−1−αgN)(x)dx in view of the dominated convergence theorem.

Since gN∈C∞[0,1]→R , it follows from the properties of fractional integrals and derivatives (see [6, 10]) that (D1−1−αgN)(x)=(I1−αgN′)(x), where (I1−αh)(x)=e−iπαΓ(α)∫x1(y−x)α−1h(y)dy is the right-sided fractional Riemann–Liouville integral of order α. Note that ∫01∫x1|(D1−αf)(x)|(y−x)α−1|gN′(y)|dydx≤CN∫01|(D1−αf)(x)|(1−x)αdx≤CN∫01|(D1−αf)(x)|ρ(x)dx<∞ thanks to our assumptions.

Hence, by the Fubini theorem, eiπα∫01(D1−αf)(x)(D1−1−αgN)(x)dx=1Γ(α)∫01∫0y(y−x)1−α(D1−αf)(x)dxgN′(y)dy=∫01(I0+αD0+αf)(y)gN′(y)dy, where (I0+αh)(x)=1Γ(α)∫0x(x−y)α−1h(y)dy is the left-sided fractional Riemann–Liouville integral of order α. Note that D0+αf∈L1[0,t] for any t∈(0,1) , so by [6, Theorem 2.4], for almost all y∈[0,t] , we have (I0+αD0+αf)(y)=f(y) . Therefore, this equality holds for almost all y∈[0,1] , and eiπα∫01(D1−αf)(x)(D1−1−αgN)(x)dx=∫01f(y)gN′(y)dy. Taking into account (12), we get that, indeed, the extended fractional integral ∫01f(s) does not depend on α.

Concerning additivity, under the assumption Dt+αf∈L1([t,1],ρ) , repeating the previous argument, we get ∫t1f(x)dg(x)=limN→∞∫t1f(y)gN′(y)dy. On the other hand, from the proof of [10, Theorem 2.5] we have ∫0tf(x)dg(x)=limN→∞∫0tf(y)gN′(y)dy. Therefore, ∫0tf(x)dg(x)+∫t1f(x)dg(x)=limN→∞∫01f(y)gN′(y)dy=∫01f(x)dg(x), as required.  □

 Appendix. Proof of Lemma 2

We have E[ΔBiHΔBjH]=ρH(|i−j|) with ρH(m)=12((m+1)2H+|m−1|2H−2m2H),m≥0. It is easy to see that ρH(m)∼H(2H−1)m2H−2,m→∞. The double sum in the question can be transformed as follows: SnH:= ∑i,j=0n−1(E[ΔBiHΔBjH])2= ∑i,j=0n−1ρH(|i−j|)2=|m=i−j|= ∑m=1−nn−1(n−|m|)ρH(|m|)2.

Let H∈(1/2,3/4) . Then SnHn= ∑m=1−nn−1(1−|m|n)ρH(|m|)2. In view of (13), the series ∑m=−∞+∞ρH(|m|)2 converges, so by the dominated convergence theorem, SnHn→ ∑m=−∞+∞ρH(|m|)2,n→∞.

Now let H∈[3/4,1) . Define an=nlogn if H=3/4 , an=n4H−2 if H∈(3/4,1) . Applying the Stolz–Cèsaro theorem, we have limn→∞SnHan=limn→∞Sn+1H−SnHan+1−an=limn→∞1an+1−an ∑m=−nnρH(|m|)2, provided that the latter limit exists. Using the Stolz–Cèsaro theorem once more, we have limn→∞1an+1−an ∑m=−nnρH(|m|)2=2limn→∞ρH(n)2an+1+an−1−2an, provided that the latter limit exists. It is easy to see that an+1+an−1−2an∼n−1,H=3/4,(4H−2)(4H−3)n4H−4,H∈(3/4,1), as n→∞ . Thus, we get the required statement thanks to (13).  □

 References Azmoodeh, E., Mishura, Y., Valkeila, E.: On hedging European options in geometric fractional Brownian motion market model. Stat. Decis. 27(2), 129–143 (2009) MR2662719. doi:10.1524/stnd.2009.1021 Li, W.V., Shao, Q.-M.: Gaussian processes: Inequalities, small ball probabilities and applications. Handb. Stat. 19, 533–597 (2001) MR1861734. doi:10.1016/S0169-7161(01)19019-X Mishura, Y.: Stochastic Calculus for Fractional Brownian Motion and Related Processes vol. 1929. Springer (2008) MR2378138. doi:10.1007/978-3-540-75873-0 Mishura, Y., Shevchenko, G.: Small ball properties and representation results. arXiv:math.PR/1508.07134 (2015) Mishura, Y., Shevchenko, G., Valkeila, E.: Random variables as pathwise integrals with respect to fractional Brownian motion. Stoch. Process. Appl. 123(6), 2353–2369 (2013) MR3038509. doi:10.1016/j.spa.2013.02.015 Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon (1993) MR1347689 Shevchenko, G., Viitasaari, L.: Integral representation with adapted continuous integrand with respect to fractional Brownian motion. Stoch. Anal. Appl. 32(6), 934–943 (2014) MR3270688. doi:10.1080/07362994.2014.948725 Shevchenko, G., Viitasaari, L.: Adapted integral representations of random variables. Int. J. Mod. Phys. Conf. Ser. 36, 1560004 (2015). World Scientific Viitasaari, L.: Integral representation of random variables with respect to Gaussian processes. To appear in Bernoulli, available at arXiv:math.PR/1307.7559 (2013) Zähle, M.: Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Relat. Fields 111(3), 333–374 (1998) MR1640795. doi:10.1007/s004400050171