We present a random walk approximation to fractional Brownian motion where the increments of the fractional random walk are defined as a weighted sum of the past increments of a Bernoulli random walk

Lindstrøm, Tom

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Dept. of Math./CMA University of OsloPure Mathematics No 11ISSN 0806–2439 August 2007A weighted random walk approximation tofractional Brownian motionTom Lindstrøm∗AbstractWe present a random walk approximation to fractional Brownian mo-tion where the increments of the fractional random walk are defined as aweighted sum of the past increments of a Bernoulli random walk.Keywords: Fractional Brownian motion, random walks, discrete approxi-mations, weak convergenceAMS Subject Classification (2000): Primary 60F17, 60G15, 60G18The purpose of this brief note is to describe a discrete approximation to frac-tional Brownian motion. The approximation works for all Hurst indices H, buttake slightly different forms for H ≤ 12 and H > 12 . There are already severaldiscrete approximations to fractional Brownian motion in the literature (see,e.g., [11], [1], [3], [10], [4], [2], [5], [8] for this and related topics), and the ad-vantage of the present approach is that the increments of the fractional randomwalk is given as a weighted sum of past increments of an ordinary (Bernoulli)random walk. This gives an excellent understanding of the dynamics of theprocess and is a good starting point for stochastic calculus with respect to frac-tional Brownian motion. A similar idea is exploited in much greater generalityby Konstantopoulos and Sakhanenko in [5], but they assume that H > 12 , whilethe present paper is mainly of interest when H < 12 .The discrete approximation is based on Mandelbrot and Van Ness’ [6] movingframe representation of fractional Brownian motion:xt = cH∫ t−∞((t− r)H− 12 − (−r)H− 12+)dbrwhere the scaling constant cH is given bycH =(∫ ∞0((1 + u)H−12 − uH− 12)2du+12H)− 12=√Γ(2H + 1) sin(piH)Γ(H + 12 )(see also [9]). This representation will be used to establish the convergence.∗Centre of Mathematics for Applications and Department of Mathematics, PO Box 1053Blindern, N-0316 Oslo, Norway. e-mail:lindstro@math.uio.no11 The main theoremTo state the main result, we need some notation. For each natural number N ,let ∆tN = 1N and think ofTN = {k∆tN | k ∈ Z}as a timeline. We let T+N denote the nonnegative part of T . It is convenient touse the following convention for sums over elements in TN :t∑r=sf(r) = f(s) + f(s+∆t) + · · ·+ f(t−∆tN )Note that the lower limit s is included in the sum, but the upper limit t is not.We shall also write ∆f(t) = f(t +∆tN ) − f(t) for the forward increment of fat t.For all t ∈ TN , let ωN (t) be independent random variables taking values ±1with probability 12 . We shall write ∆BN (t) =√∆tNωN (t) and think of BN asa Bernoulli random walk approximating Brownian motion. For 0 < H < 1 andN ∈ N, define a process XH,N : ΩN × T+N → R by XN,H(0) = 0 and∆XH,N (s) = KH∆tH− 12N ∆BN (s) +s∑r=−∞(H − 12)(s− r)H− 32∆tN∆BN (r)(using, e.g., Kolmogorov’s one series theorem, see [12], one easily checks thatthe sum converges a.s.) where the constant KH is defined byKH = −(H −12 )ζ(32 −H) for H < 121 for H ≥ 12(as usual, ζ(s) =∑∞n=1 n−s when s > 1). Except for the Mandelbrot-Van Nessscaling factor cH , XH,N will be our random walk approximation to fractionalBrownian motion. For convergence puposes it will be convenient to think ofXH,N as a ca`dla`g process defined on [0,∞), and we do this simply by assumingthat XH,N is constant between points in TN .Remark: Note that the increment ∆XH,N (s) is a weighted sum of incrementsof the Bernoulli random walk BN — it is a linear combination of the currentcoin toss ωN (s) and all previous coin tosses ωN (r), r < s. Observe also thatsince limH↑ 12 −(H −12 )ζ(32 − H) = lims↓1(s − 1)ζ(s) = 1, the two cases meetcontinuously at H = 12 . For H >12 , we may actually choose KH as we pleasesince the term will vanish in the limit (see below), but KH = 1 is the naturalvalue and probably the one that gives best results in numerical work.We are now ready to state the main result. Note that when H = 12 ,∆X 12 ,N(t) = ∆BN (t) and the theorem just reduces to the classical convergenceof a Bernoulli random walk to Brownian motion.2Theorem 1 (Main Theorem) For all real numbers H, 0 < H < 1, the pro-cesses cHXH,N converge weakly in D([0,∞)) to fractional Brownian motion withHurst index H.Notation: In the rest of the paper, we drop the notational dependence on Nand H, and write simply X, B, T , ∆t for XH,N , BN , TN , ∆tN etc. when noconfusion can arise.As we are interested in understanding the dynamics of fractional Brownianmotion, we have defined X by specifying its increments ∆X(s). To prove themain theorem, we need an expression for X(t). This is just a small calculation:X(t) =t∑s=0∆X(s) =t∑s=0KH∆tH−12∆Bs+t∑s=0s∑r=−∞(H − 12)(s− r)H− 32∆t∆BrChanging the order of summation, we haveX(t) = KH∆tH−12Bt +t∑r=0t∑s=r+∆t(H − 12)(s− r)H− 32∆t∆Br+0∑r=−∞t∑s=0(H − 12)(s− r)H− 32∆t∆Brwhere Bt =∑tr=0∆Br is a random walk converging to Brownian motion. Ob-serve that when H > 12 , the first term KH∆tH− 12Bt vanishes when N → ∞(this is why the choice of KH is irrelevant in this case), but when H < 12 , theterm explodes. In this case we have a delicate balance between two terms goingto infinity, and a correct choice of KH is crucial.The idea is now to simplify the expression for X by replacing the sums∑(H− 12 )(s− r)H−32∆t by the corresponding integrals∫(H− 12 )(s− r)H−32 ds,and then performing the integration. This works nicely for H > 12 , but whenH < 12 , one of the integrals diverges, and we have to be more careful. Putcrudely, it is the divergence of this integral that will cancel the divergence ofthe term KH∆tH−12Bt.We are ready to prove the main theorem, and start with the simplest case.2 The case H > 12We start from the expressionX(t) = ∆tH−12Bt +t∑r=0t∑s=r+∆t(H − 12)(s− r)H− 32∆t∆Br+3+0∑r=−∞t∑s=0(H − 12)(s− r)H− 32∆t∆Brabove (remember that KH = 1 in this case). Since H > 12 , we have no problemwith convergence, and if we let N (r, t) be the error term:N (r, t) :=t∑s=r+∆t(H − 12)(s− r)H− 32∆t−∫ tr+∆t(H − 12)(s− r)H− 32 ds,we gett∑s=r+∆t(H − 12)(s− r)H− 32∆t =∫ tr+∆t(H − 12)(s− r)H− 32 ds+ N (r, t) == (t− r)H− 12 −∆tH− 12 + N (r, t)Similarly, withδN (r, t) :=t∑s=0(H − 12)(s− r)H− 32∆t−∫ t0(H − 12)(s− r)H− 32 ds,we gett∑s=0(H − 12)(s− r)H− 32∆t =∫ t0(H − 12)(s− r)H− 32 ds+ δN (r, t) == (t− r)H− 12 − (−r)H− 12 + δN (r, t)This means thatX(t) =t∑r=0((t− r)H− 12 + N (r))∆Br++0∑r=−∞((t− r)H− 12 − (−r)H− 12 + δN (r))∆Br ==t∑r=−∞((t− r)H− 12 − (−r)H− 12+)∆Br +t∑r=0N (r, t)∆Br +0∑r=−∞δN (r, t)∆BrWe want to prove that X converges weakly to fractional Brownian motion.According to Theorem 1 in [5], it suffices to show that E(c2HX(t)2) → t2H .This follows immediately from the Mandelbrot-Van Ness representation andthe following lemma.Lemma 2 For 12 < H < 1:(i) E((∑tr=0 N (r, t)∆Br)2)≤ (H − 12 )2t∆t2H−14(ii) E((∑0r=−∞ δN (r, t)∆Br)2)≤ (H − 12 )2ζ(3− 2H)∆t2HProof: (i) We first observe thatN (r, t) =t∑s=r+∆t(H − 12)(s− r)H− 32∆t−∫ tr+∆t(H − 12)(s− r)H− 32 ds > 0since∑ts=r+∆t(H − 12 )(s− r)H−32∆t is an upper Riemann sum for the integral.Since∑ts=r+2∆t(H − 12 )(s− r)H−32∆t is a lower Riemann sum, we also have0 ≤ N (r, t) ≤ (H − 12)∆tH− 32∆t = (H − 12)∆tH−12ThusE((t∑r=0N (r, t)∆Br)2)=t∑r=0N (r, t)2∆t ≤≤t∑r=0(H − 12)2∆t2H−1∆t ≤ (H − 12)2t∆t2H−1(ii) Using approximating Riemann sums as in part (i), we see that0 ≤ δN (r, t) ≤ (H − 12)(−r)H− 32∆t,and thusE((0∑r=−∞δN (r, t)∆Br)2)=0∑−∞δN (r, t)2∆t ≤0∑r=−∞(H − 12)2(−r)2H−3∆t3Letting r = −k∆t, we getE((0∑r=−∞δN (r, t)∆Br)2)≤∞∑k=0(H − 12)2k2H−3∆t2H == (H − 12)2ζ(3− 2H)∆t2HThis completes the proof of the lemma (and also the proof of the Main Theoremfor the case H > 12 ). 3 The case H < 12Again we start from the expressionX(t) = KH∆tH−12Bt +t∑r=0t∑s=r+∆t(H − 12)(s− r)H− 32∆t∆Br+5+0∑r=−∞t∑s=0(H − 12)(s− r)H− 32∆t∆BrIn this case, one of the integrals we worked with above diverges, and we haveto be more careful. Let us start with a closer look at the term∑ts=r+∆t(H −12 )(s− r)H−32∆t. We obviously havet∑s=r+∆t(H − 12)(s− r)H− 32∆t =∞∑s=r+∆t(H − 12)(s− r)H− 32∆t−∞∑s=t(H − 12)(s− r)H− 32∆tand if we let r = N∆t, s = k∆t, we get∞∑s=r+∆t(H − 12)(s− r)H− 32∆t =∞∑k=N+1(H − 12)(k∆t−N∆t)H− 32∆t= (H − 12)∆tH−12∞∑k=N+1(k −N)H− 32 = (H − 12)∆tH−12∞∑n=1nH−32= (H − 12)∆tH−12 ζ(32−H) = −KH∆tH− 12Substituting this into the expression for X(t), we getX(t) =t∑r=0∞∑s=t−(H − 12)(s− r)H− 32∆t∆Br+0∑r=∞t∑s=0(H − 12)(s− r)H− 32∆t∆BrThe two sums in this expression have less dangerous limits than the one we justgot rid of, and can be approximated by integrals. If we let˜N (r, t) :=∞∑s=t−(H − 12)(s− r)H− 32∆t−∫ ∞t−(H − 12)(s− r)H− 32 ds,we get (remember that H < 12 ):∞∑s=t−(H − 12)(s− r)H− 32∆t =∫ ∞t−(H − 12)(s− r)H− 32 ds+ ˜N (r, t)=[−(s− r)H− 12]s=∞s=t+ ˜N (r, t) = (t− r)H− 12 + ˜N (r, t)6Similarly, if we letδ˜N (r, t) :=t∑s=0−(H − 12)(s− r)H− 32∆t−∫ t0−(H − 12)(s− r)H− 32 ds,we gett∑s=0(H − 12)(s− r)H− 32∆t =∫ t0(H − 12)(s− r)H− 32 ds− δ˜N (r, t)=[(s− r)H− 12]s=ts=0− δ˜N (r, t) = (t− r)H− 12 − (−r)H− 12 − δ˜N (r, t)We thus haveX(t) =t∑r=0((t− r)H− 12 + ˜N (r, t))∆Br++0∑r=−∞((t− r)H− 12 − (−r)H− 12 − δ˜N (r, t))∆Br=t∑r=−∞((t− r)H− 12 − (−r)H− 12+)∆Br++t∑r=0˜N (r, t)∆Br −0∑r=−∞δ˜N (r, t)∆BrTo prove that cHX converges weakly to fractional Brownian motion, we cannow longer use Theorem 1 of [5] as in the previous case since this theoremrequires that H > 12 . However, the first term in the expression above obviouslyconverges weakly to ∫ tr=−∞((t− r)H− 12 − (−r)H− 12+)dbr,and the next lemma shows that error terms go uniformly to zero. Using theMandelbrot-Van Ness representation, we then get the Main Theorem for H < 12 .Lemma 3 For each H, 0 < H < 12 , there is a constant KH ∈ R+ (independentof N and t) such that∣∣∣∣∣X(t)−t∑r=−∞((t− r)H− 12 − (−r)H− 12+)∆Br∣∣∣∣∣ ≤ KH∆tHProof: It clearly suffices to show that there are constants CH , DH ∈ R+ (inde-pendent of N and t) such that∣∣∣∣∣t∑r=0˜N (r, t)∆Br∣∣∣∣∣ ≤ CH∆tH and∣∣∣∣∣0∑r=−∞δ˜N (r)∆Br∣∣∣∣∣ ≤ DH∆tH7We begin with the ˜N -case. By definition˜N (r, t) =∞∑s=t−(H − 12)(s− r)H− 32∆t−∫ ∞t−(H − 12)(s− r)H− 32 dsSince∑∞s=t−(H − 12 )(s − r)H−32∆t is an upper Riemann sum for the integral∫∞t−(H − 12 )(s − r)H−32 ds, and∑∞s=t+∆t−(H − 12 )(s − r)H−32∆t is a lowerRiemann sum, we have0 ≤ ˜N (r, t) ≤ −(H − 12)(t− r)H− 32∆tHence (remember that |∆Br| = ∆t 12 )∣∣∣∣∣t∑r=0˜N (r, t)∆Br∣∣∣∣∣ ≤t∑r=0−(H − 12)(t− r)H− 32∆t 32If we let t = K∆t, r = k∆t, we can rewrite the last sum asK−1∑k=0−(H − 12)(K − k)H− 32∆tH ≤ −(H − 12)ζ(32−H)∆tHThis completes the ˜N -part of the argument.Turning to the term∑0r=−∞ δ˜N (r)∆Br, we first observe that by definitionδ˜N (r) =t∑s=0−(H − 12)(s− r)H− 32∆t−∫ t0−(H − 12)(s− r)H− 32 dsAgain,∑ts=0−(H − 12 )(s− r)H−32∆t is an upper Riemann sum, and we easilysee that0 ≤ δ˜N (r) ≤ −(H − 12)(−r)H− 32∆tLetting r = −k∆t, we getE∣∣∣∣∣0∑r=−∞δ˜N (r)∆Br∣∣∣∣∣ ≤0∑r=−∞−(H − 12)(−r)H− 32∆t 32 ≤≤ −(H − 12)∆tH∞∑k=0kH−32 = −(H − 12)ζ(32−H)∆tHThis proves the lemma (and hence the Main Theorem for the remaining caseH < 12 ). 8References[1] N.J. 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A weighted random walk approximation to fractional Brownian motion

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