on some maximal inequalities for fractional brownian motions

$: On some maximal inequalities for fractional Brownian motions$
Statistics & Probability Letters 44 (1999) 47–54

On some maximal inequalities for fractional Brownian motions

Alexander Novikova ; 1, Esko Valkeilab;∗

aDepartment of Statistics, University of Newcastle, Newcastle, 2308 AustraliabDepartment of Mathematics, University of Helsinki, P.O. Box 4, 00014 Helsinki, Finland

Received April 1998; received in revised form October 1998

Abstract

We prove some maximal inequalities for fractional Brownian motions. These extend the Burkholder–Davis–Gundyinequalities for fractional Brownian motions. The methods are based on the integral representations of fractional Brownianmotions with respect to a certain Gaussian martingale in terms of beta kernels. c© 1999 Elsevier Science B.V. All rightsreserved

MSC: 60G15; 60G40; 60H99

Keywords: Fractional Brownian motions; Maximal inequalities

1. Introduction

1. (Fractional Brownian motions): We work in probability space (; F; P). Fractional Brownian motionZ =ZH with Hurst index H is a continuous Gaussian process with stationary increments and with the follow-ing properties:

(i) Z0 = 0.(ii) EZt = 0 for all t¿0.(iii) EZtZs = 1

2

(t2H + s2H − |s− t|2H) for all s; t¿0.

The standard Brownian motion B is a fractional Brownian motion with Hurst index H = 12 .

2. (Some maximal inequalities): Denote by F =(Ft)t¿0 the �ltration generated by the fractional motion Z :Ft = �(ZHs : s6t). For any process X denote by X ∗ the supremum process: X ∗

t = sups6t |Xs|.

∗ Corresponding author.1 Supported by Suomalainen Tiedeakatemia, the Research Grants Committee of the University of Helsinki and RMS New Sta� Grant.

0167-7152/99/$ – see front matter c© 1999 Elsevier Science B.V. All rights reservedPII: S0167 -7152(98)00290 -9

48 A. Novikov, E. Valkeila / Statistics & Probability Letters 44 (1999) 47–54

Fractional Brownian motion is a self-similar process: Zatd= aHZt , where a¿ 0 is a constant and the notation

X d=Y means that the random variable X has the same distribution as the random variable Y . Even more is

true: from the self-similarity it follows for the supremum process Z∗ that Z∗atd= aHZ∗

t . For any p¿ 0 we have

then the following result using self-similarity:

Theorem 1.1. Let T ¿ 0 be a constant and Z a fractional Brownian motion with Hurst index H . Then

E (Z∗T )p = K(p;H)TpH ; (1)

where K(p;H) = E(Z∗1

)p.

The value of the constant K(p;H) is not known to us. The following generalizes the above for fractionalBrownian motions indexed with stopping times. Note that in the results below we do not have any upperbound for the case H ¡ 1

2 , but we conjecture that we have a similar result for the upper bound as for theother cases, and the proof is similar to the proof of the lower bound for the case H ¿ 1

2 .

Theorem 1.2. Let � be a stopping time with respect to the �ltration F . Then for any p¿ 0 and H ∈ ( 12 ; 1)we have that

c(p;H)E(�pH)6E ((Z∗

� )p)6C(p;H)E

(�pH); (2)

and for any p¿ 0 and H ∈ (0; 12 ) we have thatc(p;H)E

(�pH)6E ((Z∗

� )p) ; (3)

where the constants c(p;H); C(p;H)¿ 0 depend only from parameters p;H .

3. (Discussion and comments): Recall the classical Burkholder–Davis–Gundy (B–D–G) inequalities for thestandard Brownian motion: for any stopping time � with respect to the �ltration generated by the Brownianmotion B and p¿ 0 we have

c(p)E�p=26E ((B∗� )p)6C(p)E�p=2: (4)

The inequalities in (2) generalize the B–D–G inequalities to fractional Brownian motions. B–D–G have along history and we cite only some works in this area. Maybe the �rst general results were due to Novikov(p¿ 1

2 ) and Burkholder (see Burkholder, 1973; Novikov, 1971).It is well known that B–D–G inequalities generalize to arbitrary martingales M : one replaces � in (4) by

the stopped angle bracket process [M;M ]�. For a discussion and more references see Dellacherie and Meyer(1982, Ch. VII) or Liptser and Shiryaev (1986, Ch. I).With an arbitrary martingale M there are also predictable moment bounds: Novikov has given upper bounds

for purely discontinuous martingales in Novikov (1975) with p¿ 1, Dzhaparidze and Valkeila have given pre-dictable upper and lower bounds for p¿ 2 for locally square integrable martingale Dzhaparidze and Valkeila(1990) and Lebedev has given predictable upper bounds for an arbitrary vector-valued locally square integrablemartingale M for p¿ 1 Lebedev (1996).It is worthwhile to note that the process Z is not a semimartingale, except when it is the standard Brownian

motion. For this see Liptser and Shiryaev (1986, p. 232), for the case H ¿ 12 .

2. Auxiliary results

1. (Integral representations): We give some recently obtained integral representation between a fractionalBrownian motion Z and the so-called fundamental martingale M . The martingale M might also be calledMolchans martingale (see Molchan and Golosov, 1969).

A. Novikov, E. Valkeila / Statistics & Probability Letters 44 (1999) 47–54 49

The integrals below can be de�ned by integration by parts, where the singularities of the kernels do notcause problems, due to the H�older continuity of the fractional Brownian motion Z (see Norros et al., 1999,Lemma 2.1).Put w(t; s) := (c=C)s−�(t − s)−� for s ∈ (0; t) and w(t; s) = 0 for s¿ t, where � := H − 1

2 ,

C :=

√H

(H − (1=2))B(H − (1=2); 2− 2H)and

c :=1

B(H + (1=2); (3=2)− H) ;

where the beta coe�cient B(�; �) for �; �¿ 0 is de�ned by

B(�; �) :=�(�)�(�)�(� + �)

:

Proposition 2.1. For H ∈ (0; 1) de�ne M by

Mt =∫ t

0w(t; s) dZs: (5)

Then M is a Gaussian martingale, FM = F with variance 〈M 〉t = (C2=4H 2 (2− 2H))t2−2H .

For the proof we refer to Norros et al. (1999, Proposition 2.1). For later use denote by c2 the constant

c2:=

√C2

4H 2 (2− 2H) :

Put Yt:=∫ t0 s(1=2)−H dZs. Then Zs =

∫ t0 sH−(1=2) dYs. Moreover, by de�nition

Mt =cC

∫ t

0(t − s)(1=2)−H dYs

and by Norros et al. (1999, Theorem 3.2)

Yt = 2H∫ t

0(t − s)H−(1=2) dMs: (6)

2. (Martingale inequalities): Let N be a continuous martingale. We have the following from the B–D–Ginequalities:

Proposition 2.2. For any p¿ 0 and stopping time � there exists constants cp; Cp¿ 0 such that

cpE〈N; N 〉p=2� 6E ((N ∗� )

p)6CpE〈N; N 〉p=2� : (7)

Corollary 2.1. For any p¿ 0 and stopping time � there exists constants cp; Cp¿ 0 such that

cpcp2 E�

p(1−H)6E((M∗� )p)6Cpc

p2 E�

p(1−H): (8)

Proof. The angle bracket of the martingale M is 〈M 〉t= c22t2−2H . Apply now (7) with 〈M 〉 and stopping time� to obtain (8).


3. Proofs, H¿ 12

1. (An upper bound for H¿ 12 ): We have that Zt=

∫ t0 sH−(1=2)dYs. Use integration by parts to get the upper

estimate for Z∗:

Z∗t 62t

�Y ∗t :

For the process Y use representation (6) to get the estimate

Y ∗t 64Ht

�M∗t : (9)

From these two upper bounds we derive the following upper bound

Zt68HM∗t t2�: (10)

Note that (10) is valid for any stopping time �:

Z∗�68HM

∗� �2�:

Hence for any p¿ 0 we have that

E (Z∗� )

p6 (8H)pE(�2�p (M∗

� )p) : (11)

With H�older’s inequality, (11) with q= H=2�= H=(2H − 1)¿ 1, and r = H=(1− H) we getE(�2�p (M∗

� )p)6 (E�2�qp)1=q (E (M∗

� )pr)1=r: (12)

Apply now (8) to obtain

E (M∗� )pr6cp2CpE�

(1−2�)pr=2 = cp2CpE�pH : (13)

To �nish, note that the right-hand side inequality in (2) for H ¿ 12 follows from (12), and (13) with constant

C := C(p;H) = (8H)pcp2Cp and Cp is the constant in the Burkholder–Davis–Gundy inequality.

2. (A lower bound for H ¿ 12 ): Pick the constants a; b ∈ (0; 1) with a¡b. Then, using Proposition 2.1 we

obtain that

Mt =∫ at

0w(t; s) dZs +

∫ bt

atw(t; s) dZs +

∫ t

btw(t; s) dZs: (14)

Integrating by parts one can estimate the middle term in (14):∣∣∣∣∣∫ bt

atw(t; s) dZs

∣∣∣∣∣ = cC

∣∣∣∣∣Zbtt−2�b−�(1− b)−� − Zatt−2�a−�(1− a)−� +∫ bt

atZsd((t − s)−� s−�)

∣∣∣∣∣6C(a; b; H)t−2�Z∗

t ;

where t is �xed and

C(a; b; H) := 2cC

(b−� (1− b)−� + a−� (1− a)−�) :

Put � t:= t2�|Mt | and from the above we can conclude that

� t6t2�|Mt(a) +Mt(1− b)|+ C(a; b; H)Z∗t ; (15)

where Mt(a):=∫ at0 w(t; s) dZs and Mt(1− b)

:=∫ tbt w(t; s) dZs.

Lemma 3.1. With the above notation and for any p¿ 0 and stopping time �

E (�∗� )p¿zpE�pH : (16)


Proof. First we prove (16) for the case p= 2. Integration by parts gives

�2t =∫ t

0s4� dM 2

s +∫ t

0M 2s ds

4�;

i.e.

�2t =∫ t

0

(c22 (1− 2�) s2� + 4�s4�−1M 2

s

)ds+ 2

∫ t

0s4�Ms dMs: (17)

Hence for bounded stopping times �

E (�∗� )2¿c22

1− 2�1 + 2�

E�2H : (18)

For general � apply �rst (18) to the stopping time �∧ n. Use now Fatou’s lemma to obtain (16) for arbitrary� with p= 2.Consider now the case p¡ 2. Note �rst that the inequality (16) with p = 2 means that the predictable

process (�∗)2 dominates the process I 2H , where It= t. Using the domination results due to Lenglart we obtainfor p¡ 1

E (�∗�)p¿zpE�pH

with

zp:=

(c22(1−2�)(1+2�)

)p(4− p)

2− p (19)

(see Dellacherie and Meyer, 1982, VI.113, for the Lenglart results).To complete the proof of the lemma consider the case p¿ 2. We modify the approach developed in

Novikov (1971). Take k ¿ 0; �¿ 0 and de�ne the process � by

�t:= �+ kt2H + �2t :

From (17) we obtain that

〈�〉t = 4 c4

C4

∫ t

0M 2s s6� ds;

use this, (17) and the Ito formula to have

�p=2t = �p=2 +∫ t

0

{p2�(p=2)−1s

[(c222− 2H2H

+ k2H)s2H−1 + 4�s4H−3M 2

s

]

+p(p− 2)

2�(p=2)−2s s6�M 2

s

}ds+ Nt; (20)

where N is a continuous martingale, N0 = 0. From (20) we obtain for any bounded stopping time �

E(�p=2�

)¿ �p=2 + E

∫ �

0

p2�(p=2)−1s

(c222− 2H2H

+ 2kH)s2H−1 ds

¿ �p=2 + E∫ �

0

p2

(ks2H

)(p=2)−1(1 + 2kH)s2H−1 ds

= �p=2 +k(p=2)−1

(c22((2− 2H)=(2H)) + 2kH

)H

E�pH : (21)


Use again Fatou’s lemma to conclude that (21) holds for arbitrary stopping time � and for �=0. So we have

E(�2� + k�

2H)p=2¿k(p=2)−1(c22((2− 2H)=(2H)) + 2kH

)H

E�pH : (22)

Use the inequality(�2� + k�

2H)p=262(p=2)−1 (�p� + kp=2�pH)in (22) to obtain

E�p�¿zp(k)E�pH

with

zp(k):= kp=2

(2−(p=2)+1

((c22(2− 2H)=k2H) + 2H

)H

− 1):

We complete the proof of Lemma 3.1 by noting that zp(k)¿ 0 for small enough k.

To continue with the proof of the lower bound note that using (15) and (16) we have the followinginequality:

CpzpE�pH6 (C (a; b; H))pE (Z∗� )p + E

(�2�|M�(a) +M�(1− b)|

)p: (23)

Next we show that for any stopping time �

|M�(a)|68H(

a1− a

)H−1=2M∗� : (24)

Note �rst that integration by parts gives

Mt(a) = (t (1− a))1=2−H Yat −∫ at

0Ys d(t − s) 1=2−H :

Hence,

|Mt(a)|62 (t (1− a)) 1=2−HY ∗at ;

the estimate (9) now gives the inequality |Mt(a)|68H (a=(1− a))H−1=2M∗at and since this holds for all t, we

have shown that (24) is true.Next we prove that

|M�(1− b)|68H(1− bb

)H−1=2M∗� : (25)

Note �rst that with �xed t the process Z̃ s:= Zt −Zt−s is a fractional Brownian motion with Hurst index H ,

s ∈ (0; t). Then, introducing the substitution u= t − s we have that

Mt(1− b) =∫ t

tb(t − s)1=2−Hs1=2−H dZs =

∫ (1−b)

0u1=2−H (t − u)1=2−H dZ̃ u:

Hence, similarly to above we obtain

|Mt(1− b)|68H(1− bb

)H−1=2M̃

∗t ;

where M̃ is obtained from Z̃ using (5). It remains to note that for each t M̃∗t =M

∗t and we have (25).


Similarly to (11) and (12) we get, using also (24) that for any stopping time �

E(�2H−1M� (a)

)p6(

a1− a

)p(H−1=2)(8H)pC(H;p)E�pH (26)

and with (25)

E(�2H−1M� (1− b)

)p6(1− bb

)p(H−1=2)(8H)pC(H;p)E�pH : (27)

Use (26) and (27) in (23) to obtain

E (Z∗� )p¿

[cpzp − 2(p=2)−1((a=(1− a))p(H−1=2) + ((1− b)=b)p(H−1=2))](C (a; b; H))p

E�pH : (28)

It is clear that by taking small enough a and by taking b close to one the coe�cient on the right-hand sideof (28) is strictly positive.

4. Proofs, H ¡ 12

1. (A lower bound for H ¡ 12 ): We use representation (5):

Mt =cC

∫ t

0s−� (t − s)−� dZs:

Integration by parts gives the estimate

M∗t 6

cCB(1=2− H; 3=2− H)Z∗

t 2t1−2H : (29)

Hence for any stopping time � and any p¿ 0 we have from (29)

(M∗� )

pH=(1−H)6( cC2B(1=2− H; 3=2− H)

)pH=(1−H)(Z∗� )pH=(1−H)�(1−2H)pH=(1−H): (30)

By Proposition 2.2 we have

E (M∗� )pH=(1−H)¿cp;HE�pH : (31)

To complete, apply H�older’s inequality with q= (1−H)=H and r = (1−H)=(1− 2H) to the right-hand sideof (30) to get

E (Z∗� )p¿c(p;H)E�pH

with

c(p;H) = c(1−H)=Hp;H 2−p (B (1=2− H; 3=2− H))−p Cp

cp:

Acknowledgements

The authors are grateful to an anonymous referee for pointing out an error in the constants of Lemma 3.1.


References

Burkholder, D. L., 1973. Distribution function inequalities for martingales. Ann. Probab. 1, 19–42.Dellacherie, C., Meyer, P.-A., 1982. Probabilities and Potential B. North-Holland, Amsterdam.Dzhaparidze, K., Valkeila, E., 1990. On the Hellinger-type distances for �ltered Experiments. Probab. Theory Related Fields 85, 105–117.Lebedev, V.A., 1996. Martingales, Convergence of Probability Measures and Stochastic Equations, MAI, Moscow (in Russian).Liptser, R.S., Shiryaev, A.N., 1986. Theory of Martingales. Nauka, Moscow (in Russian).Molchan, G., Golosov, J., 1969. Gaussian stationary processes with asymptotic power spectrum. Soviet Math. Dokl. 10, 134–137.Norros, I., Valkeila, E., Virtamo, J., 1999. An elementary approach to a Girsanov formula and other analytical results on fractionalBrownian motions. Bernoulli, June issue, to appear.

Novikov, A.A., 1971. On moment inequalities for stochastic integrals.. Theory Probab. Appl. 16, 538–541.Novikov, A.A., 1975. On discontinuous martingales. Theory of Probab. and its Appl. 20, 11–26.

on some maximal inequalities for fractional brownian motions

Documents