A decomposition of the bifractional Brownian motion and some applications

Statistics and Probability Letters 79 (2009) 619–624

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A decomposition of the bifractional Brownian motion and some applications Pedro Lei ∗ , David Nualart Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

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Article history: Received 24 March 2008 Received in revised form 8 October 2008 Accepted 10 October 2008 Available online 19 October 2008

a b s t r a c t In this paper we have shown a decomposition of the bifractional Brownian motion with parameters H , K into the sum of a fractional Brownian motion with Hurst parameter HK plus a stochastic process with absolutely continuous trajectories. Some applications of this decomposition are discussed. © 2008 Elsevier B.V. All rights reserved.

MSC: 60G15

1. Introduction The bifractional Brownian motion is a generalization of the fractional Brownian motion, defined as a centered Gaussian H ,K process BH ,K = (Bt , t ≥ 0), with covariance RH ,K (t , s) = 2−K ((t 2H + s2H )K − |t − s|2HK ),

(1) H ,1

where H ∈ (0, 1) and K ∈ (0, 1]. Note that, if K = 1 then B is a fractional Brownian motion with Hurst parameter H ∈ (0, 1), and we denote this process by BH . Some properties of the bifractional Brownian motion have been studied by Houdré and Villa (2003) and Russo and Tudor (2006). In particular, Russo and Tudor (2006) show that the bifractional Brownian motion behaves as a fractional Brownian motion with Hurst parameter HK . The stochastic calculus with respect to the bifractional Brownian motion has been recently developed by Kruk, Russo and Tudor (2007) and Es-Sebaiy and Tudor (2007). The purpose of this note is to show a decomposition of the bifractional Brownian motion as the sum of a fractional Brownian motion with Hurst parameter HK plus a process with absolutely continuous trajectories. This decomposition leads to a better understanding as well as simple proofs of some of the properties of the bifractional Brownian motion that have been obtained in the literature. 2. Preliminaries Suppose that BH ,K is a bifractional Brownian motion with covariance (1). The following properties have been proved by Houdré and Villa (2003) and summarized by Russo and Tudor (2006). (i) The bifractional Brownian motion with parameters (H , K ) is HK -self-similar, that is, for any a > 0, the processes H ,K (a−HK Bat , t ≥ 0) and (BHt ,K , t ≥ 0) have the same distribution. This is an immediate consequence of the fact that the covariance function is homogeneous of order 2HK .

∗

Corresponding author. E-mail addresses: [email protected] (P. Lei), [email protected] (D. Nualart).

0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.10.009

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(ii) For every s, t ∈ [0, ∞), we have



H ,K

2−K |t − s|2HK ≤ E (Bt

 − BHs ,K )2 ≤ 21−K |t − s|2HK .

(2)

This inequality shows that the process BH ,K is a quasi-helix in the sense of Kahane (1981, 1985). Applying Kolmogorov’s continuity criterion, it follows that BH ,K has a version with Hölder continuous trajectories of order δ for any δ < HK . Note that the bifractional Brownian motion does not have stationary increments, except in the case K = 1. It turns out that the bifractional Brownian motion is related to some stochastic partial differential equations. For example, suppose that (u(t , x), t ≥ 0, x ∈ R) is the solution of the one-dimensional stochastic heat equation on R with initial condition u(0, x) = 0

∂u 1 ∂ 2u ∂ 2W = + , 2 ∂t 2 ∂x ∂t∂x where W = {W (t , x), t ≥ 0, x ∈ R} is a two-parameter Wiener process. In other words, W is a centered Gaussian process with covariance E (W (t , x)W (s, y)) = (t ∧ s)(|x| ∧ |y|). Then, for any x ∈ R, the process (u(t , x), t ≥ 0) is a bifractional Brownian motion with parameters H = K = − 81

1 4

by the constant (2π ) 2 u(t , x) =

Z tZ 0

1 , 4

multiplied

. In fact,

pt −s (x − y)W (ds, dy), R

where pt (x) = √1 e− 2π

x2 2

, and the covariance of u(t , x) is given by

E(u(t , x)u(s, x)) =

t ∧s

Z

Z

0

pt −r (x − y)ps−r (x − y)dydr R

t ∧s

Z

pt +s−2r (0)dr

= 0

1

= √

2π

p √ ( t + s − |t − s|).

3. A decomposition of the bifractional Brownian motion Consider the following decomposition of the covariance function of the bifractional Brownian motion: RH ,K (t , s) =

1 2K

[(t 2H + s2H )K − t 2HK − s2HK ] +

1 2K

[t 2HK + s2HK − |t − s|2HK ].

(3)

The second summand in the above equation is the covariance of a fractional Brownian motion with Hurst parameter HK . The first summand turns out to be non-positive definite. As a consequence, the opposite of this summand will be the covariance of a Gaussian process. In order to define this process, consider a standard Brownian motion (Wθ , θ ≥ 0). For any 0 < K < 1, define the process X K = (XtK , t ≥ 0) by XtK =

∞

Z

(1 − e−θ t )θ −

1+K 2

dWθ .

(4)

0

Then, X K is a centered Gaussian process with covariance:

γ (t , s) = E[ K

XtK XsK

∞

Z

(1 − e−θ t )(1 − e−θ s )θ −1−K dθ

]= 0

=

Γ (1 − K ) K

[t K + sK − (t + s)K ].

(5)

In this way we obtain the following result. Proposition 1. Let BH ,K be a bifractional Brownian motion, and suppose that (Wθ , θ ≥ 0) is a Brownian motion independent H ,K of BH ,K . Let X K be the process defined in (4). Set Xt = XtK2H . Then, the processes (C1 XtH ,K + BHt ,K , t ≥ 0) and (C2 BHK t , t ≥ 0) have the same distribution, where C1 =

q

2−K K

Γ (1−K )

and C2 = 2

1−K 2

.

P. Lei, D. Nualart / Statistics and Probability Letters 79 (2009) 619–624 H ,K

Proof. Let Yt = C1 Xt

+ BHt ,K . Then, from (3) and (5) for s, t ≥ 0, we have H ,K

E(Ys Yt ) = C12 E(XsK2H XtK2H ) + E(BHs ,K Bt 1

=

2K 1

=

2K

621

)

(t 2HK + s2HK − (t 2H + s2H )K ) +

1

((t 2H + s2H )K − |t − s|2HK )

2K

(t 2HK + s2HK − |t − s|2HK ),

which completes the proof.



The next result provides some regularity properties for the process X K . Theorem 2. The process X K has a version with trajectories which are infinitely differentiable trajectories on (0, ∞) and absolutely continuous on [0, ∞). Proof. Note that E[(XtK )2 ] = C3 t K , where C3 = exists because

E[Yt2 ] =

∞

Z

Γ (1−K ) K

(2 − 2K ). For any t > 0, define Yt =

R∞ 0

θ

1−K 2

e−θ t dWθ . This integral

θ 1−K e−2θ t dθ = Γ (K )2K −2 t K −2 .

0

Applying Fubini’s theorem and the fact that Ys is a normal random variable, we have t

Z

r

 |Ys |ds =

E

2

Z tp

π

0

r 2

E[|Ys | ]ds =

0

2

K

Γ (K ) 2 2 −1 π

t

Z

s

K −2 2

ds < ∞.

0

On the other hand, applying stochastic Fubini’s theorem, we have t

Z

∞

Z t Z

θ

Ys ds = 0

0

e

−θ s

 dWθ

θ−

=

∞

Z

θ

ds =

0

∞

Z

1−K 2

0 1+K 2

1−K 2

t

Z

e

−θ s

 ds

dWθ

0

(1 − e−θ t ) dWθ = X K (t ).

0

This implies that X K is absolutely continuous and Yt = (XtK )0 on (0, ∞). Similarly, the nth derivative of X K exists on (0, ∞) and it is given by

(XtK )(n) =

∞

Z

1

K

(−1)n−1 (θ )n− 2 − 2 e−θ t dWθ .  0

The next proposition provides some information about the behavior of X K at the origin. Proposition 3. There exists a non-negative random variable G(ω) such that for all 0 < t <

1 , e

p |Xt | ≤ G(ω) t K log log t −1 . Proof. Applying an integration by parts yields ∞

Z

ϕ(θ , t )Wθ dθ ,

Xt = 0

where

ϕ(θ , t ) = te−θ t θ −

1+K 2

−

1 + K − 3+K θ 2 (1 − e−θ t ). 2

(6)

By the law of iterated logarithm for the Brownian motion, given c > 1 we can find two random points 0 < t0 < t1 , with t0 < 1e , t1 > e, such that almost surely, for all θ ≤ t0 ,

p |Wθ | ≤ c 2θ log log θ −1 ,

(7)

and for all θ ≥ t1 ,

p |Wθ | ≤ c 2θ log log θ .

(8)

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Then we make the decomposition t0

Z

Wθ ϕ(θ , t )dθ +

Xt =

t1

Z

Wθ ϕ(θ , t )dθ +

Wθ ϕ(θ , t )dθ . t1

t0

0

∞

Z

For the first term, using (7) and the estimate

|ϕ(θ , t )| ≤ t θ −



1+K 2

e−θ t +

1+K



≤ 2t θ −

2

1+K 2

,

(9)

we obtain

Z

t0 0

Z Wθ ϕ(θ , t )dθ ≤ c

t0

p

2θ log log θ −1 |ϕ(θ , t )|dθ

0 t0

Z ≤ 2ct

K

θ2

p

2 log log θ −1 dθ

0

= tG1 , for some non-negative random variable G1 . For the second term, using (9), we have

Z

t1 t0



Wθ ϕ(θ , t )dθ ≤ 2t

t1

Z

|Wθ |θ −

1+K 2

dθ = tG2 .

t0

With the change of variables η = θ t, (6) yields

|ϕ(θ , t )| ≤ te−η =t

3+K 2

 η − 1+2 K t



1+K 2

e−η η−

 η − 3+2 K

(1 ∧ η)  3+K + η− 2 (1 ∧ η) .

+

t

(10)

Using (8) and (10), we obtain the following estimate for the third term

Z

∞ t1

Z Wθ ϕ(θ , t )dθ ≤ c

∞

p

√ ≤

2θ log log θ|ϕ(θ , t )|dθ

t1

2ct

K 2

Z

∞

r log log

tt1

η t

K

K



e−η η− 2 + η−1− 2 (1 ∧ η) dη.

Applying the inequality log | log η| + log t −1 ≤ log 2 max(| log η|, log t −1 )







≤ log 2 + | log | log η|| + log log t −1 ), we obtain

Z

∞ t1

K p Wθ ϕ(θ , t )dθ ≤ G3 t 2 log log t −1 ,

for some constant G3 . This completes the proof.



4. Applications We first describe the space of integrable functions with respect to the bifractional Brownian motion. Suppose that X = (Xt , t ∈ [0, T ]) is a continuous zero mean Gaussian process. Denote by E the set of step functions on [0, T ]. Let HX be the Hilbert space defined as the closure of E with respect to the scalar product

h1[0,t ] , 1[0,s] iH = E(Xt Xs ). The mapping 1[0,t ] → Xt can be extended to a linear isometry between HX and the Gaussian space H1 (X ) associated with X . We will denote this isometry by ϕ → X (ϕ). The problem is to find HX for a particular process X . In the case of the standard Brownian motion B, the space HB is L2 ([0, T ]). For the fractional Brownian motion BH with Hurst parameter H ∈ (0, 21 ), it is known (see Decreusefond and Üstünel (1999)) that HBH coincides with the fractional 1

−H

Sobolev space I02+ (L2 ([0, T ])). In the case H >

1 , 2

the space HBH contains distributions, according to the work by Pipiras

and Taqqu (2001). In a recent work, Jolis (2007) has proved that if H > 12 , the space HBH is the set of restrictions to the space of smooth functions D (0, T ) of the distributions of W 1/2−H ,2 (R) with support contained in [0, T ]. H ,K For the bifractional Brownian motion we can prove the following result. As before, we denote by Xt the process XtK2H .

P. Lei, D. Nualart / Statistics and Probability Letters 79 (2009) 619–624

623

Proposition 4. For H ∈ (0, 1) and K ∈ (0, 1], the equality HX H ,K ∩ HBH ,K = HBHK holds. Proof. For any step function ϕ ∈ E the following equality is a consequence of the decomposition proved in Proposition 1 and the independence of X K and BH ,K

Z C1 E

T

ϕ(t )

H ,K dXt

0

2 ! +E

Z

T

H ,K dBt

ϕ(t ) 0

2 ! = C2 E

Z

T

ϕ(t )

dBHK t

0

2 ! ,

where C1 and C2 are positive constants. The equality HX H ,K ∩ HBH ,K = HBHK follows immediately.



On the other hand, for any step function ϕ ∈ E , it holds that

E(X

H ,K

(ϕ) ) ≤ CH ,K 2

2

T

Z

|ϕ(t )|t

HK −1

dt

,

0

where CH ,K is a constant depending only on H and K . As a consequence, L1 ([0, T ]; t HK −1 dt ) ⊂ HX H ,K . The proof is sketched as follows. By taking partial derivative of the covariance function γ K given in (5) it follows that

∂ 2 γ K (s2H , t 2H ) = CH ,K (t 2H + s2H )K −2 t 2H −1 s2H −1 , ∂ s∂ t for some constant CH ,K . Then, for any ϕ ∈ E it holds that Z TZ T |ϕ(s)ϕ(t )|(st )2H −1 (t 2H + s2H )K −2 dsdt 0

0 T

Z

T

Z

|ϕ(s)ϕ(t )|(st )2H −1 (s2H t 2H )

≤ 0

K −2 2

dsdt

0

Z

T

=

|ϕ(t )|t HK −1 dt

2

.

0

By Hölder’s inequality this implies that Lp ([0, T ]) ⊂ HX H ,K for any p >

1 HK

. As a consequence,

L1 ([0, T ]; t HK −1 dt ) ∩ HBH ,K ⊂ HX H ,K ∩ HBH ,K = HBHK . In the case HK < 1 −HK 2

I0+

1 , 2

this implies that a function in HBH ,K which is in L1 ([0, T ]; t HK −1 dt ) must belong to the Sobolev space

(L2 ([0, T ])). Consider now the notion of α -variations for a continuous process X = (Xt , t ≥ 0). The process X admits an α -variation if n,α

Vt

(X ) =

n−1 X

|∆Xti |α

(11)

i=0

converges in probability as n tends to infinity for all t ≥ 0, where ti = itn and ∆Xti = Xti+1 − Xti . As a consequence of the Ergodic Theorem and the scaling property of the fractional Brownian motion, it is easy to show (see, for instance, Roger (1997)) that the fractional Brownian motion with Hurst parameter H ∈ (0, 1) has an H1 -variation equal to CH t, where CH = E(|ξ |H ) and ξ is a standard normal random variable. Then, Proposition 1 allows us to obtain the 1 -variation of bifractional Brownian motion. This provides a simple proof of a similar result in Russo and Tudor (2006). HK Proposition 5. The bifractional Brownian motion with parameters H and K has a CHK = E(|ξ |HK ) and ξ is a standard normal random variable.

1 HK

1

-variation equal to C2HK CHK t, where

Proof. We have to show that for all ε > 0 and δ > 0, there exists n0 such that for all n ≥ n0 ,

! n −1 X 1 H ,K 1 HK P |∆Bti | HK − C2 CHK t > ε < δ. i =0 H ,K

Clearly, these probabilities are the same if we replace BH ,K by the process C2 BHK − C1 Xt t suffices to show the result for this process. Applying Minkowski’s inequality, n −1 X i =0

|

1 ∆BtHi ,K HK

|

!HK ≤ C2

n−1 X i=0

!HK 1 HK ∆BHK ti

|

|

+ C1

n −1 X i=0

1 ∆XtHi ,K HK

|

|

!HK .

given in Proposition 1, and it

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P. Lei, D. Nualart / Statistics and Probability Letters 79 (2009) 619–624

On the other hand, C2

n −1 X

!HK |

1 HK ∆BHK ti

− C1

|

n−1 X

1 ∆XtHi ,K HK

!HK

|

|

≤

i=0

i =0

n −1 X

|

1 ∆BHti ,K HK

!HK .

|

i =0

From the results for the fractional Brownian motion we know that 1 n, HK

lim Vt

n→∞

(BHK ) = CHK t

almost surely and in L1 . To complete the proof, it is enough to show that n −1 X

|∆XtHi ,K | HK ≤ sup |∆XtHi ,K | HK −1 1

1

i

i=0

n −1 X

Pn−1 i =0

|∆XtHi ,K | HK converges to zero. We can write 1

|∆XtHi ,K |.

i =0

The first factor in the above expression converges to zero by continuity, and the second factor is bounded by the total variation of X H ,K on [0, T ] since X H ,K is absolutely continuous on [0, T ] by Theorem 2. The proof is complete.  Similarly, for the lim

ε→0

1

t

Z

ε

1 HK

-strong variation of the process BH ,K we can show that, in probability, 1

,K |BsH+ε − BHs ,K | HK ds = C2HK CHK t . 1

0

Tudor and Xiao (2007) have proved the Chung’s law of the iterated logarithm for the bifractional Brownian motion: H ,K

H ,K

max |Bt +t0 − Bt0 |

lim inf r →0

t ∈[0,r ]

r HK /(log log(1/r ))HK

= C0 (HK ),

(12)

where C0 is a positive and finite constant depending on HK, for all t0 ≥ 0. A similar result for the fractional Brownian motion was obtained by Monrad and Rootzén (1995). The decomposition obtained in this paper allows us to deduce Chung’s law of iterated logarithm for t0 > 0 for the bifractional Brownian motion, from the same result for the fractional Brownian motion with Hurst parameter HK , with the same constant. The proof will be based on the fact that the process C2 BHK − C1 XtH ,K , t introduced in Proposition 1, is a bifractional Brownian motion. Let us finally remark that the decomposition established in this paper permits developing a stochastic calculus for the bifractional Brownian motion if we use the well-known results in the literature on the stochastic integration with respect to the fractional Brownian motion, and take into account that the process X H ,K has absolutely continuous trajectories (see Nualart (2003, 2006) and the references therein). Acknowledgement D. Nualart is supported by the NSF grant DMS0604207. References Decreusefond, L., Üstünel, A.S., 1999. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214. Es-Sebaiy, K., Tudor, C.A., 2007. Multidimensional bifractional Brownian motion: Itô and Tanaka’s formulas. Stoch. Dyn. 3, 365–388. Houdré, C., Villa, J., 2003. An example of infinite dimensional quasi-helix. Contemp. Math. 366, 195–201. Jolis, M., 2007. On the Wiener integral with respect to the fractional Brownian motion on an interval. J. Math. Anal. Appl. 330, 1115–1127. Kahane, J.P., 1981. Helices at quasi-helices. Adv. Math. 7B, 417–433. Kahane, J.P., 1985. Some Random Series of Functions. Cambridge University Press. Kruk, I., Russo, F., Tudor, C.A., 2007. Wiener integrals, Malliavin calculus and covariance structure measure. J. Funct. Anal. 249, 92–142. Monrad, D., Rootzén, H., 1995. Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Theory Related Fields 101, 173–192. Nualart, D., 2003. Stochastic integration with respect to fractional Brownian motion and applications. Contemp. Math. 336, 3–39. Nualart, D., 2006. The Malliavin Calculus and Related Topics. Springer Verlag. Pipiras, V., Taqqu, M.S., 2001. Are classes of deterministic integrands for fractional Brownian motion on an interval complete. Bernoulli 6, 873–897. Rogers, L.C.G., 1997. Arbitrage with fractional Brownian motion. Math. Finance 7, 95–105. Russo, F., Tudor, C.A., 2006. On the bifractional Brownian motion. Stoch. Process. Appl. 5, 830–856. Tudor, C.A., Xiao, Y., 2007. Sample path properties of bifractional Brownian motion. Bernoulli 13, 1023–1052.