Mixed Generalized Fractional Brownian Motion
aa r X i v : . [ m a t h . P R ] F e b MIXED GENERALIZED FRACTIONAL BROWNIAN MOTION
SHAYKHAH ALAJMI AND EZZEDINE MLIKI , Abstract.
To extend several known centered Gaussian processes, we introduce a new centred mixedself-similar Gaussian process called the mixed generalized fractional Brownian motion, which couldserve as a good model for a larger class of natural phenomena. This process generalizes both the well-known mixed fractional Brownian motion introduced by Cheridito [10] and the generalized fractionalBrownian motion introduced by Zili [31]. We study its main stochastic properties, its non-Markovianand non-stationarity characteristics and the conditions under which it is not a semimartingale. Weprove the long range dependence properties of this process. Introduction
Fractional Brownian motion on the whole real line (fBm for short) B H = { B Ht , t ∈ R } of Hurstparameter H is the best known centered Gaussian process with long-range dependence. Its covariancefunction is Cov( B Ht , B Hs ) = 12 [ | t | H + | s | H − | t − s | H ] (1.1)where H is a real number in (0 ,
1) and the case H = corresponds to the Brownian motion. Itis the unique continuous Gaussian process starting from zero, the self-similarity and stationarity ofthe increments are two main properties for which fBm enjoyed successes as modeling tool in financeand telecommunications. Researchers have applied fractional Brownian motion to a wide range ofproblems, such as bacterial colonies, geophysical data, electrochemical deposition, particle diffusion,DNA sequences and stock market indicators [2] and [23]. In particular, computer science applicationsof fBm include modeling network traffic and generating graphical landscapes [22] and [26]. The fBmwas investigated in many papers (e.g. [3, 4, 14, 17, 19]). The main difference between fBm and regularBrownian motion is that the increments in Brownian motion are independent, increments for fBm arenot.In [6], the authors suggested another kind of extension of the Brownian motion, called the sub-fractional Brownian motion (sfBm for short), which preserves most properties of the fBm, but not thestationarity of the increments. It is a centered Gaussian process ξ H = (cid:8) ξ Ht , t ≥ (cid:9) , defined by: ξ Ht = B Ht + B H − t √ , t ≥ , (1.2)where H ∈ (0 , H = corresponds to the Brownian motion. Mathematics Subject Classification.
Key words and phrases.
Mixed fractional Brownian motion, generalized fractional Brownian motion, long-rangedependence, stationnarity, Markovity, semimartingale.
The sfBm is intermediate between Brownian motion and fractional Brownian motion in the sensethat it has properties analogous to those of fBm, self-similarity, not Markovian but the incrementson nonoverlapping intervals are more weakly correlated, and their covariance decays polynomially ata higher rate in comparison with fBm (for this reason in [6] is called sfBm). So the sfBm does notgeneralize the fBm. The sfBm was investigated in many papers (e.g. [5, 6, 25, 28]).An extension of the sfBm was introduced by Zili in [30] as a linear combination of a finite numberof independent sub-fractional Brownian motions. It was called the mixed sub-fractional Brownianmotion (msfBm for short). The msfBm is a centered mixed self-similar Gaussian process and does nothave stationary increments. The msfBm do not generalize the fBm.In [31] Zili introduced new model called the generalized fractional Brownian motion (gfBm forshort) which is an extension of both sub-fractional Brownian motion and fractional Brownian motion.A gfBm with parameters a, b, and H , is a process Z H = (cid:8) Z Ht ( a, b ) , t ≥ (cid:9) defined by Z Ht ( a, b ) = aB Ht + bB H − t , t ≥ H . It was introducedby Cheridito [10] to present a stochastic model of the discounted stock price in some arbitrage-freeand complete financial markets. The mfBm is a centered Gaussian process starting from zero withcovariance functionCov( N Ht ( a, b ) , N Hs ( a, b )) = a ( t ∧ s ) + b (cid:0) t H + s H − | t − s | H (cid:1) , (1.4)with H ∈ (0 , . When a = 1 and b = 0 , the mfBm is the Brownian motion and when a = 0 and b = 1 , is the fBm. We refer also to [11, 10, 29, 27] for further information on this process.In this paper, we introduce a new stochastic model, which we call the mixed generalized fractionalBrownian motion. Definition 1.1.
A mixed generalized fractional Brownian motion (mgfBm for short) of parameters a, b, c and H ∈ (0 , is a centred Gaussian process M H ( a, b, c ) = { M Ht ( a, b, c ) , t ≥ } , defined on aprobability space (Ω , F , P ) , with the covariance function C ( t, s ) = a ( t ∧ s ) + ( b + c ) t H + s H ) − bc ( t + s ) H − ( b + c )2 | t − s | H (1.5) where t ∧ s = ( t + s − | t − s | ) . The mgfBm is completely different from all the extensions mentioned above. The process M H ( a, b, c )is motivated by the fact that this process already introduced for specific values of a , b and c . Indeed M H ( a, b,
0) is the mixed fractional Brownian motion and M H (0 , b, c ) , is the generalized fractionalBrownian motion. This why we will name M H ( a, b, c ) the mixed generalized fractional Brownian IXED GENERALIZED FRACTIONAL BROWNIAN MOTION 3 motion. It allows to deal with a larger class of modeled natural phenomena, including those withstationary or non-stationary increments.Our goal is to study the main stochastic properties of this new model, paying attention to the long-range dependence, self-similarity, increment stationary, Markovity and semi-martingale properties.2.
The main properties
Existence of the mixed generalized fractional Brownian motion M H ( a, b, c ) for any H ∈ (0 ,
1) canbe shown in the following way, consider the process M Ht ( a, b, c ) = aB t + bB Ht + cB H − t , t ≥ , (2.1)where B = { B t , t ∈ R } is a Brownian motion and B H = { B Ht , t ∈ R } is an independent fractionalBrownian motion with Hurst parameter H ∈ (0 , . Using (1.1) and since B and B H are independent we obtain the following lemma. Lemma 2.1.
For all s, t ≥ , the process (2.1) is a centered Gaussian process with covariancefunction given by (1.5).Proof. Let s, t ≥ C ( t, s ) = Cov (cid:0) M Ht ( a, b, c ) , M Hs ( a, b, c ) (cid:1) . Then C ( t, s ) = Cov (cid:2)(cid:0) aB t + bB Ht + cB H − t (cid:1) , (cid:0) aB s + bB Hs + cB H − s (cid:1)(cid:3) = a ( t ∧ s ) + b Cov ( B Ht , B Hs ) + bc [ Cov ( B Ht , B H − s )] + cb [ Cov ( B H − t , B Hs )]+ c Cov ( B H − t , B H − s )= a ( t ∧ s ) + b (cid:0) t H + s H − | t − s | H (cid:1) + bc (cid:0) t H + s H − | t + s | H (cid:1) + cb (cid:0) t H + s H − | − ( t + s ) | H (cid:1) + c (cid:0) t H + s H − | − ( t − s ) | H (cid:1) = a ( t ∧ s ) + b t H + b s H − b | t − s | H + bc t H + bc s H = a ( t ∧ s ) + ( b + c ) t H + s H ) − bc | t + s | H − ( b + c )2 | t − s | H . Hence the covariance function of the process (2.1) is precisely C ( t, s ) given by (1.5). Therefore the M H ( a, b, c ) exists. (cid:3) Remarks 2.1.
Some special cases of the mixed generalized fractional Brownian motion: (1) If a = 0 , b = 1 , c = 0 , then M H (0 , , is a fBm. (2) If a = 0 , b = c = √ , then M H (0 , √ , √ ) is a sfBm. (3) If a = 1 , b = 0 , c = 0 , then M H (1 , , is a Bm. (4) If a = 0 , then M H (0 , b, c ) , is a gfBm. (5) If c = 0 , then M H ( a, b, , is a mfBm. (6) If b = c, then M H ( a, b √ , b √ ) , is the smfBm. S. ALAJMI, E. MLIKI
So the mixed generalized fractional Brownian motion is, at the same, a generalization of the frac-tional Brownian motion, sub-fractional Brownian motion, the sub-mixed fractional Brownian motion,generalized fractional Brownian motion, mixed fractional Brownian motion and of course of the stan-dard Brownian motion.
Proposition 2.1.
The mgfBm satisfies the following properties: (1)
For all t ≥ , E (cid:0) M Ht ( a, b, c ) (cid:1) = a t + (cid:0) b + c − (2 H − bc (cid:1) t H . (2) Let ≤ s < t . Then E (cid:0) M Ht ( a, b, c ) − M Hs ( a, b, c ) (cid:1) = a | t − s | − H bc ( t H + s H )+ ( b + c ) | t − s | H + 2 bc | t + s | H . (3) We have for all ≤ s < t , a ( t − s ) + γ ( b,c,H ) ( t − s ) H ≤ E (cid:0) M Ht ( a, b, c ) − M Hs ( a, b, c ) (cid:1) ≤ a ( t − s ) + ν ( b,c,H ) ( t − s ) H where γ ( b,c,H ) = ( b + c − bc (2 H − − C ( b, c, H ) + ( b + c ) D ( b, c, H ) ,ν ( b,c,H ) = ( b + c ) C ( a, b, H ) + ( b + c − bc (2 H − − D ( b, c, H ) , C = { ( b, c, H ) ∈ R × ]0 , H > , bc ≥ or ( H < , bc ≤ } , and D = { ( b, c, H ) ∈ R × ]0 , H > , bc ≤ or ( H < , bc ≥ } . Proof. (1) It is a direct consequence of (1.5).(2) Let 0 ≤ s < t and α ( t, s ) = E (cid:0) M Ht ( a, b, c ) − M Hs ( a, b, c )) (cid:1) . Then α ( t, s ) = E (cid:0) M Ht ( a, b, c ) (cid:1) + E (cid:0) M Hs ( a, b, c ) (cid:1) − E (cid:0) M Ht ( a, b, c ) M Hs ( a, b, c ) (cid:1) = a t + b t H + 2 bct H − H bct H + c t H + a s + b s H + 2 bcs H − H bcs H + c s H − a ( t ∧ s ) − b t H − b s H + b | t − s | H − bct H − bcs H + bc | t + s | H − cbt H − cbs H + cb | t + s | H − c t H − c s H + c | t − s | H = a ( t + s ) − H bc ( t H + s H ) − a ( t ∧ s ) + ( b + c ) | t − s | H + 2 bc | t + s | H = a | t − s | − H bc ( t H + s H ) + ( b + c ) | t − s | H + 2 bc | t + s | H . (3) It is a direct consequence of the second item of Proposition 2.1 and Lemma 3 in [31]. (cid:3) Proposition 2.2.
For all ( a, b, c ) ∈ R \{ (0 , , } and H ∈ (0 , \{ } the mgfBm is not a self-similarprocess. IXED GENERALIZED FRACTIONAL BROWNIAN MOTION 5
Proof.
This follows from the fact that, for fixed h >
0, the processes (cid:8) M Hht ( a, b, c ) , t ≥ (cid:9) and (cid:8) h H M Ht ( a, b, c ) , t ≥ (cid:9) are Gaussian, centered, but don’t have the same covariance function. In-deed C ( ht, hs ) = a ( ht ∧ hs ) + b (cid:0) ( ht ) H + ( hs ) H − | ht − hs | H (cid:1) + bc (cid:0) ( ht ) H + ( hs ) H − | ht + hs | H (cid:1) + cb (cid:0) ( ht ) H + ( hs ) H − | − ( ht + hs ) | H (cid:1) + c (cid:0) ( ht ) H + ( hs ) H − | − ( ht − hs ) | H (cid:1) = a ( ht ∧ hs ) + b ht ) H + b hs ) H − b | ht − hs | H + bc ht ) H + bc hs ) H − bc | ht + hs | H + bc ht ) H + bc hs ) H − bc | ht + hs | H + c ht ) H + c hs ) H − c | ht − hs | H = a h ( t ∧ s ) + h H ( b + c ) (cid:0) ( t ) H + ( s ) H (cid:1) − bch H | t + s | H − h H ( b + c )2 | t − s | H . On the other hand,
Cov (cid:0) h H M Ht ( a, b, c ) , h H M Hs ( a, b, c ) (cid:1) = h H Cov (cid:0) M Ht ( a, b, c ) , M Hs ( a, b, c ) (cid:1) = a h H ( t ∧ s ) + h H ( b + c ) (cid:0) ( t ) H + ( s ) H (cid:1) − bch H | t + s | H − h H ( b + c )2 | t − s | H . Then the mgfBm is not a self-similar process for all ( a, b, c ) ∈ R \ { (0 , , } . (cid:3) Remarks 2.2.
As a consequence of Proposition 2.2, we see that: (1) M H (0 , b, c ) is a self-similar process for all ( b, c ) ∈ R . (2) M ( a, b, c ) is a self-similar process for all ( a, b, c ) ∈ R . Now, we will study the Markovian property.
Theorem 2.1.
Assume H ∈ (0 , \ (cid:8) (cid:9) , a ∈ R and ( b, c ) ∈ R \ { (0 , } . Then M H ( a, b, c ) is not aMarkovian process.Proof. The process M H ( a, b, c ) is a centered Gaussian. Then, if M Ht ( a, b, c ) is a Markovian process,according to Revuz and Yor [24], for all s < t < u , we would have C ( s, u ) C ( t, t ) = C ( s, t ) C ( t, u ) . We will only prove the theorem in the case where a = 0, the result with a = 0 is known in [31]. Forthe proof we follow the proof of Proposition 1 given in [31]. S. ALAJMI, E. MLIKI
Using Proposition 2.1 we get C ( s, u ) = a s + ( b + c ) u H + s H ) − bc | u + s | H − ( b + c )2 | u − s | H ,C ( t, t ) = a t + (cid:0) b + c − (2 H − bc (cid:1) t H ,C ( s, t ) = a s + ( b + c ) t H + s H ) − bc | t + s | H − ( b + c )2 | t − s | H ,C ( t, u ) = a t + ( b + c ) u H + t H ) − bc | u + t | H − ( b + c )2 | u − t | H . In the particular case where 1 < s = √ t < t < u = t , we have C ( √ t, t ) = a t + ( b + c ) t H + t H ) − bc | t + t | H − ( b + c )2 | t − t | H ,C ( t, t ) = a t + (cid:0) b + c − (2 H − bc (cid:1) t H ,C ( √ t, t ) = a t + ( b + c ) t H + t H ) − bc | t + t | H − ( b + c )2 | t − t | H ,C ( t, t ) = a t + ( b + c ) t H + t H ) − bc | t + t | H − ( b + c )2 | t − t | H . Then by using that, C ( √ t, t ) C ( t, t ) = C ( √ t, t ) C ( t, t ) , we have (cid:20) a t + ( b + c ) t H + t H ) − bc | t + t | H − ( b + c )2 | t − t | H (cid:21) × (cid:2) a t + (cid:0) b + c − (2 H − bc (cid:1) t H (cid:3) = (cid:20) a t + ( b + c ) t H + t H ) − bc | t + t | H − ( b + c )2 | t − t | H (cid:21) × (cid:20) a t + ( b + c ) t H + t H ) − bc | t + t | H − ( b + c )2 | t − t | H (cid:21) . It follows that (cid:20) a t + t H (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19)(cid:21) × (cid:2) a t + (cid:0) b + c − (2 H − bc (cid:1) t H (cid:3) = (cid:20) a t + t H (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19)(cid:21) × (cid:20) a t + t H (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19)(cid:21) . IXED GENERALIZED FRACTIONAL BROWNIAN MOTION 7
Hence a t + a (cid:0) b + c − (2 H − bc (cid:1) t H + + a (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) t H +1 + (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) × (cid:0) b + c − (2 H − bc (cid:1) t H = a t + a (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) t H + + a (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) t H +1 + (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) × (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) t H Take t H as a common factor, we get a (cid:0) b + c − (2 H − bc (cid:1) t − H + + a (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) t − H +1 + (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) × (cid:0) b + c − (2 H − bc (cid:1) = a (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) t − H + + a (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) t − H +1 + (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) × (cid:18) ( b + c ) t − H ) − bc | t − | H − ( b + c )2 | − t − | H (cid:19) . S. ALAJMI, E. MLIKI
Therefore a (cid:0) b + c − (2 H − bc (cid:1) t − H + + a [ 12 ( b + c ) (cid:0) t − H (cid:1) − bc (cid:16) Ht − + H (2 H − t − + ◦ ( t − ) (cid:17) −
12 ( b + c ) (cid:16) − Ht − + H (2 H − t − + ◦ ( t − ) (cid:17) ] t − H +1 + [ 12 ( b + c ) (cid:0) t − H (cid:1) − bc (cid:16) Ht − + H (2 H − t − + ◦ ( t − ) (cid:17) −
12 ( b + c ) (cid:16) − Ht − + H (2 H − t − + ◦ ( t − ) (cid:17) ] × [ b + c − (2 H − bc ]= a [ 12 ( b + c ) (cid:0) t − H (cid:1) − bc (cid:0) Ht − + H (2 H − t − + ◦ ( t − ) (cid:1) −
12 ( b + c ) (cid:0) − Ht − + H (2 H − t − + ◦ ( t − ) (cid:1) ] t − H + + a [ 12 ( b + c ) (cid:0) t − H (cid:1) − bc (cid:16) Ht − / + H (2 H − t − + ◦ ( t − ) (cid:17) −
12 ( b + c ) (cid:16) − Ht − + H (2 H − t − + ◦ ( t − ) (cid:17) ] t − H +1 + [ 12 ( b + c ) (cid:0) t − H (cid:1) − bc (cid:16) Ht − + H (2 H − t − + ◦ ( t − ) (cid:17) −
12 ( b + c ) (cid:16) − Ht − + H (2 H − t − + ◦ ( t − ) (cid:17) ] × [ 12 ( b + c ) (cid:0) t − H (cid:1) − bc (cid:0) Ht − + H (2 H − t − + ◦ ( t − ) (cid:1) −
12 ( b + c ) (cid:0) − Ht − + H (2 H − t − + ◦ ( t − ) (cid:1) ] . First case: 0 < H < , a = 0 and b + c = 0. By Taylor’s expansion we get, as t → ∞ , a (cid:0) b + c − (2 H − bc (cid:1) t − H + + a
12 ( b + c ) t − H +1 + 12 ( b + c ) [ b + c − (2 H − bc ] t − H ≈ a
12 ( b + c ) t − H + + a
12 ( b + c ) t − H +1 + 14 ( b + c ) t − H . Therefore a (cid:0) b + c − (2 H − bc (cid:1) t − H + + 12 ( b + c ) [ b + c − (2 H − bc ] t − H ≈ a
12 ( b + c ) t − H + + 14 ( b + c ) t − H , which is true if and only if( b − c ) − (2 H − bc = 0 and a ( b − c ) − a (2 H − bc = 0 . However, it is easy to check that ( b − c ) − (2 H − bc > a b − c ) − a (2 H − bc > c, b and every real a . IXED GENERALIZED FRACTIONAL BROWNIAN MOTION 9
Second case: 0 < H < , a = 0 and b + c = 0. By Taylor’s expansion we get, as t → ∞ , a [ b + c − (2 H − bc ] t − H + + a (cid:2) − Hbc + ( b + c ) H (cid:3) t − H − + (cid:2) − Hbc + ( b + c ) H (cid:3) × (cid:2) b + c − (2 H − bc (cid:3) t − ≈ a (cid:2) − Hbc + ( b + c ) H (cid:3) t − H − + a (cid:2) − Hbc + ( b + c ) H (cid:3) t − H + + (cid:2) − Hbc + ( b + c ) H (cid:3) t − . Hence a [ b + c − (2 H − bc ] t − H + + (cid:2) − Hbc + ( b + c ) H (cid:3) × (cid:2) b + c − (2 H − bc (cid:3) t − ≈ a (cid:2) − Hbc + ( b + c ) H (cid:3) t − H + + (cid:2) − Hbc + ( b + c ) H (cid:3) t − , which is true if and only if b = c = 0. This is a contradiction.Third case: < H < a = 0 and b − c = 0. By Taylor’s expansion we get, as t → ∞ , a [ b + c − (2 H − bc ] t − H + + a H ( b − c ) t − H − + H ( b − c ) [ b + c − (2 H − bc ] t − ≈ a H ( b − c ) t − H − + a H ( b − c ) t − H + + H ( b − c ) t − . Then a [ b + c − (2 H − bc ] t − H + + H ( b − c ) [ b + c − (2 H − bc ] t − ≈ a H ( b − c ) t − H + + H ( b − c ) t − , which is true if and only if (cid:2) b (1 − H ) + c (1 − H ) + (2 − H + 2 H ) bc (cid:3) = 0 . However, it is easy to check that b (1 − H ) + c (1 − H ) + (2 − H + 2 H ) bc > c, b and everyreal a .Fourth case: < H < a = 0 and b − c = 0. By Taylor’s expansion we get, as t → ∞ , a (cid:0) b + c − (2 H − bc (cid:1) t − H + + 12 ( b + c ) [ b + c − (2 H − bc ] t − H ≈ a
12 ( b + c ) t − H + + 14 ( b + c ) t − H , which is true if and only if 2 − H = 0 . This contradicts the fact that H = . (cid:3) Let us check the mixed-self-semilarity property of the mgfBm. This property was introduced in[29] for the mfBm and investigated to show the H¨older continuity of the mfBm. See also [12] for thesfBm case.
Proposition 2.3.
For any h > , (cid:8) M Hht ( a, b, c ) (cid:9) △ = n M Ht ( ah , bh H , ch H ) o . where △ = ”to have the same law”. Proof.
For fixed h >
0, the processes { M Hht ( a, b, c ) } and { M Ht ( ah , bh H , ch H ) } are Gaussian andcentered. Therefore, one only have to prove that they have the same covariance function. But, forany s and t in R + , since B and B H are independent, then Cov (cid:0) M Hht ( a, b, c ) , M Hhs ( a, b, c ) (cid:1) = a h ( t ∧ s ) + ( b + c )2 (cid:2) h H (cid:0) t H + s H − | t − s | H (cid:1)(cid:3) + bc (cid:2) h H (cid:0) t H + s H − | t + s | H (cid:1)(cid:3) = Cov (cid:16) M Ht ( ah , bh H , ch H ) , M Hs ( ah , bh H , ch H ) (cid:17) . (cid:3) Proposition 2.4.
For all ( a, b, c ) ∈ R \ { (0 , , } , the increments of the M H ( a, b, c ) are not station-ary.Proof. Let ( a, b, c ) ∈ R \ { (0 , , } . For a fixed t ≥ { P t , t ≥ } define by P t = M Ht + s ( a, b, c ) − M Hs ( a, b, c ) . Using Proposition 2.1 we get
Cov ( P t , P t )] = E (cid:2) ( M Ht + s ( a, b, c ) − M Hs ( a, b, c )) (cid:3) = a ( t + s + s ) − H bc (( t + s ) H + s H ) − a s + ( b + c ) | t + s − s | H + 2 bc | t + s + s | H = a ( t + 2 s ) − H bc (( t + s ) H + s H ) − a s + ( b + c ) t H + 2 bc | t + 2 s | H . Using Proposition 2.1 we get
Cov ( M Ht ( a, b, c ) , M Ht ( a, b, c )) = a t + (cid:0) b + c − (2 H − bc (cid:1) t H . Since both processes are centered Gaussian, the inequality of covariance functions implies that P t doesnot have the same distribution as M Ht ( a, b, c ). Thus, the incremental behavior of M H ( a, b, c ) at anypoint in the future is not the same. Hence the increments of M H ( a, b, c ) are not stationary. (cid:3) Remarks 2.3.
As a consequence of Proposition 2.4, we see that: (1) the increments of M H (0 , b, c ) are not stationary for all ( b, c ) ∈ R \ { (0 , } . (2) the increments of M H ( a, b, are stationary for all ( a, b ) ∈ R . Proposition 2.5. (1)
Let H ∈ (0 , . The mgfBm admits a version whose sample paths are almostH¨older continuous of order strictly less than ∧ H. (2) When b or c not zero and H ∈ (0 , \ { } the mgfBm is not a semi-martingale.Proof. (1) Let s and t in R + and α = 2 . The proof follows by Kolmogorov criterion from Lemma3 in [31] and using Proposition 2.1 we get E (cid:0) | M Ht ( a, b, c ) − M Hs ( a, b, c ) | α (cid:1) = a | t − s | − H bc ( t H + s H )+ ( b + c ) | t − s | H + 2 bc | t + s | H ≤ C α | t − s | α ( ∧ H ) where C α = (cid:0) a + ν ( b, c, H ) (cid:1) and ν ( b,c,H ) is given in Lemma 3 in [31]. IXED GENERALIZED FRACTIONAL BROWNIAN MOTION 11 (2) Suppose first that
H < . We get from Proposition 2.1 E (cid:0) M Ht ( a, b, c ) − M Hs ( a, b, c ) (cid:1) ≥ γ ( b,c,H ) ( t − s ) H . Since 2
H < γ ( b,c,H ) > H > . We get from Properties 2.1 a ( t − s ) + γ ( b,c,H ) ( t − s ) H ≤ E (cid:0) M Ht ( a, b, c ) − M Hs ( a, b, c ) (cid:1) ≤ ( a + ν ( b,c,H ) )( t − s ) ∧ H then γ ( b,c,H ) ( t − s ) H ≤ E (cid:0) M Ht ( a, b, c ) − M Hs ( a, b, c ) (cid:1) ≤ ( a + ν ( b,c,H ) )( t − s ) H . Since 1 < H < ν ( b,c,H ) > (cid:3) Long range dependence of the mgfBm
Definition 3.1.
We say that the increments of a stochastic process X are long-range dependent if forevery integer p ≥ , we have X n ≥ R X ( p, p + n ) = ∞ , where R X ( p, p + n ) = E (( X p +1 − X p )( X p + n +1 − X p + n )) . This property was investigated in many papers (e.g. [2, 8, 9, 5, 14]).
Theorem 3.1.
For every a ∈ R and ( b, c ) ∈ R \ { (0 , } the increments of M H ( a, b, c ) are long-rangedependent if and only if H > and b = c .Proof. For all n ∈ N \ { } and p ≥ R M ( p, p + n ) = E (cid:0) ( M Hp +1 ( a, b, c ) − M Hp ( a, b, c ))( M Hp + n +1 ( a, b, c ) − M Hp + n ( a, b, c )) (cid:1) = E (cid:0) M Hp +1 ( a, b, c ) M Hp + n +1 ( a, b, c ) (cid:1) − E (cid:0) M Hp +1 ( a, b, c ) M Hp + n ( a, b, c ) (cid:1) − E (cid:0) M Hp ( a, b, c ) M Hp + n +1 ( a, b, c ) (cid:1) + E (cid:0) M Hp ( a, b, c ) M Hp + n ( a, b, c ) (cid:1) = C ( p + 1 , p + n + 1) − C ( p + 1 , p + n ) − C ( p, p + n + 1) + C ( p, p + n )= a ( p + 1) + ( b + c ) (cid:0) ( p + 1) H + ( p + n + 1) H (cid:1) − bc (2 p + n + 2) H − ( b + c )2 n H − a ( p + 1) − ( b + c ) (cid:0) ( p + 1) H + ( p + n ) H (cid:1) + bc (2 p + n + 1) H + ( b + c )2 | n − | H − a p − ( b + c ) (cid:0) ( p ) H + ( p + n + 1) H (cid:1) + bc (2 p + n + 1) H + ( b + c )2 | n + 1 | H . Hence R M ( p, p + n ) = ( b + c )2 (cid:0) ( n + 1) H − n H + ( n − H (cid:1) − bc (cid:0) (2 p + n + 2) H − p + n + 1) H + (2 p + n ) H (cid:1) . Then for every integer p ≥
1, by Taylor’s expansion, as n → ∞ , we have R M ( p, p + n ) = b + c n H "(cid:18) n (cid:19) H − (cid:18) − n (cid:19) H − bcn H "(cid:18) p + 2 n (cid:19) H − (cid:18) p + 1 n (cid:19) + (cid:18) pn (cid:19) H = H (2 H − n H − ( b − c ) − H (2 H − H − bc (2 p + 1) n H − (1 + ◦ (1)) . If b = c , we see that as n → ∞ , R M ( p, p + n ) ≈ H (2 H − n H − ( b − c ) . Then X n ≥ R M ( p, p + n ) = ∞ ⇔ H − > − ⇔ H > . If b = c , then, as n → ∞ , R M ( p, p + n ) ≈ H (2 H − H − a (2 p + 1) n H − . For every H ∈ (0 , H − < − X n ≥ R M ( p, p + n ) < ∞ . (cid:3) Remarks 3.1. (1)
For all a ∈ R and b ∈ R \ { } the increments of M H ( a, b, are long-rangedependent if and only if H > . (2) If b = c = √ the increments of M H (0 , √ , √ ) are short-range dependent if and only if H ∈ (0 , . But if b = c the increments of M H (0 , b, c ) are long-range dependent if and only if H > . (3) From [6] , the increments of M H (0 , √ , √ ) on intervals [ u, u + r ] , [ u + r, u + 2 r ] are moreweakly correlated than those of M H (0 , , . (4) From [32] , If
H > , b + c = 1 and bc ≥ , the increments of M H (0 , b, c ) are more weaklycorrelated than those of M H (0 , , , but more strongly correlated than those of M H (0 , √ , √ ) . (5) From [32] , If H ≥ , ( bc ≤ and ( b − c ) ≤ or ( bc ≥ and b + c ≤ , the increments of M H (0 , b, c ) are more strongly correlated than those of both M H (0 , , and M H (0 , √ , √ ) . IXED GENERALIZED FRACTIONAL BROWNIAN MOTION 13
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