On the long range dependence of time-changed mixed fractional Brownian motion model
OON THE LONG RANGE DEPENDENCE OF TIME-CHANGED MIXEDFRACTIONAL BROWNIAN MOTION MODEL
SHAYKHAH ALAJMI AND EZZEDINE MLIKI , Abstract.
A time-changed mixed fractional Brownian motion is an iterated process constructed asthe superposition of mixed fractional Brownian motion and other process. In this paper we considermixed fractional Brownian motion of parameters a, b and H ∈ (0 ,
1) time-changed by two processes,gamma and tempered stable subordinators. We present their main properties paying main attentionto the long range dependence. We deduce that the fractional Brownian motion time-changed bygamma and tempered stable subordinators has long range dependence property for all H ∈ (0 , Introduction
The fractional Brownian motion (fBm) B H = { B Ht , t ≥ } with parameter H , is a centeredGaussian process with covariance functionCov( B Ht , B Hs ) = 12 [ t H + s H − | t − s | H ] , s, t ≥ , (1.1)where H is a real number in (0 , , called the Hurst index or Hurst exponent. The case H = corresponds to the Brownian motion (Bm).An extension of the fBm was introduced by Cheridito [5], called the mixed fractional Brownianmotion (mfBm for short) which is a linear combination between a Brownian motion and an independentfractional Brownian motion of Hurst exponent H , with stationary increments exhibit a long-rangedependent for H > . A mfBm of parameters a, b and H is a process N H ( a, b ) = { N Ht ( a, b ) , t ≥ } ,defined on the probability space (Ω , F , P ) by N Ht ( a, b ) = aB t + bB Ht , where B = { B t , t ≥ } is a Brownian motion and B H = (cid:8) B Ht , t ≥ (cid:9) is an independent fractionalBrownian motion of Hurst exponent H ∈ (0 , Y Hβ ( a, b ) = { Y Hβ t ( a, b ) , t ≥ } = { N Hβ t ( a, b ) , t ≥ } , where the parent process N H ( a, b ) is a mfBm with parameters a, b, H ∈ (0 ,
1) and the subordinator β = { β t , t ≥ } is assumed to be independent of both the Brownian motion and the fractionalBrownian motion. If H = , the process Y β (0 ,
1) is called subordinated Brownian motion, it was
Mathematics Subject Classification.
Key words and phrases.
Mixed fractional Brownian motion, long-range dependence, subordination, Temperedstable subordinator, Gamma subordinator . a r X i v : . [ m a t h . P R ] F e b S. ALAJMI, E. MLIKI investigated in [17, 24]. Also, the process Y Hβ (0 ,
1) is called subordinated fractional Brownian motionit was investigated in [15, 16].Time-changed process is constructed by taking superposition of tow independent stochastic sys-tems. The evolution of time in external process is replaced by a non-decreasing stochastic process,called subordinator. The resulting time-changed process very often retain important properties ofthe external process, however certain characteristics might change. This idea of subordination wasintroduced by Bochner [4] and was explored in many papers (e.g. [1, 11, 12, 15, 21, 22, 25, 27, 30]).The time-changed mixed fractional Brownian motion has been discussed in [10] to present a sto-chastic model of the discounted stock price in some arbitrage-free and complete financial markets.This model is the process X Ht ( a, b ) = X H ( a, b ) exp { µβ t + σN Hβ t ( a, b ) } , where µ is the rate of the return and σ is the volatility and β t is the α -inverse stable subordinator.The time-changed processes have found many interesting applications, for example in finance [20,10, 13, 25, 28], in statistical inference [14] and in physics [9].Our goal in this parer is to study the main properties of the time-changed mixed fractional Brownianmotion model paying attention to the long range dependence property. In the first case the internalprocess, which plays role of time, is the tempered stable subordinator while in the second case theinternal process is the gamma subordinator.2. MfBm time-changed by Tempered Stable Subordinator
Definition 2.1.
Tempered Stable Subordinator with index α ∈ (0 , and tempering parameter λ > (TSS) is the non-decreasing and non-negative L´evy process S λ,α = { S λ,αt , t ≥ } with density function: f λ,α ( x, t ) = exp ( − λx + λ α t ) f α ( x, t ) , λ > , α ∈ (0 , , where f α ( x, t ) = 1 π (cid:90) ∞ e − xy e − ty α cos απ sin( ty α sin απ ) dy. More detail about TSS can be founded in [15] . Lemma 2.1. (see [15] for the proof )For q > , the asymptotic behavior of q-th order moments of S λ,αt satisfies E ( S λ,αt ) q ∼ ( αλ α − t ) q , as t → ∞ . Definition 2.2.
Let N H ( a, b ) = { N Ht ( a, b ) , t ≥ } be a mfBm and let S λ,α = { S λ,αt , t ≥ } be aTSS with index α ∈ (0 , and tempering parameter λ > . The time-changed process of N H ( a, b ) bymeans of S λ,α is the process Y HS λ,α ( a, b ) = { Y HS λ,αt , t ≥ } defined by: Y HS λ,αt = N HS λ,αt ( a, b ) = aB S λ,αt + bB HS λ,αt , ( a, b ) ∈ R × R \{ } , (2.1) where the subordinator S λ,αt is assumed to be independent of both the Bm and the fBm. N THE LRD OF TIME-CHANGED MFBM MODEL 3
Notation 2.1.
Let X and Y be two random variables defined on the same probability space (Ω , F , P ) . We denote the correlation coefficient
Corr ( X, Y ) by Corr ( X, Y ) =
Cov ( X, Y ) (cid:112) V ar ( X ) V ar ( Y ) . (2.2) Proposition 2.1.
Let Y HS λ,α ( a, b ) be the mfBm time-changed by S λ,α . Then by Taylor’s expansion weget, for fixed s and large t , Cov ( Y HS λ,αt , Y HS λ,αs ) ∼ a s ( αλ α − ) + b Hs ( αλ α − ) H t H − , as t → ∞ . (2.3) Proof.
For fixed s and using ([15], pp 195), the process Y HS λ,αt follows Cov ( Y HS λ,αt , Y HS λ,αs ) = 12 E (cid:104) ( Y HS λ,αt ) + ( Y HS λ,αs ) − ( Y HS λ,αt − Y HS λ,αs ) (cid:105) = 12 E (cid:104) ( N HS λ,αt ( a, b )) + ( N HS λ,αs ( a, b )) − ( N HS λ,αt ( a, b ) − N HS λ,αs ( a, b )) (cid:105) = 12 E (cid:104) ( aB S λ,αt + bB HS λ,αt ) + ( aB S λ,αs + bB HS λ,αs ) (cid:105) − E (cid:20)(cid:16) a ( B S λ,αt − B S λ,αs ) + b ( B HS λ,αt − B HS λ,αs ) (cid:17) (cid:21) . Since B H has stationary increments, then Cov ( Y HS λ,αt , Y HS λ,αs ) = 12 E (cid:104) ( aB S λ,αt + bB HS λ,αt ) + ( aB S λ,αs + bB HS λ,αs ) − ( aB S λ,αt − s + bB HS λ,αt − s ) (cid:105) = 12 E (cid:104) ( aB S λ,αt ) + ( bB HS λ,αt ) + 2 abB S λ,αt B HS λ,αt (cid:105) + 12 E (cid:104) ( aB S λ,αs ) + ( bB HS λ,αs ) + 2 abB S λ,αs B HS λ,αs (cid:105) − E (cid:104) ( aB S λ,αt − s ) + ( bB HS λ,αt − s ) + 2 abB S λ,αt − s B HS λ,αt − s (cid:105) . (2.4)By the independence of B t and B Ht , we get Cov ( Y HS λ,αt , Y HS λ,αs ) = a (cid:104) E ( B S λ,αt ) + E ( B S λ,αs ) − E ( B S λ,αt − s ) (cid:105) + b (cid:104) E ( B HS λ,αt ) + E ( B HS λ,αs ) − E ( B HS λ,αt − s ) (cid:105) = a E ( B (1)) (cid:104) E ( S λ,αt ) + E ( S λ,αs ) − E ( S λ,αt − s ) (cid:105) + b E ( B H (1)) (cid:104) E ( S λ,αt ) H + E ( S λ,αs ) H − E ( S λ,αt − s ) H (cid:105) . S. ALAJMI, E. MLIKI
Hence for large t and using Lemma 2.1, we have Cov ( Y HS λ,αt , Y HS λ,αs ) ∼ a (cid:2) ( αλ α − ) t + E ( S λ,αs ) − ( αλ α − )( t − s ) (cid:3) + b (cid:104) ( αλ α − ) H t H + E ( S λ,αs ) H − ( αλ α − ) H ( t − s ) H (cid:105) = a αλ α − ) t (cid:16) st + E ( S λ,αs ) t − + O ( t − ) (cid:17) + b αλ α − ) H t H (cid:16) H st + E ( S λ,αs ) H t − H + O ( t − ) (cid:17) ∼ a αλ α − ) s + b Hs ( αλ α − ) H t H − . (cid:3) Proposition 2.2.
Let N H ( a, b ) = { N Ht ( a, b ) , t ≥ } be the mfBm of parameters a, b and H . Let S λ,α = { S λ,αt , t ≥ } be the TSS with index α ∈ (0 , and tempering parameter λ > and let Y HS λ,α ( a, b ) be the mfBm time-changed process by means of S λ,α . Then for fixed s > and t → ∞ , weget E [( Y HS λ,αt − Y HS λ,αs ) ] ∼ a t ( αλ α − ) + b H ( αλ α − ) H t H − a s ( αλ α − ) − b Hs ( αλ α − ) H t H − + 12 a s ( αλ α − ) + b H ( αλ α − ) H s H . Proof.
Let s > t → ∞ . Then by using Eq. (2.3), we have E [( Y HS λ,αt − Y HS λ,αs ) ] = E (cid:104) ( Y HS λ,αt − Y HS λ,αs )( Y HS λ,αt − Y HS λ,αs ) (cid:105) = E (cid:104) ( Y HS λ,αt ) − Y HS λ,αt Y HS λ,αs − Y HS λ,αt Y HS λ,αs + ( Y HS λ,αs ) (cid:105) = E (cid:104) ( Y HS λ,αt ) − Y HS λ,αt Y HS λ,αs + ( Y HS λ,αs ) (cid:105) ∼ a t ( αλ α − ) + b H ( αλ α − ) H t H − a s ( αλ α − ) − b Hs ( αλ α − ) H t H − + 12 a s ( αλ α − ) + b H ( αλ α − ) H s H . (cid:3) Now we discuss the long range dependence behavior of Y HS λ,α ( a, b ) . Definition 2.3.
Note that, a finite variance stationary process { X t , t ≥ } is said to have long rangedependence property (Cont and Tankov [6] ), if (cid:80) ∞ k =0 γ k = ∞ , where γ k = Cov ( X k , X k +1 ) . In the following definition we give the equivalent definition for a non-stationary process { X t , t ≥ } . Definition 2.4.
Let s > be fixed and t > s . Then process { X t , t ≥ } is said to have long rangedependence property property if Corr ( X t , X s ) ∼ c ( s ) t − d , as t → ∞ , N THE LRD OF TIME-CHANGED MFBM MODEL 5 where c ( s ) is a constant depending on s and d ∈ (0 , . Theorem 2.1.
Let N H ( a, b ) = { N Ht ( a, b ) , t ≥ } be the mfBm of parameters a, b and H . Let S λ,α = { S λ,αt , t ≥ } be the TSS with index α ∈ (0 , and tempering parameter λ > . Then thetime-changed mixed fractional Brownian motion by means of S λ,α has long range dependence propertyfor every H > .Proof. The process Y HS λ,α ( a, b ) is not stationary, hence the Definition 2.4 will be used to establish thelong range dependence property.Let < H <
1. Using Eqs. (2.2), (2.3) and by Taylor’s expansion we get, as t → ∞ Corr ( Y HS λ,αt , Y HS λ,αs ) ∼ a s ( αλ α − ) + b Hs ( αλ α − ) H t H − (cid:113)(cid:0) a ( αλ α − ) t + b H ( αλ α − ) H t H (cid:1)(cid:113) E ( Y HS λ,αs ) = a s ( αλ α − ) + b Hs ( αλ α − ) H t H − (cid:113) b H ( αλ α − ) H t H (cid:2) a b H ( αλ α − ) − H t − H + 1 (cid:3)(cid:113) E ( Y HS λ,αs ) ∼ a H − s ( αλ α − ) − H | b | (cid:113) E ( Y HS λ,αs ) t − H + | b | H s ( αλ α − ) H (cid:113) E ( Y HS λ,αs ) t H − . Then the correlation function of Y HS λ,αt decays like a mixture of power law t − H + t − (1 − H ) and thetime-changed process Y HS λ,α ( a, b ) exhibits long range dependence property for all H > . (cid:3) Figure 1.
The correlation function of mixed fractional Brownian motion time-changed by (TSS) for s = 1, a = b = 1, λ = 0 . α = 0 . H = 0 . Remark 2.1.
When a = 0 and b = 1 in Eqs. (2.3) and (2.2) , we obtain Cov ( Y HS λ,αt , Y HS λ,αs ) ∼ Hs ( αλ α − ) H t H − , as t → ∞ ,Corr ( Y HS λ,αt , Y HS λ,αs ) ∼ Hs − H t H − , as t → ∞ . S. ALAJMI, E. MLIKI
Hence we obtain the result proved in [15]
Corollary 2.1.
The fractional Brownian motion time-changed by TSS has long range dependenceproperty for every H ∈ (0 , . MfBm time-changed by the gamma subordinator
Definition 3.1.
Gamma process
Γ = { Γ t , t ≥} is a Stationary independent increments process withgamma distribution. More precisely, the increment Γ t + s − Γ s have density function f ( x, t ) = 1Γ( t/ν ) x ( t/ν ) − e − x , x > , ν > . More detail about gamma subordinator can be founded in [16] . Lemma 3.1. (see [16] for the proof )For q > , the asymptotic behavior of q-th order moments of Γ t satisfies E (Γ t ) q ∼ (cid:18) tν (cid:19) q , as t → ∞ . Definition 3.2.
Let N H ( a, b ) = { N Ht ( a, b ) , t ≥ } be a mfBm and let Γ = { Γ t , t ≥ } be a gammasubordinator. The time-changed process of N H ( a, b ) by means of Γ is the process Y H Γ ( a, b ) = { Y H Γ t , t ≥ } defined by: Y H Γ t = N H Γ t ( a, b ) = aB Γ t + bB H Γ t , ( a, b ) ∈ R × R \{ } , (3.1) where the subordinator Γ t is assumed to be independent of both the Bm and the fBm. Proposition 3.1.
Let Y H Γ ( a, b ) be the mfBm time-changed by Γ . Then we have (1) For s < t , the covariance function for the process Y H Γ t follows Cov ( Y H Γ t , Y H Γ s ) = a (cid:20) Γ(1 + t/ν )Γ( t/ν ) + Γ(1 + s/ν )Γ( s/ν ) − Γ(1 + ( t − s ) /ν )Γ(( t − s ) /ν ) (cid:21) + b (cid:20) Γ(2 H + t/ν )Γ( t/ν ) + Γ(2 H + s/ν )Γ( s/ν ) − Γ(2 H + ( t − s ) /ν )Γ(( t − s ) /ν ) (cid:21) . (2) For fixed s and large t , the process Y H Γ t follows Cov ( Y H Γ t , Y H Γ s ) ∼ a sν + 2 b Hsν H t H − . (3.2) Proof. (1) Let s < t . Using similar procedure as the proof of Eq. (2.4), we get
Cov ( Y H Γ t , Y H Γ s ) = E ( Y H Γ t Y H Γ s )= a (cid:2) E ( B Γ t ) + E ( B Γ s ) − E ( B Γ t − s ) (cid:3) + b (cid:104) E ( B H Γ t ) + E ( B H Γ s ) − E ( B H Γ t − s ) (cid:105) = a (cid:20) Γ(1 + t/ν )Γ( t/ν ) + Γ(1 + s/ν )Γ( s/ν ) − Γ(1 + ( t − s ) /ν )Γ(( t − s ) /ν ) (cid:21) + b (cid:20) Γ(2 H + t/ν )Γ( t/ν ) + Γ(2 H + s/ν )Γ( s/ν ) − Γ(2 H + ( t − s ) /ν )Γ(( t − s ) /ν ) (cid:21) . N THE LRD OF TIME-CHANGED MFBM MODEL 7 (2) Let g ( x ) = Γ( x + 2 H ) / Γ( x ) and f ( x ) = Γ( x + 1) / Γ( x ). By Taylor expansion and [16] we have g ( x + h ) g ( x ) = 1 + 2 H ( h/x ) + H (2 H − h/x ) + O ( x − ) , (3.3)and f ( x + h ) f ( x ) = 1 + ( h/x ) + O ( x − ) . (3.4)For fixed s and large t , using Eqs. (3.2), (3.3) and (3.4), Y H Γ t follows Cov ( Y H Γ t , Y H Γ s ) = a f ( t/ν ) (cid:20) f ( s/ν ) f ( t/ν ) − f (( t − s ) /ν ) f ( t/ν ) (cid:21) + b g ( t/ν ) (cid:20) g ( s/ν ) g ( t/ν ) − g (( t − s ) /ν ) g ( t/ν ) (cid:21) = a t/ν ) (cid:20) f ( s/ν ) f ( t/ν ) − (cid:16) − st + O ( t − ) (cid:17)(cid:21) + b t/ν ) H (cid:20) g ( s/ν ) g ( t/ν ) − (cid:18) − H (cid:16) st (cid:17) + H (2 H − (cid:18) s t (cid:19) + O ( t − ) (cid:19)(cid:21) ∼ a sν + 2 b Hsν H t H − . (cid:3) Proposition 3.2.
Let N H ( a, b ) = { N Ht ( a, b ) , t ≥ } be the mfBm and let Γ = { Γ t , t ≥ } be agamma subordinator. Let Y H Γ ( a, b ) = { Y H Γ t , t ≥ } be the mfBm time-changed by means of Γ . Thenfor fixed s > and t → ∞ , we have (1) E [( Y H Γ t − Y H Γ s ) ] ∼ a tν + 2 b Hν H t H − a sν − b Hsν H t H − + a sν + 2 b Hν H s H . (2) For < H < . The correlation function is given by Corr ( Y H Γ t , Y H Γ s ) ∼ a (2 H ) − sν − H | b | (cid:113) E ( Y H Γ s ) t − H + | b | (2 H ) sν H (cid:113) E ( Y H Γ s ) t H − . (3.5) Proof.
Let s > t . Then(1) Using Eq. (3.2), we have E [( Y H Γ t − Y H Γ s ) ] = E (cid:2) ( Y H Γ t ) − Y H Γ t Y H Γ s + ( Y H Γ s ) (cid:3) ∼ a tν + 2 b Hν H t H − a sν − b Hsν H t H − + a sν + 2 b Hν H s H . S. ALAJMI, E. MLIKI (2) Let < H <
1. Using Eqs. (2.2), (3.2) and by Taylor’s expansion we get, as t → ∞ Corr ( Y H Γ t , Y H Γ s ) ∼ a sν + b Hsν H t H − (cid:113) ( a tν + b Hν H t H ) (cid:113) E ( Y H Γ s ) = 1 | b | (2 H ) ( tν ) H (cid:113)(cid:2) a b Hν − H t − H (cid:3) [ a sν + b Hsν H t H − ] (cid:113) E ( Y H Γ s ) = (2 H ) − ( tν ) − H | b | (cid:2) a b Hν − H t − H (cid:3) [ a sν + b Hsν H t H − ] (cid:113) E ( Y H Γ s ) ∼ a (2 H ) − sν − H | b | (cid:113) E ( Y H Γ s ) t − H + | b | (2 H ) sν H (cid:113) E ( Y H Γ s ) t H − . Hence the correlation function of Y H Γ t decays like a mixture of power law t − H + t − (1 − H ) . (cid:3) Figure 2.
The correlation function of mixed fractional Brownian motion time-changed by Gamma for s = 1, a = b = 1, v = 0 .
75 and H = 0 . Theorem 3.1.
Let N H ( a, b ) = { N Ht ( a, b ) , t ≥ } be the mfBm of parameters a, b and H . Let Γ = { Γ t , t ≥ } be a gamma subordinator with parameter ν > . Then the time-changed mixedfractional Brownian motion by means of Γ has long range dependence property for every H > . Remark 3.1.
When a = 0 and b = 1 in Eqs. (3.2) and (2.2) , we get Cov ( Y H Γ t , Y H Γ s ) ∼ Hsν H t H − , as t → ∞ ,Corr ( Y H Γ t , Y H Γ s ) ∼ Hsν H (cid:112) E ( B Γ s ) t H − , as t → ∞ . N THE LRD OF TIME-CHANGED MFBM MODEL 9
Hence we obtain the result proved in [16]
Corollary 3.1.
The fractional Brownian motion time-changed by gamma subordinator has long rangedependence property for every H ∈ (0 , . References [1] S. Alajmi, E. Mliki,
On the Mixed Fractional Brownian Motion Time Changed by Inverse α -Stable Subordinator ,Applied Mathematical Sciences, , (2020), 755-763. 2[2] S. Alajmi, E. Mliki, Mixed Generalized Fractional Brownian Motion , arXiv (2021). 1[3] J. Beran, Y. Feng, S. Ghosh and R. Kulik,
Long-Memory Processes , Springer, (2016).[4] S. Bochner,
Diffusion equation and stochastic processes , Proc. Nat. Acad. Sci. U. S. A , (1949), 368-370. 2[5] P. Cheridito, Mixed fractional Brownian motion , Bernoulli 7, , (2001), 913-934. 1[6] R. Cont, P. Tankov Financial modeling with jump processes , CRC press, (2003). 4[7] C. El-Nouty,
The fractional mixed fractional Brownian motion , Statist Prob Lett, , (2003), 111-120. 1[8] D. Filatova, Mixed fractional Brownian motion: some related questions for computer network traffic modeling ,International Conference on Signals and Electronic Systems, (2008), 393-396. 1[9] V. Ganti, A. Singh, P. Passalacqua and E. Foufoula
Subordinated Brownian motion model for sediment transport ,Physical Review E, , (2009). 2[10] Z. Guo and H. Yuan, Pricing European option under the time-changed mixed Brownian-fractional Brownian model ,Physica A: Statistical Mechanics and its Applications, , (2014), 73-79. 2[11] M. Hmissi and E. Mliki,
On exit law for subordinated semigroups by means of C -subordinators , Comment. Math.Univ. Crolin. 51, , (2010), 605-617. 2[12] M. Hmissi, H. Mejri and E. Mliki, On the fractional powers of semidynamical systems , Grazer MathematisheBerichte, (2007), 66-78. 2[13] G. Hui, J.R Liang and Y. Zhang
The time changed geometric fractional Brownian mtion and option pricing withtransaction costs , Physica A: Statistical Mechanics and its Applications, , (2012), 3971-3977. 2[14] A. Kukush, Y. Mishura and E. Valkeila,
Statistical Inference with Fractional Brownian Motion , Statistical Inferencefor Stochastic Processes, , (2005), 71-93. 2[15] A. Kumar, J. Gajda and A. Wyloma´nska, Fractional Brownian Motion Delayed by Tempered and Inverse TemperedStable Subordinators , Methodol Comput Appl Probab , (2019), 185-202. 2, 3, 6[16] A. Kumar, A. Wyloma´nska, R. Poloza´nski and S. Sundar, Fractional Brownian motion time-changed by gammaand inverse gammam process , Pysica A: Statistical Mechanics and its Applications , (2017), 648-667. 2, 6, 7, 9[17] M. Magdziarz,
Stochastic Path properties of subdiffusion, a martingale approach.
Stoch. Models , (2010), 256-271. 2[18] M. Magdziarz, Black-Scholes formula in subdiffusive regime , J. Stat. Phys., , (2009), 553-564.[19] M. Majdoub and E. Mliki, Well-posedness for Hardy-H´enon parabolic equations with fractional Brownian noise ,Analysis and Mathematical Physics, , (2021), 1-12.[20] A. Melnikov, Y. Mishura, On pricing in financial markets with long-range dependence , Math Finan Econ, , (2011),29-46. 1[21] H. Mejri and E. Mliki, On the abstract exit equation , Grazer Mathematishe Berichte, , (2009), 84-98. 2[22] H. Mejri and E. Mliki,
On the abstract subordinated exit equation , Abstract and Applied Analysis, , (2010). 2[23] J. B. Mijena and E. Nane,
Correlation structure of time-changed Pearson diffusions.
Statistics and ProbabilityLetters. , (2014), 68-77. 2[24] E. Nane, Laws of the iterated logarithm for a class of iterated processes.
Statist. Probab. Lett. , (2009), 1744-1751.2[25] ¨O. ¨Onalan, Time-changed generalized mixed fractional Brownian motion and application to arithmetic averageAsian option pricing.
International journal of Applied Mathematical Research, , (2017), 85-92. 2 [26] K. Peter, A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices , Econometrica, ,(1973), 135-155.[27] R. Schilling, Subordination in the sense of Bochner and a relaited functional calculs , J. Austral. Math. Soc. (Ser.A), , (1998), 368-396. 2[28] F. Shokrollahi, The evaluation of geometric Asian power options under time changed mixed fractional Brownianmotion , Journal of Computational and Applied Mathematics, , (2018), 716-724. 2[29] M. Sokolov,
L´evy flights from a continuous-time process. , Phys. Rev. , (2003), 469-474.[30] M. Teuerle, A. Wy(cid:32)loma´nska and G. Sikora, Modeling anomalous diffusion by a subordinated frac-tional Levy-stableprocess , J Stat Mech: Theory Exp, (2013), 5-16. 2[31] C. Th¨ a le, Further Remarks on Mixed Fractional Brownian motion , Applied Mathematical Sciences, , (2009),1885-1901. 1 Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P. O. Box1982, Dammam, Saudi Arabia. Basic and Applied Scientific Research Center, Imam Abdulrahman Bin FaisalUniversity, P.O. Box 1982, Dammam, 31441, Saudi Arabia.
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