Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈ ] 12 , 3 4 [

The classic Black-Scholes-Merton model was introduced in 1973. Option pricing problems have been one of the hotest issues for researchers and practitioners from the academia and industry. It is well known that the basis of Option pricing problems is how to describe the behavior of the underlying asset’s price. In (Black, F. & Scholes, M ,1993), the underlying asset’s price is assumed to follow the geometric Brownian motion. However, extensive empirical studies shows that the distribution of the logarithmic and distribution is strictly not permitted, except for Open Access articles. Returns of financial asset usually exhibit properties of self-similarity and long-range dependence in both auto-correlations and cross-correlations. Since the fractional Brownian motion has two important properties (self-similarity and long-range dependence), it also has the ability to capture the behavior of underlying asset price. There are many scholars in the study of option pricing based on a fractional Brownian motion such as (Duncan, T.E., Hu, Y., & Duncan, B, 2000); (Necula, C, 2004); (Cheridito, P, 2003). However, the fractional Brownian motion is neither a Markov process nor a semi-martingale as well as it cannot use the usual stochastic calculus to analyze it. Thereby making the fractional Brownian motion not suitable for the behavior of stock price. To eliminate the arbitrage opportunities and to reflect the long memory of the financial time series, many scholars have proposed the use of mixed fractional Brownian motion. The mixed fractional Brownian motion is a family of Gaussian processe, comprised of a linear combination of the Brownian motion and the fractional Brownian motion. Its defined on the probability space (Ω,F ,P) for any t ∈ [0,T ] by


Introduction
The classic Black-Scholes-Merton model was introduced in 1973.Option pricing problems have been one of the hotest issues for researchers and practitioners from the academia and industry.It is well known that the basis of Option pricing problems is how to describe the behavior of the underlying asset's price.In (Black, F. & Scholes, M ,1993), the underlying asset's price is assumed to follow the geometric Brownian motion.However, extensive empirical studies shows that the distribution of the logarithmic and distribution is strictly not permitted, except for Open Access articles.Returns of financial asset usually exhibit properties of self-similarity and long-range dependence in both auto-correlations and cross-correlations.Since the fractional Brownian motion has two important properties (self-similarity and long-range dependence), it also has the ability to capture the behavior of underlying asset price.There are many scholars in the study of option pricing based on a fractional Brownian motion such as (Duncan, T.E., Hu, Y., & Duncan, B, 2000); (Necula, C, 2004); (Cheridito, P, 2003).However, the fractional Brownian motion is neither a Markov process nor a semi-martingale as well as it cannot use the usual stochastic calculus to analyze it.Thereby making the fractional Brownian motion not suitable for the behavior of stock price.To eliminate the arbitrage opportunities and to reflect the long memory of the financial time series, many scholars have proposed the use of mixed fractional Brownian motion.The mixed fractional Brownian motion is a family of Gaussian processe, comprised of a linear combination of the Brownian motion and the fractional Brownian motion.Its defined on the probability space (Ω, F , P) for any t ∈ [0, T ] by such that (a, b) (0, 0), where B = (B t , t ≥ 0) is a standard Brownian motion and B H = (B H t , t ≥ 0) is a fractional Brownian motion with the Hurst parameter H ∈ (0, 1).It is an important class of long memory processes when the Hurst parameter H ∈] 1 2 , 1[. (Cheridito, P, 2001) has proved that for H ∈] 3 4 , 1[, the mixed model with the dependent Brownian motion and fractional Brownian motion is equivalent to the one with the Brownian motion and is a semi-martingale.Hence, it is arbitrage-free.For H ∈] 1 2 , 1[, (Mishura, Y.S, 2008) proved that, the model is arbitrage-free.However the arbitrage problem still exists for H ∈] 1 2 , 3 4 [.The process X H,a,b   t   is not a semi-martingale except for H 1 2 .The issue with this is that the extensively used Itô calculus, developed from semi-martingales to solve stochastic integral, does not apply here.Similarly, the non semi-martingale property of mixed fractional Brownian motion indicates that arbitrage opportunities are possible.The stochastic differential equation of stock price S t assuming X H,a,b t is defined on a probability space (Ω, F , P) for all t ∈ [0, T ] by: where µ, σ, a and b are constants, the Hurst parameter H ∈] 1 2 , 3 4 [.The analytical solution based on wick-product integration approach is not a semi-martingale and present the arbitrage opportunities on the financial market.
In this paper, to capture the long range property and to exclude the arbitrage in the environment of mixed fractional Brownian motion, we use the Liouville form of the fractional Brownian motion on the space L 2 (Ω, F , P).In other to do this, we define for all λ > 0, the process X H,a,b,λ on the same probability space , for any t ∈ [0, T ] by and the stochastic differential equation given by (2) becomes Hence, we use the idea of (Thao, T. H, 2003) to construct the prove of existence and uniqueness solution of (4).
We show that X H,a,b,λ converges to X H,a,b .Our motivation is that the process defined by B H,a,b,λ can be seen as a semimartingale for all λ > 0 and therefore the process X H,a,b,λ t is a semi-martingale when H ∈] 1 2 , 3 4 [.The rest of the paper is organized as follows.In section 2, we briefly introduce the definition and main properties related to mixed fractional Brownian motion.Also, some necessary properties are provided.In section 3, we study an approximation of the process X H,a,b .In section 4, it shows the existence and uniqueness solution of (2).In section 5, we study the modification of the mixed fractional model.In the section 6, we study the convergence of the solution of (4), while section 7, present the solution of mixed fractional equation and section 8 present an application on the financial market.Conclusion is given in the last section.

Preliminaries
In this sub-section, we shall briefly review the definition and some main properties of the mixed fractional Brownian motion.These properties can help us to prove the existence and uniqueness theorem of the solution of (2).These result can be found in (Cheridito, P, 2003).

Definition 1 (Local martingale)
The process M t is a local martingale with respect to the filtration F t if there exists a non-decreasing sequence of stopping times τ k → ∞ a.s.such that the processes M (τ k ) t = M t∧τ k − M 0 are martingales with respect to the same filtration.

Definition 2 (Bounded variation function ) A continuous function
where M is a local continuous martingale and A is a process locally bounded variation.
Proposition 1 For all real γ, a process exp is a martingale.B t is an standard Brownian motion.
Definition 5 The mixed fractional Brownian motion X H,a,b is a continuous centered Gaussian process with variance a 2 t + b 2 t 2H and covariance function defined by: where E(.) denotes the expectation with respect to the probability measure P.
Proposition 2 1.The increments of X H,a,b   t are stationary and these increments are correlated if and only if H = 1 2 .
2. The process X H,a,b t is also mixed self similar.
3. The process X H,a,b is neither a Markov process nor a semi-martingale when b 0, unless H = 1 2 .
4. The process X H,a,b exhibits a long rang dependance if and only if H > 1 2 .
5. For all T > 0, with probability one X H,a,b has a version, the sample paths of which are Hölder continuous of order γ ≤ H on the interval [0, T ].Every sample path of X H,a,b is almost surely nowhere differentiable.

Basic Properties
We recall the following lemma which will use to show the proof of the existence and uniqueness of the solution of (2).
Lemma 3 (Doob's L p Inequality: (Oksendal, B, 2003)) If M t is a martingale such that t → M t (ω) is almost continuous surely, then for all p ≥ 0, T ≥ 0 and for all ξ > 0, Lemma 4 (Borel-Cantelli's lemma: (Chandra, T. K, 2012)) Let (Ω, F , P) be a probability space.Let {A k } k≥1 be a sequence of events in F such that Lemma 5 (Fatou's Lemma: (Knapp, A. W, 2005)) Let f n ≥ 0 be a sequence of measurable function and S is a measurable set.Then,

Approximation of Mixed Fractional Model
In this section, we approximate the process X H,a,b t by a semi-martingale when 1 2 < H < 3 4 .To do this, we begin to find the asymptotic solution of model defined by (4).For all λ > 0, we have According to the Liouville form of the fractional Brownien motion, we have Where , where K is the fractional kernel Lévy's function.
By applying the derivation of fractional Brownian motion, we obtain: in other words: According to the Fubini's theorem, we have: From where We deduce the following result: Lemma 7 The process X H,a,b,λ t is a continuous semi-martingale.
Proof 2 We have X H,a,b,λ t = aB t + bB H,λ t and from lemma 6, B H,λ t is a semi-martingale.
The process X H,a,b,λ t is a linear combination of continuous semi-martingale , from where X H,a,b,λ t is a semi-martingale.
We recall the following estimation: Proof 3 Applying the mean value theorem on the function u → u α , we have According to the isometric Itô's integration lemma, we have: By combining (20) and (21), we obtain: where C(α) depend only α: where ∥.∥ is a standard norm in L 2 (Ω).
According to the lemma 8, we deduce the following result: Lemma 9 The process X

Theorem of Existence and Uniqueness
In this section, we use the approximation approach which is given in terms of practical approach of the theory by (Thao, T. H ,2003) (Intarasit, A., & Sattayatham, P, 2010).We study the Liouville form of fractional Brownian motion.We approximate the process B H t in space L 2 (Ω, F, P) by a semi-martingale, So, we use the idea of (Alos, E., Mazet, O., & Nualart, D, 2000) to introduce the semi-martingale where α = H − 1 2 and B t a standard Brownian motion.Furthermore, where We shall prove in section 3 that B H,λ t converges uniformly with respect to t ∈ [0, T ] to B H t and further lead to Now, we construct the proof of the existence and uniqueness solution of (2).We define (2) under the following stochastic differential equation: where µX t (x) and σ(s)X t (x) are two continous functions.X 0 is a random variable such that E(X 2 0 ) < ∞.
Let L 2 (Ω, F , P) be the probability space where F is the σ-algebra of the set Ω, and P is a probability measure.
Let {X t } t∈[0,∞[ be a stochastic process defined on (Ω, F , P) such that ω → X t (ω) is a continous function represented by the trajectories of process We can write X t (ω) = X(t, ω) and defined a function T × Ω → R n as (t, ω) → X(t, ω).
To solve (29), we need the following assumptions Theorem 1 Let T > 0, under assumption H.1 and H.2. Then the mixed geometric fractional Brownian motion defined by (29) has a unique solution in t ∈ [0, T ].
Proof 5 Let X and Y are two solutions of (29), suppose that (30) We have As However, By putting (26) in (35), we have: We have the following estimation: By Lemma 2 and using (4), we have In the last expression, we show that E [ (∫ t∧s n 0 β 3 ϕ λ s ds ) 2 ] = 0.
Since β 3 is bounded, for all constant M, we have and from (28) we have the following inequality Finally, we have We define , thus for all t ∈ [0, T ], we have By applying the Lemma 1 (C = 0 and u(s) = Since t → X t and t → Y t are continious, this implies the result for t ∈ [0, s n ] whereas n → ∞, so we obtain the uniqueness solution on [0, T ].Now we prove the existence of the solution of (29), consider the stochastic differential equation define by (29) when X λ 0 = X 0 and the corresponding approximation equation of (29) become By using (26), we can write (39) as By replacing (41) into (40), we obtain The equation (42) can be written as The equation ( 42) and (43) represent the stochastic differential equation driven by B s , where b(s, X λ s ) and σ(s, X λ s ) satisfy the hypotheses H.1 and H.2.
If the solution of (43) exist, this implies that the solution of (39) also exist.
Hence, the solution of (29) exist.To show the existence of the solution of (43), we follow the approach of (Oksendal, B,2003).
We define Z 0 t = Z 0 and Z p t = Z p t (ω) such that By a similar calculus as in the case of uniqueness, we have Let's apply the principle of induction on (45).Let p > 1 and t ≤ T , we have By applying the Lemma 2 and the hypothesis H.2, we have Hence, we have where R 1 is a constant dependent of T, N and E Now by induction on p ≥ 0, for all t ≤ T , we have Let's prove the above inequality (49) by mathematical induction.
1.For p = 0, the statement reduces to and is obviously true according the inequality (48).

Assuming the statement is true for p
We will prove that the statement must be true for p = k + 1: The left-hand side of (52) can be written from (45) as From (51), we have This implies When we evaluate the right-hand side of (49) for p = k + 1, we obtain the same value with the right-hand side of (52).This proves the inductive step.Therefore , by the principle of mathematical induction, the given statement is true for ever positive integer p.
We use the Lemma 4 as follow It follows that for almost every ω, there exist p 0 such that Hence, the sequence (Z n t (ω)) define by: for almost all ω.
If we set which X λ t is continuous in t for almost all ω, since Z n t (ω) have the same properties for all n.As we know that every Cauchy sequence is convergent, by using (61) we have for m > n ≥ 0, (67) show that the sequence of {Z n t } converge in L 2 (P) towards Z t .Since the subsequence (Z n t ) converge to Z t for all ω, we have to Z t = X λ t almost surely that is.
Now, we show that X λ t satisfy (29) and (42).For all n, From (76), we obtain for all t ∈ [0, T ], Z n+1 t = X λ t when n → ∞, that is uniformly convergent for almost all ω.From (72) and by application of Lemma 5, we have From Lemma 2, we have this implies X λ t − Z n t → 0.
By taking the limit for (76) when n → ∞ we have

Modification of the Mixed Fractional Model
In this section, we use the theorem 8 to study the mixed modification model.By this modified model, we can use the stochastic Itô's calculus when we consider the consequence of the stock price on the financial market using long memory property.For each λ > 0, we associate to (2) the following asymptotic model: From ( 16), we deduce that (79) become Let G λ t is an absolutely continuous trajectory process (80) become: We have with The equation ( 83) is a stochastic Itö differential equation and solving this equation, we obtain the following result which represent the solution of (79).
Theorem 2 The solution of mixed modified fractional stochastic equation defined by (79) is given by S λ 0 is the initial condition Proof 6 From (81) and (82), we have By applying the Itô's lemma to the function u → log(u) with u = S λ t > 0, where S λ t is the solution of (79).We obtain The  This figure 1, shows that increasing the hurst parameters affects the future price of the asset so that, by increasing the Hurst parameter, the difference between expected lowest price and the highest price be increased and the paths are almost convegent.The simulation of the mixed modified fractional Brownian motion model have been given by the following algorithm Algorithm.MMFBM model simulation process.
T is the maturity of the option.2. For j = 1 to number of simulation n.
By applying this algorithm and by using the parameter of the option given by table 1, we have the following results for differents values of the Hurst parameter H.In the table 2, 4 and 3, we observe that when the value of the Hurst parameter increases and the number of paths increases, the value of the stock price under MFBM and MMFBM are almost the same on the financial market.We can say that through this process the market is stable and balanced.It means that the process S λ t is well-defined as a semi-martingale when H ∈] 1 2 , 3 4 [, we can conclude that the arbitrage problem is almost dismissed.

Conclusion
This study provided a process to obtain the price of the underlying asset on the financial market having a long term memory.As this is not always the property of the master mind.As this is not always the case in the traditional (Black, F.& Scholes, M, 1973).After showing the existence and uniqueness of the solution of (2) describing the mixed fractional model, we consider for each λ > 0, the process X H,a,b,λ t which represents a modification of the fractional mixed process (29) and we have shown that the process (79) has a unique solution that converges uniformly to the process checking the (29) statement in the space L 2 (Ω).

Figure 1 .
Figure 1.Simulated asset paths of Mixed modified fractional Brownian motion model

Table 1 .
Simulation of the Stock Price Under the Modified Mixed Fractional model In this section, we notice that, BM denotes the results of the Black-Scholes equation driven by classical Brownian motion.FBM denotes the results of the Black-Scholes equation driven by fractional Brownian motion.MFBM denotes the results of the Black-Scholes equation driven by the mixed fractional Brownian motion and MMFBM denotes the results of the Black-Scholes equation driven by the mixed modified fractional Brownian motion.The asset price has been estimated under the Mixed modified fractional Brownian motion model, where the parameter of the option model is given by the table 1 and in figure 1 below, we observe 7 simulated paths for the asset price with different Hurst parameter H such that H ∈] 1 2 , 3 4 [.Parameter of MFBM model 0 (t − s + λ) α−1 dB s ] + (σa + σbλ α )dB t .(90)8.