Fractional G-White Noise Theory, Wavelet Decomposition For

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0 The stochastic process BH (t), which is continuous Gaussian process with stationary increments 1 E[B0 (t)B0 (s)] = [ t 2H + s 2H t s 2H ], H (0,1), (E[ ] is some linear expectation) H H 2 | | | | − | − | ∈ · was originally introducedby Kolmogorov[28] (1940) in study of turbulence under the name ”Wiener Spiral”, and the process has the self-similarity property: for a > 0

Law(B0 (at), t 0)= Law(aH B0 (t), t 0). H ≥ H ≥ Later, when the papers of Hurst [23] (1951) and Hurst, Black and Simaika [24] (1965) devoted to long-term storage capacity in Nile river, were published, the parameter H got the name ”Hurst parameter”. The current name fractional Brownian motion (fBm) comes from the other pioneering paper by Mandelbrot and Van Ness [33] (1968), in which the stochastic calculus with respect to the fBm was considered. The fractional Brownian motion has similarity property and long rang dependence property, which was leaded to describe a great variety of natural and physical phenomena, such as, hydrodynamics, natural images, traffic modelling in broadband networks, telecommunications, and fluctuations of the stock market. The first continuous-timestochastic model for a financial asset appeared in the thesis of Bachelier [4] (1900). He proposed modelling the price of a stock with Brownian motion plus a linear drift.The drawbacks of this model are that the asset price could become negative and the relative returns are lower for higher stock prices. Samuelson [45] (1965) introduced the more realistic model σ2 0 St = S0 exp((µ )+ σB ), − 2 t which have been the foundation of financial engineering. Black and Scholes [7] (1973) derived an explicit formula for the price of a European call option by using the Samuelson model with S0 = exp(rt) through the continuous replicate trade. Such models exploded in popularity because of the successful option pricing theory, as well as the simplicity of the solution of associated optimal investment problems given by Merton [34] (1973). However, the Samuelson model also has deficiencies and up to now there have been many efforts to build better models. Cutland et al. [11] (1995) discuss the empirical evidence that suggests that long-range dependence should be accounted for when modelling stock price movements and present 1 0 a fractional version of the Samuelson model. For H ( 2 ,1) the fractional Gaussain noise BH (k + 0 ∈ 1) BH (k) exhibits long-range dependence, which is also called the Joseph effect in mandelbrot’s − 1 terminology [32] (1997), for H = 2 the fBm is semimartingale and all correlations at non-zero lags 1 are zero, and for H (0, 2 ) the correlations sum up to zero which is less interesting for financial applications [11] (1995).∈ However, empirical evidence is given of a Hurst parameter with values in 1 (0, 2 ) for foreign exchange rates [30]. Hu and Øksendal [25] (2003) develop fractional white noise theory in a white noise probability space (S′(R),F ) with F the Borel field, modelled the financial market by Wick-Itôtype of stochas- 1 tic differential equations driven by fractional Brownian motions with Hurst index H ( 2 ,1), and compute explicit the price and replicating portfolio of a European option in this market.∈ Elliott and Hoek [15] (2003) present an extension of the work of Hu and Øksendal [25] for fractional Brownian motion in which processes with all indices include H (0,1) under the same probability measure, describe the financial market by a SDE driven by a sum∈ of fractional Brownian motion with various Hurst indices and develop the European option pricing in such a market. In an uncertainty financial market, the uncertainty of the fluctuation of the asset price comes from the drift uncertainty and the volatility uncertainty. For the drift uncertainty, in the probability

2 framework Chen and Epstein [9] (2002) propose to use g-expectation introduced by Peng in [36] (1997) for a robust valuation of stochastic utility. Karoui, Peng and Quenez in [27] and Peng in [37] (1997) propose to use time consistent condition g-expectation defined by the solution of a BSDE, as bid-ask dynamic pricing mechanism for the European contingent claim. Delbaen, Peng and Gianin ([12]) (2010) prove that any coherent and time consistent risk measure absolutely continuous with respect to the reference probability can be approximated by a g-expectation. In the probability framework, the volatility uncertainty model was initially studied by Avel- laneda, Levy and Paras [3] (1995) and Lyons [31] (1995) in the risk neutral probability measures, they intuitively give the bid-ask prices of the European contingent claims as superior and inferior expectations corresponding with a family of equivalent probability. There is uncertainty in economics, and no one knows its probability distribution. Almost all the financial market fluctuations show volatility uncertainty (VIX, S&P 500, Nasdaq, Dow Jones, Eurodollar, and DAX, etc), and the volatility uncertainty is the most important, interesting and open problem in valuation (see [46] (2011)). Motivated by the problem of coherent risk measures under the volatility uncertainty (see [2] (1999)), Peng develops the process with volatility uncertainty, which is called G-Brownian motion in sublinear expectation space (Ω,H ,Eˆ). He constructs the G- framework which is a very powerful and beautiful tool to analyse the uncertainty risk (see [39], [41], and [42]).In the sublinear expectation space the G-Brownian motion is a G-martingale under the G-expectation, the market modelled by the G-Brownian motion is incomplete. Using G-framwork, Epstein and Ji [16] study the utility uncertainty application in economics, and Chen [9] gives a time consistent G-expectation bid-ask dynamic pricing mechanism for the European contingent claim in the uncertainty financial market modelled by SDE driven by the G-Brownian motion. In this paper we consider to develop a fractional G-white noise theory under uncertainty. We define fractional G-Brownian motion (fGBm) BH (t) with Hurst index H (0,,1) in a G-white noise space, which is a centered G-Gausian process (see Peng [43]) with stationary∈ increment in the sense of sub-linearity,and it is more realistic to model the financial market by using the fGBm. Meanwhile, we construct wavelet decomposition of fGBm on the family of wavelet with compactly support. We develop a fractional G-white noise theory in a sublinear expectation space (or G-white noise space) (S′(R),S(R),Eˆ), consider fGBm on the G-white noise space, define fractional G-noise and set up fractional G-Itô-Wick stochastic integral with respect to fGBm. We derive the fractional G-Itô formula, define the fractional Malliavin differential derives, and prove the fractional G-Clark-Ocone formula. Furthermore, we present the G-Girsanov’s Theorem. Applying our theory in the financial market modelled by G-Wick-Itôtype stochastic differential equations driven by fGBm BH (t), we prove that the sublinear expectation of the discounted European contingent claim is the bid-ask price of the European claim. Our paper is organized as follows: In Sec. 2 we define the fGBm with Hurst index H (0,1) in the sublinear space. We prove that the fGBm is a continuous stochastic path with Hölder∈ exponent in [0,H), and has self-similarity property and long rang dependence property in the sense of the sub-linearity. Furthermore, we establish the wavelet decomposition for the fGBm by using wavelets with compactly support. In Sec. 3 we present fractional G-white noise theory. In Sec. 4 we present the G-Girsanov’s Theorem. In Sec. 5, we apply our theory in the financial market modelled by G- Wick-Itôtype stochastic differential equations driven by fGBm BH (t), and derive the bid-ask price for the European contingent claim.

3 2 Fractional G-Brownian Motion 2.1 Sublinear Expectation and Fractional G-Brownian Motion Let Ω be a given set and let H be a linear space of real valued functions deﬁned on Ω containing constants. The space H is also called the space of random variables.

Deﬁnition 1 A sublinear expectation Eˆ is a functional Eˆ : H R satisfying (i) Monotonicity: −→ Eˆ[X] Eˆ[Y] if X Y. ≥ ≥ (ii) Constant preserving: Eˆ[c]= c for c R. ∈ (iii) Sub-additivity: For each X,Y H , ∈ Eˆ[X +Y] Eˆ[X]+ Eˆ[Y ]. ≤ (iv) Positive homogeneity: Eˆ[λX]= λEˆ[X] for λ 0. ≥ The triple (Ω,H ,Eˆ) is called a sublinear expectation space.

In this paper, we mainly consider the following type of sublinear expectation spaces (Ω,H ,Eˆ): n n if X1.X2,...,Xn H then ϕ(X1.X2,...,Xn) H for ϕ Cb,Lip(R ), where Cb,Lip(R ) denotes the linear space of functions∈ φ satisfying ∈ ∈

φ(x) φ(y) C(1 + x m + y m) x y for x,y R, | − | ≤ | | | | | − | ∈ some C > 0,m N is depending on φ. ∈ p ˆ p p For each ﬁxed p 1, we take H0 = X H ,E[ X ]= 0 as our null space, and denote H /H0 ≥ { ∈ p 1/p | | } p as the quotient space. We set X p := (Eˆ[ X ]) , and extend H /H to its completion Hp under k k | | 0 p. Under p the sublinear expectation Eˆ can be continuously extended to the Banach space k·k k·k p Ω bˆ (Hp, p). Without loss generality, we denote the Banach space (Hp, p) as LG( ,H ,E). For the G-frameworkk·k of sublinear expectation space, we refer to [38], [39],k·k [40], [41], [42] and [43]. In thisb paper we assume that µ,µ,σ and σ are nonnegative constants suchb that µ µ and σ σ. ≤ ≤

Deﬁnition 2 Let X1 and X2 be two random variables in a sublinear expectation space (Ω,H ,Eˆ),X1 d and X2 are called identically distributed, denoted by X1 = X2 if

n Eˆ[φ(X1)] = Eˆ[φ(X2)] for φ Cb Lip(R ). ∀ ∈ , Deﬁnition 3 In a sublinear expectation space (Ω,H ,Eˆ), a random variable Y is said to be independent of another random variable X, if

Eˆ[φ(X,Y )] = Eˆ[Eˆ[φ(x,Y )] x=X ]. | Deﬁnition 4 (G-normaldistribution) A randomvariableX on a sublinearexpectation space (Ω,H ,Eˆ) is called G-normal distributed if

aX + bX¯ = a2 + b2X fora,b 0, ≥ where X¯ is an independentcopy of X. p

4 Remark 1 For a random variable X on the sublinear space (Ω,H ,Eˆ), there are four typical pa- rameters to character X

µ = EXˆ , µ = Eˆ[ X], − − σ2 = EXˆ 2, σ2 = Eˆ[ X 2], − − where [µ,µ] and [σ2,σ2] describe the uncertainty of the mean and the variance of X, respectively. It is easy to check that if X is G-normal distributed, then

µ = EXˆ = µ = Eˆ[ X]= 0, − − and we denote the G-normal distribution as N( 0 ,[σ2,σ2]). If X is maximal distributed, then { } σ2 = EXˆ 2 = σ2 = Eˆ[ X 2]= 0, − − and we denote the maximal distribution as N([µ,µ], 0 ). { }

Deﬁnition 5 We call (Xt )t R a d-dimensional stochastic process on a sublinear expectation space ∈ (Ω,H ,Eˆ), if for each t R, Xt is a d-dimensional random vector in H . ∈

Deﬁnition 6 Let (Xt )t R and (Yt )t R be d-dimensional stochastic processes deﬁned on a sublinear ∈ ∈ expectation space (Ω,H ,Eˆ), for each t = (t1,t2,...,tn) T , ∈ X n d F [ϕ] := Eˆ[ϕ(Xt )], ϕ Cl Lip(R × ) t ∀ ∈ ,

is called the ﬁnite dimensional distribution of Xt . X and Y are said to be indentically distributed, d i.e., X = Y, if X Y n d F [ϕ]= F [ϕ], t T and ϕ Cl Lip(R × ) t t ∀ ∈ ∀ ∈ . where T := t = (t1,t2,...,tn) : n N,ti R,ti = t j,0 i, j n,i = j . { ∀ ∈ ∈ 6 ≤ ≤ 6 }

Deﬁnition 7 A process (Bt )t 0 on the sublinear expectation space (Ω,H ,Eˆ) is called a G-Brownian motion if the following properties≥ are satisﬁed: (i) B0(ω)= 0; 2 2 (ii) For each t,s > 0, the increment Bt+s Bt is G-normal distributed by N( 0 ,[sσ ,sσ ] and is − { } independent of (Bt ,Bt ,...,Bt ), for each n N andt1,t2,...,tn (0,t]; 1 2 n ∈ ∈

Deﬁnition 8 A process (Xt )t R on a sublinear expectation space (Ω,H ,Eˆ) is called a centered G- ∈ 2 2 Gaussian process if for each ﬁxed t R, Xt is G-normal distributed N( 0 ,[σ ,σ ]), where 0 σ ∈ { } t t ≤ t ≤ σt .

Remark 2 Peng in [41] constructs G-framework, which is a powerful and beautiful analysis tool for risk measure and pricing under uncertainty. In [43], Peng deﬁnes G-Gaussian processes in a nonlinear expectation space, q-Brownian motion under a complex-valued nonlinear expectation space, and presents a new type of Feynman-Kac formula as the solution of a Schrodinger¨ equation.

From now on, in this section we start to deﬁne a two-sided G-Brownian motion and a fractional G-Brownian motion, furthermore we construct the fractional G-Brownian motion and present the similarity property and long rang dependent property for the fractional G-Brownian motion in the sense of linearity.

5 Deﬁnition 9 A process (B 1 (t))t R Ω on the sublinear expectation space (Ω,H ,Eˆ) is called a 2 ∈ ∈ (1) (2) two-sided G-Brownian motion if for two independent G-Brownian motions (Bt )t 0 and (Bt )t 0 ≥ ≥ B(1)(t) t 0 B 1 (t)= ≥ (1) 2 B ( t) t 0 (2) − ≤ We consider a family of continuous process under uncertainty which is corresponding with the fractional Brownian motion (fBm) provided by Kolmogorov (see [28]) and Manbrot (see [33]), and we deﬁne it as fractional G-Brownian motion (fGBm):

Deﬁnition 10 Let H (0,1), a centered G-Gaussian process (BH (t))t R on the sublinear space (Ω,H ,Eˆ) is called fractional∈ G-Brownian motion with Hurst index H if ∈ (i) BH (0)= 0; (ii) ˆ 1 σ2 2H 2H 2H + E[BH (s)BH (t)] = 2 ( t + s t s ), s,t R , 1 2 | |2H | |2H − | − |2H ∈ + (2) Eˆ[ BH (s)BH (t)] = σ ( t + s t s ), s,t R , − − 2 | | | | − | − | ∈ we denote the fractional G-Brownian motion as fGBm.

We can easily check that (B 1 (t))t R is G-Brownian motion, and we denote B(t)= B 1 (t). 2 ∈ 2 2.2 Moving Average Representation Similar with the Mandelbrot-Van Ness representation of fBm, we give the moving average representation of fGBm with respect to the G-Brownian motion as follow Theorem 1 Let H (0,1), for t R the Fractional G-Brownian Motion with Hurst index H is represented as ∈ ∈

ω w H 1/2 H 1/2 ω BH (t, )= CH [(t s)+− ( s)+− ]dB(s, ), (3) ZR − − − π Γ 1/2 w (2H sin H (2H)) where CH = and (Bt )t R is a two-sided G-Brownian motion. Γ(H + 1/2) ∈

Proof. It is clear that BH (0)= Eˆ[BH (t)] = 0, and it is trivial to prove the equations in (2) for s = t. From the Deﬁnition 7 and 9, and by using G-Itˆostochastic integral ([41]) and the integral transform we have that, for s > t

Eˆ[BH (s)BH (t)] π π Γ 0 2 sin H (2H) 2 H 1/2 H 1/2 H 1/2 H 1/2 = σ [(s u) − ( u) − ][(t u) − ( u) − ]du Γ2(H + 1 ) {Z ∞ − − − − − − 2 − t H 1/2 H 1/2 + (s u) − (t u) − du Z0 − − } = σ2( s 2H + t 2H s t 2H) | | | | − | − | π π Γ 0 2 sin H (2H) 2 H 1/2 H 1/2 H 1/2 H 1/2 σ [(s u) − ( u) − ][(t u) − ( u) − ]du − Γ2(H + 1 ) {Z ∞ − − − − − − 2 − t H 1/2 H 1/2 + (s u) − (t u) − du , Z0 − − } thus we prove the ﬁrst equation in (2), and other cases can be proved in a similar way. We can prove the second equation in (2) with Eˆ[ ] replaced by Eˆ[ ] in the above equation, hence we prove (2). · − −·

6 2.3 Properties of Fractional G-Brownian Motion

Deﬁnition 11 In the sublinear expectation space (Ω,H ,Eˆ), a process (Xt )t R is called to have H- self-similarity property, if ∈

d X(at) = aH X(t) for a > 0. (4) Ω p Ω ˆ Theorem 2 AfGBmBH (t) with Hurst index H (0,1) in ( ,LG( ),E) has the following properties (i) H-self-similar property ∈

d H BH (at) = a BH (t) for a > 0.

(ii) The fGBm (BH (t))t R is a continuous path with stationary increment, with β [0,H) order Holder¨ continuous and almost∈ nowhere Holder¨ continuous with order γ > H, i.e., for ∈α 0, ≥ α α αH Eˆ[ BH (s) BH (t) ]= Eˆ[ BH (1) ] t s , | − α | | | α | − | αH Eˆ[ BH (s) BH (t) ]= Eˆ[ BH (1) ] t s . − −| − | − −| | | − | Proof. ˆ 2 (i) From the Deﬁnition 10, the fGBm is a centered G-Gaussian process, and from E[BH (at)] = a2H Eˆ[B2 (t)] and Eˆ[ B2 (at)] = a2H Eˆ[ B2 (t)] we prove the H-self-similar property. H − − H − − H (ii) It is easy to check that BH (s) BH (t) and BH (s t) is identity distributed with G-normal dis- 2 2H 2 −2H − tribution N( 0 ,[σ (t s) ,σ (t s) ]), andthe fGBm (BH (t))t>0 has the self similarity property, therefor, we{ derive} that− − α α Eˆ[ BH (s) BH (t) ] = Eˆ[ BH (s t) ] | − | | −α | αH = Eˆ[ BH (1) ] s t , | | | − | with the similar argument for Eˆ[ ], we prove the theorem. − −· Theorem 3 (Long range dependence) For the fGBm (BH (t))t R with Hurst index H in sublinear expectation space (Ω,H ,Eˆ) ∈ (i) For H (0,1) ∈ 1 2 2H 2H 2H Eˆ[(BH (n + 1) BH(n))BH (1)] = σ [(n + 1) 2n + (n 1) ], (5) − 2 − −

1 2 2H 2H 2H Eˆ[ (BH (n + 1) BH(n))BH (1)] = σ [(n + 1) 2n + (n 1) ]. (6) − − − 2 − − (ii) If H (1/2,1), there exhibits long rang dependence , i.e., ∈ 0 < r(n) < r(n), n N, ∀ ∈ if H = 1/2 there exhibits uncorrelated , i.e., r(n)= r(n)= 0, and if H (0,1/2) ∈ lim r(n)= lim r(n)= 0, n ∞ n ∞ −→ −→ where r(n)= Eˆ[(BH (n + 1) BH(n))BH (1)] − and r(n)= Eˆ[ (BH (n + 1) BH(n))BH (1)] − − − are upper and lower auto-correlation function of BH (n + 1) BH(n), respectively. −

7 Proof. (i) From the construction of fGBm in the next section (see (28) in the next section) we have

ˆ σ2 E[(BH (n + 1) BH(n))BH (1)] = [MH I[n,n+1](x)MH I[0,1](x)]dx, (7) − ZR

σ2 Eˆ[ (BH (n + 1) BH(n))BH (1)] = [MH I[n,n+1](x)MH I[0,1](x)]dx. (8) − − − ZR Define π Γ 1/2 CH′ = [sin( H) (2H + 1)]− CH , where 1 1 1 1 1/2 CH = [2Γ(H )cos( π(H ))]− [sin(πH)Γ(2H + 1)] , − 2 2 − 2 similar with the Definition 12 in the next section for the operator MH , we denote MH′ as the operator with replace CH by C in the definition MH . For0 a < b and 0 < H < 1, by Parseval’s Theorem H′ ≤ 2 1 \ ξ 2 ξ [MH′ I[a,b](x)] dx = [MH′ I[a,b]( )] d ZR 2π ZR 1 ξ 1 2H ξ 2 ξ = − I[a,b]( ) d 2π ZR | | ibξ iaξ 1 1 2H e− e− 2 = ξ − [d − ] dξ 2π ZR | | iξ − 1 = (b a)2H, sinπHΓ(2H + 1) −

from which and notice (7) and (8) we can prove (i). (ii) From (5) we have that

2H 2 2 rn H(2H 1)n − Eˆ[B (1)], n ∞, H = 1/2, ∼ − H −→ 6 rn = 0, H = 1/2. and

∞ < ∞, H (0.1/2); 2 2H 2H ∈ ∑ rn = Eˆ[B (1)] lim ((n + 1) n 1) = 0; H = 1/2; H n ∞ − −  n=1 −→ = ∞, H (1/2,1),  ∈ we can also derive the similar expressions for rn, thus we ﬁnish the proof.

2.4 Wavelet Decomposition of Fractional G-Brownian Motion We consider to expand the fGBm on the periodic compactly supported wavelet family (see [14] and [19]):

j j ψ j,k : x ∑ ψ(2 (x l) k), j 0, 0 k 2 1 (9) { −→ l Z − − ≥ ≤ ≤ − } ∈ where ψ is a mother wavelet, and we denote φ(x) as its periodic scaling function. We assume that

the wavelet ψ belongs to the Schwartz class S(R); •

8 the ψ has N( 2) vanishing moments, i.e., • ≥ ∞ tN ψ(t)dt = 0. Z ∞ − By convention, if j = 1, 0 k 2 1 1 means k = 0, we denote − ≤ ≤ − − 1 1 2− 2 ψ 1,k(t)= φ(x k),0 k 2− 1. − − ≤ ≤ − Then the periodized wavelet family

j/2 j 2 ψ j k(t), j 1, 0 k 2 1 (10) { , ≥− ≤ ≤ − } form an orthonormal basis of L2(T ), where T := R/Z (1-period). Without loss of generality, we consider T = [0,1]. For α > 0, we denote Liouville fractional integral as

t α 1 α 1 (I f )(x) := (t x) − f (x)dx, (11) Γ(α) Z0 − and deﬁne Riemann-Liouville fractional integral coincide with the Marchaud fractional integral as follows +∞ α 1 α 1 α 1 (IM f )(x) := [(t x) − ( x) − ] f (x)dx, (12) Γ(α) Z ∞ − + − − + − Theorem 4 There exists a wavelet expansion for a fGBm process BH (t), i.e., for H (0,1) ∈ ∞ 2 j 1 w − α ψ BH (t)= CH ∑ ∑ µ j,k(IM j,k)(t) (13) j= 1 k=0 − where 1 α = H + , 2 (2H sinπHΓ(2H))1/2 Cw = , H Γ(H + 1/2) (H 1 ) j µ j,k = 2− − 2 ε j,k

2 2 and ε j k are i.i.d. G-normal distributed with BH (1) N( 0 ,[σ ,σ ]). , ∼ { } Proof. (i) We denote the right hand side of (13) as

∞ 2 j 1 w − α ψ F(t)= CH ∑ ∑ µ j,k(IM j,k)(t) j= 1 k=0 − . Without loss generality, we can rewrite F(t) as follows ∞ w ε F(t)= CH ∑ fn(t) n, (14) n= 1 − ∞ (H+1) j α ψ 2 where fn n= 1 denotes the countable Riesz basis 2− (IM j,k)(x) j 1,k=0,1, ,2 j 1 of L (R) (see [47]).{ } − { } ≥− ··· −

9 For proving that the right-hand side of above equation deﬁnes a generalized process, i.e., as a linear functional ∞ F(u)= F(t)u(t)dt, for u S(R), (15) Z ∞ ∀ ∈ − we only need to prove

∞ w F 1 = C ∑ fn(t)εn 1 < ∞. (16) H− H H− k k k n= 1 k − By the representation theorem of a sublinear expectation (see [41]), there exists a family of linear expectations Eθ : θ Θ such that { ∈ } Eˆ[X]= sup Eθ[X], for X H . (17) θ Θ ∈ ∈ Thus, by Kolmogrov’s convergence critera we conclude that

∞ w C ∑ fn(t)εn 1 < ∞. (18) H H− k n= 1 k − Consequently, by Plancherel theorem 1 ∞ < F,u > = F(ξ)u(ξ)dξ | | 2π|Z ∞ | −∞ 1 2 1/2 2 1/2 = Fbˆ(ξ)(b1 + ξ )− u(ξ)(1 + ξ ) dξ 2π|Z ∞ | − 1 F 1 u 1 b ≤ 2πk kH− k kH− < ∞.

∞ itξ whereu ˆ(ξ)= ∞ u(t)e− dt is Fourier transform of u. − (ii) We proveR that BH (t),t R is a centered generalized G-Gaussian process with stationary increment, i.e., with zero{ mean and∈ } σ ∞ ∞ 2H 2H 2H EBˆ H (u)BH (v)= ( t + s t s )u(t)v(s)dtds, 2 Z ∞ Z ∞ σ ∞ | | ∞ | | − | − | (19) − − 2H 2H 2H Eˆ[ BH (u)BH (v)] = ( t + s t s )u(t)v(s)dtds. − − 2 Z ∞ Z ∞ | | | | − | − | − − From the deﬁnition of the fractional integral (12), we have

∞ ∞ 2 j 1 − 1 α w (H+ 2 ) jε ψ BH (u) = CH ∑ ∑ 2− j,k(IM j,k(t)u(t)dt Z ∞ j= 1 k=0 − − ∞ 2 j 1 ∞ ∞ w − (H+ 1 ) j α α = C ∑ ∑ 2− 2 ε j,k u(t) [((I δ)(t x))+ ((I δ)( x))+]ψ j,k(x)dxdt, H Z ∞ Z ∞ j= 1 k=0 − − − − − − α 1 α (t s) where (I δ)(t s)= − − . − Γ(α)

10 j G-normal distributed ε j k ( j = 1,0,1,...,;k = 0,...,2 1) are independent, we derive , − −

EBˆ H (u)BH (v) ∞ 2 j 1 α α = Eˆ[B2 (1)](Cw )2 ∑ ∑− 2 (2H+1) j [((I δ)(t s)) ((I δ)( s)) ]u(t)ψ (s)dsdt H H − Z Z + + j,k j= 1 k=0 R R − − − − α α [((I δ)(t s))+ ((I δ)( s))+]v(t)ψ j,k(s)dsdt ZR ZR − − − σ2 w 2 αδ αδ = (CH ) [((I )(t s))+ ((I )( s))+]u(t)dt ZR ZR − − − ∞ h 2 j 1 − α α (H+ 1 ) j (H+ 1 ) j v(t) ∑ ∑ < ((I δ)(t s )) ((I δ)( s )) ,2 2 ψ (s ) > 2 2 2 ψ (s)dt ds Z ′ + ′ + − j,k ′ L (R) − j,k R j= 1 k=0 − − − − i σ2 w 2 αδ αδ αδ αδ = (CH ) [((I )(t s))+ ((I )( s))+]u(t)dt v(t)[((I )(t s))+ ((I )( s))+]dt ds ZR ZR − − − ZR − − − h i σ2 w 2 αδ αδ αδ αδ = (CH ) u(t)v(s) ((I )(t t′))+ ((I )( t′))+ ((I )(s t′))+ ((I )( t′))+ dt′ dsdt. ZR ZR ZR − − − − − − h i Following from the proof of Theorem 1, we have

σ2 w 2 αδ αδ αδ αδ (CH ) ((I )(t t′))+ ((I )( t′))+ ((I )(s t′))+ ((I )( t′))+ dt′ ZR − − − − − − σ2 = [ t 2H + s 2H t s 2H ], 2 | | | | − | − | we finish the first equation in (19). With the similar argument, we can prove the second part in (19). Thus, we prove that the right hand of (13) is a generalized fGBm, we finish the proof of the theorem.

Remark 3 For construct the G-normal distributed random vector, for example BH (1), Peng in [41] proposed the central limit theorem with zero-mean. ∞ d Let Xi i=1 be a sequence of R valued random variables on a sublinear expectation space { } − d (Ω,H ,Eˆ), Eˆ[X1]= Eˆ[ X1]= 0,andassumethatXi+1 = Xi andXi+1 is independencefrom X1,...,Xi . Then − − { } law Sn X, −→ where 1 n Sn := ∑ Xi √n i=1 and X is G-normal distributed.

3 Fractional G-Noise and Fractional G-Itoˆ Formula 3.1 Fractional G-Brownian Motion on the G-White Noise Space Let S(R) denotes the Schwartz space of rapidly decreasing inﬁnitely differentiable real valued func- 2 tions, let S′(R) be the dual space of S(R), and < , > denotes the dual operation, for f L (R) by approximating by step functions · · ∈

< f ,ω >:= fdB(ω), (20) Z

11 where B(ω)= B( ,ω) is the two-sided G-Brownian motion with B(1) N( 0 ,[σ2,σ2]). Then · ∼ { } (S′(R),S(R),Eˆ) is a sublinear expectation space. Remark 4 Concerning the G-framework and G-Itoˆ stochastic integral theorem, we refer to Peng’s paper [40], book [41] and references therein.

Denote I[0,t](s) as the indicator function

1 if0 s t ≤ ≤ I (s)= 1 if t s 0 (21) [0,t]  − ≤ ≤  0 otherwise Deﬁne the following process 

B˜t (ω) :=< I ( ),ω >, (22) [0,t] · 2 2 then (B˜t )t R is two-sided G-Brownian motion with B˜t N( 0 ,[σ t ,σ t ]). Without loss general- ∈ ∼ { } | | | | ity, for t R we denote Bt as two-sided G-Brownian motion B˜t . ∈ For H (0,1), we define the following operator MH ∈ Definition 12 The operator MH is defined on functions f S(R) by ∈ [ 1/2 H MH f (y)= y − fˆ(y), y R, (23) | | ∈ where ixy gˆ := e− g(x)dx ZR denotes the Fourier transform.

1 For 0 < H < 2 we have f (x t) f (x) MH f (x)= CH − − dt, (24) ZR t 3/2 H | | − where 1 1 1 1 1/2 CH = [2Γ(H )cos( π(H ))]− [sin(πH)Γ(2H + 1)] . − 2 2 − 2 1 For H = 2 we have

MH f (x)= f (x). (25)

1 For 2 < H < 1 we have f (t) MH f (x)= CH dt. (26) ZR t x 3/2 H | − | − We deﬁne

2 2 L (R) := f : MH f L (R) H { ∈ } 1 H 2 = f : y 2 − fˆ(y) L (R) (27) { | | ∈ } ∞ = f : f 2 < , where f 2 = MH f L2(R), { k kLH (R) } k kLH (R) k k 2 then the operator MH can be extended from S(R) to LH (R).

12 For H (0,1), consider the following process ∈

B˜H (t,ω) :=< MH I ( ),ω > (28) (0,t) · then, for t R it is a centered G-Gaussian process (see Peng (2011)[43]) with B˜H (0)= Eˆ[B˜H (t)] = 0, and ∈

σ2 Eˆ[B˜H (s)B˜H (t)] = [ MH I(0,s)(x)MH I(0,t)(x)dx] ZR 1 = σ2[ t 2H + s 2H s t 2H], 2 | | | | − | − | σ2 Eˆ[ B˜H (s)B˜H (t)] = [ MH I(0,s)(x)MH I(0,t)(x)dx] − − ZR 1 = σ2[ t 2H + s 2H s t 2H], 2 | | | | − | − |

then the continuous process B˜H (t) is a fGBm with Hurst index H , we denote B˜H (t) as BH (t) . ∑ Let f (x)= j a jI[t j,t j+1](x) be a step function, then

< MH f ,ω >= f (t)dBH (t), (29) ZR and can be extended to all f L2 (R). And we also have ∈ H 2 f (t)dBH (t)= MH f (t)dB(t), f LH (R). (30) ZR ZR ∈

3.2 Fractional G-Noise Recall the Hermite polynomials

2 n 2 n x d x hn(x) = ( 1) e 2 e− 2 , n = 0,1,2, − dxn ···· We denote the Hermite functions as follows:

1 1 x2 hñ(x)= π− 4 ((n 1)!)− 2 hn 1(√2x)e− 2 , n = 1,2, − − ···· 2 Then hñ,n = 1,2, is an orthonormal basis of L (R) and { ···} 1 Cn 12 if x 2√n h˜ x − n( ) γx2 | |≤ | |≤ ( Ce− if x > 2√n, | | where C and γ are constants independent of n. Define

1 ei(x) := M− h˜i(x), i = 1,2, H ···· 2 Then ei,i = 1,2, is an orthonomal basis of L (R). { ···} H We denote J as the set of all ﬁnite multi-indices α = (α1, ,αn) for some n 1 and αi N0 = 0,1,2, , and for α J ··· ≥ ∈ { ···} ∈ ω Πn ˜ ω Hα( ) = i=1hαi (< hi, >) Πn ˜ ω = i=1hαi ( hidB( )), ZR

13 where (B(t,ω))t R is the two-sided G-Brownian motion. ˆ α ∈ Denote h˜⊗ as the symmetric product with factors h˜1,...,h˜n with each h˜i being taken αi times, similar with the statement for the fundamental result of Itˆo [26] (1951) we denote

ˆ α α h˜⊗ dB⊗| | := Hα(ω). (31) Z α R| |

ˆ 2 n Deﬁnition 13 We deﬁne space LH (R ) as follows

2 n n 2 n Lˆ (R ) := f (x1,...,xn) be symmetric function of (x1,...,xn) M f L (R ) , H { | H ∈ } n where MH f means the operator MH is applied to each variable of f, and we denote

2 n 2 f Lˆ 2 Rn := (MH f ) ds. (32) k k H ( ) ZRn

For f Lˆ 2 (Rn), we deﬁne ∈ H n n n fdBH⊗ := (MH f )dB⊗ . ZRn ZRn Deﬁnition 14 The random variable

2 2 F L (S′(R),S(R),Eˆ), if and only if F MH L (S′(R),S(R),Eˆ). ∈ G,H ◦ ∈ G

The expansion of F MH in terms of the Hermite functions: ◦

F(MH ω) = ∑cαHα(ω) α ω ω = ∑cαhα1 (< MH e1, >)...hαn (< MH en, >) α ω ω = ∑cαhα1 (< e1,MH >)...hαn (< en,MH >). α

Consequently, ω ω ω F( ) = ∑cαhα1 (< e1, >)...hαn (< en, >) α = ∑cαHα(ω). α

For giving F L2 (S (R),S(R),Eˆ), we have ∈ G ′ F(ω) = ∑cαHα(ω) α

ˆ α n = ∑ ∑ cα h˜⊗ dB⊗ Z n n α =n R | | ˆ α n = ∑ ∑ cα e⊗ dB⊗ Z n MH n α =n R | | Aase et al. [1] (2000), Holden et al. [21], and Elliott & Hoek [15] (2003) gave the deﬁnitions of Hida space (S) and (S)∗ under the probability framework. Here we deﬁne G-Hida space (S) and (S)∗ as follows

14 Deﬁnition 15 (i) We deﬁne the G-Hida space (S) to be all functions ψ with the following expansion

Ψ(ω)= ∑ aαHα(ω) (33) α J ∈ satisﬁes Ψ 2 α kα ∞ k,(S) := ∑ aα !(2N) < for all k = 1,2, , k k α J ··· ∈ where γ γ γ γm (2N) := (2 1) 1 (2 2) 2 (2 m) if γ = (γ1,γ2, ,γm) J. · · ··· · ··· ∈ (ii) We deﬁne the G-Hida space (S)∗ to be the set of the folowing expansions Φ(ω)= ∑ bαHα(ω) (34) α J ∈ such that 2 qα Φ := ∑ bαα!(2N) < ∞ for some integer q (0,+∞). q,(S)∗ − k k− α J ∈ ∈ The duality between (S) and (S)∗ is given as follows: for Ψ(ω)= ∑ aαHα(ω) (S), α J ∈ ∈ Φ(ω)= ∑ bαHα(ω) (S)∗ α J ∈ ∈

<< Ψ,Φ >>:= ∑ α!aαbα. α J ∈ The space (S)∗ is convenient for the wick product

Deﬁnition 16 If for Fi( ) (S) · ∈ ∗ (i) Fi(ω)= ∑ cα Hα(ω), i=1,2, α J ∈ we deﬁne their Wick product (F1 F2)(ω) by ⋄ ω (1) (2) ω (1) (2) ω (F1 F2)( )= ∑ cα cβ Hα+β( )= ∑ ∑ cα cβ Hγ( ). (35) ⋄ α,β J γ J α+β=γ ! ∈ ∈ From the chaos expansion of the fGBm BH (t), we have ω BH (t) = < MH I[0,t], > ω = < I[0,t],MH > ∞ ω = < ∑ (I[0,t],ek) 2 ek,MH > LH (R) k=1 ∞ ω = < ∑ (MH I[0,t],MH ek)L2(R)ek,MH > k=1 ∞ ˜ ω = ∑ (MH I[0,t],hk)L2(R) < ek,MH > k=1 ∞ ˜ ω = ∑ (I[0,t],MH hk)L2(R) < ek,MH > k=1 ∞ t = ∑ M h˜ (s)dsH (k) (ω), Z H k ε k=1 0

15 where ε(k) = (0,0, ,0,1,0, ,0) with 1 on the kth entry, 0 otherwise, and k = 1,2, . We deﬁne ··· ··· ··· the H-fractional G-noise WH (t) by the expansion as follows

Deﬁnition 17 H-fractional G-noise with respect to the fGBm BH (t) is deﬁne as following ∞ ˜ WH (t)= ∑ MH hk(t)Hε(k) (ω). (36) k=1 1 1 For H = , we call -fractional G-noise W1 (t) with respect to the G-Brownian motion B(t) as 2 2 2 G-white noise. 1 For 2 < H < 1, H-fractional G-noise WH (t) is called fractional G-black noise. 1 For 0 < H < 2 , H-fractional G-noise WH (t) is called fractional G-pink noise.

Then it can be shown that WH (t) for all t as dB (t) H = W (t). (37) dt H Remark 5 Long Range Dependence Theorem 3 characterizes the H-fractional G-noise, especially, its uncertainty. The fractional G-black noise has persistence long memory, the fractional G-pink noise has negative correlations in the sense of the sub-linearity and quickly alternatively change its value, and the fractional G-white noise is i.i.d. G-Gaussian sequence. The characters of fractional G-noise make it has strong natural background ([29], [32], [44] and [18]).

Lemma 1 The H-fractional G-noise WH (t) is in the G-Hida space (S) for allt, for H (0,1). ∗ ∈ ˆ k 1 Proof. By the deﬁnition 12, and notice that h˜k(y)= √2π( i) h˜k(y) we derive that − − 1 iyt 1 H ˆ Mhh˜k(t) = e y 2 − h˜k(y)dy 2π ZR | | k 1 ( i) − iyt 1 H = − e y 2 − h˜k(y)dy. √2π ZR | | From Hille [20] (1958) and Thangavelu [35] (1993),

2√k 2√k +∞ 1 H 1 − 1 H γy2 Mhh˜k(t) C y 2 − dy k− 12 + [( + ) y 2 − e− dy] | | ≤ Z 2√k | | · Z ∞ Z2√k | | " − − # 1 + 3 H Ck− 12 4 − 2 . ≤ Therefore, we have ∞ 2 2 q WH (t) = ∑ MH h˜k(t) (2k)− q,(S)∗ k k− k=1 | | ∞ 1 + 3 H q C ∑ k− 6 2 − − ≤ k=1 4 C, bounded uniformly for all t, and q > . ≤ 3 Hence, we ﬁnish the proof of the Lemma.

16 Deﬁnition 18 SupposeY (S) is such that Y(t) WH (t) is integrable in (S) ,wesaythatY isdBH - ∈ ∗ ⋄ ∗ integrable and we deﬁne the fractional G-Ito-Wickˆ integral of Y(t)= Y(t,ω) with respect to BH (t) by

Y(t)dBH (t) := Y(t) WH (t)dt. (38) ZR ZR ⋄ Using Wick calculus we derive Lemma 2

T 1 2 1 2H BH (t)dBH (t)= (BH (T )) T . (39) Z0 2 − 2

3.3 Fractional G-Itoˆ Formula

Consider the fractional stochastic differential equation driven by fGBm BH (t)

dX(t) = X(t) (α(t)dt + β(t)dB (t)), t 0, H (40) X(0) = x. ⋄ ≥

We write the above fractional stochastic differential equation in (S)∗ dX(t) = X(t) [α(t)+ β(t)WH(t)]dt, (41) dt ⋄ by Wick product formula we have

t t X(t) = X(0) exp⋄ α(s)ds + β(s)dBH (s) , (42) ⋄ Z0 Z0 i.e.

t t β α 1 β 2 X(t) = xexp (s)dBH (s)+ (s)ds (MH ( (s)I[0,t](s))) ds . (43) Z0 Z0 − 2 ZR is the solution of the fractional SDE (40). Theorem 5 Assume that f (s,x) C1,2(R R), the fractional G-Itoˆ formula is as follows: ∈ × t ∂ t ∂ t ∂2 f f f 2H 1 f (t,BH (t)) = f (0,0)+ (s,BH (s))ds + (s,BH (s))dBH (s)+ H 2 (s,BH (s))s − ds. (44) Z0 ∂s Z0 ∂x Z0 ∂x Proof. Let

g(t,x)= exp(αx + β(t)) (45) where α R be a constant and β : R R be a deterministic differentiable function. Set ∈ −→

Y(t)= g(t,BH (t)).

Then 1 Y(t)= exp (β(t)+ αB (t)+ α2t2H ), ⋄ H 2

17 by Wick calculus in (S)∗, Y (t) is the solution of the following fractional SDE

d 2 2H 1 Y(t) = Y (t)β (t)+Y(t) (αWH (t)) +Y(t)Hα t , dt ′ ⋄ − ( Y (0) = exp(β(0)). Hence t t t 2 2H 1 Y (t)= Y (0)+ Y (s)β′(s)ds + Y (s)αdBH (s)+ H Y (s)α s − ds, Z0 Z0 Z0 which means that t ∂ t ∂ t ∂2 g g g 2H 1 g(t,BH (t)) = g(0,0)+ (s,BH (s))ds + (s,BH (s))dBH (s)+ H 2 (s,BH (s))s − ds. (46) Z0 ∂s Z0 ∂x Z0 ∂x 1,2 For f (t,x) C (R R), we can ﬁnd a sequence fn(t,x) of a linear combinations of function g(t,x) ∈ × ∂ f (t,x) ∂ f (t,x) ∂2 f (t,x) ∂ f (t,x) ∂ f (t,x) in (45) such that f (t,x), n , n , and n pointwisely convergence to f (t,x), , , n ∂t ∂x ∂x2 ∂t ∂x ∂2 f (t,x) 1,2 and , respectively, in C (R R), then fn(t,x) satisfying (46) for all n. Since ∂x2 × t ∂ fn (s,BH (s))dBH (s) Z0 ∂x t ∂ fn = (s,BH (s)) WH (s)d(s) Z0 ∂x ⋄ t ∂ f (s,BH (s)) WH (s)d(s) in (S)∗, as n ∞. −→ Z0 ∂x ⋄ −→ We prove the theorem.

3.4 Fractional Differentiation Similar with the approach to differentiation in Aase et al. [1] (2000), Hu & Øksendal [25] (2003), and Elliott & Van der Hoek [15] (2004), we deﬁne the fractional differentiation of the function deﬁned on the white noise space (S′(R),S(R),Eˆ)

Deﬁnition 19 Suppose F : S′(R) R and suppose γ S′(R). We say F has a directional MH - derivative in the direction γ if −→ ∈

(H) F(ω + εMH γ) F(ω) Dγ F(ω) := lim − ε 0 ε −→ (H) exists in G-Hida space (S)∗, and Dγ F(ω) is called the directional MH -derivative of F in the direction γ.

Deﬁnition 20 We say that F : S (R) R is MH -differentiable if there is a map ′ −→ Ψ : R (S)∗ −→ such that (MH Ψ(t)) (MH γ)(t) is (S) integrable and · ∗ (H) 2 Dγ F(ω)=< F,γ >M , for all γ L (R), H ∈ H γ Ψ γ where < F, >MH := R(MH (t)) (MH )(t)dt. We then deﬁne R · ∂H (H) ω ω Ψ ω Dt F( ) := ∂ωF(t, )= (t, ) (H) and we call D F(ω) the Malliavin derivative or stochastic MH gradient of F at t. t −

18 2 n Deﬁnition 21 Suppose k 1,2,... and for n = 0,1,2,... fn L (R ). We say that ∈{ } ∈ H ∞ n b Ψ = ∑ fndB⊗ (x) Z n H n=0 R belongs to the space Gk(MH ) if ∞ Ψ 2 : ∑ n! Mn f 2 e2kn ∞ Gk = H n L2(Rn) < . k k n=0 k k

Deﬁne G(MH )= k Gk(MH ) and give G(MH ) the projective topology. T Deﬁnition 22 Suppose q 1,2,... . We say the formal expansion ∈{ } ∞ n Q = ∑ gndB⊗ (x), Z n H n=0 R

2 n with gn LH (R ),n = 0,1,2,..., belongs to the space G q(MH ) if ∈ − ∞ b Q 2 ∑ n! Mn g 2 e 2qn ∞ G q = H n L2(Rn) − < . k k − n=0 k k ∞ Deﬁne G ∗(MH )= q=1 G q(MH ) and given G ∗(MH ) the inductive topology. Then G ∗(MH ) is the − dual space of G(MHS). For Q G (MH ) and Ψ G(MH ) deﬁne ∈ ∗ ∈ ∞ Ψ n n << Q, >>G = ∑ n! < MH gn,MH fn >L2(Rn) . n=0

Deﬁnition 23 Suppose ∞ n Q = ∑ gn(s)dB⊗ , Z n H n=0 R

MH is in G ∗(MH ). We deﬁnethe quasi-G-conditional expectation of Q with respect to Ft = σ BH(s),0 s t as { ≤ ≤ } ∞ ˜ MH n EMH [Q Ft ] := ∑ gn(s)In,(0,t)(s)dB⊗ (s), Z n H | n=0 R

MH where I (s)= I (s1) I (sn). We say that Q G (MH ) is F measurable if n,(0,t) (0,t) ··· (0,t) ∈ ∗ t

MH E˜M [Q F ]= Q H | t 1 MH MH Note that, if H = the quasi-G-conditional expectation E˜M [BH Fs ]= BH (s) = Eˆ[BH Fs ]. 6 2 H | 6 | Deﬁnition 24 Suppose F = ∑α cαHα G ∗(MH ). Then the stochastic MH gradient of F at t is deﬁned by: ∈ −

H ω α ω Dt F( ) = ∑cα(∑ iHα εi ( )ei(t)) α i − β ω = ∑(∑cβ+εi ( i + 1)ei(t))Hβ( ). β i

19 H H MH From now on we denote ∇ Q := E˜M [D Q F ]. t H t | t α Theorem 6 (Fractional G-Clark-Ocone formula for polynomials) Suppose P(x)= ∑α cαx ,x = 2 (x1,...,xn), is a polynomial. Consider fi L (R),1 i n, and ∈ MH ≤ ≤ t (t) ω ω ω Xi ( )= fidBH ( )=< MH (I(0,t) fi), > . Z0

t (t) (t) With X = (X1 ,...,Xn ) suppose that

(t) F = P⋄(X ).

Then

T ˆ ˜ H MH F = E(F)+ EMH [Dt F Ft ]dBH (t). (47) Z0 |

MH Proof. From the Deﬁnition 23 of the quasi-G-conditional expectation, we see that F is FT measurable, so

MH F(ξ) = E˜M [F F ] H | T ˜ MH α = ∑cαEMH [X FT ]⋄ α | T = P⋄(X ).

Thus we have T T n ∂P E˜ [F F MH ]dB (t) = E˜ [∑( ) (X) f (t) F MH ]dB (t) Z MH T H Z MH ∂ ⋄ i T H 0 | 0 i=1 xi | T d (t) = P⋄(X )dt Z0 dt (T) (0) = P⋄(X ) P⋄(X ) − = F Eˆ[F]. − Using the similar argument in Aase et al. [1] (2000), Hu & Øksendal [25] (2003), and Elliott & Van der Hoek [15] (2004), we can establish the following fractional G-Clark-Ocone Theorem

MH Theorem 7 (Fractional G-Clark-Ocone Theorem) (a) Suppose Q G ∗(MH ) is FT measurable. H H MH ∈ H MH Then D Q G (MH ) and E˜M [D Q F ] G (MH ) for almost all t. E˜M [D Q F ] WH (t) is t ∈ ∗ H t | T ∈ ∗ H t | T ⋄ integrable in S∗ and

T ˆ ˜ H MH Q = E[Q]+ EMH [Dt Q FT ] WH(t)dt. (48) Z0 | ⋄

2 MH H MH 1,2 (b) Suppose Q L is F measurable. Then E˜M [D Q F ](ω) L (R) (the deﬁnition of ∈ MH T H t | T ∈ MH L1,2 (R) is the analogue in Elliott & Hoek [15]) and MH

T ˆ ˜ H MH Q = E[Q]+ EMH [Dt Q FT ]dBH (t). (49) Z0 |

20 4 G-Girsanov’s Theorem

In this section we will construct G-Girsanov’s Theorem, a analogue in a special situation was proposed in [9] deﬁned from PDE and [25] in a view of SDE. For φ L2(R), setting ∈ t BG(t) := B(t) φ(s)ds, (50) − Z0

where B(t) is a G-Brownian motion in sublinear space (Ω,H ,F ,Eˆ), Ft = σ B(s),s t . We will construct a time consistent G expectation EG, and transfer the sublinear{ expectation≤ } space (Ω,H ,Eˆ) to the sublinear expectation space (Ω,H ,EG) such that BG(t) is a G-Brownian motion on (Ω,H ,EG). Deﬁne a sublinear function G( ) as follows ·

1 2 + 2 G(α)= (σ α σ α−), α R. (51) 2 − ∀ ∈

For given ϕ Cb lip(R), we denote u(t,x) as the viscosity solution of the following G-heat equation ∈ ,

∂t u G(∂xxu)= 0, (t,x) (0,∞) R, (52) − ∈ × u(0,x)= ϕ(x).

Remark 6 The G-heat equation (52) is a special kind of Hamilton-Jacobi-Bellman equation, also the Barenblatt equation except the case σ = 0 (see [5] and [6]). The existence and uniquenessof (52) in the sense of viscosity solution can be found in, for example [17], [10], and [40] for C1,2-solution if σ > 0. The stochastic path information of BG(t) up to t is the same as B(t), without loss of generality we still denote F as the path information of BG(t) up to t. Consider the process BG(t), we define EG[ ] : H R as · −→ EG[ϕ(BG(t)] = u(t,0),t (0,+∞) ∈ and for each t,s 0 and 0 < t1 < < tN t ≥ ··· ≤ G G G G G G G G E [ϕ(B (t1), ,B (tN ),B (t + s) B (t)] := E [ψ(B (t1), ,B (tN ))] ··· − ··· G G where ψ(x1, ,xN)= E [ϕ(x1, ,xN ,B (s)]. ··· ··· For 0 < t1 < t2 < < ti < ti+1 < < tN < +∞, we define G conditional expectation with ··· ··· respect to Fti as G ϕ G G G G G G G E [ (B (t1),B (t2) B (t1), ,B (ti+1) B (ti), ,B (tN ) B (tN 1) Fti ] − G G −G ···G G − ··· − | := ψ(B (t1),B (t2) B (t1), ,B (ti) B (ti 1)), − ··· − − ψ G ϕ G G G G where (x1, ,xi)= E [ (x1, ,xi,B (ti+1) B (ti), ,B (ttN ) B (tN 1). By the comparisontheorem··· of··· the G-heat equation− (52)··· and the sublinear− property− of the function G( ), we consistently define a sublinear expectation EG on H . Under sublinear expectation EG we define· above BG(t) is a G-Brownian motion and BG(t) is N( 0 ,[σ2t,σ2t]) distributed. We call EG[ ] as G-expectation on (Ω,H ). Without loss generality. we denote{ } BG(t)) as two-sided G-Brownian· motion.

21 ˆ p Ω ˆ G With a similar argumentused in Section 2, we denotethe Banach space (Hp, p) as LG( ,Hp,E ), G p Ω ˆ G k·k and B (t) is a G-Brownian motion in LG( ,Hp,E ), consequently,

G G BH (t) : = MH I(0,t)(s)dB (s) ZR φ = MH I(0,t)(s)dB(s) MH I(0,t)(s) (s)ds ZR − ZR is a fGBm under G-expectation EG. To eliminate a drift in the fGBm we need to solve equations of the form

φ MH I(0,t)(s) (s)ds = g(t). ZR That is t MH φ(s)ds = g(t). Z0

Consequently, with (MH φ)(t)= g′(t) φ 1 (t) = (MH− g′)(t).

If g′(t)= A on [0,T ] φ (t) = AM1 H I(0,T)(t) − 1 A(sin(π(1 H))Γ(3 2H)) 2 T t t = − π − − + , Γ 1 1 1 +H 1 +H 2 ( 2 H)cos( 2 ( 2 H)) " T t 2 t 2 # − − | − | | | and for 0 t T ≤ ≤ 1 A(sin(π(1 H))Γ(3 2H))2 1 1 φ 2 H 2 H (t) = − π − (T t) − + (t) − . 2Γ( 1 H)cos( ( 1 H)) − 2 − 2 2 − h i

Theorem 8 (G-Girsanov Theorem) Assume that in the sublinear expectation space (Ω,H ,Eˆ) the G stochastic process BH (t) is a fGBm with Hurst index H (0,1), then there exists G-expectation E , such that, in the G-expectation space (Ω,H ,EG) ∈ t BG(t) := B(t) φ(s)ds, − Z0

is a G-Brownian motion with notation B(t)= B 1 (t), and 2

G φ BH (t) : = MH I(0,t)(s)dB(s) MH I(0,t)(s) (s)ds ZR − ZR G φ φ is a fGBm under G-expectation E with drift R MH I(0,t)(s) (s)ds = g(t), and (t) = (M1 H g′)(t). R − 5 Financial Application

θ We start from a family of probabilityspace with equivalentprobability measure (S′(R),S(R),F ,P ) : θ Θ where Pθ is an induced probability by t σθdB0(t) with B0(t) be a standard{ Brownian mo- ∈ } 0 t tion in reference space ((S′(R),S(R),,F,P0), F Ris the augment ﬁlter constructed from the Brownian 0 σθ θ Θ Θ motion B (t), and ( t )t 0 are unknown processes parameterized by , and is a nonempty σθ ≥ ∈σθ 0 convex set. Here t means the uncertainty of the volatility of the process t B (t), we assume that

22 σθ Assumption 1 (H1) Assume that ( t )t 0 is an adapt process for Ft , and satisfying ≥ θ σ [σ,σ],for all θ Θ. t ∈ ∈ We set E[X]= sup Eθ[X], θ Θ ∈ θ where Eθ is the correspond linear expectationb of probability P . Thus ((S′(R),S(R),F ,E) is a σθ 0 sublinear expectation space, Denis, Hu and Peng [13] prove that t B (t) is a G-Brownian motion in σθ 0 0 sublinear expectation space ((S′(R),S(R),F ,E), we denote B 1 := t B (t). Set B (t) is fBmb with 2 H σθ 0 Hurst index H (0,1) in ((S′(R),S(R),,F ,P0), then it is easy to prove that t BH (t) is a fGBm with ∈ b Hurst index H in (Ω,F ,E), we denote it as BH (t). We consider an incompleted ﬁnancial market which contains a bond P(t) with b dP(t) = rP(t)dt, 0 t T, (53) ≤ ≤ P(0) = 1,

and a stock whose price S(t) with uncertain volatility and satisfying the following SDE driven by a fGBm:

dS(t) = S(t) [µdt + dBH(t)] ⋄ = S(t) [µ +WH(t)]dt, 0 t T, (54) ⋄ ≤ ≤ S(0) = x.

which equivalent to a family of SDE with θ Θ ∈ σθ 0 dS(t) = S(t) [µdt + t dBH (t)] ⋄ θ = S(t) [µ + σ W 0(t)]dt, 0 t T, (55) ⋄ t H ≤ ≤ S(0) = x,

0 0 where WH (t) is fractional noise with respect to fBm BH (t). By G-Girsanov’s Theorem, there exists G-expectation EG[ ] and G-Brownian motion BG(t) in the sublinear expectation space L p(S (R),S(R),EG) (p 1), such· that G ′ ≥ BG(t) = (µ r)t + B(t). (56) − Under EG

G G B (t) := (µ r)t + BH (t), (or W := (µ r)+WH(t)), (57) H − H − is a fGBm with Hust index H, and φ (t) = (r µ)M1 H I(0,T)(t). − − Then we have

dS(t) = S(t) [rdt + dBG(t)] ⋄ H = S(t) [r +W G(t)]dt, 0 t T. (58) ⋄ H ≤ ≤ By fractional G-Itˆoformula 1 S(t)= xexp(rt + BG (t) t2H ) H − 2

23 is the solution to the SDE (58) π H H Suppose that a portfolio (t) = (u(t),v(t)) be a pair of Ft adapted processes, where Ft is the G augment ﬁlter of fGBm BH (t). The corresponding wealth process is π W (t)= u(t)P(t)+ v(t)S(t), where π is called admissible if W θ(t) is bounded below for all t [0,T ] and π is self-ﬁnancing if ∈ π G dW (t) = u(t)dP(t)+ v(t)S(t) [rdt + dBH(t)] π ⋄ = rW (t)dt + v(t)S(t) WG(s)ds. ⋄ H

t rt π π rs G e− W (t)= W (0)+ e− v(s)S(s) WH (s)ds. Z0 ⋄ H ξ 2 G Suppose there is a negative and FT measurable contingent claim LG(S′(R),S(R),E ) with − ∈ rt maturity T > 0 in the market, applying our fractional G-Clark-Ocone Theorem to e− ξ(ω), we have

T rT ξ G rT ξ ˜ rT H ξ H G e− = E [e− ]+ EMH [e− Dt Ft ] WH (t)dt, (59) Z0 | ⋄ ˜ H where EMH is the quasi-G-conditional expectation deﬁned in Deﬁnition 23, Dt is the G-fractional Malliavin differential operator. On the Probability framework, by Girsanov’s Theorem given in [15]

µ r 0 µ r 0 BH (t) := −θ t + BH(t), (or WH (t) := −θ t +WH (t)) (60) σt σt is a fBm with respectb to the measure Pθ by b

θ dP θ 1 θ (ωb)= exp[< φ ,ω > φ 2] Pθ −2k k b φθ φ σθ where (t)= (t)/ t . We can rewrite the equation (60) as θ θ 0 θ θ 0 σ BH (t) := (µ r)t + σ B (t), (or σ WH (t) := (µ r)t + σ W (t)). t − t H t − t H Deﬁne b b E[X] := supEPθ [X], X H , θ Θ ∈ ∈ b θ where EPθ [ ] is the linear expectation corresponding to the probability measure P . Denis, Hu and · σθ Peng [13] (2010) prove that under the sublinear expectation E[ ], t B1/2(t) is a G-Brownian motion b G · on the sublinear expectation space (S′(R),S(R),E ). Notice that b b G d σθ B (t) = t B1/2(t)

G G and B (t) is G-Brownian motion on the sublinear expectationb space (S′(R),S(R),E ), thus EG[ ] = E[ ], · · i.e.,

G E [X] = supEPθ [X], X H . (61) θ Θ ∈ ∈ b 24 σθ G It is easy to proof that t BH (t) is fGBm on the sublinear expectation space (S′(R),S(R),E ). By using the Girsnov’s transform (60), the stock price process (55) can be rewritten as b θ dS(t) = S(t) [rdt + σ dBH (t)] ⋄ t σθ = S(t) [r + t WH (t)]dt, 0 t T, ⋄ b ≤ ≤ S(0) = x. b Elliott and Hoek [15], and Hu and Øksendal [25] prove that rT ξ EPθ [e− ] is the price of the claim and b

1 r(T t) θ 1 θ H H v(t)= S(t)− e− − (σ )− E [D˜ ξ F ] t H t | t determines the portfolio, such that e T rT ξ rT ξ θ rT H ξ H G e− = EPθ [e− ]+ EH [e− Dt Ft ] WH (t)dt, (62) Z0 | ⋄ θ b e ˜ H where EH is the quasi-conditional expectation, and Dt is the fractional Malliavin differential operator deﬁned in [15]. From (61e ), we have G rT ξ rT ξ E [e− ]= supEPθ [e− ], (63) θ Θ ∈ G rT b thus, notice (59), E [e− ξ] is the bid price of the claim ξ with

1 r(T t) H H v(t)= S(t)− e− − E˜M [D ξ F ] H t | t determines the portfolio in the super hedging. Similarly, we can derive that

G rT rT E [ e− ξ]= inf E θ [e− ξ], (64) − − θ Θ P ∈ is the ask price of the claim. b

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