Towards a Better Understanding of Fractional Brownian Motion and Its Application to Finance

The aim of this work is to first build the underlying theory behind fractional Brownian motion and applying fractional Brownian motion to financial market. By incorporating the Hurst parameter into geometric Brownian motion in order to characterize the long memory among disjoint increments, geometric fractional Brownian motion model is constructed to model S &P 500 stock price index. The empirical results show that the fitting effect of fractional Brownian motion model is better than ordinary Brownian motion.


Introduction
The main feature of the Black-Scholes market is that the price of a share of stock is given by the exponential Brownian motion. Thus, in particular, the logarithm of the price is a process with independent increments. However, this assumption is quite unrealistic, since the previous history of the market strongly influences its future Communicated 1 behaviour. Therefore, there were several attempts to describe a market whose logprice is a stochastic process with dependent increments.
From the point of view of the theory of stochastic processes, the fractional Brownian motion is one of the most studied stochastic processes with dependent increments [1][2][3]. This process is, in particular, a Gaussian process. So, it was natural to study a Black-Scholes market in which the standard Brownian motion is replaced by a fractional Brownian motion. However, it was proved by Shiryaev [4] that such a market admits arbitrage opportunities.
Hu and Øksendal [5] proposed a model of Black-Scholes type which involves fractional Brownian motion and in which usual multiplication is replaced by the socalled Wick product. They proved that such a market has no arbitrage opportunities, which is complete, and there is an analogue of the Black-Scholes pricing formula for a European call option. We also refer to [6] for an extension of the result of Hu and Øksendal. In recent years, the fractional Brownian model has been widely used in finance, for example, many consider fractional Brownian motion model in option pricing [7][8][9][10][11][12][13][14]. But none of them balance the rigorousness of the theory and application of fractional Brownian motion well. The aim of this study is to make a systematic introduction into the fractional white noise calculus and to explain the main ideas of Hu and Øksendal's result [5]. In particular, general Gaussian measures on the dual of a nuclear space are studied. Additionally to [5], some existing results from the book by Berezansky and Kondratiev [15] are taken. Moreover, correspondent proofs of the above literature are strongly expanded and the asset price movement is simulated based on fractional Brownian motion.
Our paper is organized as follows: in Sect. 2, some results of fractional white noise calculus are briefly reviewed. In Sect. 3, the fractional Brownian motion is used to model the financial market and a pricing formula for European call option is derived based on fractional Brownian motion. In Sect. 4, empirical analysis between fractional Brownian motion and Brownian motion will be compared using Monte Carlo simulation. In Sect. 5, the results above are concluded. And in "Appendix", fractional stochastic integral of Itô type is defined using wick product from Gel'fand triple.

Review of Fractional White Noise Calculus
In this section, we will shortly review some results of fractional white noise calculus. Some known results and detailed account of the underlying theory can be found in "Appendix". The following two definitions are taken from [16]. Definition 1 Let F : S (R) → R and let γ ∈ S (R). We say that F has a directional derivative in the direction γ if exists in (S) .

Example 1.1 If F(x)
= x, f for some f ∈ S(R) and γ ∈ L 2 (R) ⊂ S (R) then Definition 2 We say that F : S (R) → R is differentiable if there exists a map : R → (S) such that is (S) -integrable and for all γ ∈ L 2 (R). In this case, we put and call it the Malliavin (or Hida) derivative of F at t.

Remark
In fact, in the above definition one assumes that the product (t, x)γ (t) is well defined.
where D φ s is the φ derivative defined in [17]. Namely, if D s F denotes the usual Malliavin derivative, then Then, by [18], Theorem 3.7 we have the following fractional Itô isometry: Moreover, we have . Then : f , x : : g, x :=: f⊗g, x ⊗2 : .
Therefore, we obtain Moreover, by the polarization identity, we have so that Therefore, by Eqs. (5), (6), the standard Wick calculus and finally the fact that we have Example 1.4 (Geometric fractional Brownian motion) Consider the fractional stochastic differential equation where x, μ and σ are constants. By (158), we get: Dividing by dt, we obtain Since μ is a constant Furthermore, since is commutative, and so d X(t) dt = (μ + σ W H (t)) X (t).
Using Wick calculus, we see that the solution of this equation is Here, Using (139) in "Appendix" and (5) we have The following results we have taken from the paper [5] without proofs. i.e.
Define a probability measure μ φ,γ on the σ -algebra F

Remark 1
Recall that, in the classical case, the Girsanov theorem allows us to find a probability measure which is equivalent to the initial probability measure under which a "shifted" Brownian motion becomes a usual Brownian motion. In the fractional case, the above theorem gives a similar result, under a new probability measure μ φ,γ (which is equivalent to the initial measure μ φ ) the shifted fractional Brownian motion B H (t) = B H (t) + t 0 γ s ds becomes a usual fractional Brownian motion.
is not a martingale, unlike in the standard Brownian motion case, the restriction of Proposition 1 (Wick products on different white noise spaces) Let P = μ φ , Q = μ φ,γ andB H (t) = B H (t)+ t 0 γ s ds be as in Theorem 1. Let the Wick products corresponding to P and Q be denoted by p and Q , respectively. Then Definition 3 Let G =: g n , ω ⊗n :, where g n ∈ H ⊗n φ . Then, we define the fractional (or quasi-) conditional expectation of G with respect to F and we extendẼ μ φ (·|F (H ) t ) by linearity.

Remark
In the classical case, this would be indeed the conditional expectation with respect to F t . In the fractional case, this is not so, i.e.Ẽ(·|F . Then belong to the space L 1,2 φ (R) and we have ] the fractional Clark derivative of G in analogy with the classical Brownian motion case. We will use the notation Remark 1 By analogy with the classical case, the fractional Clark-Ocone formula gives a representation of a square-integrable random variable G which is defined through the information available up to time T (i.e. F

(H )
T -measureable) as the expectation of G plus the integral from 0 till T with respect to the fractional Brownian motion in which the integrand for each t ∈ [0, T ] depends only on the information available up to time t (i.e. F (H ) t -measurable).
where ρ > 0 is constant, i.e. A(t) = e ρt and a stock whose price X (t) at time t satisfies the equation where μ and σ = 0 are constants, 0 ≤ t ≤ T . By Example 1.4 we have A portfolio or trading strategy -adapted 2-dimensional process giving the number of units u(t), v(t) held at time t of the bond and the stock, respectively.
We assume that the corresponding value Z (t) = Z θ (t, ω) is given by Remark Note that in the above formula, one replaces the usual multiplication by the Wick product. This will later on allow no arbitrage opportunities. However, it is not completely clear whether the above definition is indeed appropriate from the point of view of finance. The same remains in force for some definitions discussed below, The portfolio is called self-financing if We assume that θ = (u, v) is a self-financing portfolio. Then, by (17) we have Substituting (14) and (146) in "Appendix" into Eq. (18) gives By the Girsanov theorem for fractional Brownian motion (Theorem 1) we see that is a fractional Brownian motion with respect to the measureμ φ defined on F (H ) where K (s) = K (T , x) is defined by the following properties: supp K ⊂ [0, T ] and By [5], the solution of (23) is given by where (·) denotes the gamma function.
In terms ofB H (t), Eq. (20) can be written as Multiplying by e −ρt and integrating, we get where z = Z θ (0)(constant) is the initial fortune.

Definition 6 The market (A(t), X (t)); t ∈ [0, T ] is called complete if for every F (H )
Tmeasurable bounded random variable F(x) there exist z ∈ R and portfolio θ = (u, v) such that By Eq. (26), (30) is equivalent to If we apply the fractional Clark-Ocone theorem (Theorem 2) to G(x) = e −ρT F(x) and with B H (t) replaced byB H (t), we get whereD t denotes the stochastic gradient with respect toμ φ . Note that the σ -algebrâ F t . Comparing (31) and (32), we conclude that our market is indeed complete. Indeed, choose and find v from the equation and finally choose u(t) according to (146) in "Appendix". If F represents a pay-off of the contingent claim at time T , then the time 0 fair price of this claim must be the time 0 value of the trading strategy which replicates this claim, i.e. z. Thus, we have proved the following theorem: where μ φ is defined in (22). Moreover, the corresponding replicating portfolio and u(t) is determined by (146).
Let us now consider a European call option [19]: where c > 0 is a constant (the exercise price). In this case we get: Corollary 1 (The fractional Black-Scholes formula) The time 0 price of the fractional European call (35) is where is the normal distribution function (the error function). The corresponding replication portfolio θ(t) = (u(t), v(t)) is given by Eq. (146) in "Appendix", (26), and where where Proof We will only prove the formulas (36) and (37), since the second part of the corollary uses one technical result which requires a long proof. So, by (16), (21) and (33) we have which is (36).
Now, let us prove that (36) is equal to (37). Note that Then, where, in the last equality, we used that the function 1 2π e − x 2 2 is symmetric with respect to the origin. Since − y 0 −σ T 2 H T H = η + 1 2 σ T H and − y 0 T H = η − 1 2 σ T H , we reach (37). Remark 2 Note that z is independent of μ. One may compare this corollary with the standard results for the classical Brownian motion case. We see that we get the fractional Black-Scholes formula from the classical Black-Scholes formula by simply changing the volatility σ in the classical formula with σ T H − 1 2 .

Monte Carlo Simulation and Numerical Results
This section provides the parameter estimation and Monte Carlo simulation of the models under the fractional Geometric Brownian motion and Geometric Brownian motion.

Parameter Estimation
We consider the Hurst index H first. If H ∈ (0, 1 2 ), then the disjoint increments are positively correlated. The process shows the short term dependency. If H = 1 2 , then the process is the classical Brownian motion. If H ∈ ( 1 2 , 1), the disjoint increments are negatively correlated. The process shows the long-term dependency.
In this study, the R/S analysis (i.e. rescaled range analysis) is implemented to estimate the Hurst parameter, H . Given a time series Mi, 1 ≤ i ≤ M of asset prices. We first transform the time series into logarithmic return series of the length N = M −1 through logarithm.
Then, we separate the new series into A subsets of the equal length and the length of every subset is n = N /A. The mean value of every subset e a , a = 1, 2, . . . , A is where M ai are the elements of subset a. Next in every subset a, we calculate the total sum of the difference of the first k values (k = 1, 2, . . . , n) corresponding to the mean value e a of the subset: Then, the range R a of every subset a is And the standard deviation S a is Then, in every subset a, we calculate R a S a and calculate the mean of the rescaled range R a S a of A subsets with time increment n: As we increase the value of n, we get the logarithmic asset value series of rescaled range (R/S) n . According to the definition of Hurst index H , (R/S) n is in proportion to n H , i.e.
where C is the coefficient. Taking the logarithm on both sides of (43), we have ln(R/S) n = ln C + H ln n. According to (44), the value of H can be calculated through regression analysis. We download the S&P 500 data from Yahoo Finance. In this paper, we select the closing index of the 251 trading days for one year from October 31, 2019 to October 30, 2020. Microsoft Excel software is used to do the regression analysis. We get H = 0.6002. And the goodness of fit is R 2 = 0.9591, which indicates that the fitting effect is good. Therefore, it is reasonable to use the fractional Brownian motion to simulate the movement of the stock index. In the following, in the simulation process, we take the value of H as 0.6, which is rounded to the nearest integer ( Fig. 1).
We consider again Eq. (15) satisfied by the stock price process {X t , t ≥ 0}: where μ and σ are the expected return of stocks and the volatility of stock prices, respectively. B H (t) in Eq. (15) represents the fractional Brownian motion, which is further expressed as where ∼ N (0, 1). If we set G t = ln X t , then by Eqs. (15), (45), and (16), we get By discretization, we have It is trivial to see that Solving the above equations, we have Equations (46) and (47) will be used in the following simulation process to estimate the mean value μ and the standard deviation σ of the stock logarithmic return.

Numerical Results
In the following, we simulate the trend of the stock prices. The following are the concrete steps: Step 1 Using the closing index of the 251 trading days from October 31, 2019 to October 30, 2020, and the closing index of October 30, 2019, which is rounded to 3048, we get the estimates of the mean value μ and the standard deviation σ of the S&P 500 index logarithmic return by Eqs. (46) and (47).
Step 2 Discretizing the S&P 500 price Index, we have Let t i = i t and t = 1, it becomes where i and σ i indicate the standard normal random variable and corresponding standard deviation generated the ith time. Namely, when i = 1, it is where 2 and σ 2 indicate the standard normal random variable and the second standard deviation generated the second time. When i = 3, it is where 3 and σ 3 indicate the standard normal random variable and the third standard deviation generated the third time. Likewise, we can obtain the 251 simulation values.
Step 3 Using the Microsoft Excel, we simulate the movement of S&P 500 index. Then, we compare the real movement with simulated movement to determine whether the price of S&P 500 index obeys the fractional Brownian motion. The results are shown in Figs. 2, 3, and 4. In the graphs, the blue track represents the real movement, and the red track represents the corresponding fractional Brownian motion simulated movement. It is easy to observe the simulation effect is quite ideal.
Furthermore, we need a measurement to evaluate the results of our simulation. Here, we use root mean square error (RMSE). The root mean square error is defined as where N is the number of the samples. In our Monte Carlo simulations, N = 251. Y i andŶ i are the actual price and the Monte Carlo simulated price of the S&P 500 index, respectively. Therefore, R value corresponding to the simulated movement in Figs. 2, 3, and 4 are 225.252, 220.420, and 243.65. It can be seen that the second simulation is the highest and the third simulation is the lowest. In addition, in order to compare the fitting goodness between the fractional Brownian motion method and the Brownian motion simulation method, we use the previous data, and we get that Brownian motion simulation's R value is 282.408. It is concluded that the proposed method is superior to the ordinary standard Brownian motion method. Therefore, it verifies the theory we construct in the previous chapters.

Conclusion
In this paper, we systematically study the fractional Brownian motion theory based on Gel'fand triples and apply the fractional Brownian motion to financial markets. During the simulation of S& P 500 index, unknown parameters need to be estimated and the goodness of fit of the models are compared between fractional Brownian motion and traditional Brownian motion. The results show that fractional Brownian motion model is more accurate than Brownian motion model. Future work can consider simulate the fractional Gaussian noise using more factors like mean-reverting jump-type model and seasonality model. Moreover, data-driven methods can also be used to replicate the financial markets more closely.
Acknowledgements The authors are grateful to the editor and anonymous reviewers for their suggestions in improving the quality of the paper. This work is supported by Scientific research fund for high-level talents, Jimei University, and Open fund of Digital Fujian big data modelling and intelligent computing institute, Jimei University.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A Gel'fand Triples
The notions of a nuclear space and its dual space with respect to a centre Hilbert space will be introduced.

A.1 Rigging by Hilbert Spaces
First, let us consider the notion of a space with negative norm. Let H 0 be a real Hilbert space with scalar product (·, ·) H 0 and norm · H 0 and we suppose that where H + is a dense subset of H 0 . We suppose that H + is a Hilbert space with respect to another scalar product (·, ·) H + and that the norm · H + in H + is such that Each element f ∈ H 0 generates a linear continuous functional f , · on H + by the formula Let us show that this is indeed so. Let us first check the linearity: Let α 1 , α 2 ∈ R, then using the linearity of inner product, we have Next, let us check that the map (49) is continuous. We suppose that u n → u 0 in H 0 as n → ∞. Then, by (48), We introduce a new norm on H 0 , denoted by · H − by taking the norm of f as the norm of the functional f , · : Since the right hand side of (50) is a norm of a linear functional, to check that Let X be a normed linear space with norm · . If any Cauchy sequence {x n } ∞ n=1 in X has a limit, then X is called complete, and thus X becomes a Banach space. In fact, if X is not complete, then one can always construct a so-called completion of this space, which is a Banach space containing X as a dense subset and the norm of the completion restricted to X coincides with the original norm on X . (Recall that the completion of X is constructed by taking all possible Cauchy sequences in X and identifying those Cauchy sequences {x n } ∞ n=1 and {x n } ∞ n=1 in X for which for any ε > 0 there exists N ∈ N such that x n − x m < ε for all n, m ≥ N .) Now we complete H 0 in the norm (50) and obtain a Banach space H − , which is called the space with negative norm. Thus, we have constructed the chain of spaces with positive, zero and negative norms. Each element α ∈ H − is clearly a linear continuous functional on H + , so that where (H + ) denotes the dual space of H + . We will write (α, u) H 0 or α, u for the action of the functional α on an element u ∈ H + . It is obvious that which is a generalization of the Cauchy inequality. Let us prove that the space H − is, in fact, a Hilbert space.
We introduce on H − a quasi-scalar product (i.e. we do not yet know whether as follows: By (50) and (54) we then have, for f ∈ H 0 By the Riesz representation theorem, we therefore have, by (55), Since · as in Eq. (50) is a norm in H − , the (·, ·) H − is indeed a scalar product and not a quasi-scalar product. By completing H 0 with respect to the norm · − , we extend the scalar product (·, ·) H − to the whole space H − , and hence H − becomes a Hilbert space.
By the first equality of (55), I is an isometric operator from H − into H + , defined on the dense subset H 0 of H − . Extending it by continuity, we obtain an isometric operator I : Proof Let u ∈ H + be such that u⊥ Ran(I). Then, for all f ∈ H 0 we have by (55) that Furthermore, Ran(I) is closed in H + as the range of an isometric operator, therefore Ran(I) = H + .

Lemma 2
We have: We will now derive the main result of this section stating that the Hilbert space H − is the dual space of H + with respect to the centre space H 0 . Proof We have to show that any functional l ∈ (H + ) is of the form l(u) = (α, u) H 0 , u ∈ H + for some α ∈ H 0 . By the Riesz theorem, there exists a ∈ H + such that l(u) = (a, u) H + , u ∈ H + . Since Ran(I) = H + , we set α = I −1 a ∈ H − and by Lemma 2, it follows then Hence, the pairing between the space H + and its dual space (H + ) = H − is determined by means of the space H 0 .
where R is a space with a measure μ given on a certain σ -algebra R of subsets of R and such that μ(R) ≤ ∞.
denote the set of all bounded infinite differentiable functions on R with bounded derivatives. The Sobolev space W l 2 (R, p(x)dx), l ∈ N, p(x) ≥ 1, x ∈ R, measurable, a.e. finite, is defined as the completion of C ∞ b (R) with respect to scalar product where ϕ (k) (x) denotes the kth derivative of ϕ (ϕ (0) := ϕ). Then, we may set H + = W l 2 (R, p(x)dx) for l ∈ N, and H 0 = L 2 (R, dx). Then, H − is called a Sobolev space with negative index −l and is denoted by

A.2 Rigging by a Nuclear Space
Let Φ be a linear topological space, Φ ⊆ H 0 , and we suppose that the embedding operator O : Φ → H 0 is continuous. As in Sect. A.1, any element f ∈ H 0 generates a linear continuous functional l f on Φ by the formula l f (u) In what follows, we will only consider the case where Φ is a nuclear space (generated by a countable family of Hilbert spaces). So, let (H n ) n∈N be a family of Hilbert spaces. We suppose that the set Φ := n∈N H n is dense in each H n , and that ∀n ∈ N H n+1 ⊂ H n and the inclusions are quasi-nuclear. On Φ we introduce the topology of a projective limit. As a basis of this topology, we take all possible open balls, In other words, the topology on Φ is the weakest topology on Φ with respect to which all inclusions O n : Φ → H n become continuous. Then, Φ is called the projective limit of Hilbert spaces (H n ) n∈N , and one writes Such a space Φ is called a nuclear space. Suppose now that for some Hilbert space H 0 , we have H 1 ⊂ H 0 densely and continuously. We can now construct the riggings Notice that since H n+1 ⊆ H n , we have This equality should be understood as follows: ∀l ∈ Φ ∃n ∈ N such that l may be extended by continuity from Φ to a linear continuous functional on H n and vice versa if l ∈ H −n for some n ∈ N, then l Φ ∈ Φ .
Proof Each element of H −n is a linear continuous functional on H n . Since Φ ⊂ H n continuously, every element of H −n generates a linear continuous functional on Φ, so that we get the inclusion H −n ⊂ Φ . Let now l ∈ Φ . Since l is continuous at 0, there exists a basic neighbourhood of 0 in Φ, U(0, n, δ), such that |l(ϕ)| < 1 for all ϕ ∈ U(0, n, δ). Consider the space H n . Then, l determines a linear functional defined on a dense set Φ in H n and bounded by 1 on the intersection of Φ with the ball { u H n < δ, u ∈ H n }. Therefore, l is bounded on Φ equipped with the norm · H n . Extending l by continuity, we obtain a continuous functional on H n , i.e. an element of H −n .
Since Φ = n∈N H −n , we can introduce on Φ the topology of the inductive limit of Hilbert spaces (H −n ) n∈N . This topology is defined by basic open sets where c.l.s. denotes the convex linear span, ξ ∈ Φ and N ∈ n → ε(n) > 0. Now, one writes Φ = ind lim n∈N H −n . Thus, we have which is called a Gel'fand (standard) triple.
We set which is a nuclear space, being densely and continuously embedded into 2 . Thus, we get the Gel'fand triple . For each j = 0, 1, 2, . . . , the jth Hermite function e j is defined by It is known that (e j ) ∞ j=0 is an orthonormal basis of L 2 (R, dx). Through this basis one gets that L 2 (R, dx) is unitarily isomorphic to 2 . Thus, using this isomorphism, we get the following Gel'fand triple (being isomorphic to (68)) Thus, for each n ∈ N, S n (R) consists of all functions of the form and the inner product of f , g ∈ S n (R), is given by where g(x) = ∞ j=0 g j e j (x). (Note that all these series converge in the Hilbert space L 2 (R), and therefore in the Lebesgue measure). Furthermore, the dual space S −n (R) may be interpreted as the Hilbert space consisting of formal series and the dual pairing between ξ and f ∈ S n (R) is given by In fact, one can give an explicit description of all functions from the space S(R). This space consists of all smooth (infinitely differentiable) functions on R which satisfy the following condition for each m = 0, 1, 2, . . . , k ∈ N, In other words, the function f , together with all its derivatives, converges at infinite to zero quicker than any function (1+|x|) −k . The space S(R) is called the Schwartz space of smooth rapidly decreasing functions. The dual space S (R) is called the Schwartz space of tempered distributions.

A.4 Minlos Theorem
Let Φ ⊆ H 0 ⊆ Φ be a Gel'fand triple. We first need a σ -algebra on Φ . Take any f 1 , . . . , f n ∈ Φ, n ∈ N and any Borel set A ∈ B(R n ). We define a cylinder set in Φ as We denote by C(Φ ) all cylinder sets in Φ and it is easy to see that C(Φ ) is an algebra. Furthermore, by C σ (Φ ) we denote the σ -algebra which is generated by C(Φ ). One can show that C σ (Φ ) coincides with the Borel σ -algebra B(Φ ), the space Φ being endowed with the inductive limit topology. The following theorem is an infinitedimensional generalization of the classical Bochner theorem on the Fourier transform of a probability measure.
Remark Let H 0 = R d with the usual Euclidian inner product. Since the space H 0 is finite-dimensional (i.e. it has exactly d elements in its orthonormal basis), the identity operator in H 0 is of Hilbert-Schmidt class. Therefore, we can set H n = H 0 , n ∈ N, and thus, R d is also a nuclear space, which is dual to itself. Then, the Fourier transform (70) becomes Thus, in the finite-dimensional case, that condition that F be continuous on R d is automatically satisfied. However, in infinite dimensions, this condition is crucial, and one can give examples of functions F : Φ → C which satisfy F(0) = 1 and positive definiteness, which are not continuous on Φ and for which there is no probability measure on Φ whose Fourier transform is F.

B.1 Definition and Moments of a Gaussian Measure
Let be a rigging of a Hilbert space H 0 by a nuclear space Φ. Let C : Φ → Φ be a linear operator. We suppose that there exists n ∈ N such that C acts continuously from H n into H −n . Furthermore, we suppose that C is positive, i.e. for any f ∈ Φ, f = 0, we have We define As easily seen is an inner product on Φ. Indeed, for every f ∈ Φ, and for any f 1 , (Note that C is symmetric since C is positive, which can be proved by analogy with the case of real Hilbert spaces). Therefore, · H C is a norm on Φ. Define the Hilbert space H C as the completion of Φ with respect to the norm · H C . We define (74)

Proposition 2
The mapping F( f ) satisfies conditions of the Minlos theorem, and therefore there exists a probability measure μ C on (Φ , C σ (Φ )) such that Proof Evidently F(0) = 1. Since the operator C is continuous, the mapping Φ f → C f , f = f H C ∈ R is continuous. Therefore, the mapping Φ f → F( f ) ∈ R is continuous. Thus, it only remains to prove that F is positive definite. To this end, let us first fix arbitrary f 1 , . . . , f n ∈ Φ. Define and define Since the operator C is positive, the matrix C is positive definite. Therefore, there exists a centred Gaussian measure on R n with correlation operator C. This measure is given by The Fourier transform of γ (n) is given by We note that, for any i, j ∈ {1, . . . , n}, by (2), Thus, by (79) and (80), we have Let μ Since u 1 f 1 + · · · + u n f n ∈ Φ, we continue as follows:

Now the statement follows from (78).
Therefore, the measure μ is called Gaussian measure with correlation operator C. Then, the nth moment of μ is defined by if the integral exists.
We take the first derivative in t at zero, i.e. ( d dt )| t=0 in order to get the first moment of the Gaussian measure: then we insert t = 0 and get Therefore, μ C is a centred measure. We now take the second derivative in order to get the second moment of the Gaussian measure: and hence Thus, we get the following formula for the covariance of ·, f : This is the reason why the operator C is called the covariance operator. We define the operator I by By Eq. (84), we get Let f ∈ H C and we approximate f by f n ∈ Φ, n ∈ N, in · H C . Then, It follows also that we can extend the operator I by continuity to the whole of H C and I then becomes an isometric operator between H C and L 2 (Φ , dμ C ). It can be easily shown by approximation that, for each f ∈ H C , we have Proof We first consider the special case when f 1 = · · · = f n = f .

Lemma 4 (Moments of Gaussian measures) For μ C , we have
For a smooth function f : R → R, we have the following formula, d n dt n e f (t) = m 1 ,...,m k ∈Z + m 1 +2m 2 +···+km k =n n!( f (1) (90) Setting we have, Thus, now (90) reduces to Therefore, it follows now that We now consider the general case by using the polarization identity: For i.e. we represent the product of a 1 , . . . , a n through a linear combination of powers of their sums. This classical formula may be proved by induction, but we will skip the proof. Now, substituting a k = x, f k , i = 1, . . . , m, f 1 , . . . , f m ∈ Φ, we have: Therefore, we have that We note that the mapping It follows from (92) that this mapping is uniquely identified by its diagonal values, i.e. by C n ( f 1 , . . . , f n ), f ∈ H C . Now, formulas on the right hand side of (88) and (89) are also linear in f i 's and they coincide with C n ( f , . . . , f ). Therefore, they are true for different f 1 , . . . , f n ∈ H C .

B.2 White Noise and Fractional White Noise
We will now discuss two examples of Gaussian measures. We consider the Schwartz triple and we set C to be the identical operator 1, which evidently satisfies conditions of a correlation operator. Let us recall the definition of a standard Brownian motion. Definition 8 A continuous-time stochastic process B t : 0 ≤ t < +∞ is called a standard Brownian Motion starting at zero if it has the following four properties: The increments of B t are independent; that is, for any finite set of times 0 ≤ t 1 < t 2 < · · · < t n the random variables are independent. (iii) For any 0 ≤ s ≤ t < T the increment B t − B s has the Gaussian distribution with mean 0 and variance t − s. (iv) For all ω in a set of probability one, B t (ω) is a continuous function of t.
For each t ≥ 0, we now set where χ [0,t] denotes the indicator function of [0, t]. Evidently χ [0,t] ∈ L 2 (R), so that X t ∈ L 2 (S (R), μ 1 ). Thus, we have the stochastic process (X t ) t≥0 . For a fixed t ≥ 0 and u ∈ R we have and therefore X t is a Gaussian random variable. Furthermore, and for any 0 < s < t, we have By (94)-(96), one can conclude that (X t ) t≥0 is a version of Brownian motions, i.e. (X t ) t≥0 satisfies the properties (i) − (iii) in Definition 3.1. It is known from the general theory of Brownian motion that one can take a version of each X t so that (X t ) t≥0 becomes a proper Brownian motion, i.e. condition (iv) is satisfied as well. Calculating informally, we next have where δ t denotes the delta function at t. Therefore, x(t) is the derivative of the Brownian motion, i.e. white noise. The measure μ is called (Gaussian) white noise measure.
Let us now consider the case of a fractional white noise. We again consider the Schwartz triple and we fix a so-called Hurst parameter H ∈ ( 1 2 , 1). Define and let (as easily seen the integral above exists). Let us show that the operator C φ can serve as a correlation operator of a Gaussian measure. Let us first show that C φ is a continuous operator from S(R) into S (R). We have, for any f , g ∈ S(R) by changing the variables u = s and v = s − t, we obtain It is known that Hence, which implies that i.e.
Therefore, C φ is a continuous operator. Furthermore, it is known the operator C φ is positive, i.e. for any f ∈ S(R), f = 0, we have Therefore, we can consider the Gaussian measure μ C φ , which we will denote by μ φ . We will also denote the Hilbert space H C φ just by H φ . As easily seen, there exists Making the change of variables s = s and t = s − t, we get where is a bounded set in R 2 . Hence, From (102), by the dominated convergence theorem, we conclude that ( f n ) n∈N is a Cauchy sequence in H φ . Therefore, there exists an f ∈ H φ such that f n → f as n → ∞ in H φ , and we can evidently identify this f with χ [0,t] . Hence, is well defined and (B H (t)) t≥0 is a stochastic process. Informally, we have that Let us prove that (B H (t)) t≥0 is a version of a fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1)). Indeed, for u ∈ R From (104), it follows that B H (t) is a Gaussian random variable. Furthermore, we have that and for any s, t > 0, Since the integral is symmetric in t and s, it is sufficient to consider only the case s ≤ t. By changing the variables t = t and t − s = s we obtain, Hence, we see that B H (t) is a Gaussian process with Therefore, (B H (t)) t≥0 is a version of a fractional Brownian motion.

C.1.1 Tensor Product of Hilbert Spaces
Assume that (H k ) n k=1 is a finite sequence of real separable Hilbert spaces, and that (e (k) j ) ∞ j=0 is some orthonormal basis in H k . We now construct a formal product as follows where α = (α 1 , . . . , α n ) ∈ Z n + = Z 1 + ×· · ·×Z 1 + (n times). Let us consider the ordered sequence (e (1) α 1 , . . . , e (n) α n ) and span a real Hilbert space by the formal vectors (106), assuming that they form the orthonormal basis of this space. The separable Hilbert space obtained as a result is called a tensor product of the spaces H 1 , . . . , H n and is denoted by H 1 ⊗ · · · ⊗ H n = n k=1 H k . Its vectors have the form Let j ∈ H k (k = 1, . . . , n) be some vectors. We define The coefficients f α = f α n of decomposition (108) satisfy (107). Therefore, the vector (108) belongs to n k=1 H k and moreover, We can easily see that this definition of a tensor product depends on the choice of the orthonormal basis (e (k) j ) ∞ j=0 in each factor H k . However, one can easily understand that after changing the bases, we obtain a tensor product isomorphic to the initial one and preserving its structure.

C.1.2 Fock Space and Its Basis
Let H be a real Hilbert space. For every n ∈ N we denote by H ⊗n = H ⊗ · · · ⊗ H the nth tensor power of H . We put our emphasis on a subspace of symmetric elements from H ⊗n . So as to define this subspace, we think about the unitary operator U σ,n in H ⊗n , introduced by the formula on a total set of elements of the form h 1 ⊗ · · · ⊗ h n ∈ H ⊗n , with each permutation σ = (σ (1), . . . , σ (n)) from the set G n of all permutations of {1, . . . , n}. We define the operator We have P 2 n = P n , P * n = P n which implies that it is an orthogonal projector in H ⊗n . The closed subspace of H ⊗n onto which P n projects is denoted by F n (H ). It is called the nth symmetric tensor power of H . Moreover, we set F 0 (H ) = R 1 . The Fock space (symmetric or Bose) F(H ) is defined as a Hilbert orthogonal sum The subspaces F n (H ) are often called n-particle subspaces, and F 0 (H ) is called the vacuum subspace. Let

The vectors
are called coherent states. It is easy to get that, for different h ∈ H , they are linearly independent and form a total set in F(H ) (i.e. the linear span of this set is dense in F(H )). From the definition of F(H ), we have The vector (0) = (1, 0, 0, . . . ) is called a vacuum. We now construct a special basis in the Fock space which is often called the occupation numbers basis. We start from the orthonormal basis (e j ) ∞ j=1 in H . Let Z ∞ +,0 ⊂ Z ∞ + = Z + × Z + × . . . be a set of finite multi-indices α = (α 1 , . . . , α v , 0, 0, . . . ) with integer nonnegative coordinates; here v = v(α) is the length of the multi-index α, i.e. the minimal k ∈ N for which α k+1 = α k+2 = · · · = 0. For each α ∈ Z ∞ +,0 such that |α| = α 1 + · · · + α v = n ∈ N, the vector belongs to F n (H ) (in the case where α k = 0, the vector e k is absent in the tensor product, and 0!=1). The vectors e (n) α , where α ∈ Z ∞ +,0 , |α| = n, form an orthonormal basis in F n (H ). This follows from the fact that vectors of the type e j 1 ⊗· · ·⊗e j n ( j k ∈ N) form an orthonormal basis in H ⊗n , and the numerical coefficient in the expression for e form an orthonormal basis in F(H ). This is exactly the occupation numbers basis.

C.2 Continuous Polynomials on 8
We define the Hermite polynomials (H n (x)) ∞ n=0 on R as the polynomials with the leading coefficient 1 which are obtained by the procedure of orthogonalization of monomials x n with respect to the standard Gaussian measure μ 1 , In other words, H n (x) = (−1) n e x 2 d n dx n (e −x 2 ). Moreover, one can prove that these polynomials satisfy the following recursion relation: We also define the normalized Hermite polynomials as follows: Then, (h n ) ∞ n=0 is an orthonormal basis in L 2 (R, dμ 1 ). As is known H n L 2 (R,μ 1 ) = √ n!, so that and by (112), (h n ) ∞ n=0 satisfy the following relation: Now, we fix a Gel'fand triple Φ ⊆ H 0 ⊆ Φ and consider polynomials on Φ . We denote by P cyl (Φ ) the set of cylinder polynomials on Φ , i.e. functions on Φ of the form where c ∈ R and f i j ∈ Φ, 1 ≤ i, j ≤ n. For a Hilbert space H , let H⊗ n denote the symmetric tensor power, i.e.
where H −n is the dual space of H n with respect to the zero space H 0 . Then, is again a Gel'fand triple. We define the set of continuous polynomials on Φ , denoted by P(Φ ), as the set of functions on Φ of the form We can easily see that every element of P(Φ ) is indeed a continuous function on Φ . Furthermore, since for any f 1 , ..., f n ∈ Φ we have that The following theorem holds: Theorem 8 Let μ be a probability measure on (Φ , C σ (Φ )). Suppose that there exists C > 0 and p ∈ N such that Let μ C be the Gaussian measure on Φ with a correlation operator C.
Proof For any f ∈ Φ, using the Cauchy inequality and Lemma 3.1 we have: and then from Theorem 8, the proposition follows.

C.3 Orthogonal Polynomials
We define P n (Φ ) as the set of all continuous polynomials on Φ of order less than or equal to n. Furthermore, we define P μ C ,n (Φ ) to be the closure of the set P n (Φ ) in L 2 (Φ , dμ C ). We set We evidently have By Proposition 3 it is known that the continuous polynomials P(Φ ) are dense in L 2 (Φ , μ C ). Since it follows that . Therefore, by the definition of dense set, we have Let P n (Φ ,μ C ) denote the orthogonal projection of L 2 (Φ , μ C ) onto n (Φ , μ C ).
(ii) For h 1 , . . . , h n ∈ Φ, which are mutually orthogonal in H C and for fixed n 1 , . . . , n m ∈ N, we have in L 2 (Φ , μ C ) that where G n denotes the group of permutations of n numbers.
since there exists k 0 such that α k 0 < n k 0 .
(iii) We first take g k = h k = h ∈ Φ, for k = 1, . . . , n. Then, we have that The general case follows from the polarization identity in L 2 (Φ , μ C ) : for h 1 , . . . , h n ∈ H C and x ∈ Φ . We consider the mapping I C : I C (0) = 1 and then extend I C by linearity. By Theorem 9(iii), we have that Hence, I C is an isometric mapping, and therefore, we can extend it by continuity to the whole F(H C ).

Theorem 10
The image of the operator I C is the whole of L 2 (Φ , μ C ) and hence I C is a unitary operator between F(H C ) and L 2 (Φ , μ C ).
Proof Since I C is an isometry, it is sufficient to show that Im(I C ) is dense in F(H C ).
We have that for any h ∈ H , H n ((h, x) H 0 ) ∈ Im(I C ). Hence, by linearity, we have (h, x) n H 0 ∈ I m(I C ) for each n ∈ N. It follows that P cyl (Φ ) ∈ I m( For any f (n) ∈ F n (H ), we define which in the case f (n) ∈ Φ⊗ n coincides with the definition of : ( f (n) , x ⊗n ) H 0 . Thus, the unitary I C can be written down in the form where |α| = α 1 + · · · + α ν . The vector e 1 , . . . , e ν are by construction orthogonal in H C and therefore which forms an orthogonal basis in L 2 (Φ , μ C ).

C.4 Fractional Stochastic Calculus of Itô Type
Let us recall that in "Appendix A" we have constructed the Schwartz triple Let and denotes the gamma function. By Lemma 2.1 in [5], φ is a unitary operator between H φ and L 2 (R). Therefore, One can show that F(Φ) is a nuclear space. Furthermore, F(H − p ) is the dual space of F(H p ) with respect to the space F(H ). Therefore, is the dual space of F(Φ) with respect to zero space F(Φ). Thus we get the Gel'fand triple Applying this general construction to the Schwartz triple (141) to (146), we get the Gel'fand triples Recall that we have unitary operators I 1 : F(L 2 (R) → L 2 (S (R), μ 1 )) (150) where we used the notation I φ = I C φ . Applying (150), respectively (6.11), to the Gel'fand triple (148), respectively (6.9), we get the following Gel'fand triples For any f (n) ∈ Φ ⊗n and g (m) ∈ Φ ⊗m , we define This mapping can be extended by linearity and continuity to the bilinear, continuous mapping which is called the Wick product. Furthermore, restricting to F(Φ), we get a bilinear continuous mapping Realizing on F(S (R)), respectively, F(S (R)), and mapping onto (S) , respectively, (S) , we get a Wick product on (S) (as well as on (S)) and on (S) (as well as on (S)). We note that for any f (n) ∈ S (R)⊗ n (respectively,S (R)⊗ n ) and g (m) ∈ S (R)⊗ m (respectively,S (R)⊗ m ), we get : f (n) , x ⊗n : : g (m) , x ⊗m :=: f (n)⊗ g (m) , x ⊗(n+m) : (where we used a natural notation : f (n) , x ⊗n : for a generalized function f (n) ). Before giving an application of the Wick calculus, let us state a lemma as below: (see [5]) We see that for q ≤ 2 we have by inequality (156). Hence, W H (t) ∈ (S) for all t. Moreover, by (113) it follows that t → W H (t) is a continuous function from R into (S) . We define Definition 9 Suppose Z : R → (S) is a given function with the property that Then, the integral R Z (t)dt is defined to be the unique element of (S) such that Therefore, t → B H (t) is differentiable in (S) and