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Fractional Brownian motion

The fractional Brownian motion was ﬁrst introduced within a Hilbert space framework by Kolmogorov in 1940 in [141], where it was calledWiener Helix.

The concept of fractional Brownian motion, named by Mandelbrot and Van Ness, was explored in depth by Yaglom In their 1968 work, they presented a stochastic integral representation of this process using standard Brownian motion.

Deﬁnition 1.1.1 LetH be a constant belonging to(0,1) A fractional Brown- ian motion (fBm)(B (H) (t)) t ≥ 0 of Hurst indexHis a continuous and centered Gaussian process with covariance function

For H = 1/2, thefBm is then a standard Brownian motion By Deﬁnition 1.1.1 we obtain that a standard fBm B (H) has the following properties:

2 B (H) has homogeneous increments, i.e., B (H) (t+s)−B (H) (s) has the same law ofB (H) (t) fors, t≥0.

6 1 Intrinsic properties of the fractional Brownian motion

The fractional Brownian motion (fBm) is derived from the general existence theorem of centered Gaussian processes with specified covariance functions In upcoming chapters, we will explore various constructions of fBm using white noise theory The fBm is categorized into three distinct families based on the Hurst parameter (H): 0 < H < 1/2, H = 1/2, and 1/2 < H < 1 The term "Hurst parameter" was introduced by Benoit Mandelbrot in honor of hydrologist Harold Hurst, who conducted a statistical analysis of annual water run-offs from the Nile River, examining the values of successive yearly run-offs and their cumulative totals over time.

Between 662 and 1469, it was found that the normalized values of the amplitude deviation from the empirical mean followed an approximate relationship of cn H, with H set at 0.7 Additionally, the distribution of ∆ n = n i=1 δ i closely resembled n H δ 1, indicating that H was greater than 1/2 This observation suggested that the phenomenon could not be represented by a process with independent increments; instead, the δ i could be interpreted as increments of a fractional Brownian motion (fBm) This research led Mandelbrot to coin the term "Hurst index."

Stochastic integral representation

Here we discuss some of the integral representations for the fBm In [156], it is proved that the process

(1.1) where B(t) is a standard Brownian motion and Γ represents the gamma function, is a fBm with Hurst index H ∈ (0,1) If B(t) is replaced by a complex-valued Brownian motion, the integral (1.1) gives the complex fBm.

By following [177] we sketch a proof for the representation (1.1) For further detail we refer also to [207] First we notice thatZ(t) is a continuous centered

Gaussian process Hence, we need only to compute the covariance functions.

In the following computations we drop the constant 1/Γ(H+ 1/2) for the sake of simplicity We obtain

=C(H)t 2H , where we have used the change of variables=tu Analogously, we have that

Hence we can conclude thatZ(t) is a fBm of Hurst indexH.

Several other stochastic integral representations have been developed in the literature By [207], we get the following spectral representation offBm

R e its −1 is |s| 1/2 − H dB(s),˜ where ˜B(s) = B 1 +iB 2 is a complex Brownian measure on R such that

, whereB i (A) : A dB i (t) Equation (1.1) provides an integral representation for fBm over the whole real line By following the approach of [172], we can also represent thefBm over aﬁnite interval, i.e.,

For the proof, we refer to [114], [172], and [177] Note that this representation is canonical in the sense that the ﬁltrations generated byB (H) andBcoincide.

In Chapter 2 a deﬁnition of stochastic integral with respect to fBm will be introduced by exploiting the stochastic integral representation of B (H) in terms of (1.2) and of (1.3).

Remark 1.2.1.Integral representations that change fBm of arbitrary Hurst index K into fBm of indexH have been studied in [191], [133] and [134] In Theorem 1.1 of [191] it is shown that for anyK∈(0,1), there exists a unique

K-fBm B˜ (K) such that for allt∈Rthere holds

In Theorem 5.1 of [133] integration is carried out on [0, t] and showed that for given K ∈(0,1), there exists a unique K-fBm B (K) (t), t≥0, such that for allt≥0 we have a.s that

(t−s) H − K ãF 1−K−H, H−K,1 +H−K,s−t s dB (K) (s), (1.5) whereF is Gauss hypergeometric function and c H,K = 1 Γ(H−K+ 1)

In [134] an analytical connection between (1.4) and (1.5) is proved.

Correlation between two increments

For H = 1/2, B (H) is a standard Brownian motion; hence, in this case the increments of the process are independent On the contrary, for H= 1/2 the

Long-range dependence

increments are not independent More precisely, by Deﬁnition 1.1.1 we know that the covariance betweenB (H) (t+h)−B (H) (t) andB (H) (s+h)−B (H) (s) withs+h≤tand t−s=nhis ρ H (n) =1

For H > 1/2, the increments B(H)(t+h) - B(H)(t) and B(H)(t+2h) - B(H)(t+h) exhibit positive correlation, indicating an aggregation behavior that can effectively describe clustering phenomena in systems characterized by memory and persistence Conversely, for H < 1/2, these increments show negative correlation, making them suitable for modeling sequences that demonstrate intermittency and anti-persistence.

Deﬁnition 1.4.1 A stationary sequence(X n ) n ∈N exhibitslong-range dependenceif the autocovariance functionsρ(n) := cov(X k , X k+n )satisfy n lim →∞ ρ(n) cn − α = 1 for some constant c and α∈(0,1) In this case, the dependence between X k andX k+n decays slowly asntends to inﬁnity and

Hence, we obtain immediately that the incrementsX k :=B (H) (k)−B (H) (k−

1) of B (H) and X k+n := B (H) (k+n)−B (H) (k+n−1) of B (H) have the long-range dependence property forH >1/2 since ρ H (n) = 1

2[(n+ 1) 2H + (n−1) 2H −2n 2H ]∼H(2H−1)n 2H − 2 as ngoes to inﬁnity In particular, n lim →∞ ρ H (n)

Long-range dependence can be defined in various ways A function $ L $ is considered slowly varying at zero or infinity if it remains bounded within a finite interval Additionally, as $ x $ approaches zero or infinity, the ratio $ L(ax)/L(x) $ approaches 1 for any $ a > 0 $.

We introduce now thespectral densityof the autocovariance functionsρ(k) f(λ) := 1

Deﬁnition 1.4.2 For stationary sequences(X n ) n∈N with ﬁnite variance, we say that(X n ) n∈N exhibits long-range dependenceif one of the following holds:

1.lim n→∞ ( n k= − n ρ(k))/(cn β L 1(n)) = 1for some constantcandβ∈(0,1).

2.lim k→∞ ρ(k)/ck − γ L 2(k) = 1for some constant c andγ∈(0,1).

3.lim λ→0 f(λ)/c|λ| − δ L 3(|λ|) = 1 for some constant candδ∈(0,1).

HereL 1 , L 2 are slowly varying functions at inﬁnity, whileL 3 is slowly varying at zero.

Lemma 1.4.3 For fBmB (H) of Hurst index H ∈(1/2,1), the three deﬁni- tions of long-range dependence of Deﬁnition 1.4.2 are equivalent They hold with the following choice of parameters and slowly varying functions:

Proof For the proof, we refer to Section 4 in [221]

For a survey on theory and applications of long-range dependence, see also[81].

Self-similarity

By following [209], we introduce the following:

Deﬁnition 1.5.1 We say that anR d -valued random processX = (X t ) t ≥ 0 is self-similar or satisﬁes the property of self-similarity if for every a >0 there existsb >0such that

Note that (1.6) means that the two processes X at and bX t have the same ﬁnite-dimensional distribution functions, i.e., for every choicet 0 , , t n inR,

P(X at 0 ≤x 0 , , X at n ≤x n ) =P(bX t 0 ≤x 0 , , bX t n ≤x n ) for every x 0 , , x n inR.

Definition 1.5.2 If b = a − H in Definition 1.5.1, then we say that X (X t ) t≥0 is a self-similar process with Hurst index H or that it satisfies the property of (statistical) self-similaritywith Hurst indexH The quantityD 1/H is called the statistical fractal dimension ofX.

The covariance function of fractional Brownian motion (fBm) is homogeneous of order 2H, indicating that B(H) is a self-similar process characterized by the Hurst index H This means that for any constant a > 0, the processes B(H)(at) and a^(-H)B(H)(t) share the same distribution law.

H¨ older continuity

We recall that according to the Kolmogorov criterion (see [228]), a process

X = (X t ) t ∈Radmits a continuous modiﬁcation if there exist constantsα≥1, β >0, andk >0 such that

Theorem 1.6.1 LetH ∈(0,1) The fBmB (H) admits a version whose sample paths are almost surely H¨older continuous of order strictly less thanH.

Proof We recall that a function f :R→Ris H¨older continuous of order α,

0< α≤1, and writef ∈C α (R), if there existsM >0 such that

|f(t)−f(s)| ≤M|t−s| α , for every s, t∈R For anyα >0 we have

|t−s| αH ; hence, by the Kolmogorov criterion we get that the sample paths ofB (H) are almost everywhere H¨older continuous of order strictly less thanH Moreover, by [9] we have lim sup t→0 +

|B (H) (t)| t H log logt −1 =c H with probability one, wherec H is a suitable constant HenceB (H) cannot have sample paths with H¨older continuity’s order greater thanH.

Path diﬀerentiability

Definition 1.5.2 If b = a − H in Definition 1.5.1, then we say that X (X t ) t≥0 is a self-similar process with Hurst index H or that it satisfies the property of (statistical) self-similaritywith Hurst indexH The quantityD 1/H is called the statistical fractal dimension ofX.

The covariance function of fractional Brownian motion (fBm) is homogeneous of order 2H, indicating that B(H) is a self-similar process characterized by the Hurst index H This means that for any constant a > 0, the processes B(H)(at) and a^(-H)B(H)(t) share the same distribution.

We recall that according to the Kolmogorov criterion (see [228]), a process

X = (X t ) t ∈Radmits a continuous modiﬁcation if there exist constantsα≥1, β >0, andk >0 such that

Theorem 1.6.1 LetH ∈(0,1) The fBmB (H) admits a version whose sample paths are almost surely H¨older continuous of order strictly less thanH.

Proof We recall that a function f :R→Ris H¨older continuous of order α,

0< α≤1, and writef ∈C α (R), if there existsM >0 such that

|f(t)−f(s)| ≤M|t−s| α , for every s, t∈R For anyα >0 we have

|t−s| αH ; hence, by the Kolmogorov criterion we get that the sample paths ofB (H) are almost everywhere H¨older continuous of order strictly less thanH Moreover, by [9] we have lim sup t→0 +

|B (H) (t)| t H log logt −1 =c H with probability one, wherec H is a suitable constant HenceB (H) cannot have sample paths with H¨older continuity’s order greater thanH

By [156] we also obtain that the processB (H) is not mean square diﬀerentiable and it does not have diﬀerentiable sample paths.

Proposition 1.7.1 Let H ∈ (0,1) The fBm sample path B (H) (.) is not diﬀerentiable.

In fact, for everyt 0 ∈[0,∞) lim sup t→t 0

Proof Here we recall the proof of [156] Note that we assumeB (H) (0) = 0. The result is proved by exploiting the self-similarity of B (H) Consider the random variable

R t,t 0:= B (H) (t)−B (H) (t 0) t−t 0 that represents the incremental ratio of B (H) SinceB (H) is self-similar, we have that the law of R t,t 0 is the same of (t−t 0) H−1 B (H) (1) If one considers the event

, then for any sequence (t n ) n∈N decreasing to 0, we have

The thesis follows since the probability of the last term tends to 1 asn→ ∞.

The fBm is not a semimartingale for H = 1/2

The non-semimartingale nature of the fractional Brownian motion (fBm) at H = 1/2 has been established by multiple researchers Notably, for H > 1/2, references such as [82], [150], and [152] provide insights This article revisits the proof presented in [195], which applies universally for H = 1/2 To demonstrate that B(H) is not a semimartingale at this value, it is sufficient to calculate the p-variation of B(H).

Deﬁnition 1.8.1 Let(X(t)) t∈[0,T] be a stochastic process and consider a partition π={0 =t 0 < t 1 < < t n =T} Put

The p-variation ofX over the interval [0, T]is deﬁned as

V p (X,[0, T]) := sup π S p (X, π), whereπ is a ﬁnite partition of[0, T] The index ofp-variationof a process is deﬁned as

Since B (H) has the self-similarity property, the sequence (Y n,p ) n ∈N has the same distribution as

By the Ergodic theorem (see, for example, [69]) the sequence ˜Y n,p converges almost surely and in L 1 to E

|B (H) (1)| p as n tends to inﬁnity; hence, it converges also in probability toE

The convergence behavior of the process B(H)i−1n p indicates that it approaches 0 in probability when pH > 1 and diverges to infinity when pH < 1 as n tends to infinity Consequently, we deduce that the index I(B(H), [0, T]) equals 1/H Given that the index I(X, [0, T]) for any semimartingale X must lie within the range [0, 1] ∪ {2}, it follows that the fractional Brownian motion B(H) can only be classified as a semimartingale if H equals 1/2.

Due to this limitation, the Itô stochastic calculus designed for semimartingales cannot be applied to define the stochastic integral with respect to B(H) In the upcoming chapters, we will explore various approaches from the literature that have been developed to address this issue.

In [53], the concept of weak semimartingale is introduced, revealing that B(H) does not qualify as a weak semimartingale when H = 1/2 A stochastic process (X(t)) for t ≥ 0 is classified as a weak semimartingale if, for every T > 0, the collection of random variables defined by n i=1 a i [X(t i )−X(t i − 1)], with n ≥ 1 and 0 = t 0 < < t n = T, where |a i | < 1 and a i ∈ F X t i−1, is bounded in L0 Here, F X denotes the natural filtration associated with the process X Additionally, the paper demonstrates that if B(t) represents a standard Brownian motion that is independent of B(H), then the process exhibits specific characteristics.

M H (t) :=B (H) (t) +B(t) is not a weak semimartingale if H ∈ (0,1/2) ∪ (1/2,3/4), while it is a semimartingale equivalent in law to B on any ﬁnite time interval [0, T] if

H ∈(3/4,1) We refer to [53] for further details.

Invariance principle

Here we present an invariance principle forfBms due to [36].

Assume that {X n , n = 1,2, } is a stationary Gaussian sequence with

X k , 0≤t≤1, where [ã] stands for the integer part We will show that if the covariance of n k=0 X k is proportional to Cn 2H for largen, Z n (t), t≥0 converges weakly to √

CB t (H) in a suitable metric space Let us ﬁrst introduce the the metric space Let I = [0,1] and denote by L p (I) the space of Lebesgue integrable functions with exponentp Forf ∈L p (I), t∈I, deﬁne ω p (f, t) = sup

, where I h = {x ∈ I, x+h ∈ I} For 0 < α < 1 and β > 0, consider the real-valued functionω β α (ã) deﬁned by ω β α (t) =t α 1 + log1 t β

The Besov space Lip p (α, β) consists of functions f in L p (I) for which the condition f ω p α β < ∞ holds This space, equipped with the norm || ã || ω p α β, is recognized as a nonseparable Banach space Within this framework, B p α,β represents a separable subspace of Lip p (α, β), comprising functions f that meet the criterion ω p (f, t) = o(ω α β (t)) as t approaches 0 For any continuous function f, the coefficients of its decomposition in the Schauder basis are denoted by {C n (f), n ≥ 0}.

The following characterization theorem proved in [55] will be used.

Theorem 1.9.1 1 If α >1/p, then Lip p (α, β) is the space of continuous functions with the following equivalence of norms:

2.f belongs toB p α,β if and only if j lim →∞

Lemma 1.9.2 Let 1 ≤ p < ∞, 1/p < α < 1, and β > 0 A set F of measurable functionsf :I→R is relatively compact inB α,β p if

Proof It is a consequence of the Frechet–Kolmogorov theorem: a subsetK⊂

L p (I) is relatively compact if and only if f∈K sup

To demonstrate that the set F is relatively compact, we need to establish that every sequence {f_n, n ≥ 1} within F has a convergent subsequence According to the Frechet–Kolmogorov theorem, any such sequence {f_n, n ≥ 1} will possess a convergent subsequence in L^p(I) For our proof, we can assume without loss of generality that f_n converges to f in L^p(I) Next, we will show that f belongs to the space B_p^{α,β} Utilizing the Fatou lemma, we can analyze the behavior of the sequence to conclude our argument.

The combination of the previous assumptions leads to the conclusion that ω p (f, t) is equivalent to o(ω α β (t)) as t approaches 0, confirming that f belongs to the space B p α,β To complete the proof, we will demonstrate that f n converges to f within the space B α,β p According to the definition of the norm in B p α,β, it is sufficient to establish that the limit as n approaches infinity of the supremum is valid.

For any 0< δ 0 By assumption (2) we can ﬁndδ >0 such that sup

2, for alln≥1 On the other hand, δmax(1/α,1/β).

Proof By the assumptions, we haveC 0(X n ) = 0 andC 1(X n ) =X 1 n To prove the lemma, by Lemma 1.9.3 it is enough to show that there exists a constant

C p >0 such that , forλ >0 and 1/p < β < β, we have P(||X n || ω p α β > λ)≤

C p λ −p for all n≥1 Applying the characterization Theorem 1.9.1 above, it suﬃces to show that

≤C p λ −p , whereM(X n ) is the maximum of the set

Now, by the Chebyshev inequality, we have

Let X i be a stationary Gaussian sequence with mean 0 and correlations

Theorem 1.9.5 Let H∈]0,1[,β >0 andp >max(1/H,1/β) Assume n k=1 n l=1 r(k−l)∼Cn 2H , a.s n→ ∞, for some positive constant C Then Z n (t) converges weakly to √

Proof First we prove that the ﬁnite-dimensional distributions of (Z n (t), t∈I) converge weakly to those of (√

CB t (H) t ∈ I) Fix any 0 < t 1 < t 2 t i We have

2 (1.7) has the same behavior as

1.9 Invariance principle 19 forn→ ∞ Hence the limit of (1.7) is

2{t 2H i +t 2H j − |t i −t j | 2H }=C(t i , t j ), as n→ ∞ It remains to show that the sequence (Z n (t), t∈I) is tight in the

Banach space B H,β p Let s, t ∈ I such that t ≥ s and p > 1/H Using the stationarity, we see that

Note that forn large enough, we have|([nt]−[ns])/n| pH ≤ |t−s| pH for all t, s∈I Hence, by Lemma 1.9.4 it suﬃces to show that there exists a positive constant C such thatE1/n H n−1 k=0 X k p

≤C for all n≥1, equivalently to show E[|Z n (1)| p ]≤C for alln≥1.

SinceS n = n−1 k=0 X k is Gaussian,E[(S n ) 2p ] is proportional to (E[(S n ) 2 ]) p for all p ≥ 1 and n ≥ 1 By our assumption, E[(S n ) 2 ] is asymptotically proportional to n 2H Thus,

E[|Z n (1)| p ]∼E[|S n | p ] n pH =O((E[(S n ) 2 ]) p/2 ) n pH =O(1), as n→ ∞ Hence, sup n ≥ 1 E[|Z n (1)| p ] 0, and p > max(1/H,1/β) Assume that {X n , n= 1,2, } is a stationary Gaussian sequence with spectral representation

−π exp(inλ)|λ| 1/2 − H B(dλ), n= 1,2, , where B(dλ) is a Gaussian random measure with E[B(dλ)| 2 ] = dλ Then there exists a positive constantC such that(Z n (t), t∈[0,1]) converges weakly to(CB t (H) , t∈[0,1]) in the spaceB p H,β

Proof Letr(k) =E[X 1 X k+1] be the covariance function of{X n , n= 1,2, }.

It suﬃces to show that n k=1 n l=1 r(k−l)∼C 2 n 2H

2 |λ| 1−2H dλ, as n→ ∞ Therefore, the proof is complete.

Stochastic calculus

Wiener integrals

This article presents an introduction to stochastic integrals concerning fractional Brownian motion (fBm), leveraging its Gaussian characteristics The concept of stochastic integrals for deterministic functions with respect to a Gaussian process, known as Wiener integrals, was previously established in [171].

In the case of Brownian motion, they coincide with Itô integrals For fBm they were defined for the first time in [76].

Fix an interval [0, T] and letB (H) (t), t∈[0, T], be afBm of Hurst index

In the probability space (Ω, F(H), F(H)t, PH), where H is within the interval (0,1), we utilize the natural filtration (F(H)t)t∈[0,T] alongside the law PH of B(H) For a detailed construction of the measure PH, please refer to Chapter 1, and for cases where H > 1/2, consult Section 3.1 Additionally, we define for s, t > 0.

=R H (t, s) By [177], we obtain the following:

1 ForH >1/2, the covariance of thef Bmcan be written as

24 2 Wiener and divergence-type integrals for fractional Brownian motion

|rưu| 2Hư2 du dr, whereα H =H(2H−1) We can rewrite

0 x α−1 e −x dx is the Gamma function, since u

=β(2ư2H, Hư1/2)(ru) 1/2ưH (rưu) 2Hư2 , where we have used the change of variablez = (r−v)/(u−v) and x r/(uz) and supposed r > u Consider now the deterministic kernel

(u−s) H − 3/2 u H − 1/2 du, (2.2) where c H = [H(2H−1)/(β(2−2H, H−1/2))] 1/2 and t > s Then we have that

K H (t, u)K H (s, u)du since by (2.1) it follows that t ∧ s

0 u 1ư2H (yưu) Hư3/2 (zưu) Hư3/2 du dz dy

Note also that with a change of variable in (2.2),K H (t, s) can be expressed equivalently as

See also [72], [76] for further details.

For a detailed proof of equation (2.4), see [76], [189], where it is proved by using the analyticity of both members as functions of the parameter

H See also [177], where a direct proof is given by using the ideas of [172] and the fact that

In order to deﬁne the Wiener integrals with respect toB (H) , we introduce the so-calledreproducing kernel Hilbert space denoted by H.

Deﬁnition 2.1.1 The reproducing kernel Hilbert space ( RKHS ), denoted by H, associated to B (H) for everyH ∈(0,1), is deﬁned as the closure of the vector space spanned by the set of functions {R H (t,ã), t∈[0, T]} with respect to the scalar product

In the case of standard Brownian motion, there exists a nice characterization ofH, which coincides with the space of absolutely continuous functions, vanishing at 0, with square-integrable derivative.

In the case offBm, it has been proved ﬁrst in [12] for H >1/2 and then in [75] for everyH ∈(0,1) that the following holds:

Proposition 2.1.2 For any H ∈ (0,1), H is the set of functions f which can be written as f(t) t

K H (t, s) ˜f(s)ds (2.6) for somef˜∈L 2 ([0, T]) By deﬁnition, f H =f˜ L 2 ([0,T ])

If K H (t, s) is of the form (2.2), by [206, p 187] we have that the integral representation (2.6) induces an isomorphism from L 2 ([0, T]) onto the space

In Definition B.1.3, we define the space I 0+ H+1/2 (L 2 ([0, T])), which indicates that H, when viewed as a vector space of functions without considering its Hilbert space structure and norm, aligns with the characteristics of a fractional space.

I 0+ H+1/2 (L 2 ([0, T])) of functionsψ of the form ψ(x) := 1 Γ(H+ 1/2) x

(x−y) H−1/2 f(y)dy, for some f ∈ L 2 ([0, T]) For further details, see also the proofs of [75] and Theorem 3.2 of [76].

Deﬁnition 2.1.3 For any H ∈ (0,1), the (abstract) Wiener integral with respect to the fBmis deﬁned as the linear extension fromH inL 2 (P H )of the isometric map I H :

By Deﬁnition 2.1.3, it follows that the abstract Wiener integral with respect to ﬁnite combinations ofR H (t,ã) is given by

For any function $ u $ in the Hilbert space $ H $, it is possible to construct a sequence $ (u_n)_{n \in \mathbb{N}} $ that consists of finite linear combinations of functions of the form $ R_H(t_i, \hat{a}) $, which converges to $ u $ in $ H $ This leads to the definition of the abstract Wiener integral of $ u $ in relation to the Borel $ \sigma $-algebra $ B(H) $.

I H (u) = lim n→∞ I H (u n ), where the limit is taken inL 2 (P H ).

Applying this construction to standard Brownian motion reveals that, according to the definition of the Reproducing Kernel Hilbert Space (RKHS), the set of admissible integrands consists of continuous deterministic functions whose first derivative is square integrable over the interval [0, T] This characterization uniquely defines the space of admissible integrands.

2.1 Wiener integrals 27 the space of integrands in the standard Brownian motion case This is a consequence of the fact that the properties of Wiener integrals don’t change if

In the construction of the Wiener integral for Brownian motion, the space H is substituted with an isometrically isomorphic space, as the bijective isometry from L2([0, T]) to H facilitates this identification To achieve a similar characterization of the integrand space for fractional Brownian motion (fBm), we also replace H with an isometrically isomorphic Hilbert space.

Deﬁnition 2.1.4 By a representationof H we mean a pair(F, i)composed of a functional space F and a bijective isometryi between FandH.

By Proposition 2.1.2, we immediately get the following:

Theorem 2.1.5 There exists a canonical isometric bijection betweenL 2 ([0, T]) andHgiven by i 1:L 2 ([0, T])−→H, h−→f(t) t

K H (t, s)h(s)ds, where H is endowed with the scalar product deﬁned in (2.5) and L 2 ([0, T]) with the usual inner product Hence (L 2 ([0, T]), i 1 )is a representation ofH.

According to this representation, for H = 1/2 we have i 1 = I 0+ 1 , i.e., i 1 (h) = t

0 h(s)ds In general, for any H ∈ (0,1), K H (t,ã) is associated by i 1 toR H (t,ã).

We now focus on the case H >1/2 From now on we denote byEthe space of step functions on [0, T] Another representation ofHis given by

Theorem 2.1.6 For any H > 1/2, consider L 2 ([0, T]) equipped with the twisted scalar product: f, g H :=H(2H−1)

Deﬁne the linear mapi 2 on the spaceEof step functions on [0, T]by i 2: (L 2 ([0, T]), H )−→H,

Then the extension of this map to the closure of (L 2 ([0, T]), H ) with respect to the scalar product deﬁned in (2.7) is a representation ofH.

The space (L 2 ([0, T]), H ) is identified as (E, H ) and is denoted with i 2 even for the extended map Research in [188] demonstrates that this space is not complete, necessitating the closure to form a Hilbert space Additionally, studies in [189] and [190] reveal that the elements of H2 := (L 2 ([0, T]), H ) may consist of distributions of negative order rather than traditional functions.

To summarize, the Wiener integrals with respect to B (H) with H >1/2 can be seen as the extensions of the following isometries:

1 Wiener integrals of ﬁrst type:

2 Wiener integrals of second type:

I [0,t] (ã)→B (H) (t), induced by the representationsi 1 andi 2 , respectively So either one keeps the original scalar product onL 2 ([0, T]) and changes the pre-image ofB (H) (t) to

In the context of fractional Brownian motion (fBm), the scalar product on L²([0, T]) is adjusted so that I[0,t](ã) serves as the predecessor of B(H)(t), highlighting a key distinction from standard Brownian motion Additionally, the first-type integrals (2.8) do not align with the isometry property essential for the abstract framework of Wiener integrals.

The process I H 1 (I [0,t] ) is identified as a centered Gaussian process characterized by a covariance kernel of min(t, s), which aligns it with standard Brownian motion This article establishes a relationship between two distinct forms of Wiener integrals.

Consider the operatorK H induced by the kernelK H (t, s) onL 2 ([0, T]) for

LetK ∗ H be the adjoint operator ofK H in L 2 ([0, T]), i.e.,

0 f(s)(K ∗ H g)(s)ds (2.10) for every f, g∈L 2 ([0, T]) By Fubini theorem we obtain that

Since by exploiting fractional calculus (see Appendix B and also [206]) we can rewrite the action of K H as

2)x 1/2−H I T H−1/2 − x H−1/2 I T 1 − , (2.12) for f ∈ L 2 ([0, T]) From a formal point of view, one can think that the two operator are linked by the relation

K H (s, t) =K ∗ H (δ t )(s), whereδ t is the Dirac measure with mass att SinceI T 1 − δ t =I [0,t] , we have

By using the characterization (2.13) we obtain another representation forH, i 3:L 2 ([0, T])−→H, h−→f(t) =c H Γ(h−1

The expression $ t^{1/2-H} (I_T H - 1/2 \times 1/2^{-H} h)(t) $ remains valid for $ H < 1/2 $ and serves as the "dual" representation of $ i_1 $ (refer to [190]) The dual space of $ H \ includes dense linear combinations of Dirac masses, indicating that \( i_3 $ represents an isometric mapping of the dual space of $ H $ Given that representations are defined up to isomorphisms, $ i_3 $ can also be interpreted as a representation of $ H $.

By (2.12) we obtain the following relation between Wiener integrals of ﬁrst and second type.

Theorem 2.1.7 Let H >1/2 For any functionu∈L 2 ([0, T]), we have

Proof Equation (2.14) is immediately veriﬁed for indicator functionsI [0,t] by the deﬁnition ofK ∗ H The result then follows by a limiting procedure See [72] for further details

In the sequel we focus on Wiener integrals of second type induced by the isometry (2.9) For the sake of simplicity, from now onwe identify the RKHS

H with H2 = (L 2 ([0, T]), H ) through the representation mapi 2, i.e., we put H =H2 Note that the map (2.9) induced by i 2 that associatesI [0,t] to

The isometry between B(H) and the first-order chaos associated with B(H) is established, specifically within the closed subspace of L²(PH) generated by B(H) Moving forward, we will no longer differentiate between the Wiener integrals of the first and second types, simplifying our notation accordingly.

In order to characterize Wiener integrals of second type, by following the approach of [7] and [177], we now introduce the linear operator K H ∗ deﬁned onψ∈Eas follows:

By equation (2.16) it follows that the operator K H ∗ is an isometry between the space E of elementary functions and L 2 ([0, T]) that can be extended to the Hilbert spaceH This is because

The operatorK H ∗ can be rewritten by using the means of fractional calculus (see Appendix B) Note that by the representation (2.2) for the square- integrable kernel K H (t, s) we get

Hence by equations (2.15) and (2.17) and by the deﬁnition of the fractional integral (B.1) of Appendix B with α=H−1/2 andb=T, we obtain immediately the following fractional representation forK H ∗ :

Moreover, by using the following relation between the fractional integral and the fractional derivative

2.1 Wiener integrals 31 for every ψ∈L 1 (0, T) [see (B.3) of Appendix B], we also have that

In particular, we obtain that the indicator function I [0,a] belongs to the image of K H ∗ for a ∈[0, T] because putting ψ = I [0,a] in (2.19) and using the characterization (B.2) of fractional derivative, we have

As a consequence of this result, we obtain that the image of the operatorK H ∗ coincides withL 2 ([0, T]), i.e.,

Remark 2.1.8.The operatorK H ∗ deﬁned in (2.15) is the adjoint of K H in the following sense:

Lemma 2.1.9 For any functionψ∈Eandh∈L 2 ([0, T])we have

Proof.For the proof, we refer to Lemma 1 in [6]

Note that the relation betweenK ∗ H introduced in (2.11) and K H ∗ is then

This follows directly by comparing (2.12) and (2.18) since

Consider now the processB(t) that is associated by the representation i 2 to (K H ∗ ) − 1 (I [0,t] ), i.e.,

B(t) :=B (H) ((K H ∗ ) − 1 I [0,t] ) (2.21) SinceB(t) is a continuous Gaussian process with covariance given by

=s∧t, we conclude that B(t) is a standard Brownian motion Analogously, the stochastic process associated to

K H ∗ I [0,t]=K H (t, s)I [0,t](s) by the isometry induced byB(t) onL 2 ([0, T]) is afBm B (H) (t) with integral representation

Remark 2.1.10.The representation (2.22) holds in law, and it is shown in

The study presents a fixed standard Brownian motion defined on the probability space (Ω, F(H), P_H), which is essential for understanding its trajectory This characterization proves beneficial for numerical simulations of the Brownian motion paths, as outlined in the findings of reference [76].

Theorem 2.1.11 Letπ n be an increasing sequence of partitions of[0, T]such that the mesh size |π n | ofπ n tends to zero asngoes to inﬁnity The sequence of processes(W n ) n∈N deﬁned by

B(t (n) i+1 )−B(t (n) i ) converges toB (H) inL 2 (P⊗ds), where herePdenotes the probability measure induced by the standard Brownian motion B.

Proof.For the proof, we refer to Proposition 3.1 of [76]

For further results about discrete approximation for f Bm for H >1/2, see also [30], [140] and [165].

According to equations (2.21) and (2.22), the processes B(t) and B(H)(t) produce the same filtration Additionally, we derive a representation of the second type Wiener integral concerning B(H) in relation to an integral involving Brownian motion B.

Since by (2.18) the Hilbert spaceH coincides with the space of distribu- tionsψsuch thats 1/2 − H (I t H−1/2 − u H − 1/2 ψ)(s) is a square-integrable function, the integral representation in (2.23) is correctly deﬁned forψ∈H.

In order to obtain a space of functions contained inH, we consider the linear space|H|generated by the measurable functionsψsuch that ψ 2 |H| :=α H

|ψ(s)||ψ(t)||s−t| 2H − 2 ds dt 0.

Proof Here we recall brieﬂy the proof of [161] by following [8] If one applies the H¨older inequality to

|ψ(u)||ψ(r)||rưu| 2Hư2 dr du withq= 1/H, then ψ 2 |H| ≤α H

By exploiting fractional calculus (see Appendix B, and also [206]) we have that

The thesis follows by the Hardy–Littlewood inequality (see [217])

I 0+ α (ψ) L q (0,∞) ≤c H,p ψ L p (0,∞) , applied to the particular case whenα= 2H−1,q= 1/(1−H) andp=H

As a consequence the following inclusions hold

The inclusionL 2 ([0, T])⊂ |H|can also be seen directly since

As we have seen, Wiener integrals are introduced for deterministic integrands.

In order to extend the deﬁnition of the Wiener integral of second type to the general case ofstochastic integrands, we follow the approach of [72] and use

Theorem 2.1.7 to give the following deﬁnition.

Deﬁnition 2.1.14 Consider H >1/2 Letu be a stochastic process u ã (ω) :

[0, T]−→Hsuch thatK H ∗ uis Skorohod integrable with respect to the standard Brownian motionB(t) Then we deﬁne theextended Wiener integralofuwith respect to the fBmB (H) as

(K H ∗ u)(s)δB(s), where the integral on the right-hand side must be interpreted as a Skorohod integral with respect to B(t)(Deﬁnition A.2.1).

Deﬁnition 2.1.14 is an extension of (2.23) to the a case of a stochastic process useen as a random variable with values inHand such thatK H ∗ uis Skorohod integrable Note that we have used the same symbol for the standard and the extended Wiener integral.

We now consider the case when the Hurst index H belongs to the interval

(0,1/2) Main references for this section are [5], [6], [54] and [177].

As forH >1/2, we focus on the following representation for the RKHS

H Consider the space Eof step functions on [0, T] endowed with the inner product

I [0,t] , I [0,s] H :=R H (t, s), 0≤t, s≤T, (2.25) and the linear mapi 2 onEgiven by i 2: (E, H )−→H,

Divergence-type integrals for f Bm

2 ForH >1/2 there exist functionsf ∈L 2 ([0, T]) for which the equation s 1/2−H (I T H−1/2 − u H−1/2 ψ(u))(s) =f(s) (2.29) cannot be solved In fact, since I T H − − 1/2 is an integral operator, the left- hand side of (2.29) will satisfy some smoothness conditions that may not hold for a generalf ∈L 2 ([0, T]).

Note also that for the space of H¨older continuous functions of orderγ in the interval [0, T] (for a deﬁnition, see also the proof of Theorem 1.6.1), it holds

C γ ([0, T])⊂H ifγ >1/2−H As in the caseH >1/2, the process

B(t) =B (H) ((K H ∗ ) − 1 (I [0,t] )) is a Wiener process and the fractional Brownian has the integral representation

Hence we can conclude with the following

Proposition 2.1.17 For H < 1/2 the Wiener-type integral B (H) (ψ) with respect to fBmcan be deﬁned for functions ψ∈H=I T 1/2 − − H (L 2 ([0, T])) and the following holds:

2.2 Divergence-type integrals for f Bm

This article explores the properties and key outcomes of a stochastic integral for fractional Brownian motion (fBm), which is defined as the dual operator of the stochastic derivative Additionally, we examine its connections to Wiener integrals and those outlined in Chapter 3, drawing on primary references [6], [8], [170], and [177].

Consider H ∈ (0,1) and H = (E, H ) Let S H be the set of smooth cylindrical random variables of the form

F =f(B (H) (ψ 1), , B (H) (ψ n )), where n ≥ 1, f ∈ C b ∞ (R n ), and ψ i ∈ H The derivative operator D (H) of

F ∈S H is deﬁned as theH-valued random variable

The derivative operator D (H) is then a closable unbounded operator from

L p (Ω,P H ) inL p (Ω;H) for anyp≥1 We denote byD (H),k the iteration of the derivative operator The iterate derivative operator D (H),k maps L p (Ω,P H ) into L p (Ω,H ⊗k ).

Deﬁnition 2.2.1 For any k ∈ N and p≥1 we denote byD k,p H the Sobolev space generated by the closure of S H with respect to the norm

(D (H) ) j F p H ⊗j and byD k,p (H)the corresponding Sobolev space ofH-valued random variables.

We introduce the adjoint operator of the derivative.

Deﬁnition 2.2.2 We say that a random variable u ∈ L 2 (Ω;H) belongs to the domain domδ H of the divergence operator if

Deﬁnition 2.2.3 Let u ∈ domδ H Then δ H (u) is the element in L 2 (P H ) deﬁned by the duality relationship

Hence thedivergence operator δ H is the adjoint of the derivative operator

D (H) Note that by Deﬁnition 2.2.3 we obtain immediately that the space

D 1,2 (H) of H-valued random variables is included in domδ H and for u ∈

By the Meyer inequalities (see, for example, [176]), we also get for all p >1 that δ H (u) L p (P H ) ≤c p u D 1,p ( H )

Ifuis a simpleH-valued random variable of the form u n j=1

F j X j , where F j ∈ D 1,2 H and X j ∈ H, then u belongs to domδ H and by Deﬁnition2.2.3 we obtain

2.2 Divergence-type integrals forf Bm 39 δ H (u) n j=1

Moreover, if F ∈ D 1,2 H and u ∈ domδ H are such that F u, and F δ H (u) +

D (H) F, u H are square integrable, thenF u∈domδ H and (2.32) extends to δ H (F u) =F δ H (u)− D (H) F, u H (2.33)

We now investigate separately the casesH >1/2 andH 1/2; forH 0 in L 1 as t → 0 Hence, it follows that there exists a measurable set ˜Ω ⊂Ω with P H ( ˜Ω) = 1 and a sequence of positive numbers (t k ) k∈N that converges to 0 such that for all ω∈Ω˜ andk∈N,

Assume that there exists an ω ∈Ω˜ such that u(ω)∈H By (6.40) of [206], the function u(ω) has the property

Butu(ω) can only satisfy both (2.39) and (2.40) at the same time ifH > α 1/2−H, which contradictsH≤1/4 Therefore,u(ω)∈/ Hfor allω∈Ω This˜ concludes the proof

By Proposition 2.2.6 it follows that processes of the form

The operator B (H) (t) defined for t in R cannot belong to the domain of the divergence δ H if H is less than or equal to 1/4 Therefore, we explore an extension of the divergence δ H to include processes with paths outside of H, following the methodologies outlined in references [54] and [177].

By (2.28) and the fractional integration by parts formula (B.5), for anyf, g∈

=d 2 H f, s H − 1/2 s 1/2 − H D 1/2−H 0+ (s 1 − 2H D T 1/2−H − s H − 1/2 g) L 2 ([0,T]) , where forH ≤1/2 the operatorK H ∗ is introduced in (2.26) This implies that the adjoint K H ∗,a of the operatorK H ∗ in L 2 ([0, T]) is

In order to introduce an extended domain for the divergence operator δ H as in the approach of [54], we introduce the space

LetS K the space of smooth cylindrical random variables of the form

F =f(B (H) (ψ 1), , B (H) (ψ n )), where n ≥1,f ∈ C b ∞ (R n ), i.e., f is bounded with smooth bounded partial derivatives, andψ i ∈K.

Deﬁnition 2.2.7 Let u(t), t ∈ [0, T] be a measurable process such that

< ∞ We say that u ∈ dom ∗ δ H if there exists a random variableδ H (u)∈L 2 (P H )such that for all F ∈S K we have

Note that ifu∈dom ∗ δ H , thenδ H (u) is unique and the mapping δ H : dom ∗ δ H → ∪ p>1 L p (P H ) is linear This extended domain satisﬁes the following natural requirements:

1 domδ H ⊂ dom ∗ δ H , and δ H restricted to domδ H coincides with the divergence operator (Proposition 3.5 of [54]).

2 In particular, domδ H = dom ∗ δ H ∩[∪ p>1 L p (Ω;H)] (Proposition 3.5 of [54]).

3 If u ∈ dom ∗ δ H such that E[u] ∈ L 2 ([0, T]), then E[u] belongs to H (Proposition 3.6 of [54]).

4 Ifuis a deterministic process, thenu∈dom ∗ δ H if and only ifu∈H.

5 The extended divergence operator δ H is closed in the following sense. Let p ∈ (1,∞] and q ∈ (2/(1 + 2H),∞] Let (u k ) k≥1 be a sequence in dom ∗ δ H ∩L p (Ω, L q ([0, T])) andu∈L p (Ω, L q ([0, T])) such that k→∞ lim u k =u in L p (Ω, L q ([0, T])).

It follows that for alln∈N0 andF ∈S H , we have k lim →∞ u k (t)K H ∗ ,a K H ∗ D (H) F=u(t)K H ∗ ,a K H ∗ D (H) F inL 1 (Ω×[0, T]) If there exists a ˆp∈(1,∞] and anX ∈L p ˆ (Ω) such that k→∞ lim δ H (u k ) =X in L p ˆ (Ω),thenu∈dom ∗ δ H andδ H (u) =X.

2.2 Divergence-type integrals forf Bm 45

By Theorem 3.7 of [54] we also get the following theorem:

Theorem 2.2.8 (Fubini Theorem) Let (Y,Y, à) be a measure space and u=u(ω, t, y)∈L 0 (Ω×[0, T]×Y)such that

Fractional Wick Itˆ o Skorohod (fWIS) integrals for fBm of Hurst index H > 1 / 2

This chapter presents the definition of the stochastic integral concerning fractional Brownian motion (fBm) for the Hurst index range of 1/2 < H < 1, utilizing the white noise analysis method We define fractional white noise and establish the stochastic integral as an element within the fractional Hida distribution space.

To derive a classical Itô formula, it is essential for the stochastic integral to function as a standard random variable This leads to the introduction of the φ-derivative, which facilitates the existence of the Wick product in L² Consequently, classical Itô-type formulas are established, and their applications are explored Key references for this chapter include [32].

Fractional white noise

FixH with 1/2< H 0, t

LetS(R) be the Schwartz space of rapidly decreasing smooth functions onR, and iff ∈S(R), denote f 2 H :

If we equipS(R) with the inner product f, g H :

The completion of S(R) leads to the separable Hilbert space L²φ(R), and we can similarly define L²φ(R+) or L²φ([0, T]) over finite intervals It's important to note that elements of L²φ(R) can include distributions, as discussed in Chapter 2.

L 2 H (R) the subspace of deterministic functions contained in L 2 φ (R).

Remark 3.1.1.We remark that in (3.3) and (3.4) we use the same notation as in Theorem 2.1.6, since the inner product (3.4) extends (2.7) to the case of functions deﬁned on the whole real axis Hence we start here with an analogous setting to the one for Wiener integrals in Chapter 2, but we provide a diﬀerent construction of the stochastic integral for fBm for H > 1/2 Moreover, by (2.7), (3.1), and (3.4) we obtain that L 2 φ ([0, T]) = H = (L 2 ([0, T]), H ) andL 2 φ (R)⊇L 2 φ ([0, T]) =Hif we identifyψ∈L 2 φ ([0, T]) withψI [0,T]

In particular we obtain the following representation ofL 2 φ (R).

(tưu) H ư 3/2 f(t)dt, where c H H(2H−1)Γ(3/2−H)/(Γ(H−1/2)Γ(2−2H)), andΓ denotes the gamma function ThenI − H − 1/2 is an isometry fromL 2 φ (R)toL 2 (R).

Proof By a limiting argument, we may assume that f andg are continuous with compact support By deﬁnition,

R 2 f(s)g(t) s∧t ư∞ (sưu) Hư3/2 (tưu) Hư3/2 du ds dt

R f(s)g(t)φ(s, t)ds dt= (f, g H , where we have used the identity (see [103, p 404]) c 2 H s∧t ư∞ (sưu) H ư 3/2 (tưu) H ư 3/2 du=φ(s, t).

Now letΩ=S (R) be the dual ofS(R) (considered as Schwartz space), i.e.,

The space of tempered distributions on R is denoted as Ω The mapping f → exp(−1/2f²_H), where f belongs to S(R), is positive definite on S(R) According to the Bochner–Minlos theorem, there exists a probability measure P_H defined on the Borel subsets B(Ω) of Ω.

Ω e iω,f dP H (ω) =e −1/2f 2 H ∀f ∈S(R), (3.5) where ω, fdenotes the usual pairing betweenω ∈S (R) and f ∈S(R) and f H is deﬁned in (3.3) It follows from (3.5) that

=f 2 H , (3.6) where E denotes the expectation under the probability measure P H Using this we see that we may deﬁne

B˜ t (H) = ˜B (H) (t, ω) =ω, I [0,t](ã) as an element of L 2 (P H ) for eacht∈R, where

By Kolmogorov’s continuity theorem ˜B t (H) has a t-continuous version, which we will denote by B t (H) , t ∈ R From (3.6) we see that B t (H) is a Gaussian process with

It follows that B t (H) is a fBm From now on we endow Ω with the natural ﬁltration F t (H) ofB (H)

The stochastic integral with respect to fBm for deterministic function is easily deﬁned (see also Section 2.1).

Lemma 3.1.3 Iff,gbelong toL 2 H (R), then

R g s dB (H) s are well-deﬁned zero mean, Gaussian random variables with variances f 2 H and g 2 H , respectively, and

Proof This lemma is veriﬁed in [103] It can be proved directly by verifying it for simple functions n i=1 a i I [t i ,t i+1 ] (s) and then proceeding with a passage to the limit

LetL p (P H ) =L p be the space of all random variablesF :Ω→Rsuch that

For anyf ∈L 2 H (R), deﬁneε:L 2 H (R)→L 1 (P H ) as ε(f) := exp

Iff ∈L 2 H (R), thenε(f)∈L p (P H ) for eachp≥1 andε(f) is called an exponential functional (e.g., [160]) LetEbe the linear span of the exponentials, that is,

Theorem 3.1.4.Eis a dense set of L p (P H )for each p≥1 In particular,E is a dense set ofL 2 (P H ).

Proof A functionalF :Ω→Ris said to be a polynomial of thefBm if there is a polynomialp(x 1 , x 2 , , x n ) such that

The function F can be expressed as F = p(B_t(H)_1, B_t(H)_2, , B_t(H)_n) for a sequence of time points 0 ≤ t_1 < t_2 < < t_n Given that (B(H)_t, t ≥ 0) is a Gaussian process, it is established that the collection of all polynomial fractional Brownian functionals is dense in L^p(P_H) for p ≥ 1 This denseness of polynomials arises from the continuity of the process and is supported by the Stone-Weierstrass theorem.

To demonstrate the theorem, it suffices to show that any polynomial can be approximated by elements in the set E Given that the Wick product of exponentials remains an exponential, it becomes clear that it is essential to establish that for any time t > 0, the space B(H) t can be approximated by elements in E.

Let f δ (s) = I [0,t] (s) δ, δ > 0 Clearly f δ is in L 2 H (R) Then ε(f δ ) c(δ)e δB t (H) for some positive constantc(δ) It is easy to see that

F δ =ε(f δ )−c(δ) c(δ)δ = e δB (H) t −1 δ is inE Ifδ→0, thenF δ →B t (H) in L p (P H ) for eachp≥1 This completes the proof

The following theorem is also interesting.

Theorem 3.1.5 If f 1 , f 2 , , f n are elements in L 2 H (R) such that f i − f j H = 0 for i=j, then ε(f 1), ε(f 2), , ε(f n ) are linearly independent in

Proof This theorem is known to be true if thefBm is replaced by a standard Brownian motion, (e.g., [160]).

Let f 1 , f 2 , , f k be distinct elements in L 2 H (R) Let λ 1 , λ 2 , , λ k be real numbers such that λ 1 ε(f 1) +λ 2 ε(f 2) +ã ã ã+λ k ε(f k ) L 2 (P H )= 0.

By an elementary computation for Gaussian random variables it follows that λ 1 e f 1 ,g H +λ 2 e f 2 ,g H +ã ã ã+λ k e f k ,g H = 0.

Replaceg byδg forδ∈Rto obtain λ 1 e δf 1 ,g H +λ 2 e δf 2 ,g H +ã ã ã+λ k e δf k ,g H = 0.

Expand the above identity in the powers ofδand compare the coeﬃcients of δ p , forp∈ {0,1, , k−1} to obtain the family of equations λ 1 f 1 , g p H +λ 2 f 2 , g p H +ã ã ã+λ k f k , g p H = 0 forp= 0,1, , k−1 This is a linear system ofkequations andkunknowns.

By the Vandermonde formula, the determinant of this linear system is det (f i , g p H ) =$ i H > \frac{1}{4} \), the process $ u_t = f(B_t(H)) $ is classified within $ D^{1,2}(H K_H) $, ensuring that $ \text{Tr}_D(H) u $ exists.

0 f (B (H) (t))t 2H−1 dt. whereδ H (f(B (H) ) is the divergence-type integral introduced in Chapter 2.

To expand the range of integrands beyond those discussed in (5.5), we leverage the properties of the extended divergence operator, as demonstrated in [54] Their findings indicate that the symmetric integral of a general smooth function of B(H) with respect to B(H) is present in L²(PH) if and only if H exceeds 1/6.

Note that ifh:R−→Ris a continuous function, then lim → 0

2h 2 (0) Hence it follows that for all H∈(0,1),

For H > 1/2, the process B(H) exhibits zero quadratic variation, allowing us to apply Theorem 2.1 from [199] to establish that for all H > 1/2 and g ∈ C¹(R), the integral b a g(B(H)(s)) dB(H)(s) equals G(B(H)(b)²) - G(B(H)(a)²), where G(x) = g(x) This relationship is also validated for H = 1/2 in [200], even when g ∈ L² loc(R) Conversely, for H < 1/2, while B(H) has infinite quadratic variation, equation (5.6) remains valid for g ∈ C¹(R) if H is within (1/4, 1/2), as demonstrated in [5] Theorem 4.1 from [102] extends this proof to H = 1/4, with g ∈ C³(R) Notably, Theorem 5.3 from [54] identifies H = 1/6 as the critical value for the existence of the symmetric integral in (5.6) when g ∈ C³(R).

Proposition 5.3.3 Let g∈C 3 (R) Then the following results hold:

130 5 Pathwise integrals for fractional Brownian motion b a g(B (H) (s))d o B (H) (s) =G(B (H) (b) 2 )−G(B (H) (a) 2 ), whereG(x) = x

2 On the other hand, ifH ∈(0,1/6], then b a

Proof The proof of this result is quite technical and we refer to [54, Theorem5.3, Lemma 5.4, 5.5, 5.6 and Proposition 5.7] for further details.

Relation with the divergence integral

In Proposition 5.3.2 we have already investigated the relation between the symmetric and the divergence-type integrals forH 1/2 following the approach of [8] and [177].

We recall that the spaceH= (L 2 ([0, T]), H ) is introduced in Theorem 2.1.6,|H|in (2.24), the stochastic derivativeD (H) in Section 2.2 and thatP H denotes the law ofB (H)

Proposition 5.4.1 Let H >1/2 Suppose(u t ) t∈[0,T] is a stochastic process in D 1,2 (|H|)and that

Then the symmetric integral exists and the following relation holds:

Under the assumptions of Proposition 5.4.1, the symmetric, backward and forward integrals coincide For example, a suﬃcient condition for (5.7) is that

Proof Here we sketch a short proof of (5.8) by following [177] For further details, we refer to the proof of Proposition 3 of [8] We start by approximating uby

5.4 Relation with the divergence integral 131 u t = 1 2 t+ t − u(s)ds.

Then u 2 D 1,2 (|H|) ≤d H u 2 D 1,2 (|H|) for some positive constant d H depending onH Since by (2.33) we have the following integration by part formula δ H (F u) =F δ H (u)− D (H) F, u H , ifF ∈D 1,2 H ,u∈domδ H , andF u, F δ H (u) +D (H) F, u H ∈L 2 (P H ), we obtain T

The proposition follows taking the limit as tends to 0

We now investigate when the symmetric integral coincides almost surely with the limit of Riemann sums, i.e., with the pathwise Riemann–Stieltjes integral Using the representation (5.8), we obtain the following:

Proposition 5.4.2 Let H > 1/2 If u is an adapted process continuous in the norm of D 1,2 (H)such that n −→∞ lim

0 u(s)d o B (H) (s), where the convergence holds inL 2 (P H )andπ: 0 =t 0 < t 1 < < t N =T is a partition of[0, T] with mesh size|π|= sup i=0, ,n

Proof Here we adapt the proof of Proposition 5 of [6] to the fBm case for

132 5 Pathwise integrals for fractional Brownian motion

|π|→0 n i=1 u(t i ) (B (H) (t i+1 )−B (H) (t i )), where the convergence is inL 2 (P H ), we have n i=1 u(t i ) (B (H) (t i+1)−B (H) (t i )) n i=1 u(t i )(B (H) (t i+1 )−B (H) (t i )) + n i=1

We need to study the convergence of the second term of the sum By using the properties of the operatorK H ∗ deﬁned in (2.15) with respect toã,ã H , we get

D r u(s i )(K H (s i+1 , r)−K H (s i , r))dr, where we have also applied (2.36) Then

The last term converges to zero since (5.9) holds.

Relation with the fWIS integral

We now investigate the relation between the fWIS integral introduced in Chap- ter 3 and the pathwise integral in the case H >1/2 Here T

5.5 Relation with the fWIS integral 133 must be interpreted as the fWIS integral deﬁned in Deﬁnition 3.4.1 for

H > 1/2 Main reference for this section is [83] For other approaches see also [68] and [149].

Consider the symmetric integral as introduced in Deﬁnition 5.1.1 and characterized in Proposition 5.4.2 We recall that in Chapter 3 the space

L φ (0, T) of integrands is deﬁned as the family of stochastic processes F on

[0, T] with the following properties: F ∈ L φ (0, T) if E

< ∞, F is φ-diﬀerentiable, the trace of D φ s F t ,0 ≤ s ≤ T, 0 ≤ t ≤ T exists, and

1/2 andF ∈L φ (0, T) Then the symmetric integral T

0 F s d o B (H) (s)exists and the following equality is satisﬁed

0 F s dB (H) (s)is the fWIS integral deﬁned in Theorem 3.6.1 andD s φ F in Deﬁnition 3.5.1.

Proof Since ifg∈L 2 H (R),F∈L 2 (P H ), andD Φg F∈L 2 (P H ), by Proposition 3.5.4 we have that

This equality easily proves the theorem

134 5 Pathwise integrals for fractional Brownian motion

These two types of stochastic integrals are both interesting and present diﬀerent characteristics The expectation of t

0 F s dB (H) (s) is 0, but the chain rule for this type of integral is more complicated than for the pathwise integral.

On the contrary, we have that E t

= 0 in general, as the following shows.

It is well known that if X is a standard normal random variable, X ∼

√2 n−1 (n+ 1)H((n−1)/2)! which is not 0 Ifnis even, then by the same computation, we obtain

In this article, we explore a fascinating phenomenon related to stochastic processes We define a partition π of the interval [0, T] as 0 = t0 < t1 < t2 < < tn = T For a continuous stochastic process (f(s), s ≥ 0), the Itô integral with respect to Brownian motion (Bt, t ∈ [0, T]) is established as the limit of the Riemann sums Σ(n−1)i=0 ft i (Bt i+1 − Bt i) as the partition mesh size |π| approaches 0 Additionally, we define the Stratonovich integral in the context of standard Brownian motion as the limit of similar Riemann sums.

5.5 Relation with the fWIS integral 135 n−1 i=0 f t i +f t i+1

As the partition |π| approaches zero, we demonstrate that the symmetric integral corresponds to a Stratonovich-type integral for fractional Brownian motion B(H)t, where t is greater than or equal to zero and H is greater than 1/2 Furthermore, we establish that the two limits converge for a wide range of stochastic processes.

Initially the following lemma is given.

Lemma 5.5.2 Let pbe a positive even integer Then

=|t−s| 2H , we have that (B (H) (t)−B (H) (s))/|t−s| H is a standard Gaussian random variable and

Corollary 5.5.3 For each α >1, there is aC α 1/2, L 2 H (R) coincides with the subspace generated by the deterministic functions in L 2 φ (R) Let

{e n , n= 1,2, }be an orthonormal basis ofL 2 φ (R) Then

{ξ n , n= 1,2, }={M e n , n= 1,2, } (6.7) is an orthonormal basis ofL 2 (R) Consider an elementF(ω) α∈J c α H% α (ω)∈ (S) ∗ H , where ifα= (α 1 , , α m )∈J, we put

The operatorM induces the function

The relationship between the WIS integral and the fWIS integral can be defined as follows: For a function Y: R → (S)*, if the product Y(t)W(H)(t) is integrable in (S)*, then the modified function M6 - 1 Y: R → (S)* H will also yield that M6 - 1 Y(t)W(H)(t) is integrable in (S)* H.

Analogously, suppose Y : R → (S) ∗ H is a given function such that Y(t)

W (H) (t) is integrable in (S) ∗ H Then M Y6 : R → (S) ∗ is such that M Y6 (t)

(6.11) For further details, we also refer to [116].

6.2.4 Relations with the pathwise integrals

We investigate here the relations among the stochastic integral of divergence, fWIS and WIS types and the symmetric (respectively, forward) integral.

In the divergence case, we need to distinguish between H > 1/2 and

Proposition 6.2.3 (Relation between the symmetric integral and the divergence I) Let H >1/2 Suppose (u t ) t ∈ [0,T] is a stochastic process in

Then the symmetric integral exists and the following relation holds:

6.2 Relations among the diﬀerent deﬁnitions of stochastic integral 159 T

Note that under the assumptions of Proposition 6.2.3, the symmetric, backward and forward integrals coincide.

Proposition 6.2.4 (Relation between the symmetric integral and the divergence II) LetH 1/2 If F ∈ L φ (0, T), then the symmetric integral

0 F s d o B (H) (s)exists and the following equality is satisﬁed

0 F s dB (H) (s)is the fWIS integral.

We note that the result of Theorem 6.2.5 also follows by Propositions 6.2.2 and 6.2.3 and by (6.16), that we prove in the sequel.

The relationship between generalized forward integrals and the WIS integral is significant for every H in the range (0,1) Additionally, the space L(H)1,2 encompasses all càdlàg processes ψ(t) represented as a sum of cαHα(ω), ensuring that ψ is contained within L(H)1,2.

M s D s ψ(t)ds exists in L 2 (P)for all t Moreover,

Theorem 6.2.7 (Relation between the (generalized) forward and WIS integral) Let H ∈(0,1) Suppose ψ ∈L (H) 1,2 and that one of the following conditions holds:

M t+ 2 D (H) t+ ψ(t)dt, holds as an identity in (S) ∗ , where T

0 ψ(t)dB (H) (t)is the WIS integral.

Remark 6.2.8.We show now that if (ψ t ) t∈[0,T] satisﬁes the hypotheses of Proposition 6.2.3, then

D s (H) ψ(t)|t−s| 2H − 2 ds dt (6.12) if we identifyψ t withψ t I [0,T] We need only note that

(6.13) whereM 2 [t, t+ ](s) =M s 2 (I [t,t+ ] ), and that by (4.3) and Lemma 3.1.2,

|s−x| 2H−2 dx, where k H = (H(2H −1)Γ(3/2−H))/(2Γ(H−1/2)Γ(2−2H)) By domi- nated convergence the limit of (6.13) exists almost surely as→0 and (6.12) holds.

Itˆ o formulas with respect to fBm

Various Itô formulas have been introduced in the literature based on different definitions of stochastic integrals for fractional Brownian motion (fBm) This article examines several of these formulas and explores their interconnections, starting with the Itô formula for functionals of B(H).

6.3 Itˆo formulas with respect tofBm 161

Theorem 6.3.1 Let H∈(0,1) Assume thatf(s, x) :R×R→Rbelongs to

C 1,2 (R×R), and assume that the random variables f(t, B (H) (t)), t

∂x 2 (s, B (H) (s))s 2H − 1 ds are square integrable for everyt Then f(t, B (H) (t)) =f(0,0) + t

The Itô formula, as expressed in equation (6.14), is applicable to WIS integrals for any H in the range (0,1) This formula is also valid for fWIS integrals and divergence-type integrals when H exceeds 1/2 However, to ensure the existence of divergence for H values below 1/2, more stringent conditions are required We will examine the relevant theorem presented in [54].

Theorem 6.3.2 Let H 0, and λ < 1/(4T 2H ) Then for all t ∈ [0, T], the process

F (B (H) (s))I [0,t] (s)belongs todom ∗ δ H , and we have

Proof Here we sketch a proof by following [177] For further details, we refer to Lemma 4.3 of [54] We have thatF (B (H) (s))I [0,t] (s)∈L 2 (Ω×[0, T]) and

Hence, it is suﬃcient to prove that for any G∈S H we have

This equality can be proved by choosing smooth cylindrical random variable of the formG=h n (B (H) (ψ)), whereh n denotes thenth Hermite polynomial

(see Appendix A), and applying the integration by parts formula

We now present Itô formulas for functionals of integrals for the different definitions of stochastic integral forfBmand clarify the relations among them.

We start with the one for fWIS integral In Theorem 3.7.2 we have proved the following result.

Theorem 6.3.3 (Itˆo formula for the fWIS integral) Let H >1/2 Let η t = t

0 F u dB u (H) , where(F u ,0≤u≤T) is a stochastic process in L φ (0, T).

Assume that there is anα >1−H such that

Let $ f: \mathbb{R}^+ \times \mathbb{R} \to \mathbb{R} $ be a function that possesses a continuous first derivative with respect to its first variable and a continuous second derivative with respect to its second variable, with both derivatives being bounded.

1/2 It is important to observe that the stochastic gradient defined in Definition 3.3.1 aligns with the derivative operator D(H) from equation (2.31), specifically concerning the representation H L2 φ ([0, T]).

Here we recall an Itˆo formula for the divergence integral following the approach of [8, Theorem 8].

Theorem 6.3.4 (Itˆo formula for the divergence integral) Let H >

1/2 Let ψ be a function of class C 2 (R) Assume that the process (u t ) t∈[0,T] belongs to D 2,2 loc (|H|) and that the integral X t = t

0 u(s)dB (H) (s) is almost surely continuous Assume that E

|u| 2 1/2 belongs to H Then for each t ∈

6.3 Itˆo formulas with respect tofBm 163

Remark 6.3.5.Since [(2H−1)/s 2H − 1 ](s−v) 2H − 2 I [0,s] (v) tends to the identity as H goes to 1/2, we can formally recover the Itˆo formula for the Skorohod integral with respect to the standard Brownian motion proved in [179] by taking the limit asH converges to 1/2 of equation (6.17).

Moreover, we remark that conditions under which the integral process admits a continuous modiﬁcation are proved in [6] and [8].

Finally, we provide an Itˆo formula for the WIS integral proved in [33].

Theorem 6.3.6 (Itˆo formula for the WIS integral) Suppose1/2< H 1/6 but limited to functions in C_b^5 Furthermore, a more comprehensive Itô formula for g ∈ C_b^3(R) is introduced in [73], valid for all H in the range (0,1) when dealing with cylindrical integrands.

Fractional Brownian motion in ﬁnance

fBm, or fractional Brownian motion, is not applicable in finance due to its tendency to create arbitrage opportunities, leading to its prohibition in this field However, it is often observed that individuals, particularly young men, are drawn to exploring things that are deemed forbidden.

Esko Valkeila, in a talk given at the workshop Applications of Partial Diﬀerential Equations, Institut Mittag-Leﬄer, November 2007.

The classical Black-Scholes model, based on Brownian motion, has demonstrated both success and limitations, prompting interest in extending it to fractional Brownian motion (fBm) where 0 < H < 1 When H > 1/2, fBm exhibits memory or persistence, which has proven useful in modeling weather derivatives Conversely, when H < 1/2, fBm displays turbulence or anti-persistence, characteristics observed in electricity prices within the liberated Nordic electricity market However, the idea of simply substituting fBm for classical Brownian motion in the Black-Scholes framework remains contentious due to inherent challenges.

1 If we deﬁne, as in Chapter 5, the corresponding integration with respect to fBm in the pathwise (forward) way (which is a natural form from a modeling point of view and which makes mathematical sense forH >1/2), then the corresponding ﬁnancial market hasarbitrage.

2 If we deﬁne the corresponding integration with respect tofBm in theWIS sense (see Chapter 4), then the corresponding market is free from (strong) arbitrage, but this integration is hard to justify from a modeling point of view.

We now discuss these two cases separately in more detail:

1 The pathwise (forward) integration (see Chapter 5).

Applications of stochastic calculus

Appendixes

Tiêu đề	Stochastic Calculus for Fractional Brownian Motion and Applications
Tác giả	Francesca Biagini, Bernt ỉksendal, Yaozhong Hu, Tusheng Zhang
Người hướng dẫn	J. Gani, C.C. Heyde, P. Jagers, T.G. Kurtz
Trường học	University of Oslo
Chuyên ngành	Mathematics
Thể loại	book
Năm xuất bản	2008
Thành phố	Munich

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