# On $\mathsf{Hilb}/\mathsf{Sym}$ correspondence for K3 surfaces

Deniz Genlik  Department of Mathematics
Ohio State University
100 Math Tower, 231 West 18th Ave.
Columbus, OH 43210
USA<EMAIL_ADDRESS>and  Hsian-Hua Tseng  Department of Mathematics
Ohio State University
100 Math Tower, 231 West 18th Ave.
Columbus, OH 43210
USA<EMAIL_ADDRESS>
###### Abstract.

We derive a crepant resolution correspondence for some genus zero reduced
Gromov-Witten invariants of Hilbert schemes of points on a K3 surface.

##  1\. Introduction

Let $S$ be a K3 surface. There are many references for K3 surfaces, e.g. [9].

###  1.1. The geometry

The symmetric group $S_{n}$ acts on the $n$-fold cartesian product $S^{n}$ by
permuting the factors. The quotient

$S^{n}/S_{n}$

is an algebraic variety with quotient singularities. The ($n$-fold) symmetric
product of $S$ is the stack quotient

$\mathsf{Sym}^{n}(S)=[S^{n}/S_{n}],$

which is a nonsingular projective Deligne-Mumford stack.

The Hilbert scheme of $n$ points on $S$,

$\mathsf{Hilb}^{n}(S),$

parametrizes $0$-dimensional closed subschemes of $S$ of length $n$. The
standard reference for Hilbert schemes of points on surfaces is [14].

$\mathsf{Hilb}^{n}(S)$ is smooth of dimension $2n$, see [14, Theorem 1.15].

The assignment $\mathsf{Hilb}^{n}(S)\ni[Z]\mapsto\sum_{x\in
S}\text{length}(Z_{x})[x]$ defines a morphism

(1) $\pi:\mathsf{Hilb}^{n}(S)\to S^{n}/S_{n},$

see [14, Theorem 1.5]. $\pi$ is an example of a Hilbert-Chow morphism. By [14,
Theorem 1.15], $\pi$ is a resolution of singularities.

Since $S$ is a K3 surface, $S$ has a holomorphic symplectic form $\omega$. It
follows tautologically that $\omega$ yields a holomorphic symplectic form on
$\mathsf{Sym}^{n}(S)$. By [14, Theorem 1.17], $\omega$ induces a holomorphic
symplectic form on $\mathsf{Hilb}^{n}(S)$. It follows that both
$\mathsf{Hilb}^{n}(S)$ and $S^{n}/S_{n}$ have trivial canonical bundles. Thus
$\pi$ is a crepant resolution of singularities111The Hilbert-Chow morphism is
a crepant resolution for any smooth surfaces..

###  1.2. Reduced Gromov-Witten theory

Let

$\overline{\mathcal{M}}_{g,N}(\mathsf{Hilb}^{n}(S),\beta)$

be the moduli stack of $N$-pointed genus $g$ stable maps to
$\mathsf{Hilb}^{n}(S)$ of class $\beta\in
H_{2}(\mathsf{Hilb}^{n}(S),\mathbb{Z})\simeq
H_{2}(S,\mathbb{Z})\oplus\mathbb{Z}$ (c.f. [15, Corollary 3.13] and [18,
Section 0.2]). It carries a perfect obstruction theory with virtual dimension
$(1-g)2n+3g-3+N$. Since the holomorphic symplectic form on
$\mathsf{Hilb}^{n}(S)$ is non-degenerate, the results of [10] implies that,
when $\beta\neq 0$, the virtual fundamental class vanishes,

$[\overline{\mathcal{M}}_{g,N}(\mathsf{Hilb}^{n}(S),\beta)]^{vir}=0.$

The geometric reason behind this vanishing is the presence of a cosection of
the obstruction sheaf of
$\overline{\mathcal{M}}_{g,N}(\mathsf{Hilb}^{n}(S),\beta)$, which is
constructed from the holomorphic symplectic form on $\mathsf{Hilb}^{n}(S)$,
see [12]. The cosection is surjective if $\beta\neq 0$. By [10], [11], [18],
$\overline{\mathcal{M}}_{g,N}(\mathsf{Hilb}^{n}(S),\beta)$ admits a reduced
virtual fundamental class

$[\overline{\mathcal{M}}_{g,N}(\mathsf{Hilb}^{n}(S),\beta)]^{red}\in
A_{(1-g)2n+3g-3+N+1}(\overline{\mathcal{M}}_{g,N}(\mathsf{Hilb}^{n}(S),\beta)).$

The reduced Gromov-Witten theory of $\mathsf{Hilb}^{n}(S)$ is defined by
integrating against reduced virtual fundamental classes. For
$\phi_{1},...,\phi_{N}\in H^{*}(\mathsf{Hilb}^{n}(S))$, we define reduced
Gromov-Witten invariants of $\mathsf{Hilb}^{n}(S)$ to be

$\langle\prod_{i=1}^{N}\tau_{m_{i}}(\phi_{i})\rangle_{g,\beta}^{\mathsf{Hilb}^{n}(S),red}:=\int_{[\overline{\mathcal{M}}_{g,N}(\mathsf{Hilb}^{n}(S),\beta)]^{red}}\prod_{i=1}^{N}\psi_{i}^{m_{i}}\text{ev}_{i}^{*}(\phi_{i}).$

Known calculations, e.g. [18], reveal very interesting structures present in
the reduced Gromov-Witten theory of $\mathsf{Hilb}^{n}(S)$.

Let

$\mathcal{K}_{g,N}(\mathsf{Sym}^{n}(S),\beta)$

be the moduli stack of $N$-pointed genus $g$ orbifold stable maps to
$\mathsf{Sym}^{n}(S)$ with extended degree $\beta\neq 0$ (see [1, Section
2.1]). The holomorphic symplectic form on $\mathsf{Sym}^{n}(S)$ plays a
similar role, implying that the virtual fundamental class vanishes,

$[\mathcal{K}_{g,N}(\mathsf{Sym}^{n}(S),\beta)]^{vir}=0.$

Similarly, the obstruction sheaf of
$\mathcal{K}_{g,N}(\mathsf{Sym}^{n}(S),\beta)$ admits a surjective cosection,
leading to a reduced virtual fundamental class

$[\mathcal{K}_{g,N}(\mathsf{Sym}^{n}(S),\beta)]^{red}\in
A_{*}(\mathcal{K}_{g,N}(\mathsf{Sym}^{n}(S),\beta)).$

We define reduced Gromov-Witten theory222We can also define reduced Gromov-
Witten invariants of $\mathsf{Sym}^{n}(S)$ using twistor families [2], [23].
of $\mathsf{Sym}^{n}(S)$ by integrating against reduced virtual fundamental
classes. For $\gamma_{1},...,\gamma_{N}\in H^{*}(\bar{I}\mathsf{Sym}^{n}(S))$,
we define reduced Gromov-Witten invariants of $\mathsf{Sym}^{n}(S)$ to be

$\langle\prod_{i=1}^{N}\tau_{m_{i}}(\gamma_{i})\rangle_{g,\beta}^{\mathsf{Sym}^{n}(S),red}:=\int_{[\mathcal{K}_{g,N}(\mathsf{Sym}^{n}(S),\beta)]^{red}}\prod_{i=1}^{N}\psi_{i}^{m_{i}}\text{ev}_{i}^{*}(\gamma_{i}).$

###  1.3. Crepant resolutions

The crepant resolution conjecture [1], [4], [5], concerns a correspondence
between Gromov-Witten theory of a Gorenstein orbifold and Gromov-Witten theory
of a crepant resolution of its underlying singular variety. For
$(\mathsf{Hilb}^{n}(\mathbb{C}^{2}),\mathsf{Sym}^{n}(\mathbb{C}^{2}))$,
crepant resolution conjecture is proven in full in [21], [22]. For
$(\mathsf{Hilb}^{n}(Y),\mathsf{Sym}^{n}(Y))$ with $Y$ a toric del Pezzo
surface, crepant resolution conjecture is proven for genus $0$ $3$-point
invariants in [17].

For $(\mathsf{Hilb}^{n}(S),\mathsf{Sym}^{n}(S))$, crepant resolution
conjecture is trivial because of the above vanishing. For these geometries, it
is natural to consider crepant resolution conjecture for reduced Gromov-Witten
theory.

Let $\mu=(\mu_{1},...,\mu_{k})$ be a partition of $n$ and let
$\\{\delta_{1},...,\delta_{\text{rank}H^{*}(S)}\\}$ be an additive basis of
$H^{*}(S,\mathbb{Q})$. The collection

$\vec{\mu}=((\mu_{1},\delta_{l_{1}}),...,(\mu_{k},\delta_{l_{k}}))$

is called a cohomology-weighted partition. Given $\vec{\mu}$, we associate a
class $\delta_{l_{1}}\otimes...\otimes\delta_{l_{k}}\in
H^{*}(S^{\mu},\mathbb{Q})$. Here $S^{\mu}=S^{\ell(\mu)}$. Using the natural
map $\bar{\pi}:\coprod_{\mu}S^{\mu}\to\bar{I}\mathsf{Sym}^{n}(S)$ (see [17,
Section 2.4]), we define a class

$\lambda(\vec{\mu}):=\bar{\pi}_{*}(\delta_{l_{1}}\otimes...\otimes\delta_{l_{k}})\in
H^{*}(\bar{I}\mathsf{Sym}^{n}(S),\mathbb{Q}).$

Given $\vec{\mu}$, we can also define a class in
$H^{*}(\mathsf{Hilb}^{n}(S),\mathbb{Q})$ using Nakajima operators,

$\theta(\vec{\mu}):=\frac{1}{\prod_{i=1}^{k}\mu_{i}}P_{\delta_{l_{1}}}[\mu_{1}]...P_{\delta_{l_{k}}}[\mu_{k}]\cdot
1\in H^{*}(\mathsf{Hilb}^{n}(S),\mathbb{Q}),$

see [17, Section 6], [14, Chapter 8], [18, Section 0.2] for details.

There is an additive isomorphism of graded vector spaces333Here
$\text{age}(\mu)=n-\ell(\mu)$.

(2) $L:H^{*}(\bar{I}\mathsf{Sym}^{n}(S),\mathbb{Q})\simeq
H^{*}(\mathsf{Hilb}^{n}(S),\mathbb{Q}),\quad
L(\lambda(\vec{\mu}))=(-\sqrt{-1})^{\text{age}(\mu)}\theta(\vec{\mu}),$

see [17, Proposition 6.1], [7, Proposition 3.5], [24].

We consider the generating functions of genus $0$ invariants:

(3)
$\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,\gamma}^{\mathsf{Hilb}^{n}(S),red}(y):=\sum_{\mathsf{m}}\langle
L(\gamma_{1}),L(\gamma_{2}),L(\gamma_{3})\rangle_{0,(\gamma,\mathsf{m})}^{\mathsf{Hilb}^{n}(S),red}y^{\mathsf{m}},$

(4)
$\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,\gamma}^{\mathsf{Sym}^{n}(S),red}(u):=u^{\sum_{i}\text{age}(\gamma_{i})-2n}\sum_{\mathsf{h}}\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,(\gamma,\mathsf{h})}^{\mathsf{Sym}^{n}(S),red}u^{2\mathsf{h}-2}.$

The following is the main result of this paper.

######  Theorem 1.

Suppose $\gamma\in H_{2}(S,\mathbb{Z})$ is a class of divisibility at most
$2$. Then the generating series
$\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,\gamma}^{\mathsf{Hilb}^{n}(S),red}(y)$
is the Taylor expansion at $y=0$ of a rational function in $y$, and under the
change of variables $-y=e^{\sqrt{-1}u}$, the following holds:

(5)
$\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,\gamma}^{\mathsf{Hilb}^{n}(S),red}(y)=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,\gamma}^{\mathsf{Sym}^{n}(S),red}(u).$

Theorem 1 is the crepant resolution correspondence for genus $0$ $3$-point
primary reduced Gromov-Witten theories of
$(\mathsf{Hilb}^{n}(S),\mathsf{Sym}^{n}(S))$. Following [22], we also call
this $\mathsf{Hilb}/\mathsf{Sym}$ correspondence for $S$.

###  1.4. Calculations

Calculations of reduced Gromov-Witten invariants of $\mathsf{Hilb}^{n}(S)$ in
[18] can be combined with Theorem 1 to yield formulas for reduced Gromov-
Witten invariants of $\mathsf{Sym}^{n}(S)$. We discuss one example.

Let $\pi:S\to\mathbb{P}^{1}$ be an elliptic K3 surface with a unique section
$s:\mathbb{P}^{1}\to S$ and $24$ rational nodal fibers. In [18, Section
2.2.1], $S$ is called a Bryan-Leung K3 surface.

Let $B$ be the class of a section of $\pi$ and $F$ the class of a fiber of
$\pi$. For all $h\geq 0$, the class $\beta_{h}=B+hF\in H_{2}(S,\mathbb{Z})$ is
a primitive and effective curve class. Furthermore, $\beta_{h}^{2}=2h-2$.

Consider cohomology-weighted partitions

$\vec{\nu}:=(\underbrace{(1,F),...,(1,F)}_{n})\quad\text{and}\quad\vec{\eta}:=((1,F),\underbrace{(1,[pt]^{\vee}),...,(1,[pt]^{\vee})}_{n-1}).$

Here $(-)^{\vee}$ indicates Poincaré dual.

######  Theorem 2.

We have

$\sum_{h\geq
0}q^{h-1}\langle\lambda(\vec{\nu}),\lambda(\vec{\eta}),\lambda(\vec{\nu})\rangle_{0,\beta_{h}}^{\mathsf{Sym}^{n}(S),red}(u)=\frac{(-\sqrt{-1})^{3n}}{n}\frac{F(z,\tau)^{2n-2}}{\Delta(\tau)},$

where

  1. (1)

$q=e^{2\pi\sqrt{-1}\tau}$ and $u=2\pi z$.

  2. (2)

$\Delta(\tau)=q\prod_{m\geq 1}(1-q^{m})^{24}$ is the modular discriminant.

  3. (3)

$F(z,\tau)=(y^{1/2}+y^{-1/2})\prod_{m\geq
1}\frac{(1+yq^{m})(1+y^{-1}q^{m})}{(1-q^{m})^{2}}$ is the Jacobi theta
function ($y=-e^{2\pi\sqrt{-1}z}$).

###  1.5. Outline

Our proof of Theorem 1 is based on the work of D. Nesterov [15], [16], [17].
In [15], [16], Nesterov obtains, among other things, a correspondence between
reduced Gromov-Witten theory of $\mathsf{Hilb}^{n}(S)$ and reduced stable pair
theory of a family of threefolds. In [17], Nesterov obtains a correspondence
between Gromov-Witten theory of a symmetric product and Gromov-Witten theory
of a family of threefolds. These correspondences are obtained by means of
wall-crossing formulas.

In Section 2.1 we review the results of [17]. In Section 2.2, we explain how
to adjust [17] to obtain wall-crossing results in the reduced case. Theorem 1
is proved in Section 3.1 by combining the reduced wall-crossing results in
[16] and Section 2.2 and the Gromov-Witten/Pairs correspondence proven in
[20]. Theorem 2 is proven in Section 3.2 by combining Theorem 1 and known
calculations [18].

######  Remark 3 (G. Oberdieck).

Since the Gromov-Witten/Pairs correspondence is proven for $\text{K3}\times C$
in curve classes of divisibility at most $2$ where $C$ is a smooth curve [20,
Theorem 1.2, Corollary 1.5], the strategy for proving Theorem 1 should be
applicable to derive a crepant resolutuon correspondence for
$(\mathsf{Hilb}^{n}(S),\mathsf{Sym}^{n}(S))$ for genus one Gromov-Witten
invariants with fixed $j$-invariant [19, Equation (1)]. Such a correspondence
can then be combined with the calculation in [19, Remark 1.2] to obtain
calculations of genus one Gromov-Witten invariants of $\mathsf{Sym}^{n}(S)$
with fixed $j$-invariants.

###  1.6. Acknowledgment

We thank G. Oberdieck and R. Pandharipande for helpful discussions. H.-H. T.
is supported in part by Simons Foundation Collaboration Grant.

##  2\. Gromov-Witten/Hurwitz wall-crossing

###  2.1. Review of Nesterov’s work

We summarize the main constructions of [17].

####  2.1.1. Admissible maps

Let $X$ be a smooth complex projective variety. Let
$\epsilon\in\mathbb{R}_{\leq 0}\cup\\{-\infty\\}$. Let

$Adm_{g,N}^{\epsilon}(X^{(n)},\beta)$

be the stack of $\epsilon$-admissible maps as defined in [17, Definition 2.7].
By [17, Proposition 2.10] and [17, Theorem 2.13],
$Adm_{g,N}^{\epsilon}(X^{(n)},\beta)$ is a proper Deligne-Mumford stack of
finite type. By [17, Proposition 2.14], $Adm_{g,N}^{\epsilon}(X^{(n)},\beta)$
admits a perfect obstruction theory.

By the construction of [17, Section 2.5], there are evaluation maps [17,
equation (26)]

(6)
$\text{ev}_{i}:Adm_{g,N}^{\epsilon}(X^{(n)},\beta)\to\bar{I}\mathsf{Sym}^{n}(X),$

taking values in the rigidified inertia stack $\bar{I}\mathsf{Sym}^{n}(X)$ of
$\mathsf{Sym}^{n}(X)$. Descendant $\epsilon$-admissible invariants can be
defined in the standard fashion, see [17, Definition 2.16].

####  2.1.2. $\epsilon=-\infty$

When $\epsilon=-\infty$, there is a map

(7) $\rho:\mathcal{K}_{g,N}(\mathsf{Sym}^{n}(X),\beta)\to
Adm_{g,N}^{-\infty}(X^{(n)},\beta),$

see [17, equation (15)]. By [17, Lemma 2.15], $\rho$ is virtually birational.

By [17, Lemma 2.17], $(-\infty)$-admissible invariants coincide with Gromov-
Witten invariants of $\mathsf{Sym}^{n}(X)$.

####  2.1.3. $\epsilon=0$

When $\epsilon=0$, there is an identification [17, equation (18)]

(8)
$Adm_{g,N}^{0}(X^{(n)},\beta)=\overline{\mathcal{M}}_{\mathsf{m}}^{\bullet}(X\times
C_{g,N}/\overline{\mathcal{M}}_{g,N},(\gamma,n)),$

where the right-hand side is a moduli space of relative stable maps with
disconnected domains to the relative geometry

$X\times C_{g,N}\to\overline{\mathcal{M}}_{g,N},$

where $C_{g,N}\to\overline{\mathcal{M}}_{g,N}$ is the universal family and
relative divisors are given by marked points. This identification also
identifies obstruction theories.

By [17, Lemma 2.17], $0$-admissible invariants coincide with relative Gromov-
Witten invariants of $X\times C_{g,N}\to\overline{\mathcal{M}}_{g,N}$.

####  2.1.4. Master space

The space $\mathbb{R}_{\leq 0}\cup\\{-\infty\\}$ of $\epsilon$ is divided into
chambers. When $\epsilon$ varies in a chamber,
$Adm_{g,N}^{\epsilon}(X^{(n)},\beta)$ stays the same. When $\epsilon$ crosses
a wall between two chambers, $Adm_{g,N}^{\epsilon}(X^{(n)},\beta)$ changes.
The goal of wall-crossing is to study how $\epsilon$-admissible invariants
change as $\epsilon$ crosses a wall. This is achieved in general in [17,
Section 4], based on the geometry of master spaces introduced in [17, Section
3].

Let $\epsilon_{0}$ be a wall. Let

(9) $MAdm_{g,N}^{\epsilon_{0}}(X^{(n)},\beta)$

be the moduli space of $N$-pointed genus $g$ $\epsilon_{0}$-admissible maps
with calibrated tails, as defined in [17, Definition 3.4]. By [17, Theorem
3.5] and [17, Theorem 3.9], $MAdm_{g,N}^{\epsilon_{0}}(X^{(n)},\beta)$ is a
proper Deligne-Mumford stack of finite type. By [17, Section 3.2],
$MAdm_{g,N}^{\epsilon_{0}}(X^{(n)},\beta)$ admits a perfect obstruction
theory, hence a virtual fundamental class.

The master space $MAdm_{g,N}^{\epsilon_{0}}(X^{(n)},\beta)$ admits a
$\mathbb{C}^{*}$-action obtained by scaling the $\mathbb{P}^{1}$ in the
domain. The wall-crossing formula for $\epsilon$-admissible invariants [17,
Theorem 4.3] is obtained by applying virtual localization with respect to this
$\mathbb{C}^{*}$-action, see [17, Section 4.2] for more details.

###  2.2. The reduced case

Now we consider the case $X=S$ is a K3 surface.

By the construction of [13, Section 4.3–4.4], the holomorphic symplectic form
on $S$ yields a cosection of the obstruction sheaf of
$Adm_{g,N}^{\epsilon}(S^{(n)},\beta)$. Furthermore, the cosection is
surjective when $\beta\neq 0$. By [10], [11], this yields a reduced virtual
fundamental class

$[Adm_{g,N}^{\epsilon}(S^{(n)},\beta)]^{red}.$

We can use the evaluation maps (6) to defined reduced $\epsilon$-admissible
invariants:

(10)
$\langle\prod_{i=1}^{N}\tau_{m_{i}}(\gamma_{i})\rangle_{g,\beta}^{\epsilon,red}:=\int_{[Adm_{g,N}^{\epsilon}(S^{(n)},\beta)]^{red}}\prod_{i=1}^{N}\psi_{i}^{m_{i}}\text{ev}_{i}^{*}(\gamma_{i}),$

where $\gamma_{1},...,\gamma_{N}\in H^{*}(\bar{I}\mathsf{Sym}^{n}(S))$.

When $\epsilon=-\infty$, we have the reduced version of [17, Lemma 2.15]:

(11)
$\rho_{*}[\mathcal{K}_{g,N}(\mathsf{Sym}^{n}(S),\beta)]^{red}=[Adm_{g,N}^{-\infty}(S^{(n)},\beta)]^{red}.$

Note that the reduced virtual classes are not constructed from perfect
obstruction theories. Still by [10], [11], reduced virtual classes are
constructed by embedding intrinsic normal cones into the kernel of the
cosection as cycles. Thus the proof of [6, Theorem 5.0.1] (see also [8]) may
be adapted to prove (11). Therefore,

(12)
$\langle\prod_{i=1}^{N}\tau_{m_{i}}(\gamma_{i})\rangle_{g,\beta}^{-\infty,red}=\langle\prod_{i=1}^{N}\tau_{m_{i}}(\gamma_{i})\rangle_{g,\beta}^{\mathsf{Sym}^{n}(S),red}$

Now we consider wall-crossing for reduced $\epsilon$-admissible invariants. As
in Section 2.1.4, let $\epsilon_{0}\in\mathbb{R}_{\leq 0}\cup\\{-\infty\\}$ be
a wall. Let $\epsilon_{+},\epsilon_{-}$ be close to $\epsilon_{0}$ such that
$\epsilon_{+}<\epsilon_{0}<\epsilon_{-}$. The master space
$MAdm_{g,N}^{\epsilon_{0}}(S^{(n)},\beta)$, as reviewed in Section 2.1.4,
admits a surjective cosection (assuming $\beta\neq 0$), again by the
construction of [13, Section 4.3–4.4]. Hence it has a reduced virtual
fundamental class $[MAdm_{g,N}^{\epsilon_{0}}(S^{(n)},\beta)]^{red}$. By [3],
we can apply virtual localization formula to the reduced virtual class. As in
[17, Section 4.2], the $\mathbb{C}^{*}$-fixed loci of
$MAdm_{g,N}^{\epsilon_{0}}(S^{(n)},\beta)$ is described as follows:

(13) $MAdm_{g,N}^{\epsilon_{0}}(S^{(n)},\beta)^{\mathbb{C}^{*}}=F_{+}\cup
F_{-}\cup\coprod_{\vec{\beta}}F_{\vec{\beta}}.$

As in [17, Section 4.2.1], we have
$F_{+}=Adm_{g,N}^{\epsilon_{+}}(S^{(n)},\beta)$ and
$[F_{+}]^{red}=[Adm_{g,N}^{\epsilon_{+}}(S^{(n)},\beta)]^{red}$. In the
notation of [17, Section 4.2.2], we have
$p_{*}[F_{-}]^{red}=[Adm_{g,N}^{\epsilon_{-}}(S^{(n)},\beta)]^{red}$.

The fixed loci $F_{\vec{\beta}}$ are described as products, see [17, Section
4.2.3] for details. We consider the reduced virtual class
$[F_{\vec{\beta}}]^{red}$. Since the reduced virtual class of a product splits
into a product of reduced and standard virtual classes (c.f. [13, Section
3.9]), and since standard virtual classes vanish whenever the cosection is
surjective, which happens for non-constant maps, we see that
$[F_{\vec{\beta}}]^{red}=0$ except for $\vec{\beta}=(\beta^{\prime}=0,\beta)$.
Thus we arrive at the reduced version of wall-crossing formula [17, Theorem
4.3]:

(14)
$\langle\prod_{i=1}^{N}\tau_{m_{i}}(\gamma_{i})\rangle_{g,\beta}^{\epsilon_{+},red}-\langle\prod_{i=1}^{N}\tau_{m_{i}}(\gamma_{i})\rangle_{g,\beta}^{\epsilon_{-},red}=\int_{[Adm_{g,N+1}^{\epsilon_{-}}(S^{(n)},0)]^{vir}}\prod_{i=1}^{N}\psi_{i}^{m_{i}}\text{ev}_{i}^{*}(\gamma_{i})\cdot\text{ev}_{N+1}^{*}\mu_{\beta}(z)|_{z=-\psi_{N+1}}.$

Here $\mu_{\beta}(z)$ is defined in [17, Definition 4.1] with reduced virtual
classes replacing the standard ones.

By a dimension argument similar to [17, Section 5], we see that
$\mu_{\beta}(z)$ is proportional to the class $1\in
H^{0}(\mathsf{Sym}^{n}(S))$. Since
$Adm_{g,N+1}^{\epsilon_{-}}(S^{(n)},0)=\overline{\mathcal{M}}_{g,N+1}(\mathsf{Sym}^{n}(S),0)$,
we see by string equation that

$\int_{[Adm_{0,N+1}^{\epsilon_{-}}(S^{(n)},0)]^{vir}}\prod_{i=1}^{N}\text{ev}_{i}^{*}(\gamma_{i})\cdot\text{ev}_{N+1}^{*}\mu_{\beta}(z)|_{z=-\psi_{N+1}}=0,$

whenever $2g-2+N>0$. Therefore, the reduced wall-crossing formula (14) without
descendant insertions takes a simple form:

(15)
$\langle\prod_{i=1}^{N}\tau_{0}(\gamma_{i})\rangle_{g,\beta}^{\epsilon_{+},red}=\langle\prod_{i=1}^{N}\tau_{0}(\gamma_{i})\rangle_{g,\beta}^{\epsilon_{-},red}.$

##  3\. Crepant resolution correspondence

###  3.1. Proof of Theorem 1

Fix $\gamma\in H_{2}(S,\mathbb{Z})$. Equations (15) and (12) imply that for
$\gamma_{1},\gamma_{2},\gamma_{3}\in H^{*}(\bar{I}\mathsf{Sym}^{n}(S))$,

(16)
$\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,\beta}^{0,red}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,\beta}^{-\infty,red}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,\beta}^{\mathsf{Sym}^{n}(S),red}.$

Here, by construction, the left-hand side is the reduced relative Gromov-
Witten invariants of $S\times\mathbb{P}^{1}$ relative to the divisors
$S\times\\{0,1,\infty\\}$.

For $\mathsf{Hilb}^{n}(S)$, [16, Corollary 4.2] gives that for
$\phi_{1},\phi_{2},\phi_{3}\in H^{*}(\mathsf{Hilb}^{n}(S))$,

(17)
$\langle\phi_{1},\phi_{2},\phi_{3}\rangle_{0,\beta}^{\mathsf{Hilb}^{n}(S),red}=\langle\phi_{1},\phi_{2},\phi_{3}\rangle_{n,\check{\beta}}^{S\times\mathbb{P}^{1},red}.$

Here the right-hand side is the reduced relative stable pair invariant of
$S\times\mathbb{P}^{1}$ relative to the divisors $S\times\\{0,1,\infty\\}$,
see e.g. [20] for more details.

The next step is to invoke proven Gromov-Witten/Pairs correspondence [20] for
the relative geometry $(S\times\mathbb{P}^{1},S\times\\{0,1,\infty\\})$.

Fix $\gamma_{1},...,\gamma_{N}\in
H^{*}(\bar{I}(\mathsf{Sym}^{n}(S),\mathbb{Q})$ which are homogeneous with
respect to the age grading. We consider the following generating function of
reduced $\epsilon$-quasimap invariants studied in [16]:

$\langle\gamma_{1},...,\gamma_{N}\rangle_{g,\gamma}^{\epsilon,red}(y):=\sum_{\mathsf{m}}\langle
L(\gamma_{1}),...,L(\gamma_{N})\rangle_{g,(\gamma,\mathsf{m})}^{\epsilon,red}y^{\mathsf{m}}.$

We also consider the following generating series of reduced
$\epsilon$-admissible invariants:

$\langle\gamma_{1},...,\gamma_{N}\rangle_{g,\gamma}^{\epsilon,red}(u):=u^{\sum_{i}\text{age}(\gamma_{i})-2n}\sum_{\mathsf{h}}\langle\gamma_{1},...,\gamma_{N}\rangle_{g,(\gamma,\mathsf{h})}^{\epsilon,red}u^{2\mathsf{h}-2}.$

In these notations, the reduced Gromov-Witten/Pairs correspondence for the
family

(18) $S\times C_{g,N}\to\overline{\mathcal{M}}_{g,N}$

can be formulated as follows, c.f. [17, Section 6.1]:
$\langle\gamma_{1},...,\gamma_{N}\rangle_{g,\gamma}^{0^{+},red}(y)$ is the
Taylor expansion at $y=0$ of a rational function, and under the change of
variables $-y=e^{\sqrt{-1}u}$, the following holds

(19)
$\langle\gamma_{1},...,\gamma_{N}\rangle_{g,\gamma}^{0^{+},red}(y)=\langle\gamma_{1},...,\gamma_{N}\rangle_{g,\gamma}^{0,red}(u).$

The correspondence (19) is still conjectural in general. When $g=0,N=3$, the
family (18) reduces to $S\times\mathbb{P}^{1}$. By [20, Theorem 1.2, Corollary
1.5], (19) holds when $g=0$, $N=3$, and $\gamma\in H_{2}(S,\mathbb{Z})$ is of
divisibility at most $2$. Together with (16) and (17), we obtain Theorem 1.

###  3.2. Proof of Theorem 2

The generating function

(20)
$\langle\theta(\vec{\nu}),\theta(\vec{\nu})\rangle^{\mathsf{Hilb}^{n}(S)}(y,q)=\sum_{h\geq
0}\sum_{k\in\mathbb{Z}}y^{k}q^{h-1}\langle\theta(\vec{\nu}),\theta(\vec{\nu})\rangle_{0,(\beta_{h},k)}^{\mathsf{Hilb}^{n}(S),red}$

is evaluated in [18, Theorem 10] as follows:

(21)
$\langle\theta(\vec{\nu}),\theta(\vec{\nu})\rangle^{\mathsf{Hilb}^{n}(S)}(y,q)=\frac{F(z,\tau)^{2n-2}}{\Delta(\tau)},$

where $q=e^{2\pi\sqrt{-1}\tau}$, $y=-e^{2\pi\sqrt{-1}z}$.

Since we have

$\int_{\beta_{h}+kA}\theta(\vec{\eta})=\frac{1}{n},$

divisor equation implies,

(22)
$\sum_{k\in\mathbb{Z}}y^{k}\langle\theta(\vec{\nu}),\theta(\vec{\eta}),\theta(\vec{\nu})\rangle_{0,(\beta_{h},k)}^{\mathsf{Hilb}^{n}(S),red}=\frac{1}{n}\sum_{k\in\mathbb{Z}}y^{k}\langle\theta(\vec{\nu}),\theta(\vec{\nu})\rangle_{0,(\beta_{h},k)}^{\mathsf{Hilb}^{n}(S),red}.$

By (5), we have

(23)
$\begin{split}&\sum_{k\in\mathbb{Z}}y^{k}\langle\theta(\vec{\nu}),\theta(\vec{\eta_{h}}),\theta(\vec{\nu})\rangle_{0,(\beta_{h},k)}^{\mathsf{Hilb}^{n}(S),red}\\\
=&\langle
L^{-1}\theta(\vec{\nu}),L^{-1}\theta(\vec{\eta}),L^{-1}\theta(\vec{\nu})\rangle_{0,\beta_{h}}^{\mathsf{Sym}^{n}(S),red}(u)\\\
=&\frac{1}{(-\sqrt{-1})^{3n}}\langle\lambda(\vec{\nu}),\lambda(\vec{\eta}),\lambda(\vec{\nu})\rangle_{0,\beta_{h}}^{\mathsf{Sym}^{n}(S),red}(u),\end{split}$

after $-y=e^{\sqrt{-1}u}$. Theorem 2 follows by combining this with (21).

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# On The Sub-Mixed Fractional Brownian Motion

Charles El-Nouty and Mounir Zili

###### Abstract

Let $\\{S_{t}^{H},\,t\geq 0\\}$ be a linear combination of a Brownian motion
and of an independent sub-fractional Brownian motion with Hurst index $0<H<1$.
Its main properties are studied and it is shown that $S^{H}$ can be considered
as an intermediate process between a sub-fractional Brownian motion and a
mixed fractional Brownian motion. Finally, we determine the values of $H$ for
which $\;S^{H}$ is not a semi-martingale.

##  1 Introduction

Let $\\{B_{t}^{H},t\in{\mathbb{R}}\\}$ be a fractional Brownian motion (fBm)
with Hurst index $0<~{}H<~{}1$, i.e. a centered Gaussian process with
stationary increments satisfying $B_{0}^{H}=0$, with probability 1, and
${\mathbb{E}}{(B_{t}^{H})}^{2}=\;{\mid t\mid}^{2H},t\in{\mathbb{R}}$. We
obviously have for any real numbers $t$ and $s$

(1.1) $cov\Bigl{(}B_{t}^{H},B_{s}^{H}\Bigr{)}=\frac{1}{2}\;\Bigl{(}\mid
t\mid^{2H}+\mid s\mid^{2H}-\mid t-s\mid^{2H}\Bigr{)}.$

Consider $\\{B_{t},t\in{\mathbb{R}}\\}$ an independent Brownian motion (Bm)
and $(a,b)$ two real numbers such that $(a,b)\neq(0,0)$.

The mixed-fractional Brownian motion (mfBm) is an extension of a Bm and a fBm.
It was introduced in [3] in order to solve some problems in mathematical
finance, such as modelling some arbitrage-free and complete markets. The mfBm
$\displaystyle M^{H}=\\{M_{t}^{H}(a,b);t\geq 0\\}=\\{M_{t}^{H};t\geq 0\\}$ of
parameters $a,b$ and $H$ is defined as follows:

$\forall t\in{\mathbb{R}}_{+},\hskip
14.22636ptM_{t}^{H}=M_{t}^{H}(a,b)=a\;B_{t}+b\;B_{t}^{H}.$

We refer also to [5] and [12] for further information on this process. Let us
recall some of its main properties.

###### Lemma 1

The mfBm $\;(M_{t}^{H}(a,b))_{t\in{\mathbb{R}}_{+}}$ satisfies the following
properties:

  * •

$M^{H}$ is a centered Gaussian process.

  * •

$\forall s\in{{\mathbb{R}}}_{+},\forall t\in{{\mathbb{R}}}_{+},$

$Cov\Big{(}M_{t}^{H}(a,b),M_{s}^{H}(a,b)\Big{)})=a^{2}(t\wedge
s)+\frac{b^{2}}{2}\Big{(}t^{2H}+s^{2H}-\mid t-s\mid^{2H}\Big{)},$

where $\displaystyle t\wedge s=\frac{1}{2}\Big{(}t+s-\mid t-s\mid\Big{)}.$

  * •

The increments of the mfBm are stationary.

In [2], the authors suggested a second extension of a Bm, called the sub-
fractional Brownian motion (sfBm), that preserves most of the properties of
the fBm, but not the stationarity of the increments. It is the stochastic
process $\displaystyle\xi^{H}=\\{\xi_{t}^{H};t\geq 0\\}$, defined by:

(1.2) $\forall t\in{\mathbb{R}}_{+},\hskip
14.22636pt\xi_{t}^{H}=\frac{B_{t}^{H}+B_{-t}^{H}}{\sqrt{2}},$

This process arises from occupation time fluctuations of branching particle
systems with Poisson initial condition (see [2]). Let us state some results on
the sfBm.

###### Lemma 2

The sfBm $\;(\xi_{t}^{H})_{t\in{\mathbb{R}}_{+}}$ satisfies the following
properties:

  * •

$\xi^{H}$ is a centered Gaussian process.

  * •

$\displaystyle\forall s\in{{\mathbb{R}}}_{+},\forall t\in{{\mathbb{R}}}_{+},$

$Cov\Big{(}\xi_{t}^{H},\xi_{s}^{H}\Big{)})=s^{2H}+t^{2H}-\frac{1}{2}\Big{(}(s+t)^{2H}+\mid
t-s\mid^{2H}\Big{)}.$

  * •

The increments of the smfBm are not stationary.

We can easily remark that, when $H=1/2,\;\xi^{1/2}$ is a Bm.

We refer to [2, 6, 11] for further information on this process.

In the spirit of [2] and [12], we introduce a new process, that we will call
the sub-mixed fractional Brownian motion (smfBm). More precisely, the smfBm of
parameters $a,b$ and $H$, is a process $\;S^{H}=\\{S_{t}^{H}(a,b);t\geq
0\\}=\\{S_{t}^{H};t\geq 0\\}$, defined by:

(1.3) $\forall t\in{\mathbb{R}}_{+},\hskip
14.22636ptS_{t}^{H}=S_{t}^{H}(a,b)=\frac{a\;(B_{t}+B_{-t})+b\;(B_{t}^{H}+B_{-t}^{H})}{\sqrt{2}}=a\;\xi_{t}+b\
\xi_{t}^{H},$

where $\xi$ is a Bm, obviously independent of $\xi^{H}$.

When $a=0$ and $b=1,\;S^{H}=\xi^{H}$ is a sfBm. When $a=1$ and
$b=0,\;S^{H}=\xi$ is a Bm.

So the smfBm is clearly an extension of the sfBm and the Bm. This is the
flavor of this process. We will show first that it has the same properties as
the sfBm. Then, we will prove that it has also some of the main properties of
the mfBm, but that its increments are not stationary; they are more weakly
correlated on non-overlapping intervals. Hence $S^{H}$ may be considered as
being intermediate between the sfBm and the mfBm. This is why we call it the
smfBm.

The aim of this paper is to study on one hand some key properties of the smfBm
and on the other hand its martingale properties. The motivation of the authors
is to measure the consequences of the lack of increments stationarity.

In section 2, the main properties of the smfBm are studied, namely:

  * •

the mixed-self-similarity property (see [12]),

  * •

the non Markovian property,

  * •

the increments non stationarity property,

  * •

the correlation coefficient and the influence of the parameters $a$ and $b$ on
it,

  * •

the comparison between the mfBm and the smfBm covariance properties.

Finally it is shown in section 3 that the smfBm is a semi-martingale if and
only if

$b=0\;\;\;\mbox{or}\;\;\ H\,\in\,\\{1/2\\}\,\cup\,]3/4,1[.$

##  2 Main properties

###  2.1 Basic properties

The following lemmas describe the basic properties of the smfBm.

###### Lemma 3

The smfBm $\displaystyle(S_{t}^{H}(a,b))_{t\in{\mathbb{R}}_{+}}$ satisfies the
following properties:

  * •

$S^{H}$ is a centered Gaussian process.

  * •

$\forall s\in{\mathbb{R}}_{+},\;\forall t\in{\mathbb{R}}_{+}$,

(2.1)
$\begin{array}[]{rcl}&&Cov\Big{(}S_{t}^{H}(a,b),S_{s}^{H}(a,b)\Big{)}=a^{2}\;\left(s\wedge
t\right)\\\ \vskip
8.53581pt\cr&+&b^{2}\;\left(t^{2H}+s^{2H}-\frac{1}{2}\left(\left(s+t\right)^{2H}+\mid
t-s\mid^{2H}\right)\right).\end{array}$

  * •

(2.2) $\forall t\in{\mathbb{R}}_{+},\hskip
8.53581pt{\mathbb{E}}\Big{(}\big{(}S_{t}^{H}(a,b)\big{)}^{2}\Big{)}=a^{2}t+b^{2}\;\left((2-2^{2H-1})\quad
t^{2H}\right).$

###### Proof.

It is a direct consequence of lemma 2. ∎

NOTATION. Let $(X_{t})_{t\in{\mathbb{R}}_{+}}$ and
$(Y_{t})_{t\in{\mathbb{R}}_{+}}$ be two processes defined on the same
probability space $(\Omega,F,{\mathbb{P}})$. The notation
$\\{X_{t}\\}\overset{\Delta}{=}\\{Y_{t}\\}$ will mean that
$(X_{t})_{t\in{\mathbb{R}}_{+}}$ and $(Y_{t})_{t\in{\mathbb{R}}_{+}}$ have the
same law.

Let us check the mixed-self-similarity property of the smfBm, which was
introduced in [12] in the mfBm case.

###### Lemma 4

For any $h>0$,
$\displaystyle\\{S_{ht}^{H}(a,b)\\}\stackrel{{\scriptstyle\Delta}}{{=}}\Big{\\{}S_{t}^{H}\Big{(}ah^{1/2},bh^{H}\Big{)}\Big{\\}}.$

###### Proof.

For fixed $h>0$ , the processes $\\{S_{ht}^{H}(a,b)\\}$, and
$\Big{\\{}S_{t}^{H}\Big{(}ah^{1/2},bh^{H}\Big{)}\Big{\\}}$ are centered
Gaussian. Therefore, one has only to prove that they have the same covariance
function. We have for any $s$ and $t$ in ${\mathbb{R}}_{+}$:

$\begin{array}[]{rcl}Cov\Big{(}S_{ht}^{H}(a,b),S_{hs}^{H}(a,b)\Big{)}&=&(h^{1/2}a)^{2}\;\left(s\wedge
t\right)\\\ \vskip
8.53581pt\cr&+&(h^{H}b)^{2}\;\left(t^{2H}+s^{2H}-\frac{1}{2}\left(\left(s+t\right)^{2H}+\mid
t-s\mid^{2H}\right)\right)\\\ \vskip
8.53581pt\cr&=&Cov\Bigg{(}S_{t}^{H}(ah^{1/2},bh^{H}),S_{s}^{H}(ah^{1/2},bh^{H})\Bigg{)}.\end{array}$

This ends the proof of the lemma.

∎

###### Lemma 5

For any $H\in\Big{]}0,1\Big{[}\setminus\Big{\\{}\frac{1}{2}\Big{\\}}$,
$a\in{\mathbb{R}}$ and $b\in{\mathbb{R}}^{\ast}$,
$(S_{t}^{H}(a,b))_{t\in{\mathbb{R}}_{+}}$ is not a Markovian process.

###### Proof.

By lemma 3, $S^{H}$ is a centered Gaussian process such that
$\mathbb{E}\left(S_{t}^{H}\right)^{2}>0$ for all $t>0$. Then, if $S^{H}$ were
a Markovian process, according to [9], for all $0<s<t<u$ we would have:

(2.3)
$Cov\Big{(}S_{s}^{H},S_{u}^{H}\Big{)}Cov\Big{(}S_{t}^{H},S_{t}^{H}\Big{)}=Cov\Big{(}S_{s}^{H},S_{t}^{H}\Big{)}Cov\Big{(}S_{t}^{H},S_{u}^{H}\Big{)}.$

We get by lemma 3,

$\displaystyle Cov\Big{(}S_{s}^{H},S_{t}^{H}\Big{)}$ $\displaystyle=$
$\displaystyle
a^{2}s+b^{2}s^{2H}+b^{2}\left(t^{2H}-\frac{1}{2}\left(t+s\right)^{2H}-\frac{1}{2}\left(t-s\right)^{2H}\right),$
$\displaystyle Cov\Big{(}S_{t}^{H},S_{t}^{H}\Big{)}$ $\displaystyle=$
$\displaystyle a^{2}t+b^{2}\left(2-2^{2H-1}\right)\quad t^{2H},$
$\displaystyle Cov\Big{(}S_{t}^{H},S_{u}^{H}\Big{)}$ $\displaystyle=$
$\displaystyle
a^{2}t+b^{2}t^{2H}+b^{2}\left(u^{2H}-\frac{1}{2}\left(u+t\right)^{2H}-\frac{1}{2}\left(u-t\right)^{2H}\right),$
$\displaystyle Cov\Big{(}S_{s}^{H},S_{u}^{H}\Big{)}$ $\displaystyle=$
$\displaystyle
a^{2}s+b^{2}s^{2H}+b^{2}\left(u^{2H}-\frac{1}{2}\left(u+s\right)^{2H}-\frac{1}{2}\left(u-s\right)^{2H}\right).$

Let $s$ be fixed and set $u=e^{t}$. When $t\rightarrow+\infty,$ Taylor
expansions yield

$t^{2H}-\frac{1}{2}\left(t+s\right)^{2H}-\frac{1}{2}\left(t-s\right)^{2H}=-H\left(2H-1\right)\frac{s^{2}}{t^{2-2H}}+o\left(\frac{s^{2}}{t^{2-2H}}\right),$

and

$u^{2H}-\frac{1}{2}\left(u+t\right)^{2H}-\frac{1}{2}\left(u-t\right)^{2H}=-H\left(2H-1\right)\frac{t^{2}}{e^{(2-2H)t}}+o\left(\frac{t^{2}}{e^{(2-2H)t}}\right).$

Therefore, for $(h,x)\in\\{(s,t),(t,u),(s,u)\\}$,

$\underset{x\rightarrow\infty}{\lim}\left(x^{2H}-\frac{1}{2}\left(x+h\right)^{2H}-\frac{1}{2}\left(x-h\right)^{2H}\right)=0.$

To verify (2.3), a necessary condition is that, when $b\neq 0$,

$\underset{t\rightarrow\infty}{\lim}\left(Cov\Big{(}S_{s}^{H},S_{u}^{H}\Big{)}Cov\Big{(}S_{t}^{H},S_{t}^{H}\Big{)}-Cov\Big{(}S_{s}^{H},S_{t}^{H}\Big{)}Cov\Big{(}S_{t}^{H},S_{u}^{H}\Big{)}\right)=0,$

that is

$\left(a^{2}s+b^{2}s^{2H}\right)\underset{t\rightarrow\infty}{\lim}\left(\left(a^{2}t+b^{2}\left(2-2^{2H-1}\right)t^{2H}\right)-\left(a^{2}t+b^{2}t^{2H}\right)\right)=0.$

The last equality is satisfied when

$2-2^{2H-1}=1\Leftrightarrow H=\frac{1}{2}.$

The proof of lemma 5 is complete. ∎

###### Proposition 6

Second moment of increments:
We have for all $(s,t)\in{\mathbb{R}}_{+}^{2},$ $s\leq t$,

  * •

(2.4)
$\begin{array}[]{rcl}&&E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2}=a^{2}(t-s)\\\
\vskip
5.69054pt\cr&+&b^{2}\Bigg{(}-2^{2H-1}(t^{2H}+s^{2H})+(t+s)^{2H}+(t-s)^{2H}\Bigg{)}.\end{array}$

  * •

(2.5) $a^{2}(t-s)+b^{2}\gamma(t-s)^{2H}\leq
E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2}\leq
a^{2}(t-s)+b^{2}\nu(t-s)^{2H},$

where

$\gamma=\left\\{\begin{array}[]{rcl}\displaystyle 2-2^{2H-1}&if&\displaystyle
H>\frac{1}{2},\\\ \vskip 5.69054pt\cr\displaystyle 1&if&\displaystyle
H\leq\frac{1}{2},\\\ &&\end{array}\right.$

and

$\nu=\left\\{\begin{array}[]{rcl}\displaystyle 1&if&\displaystyle
H\geq\frac{1}{2},\\\ \vskip 5.69054pt\cr\displaystyle
2-2^{2H-1}&if&\displaystyle H<\frac{1}{2}.\\\ &&\end{array}\right.$

###### Proof.

Equality (2.4) is a direct consequence of equalities (2.1) and (2.2). So let
us check the inequalities (2.5). Setting

(2.6) $A(s,t)=\Bigg{(}\frac{t+s}{2}\Bigg{)}^{2H}-\frac{t^{2H}+s^{2H}}{2},$

we can write

(2.7)
$E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2}-a^{2}(t-s)=b^{2}\Bigg{(}(t-s)^{2H}+2^{2H}A(s,t)\Bigg{)}.$

We get by convexity that, if $H\leq\frac{1}{2}$, then $A(s,t)\geq 0$ and
consequently

(2.8) $a^{2}(t-s)+b^{2}(t-s)^{2H}\leq
E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2},$

and if $\displaystyle H\geq\frac{1}{2}$, then $A(s,t)\leq 0$ and consequently

(2.9) $a^{2}(t-s)+b^{2}(t-s)^{2H}\geq
E\Big{(}S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}^{2}.$

To complete the proof of proposition 6, we need a technical lemma.

###### Lemma 7

Consider, for any $s>0$, the function $f$ defined as follows

$f(x)=-2^{2H-1}((x+s)^{2H}+s^{2H})+(x+2s)^{2H}-(1-2^{2H-1})\ x^{2H},\quad
x\geq 0.$

If $H<\frac{1}{2}$, $f$ is a negative decreasing function, whereas, if
$H>\frac{1}{2}$, $f$ is a positive increasing one.

###### Proof.

(of lemma 7 ) It is clear that $\ f(0)=0$. We get for $x>0$

$f^{\prime}\left(x\right)=H\ x^{2H-1}g(x),$

where

$g(x)=-2^{2H}\Big{(}\frac{s}{x}+1\Big{)}^{2H-1}+2\Big{(}\frac{2s}{x}+1\Big{)}^{2H-1}-(2-2^{2H}).$

We have

$g^{\prime}(x)=\frac{(2H-1)s}{x^{2}}\Bigg{(}2^{2H}\Big{(}\frac{s}{x}+1\Big{)}^{2H-2}-4\Big{(}\frac{2s}{x}+1\Big{)}^{2H-2}\Bigg{)}.$

Let us consider the two following cases:

Case $1$: $H<\frac{1}{2}$. Since $2H-1<0$, $2-2^{2H}>0$ and consequently

$\lim_{x\rightarrow
0^{+}}g(x)=-(2-2^{2H})<0\;\;\;\mbox{and}\;\;\;\lim_{x\rightarrow+\infty}g(x)=0.$

Set

$\ell(x)=\frac{s+x}{2s+x}=\frac{\frac{s}{x}+1}{\frac{2s}{x}+1}.$

Since $\ell$ increases from $\frac{1}{2}$ to $1$, $\ell^{2H-2}$ decreases from
$2^{2-2H}$ to $1$. Then $\;\ell(x)^{2H-2}\leq~{}2^{2-2H}$, which is equivalent
to

$2^{2H}\Big{(}\frac{s}{x}+1\Big{)}^{2H-2}-4\Big{(}\frac{2s}{x}+1\Big{)}^{2H-2}\leq
0,$

and consequently $g^{\prime}(x)\geq 0$. Since $g$ increases from $-(2-2^{2H})$
to $0$, $g(x)\leq 0$ and therefore $f^{\prime}(x)\leq 0$. Hence $f$ decreases
and $f(x)\leq 0$.

Case $2$: $H>\frac{1}{2}$. Following the same lines as in case $1$, we get
$g^{\prime}(x)\leq 0$. Since the function $g$ decreases from $-(2-2^{2H})$ to
$0$, $f$ increases and $f(x)\geq 0$. This completes the proof of lemma 7.

∎

Combining $(\ref{eq17})$ and $(\ref{eq18})$ with (2.7) and lemma 7, we
complete the proof of proposition 6.

∎

###### Remark 8

As a consequence of proposition 6, we insist on the fact that the smfBm does
not have stationary increments, but this property is replaced by inequalities
(2.5).

###  2.2 Study of the correlation coefficient of the smfBm increments

NOTATION. Let $\displaystyle X$ and $\displaystyle Y$ be two random variables
defined on the same probability space $(\Omega,F,{\mathbb{P}})$ such that
$V(X)\times V(Y)\neq 0$. We denote the correlation coefficient $\rho(X,Y)$ by:

$\rho(X,Y)=\frac{Cov(X,Y)}{\sqrt{V(X)}\sqrt{V(Y)}}.$

###### Lemma 9

We have for
$a\in{\mathbb{R}},b\in{\mathbb{R}}^{*},s\in{\mathbb{R}}_{+},t\in{\mathbb{R}}_{+}$
and $h\in{\mathbb{R}}_{+}$ such that $\displaystyle 0<h\leq t-s$,

(2.10)
$\rho\Big{(}S_{t+h}^{H}-S_{t}^{H},S_{s+h}^{H}-S_{s}^{H}\Big{)}=\frac{\gamma(s,t,h)}{\sqrt{\Big{(}2\frac{a^{2}}{b^{2}}h+\alpha(s,h)\Big{)}\Big{(}2\frac{a^{2}}{b^{2}}h+\alpha(t,h)\Big{)}}},$

where

$\begin{array}[]{rcl}\displaystyle\gamma(s,t,h)&=&\displaystyle\Bigg{(}(t-s+h)^{2H}-2(t-s)^{2H}+(t-s-h)^{2H}\\\
\vskip
8.53581pt\cr&-&\displaystyle(t+s)^{2H}+2(t+s+h)^{2H}-(t+s+2h)^{2H}\Bigg{)},\end{array}$

and
$\;\alpha(s,h)=-2^{2H}\Bigl{(}(s+h)^{2H}+s^{2H}\Bigr{)}+2\,(2s+h)^{2H}+2h^{2H}.$

###### Proof.

We have by equality (2.4)

(2.11)
$\begin{array}[]{rcl}{\mathbb{E}}\Bigl{(}S_{t+h}^{H}-S_{t}^{H}\Bigr{)}^{2}&=&a^{2}\,h+b^{2}\;\Biggl{(}-2^{2H-1}\;\Bigl{(}(t+h)^{2H}+t^{2H}\Bigr{)}\\\
&+&(2t+h)^{2H}+h^{2H}\Biggr{)}\\\
&=&a^{2}\,h+\frac{b^{2}}{2}\;\alpha(t,h).\end{array}$

Recall that a Bm has independent increments and that the processes $\xi^{H}$
and $\xi$ are independent. Then, we have

$Cov\Big{(}S_{t+h}^{H}-S_{t}^{H},S_{s+h}^{H}-S_{s}^{H}\Big{)}=b^{2}\;Cov\Big{(}\xi_{t+h}^{H}-\xi_{t}^{H},\xi_{s+h}^{H}-\xi_{s}^{H}\Big{)}$,

and we get by using lemma 2

(2.12)
$Cov\Big{(}S_{t+h}^{H}-S_{t}^{H},S_{s+h}^{H}-S_{s}^{H}\Big{)}=\frac{b^{2}}{2}\;\gamma(s,t,h).$

Combining (2.11) with (2.12), we complete the proof of lemma 9.

∎

###### Corollary 10

Let $a\in{\mathbb{R}}$ and $b\in{\mathbb{R}}^{*}$. Then, the increments of
$\displaystyle(S_{t}^{H}(a,b))_{t\in{\mathbb{R}}_{+}}$ are positively
correlated for $\displaystyle\frac{1}{2}<H<1$, uncorrelated for $\displaystyle
H=\frac{1}{2}$, and negatively correlated for $\displaystyle 0<H<\frac{1}{2}$.

###### Proof.

Let us write the function $\gamma$ given in (2.10) as

$\displaystyle\gamma(s,t,h)=f(t-s)-f(t+s+h),$

where $f:x\longmapsto(x+h)^{2H}-2x^{2H}+(x-h)^{2H}$. We have for every $x>0$

$f^{\prime}(x)=2H\Big{(}(x+h)^{2H-1}-2x^{2H-1}+(x-h)^{2H-1}\Big{)}.$

The study of the convexity of the function $\displaystyle x\longmapsto
x^{2H-1}$ enables us to determine the sign of $f^{\prime}$ and therefore the
monotony of $f$. This ends the proof of corollary 10.

∎

As a direct consequence of lemma 9, we get the following corollary.

###### Corollary 11

Assume that $b\neq 0$. Then,
$\mid\rho\Big{(}S_{t+h}^{H}-S_{t}^{H},S_{s+h}^{H}-S_{s}^{H}\Big{)}\mid$ is a
decreasing function of $\frac{a^{2}}{b^{2}}$.

Thus, to model some phenomena, we can choose the parameters $H,a$ and $b$ in
such a manner that $\\{S_{t}^{H}(a,b),\,t\geq 0\\}$ yields a good model,
taking the sign and the level of correlation of the phenomenon of interest
into account. For example, let us assume that the parameters $H$ and $a$ are
known with $H>1/2$, and $b\neq 0$ is not known. Combining corollary 10 with
corollary 11, we obtain that the correlation of the increments of $S_{H}$
increases with $\mid b\mid$.

###  2.3 Some comparisons between mfBm and smfBm

Set for any $\displaystyle s,t>0$

$R_{H}(s,t)=Cov\Big{(}M_{t}^{H}(a,b),M_{s}^{H}(a,b)\Big{)}\hskip
5.69054pt\mathrm{and}\hskip
5.69054ptC_{H}(s,t)=Cov\Big{(}S_{t}^{H}(a,b),S_{s}^{H}(a,b)\Big{)}.$

Let us compare $R_{H}$ and $C_{H}$.

###### Lemma 12

  * •

$\displaystyle C_{H}(s,t)\geq 0.$

  * •

If $\displaystyle H>\frac{1}{2},$ $\displaystyle C_{H}(s,t)<R_{H}(s,t)$.

  * •

If $\displaystyle H=\frac{1}{2},$ $\displaystyle C_{1/2}(s,t)=R_{1/2}(s,t)$.

  * •

If $\displaystyle H<\frac{1}{2},$ $\displaystyle C_{H}(s,t)>R_{H}(s,t)$.

###### Proof.

Let us show the first assertion. We have by equality (2.4)

$\frac{1}{2}\Bigg{(}-2^{2H}(t^{2H}+s^{2H})+2\,(t+s)^{2H}+2\mid
t-s\mid^{2H}\Bigg{)}=E\Big{(}S_{t}^{H}(0,1)-S_{s}^{H}(0,1)\Big{)}^{2}\geq 0.$

Thus, we get for every $0<s^{{}^{\prime}}<t^{{}^{\prime}}$

$2\,(t^{{}^{\prime}}+s^{{}^{\prime}})^{2H}+2\,(t^{{}^{\prime}}-s^{{}^{\prime}})^{2H}\geq
2^{2H}(t^{{}^{\prime}2H}+s^{{}^{\prime}2H}).$

By applying this inequality with $t^{{}^{\prime}}=t+s$ and
$s^{{}^{\prime}}=t-s$, we obtain

$2\,(t^{2H}+s^{2H})\geq(t+s)^{2H}+(t-s)^{2H}.$

This implies by equality (2.1) that $\displaystyle C_{H}(s,t)\geq 0.$

For the next three assertions, we observe that, by using the expressions of
$C_{H}$ and $R_{H}$,

$C_{H}(s,t)-R_{H}(s,t)=\frac{b^{2}}{2}\;\Big{(}t^{2H}+s^{2H}-(s+t)^{2H}\Big{)}.$

When $H=\frac{1}{2},C_{1/2}=R_{1/2}$. When $H\neq\frac{1}{2}$, set
$u=\frac{s}{t},\;0\leq u\leq 1$. We get

$C_{H}(s,t)-R_{H}(s,t)=\frac{b^{2}}{2}\;t^{2H}\;g(u),$

where $g(u)=1+u^{2H}-(1+u)^{2H}$.

The study of the function $g$ completes the proof of the lemma.

∎

Let us turn to the expressions of the covariances of the mfBm and the smfBm
increments on non-overlapping intervals. To this aim, denote for $0\leq
u<v\leq s<t,$

$R_{u,v,s,t}=Cov\Big{(}M_{v}^{H}(a,b)-M_{u}^{H}(a,b),M_{t}^{H}(a,b)-M_{s}^{H}(a,b)\Big{)}$

and

$C_{u,v,s,t}=Cov\Big{(}S_{v}^{H}(a,b)-S_{u}^{H}(a,b),S_{t}^{H}(a,b)-S_{s}^{H}(a,b)\Big{)}.$

We deduce easily from lemma 1 and lemma 3 the following result.

###### Lemma 13

We have

(2.13)
$R_{u,v,s,t}=\frac{b^{2}}{2}\;\Big{(}(t-u)^{2H}+(s-v)^{2H}-(t-v)^{2H}-(s-u)^{2H}\Big{)},$

(2.14) $\begin{array}[]{rcl}\displaystyle
C_{u,v,s,t}&=&\displaystyle\frac{b^{2}}{2}\;\Big{(}(t+u)^{2H}+(t-u)^{2H}+(s+v)^{2H}+(s-v)^{2H}\\\
\vskip
5.69054pt\cr&-&\displaystyle(t+v)^{2H}-(t-v)^{2H}-(s+u)^{2H}-(s-u)^{2H}\Big{)}.\end{array}$

Let us show that the covariances of the mfBm and the smfBm increments on non-
overlapping intervals have the same sign but, those of the smfBm are smaller
in absolute value than those of the mfBm.

###### Corollary 14

We have for $0\leq u<v\leq s<t,$ that $R_{u,v,s,t}$ and $C_{u,v,s,t}$ are
strictly positive (respectively strictly negative) for $H>1/2$ (respectively
$H<1/2$). Moreover, $C_{u,v,s,t}<R_{u,v,s,t}$ (respectively $>$).

###### Proof.

First, we have $0\leq u<v\leq s<t$

$R_{u,v,s,t}=\frac{b^{2}}{2}\;\Bigl{(}g_{1}(v)-g_{1}(u)\Bigr{)},$

where $g_{1}(x)=(s-x)^{2H}-(t-x)^{2H},\;u\leq x\leq v$.

We have

$g_{1}^{{}^{\prime}}(x)=2H\;\Bigl{(}-(s-x)^{2H-1}+(t-x)^{2H-1}\Bigr{)}.$

When $H<1/2,g_{1}^{{}^{\prime}}\leq 0$. Then $g_{1}$ decreases and therefore
$R_{u,v,s,t}\leq 0$. When $H>1/2,g_{1}^{{}^{\prime}}\geq 0$. Then $g_{1}$
increases and therefore $R_{u,v,s,t}\geq 0$.

Next we have for $0\leq\ u<v\leq s<t$

$C_{u,v,s,t}=\frac{b^{2}}{2}\Bigl{(}g_{2}(t)-g_{2}(s)\Bigr{)}$

where $g_{2}(x)=-(x+v)^{2H}-(x-v)^{2H}+(x+u)^{2H}+(x-u)^{2H},\;s\leq x\leq t$.

We have

$g_{2}^{{}^{\prime}}(x)=2H\;\Bigl{(}g_{3}(u)-g_{3}(v)\Bigr{)},$

where $g_{3}(y)=(x+y)^{2H-1}+(x-y)^{2H-1},\;u\leq y\leq v$.

We have

$g_{3}{{}^{\prime}}(y)=(2H-1)\;\Bigl{(}(x+y)^{2H-2}-(x-y)^{2H-2}\Bigr{)}.$

When $H<1/2,g_{3}^{{}^{\prime}}>0$. Since $g_{3}$ increases,
$g_{2}^{{}^{\prime}}<0$ and therefore $g_{2}$ decreases. Thus $C_{u,v,s,t}\leq
0$. When $H>1/2,g_{3}^{{}^{\prime}}<0$. Since $g_{3}$ decreases,
$g_{2}^{{}^{\prime}}>0$ and therefore $g_{2}$ increases. Thus $C_{u,v,s,t}\geq
0$.

Finally let us denote by $D_{(}u,v,s,t)$ the quantity defined as follows

(2.15) $\begin{array}[]{crl}D_{u,v,s,t}&=&C_{u,v,s,t}-R_{u,v,s,t}\\\
&=&\frac{b^{2}}{2}\;\Bigl{(}(t+u)^{2H}-(t+v)^{2H}+(s+v)^{2H}-(s+u)^{2H}\Bigr{)}\\\
&=&\frac{b^{2}}{2}\;\Bigl{(}g_{4}(t)-g_{4}(s)\Bigr{)},\end{array}$

where $g_{4}(x)=(x+u)^{2H}-(x+v)^{2H},\;s\leq x\leq t.$

Let us remark that, when $H>1/2,\;g_{4}$ decreases, and when $H<1/2,\;g_{4}$
increases. This ends the proof of the lemma.

∎

###### Corollary 15

We have

  * •

$\displaystyle\lim_{s,t\rightarrow+\infty}R_{u,v,s,t}=0$ if and only if
$0<H\leq\frac{1}{2}$.

  * •

For every $\displaystyle 0<H<1$,
$\displaystyle\lim_{s,t\rightarrow+\infty}C_{u,v,s,t}=0$.

###### Proof.

Combining (2.13) with Taylor expansions, we have as $s,t\rightarrow+\infty$,

$R_{u,v,s,t}=b^{2}\;H\;(v-u)\;\Bigl{(}\frac{1}{t^{1-2H}}-\frac{1}{s^{1-2H}}\Bigr{)}+o\Bigl{(}\frac{1}{t^{1-2H}}\Bigr{)}+o\Bigl{(}\frac{1}{s^{1-2H}}\Bigr{)},$

which proves the first assertion of the corollary.

Let us turn to $C_{u,v,s,t}$. Combining (2.14) with Taylor expansions, we have
as $s,t\rightarrow~{}+\infty$,

$C_{u,v,s,t}=b^{2}\;H\;(2H-1)\;(v^{2}-u^{2})\;\Bigl{(}\frac{1}{s^{2-2H}}-\frac{1}{t^{2-2H}}\Bigr{)}+o\Bigl{(}\frac{1}{s^{2-2H}}\Bigr{)}+o\Bigl{(}\frac{1}{t^{2-2H}}\Bigr{)},$

which completes the proof of corollary 15.

∎

In the next lemma, we will show that the increments of the smfBm on intervals
$[u,u+r]$ and $[u+r,u+2r]$ are more weakly correlated than those of the mfBm.

###### Lemma 16

Assume $H\neq 1/2$. We have for $u\geq 0$ and $r>0$,

(2.16)
$\Big{|}\rho\Bigl{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Bigr{)}\Big{|}\leq\Big{|}\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}\Big{|}.$

###### Proof.

Combining the definition of $R_{u,v,s,t}$ with (2.13), we get

(2.17)
$\begin{array}[]{crl}\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}&=&\frac{R_{u,u+r,u+r,u+2r}}{\sqrt{V(M_{u+r}^{H}-M_{u}^{H})\;V(M_{u+2r}^{H}-M_{u+r}^{H})}}\\\
&&\\\
&=&\frac{b^{2}(2^{2H-1}-1)r^{2H}}{\sqrt{V(M_{u+r}^{H}-M_{u}^{H})\;V(M_{u+2r}^{H}-M_{u+r}^{H}\Bigr{)}}}.\end{array}$

Moreover, we get by lemma 1

(2.18)
$V(M_{u+r}^{H}-M_{u}^{H})=V(M_{u+2r}^{H}-M_{u+r}^{H})=V(M_{r}^{H})=a^{2}\;r+b^{2}\;r^{2H}.$

Then, combining (2.17) with (2.18), we have

(2.19)
$\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}=\frac{b^{2}(2^{2H-1}-1)r^{2H}}{a^{2}\;r+b^{2}\;r^{2H}}.$

Let us turn to
$\rho\Bigl{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Bigr{)}$. We have

(2.20)
$\rho\Big{(}S_{u+2r}^{H}-S_{u+r}^{H},S_{u+r}^{H}-S_{u}^{H}\Big{)}=\frac{C_{u,u+r,u+r,u+2r}}{\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})V(S_{u+r}^{H}-S_{u}^{H})}}.$

Let us consider the two following cases.

Case 1. $H<\frac{1}{2}$

By using (2.19) and (2.20), we can rewrite inequality (2.16) as follows:

(2.21)
$\Big{|}\frac{C_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}\Big{|}\leq\frac{\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})V(S_{u+r}^{H}-S_{u}^{H})}}{a^{2}\;r+b^{2}\;r^{2H}}.$

Note that by corollary 14 and equality (2.15)

$\Big{|}\frac{C_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}\Big{|}=\frac{C_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}=1+\frac{D_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}.$

Then, (2.21) can be rewritten as follows

(2.22) $0\leq
1+\frac{D_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}\leq\frac{\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})V(S_{u+r}^{H}-S_{u}^{H})}}{a^{2}\;r+b^{2}\;r^{2H}}.$

The second part of proposition 6 implies that

$a^{2}r+b^{2}r^{2H}\leq\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})\;V(S_{u+r}^{H}-S_{u}^{H})}.$

Then, to prove (2.22), it suffices to show that

$1+\frac{D_{u,u+r,u+r,u+2r}}{R_{u,u+r,u+r,u+2r}}\leq 1.$

By corollary 14, $R_{u,u+r,u+r,u+2r}<0$ and $D_{u,u+r,u+r,u+2r}>0$. The proof
of case 1 is complete.

Case 2. $H>\frac{1}{2}$

Combining (2.19) with (2.20), we get

$\frac{\rho\Big{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Big{)}}{\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}}=\frac{C_{u,u+r,u+r,u+2r}}{\sqrt{V(S_{u+2r}^{H}-S_{u+r}^{H})V(S_{u+r}^{H}-S_{u}^{H})}}\;\;\frac{a^{2}\;r+b^{2}\;r^{2H}}{b^{2}(2^{2H-1}-1)r^{2H}}.$

Recall that we have precise expressions of $V(S_{u+r}^{H}-S_{u}^{H})$ and of
$V(S_{u+2r}^{H}-S_{u+r}^{H})$ by the first part of proposition 6 and of
$C_{u,v,s,t}$ by equality (2.14).

Set $x=\frac{2u}{r}$ and denote by $A,B$ and $C$ the functions defined as
follows :

$A(x)=2(x+2)^{2H}+(2^{2H}-2)-(x+3)^{2H}-(x+1)^{2H},$
$B(x)=2-x^{2H}-(x+2)^{2H}+2(x+1)^{2H}$

and $C(x)=2-(x+2)^{2H}-(x+4)^{2H}+2(x+3)^{2H}.$

Easy computations yield

$C_{u,u+r,u+r,u+2r}=\frac{b^{2}}{2}\;r^{2H}\;A(x)$
$V(S_{u+r}^{H}-S_{u}^{H})=\frac{1}{2}\;\Bigl{(}2a^{2}r+b^{2}r^{2H}B(x)\Bigr{)}$
$V(S_{u+2r}^{H}-S_{u+r}^{H})=\frac{1}{2}\;\Bigl{(}2a^{2}r+b^{2}r^{2H}C(x)\Bigr{)}$

Then, we have

$\frac{\rho\Big{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Big{)}}{\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}}=\frac{A(x)\;(a^{2}\;r+b^{2}\;r^{2H})}{(2^{2H-1}-1)\;\sqrt{(2a^{2}r+b^{2}r^{2H}B(x))(2a^{2}r+b^{2}r^{2H}C(x))}}$
$=\frac{A(x)}{(2^{2H-1}-1)\;\sqrt{B(x)\,C(x)}}\;\frac{a^{2}\;r+b^{2}\;r^{2H}}{\sqrt{(2a^{2}r/B(x)+b^{2}r^{2H})(2a^{2}r/C(x)+b^{2}r^{2H})}}.$

Since it has been proved in [[, see]p. 412]TB, that

$\frac{A(x)}{(2^{2H-1}-1)\;\sqrt{B(x)\,C(x)}}\leq 1,$

we get

$\frac{\rho\Big{(}S_{u+r}^{H}-S_{u}^{H},S_{u+2r}^{H}-S_{u+r}^{H}\Big{)}}{\rho\Big{(}M_{u+r}^{H}-M_{u}^{H},M_{u+2r}^{H}-M_{u+r}^{H}\Big{)}}\leq\frac{a^{2}\;r+b^{2}\;r^{2H}}{\sqrt{(2a^{2}r/B(x)+b^{2}r^{2H})(2a^{2}r/C(x)+b^{2}r^{2H})}}.$

Therefore it suffices to show

$a^{2}\;r+b^{2}\;r^{2H}\leq
2a^{2}r/B(x)+b^{2}r^{2H}\;\;\;\mbox{and}\;\;\;a^{2}\;r+b^{2}\;r^{2H}\leq
2a^{2}r/C(x)+b^{2}r^{2H},$

that is

$0<B(x)\leq 2\;\;\;\mbox{and}\;\;\;0<C(x)\leq 2.$

Let us show the first double inequality. Since by lemma 2

$b^{2}\;r^{2H}\;B(x)=2\;V\Bigl{(}b(\xi^{H}(u+r)-\xi^{H}(u))\Bigr{)},$

$B(x)>0$. Moreover, since the function $x\rightarrow x^{2H}$ is convex for
$H>1/2,\;B(x)\leq 2$. Similarly, we can establish $0<C(x)\leq 2$.

The proof of the lemma is complete.

∎

In [12], it was proved that the increments of the mfBm $(M_{t}^{H}(a,b))$ are
short-range dependent if, and only if $H<\frac{1}{2}$. To end this subsection,
let us show that for every $\displaystyle H\in]0,1[$, the increments of
$\displaystyle(S^{H}_{t}(a,b))_{t\in{{\mathbb{R}}_{+}}}$ are short-range
dependent. For convenience, let us introduce the following notation

$C(p,n)=C_{p,p+1,p+n,p+n+1},$

where $p$ and $n$ are integers with $n\geq 1$.

We get by (2.14)

$C(p,n)=\frac{b^{2}}{2}\Bigg{(}(n+1)^{2H}-2n^{2H}+(n-1)^{2H}-(2p+n+2)^{2H}+2(2p+n+1)^{2H}-(2p+n)^{2H}\Bigg{)}.$

A third-order Taylor expansion enables us to state the following lemma.

###### Lemma 17

For any $0<H<1$ and $p\in{\mathbb{N}}$, we have when $n\rightarrow+\infty$

$C(p,n)=\Big{(}2(1-H)H(2H-1)(2p+1)b^{2}\Big{)}\;n^{2H-3}+o(n^{2H-3}),$

and consequently

$\sum_{n\geq 1}\mid C(p,n)\mid<+\infty.$

##  3 Semi-martingale properties

In the sequel, we assume $b\neq 0$. For any process $X$, set

$\Delta_{j}^{n}X(t)=X(jt/n)-X((j-1)t/n),\;j\in\\{1,..n\\}.$

Denote by $A_{n}$ the quantity defined as follows :

$A_{n}={\mathbb{E}}\Biggl{(}\sum_{j=1}^{n}\;\Biggl{(}\Delta_{j}^{n}S(t)\Biggr{)}^{2}\Biggr{)}=\sum_{j=1}^{n}\;{\mathbb{E}}\Biggl{(}\Delta_{j}^{n}S(t)\Biggr{)}^{2}.$

###### Lemma 18

  * •

If $H<\frac{1}{2}$, then
$\displaystyle\lim_{n\rightarrow+\infty}\;A_{n}=+\infty$.

  * •

If $H=\frac{1}{2}$, then $\;A_{n}=(a^{2}+b^{2})\;t$.

  * •

If $H>\frac{1}{2}$, then
$\displaystyle\lim_{n\rightarrow+\infty}\;A_{n}=a^{2}\;t$.

###### Proof.

Since the processes $B$ and $B_{H}$ are independent, we have

${\mathbb{E}}\Biggl{(}\Delta_{j}^{n}S(t)\Biggr{)}^{2}=\frac{a^{2}}{2}\;{\mathbb{E}}\Biggl{(}\Delta_{j}^{n}B(t)+\Delta_{j}^{n}B(-t)\Biggr{)}^{2}+\frac{b^{2}}{2}\;{\mathbb{E}}\Biggl{(}\Delta_{j}^{n}B_{H}(t)+\Delta_{j}^{n}B_{H}(-t)\Biggr{)}^{2}.$

Using equality (1.1), direct computations imply

${\mathbb{E}}\Biggl{(}\Delta_{j}^{n}S(t)\Biggr{)}^{2}=a^{2}\;\frac{t}{n}+\;b^{2}\;\frac{t^{2H}}{n^{2H}}+2^{2H}\;b^{2}\;\frac{t^{2H}}{n^{2H}}\;\Biggl{(}\Biggl{(}\frac{2j-1}{2}\Biggr{)}^{2H}-\frac{j^{2H}+(j-1)^{2H}}{2}\Biggr{)},$

and hence

$A_{n}=a^{2}\;t+\;b^{2}\;t^{2H}\;n^{1-2H}+2^{2H}\;b^{2}\;\frac{t^{2H}}{n^{2H}}\;\sum_{j=1}^{n}\;\Biggl{(}\Biggl{(}\frac{2j-1}{2}\Biggr{)}^{2H}-\frac{j^{2H}+(j-1)^{2H}}{2}\Biggr{)}.$

Let us consider the function $f$ defined as follows :

$f(x)=\Biggl{(}\frac{2x-1}{2}\Biggr{)}^{2H}-\frac{x^{2H}+(x-1)^{2H}}{2},\;\;\;x\geq
0.$

We deduce from convexity properties that, when $H<1/2,\;f(x)>0$, when
$H=1/2,\;f(x)=0$ and when $H>1/2,\;f(x)<0$. We have also

$f^{{}^{\prime}}(x)=2\;H\;\Biggl{(}\Biggl{(}\frac{2x-1}{2}\Biggr{)}^{2H-1}-\frac{x^{2H-1}+(x-1)^{2H-1}}{2}\Biggr{)},\;\;\;x\geq
0.$

To determine $\displaystyle\lim_{n\rightarrow+\infty}\;A_{n}$, we have to
consider the following three cases.

Case 1. $H<1/2$

Since $f^{{}^{\prime}}\leq 0$, for every $j\in\\{1,..,n\\}$,

$f(j)\geq
f(n)=\Biggl{(}\frac{2n-1}{2}\Biggr{)}^{2H}-\frac{n^{2H}+(n-1)^{2H}}{2}>0.$

When $n$ is large enough, we get

$a^{2}\;t+\;b^{2}\;t^{2H}\;n^{1-2H}+2^{2H}\;b^{2}\;t^{2H}\;\Biggl{(}-\frac{H(2H-1)}{4n}+o\Bigl{(}\frac{1}{n}\Bigr{)}\Biggr{)}\leq
A_{n},$

and therefore, since $b\neq 0$,

$\lim_{n\rightarrow+\infty}\;A_{n}=+\infty.$

Case 2. $H=1/2$

We obviously have

$A_{n}=(a^{2}\;+\;b^{2})\;t.$

Case 3. $H>1/2$

Since $f^{{}^{\prime}}\geq 0,\;f$ increases from
$f(1)=\frac{1}{2^{2H}}-\frac{1}{2}$ to

$f(n)=\Biggl{(}\frac{2n-1}{2}\Biggr{)}^{2H}-\frac{n^{2H}+(n-1)^{2H}}{2}<0.$

When $n$ is large enough, we get

$\displaystyle
a^{2}\;t+\;b^{2}\;t^{2H}\;n^{1-2H}+2^{2H}\;b^{2}\;t^{2H}\;n^{1-2H}\;\Biggl{(}\frac{1}{2^{2H}}-\frac{1}{2}\Biggr{)}$

$\leq A_{n}\leq
a^{2}\;t+\;b^{2}\;t^{2H}\;n^{1-2H}+2^{2H+}\;b^{2}\;t^{2H}\;\Biggl{(}-\frac{H(2H-1)}{4n}+o\Bigl{(}\frac{1}{n}\Bigr{)}\Biggr{)}$

and therefore

$\lim_{n\rightarrow+\infty}\;A_{n}=a^{2}\;t.$

This completes the proof of the lemma.

∎

Let us now recall the Bichteler-Dellacherie theorem [[, see]section
VIII.4]DELL.

###### Theorem 19

Assume that a filtration $\displaystyle\mathcal{F}=({\cal F}_{t})_{0\leq t\leq
T},0<T<+\infty,$ satisfies the usual assumptions, i.e. it is right-continuous,
${\cal F}_{T}$ is complete and ${\cal F}_{0}$ contains all null sets of ${\cal
F}_{T}$. An a.s. right-continuous, $\displaystyle{\cal F}$-adapted stochastic
process $\\{X_{t},0\leq t\leq T\\}$ is a $\displaystyle{\cal F}$-semi-
martingale if and only if

$I_{X}\Bigl{(}\beta(\mathcal{F})\Bigr{)}$

is bounded in $L^{0}$, where

$\beta(\mathcal{F})=\Biggl{\\{}\sum_{j=0}^{n-1}\;f_{j}\;\mathbf{1}_{]t_{j},t_{j+1}]},\;\;n\in\mathbb{N},\;0\leq
t_{0}\leq..\leq t_{n}\leq T,\;$ $\forall\,j,\,f_{j}\hskip
2.84526pt\mathrm{is}\hskip 2.84526pt\mathcal{F}_{t_{j}}\hskip
2.84526pt\mathrm{measurable}\hskip 2.84526pt\mathrm{and}\hskip 2.84526pt\mid
f_{j}\mid\leq 1,\mathrm{with}\hskip 2.84526pt\mathrm{probability}\hskip
2.84526pt1\Biggr{\\}},$

and

$I_{X}(\theta)=\sum_{j=0}^{n-1}\;f_{j}\;\Bigl{(}X_{t_{j+1}}-X_{t_{j}}\Bigr{)},\;\;\theta\in\beta(\mathcal{F}).$

Following the same lines as those of [3], we introduce two definitions.

###### Definition 20

A stochastic process $\\{X_{t},0\leq t\leq T\\}$ is a weak semi-martingale
with respect to a filtration $\displaystyle\mathcal{F}=({\cal F}_{t})_{0\leq
t\leq T}$ if $X$ is $\displaystyle{\cal F}$-adapted and $\displaystyle
I_{X}\Bigl{(}\beta(\mathcal{F})\Bigr{)}$ is bounded in $L^{0}$.

We insist on the fact that if a process $X$ is not a weak semi-martingale with
respect to its own filtration, then it is not a weak semi-martingale with
respect to any other filtration.

###### Definition 21

Let $\\{X_{t},0\leq t\leq T\\}$ be a stochastic process. We call $X$ a weak
semi-martingale if it is a weak semi-martingale with respect to its own
filtration $\displaystyle\mathcal{F}^{X}=({\cal F}_{t}^{X})_{0\leq t\leq T}$.
We call $X$ a semi-martingale if it is a semi-martingale with respect to the
smallest filtration that contains $\displaystyle\mathcal{F}^{X}$ and satisfies
the usual assumptions.

Let us determine now the values of $H$ for which the smfBm is not a semi-
martingale.

###### Corollary 22

If $\;0<H<\frac{1}{2}$, then the smfBm $S^{H}(a,b)$ is not a weak semi-
martingale.

###### Proof.

A direct consequence of lemma 18 is that since $0<H<\frac{1}{2}$ and $b\neq
0$, the quadratic variation of the smfBm is infinity. To complete the proof of
the corollary, it suffices to apply proposition 2.2 of [3, pp. 918-919].

∎

The study of the case $H>3/4$ is based on a result of [1, p. 348]. We insist
on the fact that this method is different from the one which was used in [3].

###### Proposition 23

For every $T>0,\;H\in]\frac{3}{4},1[$, and $\;a\neq 0$, the smfBm

$S^{H}(a,b)=\\{S_{t}^{H}(a,b),t\in[0,T]\\}$

is a semi-martingale equivalent in law to $a\times B_{t}$, where
$\\{B_{t},t\in[0,T]\\}$ is a Bm.

###### Proof.

The smfBm $S^{H}$ can be rewritten as follows

$\forall\;t\in{\mathbb{R}}^{+}\;\;\;S_{t}^{H}(a,b)=a\Big{(}\xi_{t}+\frac{b}{a}\xi_{t}^{H}\Big{)}$

where $\xi$ and $\xi^{H}$ have been introduced by equation (1.3). Recall that
the processes $\xi$ and $\xi^{H}$ are independent.

The covariance function of the Gaussian process
$\displaystyle\frac{b}{a}\;\xi^{H}$

$R(s,t)=\frac{b^{2}}{a^{2}}\Bigg{(}t^{2H}+s^{2H}-\frac{1}{2}\Big{(}(s+t)^{2H}+\mid
t-s\mid^{2H}\Big{)}\Bigg{)},$

is twice continuously differentiable on
$\displaystyle[0,T]^{2}\setminus\\{(s,t);t=s\\}$.

According to [1, p. 348], it suffices to verify
$\displaystyle\frac{\partial^{2}R}{\partial s\partial t}\in L^{2}([0,T]^{2})$,
in order to show that the process

$\\{\xi_{t}+\frac{b}{a}\xi_{t}^{H},t\in[0,T]\\}$

is a semi-martingale equivalent in law to a Bm.

We have for any $\displaystyle(s,t)\in[0,T]^{2}\setminus\\{(s,t);t=s\\}$

$\frac{\partial^{2}R(s,t)}{\partial s\partial
t}=\frac{b^{2}}{a^{2}}\;H(2H-1)\;\Big{(}\mid
t-s\mid^{2H-2}-(s+t)^{2H-2}\Big{)}.$

It is easy to check that if $\displaystyle H>\frac{3}{4},$ then
$\displaystyle\frac{\partial^{2}R}{\partial s\partial t}\in L^{2}([0,T]^{2})$.
This completes the proof of the proposition.

∎

To study the case $H\in]1/2,3/4]$, we follow the same lines as those of [3].
But many technical results have to be proved. Let us first recall the
definition of a quasi-martingale.

###### Definition 24

A stochastic process $\\{X_{t},0\leq t\leq T\\}$ is a quasi-martingale if
$\displaystyle X_{t}\in L^{1}$ for all $t\in[0,T],$ and

$\sup_{\tau}\sum_{j=0}^{n-1}\left\|{\mathbb{E}}\Big{(}X_{t_{j+1}}-X_{t_{j}}|{\cal
F}_{t_{j}}^{X}\Big{)}\right\|_{1}<+\;\infty,$

where $\tau$ is the set of all finite partitions

$0=t_{0}<t_{1}<...<t_{n}=T\hskip 2.84526ptof\hskip 2.84526pt[0,T].$

In the following key lemma, we will specify the relation between quasi-
martingale and weak semi-martingale in the case of our process $S^{H}$.

###### Lemma 25

If $S^{H}$ is not a quasi-martingale, then it is not a weak semi-martingale.

###### Proof.

Let us assume that $S^{H}$ is a weak semi-martingale. Then, by theorem 1 of
[10], we have

$I_{S^{H}}\Bigl{(}\beta(\mathcal{F}^{S^{H}})\Bigr{)},$

which was defined in theorem 19, is bounded in $L^{2}$, and therefore in
$L^{1}$.

But, for any partition $0=t_{0}<t_{1}<..<t_{n}=T$,

$\sum_{j=0}^{n-1}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\mathbf{1}_{]t_{j},t_{j+1}]}\in\beta(\mathcal{F}^{S^{H}}),$

and

$\begin{array}[]{rcl}&&\displaystyle\Bigg{|}\Bigg{|}I_{S^{H}}\Biggl{(}\sum_{j=0}^{n-1}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\mathbf{1}_{]t_{j},t_{j+1}]}\Biggr{)}\Bigg{|}\Bigg{|}_{1}\\\
\vskip
8.53581pt\cr&=&\displaystyle\Bigg{|}\Bigg{|}\sum_{j=0}^{n-1}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\Bigl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\Bigr{)}\Bigg{|}\Bigg{|}_{1}\\\
\vskip
8.53581pt\cr&\geq&\displaystyle\sum_{j=0}^{n-1}\;{\mathbb{E}}\Biggl{(}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\Bigl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\Bigr{)}\Biggr{)}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sum_{j=0}^{n-1}\;{\mathbb{E}}\Biggl{(}{\mathbb{E}}\Biggl{(}\;sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\;\Bigl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\Bigr{)}\Bigg{|}\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sum_{j=0}^{n-1}\;{\mathbb{E}}\Biggl{(}sgn\Biggl{(}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}{\mathbb{E}}\Biggl{(}\;\;\Bigl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\Bigr{)}\Bigg{|}\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Biggr{)}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sum_{j=0}^{n-1}\;\Bigg{|}\Bigg{|}{\mathbb{E}}\Biggl{(}S_{t_{j+1}}^{H}-S_{t_{j}}^{H}\mid\mathcal{F}_{t_{j}}^{S^{H}}\Biggr{)}\Bigg{|}\Bigg{|}_{1}.\end{array}$

Then, $S^{H}$ is a quasi-martingale. The proof of the lemma is complete.

∎

The following lemmas deal with the two last cases $1/2<H<3/4$ and $H=3/4$.

###### Proposition 26

If $\displaystyle H\in\Big{]}\frac{1}{2},\frac{3}{4}\Big{[}$, then the smfBm
$S^{H}(a,b)$ is not a quasi-martingale.

###### Proof.

For $n\in{\mathbb{N}}$ and $j\in\\{1,2,...,n\\}$, let us denote

$\Delta_{j}^{n}S^{H}=S^{H}_{\frac{Tj}{n}}-S^{H}_{\frac{T(j-1)}{n}}.$

Since conditional expectation is a contraction with respect to the $L^{1}-$
norm, we have for all $n\in{\mathbb{N}}$ and all $j=1,...,n-1$,

$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}\right\|_{1}\leq\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\mathcal{F}_{\frac{Tj}{n}}^{S^{H}}\Big{)}\right\|_{1}.$

Moreover, since
$\displaystyle{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}$
is a centered Gaussian random variable,

$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}\right\|_{1}=\sqrt{\frac{2}{\pi}}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}\right\|_{2}.$

Consequently,

(3.1)
$\begin{array}[]{rcl}\displaystyle\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\mathcal{F}_{\frac{Tj}{n}}^{S^{H}}\Big{)}\right\|_{1}&\geq&\displaystyle\sqrt{\frac{2}{\pi}}\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{H}|\Delta_{j}^{n}S^{H}\Big{)}\right\|_{2}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sqrt{\frac{2}{\pi}}\sum_{j=1}^{n-1}\left\|\frac{Cov\Big{(}\Delta_{j+1}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}}{Cov\Big{(}\Delta_{j}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}}\Delta_{j}^{n}S^{H}\right\|_{2}\\\
\vskip
8.53581pt\cr&=&\displaystyle\sqrt{\frac{2}{\pi}}\sum_{j=1}^{n-1}\frac{Cov\Big{(}\Delta_{j+1}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}}{\sqrt{Cov\Big{(}\Delta_{j}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}}}:=\displaystyle\sqrt{\frac{2}{\pi}}\;I_{n}.\end{array}$

We have by lemma 13,

$\begin{array}[]{rcl}\displaystyle
Cov\Big{(}\Delta_{j+1}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}&=&\displaystyle
C_{\frac{T(j-1)}{n},\frac{Tj}{n},\frac{Tj}{n},\frac{T(j+1)}{n}}\\\ \vskip
8.53581pt\cr&=&\displaystyle\frac{b^{2}\;T^{2H}}{2\;n^{2H}}\;\Big{(}2^{2H}(2j^{2H}+1)-2-(2j+1)^{2H}-(2j-1)^{2H}\Big{)}.\end{array}$

Combining proposition 6 with the fact that $2H>1$, we get

$\begin{array}[]{rcl}&&\displaystyle
Cov\Big{(}\Delta_{j}^{n}S^{H},\Delta_{j}^{n}S^{H}\Big{)}\\\ \vskip
5.69054pt\cr&=&\displaystyle
a^{2}\frac{T}{n}+\frac{b^{2}\;T^{2H}}{n^{2H}}\Bigg{(}-2^{2H-1}(j^{2H}+(j-1)^{2H})+(2j-1)^{2H}+1\Bigg{)}\\\
\vskip
5.69054pt\cr&\leq&\displaystyle\displaystyle\frac{1}{n}\Bigg{(}a^{2}\;T+b^{2}\;T^{2H}\Big{(}-2^{2H-1}(j^{2H}+(j-1)^{2H})+(2j-1)^{2H}+1\Big{)}\Bigg{)}.\\\
&&\end{array}$

Then,

$I_{n}\geq\frac{b^{2}\;T^{2H}}{2\;n^{2H-\frac{1}{2}}}\sum_{j=1}^{n-1}\frac{u_{j}}{v_{j}}=\frac{b^{2}\;T^{2H}}{2\;n^{2H-\frac{3}{2}}}\times\Biggl{(}\frac{1}{n}\sum_{j=1}^{n-1}\frac{u_{j}}{v_{j}}\Biggr{)},$

where we have for any $n\in{\mathbb{N}}^{*}$,

$u_{n}=2^{2H}(2n^{2H}+1)-2-(2n+1)^{2H}-(2n-1)^{2H}$

and

$v_{n}=\sqrt{a^{2}\;T+b^{2}\;T^{2H}\Big{(}-2^{2H-1}(n^{2H}+(n-1)^{2H})+(2n-1)^{2H}+1\Big{)}}.$

Since

$\lim_{n\rightarrow+\infty}\frac{u_{n}}{v_{n}}=\frac{2^{2H}-2}{\sqrt{a^{2}\;T+b^{2}\;T^{2H}}},$

we have by Césaro theorem that

$\lim_{n\rightarrow+\infty}\frac{1}{n}\sum_{j=1}^{n-1}\frac{u_{j}}{v_{j}}=\frac{2^{2H}-2}{\sqrt{a^{2}\;T+b^{2}\;T^{2H}}}.$

Hence, since $\displaystyle\frac{1}{2}<H<\frac{3}{4}$ and
$\frac{2^{2H}-2}{\sqrt{a^{2}\;T+b^{2}\;T^{2H}}}>0$, we have
$\displaystyle\lim_{n\rightarrow\infty}I_{n}=+\infty$. Then, we get by using
(3.1) that

$\sup_{\tau}\sum_{j=0}^{n-1}\left\|{\mathbb{E}}\Big{(}S^{H}_{t_{j+1}}-S^{H}_{t_{j}}|\mathcal{F}_{t_{j}}^{S^{H}}\Big{)}\right\|_{1}=+\;\infty.$

This completes the proof of the lemma.

∎

###### Proposition 27

The smfBm $\displaystyle S^{\frac{3}{4}}(a,b)$ is not a quasi-martingale.

###### Proof.

Since conditional expectation is a contraction with respect to the $L^{1}-$
norm, we have for all $n\in{\mathbb{N}}$ and all $j=1,...,n-1$,

$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{1}\leq\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\mathcal{F}_{\frac{Tj}{n}}^{S^{\frac{3}{4}}}\Big{)}\right\|_{1}.$

Moreover, since
$\displaystyle{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}$
is a centered Gaussian random variable,

$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{1}=\sqrt{\frac{2}{\pi}}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{2}.$

Consequently,

$\begin{array}[]{rcl}\displaystyle\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\mathcal{F}_{\frac{j}{n}}^{S^{\frac{3}{4}}}\Big{)}\right\|_{1}&\geq&\displaystyle\sqrt{\frac{2}{\pi}}\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{2},\end{array}$

and the lemma is proved if we show that

(3.2)
$\lim_{n\rightarrow\infty}\sum_{j=1}^{n-1}\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{2}=+\;\infty.$

For $n\in{\mathbb{N}}$ and $j=1,...,n-1$,

$\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}},\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}$

is a Gaussian vector. Therefore,

(3.3)
${\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}=\sum_{k=1}^{j}\;b_{k}\;\Delta_{k}^{n}\;S^{\frac{3}{4}},$

where the vector $\displaystyle b=\left(\begin{array}[]{c}b_{1}\\\ \vdots\\\
b_{j}\end{array}\right)$ solves the system of linear equations

(3.4) $m=Ab,$

in which $m$ is a $j-$vector whose $k-th$ component $m_{k}$ is

$Cov\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}},\Delta_{k}^{n}S^{\frac{3}{4}}\Big{)}$

and $A$ is the covariance matrix of the Gaussian vector

$\Big{(}\Delta_{1}^{n}S^{\frac{3}{4}},...,\Delta_{j}^{n}S^{\frac{3}{4}}\Big{)}.$

Note that $A$ is symmetric and, since the random variables

$\Delta_{1}^{n}S^{\frac{3}{4}},...,\Delta_{j}^{n}S^{\frac{3}{4}}$

are lineary independent, $A$ is also positive definite. It follows from (3.3)
and (3.4) that

(3.5)
$\left\|{\mathbb{E}}\Big{(}\Delta_{j+1}^{n}S^{\frac{3}{4}}|\Delta_{j}^{n}S^{\frac{3}{4}},...,\Delta_{1}^{n}S^{\frac{3}{4}}\Big{)}\right\|_{2}^{2}=b^{T}Ab=m^{T}A^{-1}m\geq\left\|m\right\|_{2}^{2}\lambda^{-1},$

where $\lambda$ is the largest eigenvalue of the matrix $A$. Set $I$ the
identity matrix and $C=(C_{i,k})_{1\leq i,k\leq j}$ the covariance matrix of
the increments of the sfBm with index $3/4$. We have

$A=\frac{a^{2}T}{n}\;I+b^{2}\;C,$

and consequently

(3.6) $\lambda=\frac{a^{2}T}{n}+b^{2}\;\mu,$

where $\mu$ is the largest eigenvalue of the matrix $C$. We deduce also from
lemma 13

$\begin{array}[]{rcl}\displaystyle
C_{ik}&=&\displaystyle\frac{T^{3/2}}{2\;n^{3/2}}\;\Biggl{(}\Bigl{(}\mid
k-i\mid+1\Bigr{)}^{3/2}-2\mid k-i\mid^{3/2}+\mid\mid k-i\mid-1\mid^{3/2}\\\
\vskip
8.53581pt\cr&&\displaystyle+2\;(k+i-1)^{3/2}-(k+i)^{3/2}-(k+i-2)^{3/2}\Biggr{)}\\\
\vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3/2}}{2\;n^{3/2}}\;(E_{ik}+F_{ik}),\end{array}$

where

$E_{ik}=2\Biggl{(}\frac{\Bigl{(}\mid k-i\mid+1\Bigr{)}^{3/2}+\mid\mid
k-i\mid-1\mid^{3/2}}{2}-\mid k-i\mid^{3/2}\Biggr{)}$

and

$F_{ik}=2\Biggl{(}(k+i-1)^{3/2}-\frac{(k+i)^{3/2}+(k+i-2)^{3/2}}{2}\Biggr{)}.$

Note that the convexity of the function $\displaystyle x\rightarrow
x^{3/2},\;x\geq 0$, implies that $\displaystyle E_{ik}\geq 0$ and
$\displaystyle F_{ik}\leq 0$. Moreover, since $H=3/4>1/2$, corollary 14 yields
$\;C_{ik}\geq 0$.

So, using the Gershgorin circle theorem [7] we obtain

$\mu\leq\max_{k=1,..,j}\;\sum_{k=1}^{j}\;\mid
C_{ik}\mid\leq\frac{T^{3/2}}{2\;n^{3/2}}\max_{k=1,..,j}\;\sum_{k=1}^{j}E_{ik},$

and consequently

$\begin{array}[]{rcl}\displaystyle\mu&\leq&\displaystyle\frac{T^{3/2}}{n^{3/2}}\;\sum_{k=1}^{j}\;\Biggl{(}2\Biggl{(}\frac{\Bigl{(}\mid
k-1\mid+1\Bigr{)}^{3/2}+\mid\mid k-1\mid-1\mid^{3/2}}{2}-\mid
k-1\mid^{3/2}\Biggr{)}\\\ \vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3/2}}{n^{3/2}}\;\sum_{k=1}^{j}\;\Biggl{(}\Bigl{(}\mid
k-1\mid+1\Bigr{)}^{3/2}+\mid\mid k-1\mid-1\mid^{3/2}-2\mid
k-1\mid^{3/2}\Biggr{)}\\\ \vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3/2}}{n^{3/2}}\;\sum_{k^{{}^{\prime}}=0}^{j-1}\;\Biggl{(}(k^{{}^{\prime}}+1)^{3/2}-2\;k^{{}^{\prime}3/2}+\mid
k^{{}^{\prime}}-1\mid^{3/2}\Biggr{)}\\\ \vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3/2}}{n^{3/2}}\;\Bigl{(}1+j^{3/2}-(j-1)^{3/2}\Bigr{)}\\\
\vskip 8.53581pt\cr&\leq&\displaystyle
T^{3/2}\Bigg{(}\frac{1}{n^{3/2}}+\frac{1}{n^{3/2}}\;\max_{j-1\leq x\leq
j}\;\frac{d(x^{3/2})}{dx}\Bigg{)}\\\ \vskip
8.53581pt\cr&\leq&\displaystyle\frac{5}{2\;n}\;T^{3/2}.\end{array}$

Hence combining equality (3.6) with the above result, we obtain

(3.7) $\lambda^{-1}\geq\alpha\,n,$

where $\displaystyle\alpha=\frac{2}{T(2a^{2}+5b^{2}T^{1/2})}.$

Next, let us determine a suitable lower bound of $\|m\|_{2}^{2}$. From the
lemma 13 we have

(3.8)
$\begin{array}[]{rcl}\displaystyle\|m\|_{2}^{2}&=&\displaystyle\sum_{k=1}^{j}\;\Bigl{(}Cov\Bigl{(}\Delta_{j+1}^{n}S^{\frac{3}{4}},\Delta_{k}^{n}S^{\frac{3}{4}}\Bigr{)}\Bigr{)}^{2}\\\
\vskip
8.53581pt\cr&=&\displaystyle\frac{T^{3}b^{4}}{4n^{3}}\sum_{k=1}^{j}\;\Bigl{(}f_{1}(k)-f_{2}(k)\Bigr{)}^{2},\end{array}$

where

$f_{1}(k)=(j-k+2)^{3/2}-2(j-k+1)^{3/2}+(j-k)^{3/2}$

and

$f_{2}(k)=(j+k+1)^{3/2}-2(j+k)^{3/2}+(j+k-1)^{3/2}.$

The functions $f_{1}$ and $f_{2}$ satisfy three properties, which we shall use
at the end of the proof. We will state them in the following technical lemma.

###### Lemma 28

For any $k\in\\{1,..,j\\}$

  * •

$f_{1}(k)\geq 0$ and $f_{2}(k)\geq 0$,

  * •

$f_{1}(k)-f_{2}(k)>0$,

  * •

(3.9)
$f_{1}(k)-f_{2}(k)\geq\frac{3}{4}\;\Bigl{(}(j-k+1)^{-1/2}-(j+k-1)^{-1/2}\Bigr{)}\geq
0.$

###### Proof.

(of lemma 28) The first assertion of the lemma is due to the fact that the
function $\displaystyle x\longmapsto x^{3/2}$ is convex on the interval
$\displaystyle[0,+\infty[$.

Now, let us prove the second assertion of the lemma. Consider the function $g$
defined by

$g(x)=(x+1)^{3/2}-2\;x^{3/2}+(x-1)^{3/2},\;\;x\geq 1.$

Since the function $\displaystyle x\longmapsto x^{1/2}$ is concave on
$\displaystyle[1,+\infty[$, $g$ decreases on this interval and consequently

$f_{1}(k)=g(j-k+1)>g(j+k)=f_{2}(k).$

Finally, let us prove inequality (3.9). For every $a\geq 1$, let us consider
the function $g_{a}$ defined by

$g_{a}(x)=(a+x)^{3/2}-2\;a^{3/2}+(a-x)^{3/2},\;\;0\leq x\leq 1\leq a.$

We have
$\displaystyle\;g_{a}(0)=0,g_{a}^{{}^{\prime}}(x)=\frac{3}{2}\;((a+x)^{1/2}-(a-x)^{1/2})$
and therefore $g_{a}^{{}^{\prime}}(0)=0$.

On the otherhand, by Taylor-Lagrange theorem, we get that there exists
$c\in]0,1[$ such that

$g_{a}(1)=g_{a}(0)+g_{a}^{{}^{\prime}}(0)+\frac{1}{2}g_{a}^{"}(c)=\frac{1}{2}g_{a}^{"}(c),$

where

$g_{a}^{"}(x)=\frac{3}{4}\;\Bigl{(}(a+x)^{-1/2}+(a-x)^{-1/2}\Bigr{)}.$

Next, it is easy to check that the function $g_{a}^{"}$ increases, and
consequently

$g_{a}^{"}(0)\leq g_{a}^{"}(c)\leq g_{a}^{"}(1).$

So, we have

$\frac{3}{4\;a^{1/2}}\leq g_{a}(1)\leq\frac{3}{4\;(a-1)^{1/2}},$

and therefore

$f_{1}(k)=g_{j-k+1}(1)\geq\frac{3}{4}\,(j-k+1)^{-1/2}\;\;\;\mbox{and}\;\;\;f_{2}(k)=g_{j+k}(1)\leq\frac{3}{4}\,(j+k-1)^{-1/2},$

which ends the proof of the lemma.

∎

Let us turn back to the proof of proposition 27. Combining (3.8) with (3.9),
we get

(3.10)
$\|m\|_{2}^{2}\geq\frac{9\,b^{4}\,T^{3}}{64\,n^{3}}\sum_{k=2}^{j}\;\Bigl{(}(j-k+1)^{-1/2}-(j+k-1)^{-1/2}\Bigr{)}^{2}.$

For every integer $j\geq 1$, let us consider the function

$f_{j}(x)=(j-x+1)^{-1/2}-(j+x-1)^{-1/2},\;1\leq x\leq j.$

Since $f_{j}$ increases, we have

(3.11)
$\sum_{k=2}^{j}\;\Bigl{(}(j-k+1)^{-1/2}-(j+k-1)^{-1/2}\Bigr{)}^{2}\geq\int_{1}^{j}\;f_{j}(x)^{2}\;dx.$

But

(3.12)
$\begin{array}[]{rcl}\displaystyle\int_{1}^{j}\;f_{j}(x)^{2}\;dx&=&\displaystyle\int_{1}^{j}\;\Biggl{(}\frac{1}{j-x+1}+\frac{1}{j+x-1}-2\frac{1}{\sqrt{j^{2}-(x-1)^{2}}}\Biggr{)}\;dx\\\
\vskip
8.53581pt\cr&=&\displaystyle\ln(2j-1)-2\;\int_{1}^{j}\;\frac{1}{\sqrt{j^{2}-(x-1)^{2}}}\;dx\\\
\vskip
8.53581pt\cr&=&\displaystyle\ln(2j-1)+2\arccos\Bigl{(}\frac{j-1}{j}\Bigr{)}-\pi.\end{array}$

Hence, combining (3.7) with (3.10), (3.11) and (3.12), we get

(3.13)
$\|m\|_{2}^{2}\;\lambda^{-1}\geq\frac{\beta}{n^{2}}\;\Bigl{(}\ln(2j-1)+2\arccos\Bigl{(}\frac{j-1}{j}\Bigr{)}-\pi\Bigr{)},$

where $\displaystyle\beta=\frac{\alpha}{64}(9\;T^{3}\;b^{4}).$

Combining (3.13) with (3.5), we have,

$\sum_{j=1}^{n-1}\;\|E\Bigl{(}\Delta_{j+1}^{n}S_{3/4}\mid\Delta_{j}^{n}S_{3/4},..,\Delta_{1}^{n}S_{3/4}\Bigr{)}\|_{2}$
$\geq\frac{\sqrt{\beta}}{n}\;\sum_{j=1}^{n-1}\;\sqrt{\ln(2j-1)+2\arccos\Bigl{(}\frac{j-1}{j}\Bigr{)}-\pi}.$

Since
$\displaystyle\lim_{n\rightarrow\infty}\sqrt{\ln(2n-1)+2\arccos\Bigl{(}\frac{n-1}{n}\Bigr{)}-\pi}=+\;\infty$,
we have by Césaro theorem

$\lim_{n\rightarrow\infty}\frac{\sqrt{\beta}}{n}\;\sum_{j=1}^{n-1}\;\sqrt{\ln(2j-1)+2\arccos\Bigl{(}\frac{j-1}{j}\Bigr{)}-\pi}=+\;\infty,$

which completes the proof of proposition 27.

∎

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Charles EL-NOUTY

LAGA, université Paris XIII, 99 avenue J-B Clément, 93430 Villetaneuse, FRANCE

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Mounir ZILI

Preparatory Institute to the Military Academies, Research unit UR04DN04,
Avenue Maréchal Tito, 4029 Sousse, TUNISIA

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