# Non-Singular Bouncing Model in Energy Momentum Squared Gravity

Z. Yousaf1 , M. Z. Bhatti1 , H. Aman1 , P.K. Sahoo2
1Department of Mathematics, University of the Punjab,
Quaid-i-Azam Campus, Lahore-54590, Pakistan
2Department of Mathematics,
Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India
zeeshan.math@pu.edu.pkmzaeem.math@pu.edu.pkhuzaifaaman971<EMAIL_ADDRESS>pilani.ac.in

###### Abstract

This work is concerned to study the bouncing nature of the universe for an
isotropic configuration of fluid $\mathcal{T}_{\alpha\beta}$ and Friedmann-
Lemaître-Robertson-Walker metric scheme. This work is carried out under the
novel $f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$
gravitation by assuming a specific model i.e,
$f(\mathcal{G},\mathcal{T}^{2})=\mathcal{G}+\alpha\mathcal{G}^{2}+2\lambda\mathcal{T}^{2}$
with $\alpha$ and $\lambda$ are constants, serving as free parameters. The
terms $\mathcal{G}$ and $\mathcal{T}^{2}$ served as an Gauss-Bonnet invariant
and square of the energy-momentum trace term as an inclusion in the
gravitational action respectively, and is proportional to
$\mathcal{T}^{2}=\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta}$. A
specific functional form of the Hubble parameter is taken to provide the
evolution of cosmographic parameters. A well known equation of state
parameter, $\omega(t)=-\frac{k\log(t+\epsilon)}{t}-1$ is used to represent the
dynamical behavior of energy density, matter pressure and energy conditions. A
detailed graphical analysis is also provided to review the bounce.
Furthermore, all free parameters are set in a way, to make the supposed Hubble
parameter act as the bouncing solution and ensure the viability of energy
conditions. Conclusively, all necessary conditions for a bouncing model are
checked.

Keywords: cosmography; Hubble Parameter;
$f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$.
PACS: 98.80.-k; 04.20.Cv; 04.50.Kd.

##  1 Introduction

According to the big bang hypothesis, the whole universe was created by a
single explosion, with all matter in the cosmos as an infinite speck [1, 2].
This hypothesis works well in order to study the beginning, but lack to define
different cosmological problems. These problems include the horizon problem,
the flatness problem, the singularity problem, etc. In order to resolve these
big cosmic challenges, different cosmic theories have been developed in
literature [3, 4, 5]. The bouncing hypothesis is one of the major independent
theories that came up with the answers related to the starting of the universe
and should be enough to resolve the major cosmic problem of singularity. The
bouncing cosmology works on the scheme of an oscillatory universe, i.e, a
universe that came into being from the pre-existing universe without
undergoing the singularity [6, 7, 8]. This whole transition of the universe
not only explains the big-bang cosmology but also reduces one of the major
issues. For the bouncing, the universe moves into the contraction phase as a
matter-dominated the era of the universe. After the contraction, the universe
starts to expand in a nonsingular manner for which gravity dominates the
matter [9, 10]. Also, density perturbations can be produced during the bounce
era. This idea of the origination of the universe is highly accepted and
appreciated in literature.

General relativity ($\mathcal{GR}$) was presented by Einstein and it was
thought to be one of the best theories to explain different cosmological
issues. It explains the gravity under the fabric of space-time. However, to
understand gravity much more effectively and to provide the answers to the
effect of gravity, dark energy, and accelerated expansion of the universe
under the addition of different scalar fields, different attempts have been
made in past to modify $\mathcal{GR}$. These modifications change the
geometric or matter or both parts of the Einstein field equations accordingly.
These could help to discuss the effects of couplings of matter and curvature
terms on the above-described items. Roshan and Shojai [11] presented the
nonlinear form of matter term i.e,
$\mathcal{T}^{2}=\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta}$, naming
it $f(\mathcal{R},\mathcal{T^{2}})$. They further indicated that the use of
nonlinear terms may provide the prevention of early time singularities. Since
the functional form of curvature terms has helped to introduce new
gravitational theories, so it was considered to be effective to modify the
generic action integral of $\mathcal{GR}$ as corrections. These modifications
give light to the $f(\mathcal{G})$ theory, for which the term $\mathcal{G}$ is
defined as
$\mathcal{G}=\mathcal{R}_{\xi\zeta\alpha\beta}\mathcal{R}^{\xi\zeta\alpha\beta}-4\mathcal{R}_{\xi\zeta}\mathcal{R}^{\xi\zeta}+\mathcal{R}^{2}$.
Nojiri and Odintsov [12] introduced this $f(\mathcal{G})$ theory for the first
time in their work. They tested solar systems for this formalism and reported
the phase change of acceleration to deceleration for the achievement of
phantom line, which cooperated to study dark energy. Odintsov and Oikonomou
[13] considered $\mathcal{R}+f(\mathcal{G})$ form of the gravitational theory
to provide their contribution to the study of gravitational baryogenesis.
Their work included the higher-order derivatives of Gauss-Bonnet terms that
work in order to produce the baryon asymmetry. Sharif and Ikram [14] gives
rise to a new theory by following the footsteps of Harko. They coupled the
matter part $\mathcal{T}$ with the geometric part of the $f(\mathcal{G})$
theory, making it $f(\mathcal{G},\mathcal{T})$ cosmology. They investigated
the validity of their theory with the help of energy conditions. Later on,
Bhatti _et al_.[15] worked on the $f(\mathcal{G},\mathcal{T})$ theory to carry
out the investigation of some physically feasible features of compact star
formation. They inferred that the compactness of a star model grows at the
core whereas the energy conditions remain constant. Yousaf and his mates [16]
inspired by [17], have recently developed a novel
$f(\mathcal{G},\mathcal{T}^{2})$ to present the complexity of structural
scalars from the use of Herrera’s method of splitting scalars. They considered
the exponential coupling of Gauss-Bonnet terms as a functional form as
$f(\mathcal{G},\mathcal{T}^{2})=\alpha\mathcal{G}^{n}(\beta\mathcal{G}^{m}+1)+\eta\mathcal{T}^{2}$,
to explore the validity of their solutions for the Darmois and Israel
conditions. They also worked on the non-static complex structures under the
same theory to describe the effects of an electromagnetic field. They used
specific model configuration i.e,
$f(\mathcal{G},\mathcal{T}^{2})=k_{1}\mathcal{G}^{m}(k_{2}\mathcal{G}^{n}+1)+\lambda\mathcal{T}^{2}$,
in their work.

Bouncing cosmology has gained much reputation over the past few years, because
of its independent hypothetical nature from different standard comic problems.
Guth [18] during $1980^{\prime}s$, had put forward his inflationary theory to
tackle early and late time cosmic evolutionary problems. He remained
successful in solving some related problems, but the answer to the initial
singularity is still under concern. One of the best hypotheses to answer the
singularity problem is the bouncing nature of the universe. The nature of the
bouncing universe allows a certain universe model to transit from a pre-big
crunch (contracted) phase into a new big bang (expanded) phase with the
exclusion of singularity during the whole event [19]. Steinhardt and Ijjas
[20] are considered to be the pioneers of the bouncing hypothesis. They
devised a wedge diagram for a smooth bouncing method to explore the
consequences of some cosmological problems. Sahoo _et al_. [21] worked on the
non-singular bouncing by assuming the specific coupling of $\mathcal{R}$ and
$\mathcal{T}$ as
$f(\mathcal{R},\mathcal{T})=\mathcal{R}+\chi\mathcal{R}\mathcal{T}$, for
$0<\chi<\frac{\pi}{4}$. They allowed such a parametric approach for the Hubble
parameter to provide no singularity during the bounce. They used quintom and
phantom scalar field configurations for the bouncing paradigm. Bamba and his
collaborators [22] inspected the singularity-free concept of bounce by
considering an exponential form of scale factor $a(t)=\sigma\exp(\lambda
t)+\tau\exp(\lambda t)$ under the effect of $f(\mathcal{G})$ gravity. They
checked the stability of their assumed solution under the restricted
parametric scheme.

Yousaf _et al_. [23, 24] explored the bouncing universe with a specific
functional form of Hubble parameter by taking exponential
$f(\mathcal{G},\mathcal{T})$ form. Different cosmic models are under
consideration for the scale factor in order to determine the value of
expansion and contraction at the current cosmic phase and also to predict the
current phase equation of state. These models predicted different results in
the literature. However, cosmography provided us a benefit in processing
cosmological data for explaining the universal kinematics without the
involvement of the gravity model and hence provided that the cosmography can
be employed with the Taylor expansions as an alternative scheme. Also, the
cosmographic analysis for the $\mathcal{FLRW}$ universe, is helpful in such a
way that it can put aside the effect of the dynamical field equations [25].
Gruber _et al_. [26] studied an alternative approach to describe cosmography
by extending the conventional methodology. They resulted from numerical values
of the cosmographic parameters by applying the $Pad\acute{e}$ approximations.
The testing of the $\Lambda$CDM model had been conveyed by Busti _et al_. [27]
with the use of cosmographical analysis. Capozziello _et al_. provided
cosmography as a non-predictive phenomenon when the redshift parameter becomes
$z\approx 1$. They used the pad$\acute{e}$ approximations for the fifth order
and resulted the divergence of data at the higher levels of the
approximations. Lobo _et al_. [28] evaluated the dynamics of the redshift
drift. They used the expanding $FLRW$ universe to produce a general matter and
low redshift model with the use of different variables. However, the cross-
correlation of large-scale quasars can be used and translated with the CMB and
BAO scale data to produce the best for Hubble parameter $H(z)$ and angular
diametric distance $S_{A}$. Also, the cosmic chronometers approach can be done
to predict the model independent $H(z)$ measurements which have been
extensively used for cosmological applications [29, 30, 31]. The low redshift
data set with the inclusion of the megamasers and chronometers had been
presented by Krishnan and others [32]. They result that the Hubble constant
$\mathcal{H}_{0}$, showed descending behavior with the redshift and having
non-zero slop when fitted on the line by statistical means. Font _et al_. [33]
studied correlation technique for quasars by using Lya absorption and produced
the best line of fit for Planck’s data. They generated different results on
the measurements of the Hubble parameter and the angular distance. One
important thing is to develop such a cosmic Hubble parameter that comes from
early to late span in such a way that it changes from a low to a high value.
The Gaussian method helped to predict but provided a non-transitional behavior
for both $\Lambda$ and $\omega$ epochs. The null energy condition also proved
to be an important restriction for the cut-off model, when compared with
Hubble parameter data [34]. King _et al_. [35] studied the future
approximations of the redshift by the inclusion of dark energy. They tested
the equation of state by the linear parametrization technique. Hu _et al_.
[34] reported different values of the Hubble constant by the Gaussian method.
Their research produced an effective reduction in the Hubble crisis and
proposed the non-transitional behavior of the Hubble constant. Different dark
energy models respective to holography and agegraphy had been conducted by
Zhang _et al_. [36]. They produced different energy conditions for different
red shift values and resulted in an effective role of energy conditions for
different cosmic ages.

In this article, we implemented a functional form of the Hubble parameter that
evolves periodically with cosmic time $t$ and investigate the bouncing nature
of the universe in
$f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$ gravity
using a flat $\mathcal{FLRW}$ peacetime. This analysis of the bouncing
universe involves one of the most important forms of $EoS$ parameter proposed
in the literature [37, 38, 39]. The outline is given as: Sect.2 provides a
brief introduction to
$f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$ gravity
with the necessary formalism of $\mathcal{FLRW}$ metric and modified field
equations. Sect.3 builds the Hubble parameter as a bouncing solution for the
produced field equations. The cosmographic parameters are also evaluated in
this section. We provide the mathematical expressions of energy density and
matter pressure for the assumed $EoS$ parameter form in Sect.4. The energy
conditions are also formulated in the same fashion. Detailed graphical
profiles of energy conditions are represented in the same section to discuss
the evolution of the universe under the influence of restricted free
parameters. Finally, the concluding remarks are made in Sect.5.

##  2 $f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$
Formalism

The modified action for the
$f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$ gravity
theory is defined as [16]

$\mathbb{A}_{f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})}=\frac{\sqrt{-g}}{2\kappa^{2}}\left(\int
d^{4}x[f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})+\mathcal{R}]+\int
d^{4}x\mathcal{L}_{m}\right),$ (1)

where $\mathcal{R}$ and $\mathcal{G}$ symbolize the Ricci scalar and the
Gauss-Bonnet scalar terms, respectively and are provided as

$\mathcal{R}\equiv
g_{\alpha\beta}\mathcal{R}^{\alpha\beta},~{}~{}~{}~{}\mathcal{G}\equiv\mathcal{R}_{\xi\zeta\alpha\beta}\mathcal{R}^{\xi\zeta\alpha\beta}-4\mathcal{R}_{\xi\zeta}\mathcal{R}^{\xi\zeta}+\mathcal{R}^{2},$
(2)

and $\kappa^{2}$ = $8\pi$G (G be the gravitational constant) and
$\mathcal{L}_{m}=-p$. Also, the term $g$ implies the trace of the metric
tensor $g_{\alpha\beta}$ with $T_{\alpha\beta}$, $R_{\xi\zeta\alpha\beta}$ and
$R_{\alpha\beta}$ indicate the stress energy-momentum tensor, the Riemannian
tensor, and the Ricci tensor respectively. The expression for
$\mathcal{T}_{\alpha\beta}$ is given as

$\mathcal{T}_{\alpha\beta}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\alpha\beta}}.$ (3)

Equation (3) yields the following expression, due the dependency of the matter
Lagrangian $\mathcal{L}_{m}$ on $g_{\alpha\beta}$ components

$\mathcal{T}_{\alpha\beta}=g_{\alpha\beta}\mathcal{L}_{m}-\frac{2\partial\mathcal{L}_{m}}{\partial
g^{\alpha\beta}}.$ (4)

Now, by taking the variation of Eq.(1) with respect to the term
$g_{\alpha\beta}$, we get the following field equations for the
$f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$ theory as

$\mathcal{R}_{\alpha\beta}-\frac{1}{2}\mathcal{R}g_{\alpha\beta}=\mathcal{T}_{\alpha\beta}^{\emph{eff}},$
(5)

where the term $\mathcal{T}_{\alpha\beta}^{\emph{eff}}$ takes the following
form

$\displaystyle\mathcal{T}_{\alpha\beta}^{\emph{eff}}$ $\displaystyle=$
$\displaystyle\kappa^{2}\mathcal{T}_{\alpha\beta}-\Theta_{\alpha\beta}f_{\mathcal{T}^{2}}(\mathcal{G},\mathcal{T}^{2})+\frac{1}{2}g_{\alpha\beta}f(\mathcal{G},\mathcal{T}^{2})-(2\mathcal{R}\mathcal{R}_{\alpha\beta}-4\mathcal{R}_{\alpha}^{\varepsilon}\mathcal{R}_{\varepsilon\beta}-4\mathcal{R}_{\alpha\varepsilon\beta\eta}\mathcal{R}^{\varepsilon\eta}$
(6)
$\displaystyle+2\mathcal{R}_{\alpha}^{\varepsilon\eta\delta}\mathcal{R}_{\beta\varepsilon\eta\delta})f_{\mathcal{G}}(\mathcal{G},\mathcal{T}^{2})-(2\mathcal{R}\nabla^{2}g_{\alpha\beta}-2\mathcal{R}\nabla_{\alpha}\nabla_{\beta}-4\mathcal{R}_{\alpha\beta}\nabla^{2}-4g_{\alpha\beta}\mathcal{R}^{\varepsilon\eta}\nabla_{\varepsilon}\nabla_{\eta}$
$\displaystyle+4\mathcal{R}^{\varepsilon}_{\alpha}\nabla_{\beta}\nabla_{\varepsilon}+4\nabla_{\varepsilon}\nabla_{\alpha}\mathcal{R}^{\varepsilon}_{\beta}+4\mathcal{R}_{\alpha\varepsilon\beta\eta}\nabla^{\varepsilon}\nabla^{\eta})f_{\mathcal{G}}(\mathcal{G},\mathcal{T}^{2}),$

where,

$\Theta_{\alpha\beta}\equiv\frac{\delta(\mathcal{T}_{\mu\nu}\mathcal{T}^{\mu\nu})}{\delta
g^{\alpha\beta}}=2\mathcal{T}_{\alpha}^{\xi}\mathcal{T}_{\beta\xi}-\mathcal{T}\mathcal{T}_{\alpha\beta}-4\mathcal{T}^{\mu\nu}\frac{\partial^{2}\mathcal{L}_{m}}{\partial
g^{\alpha\beta}g^{\mu\nu}}-2\mathcal{L}_{m}(\mathcal{T}_{\alpha\beta}-\frac{1}{2}\mathcal{T}g_{\alpha\beta})$
(7)
$\mathcal{T}^{2}=\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta},~{}~{}~{}~{}~{}\nabla^{2}=\nabla_{\alpha}\nabla^{\alpha}$
(8)

The terms $f_{\mathcal{G}}(\mathcal{G},\mathcal{T}^{2})$ and
$f_{\mathcal{T}^{2}}(\mathcal{G},\mathcal{T}^{2})$ used above are defined as

$~{}~{}f_{\mathcal{G}}(\mathcal{G},\mathcal{T}^{2})\equiv\frac{df(\mathcal{G},\mathcal{T}^{2})}{d\mathcal{G}},~{}~{}~{}~{}and~{}~{}f_{\mathcal{T}^{2}}(\mathcal{G},\mathcal{T}^{2})\equiv\frac{df(\mathcal{G},\mathcal{T}^{2})}{d\mathcal{T}^{2}}.$
(9)

The trace of the above-defined field equations is produced as

$\mathcal{T}-\Theta
f_{\mathcal{T}^{2}}(\mathcal{G},\mathcal{T}^{2})+2\mathcal{G}f_{\mathcal{G}}(\mathcal{G},\mathcal{T}^{2})-2\mathcal{R}\nabla^{2}f_{\mathcal{G}}(\mathcal{G},\mathcal{T}^{2})+4\mathcal{R}_{\alpha\beta}\nabla^{\alpha}\nabla^{\beta}f_{\mathcal{G}}(\mathcal{G},\mathcal{T}^{2})=0.$
(10)

Equation (10) shows the non-conversed situation of the stress energy-momentum
tensor. Also, the properties of $\mathcal{GR}$ can be recovered for
$f(\mathcal{G},\mathcal{T}^{2})=0$. Similarly if we put
$f(\mathcal{G},\mathcal{T}^{2})=f(\mathcal{G})$, we get the properties of
$f(\mathcal{G})$ gravity.

Now, as we are concerned to study the bouncing nature of the universe, so we
consider the fluid distribution to be perfect throughout the cosmic evolution.
For this, we take

$\mathcal{T}_{\alpha\beta}=(\rho+p)V_{\alpha}V_{\beta}-pg_{\alpha\beta},$ (11)

here, the four-vector velocity is defined by $V^{\beta}$ with

$V^{\beta}=(1,0,0,0),~{}~{}V^{\beta}V_{\beta}=1~{},~{}V^{\beta}\nabla_{\zeta}V_{\zeta}=0.$
(12)

In addition, $\rho$ defines the energy density part and $p$ defines the
pressure part of the stress energy-momentum tensor. Also the geometric
background considered to be in a $\mathcal{FLRW}$ space time[40], so it
implies

$ds^{2}=dt^{2}-a^{2}(t)\Sigma_{i}dx_{i}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1,2,3.$
(13)

The metric component $a(t)$ symbolizes the scale factor, that contributes to
the Hubble parameter as $\mathcal{H}=\frac{\dot{a}(t)}{a(t)}$. Using Eq.(13)
and Eq.(7) in Eq.(5), we get the following field equations

$6\left(\frac{\dot{a}}{a}\right)^{2}-24\left(\frac{\dot{a}}{a}\right)^{3}\dot{f_{\mathcal{G}}}+24\left(\frac{\ddot{a}}{a}\right)\left(\frac{\dot{a}}{a}\right)^{2}f_{\mathcal{G}}-f-2(\rho^{2}+3p^{2}+4\rho
p)f_{\mathcal{T}^{2}}=2\rho\kappa^{2},$ (14)
$-2\left(2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}\right)+16\left(\frac{\ddot{a}\dot{a}}{a^{2}}\right)\dot{f_{G}}+8\left(\frac{\dot{a}}{a}\right)^{2}\ddot{f_{G}}-24\left(\frac{\ddot{a}}{a}\right)\left(\frac{\dot{a}}{a}\right)^{2}f_{G}+f=2p\kappa^{2}.$
(15)

To draw the conclusions on the field equations, we just need some functional
form of $f(\mathcal{G},\mathcal{T}^{2})$. As, there are many functional forms
regarding the interaction of matter with the curvature terms, in order to deal
with the issues of cosmic evolution. Various coupling models can be used to
evaluate the formations of both energy density and matter pressure, like one
can take $f(\mathcal{G},\mathcal{T}^{2})=\mathcal{G}+2f(\mathcal{T}^{2})$ that
may help to provide an analysis about $\Lambda CDM$ epoch. However, the other
choice is
$f(\mathcal{G},\mathcal{T}^{2})=f_{1}(\mathcal{G})+f_{2}(\mathcal{T}^{2})$
that may be worked as a correction to $f(\mathcal{G})$ gravity theory because
of $f_{2}(\mathcal{T}^{2})$. Similar forms have been explored in [41, 42] and
provided some distinct results due to the direct minimal curvature matter
coupling. Also,
$f(\mathcal{G},\mathcal{T}^{2})=f_{1}(\mathcal{G})+f_{2}(\mathcal{G})f_{3}(\mathcal{T}^{2})$
can be taken because of an explicit non-minimally coupling nature between
geometric parameters and matter variables [43]. So, we considered the
following form to produce the validating results.

$f(\mathcal{G},\mathcal{T}^{2})=f_{1}(\mathcal{G})+f_{2}(\mathcal{T}^{2}).$
(16)

To produce a bouncing universe, we need some functional forms of $f_{1}$ and
$f_{2}$ that not only describe the accelerating expansion of the universe but
also explain inflation to a great extent. For this, the higher power curvature
terms perform well to eliminate such issues. Elizalde [44] introduced the
power forms of the curvature scalar as $\eta\mathcal{R}^{n}$ ($n\geq 1$) and
produced the cosmological dynamics, so we consider the specific form of the
$f_{1}$ as the quadratic power model, so

$f_{1}(\mathcal{G})=\mathcal{G}+\alpha\mathcal{G}^{2}.$ (17)

Also, we take $\chi_{2}$ as

$f_{2}(\mathcal{T}^{2})=2\lambda\mathcal{T}^{2}.$ (18)

So, by using Eq.s (17) and (18) in the field equations, we get

$\displaystyle
6\mathcal{H}^{2}-48\alpha\mathcal{H}^{3}\mathcal{G}\dot{\mathcal{G}}+\alpha\mathcal{G}^{2}=2\kappa^{2}\rho+6\lambda\rho^{2}+18\lambda
p^{2}+16\lambda\rho p,$ (19)

and

$\displaystyle-2(2\dot{\mathcal{H}}+3\mathcal{H}^{2})+32(\dot{\mathcal{\mathcal{H}}}+\mathcal{H}^{2})\alpha\mathcal{G}\dot{\mathcal{G}}+16\alpha\mathcal{H}^{2}(\dot{\mathcal{G}}^{2}+\mathcal{G}\ddot{\mathcal{G}})-\alpha\mathcal{G}^{2}=2\kappa^{2}p-2\lambda\rho^{2}-6\lambda
p^{2}.$ (20)

In order to reduce the complexity of the Eq.(19) and Eq.(20), we utilize
$p=\omega\rho$, as the _EoS_ used in [37, 38, 39]. So we get the relations,

$\displaystyle(3\lambda+9\lambda\omega^{2}+8\lambda\omega)\rho^{2}+\kappa^{2}\rho-(3\mathcal{H}^{2}-24\alpha\mathcal{H}^{3}G\dot{\mathcal{G}}+\frac{\alpha}{2}\mathcal{G}^{2})=0$
(21)

and

$\displaystyle(-\frac{\lambda}{\omega^{2}}-3\lambda)p^{2}+\kappa^{2}p+((2\dot{\mathcal{H}}+3H^{2})-16(\dot{\mathcal{H}}+\mathcal{H}^{2})\alpha\mathcal{G}\dot{\mathcal{G}}-8\alpha\mathcal{H}^{2}(\dot{\mathcal{G}}^{2}+\mathcal{G}\ddot{\mathcal{G}})+\frac{\alpha}{2}\mathcal{G}^{2})=0.$
(22)

where, $\mathcal{G}=24\mathcal{H}^{2}(\dot{\mathcal{H}}+\mathcal{H}^{2})$.
Yousaf and his collaborators checked the stability of cosmic models in various
modified gravity theories [45, 46, 47, 48].

##  3 Hubble Parameter and Cosmography

This section mainly focuses on describing the evolutionary behavior of these
above-described dynamical terms. Hence, we consider a trigonometric form of
the $\mathcal{H}(t)$ which feasibly provides a bounce solution [49, 44], as
follows

$\mathcal{H}(t)=\zeta\sin(\phi t)h(t).$ (23)

This parameterized form of $\mathcal{H}(t)$ includes $\zeta$ and $\phi$, which
are considered to be constants here. The choice of $h(t)$ depends on the
periodic values of the function $\sin(\phi t)$, so the form of $h(t)$ can be
chosen as periodic, that cooperates with the non-vanishing values of the above
trigonometric function. This artificial approach of choosing such an ansatz
can be considered as a numerical analysis of making the bouncing solution. One
interesting feature is possessed by the term $\zeta$, which can work well as a
phase changer for the value of $\mathcal{H}(t)$. We consider $h(t)$ as

$h(t)=\exp(\varphi t),$ (24)

where $\varphi$ acts as a constant. Finally, we have

$\mathcal{H}(t)=\zeta\sin(\phi t)\exp(\varphi t).$ (25)

This functional form of the Hubble parameter is helpful to study cosmic
evolutionary expansion and contraction. This form of the Hubble parameter
gives us the bounce at $t=313$, depending upon the values of $\varphi=0.001$
and $\phi=0.01$ provided in Fig.1. We have restricted the values of
$\mathcal{H}(t)$ in the positive era of time. The basic scale factor form for
this parameterized Hubble parameter becomes

$a(t)=\exp\left(\frac{\zeta\exp(\varphi t)(\varphi\sin(\phi t)-\phi\sin(\phi
t))}{\varphi^{2}+\phi^{2}}\right).$ (26)

Similarly, the set of dynamical parameters that are derived from the Taylor
series expansion of the scale factor is termed as cosmographic factors. These
factors helped to obtain the cosmological concordance with the assumptions of
the universal homogeneity and isotropy on large cosmic scales [27, 50]. These
include deceleration, jerk and snap parameters. These factors allow us to
check the compatibility of the scale factor and the Hubble parameter. The
negative value of the deceleration parameter $q$ describes the accelerated
expansion of the universe. Similarly, jerk $j$ and snap $s$ determine the
expansion rate of the toy universe model. The mathematical interpretation for
these cosmography elements are defined as

$q=-\frac{1}{\mathcal{H}^{2}}\frac{1}{a}\frac{d^{2}a}{dt^{2}}=-1-\frac{1}{\zeta}(e^{-\varphi
t}\csc(\phi t)(\phi\cot(\phi t)+\varphi)),$ (27) $\displaystyle
j=\frac{1}{\mathcal{H}^{3}}\frac{1}{a}\frac{d^{3}a}{dt^{3}}$ $\displaystyle=$
$\displaystyle 1+\frac{1}{\zeta^{2}}(e^{-2\varphi t}\csc(\phi t)(3\zeta
e^{\varphi t}(\phi\cot(\phi t)+\varphi)$ (28) $\displaystyle+\csc(\phi
t)(2\varphi\phi\cot(\phi t)+\varphi^{2}-\phi^{2}))),$

and

$\displaystyle
s=\frac{1}{\mathcal{H}^{4}}\frac{1}{a}\frac{d^{4}a}{dt^{4}}=-\frac{1}{3\zeta(3\zeta
e^{\varphi t}+2\csc(\phi t)(\phi\cot(\phi t)+\varphi))}(2e^{-\varphi
t}\csc(\phi t)(\csc(\phi t)$ $\displaystyle(3\zeta\varphi e^{\varphi
t}\sin(\phi t)+\varphi^{2}-\phi^{2})+\phi\cot(\phi t)(3\zeta e^{\varphi
t}+2\varphi\csc(\phi t)))).$ (29)

Fig.1 shows the progression of the Hubble (left panel) and scale parameters
(right panel) along the positive time axis. Similarly, the development of jerk
(left panel) and snap factors (right panel) are provided in the fig.2. The
evolution of the deceleration parameter towards the negative value i.e,
$q\rightarrow-1$, before the bouncing point, provided in fig.6, shows the
accelerating universe.

Figure 1: The illustrations of Hubble parameter and scale factor with fixed
values of $\varphi=0.001$ and $\phi=0.01$.

Figure 2: The illustration of jerk and snap factors with fixed values of
$\varphi=0.001$ and $\phi=0.01$.

##  4 Energy Conditions under the EoS Parameter

For a specific cosmology model, energy conditions play an important role to
make its validation for the restricted free parameters. These energy
conditions help to maintain the specifications of the certain cosmic model
[51, 52, 53, 54, 55]. Similarly, these energy conditions also work for the
bouncing cosmology and provide a reasonable approach to validate the procedure
for our toy bouncing model. These conditions are described as

  * •

Dominant energy condition ($\mathcal{DEC}$)$\Leftrightarrow$ $\rho\geq 0$ ,
$\rho\pm p\geq 0$.

  * •

Strong energy condition ($\mathcal{SEC}$)$\Leftrightarrow$ $\rho+3p\geq 0$ ,
$\rho+p\geq 0$.

  * •

Weak energy condition ($\mathcal{WEC}$)$\Leftrightarrow$ $\rho\geq 0$ ,
$\rho+p\geq 0$.

  * •

Null energy condition ($\mathcal{NEC}$)$\Leftrightarrow$ $\rho+p\geq 0$.

  * •

Trace energy condition ($\mathcal{TEC}$)$\Leftrightarrow$ $\rho-3p\geq 0.$

The positivity of $\mathcal{DEC}$, $\mathcal{SEC}$ and $\mathcal{WEC}$ passes
on the validity and necessity of the bouncing concept. However, the violation
of $\mathcal{NEC}$ has a major role. This violation is different in the
$\mathcal{GR}$ context. Universal bouncing scenario is one of those ideas that
provides a chance to discuss the singularity-free universal beginning. Many
proposals in the literature suggested avoiding this singularity through
quantum aspects, but these don’t have such reliability to fit in the
gravitational theory. So, at this point gravitational theories allow a
specific mechanism to check the validity of the bounce model and as well its
own. Null energy condition is one such tool to help achieve the task. Also, it
has been proved that in the context of $\mathcal{GR}$, the violation of
$\mathcal{NEC}$ is extremely difficult to be achieved for local-field models.
So, effective field theories provide a chance to recognize the violation of
the $\mathcal{NEC}$ and to allow a non-singular bounce [56, 57, 58, 59]. One
such effective field is
$f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$ theory
that provides a chance to study the quadratic nature of the energy terms i.e,
energy density and matter pressure[60, 16]. However, it also allows getting a
non-singular bounce for the assumed gravity model form. For an excellent
bouncing model, the value of $\mathcal{H}(t)$ turns out to be
$\mathcal{\dot{H}}=-4G\rho\pi(1+\omega)>0$ for the formulation of
$\mathcal{GR}$. However, if the $\mathcal{NEC}$ gets violated, we have the
surety to get a bouncing scenario. To provide the mathematical formulation of
the energy conditions, we consider Eqs. (21) and (22). Also, the $EoS$
parameter in the negative regime provides the present cosmic evolution [61,
62, 63] and becomes favorable in the bouncing context with
$\omega(t)\approx-1$. However, bouncing cosmology provides the possible
geodesic evolution of the universe by avoiding the singularity along with the
resolution of the horizon problem, flatness problem, entropy problem and many
more [5]. For the modified gravity, $EoS$ parameter enables us to study the
universal dynamics. In this study, we used $EoS$ parameter [44] to obtain the
possible chance of obtaining a bounce solution in
$f(\mathcal{G},\mathcal{T}^{2})$ as

$\displaystyle\omega(t)=-\frac{k\log(t+\epsilon)}{t}-1,$ (30)

here $k$ is assumed to be a constant. This particular form of the $EoS$
parameters allows us to study the contracting and expanding behavior without
involving the Hubble parameter as well as the scale factor. Elizalde _et al_.
[44] produces cosmological dynamics by considering $\mathcal{R}^{2}$ gravity
and logarithmic trace terms. They checked the effects of the $\lambda$
parameter in the gravity model
$f(\mathcal{R},\mathcal{T})=\mathcal{R}+\lambda\mathcal{R}^{2}+2\beta\ln(\mathcal{T})$
along with the bouncing solution depending on the two $EOS$ parameters. Our
work first described the choice of Hubble parameter and its effects on the
dynamical field equations and then involves the $EOS$ parameter. We only took
one of the $\omega(t)$ value, because this state factor after the bouncing
point remains negative and becomes $\omega(t)\approx-1$. Also, the current
cosmic expansion and $\Lambda-CDM$ can be verified by this state factor.
However, the dynamic properties are greatly affected under the influence of
this $EoS$ parameter form. Hence, the general forms of the Eqs.(21) and (22),
under the influence of Eq.30, are presented as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{1}{2\lambda(9\omega^{2}+8\omega+3)}(\kappa^{2}+(\kappa^{4}-12\zeta^{2}\lambda(9\omega^{2}+8\omega+3)e^{2\varphi
t}\sin^{2}(\phi t)(2304\alpha\zeta^{7}e^{7\varphi t}\sin^{4}(\phi t)$ (31)
$\displaystyle(\sin(\phi t)(\zeta e^{\varphi t}\sin(\phi
t)+\varphi)+\phi\cos(\phi t))(4\zeta\varphi e^{\varphi t}\sin^{3}(\phi
t)+2\zeta\phi e^{\varphi t}\sin(2\phi t)\sin(\phi t)$ $\displaystyle-$
$\displaystyle(\phi^{2}-3\varphi^{2})\sin^{2}(\phi t)+2\phi^{2}\cos^{2}(\phi
t)+3\varphi\phi\sin(2\phi t))-96\alpha\zeta^{4}e^{4\varphi t}\sin^{2}(\phi t)$
$\displaystyle(\sin(\phi t)(\zeta e^{\varphi t}\sin(\phi
t)+\varphi)+\phi\cos(\phi t))^{2}-1))^{\frac{1}{2}}$ $\displaystyle p$
$\displaystyle=$
$\displaystyle\frac{1}{2(3\lambda\omega^{2}+\lambda)}(\kappa^{2}\omega^{2}+(\kappa^{4}\omega^{4}+4\zeta\omega^{2}(3\lambda\omega^{2}+\lambda)e^{\varphi
t}(18432\alpha\zeta^{9}e^{9t\varphi}(2\varphi(2\varphi+1)-\phi^{2})$ (32)
$\displaystyle\sin^{10}(t\phi
t)+4608\alpha\zeta^{8}e^{8t\varphi}(\varphi^{2}(25\varphi+22)-(13\varphi+2)\phi^{2})\sin^{9}(\phi
t)+9216\alpha\zeta^{7}\phi^{3}e^{7\varphi t}$ $\displaystyle\sin^{5}(\phi
t)\cos^{3}(\phi t)(7\zeta e^{\varphi t}\sin(\phi
t)10\varphi+8)+288\alpha\zeta^{7}e^{7\varphi
t}(144\varphi^{4}+320\varphi^{3}-16(9\varphi+4)$
$\displaystyle\varphi\phi^{2}-1)\sin^{8}(\phi
t)+576\alpha\zeta^{6}\phi^{4}e^{6\varphi t}\sin^{4}(2\phi t)(\zeta e^{\varphi
t}+2\csc(\phi t))+576\alpha\zeta^{6}\varphi e^{6\varphi t}$
$\displaystyle(48\varphi^{3}-16\varphi\phi^{2}-1)\sin^{7}(\phi
t)-96\alpha\zeta^{5}\varphi(3\varphi-8)e^{5\varphi t}\sin^{6}(\phi
t)+192\alpha\zeta^{4}e^{4\varphi t}(3\varphi^{2}-\phi^{2})$
$\displaystyle\sin^{5}(\phi t)+2\sin(\phi
t)(5760\alpha\zeta^{6}\varphi\phi^{3}e^{6\varphi t}\sin^{3}(2\phi
t)+48\alpha\zeta^{4}\phi^{2}e^{4\varphi t}\sin^{2}(2\phi
t)-\varphi)+288\alpha$ $\displaystyle\zeta^{5}\phi^{2}e^{5\varphi
t}\sin^{4}(\phi t)\cos^{2}(\phi t)(192\zeta^{4}e^{4\varphi t}\sin^{4}(\phi
t)+32\zeta^{3}(31\varphi+10)e^{3\varphi t}\sin^{3}(\phi t)$
$\displaystyle+16\zeta^{2}e^{2\varphi
t}(\varphi(45\varphi+56)-7\phi^{2})\sin^{2}(\phi t)+32\zeta e^{\varphi
t}(17\varphi^{2}-\phi^{2})\sin(\phi t)-1)-2\phi\cos(\phi t)$
$\displaystyle(-18432\alpha\zeta^{9}(4\varphi+1)e^{9\varphi t}\sin^{9}(\phi
t)-2304\alpha\zeta^{8}e^{8\varphi
t}(\varphi(75\varphi+44)-11\phi^{2})\sin^{8}(\phi t)-9216\alpha\zeta^{7}$
$\displaystyle e^{7\varphi
t}(3\varphi^{2}(3\varphi+5)-(4\varphi+1)\phi^{2})\sin^{7}(\phi
t)-288\alpha\zeta^{6}e^{6\varphi t}(192\varphi^{3}-32\varphi\phi^{2}-1)$
$\displaystyle\sin^{6}(\phi t)+96\alpha\zeta^{5}(3\varphi-4)e^{5\varphi
t}\sin^{5}(\phi t)-576\alpha\zeta^{4}\varphi e^{4\varphi t}\sin^{4}(\phi
t)+1)$ $\displaystyle-3\zeta e^{\varphi t}\sin^{2}(\phi t)))^{\frac{1}{2}}$

Now, the profiles of energy density and pressure under the presence of
Eq.(30), are provided in fig.3. The plots indicate that the energy density
suffers a positive behavior for the assumed values of free parameters.
Similarly, the negative behavior for the pressure term indicates that the
universe is in the accelerated expansion phase. However, the positive density
proves a strong validation for the verification of the energy conditions.
Also, one can get the positive and alternate trends of the both terms for
different time periods due to the oscillatory behavior of the assumed Hubble
parameter. We restrict our work for the positive density and negative pressure
behavior to ascertain the energy conditions. The evolutionary profiles of the
energy conditions are provided in the figs. 4 and 5. The $\mathcal{NEC}$ plot
shows the violation with in the bouncing regime and confirms the major
verification for the universe to attain a bounce with in the framework of
$\mathcal{FLRW}$ spacetime. The violated $\mathcal{WEC}$ and $\mathcal{SEC}$
are given in the left plots of the figs. 3 and 4. The violated $\mathcal{SEC}$
also maintains the recent observations for the accelerating universe [52]. One
important energy condition i.e, $\mathcal{TEC}$ has also been given in this
recent study. The positive profiles for the $\mathcal{DEC}$ and
$\mathcal{TEC}$ are given in the fig.5. The evolution of these energy
conditions is strictly dependent on the values of the free parameters used in
this study. However, one can get another configuration of these physical
factors by implementing the different free parameters. The evolution of EoS
parameter is provided in fig.6 to encounter the negative value i.e,
$\omega(t)\approx-1$, for the current expansion phase of the universe.

Figure 3: The illustration of energy density and matter pressure with fixed
values of $\alpha=0.005$, $k=0.5$, $\varphi=0.001$, $\epsilon=0.001$,
$\phi=0.01$, $\kappa=1$ and $\lambda=-0.005$.

Figure 4: The illustration of $\mathcal{NEC}$ and $\mathcal{SEC}$ with fixed
values of $\alpha=0.005$, $k=0.5$, $\varphi=0.001$, $\epsilon=0.001$,
$\phi=0.01$, $\kappa=1$ and $\lambda=-0.005$.

Figure 5: The illustration of $\mathcal{DEC}$ and $\mathcal{TEC}$ with fixed
values of $\alpha=0.005$, $k=0.5$, $\varphi=0.001$, $\epsilon=0.001$,
$\phi=0.01$, $\kappa=1$ and $\lambda=-0.005$.

Figure 6: The illustration of EoS and deceleration parameters with fixed
values of $k=0.5$ and $\epsilon=0.001$.

##  5 Discussions

This work involves the study of bouncing cosmology for an isotropic
configuration of fluid $\mathcal{T}_{\alpha\beta}$ and $\mathcal{FLRW}$
metric. We comprehend this work under
$f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$ theory of
gravitation by assuming a specific model i.e,
$f(\mathcal{G},\mathcal{T}^{2})=\mathcal{G}+\alpha\mathcal{G}^{2}+2\lambda\mathcal{T}^{2}$
with $\alpha$ and $\lambda$ are constants, serving as free parameters. This is
the first-ever attempt to cover bouncing cosmology in the
$f(\mathcal{G},\mathcal{T}_{\alpha\beta}\mathcal{T}^{\alpha\beta})$ theory. By
the consideration of a specific functional form of the Hubble parameter, we
discuss the evolution of cosmographic parameters. The assumption of a well-
known equation of state (EoS) parameter,
$\omega(t)=-\frac{k\log(t+\epsilon)}{t}-1$, is used as a direct implementation
to represent the dynamical behavior of energy density, matter pressure, and
energy conditions. The free parameters are restricted to the special values
provided in each graph plot and are used for $\mathcal{H}(t)$ to act as the
bouncing solution. The viability of energy conditions is studied with the help
of a graphical approach. Following are the concluding remarks for this present
work.

  * •

The Hubble parameter $\mathcal{H}(t)$ used in this study is considered to have
a trigonometric functional form. The evolutionary behavior of different
cosmographic factors is described under the same form of $\mathcal{H}(t)$.
This parameterized form of $\mathcal{H}(t)$ depends on the periodic values of
the function $\sin(\phi t)$ and $h(t)$. We considered this $h(t)$ as a
nonvanishing function for the periodic values of $\sin(\phi t)$. A perfect
bouncing model allows the Hubble parameter to show the contraction phase i.e,
$\mathcal{H}<0$, and when the universe expands it becomes $\mathcal{H}>0$.
During this expansion and contraction phase, there is the point in between, at
where $\mathcal{H}(t)$ becomes zero. So, in order to produce such a scenario,
we have arranged the constants ($\phi$ and $\varphi$) in the Hubble parameter
$(\mathcal{H}(t)=\zeta\sin(\phi t)\exp(\varphi t))$ to some specific values
and notice the bounce at $t=313$. However, $t=313$ is significant in such a
way that all the energy conditions necessary for the bounce, get satisfied
accordingly till $t=313$, depending on the values of $\phi$ and $\varphi$. One
can also produce other values of $t$ for bounce by restricting other values of
$\phi$ and $\varphi$. The plot of $\mathcal{H}(t)$ is given in fig.1. The
Hubble parameter gives us the bounce at $t=313$ which is the future
singularity in the scale factor, see fig.1. The mathematical forms of
deceleration, jerk, and snap are evaluated with the same $\mathcal{H}(t)$. The
deceleration parameter tends to have a negative trend i.e, $q(t)$ approaches
$-1$, which can be seen in fig.6. Similarly, the trends of jerk and snaps are
given in fig.2 with $j(t)$ approaches to $1$ and $s(t)$ approaches to $0$. All
these values show a deflection at the bouncing point, that fits in for the
bouncing universe.

  * •

We ensure the configuration of the bouncing cosmology by studying energy
conditions. These energy conditions are provided in terms of energy density
and matter pressure derived from the modified field equations. We assumed a
specific EoS parameter in the form $\omega(t)=-\frac{k\log(t+\epsilon)}{t}-1$.
This EoS parameter helped to maintain the positive and negative growth of
energy density and matter pressure for the limited bouncing time period. The
profiles of $\rho$ and $p$ are provided in the fig.3. However, the
mathematical expression for these terms is evaluated in Eqs.(31) and (32).

  * •

Under the restricted values of the free parameters, $\alpha=0.005$, $k=0.5$,
$\varphi=0.001$, $\epsilon=0.001$, $\phi=0.01$, $\kappa=1$ and
$\lambda=-0.005$, we get the violation of the $\mathcal{NEC}$ and
$\mathcal{SEC}$. The violated $\mathcal{NEC}$ derives the bouncing nature of
the universe. However, the violated $\mathcal{SEC}$ and $\mathcal{WEC}$
provide the phase of cosmic expansion. suitable with the observational data.
The left plots of figs.3 and 4 shows the violated $\mathcal{SEC}$ and
$\mathcal{WEC}$. Similarly, the positive behavior of $\mathcal{DEC}$ and
$\mathcal{TEC}$ assure that the assumed model configuration is valid. Figure 5
represents the illustration of $\mathcal{DEC}$ and $\mathcal{TEC}$. Also, the
evolution of EoS can be seen in fig.6, showing that $\omega(t)\rightarrow-1$.
This value of $\omega(t)$ favors the current accelerated expansion phase of
the universe [61, 62, 63].

  * •

The above discussion provides that the bouncing evolution of the universe,
studied in the framework of
$f(\mathcal{G},\mathcal{T}^{2})=\mathcal{G}+\alpha\mathcal{G}^{2}+2\lambda\mathcal{T}^{2}$
and agrees with the recent astronomical observations [64, 65] i.e, all the
energy conditions are fully satisfied, a great negative pressure behavior had
been observed and provided help to study the late time accelerated universe
[44]. However, this study can be used in the future for different models of
the scale factors and Hubble parameters.

  * •

We finally conclude that the bouncing evolution of the universe can be studied
effectively with the oscillating nature of the scale factor under the flat
$\mathcal{FLRW}$ regime.

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# A Note on Optimal Liquidation with Linear Price Impact

Yan Dolinsky and Doron Greenstein  Hebrew University, Department of
Statistics, Mount Scopus, Jerusalem, Israel.

###### Abstract.

In this note we consider the maximization of the expected terminal wealth for
the setup of quadratic transaction costs. First, we provide a very simple
probabilistic solution to the problem. Although the problem was largely
studied, as far as we know up to date this simple and probabilistic form of
the solution has not appeared in the literature. Next, we apply the general
result for the study of the case where the risky asset is given by a
fractional Brownian Motion and the information flow of the investor can be
diversified.

Keywords: Linear Price Impact, Optimal Liquidation, Fractional Brownian Motion

##  1\. Preliminaries and the General Result

Consider a model with one risky asset which we denote by $S=(S_{t})_{0\leq
t\leq T}$ , where $T<\infty$ is the time horizon. We assume that the investor
has a bank account that, for simplicity, bears no interest. The risky asset
$S$ is RCLL (right continuous with left limits) and adapted process defined on
a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{0\leq
t\leq T},\mathbb{P})$. The filtration $(\mathcal{F}_{t})_{0\leq t\leq T}$
satisfies the usual assumptions (right continuity and completeness). Let us
emphasis that we do not assume that the $\sigma$-algebra $\mathcal{F}_{0}$ is
the trivial $\sigma$-algebra.

In financial markets, trading moves prices against the trader: buying faster
increases execution prices, and selling faster decreases them. This aspect of
liquidity, known as market depth (see [2]) or price-impact, has received large
attention in optimal liquidation problems, see, for instance, [1, 8, 4, 7] and
the references therein.

Following [1], we model the investor’s market impact in a temporary linear
form and thus, when at time $t$ the investor turns over her position
$\Phi_{t}$ at the rate $\phi_{t}=\dot{\Phi}_{t}$ the execution price is
$S_{t}+\frac{\Lambda}{2}\phi_{t}$ for some constant $\Lambda>0$. In our setup
the investor has to liquidate his position, namely
$\Phi_{T}=\Phi_{0}+\int_{0}^{T}\phi_{t}dt=0$.

As a result, the profits and losses from trading are given by

(1.1)
$V^{\Phi_{0},\phi}_{T}:=-\Phi_{0}S_{0}-\int_{0}^{T}\phi_{t}S_{t}dt-\frac{\Lambda}{2}\int_{0}^{T}\phi^{2}_{t}dt$

where $\Phi_{0}$ is the initial number (deterministic) of shares. Observe that
all the above integrals are defined pathwise. In particular we do not assume
that $S$ is a semi–martingale.

Let us explain in more detail formula (1.1). At time $0$ the investor has
$\Phi_{0}$ stocks and the sum $-\Phi_{0}S_{0}$ on her savings account. At time
$t\in[0,T)$ the investor buys $\phi_{t}dt$, an infinitesimal number of stocks
or more intuitively sell $-\phi_{t}dt$ number of shares and so the
(infinitesimal) change in the savings account is given by
$-\phi_{t}\left(S_{t}+\frac{\Lambda}{2}\phi_{t}\right)dt$. Since we liquidate
the portfolio at the maturity date, the terminal portfolio value is equal to
the terminal amount on the savings account and given by
$-\Phi_{0}S_{0}-\int_{0}^{T}\phi_{t}\left(S_{t}+\frac{\Lambda}{2}\phi_{t}\right)dt$.
We arrive at the right-hand side of (1.1). For the case where $S$ is a
semi–martingale, by applying the integration by parts formula we get that the
right-hand side of (1.1) is equal to
$\int_{0}^{T}\Phi_{t}dS_{t}-\frac{\Lambda}{2}\int_{0}^{T}\phi^{2}_{t}dt$.

For a given $\Phi_{0}$, the natural class of admissible strategies is

$\mathcal{A}_{\Phi_{0}}:=\left\\{\phi\ :\phi\ \mbox{is}\
(\mathcal{F}_{t})_{t\in[0,T]}-\mbox{optional}\ \mbox{with}\
\int_{0}^{T}\phi^{2}_{t}dt<\infty\ \mbox{and}\
\Phi_{0}+\int_{0}^{T}\phi_{t}dt=0\right\\}.$

As usual, all the equalities and the inequalities are understood in the almost
surely sense.

We are interested in the following optimal liquidation problem

(1.2) $\mbox{Maximize}\ \mathbb{E}\left[V^{\Phi_{0},\phi}_{T}\right]\ \
\mbox{over}\ \ \phi\in\mathcal{A}_{\Phi_{0}}$

where $\mathbb{E}$ denotes the expectation with respect to $\mathbb{P}$.

The following theorem provides a complete probabilistic solution to the
optimization problem (1.2).

######  Theorem 1.1.

Assume that

(1.3) $\mathbb{E}\left[\int_{0}^{T}S^{2}_{t}dt\right]<\infty.$

Introduce the martingale

(1.4)
$M_{t}:=\mathbb{E}\left[\int_{0}^{T}S_{u}du\left|\right.\mathcal{F}_{t}\right]\
\ t\in[0,T].$

The unique ($dt\otimes\mathbb{P}$ a.s) solution to the optimization problem
(1.2) is given by

(1.5)
$\hat{\phi}_{t}:=-\frac{\Phi_{0}}{T}+\frac{M_{0}}{T\Lambda}+\frac{1}{\Lambda}\left(\int_{0}^{t}\frac{dM_{u}}{T-u}-S_{t}\right),\
\ t\in[0,T)$

and the corresponding value is equal to

(1.6)
$\max_{\phi\in\mathcal{A}_{\Phi_{0}}}\mathbb{E}\left[V^{\Phi_{0},\phi}_{T}\right]=\mathbb{E}\left[V^{\Phi_{0},\hat{\phi}}_{T}\right]=-\frac{\Phi^{2}_{0}\Lambda}{2T}+\Phi_{0}\mathbb{E}\left[\frac{M_{0}}{T}-S_{0}\right]+\frac{1}{2\Lambda}\mathbb{E}\left[\int_{0}^{T}\left(S_{t}-\frac{M_{0}}{T}-\int_{0}^{t}\frac{dM_{u}}{T-u}\right)^{2}dt\right].$

A slightly more general form of the linear-quadratic optimization problem
(1.2) has been considered in [3], however for the relatively simple setup of
optimal liquidation Theorem 1.1 provides a much simpler solution than [3]. As
far as we know, up to date this simple and probabilistic form of the solution
has not appeared in the literature.

Before, we prove Theorem 1.1 let us briefly collect some observations from
this result. First, let us notice that it is sufficient to define the optimal
portfolio on the half-open interval $[0,T)$ (as we do in (1.5). We can just
set $\phi_{T}:=0$.

Next, observe that the optimal value given by the right hand side of (1.6) can
be decomposed into three terms, the first $-\frac{\Phi^{2}_{0}\Lambda}{2T}$
does not depend on the risky asset, the second term is a product of the
initial number of shares $\Phi_{0}$ and the term
$\mathbb{E}\left[\frac{M_{0}}{T}-S_{0}\right]$ which can be interpreted as the
average drift of the risky asset $S$ (recall that we do not assume that $S$ is
a semi-martingale). The last term
$\frac{1}{2\Lambda}\mathbb{E}\left[\int_{0}^{T}\left(S_{t}-\frac{M_{0}}{T}-\int_{0}^{t}\frac{dM_{u}}{T-u}\right)^{2}dt\right]$
is a product of the market depth $\frac{1}{2\Lambda}$ and the distance of the
risky asset $S$ from a martingale. In particular if $S$ is a martingale the
last term is equal to zero.

Next, we prove Theorem 1.1.

###### Proof.

The proof will be done in three steps.
Step I: Introduce the process $N_{t}:=\int_{0}^{t}\frac{dM_{u}}{T-u}$,
$t\in[0,T).$ In this step we show that

(1.7)
$\mathbb{E}\left[\int_{0}^{T}S_{t}N_{t}dt\right]=\mathbb{E}\left[\int_{0}^{T}N^{2}_{t}dt\right]\leq\mathbb{E}\left[\int_{0}^{T}S^{2}_{t}dt\right].$

Fix $n\in\mathbb{N}$ and define the process $N^{n}=(N^{n}_{t})_{0\leq t\leq
T}$ by $N^{n}_{t}:=N_{t\wedge(T-1/n)}$, $t\in[0,T]$. From (1.3) it follows
that $M$ and $N^{n}$ are square integrable martingales.

Next, for any stochastic processes $X,Y$ we denote by $[X]_{\cdot}$ the
quadratic variation of $X$ and by $[X,Y]_{\cdot}$ the covariation of $X$ and
$Y$ (provided that these terms are well defined). Also, denote by
$\mathbb{I}_{\cdot}$ the indicator function.

From the Itô Isometry and the Fubini Theorem

$\displaystyle\mathbb{E}\left[\int_{0}^{T}S_{t}N^{n}_{t}dt\right]=\mathbb{E}\left[M_{T}N^{n}_{T}\right]=\mathbb{E}\left[[M,N^{n}]_{T}\right]$
(1.8)
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\frac{\mathbb{I}_{s<T-1/n}}{T-s}d[M]_{s}\right]=\mathbb{E}\left[\int_{0}^{T}\int_{0}^{t}\frac{\mathbb{I}_{s<T-1/n}}{(T-s)^{2}}d[M]_{s}dt\right]=\mathbb{E}\left[\int_{0}^{T}|N^{n}_{t}|^{2}dt\right].$

Hence,

(1.9)
$0\leq\mathbb{E}\left[\int_{0}^{T}|S_{t}-N^{n}_{t}|^{2}dt\right]=\mathbb{E}\left[\int_{0}^{T}S^{2}_{t}dt\right]-\mathbb{E}\left[\int_{0}^{T}|N^{n}_{t}|^{2}dt\right].$

From (1.3) and (1)–(1.9) we obtain

$\mathbb{E}\left[\int_{0}^{T}S_{t}N_{t}dt\right]=\lim_{n\rightarrow\infty}\mathbb{E}\left[\int_{0}^{T}S_{t}N^{n}_{t}dt\right]=\lim_{n\rightarrow\infty}\mathbb{E}\left[\int_{0}^{T}|N^{n}_{t}|^{2}dt\right]=\mathbb{E}\left[\int_{0}^{T}N^{2}_{t}dt\right]\leq\mathbb{E}\left[\int_{0}^{T}S^{2}_{t}dt\right]$

and (1.7) follows.

Step II: Let $\phi\in\mathcal{A}_{\Phi_{0}}$. In this step we prove that
$\mathbb{E}\left[V^{\Phi_{0},\phi}_{T}\right]$ is not bigger than the right
hand side of (1.6). Without loss of generality we assume that
$\mathbb{E}\left[V^{\Phi_{0},\phi}_{T}\right]>-\infty$.

From (1.1) and the Cauchy–Schwarz inequality it follows that

$\sqrt{\int_{0}^{T}S^{2}_{t}dt}\sqrt{\int_{0}^{T}\phi^{2}_{t}dt}-\frac{\Lambda}{2}\int_{0}^{T}\phi^{2}_{t}dt\geq
V^{\Phi_{0},\phi}_{T}+\Phi_{0}S_{0}.$

By exploiting the behaviour of the (random) parabola $x\rightarrow
x\sqrt{\int_{0}^{T}S^{2}_{t}dt}-\frac{\Lambda}{2}x^{2}$, we get that

$\int_{0}^{T}\phi^{2}_{t}dt\leq
c\left(1+\max\left(-V^{\Phi_{0},\phi}_{T},0\right)+\int_{0}^{T}S^{2}_{t}dt\right)$

for some constant $c>0$, and so
$\mathbb{E}\left[\int_{0}^{T}\phi^{2}_{t}dt\right]<\infty.$

Next, set $Z:=-\frac{\Phi_{0}\Lambda}{T}+\frac{M_{0}}{T}$ and choose
$n\in\mathbb{N}$. From the estimate
$\mathbb{E}\left[\int_{0}^{T}\phi^{2}_{t}dt\right]<\infty$ and the fact that
$N^{n}$ is square integrable martingale we obtain

$\mathbb{E}\left[\int_{0}^{T}\phi_{t}N^{n}_{t}dt\right]=\mathbb{E}\left[N^{n}_{T}\int_{0}^{T}\phi_{t}dt\right]=-\Phi_{0}\mathbb{E}\left[N^{n}_{T}\right]=0.$

This together with (1.1) and the simple inequality
$xy-\frac{\Lambda}{2}x^{2}\leq\frac{y^{2}}{2\Lambda}$, $x,y\in\mathbb{R}$
yields

$\displaystyle\mathbb{E}\left[V^{\Phi_{0},\phi}_{T}\right]=\mathbb{E}\left[-\Phi_{0}(S_{0}-Z)-\int_{0}^{T}\phi_{t}(S_{t}-Z-N^{n}_{t})dt-\frac{\Lambda}{2}\int_{0}^{T}\phi^{2}_{t}dt\right]$
$\displaystyle\leq\mathbb{E}\left[-\Phi_{0}(S_{0}-Z)+\frac{1}{2\Lambda}\int_{0}^{T}|S_{t}-Z-N^{n}_{t}|^{2}dt\right].$

By taking $n\rightarrow\infty$ in the above inequality and applying (1.7) we
obtain

(1.10)
$\mathbb{E}\left[V^{\Phi_{0},\phi}_{T}\right]\leq-\frac{\Phi^{2}_{0}\Lambda}{2T}+\Phi_{0}\mathbb{E}\left[\frac{M_{0}}{T}-S_{0}\right]+\frac{1}{2\Lambda}\mathbb{E}\left[\int_{0}^{T}\left(S_{t}-\frac{M_{0}}{T}-N_{t}\right)^{2}dt\right]$

as required.

Step III: In this step we complete the proof. Consider the trading strategy
given by (1.5). From the Fubini theorem it follows that

$\int_{0}^{T}\hat{\phi}_{t}dt=-\Phi_{0}+\frac{1}{\Lambda}\left(M_{0}+M_{T}-M_{0}-\int_{0}^{T}S_{t}dt\right)=-\Phi_{0}.$

Moreover, from (1.7) it follows that
$\mathbb{E}\left[\int_{0}^{T}\hat{\phi}^{2}_{t}dt\right]<\infty$. Thus,
$\hat{\phi}\in\mathcal{A}_{\Phi_{0}}$.

Next, choose $n\in\mathbb{N}$. By using the same arguments as in Step II we
get $\mathbb{E}\left[\int_{0}^{T}\hat{\phi}_{t}N^{n}_{t}dt\right]=0.$ Observe
that for $t\leq T-1/n$ we have
$\hat{\phi}_{t}=\frac{Z+N^{n}_{t}-S_{t}}{\Lambda}$, where (recall)
$Z=-\frac{\Phi_{0}\Lambda}{T}+\frac{M_{0}}{T}$. Hence,

$\displaystyle\mathbb{E}\left[V^{\Phi_{0},\hat{\phi}}_{T}\right]=\mathbb{E}\left[-\Phi_{0}(S_{0}-Z)-\int_{0}^{T}\hat{\phi}_{t}(S_{t}-Z-N^{n}_{t})dt-\frac{\Lambda}{2}\int_{0}^{T}\hat{\phi}^{2}_{t}dt\right]$
$\displaystyle=\mathbb{E}\left[-\Phi_{0}(S_{0}-Z)+\frac{1}{2\Lambda}\int_{0}^{T-1/n}|S_{t}-Z-N^{n}_{t}|^{2}dt\right]$
$\displaystyle-\mathbb{E}\left[\int_{T-1/n}^{T}\hat{\phi}_{t}(S_{t}-Z-N^{n}_{t})dt+\frac{\Lambda}{2}\int_{T-1/n}^{T}\hat{\phi}^{2}_{t}dt\right].$

By taking $n\rightarrow\infty$ in the above inequality and applying (1.7) we
obtain

(1.11)
$\mathbb{E}\left[V^{\Phi_{0},\hat{\phi}}_{T}\right]=-\frac{\Phi^{2}_{0}\Lambda}{2T}+\Phi_{0}\mathbb{E}\left[\frac{M_{0}}{T}-S_{0}\right]+\frac{1}{2\Lambda}\mathbb{E}\left[\int_{0}^{T}\left(S_{t}-\frac{M_{0}}{T}-N_{t}\right)^{2}dt\right].$

By combining (1.10)–(1.11) we conclude (1.6).

Finally, the uniqueness of the optimal trading strategy follows from the
strict convexity of the map $\phi\rightarrow V^{\Phi_{0},\phi}_{T}$. ∎

##  2\. Optimal Liquidation of Fractional Brownian Motion.

Fractional Brownian motion $B^{H}=(B^{H}_{t})_{t=0}^{\infty}$ with Hurst
parameter $H\in(0,1)$, is a continuous, zero-mean Gaussian process such that

$cov\left(B^{H}_{t},B^{H}_{u}\right)=\frac{t^{2H}+u^{2H}-|t-u|^{2H}}{2},\ \
t,u\geq 0.$

The process $B^{H}$ is self similar $B^{H}_{at}\sim a^{H}B^{H}_{t}$ and have
stationary increments. Moreover, the successive increments of $B^{H}$ are
positively correlated for $H>1/2$, negatively correlated for $H<1/2$, while
$H=1/2$ recovers the usual Brownian motion with independent increments.

Fractional Brownian motion which displays the long-range dependence observed
in empirical date (see [6, 12, 14] and the references therein) is not a semi-
martingale when $H\neq\frac{1}{2}$ and so, in the frictionless case it leads
to arbitrage opportunities (see, for instance, [13, 5]). In the presence of
market price impact arbitrage opportunities disappear and the expected profits
are finite (see [9, 10]). In [10] the authors studied the asymptotic behaviour
as the maturity date goes to infinity, of the optimal liquidation problem for
fractional Brownian motion with temporary price impact.

In this section, for the financial model where the risky asset is given by a
fractional Brownian motion, we study the dependence of the optimal liquidation
problem as a function of the investor’s information. We deal with three types
of investors. The first one, is the ”usual” investor with information flow
which is given by the filtration generated by the risky asset. The second type
is an investor which receives the information with a delay. The last type is a
”frontrunner” which is able to peek some time units into the future. Of course
the ”frontrunner” cannot freely take advantage of her extra knowledge due to
the linear price impact which leads to quadratic transaction costs. For the
above three cases we solve the corresponding optimal liquidation problem and
derive numerical results for the value (see Figure 1) and for the optimal
strategy (see Figure 2).

Let $H\in(0,1)$ and consider the optimization problem (1.2) for the case where
the risky asset is of the form $S_{t}=S_{0}+\sigma B^{H}_{t}+\mu t$ where
$\sigma>0$ and $\mu\in\mathbb{R}$ are constants. From Theorem 1.1 and the
discussion afterwards it follows that (for simplicity) we can take
$\mu=S_{0}=0$ and $\sigma=\Lambda=1$. Thus, $S=B^{H}$ for some $H\in(0,1)$ and
$\Lambda=1$.

For $H\in(0,1)$ introduce the Volterra kernel

$Z_{H}(t,s)=c_{H}\left(\left(\frac{t}{s}\right)^{H-\frac{1}{2}}(t-s)^{H-\frac{1}{2}}-\left(H-\frac{1}{2}\right)s^{\frac{1}{2}-H}\int_{s}^{t}u^{H-\frac{3}{2}}(u-s)^{H-\frac{1}{2}}du\right),\
\ 0<s<t$

where
$c_{H}:=\left(\frac{2H\Gamma\left(\frac{3}{2}-H\right)}{\Gamma\left(H+\frac{1}{2}\right)\Gamma(2-2H)}\right)^{1/2}.$
Then, taking an ordinary Brownian motion $W=(W_{t})_{t=0}^{\infty}$ the
formula

(2.1) $B^{H}_{t}=\int_{0}^{t}Z_{H}(t,s)dW_{s},\ \ t\geq 0.$

defines a fractional Brownian motion with Hurst parameter $H$, which generates
the same filtration as $W$ (see [11]). Moreover, given $B^{H}$, the Wiener
process $W$ can be recovered by the relations
$W_{t}:=\frac{2H}{c_{H}}\int_{0}^{t}s^{H-\frac{1}{2}}d\mathcal{M}_{s}$, $t\geq
0$ where
$\mathcal{M}_{t}:=\frac{1}{2H\Gamma\left(\frac{3}{2}-H\right)\Gamma\left(H+\frac{1}{2}\right)}\int_{0}^{t}s^{\frac{1}{2}-H}(t-s)^{\frac{1}{2}-H}dB^{H}_{s}$,
$t\geq 0.$ Denote by $(\mathcal{F}^{W}_{t})_{t\geq 0}$ the augmented
filtration which is generated by $W$.

###  2.1. Standard Information

Consider the case where the filtration $(\mathcal{F}_{t})_{0\leq t\leq T}$
(which represent the investor’s flow of information) is equal to
$(\mathcal{F}^{W}_{t})_{0\leq t\leq T}$. From the Fubini theorem and (2.1) it
follows that the martingale defined in (1.4) is equal to

$M^{H}_{t}=\int_{0}^{t}\left(\int_{s}^{T}Z_{H}(u,s)du\right)dW_{s},\ \
t\in[0,T].$

Hence, (1.5) and (2.1) yield that the optimal strategy is given by

$\hat{\phi}^{H}_{t}:=\int_{0}^{t}\left(\frac{\left(\int_{s}^{T}Z_{H}(u,s)du\right)}{T-s}-Z_{H}(t,s)\right)dW_{s},\
\ t\in[0,T].$

From the Itô Isometry and (1.6) we obtain that the corresponding value is
given by

$\displaystyle\mathbb{E}\left[V^{0,\hat{\phi}^{H}}_{T}\right]=\int_{0}^{T}\int_{0}^{t}Z^{2}_{H}(t,s)dsdt-\int_{0}^{T}\frac{\left(\int_{s}^{T}Z_{H}(u,s)du\right)^{2}}{T-s}ds$
$\displaystyle=\frac{1}{2H+1}-\int_{0}^{T}\frac{\left(\int_{s}^{T}Z_{H}(u,s)du\right)^{2}}{T-s}ds.$

###  2.2. Delayed Information

We fix a positive number $\Delta\in(0,T]$ and consider a situation where the
risky asset $S$ is observed with a delay $\Delta>0$. Namely, the filtration is
$\mathcal{F}_{t}=\mathcal{F}^{W}_{(t-\Delta)^{+}}$, $t\in[0,T]$. In particular
the underlying process $S=B^{H}$ is no longer adapted to the above filtration.

Observe that the process

$\hat{S}_{t}:=\mathbb{E}\left[B^{H}_{t}|\mathcal{F}^{W}_{(t-\Delta)^{+}}\right]=\int_{0}^{(t-\Delta)^{+}}Z_{H}(t,s)dW_{s},\
\ t\in[0,T],$

is adapted to the filtration $\mathcal{F}^{W}_{(t-\Delta)^{+}}$, $t\in[0,T]$
and satisfies

$\mathbb{E}\left[\int_{0}^{T}\gamma_{t}B^{H}_{t}dt\right]=\mathbb{E}\left[\int_{0}^{T}\gamma_{t}\hat{S}_{t}dt\right]\
\ \forall\gamma\in L^{2}(dt\otimes\mathbb{P}).$

Hence, we can apply Theorem 1.1 for the process $\hat{S}$.

From the Fubini theorem

$\int_{0}^{T}\hat{S}_{t}dt=\int_{0}^{T-\Delta}\left(\int_{s+\Delta}^{T}Z_{H}(u,s)du\right)dW_{s}.$

Thus, the martingale $M$ defined in (1.4) is equal to

$M^{H,\Delta,-}_{t}=\int_{0}^{(t-\Delta)^{+}}\left(\int_{s+\Delta}^{T}Z_{H}(u,s)du\right)dW_{s},\
\ t\in[0,T]$

and so, the optimal strategy is given by

$\hat{\phi}^{H,\Delta,-}_{t}=\int_{0}^{(t-\Delta)^{+}}\left(\frac{\int_{s+\Delta}^{T}Z_{H}(u,s)du}{T-\Delta-s}-Z_{H}(t,s)\right)dW_{s},\
\ t\in[0,T].$

Finally, the corresponding value is given by

$\mathbb{E}\left[V^{0,\hat{\phi}^{H,\Delta,-}}_{T}\right]=\int_{0}^{T}\int_{0}^{(t-\Delta)^{+}}Z^{2}_{H}(t,s)dsdt-\int_{0}^{T-\Delta}\frac{\left(\int_{s+\Delta}^{T}Z_{H}\left(u,s\right)du\right)^{2}}{T-\Delta-s}ds.$

###  2.3. Insider Information

Rather than having access to just the natural augmented filtration
$(\mathcal{F}^{W}_{t})_{t\geq 0}$ for making decisions the investor can peek
$\Delta>0$ time units into the future, and so her information flow is given by
the filtration $(\mathcal{F}^{W}_{t+\Delta})_{t\geq 0}$.

The martingale $M$ defined in (1.4) is equal to

$M^{H,\Delta,+}_{t}=\int_{0}^{(t+\Delta)\wedge
T}\left(\int_{s}^{T}Z_{H}(u,s)du\right)dW_{s},\ \ t\in[0,T].$

Hence, the optimal strategy is given by

$\displaystyle\hat{\phi}^{H,\Delta,+}_{t}=\frac{1}{T}\int_{0}^{\Delta\wedge
T}\left(\int_{s}^{T}Z_{H}(u,s)du\right)dW_{s}$
$\displaystyle+\int_{\Delta}^{(t+\Delta)\wedge
T}\frac{\int_{s}^{T}Z_{H}(u,s)du}{T+\Delta-s}dW_{s}-\int_{0}^{t}Z_{H}(t,s)dW_{s},\
\ t\in[0,T]$

and the corresponding value is given by

$\displaystyle\mathbb{E}\left[V^{0,\hat{\phi}^{H,\Delta,+}}_{T}\right]=\int_{0}^{T}\int_{0}^{t}Z^{2}_{H}(t,s)dsdt-\frac{\left|M^{H,\Delta,+}_{0}\right|^{2}}{T}-\int_{\Delta\wedge
T}^{T}\frac{\left(\int_{s}^{T}Z_{H}(u,s)du\right)^{2}}{T+\Delta-s}ds$
$\displaystyle=\frac{1}{2H+1}-\frac{1}{T}\int_{0}^{\Delta\wedge
T}\left(\int_{s}^{T}Z_{H}(u,s)du\right)^{2}ds-\int_{\Delta\wedge
T}^{T}\frac{\left(\int_{s}^{T}Z_{H}(u,s)du\right)^{2}}{T+\Delta-s}ds.$

Figure 1. The value of the liquidation problem for different flows of
information (shown in different colors) as a function of the Hurst parameter
$H$. Observe that for delayed information the value function is no longer
decreasing for $H<0.5$. The reason is that for very low $H$ values the
correlation between the increments decays faster to $0$ with their time
distance, hence a delay results in almost complete loss of information
regarding the current price. Figure 2. In this figure we simulate a sample
path of a fractional Brownian motion with Hurst parameter $H=0.7$ and the
corresponding optimal trading strategies (we take maturity date $T=5$). We
observe that the Regular Information graph, is a “lagged version” of the
Insider Information graph and the Delayed Information graph is a “lagged
version” of the Regular Information graph. As expected, the information time-
gap is reflected geometrically as a shifting transformation along the
horizontal axis.

## Acknowledgements

Supported in part by the ISF grant 230/21.

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