solutionSolutionsolutionfile solutionfile[LargeSIMRateDouble]

# Achievable Rate Optimization for Large Stacked Intelligent Metasurfaces
Based on Statistical CSI

Anastasios Papazafeiropoulos, Pandelis Kourtessis, Symeon Chatzinotas, Dimitra
I. Kaklamani, Iakovos S. Venieris A. Papazafeiropoulos is with the
Communications and Intelligent Systems Research Group, University of
Hertfordshire, Hatfield AL10 9AB, U. K., and with SnT at the University of
Luxembourg, Luxembourg. P. Kourtessis is with the Communications and
Intelligent Systems Research Group, University of Hertfordshire, Hatfield AL10
9AB, U. K. S. Chatzinotas is with the SnT at the University of Luxembourg,
Luxembourg. Dimitra I. Kaklamani is with the Microwave and Fiber Optics
Laboratory, and Iakovos S. Venieris is with the Intelligent Communications and
Broadband Networks Laboratory, School of Electrical and Computer Engineering,
National Technical University of Athens, Zografou, 15780 Athens, Greece.
Corresponding author’s email<EMAIL_ADDRESS>
###### Abstract

Stacked intelligent metasurface (SIM) is an emerging design that consists of
multiple layers of metasurfaces. A SIM enables holographic multiple-input
multiple-output (HMIMO) precoding in the wave domain, which results in the
reduction of energy consumption and hardware cost. On the ground of multiuser
beamforming, this letter focuses on the downlink achievable rate and its
maximization. Contrary to previous works on multiuser SIM, we consider
statistical channel state information (CSI) as opposed to instantaneous CSI to
overcome challenges such as large overhead. Also, we examine the performance
of large surfaces. We apply an alternating optimization (AO) algorithm
regarding the phases of the SIM and the allocated transmit power. Simulations
illustrate the performance of the considered large SIM-assisted design as well
as the comparison between different CSI considerations.

###### Index Terms:

Reconfigurable intelligent surface (RIS), stacked intelligent metasurfaces
(SIM), 6G networks.

##  I Introduction

The technology of reconfigurable intelligent surfaces (RISs) has recently
emerged to increase coverage and enhance spectral and energy efficiencies in
various communication environments [1, 2]. In general terms, an RIS includes a
surface that includes a large number of elements, which are nearly passive and
have low cost. The purpose of these elements is to adjust the phases of the
incident electromagnetic (EM) waves by using a smart controller, and hence,
shape the propagation environment dynamically [3, 4, 5].

However, most existing works on RIS assume single-layer metasurface structure
[3, 4, 6], which imposes a constraint on the adjustment of the beam patterns.
Also, the single-layer structures of RISs do not have the capability of inter-
user interference suppression as shown in [6]. These observations led the
authors in [7, 8] to propose a stacked intelligent metasurface (SIM), which
consists of an array of programmable metasurfaces similar to artificial neural
networks (ANNs). Among the processing capabilities of a SIM, we that the
forward propagation takes place at the speed of light.

On this ground, in [7], authors proposed a SIM-based design for the
transceiver of point-to-point multiple-input multiple-output (MIMO)
communication systems, where the combining and the precoding take place as the
EM waves propagate along the SIM. In [8], we observe the integration of a SIM
to the transmitter, i.e., the base station (BS) towards enabling beamforming
in the EM domain based on instantaneous channel state information (CSI).
Contrary to [7] and [8], in [9] and [10], we proposed more general hybrid
digital wave designs, where all element parameters are optimised
simultaneously through more efficient algorithms.

In this work, we focus on a SIM-enabled multiuser architecture operating
solely in the wave domain. Note that [10] assumes a hybrid digital wave
design, and [11] focuses on satellite communication systems. Also, contrary to
previous works [7, 8, 9], we consider a SIM that consists of large
metasurfaces, since we apply the use-and-then-forget (UatF) bound [12]. Most
importantly, we obtain the downlink rate and perform its optimization
regarding the phase shifts and transmit power in terms of statistical CSI.
Notably, this approach enables the optimization at every several coherence
intervals rather than optimizing at each interval. Hence, we achieve lower
overhead, which is one of the main challenges in SIM-assisted systems.

Notation: Matrices and vectors are represented by boldface upper and lower
case symbols, respectively. The notations
$(\cdot)^{\scriptscriptstyle\mathsf{T}}$,
$(\cdot)^{\scriptscriptstyle\mathsf{H}}$, and $\tr\\!\left({\cdot}\right)$
denote the transpose, Hermitian transpose, and trace operators, respectively.
Also, the symbol $\mathbb{E}\left[\cdot\right]$ denotes the expectation
operator. The floor function $\lfloor x\rfloor$ gives as output the greatest
integer less than or equal to $x$. The notation
$\text{diag}\left(\right)\left({\mathbf{A}}\right)$ represents a vector with
elements equal to the diagonal elements of ${\mathbf{A}}$. The notation
${\mathbf{b}}\sim{\cal C}{\cal N}{({\mathbf{0}},\mathbf{\Sigma})}$ represents
a circularly symmetric complex Gaussian vector with zero mean and a covariance
matrix $\mathbf{\Sigma}$.

##  II System Model

We consider a SIM-aided MIMO communication system as depicted in Fig. 1. In
particular, a BS, which includes $N_{t}$ antennas, communicates with $K$
single-antenna user equipments (UEs) through a SIM performing wave-based
processing. The SIM is implemented by $L$ metasurfaces, where each one has a
large number of $N$ meta-atoms. Let $\mathcal{K}=\\{1,\ldots,K\\}$,
$\mathcal{L}=\\{1,\ldots,L\\}$, and $\mathcal{N}=\\{1,\ldots,N\\}$ denote the
sets of UEs, metasurfaces, and meta-atoms, respectively. Note that an
intelligent controller adjusts the shifts of the phases of the electromagnetic
(EM) waves that impinge on the metasurface layers.

Figure 1: A SIM-aided MIMO system.

On this basis, let $\theta_{n}^{l}\in[0,2\pi),n\in\mathcal{N},l\in\mathcal{L}$
be the phase shift by the $n$th meta-atom on the surface layer $l$. Also, we
denote $\phi_{n}^{l}=e^{j\theta_{n}^{l}}$, and
${\bm{\Phi}}_{l}=\text{diag}\left(\right)({\bm{\phi}}^{l})\in\mathbb{C}^{N\times
N}$, where
${\bm{\phi}}^{l}=[\phi^{l}_{1},\dots,\phi^{l}_{N}]^{{\scriptscriptstyle\mathsf{T}}}\in\mathbb{C}^{{\color[rgb]{0,0,0}N}\times
1}$.111Herein, we consider phase shifts, which are continuously-adjustable and
their modulus equals to $1$ to evaluate large SIM-aided MIMO communications.
Practical issues such as the consideration of discrete phase shifts [13] is
the topic of future work. In addition, ${\mathbf{W}}^{l}\in\mathbb{C}^{N\times
N},l\in\mathcal{L}/\\{1\\}$ denotes the coefficient matrix between layer
$(l-1)$ and layer $l$. In particular, its entries from meta-atom $\tilde{n}$
on layer $(l-1)$ to meta-atom $n$ on layer $l,\forall l\in\mathcal{L}$ are
given by

$\displaystyle
w_{n,\tilde{n}}^{l}=\frac{A_{t}cosx_{n,\tilde{n}}^{l}}{r_{n,\tilde{n}}^{l}}\left(\frac{1}{2\pi
r^{l}_{n,\tilde{n}}}-j\frac{1}{\lambda}\right)e^{j2\pi
r_{n,\tilde{n}}^{l}/\lambda},$ (1)

where $A_{t}$ is the area of each meta-atom at the SIM, $x_{n,\tilde{n}}^{l}$
denotes the angle between the normal direction of the transmit metasurface
layer $(l-1)$ and the propagation direction, $r_{n,\tilde{n}}^{l}$, is the
respective transmission distance. Moreover, let
${\mathbf{w}}^{1}_{k}\in\mathbb{C}^{N\times 1}$ express the coefficient from
the transmit antenna array. Thus, the impact of the SIM can be expressed as

$\displaystyle{\mathbf{G}}={\bm{\Phi}}_{L}{\mathbf{W}}^{L}\cdots{\bm{\Phi}}_{2}{\mathbf{W}}^{2}{\bm{\Phi}}_{1}\in\mathbb{C}^{N\times
N}.$ (2)

Let ${\mathbf{h}}_{k}\in\mathbb{C}^{N\times 1},\forall k\in\mathcal{K}$
express the channel between the last layer and UE $k$ that is described by the
correlated Rician fading distribution as

$\displaystyle{\mathbf{h}}_{k}=\sqrt{\beta_{k}}\left(\sqrt{\frac{\kappa_{k}}{1+\kappa_{k}}}{\mathbf{h}}_{k,\mathrm{LoS}}+\sqrt{\frac{1}{1+\kappa_{k}}}{\mathbf{h}}_{k,\mathrm{NLoS}}\right)~{}~{}\forall
k\in\mathcal{K}.$ (3)

In (3), $\kappa_{k}$ is the Rician factor, $\beta_{k}$ is the channel gain,
${\mathbf{h}}_{k,\mathrm{NLoS}}\in\mathbb{C}^{N\times 1}$ is the LoS
component, and
${\mathbf{h}}_{k,\mathrm{NLoS}}\sim\mathcal{CN}({\mathbf{0}},{\mathbf{R}})\in\mathbb{C}^{N\times
1}$ is the NLoS component with ${\mathbf{R}}\in\mathbb{C}^{N\times N}$
representing the spatial correlation of each surface. This correlation is
obtained $\forall n\in\mathcal{N},\tilde{n}\in\mathcal{N}$ as [14]

$\displaystyle[{\mathbf{R}}_{\mathrm{SIM}}]_{\tilde{n},n}$
$\displaystyle=\mathrm{sinc}(2\|{\mathbf{u}}_{n}-{\mathbf{u}}_{\tilde{n}}\|/\lambda),n,\tilde{n}=1,\ldots,N$
(4)

where
${\mathbf{u}}_{n}=[0,i(n)d_{\mathrm{H}},j(n)d_{\mathrm{V}}]^{{\scriptscriptstyle\mathsf{T}}}$
with $i(n)=\mod(n-1,N_{x})$ and $j(n)=\lfloor(n-1)/N_{x}\rfloor$ being the
horizontal and vertical indices of element $n$, respectively. $N_{x}$ and
$N_{y}$ are the elements per row and column, while $d_{\mathrm{H}}$ and
$d_{\mathrm{V}}$ denote the horizontal width and the vertical height.

##  III Downlink Data Transmission

During the downlink transmission and based on wave-based beamforming [8], the
received signal at the $k$-th UE is written as

$\displaystyle
y_{k}={\mathbf{h}}_{k}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}\sum_{i=1}^{K}{\mathbf{w}}_{i}^{1}\sqrt{p}_{i}x_{i}+n_{k},~{}~{}~{}\forall
k\in\mathcal{K}$ (5)

where $x_{i}$ is the information symbol intended for the $k$-th UE, which has
a zero mean and unit variance. Also, $p_{i}$ is the power corresponding to the
$k$-th UE with $\sum_{i=1}^{K}p_{i}\leq P_{\mathrm{T}}$, where
$P_{\mathrm{T}}$ is the total transmit power at the BS. Also
$n_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})$ denotes the additive white Gaussian
noise (AWGN) with $\sigma_{k}^{2}$ expressing its variance at UE $k$.

The downlink achievable SE of UE $k$ is given by

$\displaystyle\mathrm{SE}=\sum_{k=1}^{K}\log_{2}\left(1+\gamma_{k}\right)\\!,$
(6)

where $\gamma_{k}$ denotes the downlink signal-to-interference-plus-noise
ratio (SINR), which is written according to the UaTF bounding technique [12]
as

$\displaystyle\gamma_{k}=\frac{p_{k}|\mathbb{E}\\{{\mathbf{h}}_{k}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}\\}|^{2}}{\sum_{i=1}^{K}p_{i}\mathbb{E}\\{|{\mathbf{h}}_{k}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{i}^{1}|^{2}\\}-p_{k}|\mathbb{E}\\{{\mathbf{h}}_{k}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}\\}|^{2}+\sigma_{k}^{2}},$
(7)

where is assumed that UE $k$ has knowledge of the average effective channel.

###### Proposition 1

The achievable SINR of UE $k$ for a given SIM during the downlink transmission
is provided by (8).

$\displaystyle\gamma_{k}=\frac{p_{k}\kappa_{k}|{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}|^{2}}{\sum_{i=1}^{K}p_{i}\tr({\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{R}})+\sum_{i\neq
k}^{K}p_{i}\kappa_{k}{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{h}}_{k,\mathrm{LoS}}+\frac{\sigma_{k}^{2}(1+\kappa_{k})}{\beta_{k}}}.$
(8)

###### Proof:

The numerator becomes

$\displaystyle|\mathbb{E}\\{{\mathbf{h}}_{k}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}\\}|^{2}=\beta_{k}\frac{\kappa_{k}}{1+\kappa_{k}}|{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}|^{2}.$
(9)

Regarding the denominator of (7), the first term is written as

$\displaystyle\mathbb{E}\\{|{\mathbf{h}}_{k}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{i}^{1}|^{2}\\}=\tr({\mathbf{h}}_{k}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{h}}_{k})$
(10)
$\displaystyle=\beta_{k}\frac{1}{1+\kappa_{k}}\tr({\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{R}})$
$\displaystyle+\beta_{k}\frac{\kappa_{k}}{1+\kappa_{k}}{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{h}}_{k,\mathrm{LoS}},$
(11)

where, in (10), we have applied that
${\mathbf{x}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{y}}=\tr({\mathbf{y}}{\mathbf{x}}^{{\scriptscriptstyle\mathsf{H}}})$
for any vectors ${\mathbf{x}}$, ${\mathbf{y}}$. By substituting (9) and (11)
in (7), we obtain the achievable SINR. ∎

##  IV Problem Formulation and Optimization

The maximization of the sum SE regarding the phase shifts of each surface and
the allocated power is of great importance.

###  IV-A Problem Formulation

The maximization problem is formulated as

$\displaystyle(\mathcal{P})~{}~{}$
$\displaystyle\max_{{\bm{\phi}}_{l},{\mathbf{p}}}\;f({\bm{\phi}}_{l},{\mathbf{p}})=\sum_{k=1}^{K}\log_{2}\left(1+\frac{D_{k}}{I_{k}}\right)$
(12a)
$\displaystyle~{}\mathrm{s.t}~{}~{}~{}{\mathbf{G}}={\bm{\Phi}}_{L}{\mathbf{W}}^{L}\cdots{\bm{\Phi}}_{2}{\mathbf{W}}^{2}{\bm{\Phi}}_{1},$
(12b)
$\displaystyle\;\quad\;\;\;\;\;\\!\\!~{}\\!|\phi^{l}_{n}|=1,n\in\mathcal{N},l\in\mathcal{L},$
(12c)
$\displaystyle\;\quad\;\;\;\;\;\\!\\!~{}\\!\sum_{i=1}^{K}p_{i}=P_{\mathrm{T}}$
(12d) $\displaystyle\;\quad\;\;\;\;\;\\!\\!~{}\\!p_{k}\geq 0,\forall
k\in\mathcal{K},$ (12e)

where $D_{k}$ and $I_{k}$ are the numerator and denominator of $\gamma_{k}$
obtained in Proposition 1. Also, we have defined the vector
${\mathbf{p}}=[p_{1},\ldots,p_{K}]^{{\scriptscriptstyle\mathsf{T}}}$. Note
that the constraint (12c) expresses that each RIS element provides only a
phase shift while (12d) corresponds to the maximum power constraint.

The non-convexity optimization problem $(\mathcal{P})$ and its dependence on
the unit-modulus constraint with respect to ${\bm{\phi}}_{l}$ make the
solution challenging. For this reason, we resort to alternating optimization
(AO). According to this technique, ${\bm{\phi}}_{l}$ and ${\mathbf{p}}$ will
be optimized individually in an iterative manner. Specifically, first, we find
the optimum ${\bm{\phi}}_{l}$ for a fixed ${\mathbf{p}}$. During the next
step, we solve for ${\mathbf{p}}$ with ${\bm{\phi}}_{l}$ fixed. The objective
converges to its optimum value by iterating this process, which leads to the
increase of $f({\bm{\phi}}_{l},{\mathbf{p}})$ after each step until a specific
point because of the upper-bound coming from the power constraint (12d).

###  IV-B SIM Optimization

Until now, ${\bm{\phi}}_{l}$ was assumed fixed. However, to exploit each
metasurface towards wave-based beamforming while maximizing (6), the
optimization of each ${\bm{\phi}}_{l}$ has to take place. Its presence is
observed inside the matrix ${\mathbf{G}}$, appearing in $D_{k}$ and $I_{k}$.
Hence, the maximization problem regarding ${\bm{\phi}}_{l}$ is described as

$\displaystyle(\mathcal{P}1)~{}~{}$
$\displaystyle\max_{{\bm{\phi}}_{l}}\;f({\bm{\phi}}_{l})$ (12ma)
$\displaystyle~{}\mathrm{s.t}~{}~{}~{}{\mathbf{G}}={\bm{\Phi}}_{L}{\mathbf{W}}^{L}\cdots{\bm{\Phi}}_{2}{\mathbf{W}}^{2}{\bm{\Phi}}_{1},$
(12mb)
$\displaystyle\;\quad\;\;\;\;\;\\!\\!~{}\\!|\phi^{l}_{n}|=1,n\in\mathcal{N},l\in\mathcal{L},$
(12mc)

where the maximization problem $(\mathcal{P}1)$ is non-convex regarding
${\bm{\phi}}_{l}$, and it obeys to a unit-modulus constraint with respect to
$\phi^{l}_{n}$. Application of the projected gradient ascent algorithm until
convergence while taking into account the unit-modulus constraint results in a
locally optimal solution to $(\mathcal{P}1)$.

The proposed algorithm suggests starting from ${\bm{\phi}}_{l}^{0}$, and then
shifting along the gradient of $f({\bm{\phi}}_{l})$. The new point
${\bm{\phi}}_{l}$ is projected onto $\Phi_{l}$ to hold the new points in the
feasible set. Specifically, the unit-modulus constraint means that
$\phi^{l}_{n}$ has to be found inside the unit circle. $P_{\Phi_{l}}(\cdot)$
is the projection onto $\Phi_{l}$. Hence, we have

$\displaystyle\bar{u}_{l,n}=\left\\{\begin{array}[]{ll}\frac{u_{l,n}}{|u_{l,n}|}&u_{l,n}\neq
0\\\ e^{j\phi^{l}_{n}},\phi^{l}_{n}\in[0,2\pi]&u_{l,n}=0\\\
\end{array},n=1,\ldots,N,\right.$ (12mp)

where the vector $\bar{{\mathbf{u}}}_{l}$ of $P_{\Phi_{l}}({\mathbf{u}}_{l})$
is a given point.

The algorithm is described by the following iteration

$\displaystyle{\bm{\phi}}_{l}^{i+1}$
$\displaystyle=P_{\Phi_{l}}({\bm{\phi}}_{l}^{i}+\mu_{i}\nabla_{{\bm{\phi}}_{l}}f({\bm{\phi}}_{l}^{i})).$
(12mq)

The Armijo-Goldstein backtracking line search method provides the step size,
which is $\mu_{i}=L_{i}\kappa^{m_{i}}$, where $\kappa\in(0,1)$ and $L_{i}>0$.
Note that $m_{i}$ is the smallest positive integer that satisfies

$\displaystyle f({\bm{\phi}}_{l}^{i+1})\geq
Q_{L_{i}\kappa^{m_{i}}}({\bm{\phi}}_{l}^{i};{\bm{\phi}}_{l}^{i+1}),$ (12mr)

where

$\displaystyle\\!\\!Q_{\mu}({\bm{\phi}}_{l};{\mathbf{x}})\\!=\\!f({\bm{\phi}}_{l})\\!+\\!\langle\nabla_{{\bm{\phi}}_{l}}f({\bm{\phi}}_{l}),{\mathbf{x}}\\!-\\!{\bm{\phi}}_{l}\rangle\\!-\\!\frac{1}{\mu}\|{\mathbf{x}}-{\bm{\phi}}_{l}\|^{2}_{2}$
(12ms)

is the quadratic approximation of $f({\bm{\phi}}_{l})$.

###### Proposition 2

The gradient of $f({\bm{\phi}}_{l})$ regarding ${\bm{\phi}}_{l}^{*}$ is
obtained in closed-form as

$\displaystyle\nabla_{{\bm{\phi}}_{l}}f({\bm{\phi}}_{l})$
$\displaystyle=\frac{1}{\log_{2}(e)}\sum_{k=1}^{K}\frac{{I}_{k}\nabla_{{\bm{\phi}}_{l}}D_{k}-D_{k}\nabla_{{\bm{\phi}}_{l}}{I}_{k}}{(1+\gamma_{k}){I}_{k}},$
(12mt)

where

with
${\mathbf{A}}_{l}={\bm{\Phi}}_{L}{\mathbf{W}}^{L}\cdots{\bm{\Phi}}_{l+1}{\mathbf{W}}^{l+1}$,
and
${\mathbf{C}}_{l}={\mathbf{W}}^{l}{\bm{\Phi}}_{l-1}{\mathbf{W}}^{l-1}\cdots{\bm{\Phi}}_{1}$.

###### Proof:

Please see Appendix A. ∎

The SIM optimization design, based on the gradient ascent, appears a
significant advantage because the gradient ascent is obtained in a closed
form. It has low computational complexity because it consists of simple matrix
operations. Specifically, the complexity of (12ma) for large SIMs is
$\mathcal{O}(N_{t}N^{2}+LN^{2}+KN^{3})$, and the complexity of (12mt) is
similar. In other words, the number of meta-atoms of each surface has a higher
impact.

###  IV-C Power Optimization

Given a fixed ${\bm{\Phi}}_{l}$, we focus on the optimization with respect to
${\mathbf{p}}$. Specifically, we have

$\displaystyle(\mathcal{P}2)~{}~{}$
$\displaystyle\max_{{\mathbf{p}}}\;f({\mathbf{p}})$ (12mua)
$\displaystyle\;\quad\;\;\;\;\;\\!\\!~{}\\!\sum_{i=1}^{K}p_{i}=P_{\mathrm{T}},~{}p_{k}\geq
0,\forall k\in\mathcal{K}.$ (12mub)

The nonconvexity of problem $(\mathcal{P}2)$ leads us to obtain a solution
which is locally optimal. For this reason, we apply a weighted minimum mean
square error (MMSE) reformulation of the sum SE. To this end, we denote
${\mathbf{c}}\\!=\\![c_{1},\ldots,c_{K}]^{{\scriptscriptstyle\mathsf{T}}}$.
Then, the SINR $\gamma_{k}$ can be written in terms of the vector
${\mathbf{p}}$ as

$\displaystyle\gamma_{k}=\frac{p_{k}q_{k}}{{\mathbf{c}}^{{\scriptscriptstyle\mathsf{T}}}{\mathbf{p}}+u_{k}^{2}},$
(12mv)

where

$\displaystyle
q_{k}=\kappa_{k}|{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}|^{2},c_{k}=\beta_{k}\frac{1}{1+\kappa_{k}}\tr({\mathbf{G}}{\mathbf{w}}_{k}^{1}{\mathbf{w}}_{k}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{R}}),$
$\displaystyle
t_{k}^{2}=\frac{\sigma_{k}^{2}(1+\kappa_{k})}{\beta_{k}},c_{i}=\beta_{k}\frac{1}{1+\kappa_{k}}\tr({\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{R}})$
$\displaystyle+\beta_{k}\frac{\kappa_{k}}{1+\kappa_{k}}{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{h}}_{k,\mathrm{LoS}},~{}\forall
i\neq k.$ (12mw)

Now, we consider the single-input and single-output (SISO) channel model that
comes from the SINR in (12mv), which is given by

$\displaystyle\tilde{y}_{k}=\sqrt{p_{k}q_{k}}s_{k}+\sum_{i=1}^{K}\sqrt{p_{i}c_{i}}s_{i}+n_{k},$
(12mx)

where $n_{k}\sim{\cal C}{\cal N}\left(0,u_{k}^{2}\right)$ while
$s_{i}\in\mathbb{C}$ is the data signal with unit variance, and
$\tilde{y}_{k}$ is the received signal.

Then, the receiver estimates $s_{k}$, i.e.,
$\hat{s}_{k}=v_{k}^{*}\tilde{y}_{k}$ with $v_{k}$ being a receiver
coefficient. The corresponding mean square error
$e_{k}({\mathbf{p}},v_{k})=[|\hat{s}_{k}-s_{k}|^{2}]$ becomes

$\displaystyle
e_{k}({\mathbf{p}},v_{k})=v_{kg}^{2}\left(p_{k}q_{k}+{\mathbf{c}}_{k}^{{\scriptscriptstyle\mathsf{T}}}{\mathbf{p}}+u_{k}^{2}\right)-2v_{k}\sqrt{q_{k}p_{k}}+1.$
(12my)

For a given ${\mathbf{p}}$, $v_{k}$ is obtained by the minimization of the MSE
as

$\displaystyle
v_{k}=\frac{\sqrt{p_{k}q_{k}}}{p_{k}q_{k}+\sum_{i=1}^{K}p_{i}c_{i}+u_{k}^{2}}.$
(12mz)

Inserting $v_{k}$ into (12my), $e_{k}$ becomes $1/\left(1+\gamma_{k}\right)$.
Based on the weighted MMSE method, let the auxiliary weight $d_{k}\geq 0$ for
the MSE $e_{k}$ and consider the problem

$\displaystyle\begin{split}(\mathcal{P}2.1)\min_{\begin{subarray}{c}{\mathbf{p}}\geq
0,\\\ \\{v_{k},d_{k}\geq
0:k=1,\ldots,K\\}\end{subarray}}&{K}\sum_{k=1}^{K}d_{k}e_{k}({\mathbf{p}},{\mathbf{v}}_{k})-\ln(d_{k})\\\
\mathrm{s.t}~{}~{}\;\\!&\sum_{i=1}^{K}p_{i}\leq P_{\mathrm{T}}.\end{split}$
(12maa)

Problems $(\mathcal{P}2)$ and $(\mathcal{P}2.1)$ are equivalent, and thus, are
subject to the same solution. The solution of $(\mathcal{P}2.1)$ can be
provided in closed form as

$\displaystyle
p_{i}=\min\left(P_{\mathrm{T}},\frac{q_{k}d_{k}^{2}v_{k}^{2}}{\left(q_{k}d_{k}v_{k}^{2}+\sum_{i=1}^{K}d_{i}v_{i}^{2}c_{i}\right)^{2}}\right).$
(12mab)

The power allocation presents a similar complexity to the SIM optimization
design since it consists of similar matrix operations to $(\mathcal{P}1)$,
i.e., its complexity is $\mathcal{O}(N_{t}N^{2}+LN^{2}+KN^{3})$.

###### Remark 1

Both algorithms, corresponding to Problems $(\mathcal{P}1)$ and
$(\mathcal{P}2)$, have low computation complexity and converge quickly. Note
that the achievement of a local optimum, obtained from this optimization, will
make different initializations result in different solutions, which will be
studied below.

##  V Numerical Results

In this section, we present and evaluate the performance of the achievable sum
SE of large SIM-assisted multiuser communications with statistical CSI by
showing both analytical results and Monte Carlo simulations. For the setup, we
assume that the SIM is parallel to the $x-y$ plane and centered along the
$z-$axis at a height $H_{\mathrm{BS}}=10~{}\mathrm{m}$. The spacing between
adjacent meta-atoms is assumed to be $\lambda/2$, and the size of each meta-
atom is $\lambda/2\times\lambda/2$. The thickness of the SIM is
$T_{\mathrm{SIM}}=5\lambda$, while the spacing is
$d_{\mathrm{SIM}}=T_{\mathrm{SIM}}/L$. Moreover, the locations of the users
are randomly distributed at a distance between $60\mathrm{m}$ and
$80\mathrm{m}$.

The distance between the $\tilde{n}-$th meta-atom of the $(l-1)-$st
metasurface and the ${n}-$th meta-atom of the $l-$st metasurface is given by
$d_{n,\tilde{n}}^{l}=\sqrt{d_{\mathrm{SIM}}^{2}+d_{n,\tilde{n}}^{2}}$, where

$\displaystyle\\!\\!d_{n,\tilde{n}}\\!=\\!\frac{\lambda}{2}\sqrt{\lfloor|n-\tilde{n}|/N_{x}\rfloor^{2}\\!+\\![\mathrm{mod}(|n-\tilde{n}|,N_{x})]^{2}}.$
(12mac)

The transmission distance between the $m$-th antenna and the $\tilde{n}$-th
meta-atom on the first metasurface layer is provided by (12mad). Note that we
have $\cos x_{n,\tilde{n}}^{l}=d_{\mathrm{SIM}}/d_{n,\tilde{n}}^{l},\forall
l\in\mathcal{L}$.

$\displaystyle{\small
d_{\tilde{n},m}^{1}\\!=\\!\sqrt{\\!d_{\mathrm{SIM}}^{2}\\!+\\!\Big{[}\\!\Big{(}\\!\mathrm{mod}(\tilde{n}\\!-\\!1,N_{x})\\!-\\!\frac{N_{x}\\!-\\!1}{2}\\!\Big{)}\frac{\lambda}{2}\\!-\\!\Big{(}m\\!-\\!\frac{N_{t}\\!+\\!1}{2}\Big{)}\frac{\lambda}{2}\Big{]}^{2}\\!+\\!\Big{(}\\!\lceil\tilde{n}/N_{x}\rceil\\!-\\!\frac{N_{y}\\!+\\!1}{2}\Big{)}^{2}\frac{\lambda_{2}}{4}}}.$
(12mad)

The path loss is given by

$\displaystyle\tilde{\beta}_{k}=C_{0}(d_{k}/\hat{d})^{-\alpha},$ (12mae)

where $C_{0}=(\lambda_{2}/4\pi\hat{d})$ is the free space path loss at the
reference distance of $\hat{d}=1~{}\mathrm{m}$, and $\alpha=2.5$ is the path-
loss exponent. The correlation matrix ${\mathbf{R}}_{\mathrm{SIM}}$ is
obtained according to (4). The carrier frequency and the system bandwidth are
$2~{}\mathrm{GHz}$ and $20~{}\mathrm{MHz}$, respectively. Moreover, we assume
$N_{t}=8$, $K=8$, $N=200$, and $L=4$.

In Fig. 2, we depict the achievable sum SE versus the number of elements $N$
of each surface while varying the number of surfaces $L$. First, it is shown
that the downlink sum SE increases with $N$ for different $L$. Moreover, an
increase in the number of surfaces results in an increase in the sum SE. In
addition, for the sake of comparison, we present the performance in the case
of instantaneous CSI for $L=4$ [8], which performs better than the case of
statistical CSI since the latter is obtained based on a lower bound that is
optimized at every several coherence intervals instead of at each coherence
interval. However, the statistical CSI modeling allows to save significant
overhead. Moreover, we show the effect of the size of each surface element. We
observe that as the size of each surface element decreases, the correlation
decreases, and the sum SE increases.  Notably, Monte Carlo (MC) simulations
corroborate the analytical results.

Figure 2: Achievable sum SE of the large SIM-aided MIMO architecture with
respect to the number of meta-atoms $N$.

Fig. 3 shows the sum SE versus the number of layers $L$ of the SIM for the
number of UEs $K$. When $N_{t}=K=8$, we observe that the sum SE improves until
$L=6$ because the SIM is able to mitigate the inter-user interference in the
EM wave domain. In particular, a significant improvement is observed compared
to the single-layer SIM. Again, we illustrate the comparison between the cases
corresponding to instantaneous and statistical CSI, where the latter exhibits
worse performance for the benefit of lower overhead.

Figure 3: Achievable sum SE of the large SIM-aided MIMO architecture with
respect to the number of meta-surfaces $L$.

In Fig. 4, we depict the convergence of the proposed algorithm. The algorithm
terminates when the difference of the objective between the two last
iterations is less than $10^{-5}$ or the number of iterations is larger than
$130$. It is shown that the algorithm converges to its maximum as the number
of iterations increases. Since Problem $(\mathcal{P})$ is non-convex, the
algorithm does not converge to a globally optimal solution. This means that
the algorithm may converge to different points starting from different initial
points. For this reason, we select the best solutions after executing the
algorithm from different initial points. Herein, we depict the sum SE versus
the iteration count for $5$ different randomly generated initial points, and
we observe that all points result in the same SE. Generally, the selection of
$5$ randomly generated initial points allows a good trade-off between
performance and complexity.

Figure 4: Achievable sum SE of the large SIM-aided MIMO architecture with
respect to the number of iterations for $5$ different randomly generated
initial points.

##  VI Conclusion

This paper provided the achievable downlink SE of large SIM-aided multiuser
communications over Ricean fading channels. In particular, we deduced a new,
tractable expression for the downlink SE for large-metasurfaces as a function
of large-scale statistics while the downlink precoding takes place in the wave
domain, in order to lower the computational burden and the processing latency.
The results were used to pursue an optimization based on statistical CSI that
achieves lower overhead with respect to instantaneous CSI. Specifically, we
proposed an AO algorithm that solves the optimization regarding the phase
shifts of each surface of the SIM and the allocated power, which contributes
to a reduction in processing latency due to lower overhead.

##  Appendix A Proof of Proposition 2

The proof starts with the derivation of $\nabla_{{\bm{\phi}}_{l}}D_{k}$. To
this end, we focus on the differential of
$|{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}|^{2}$.
We have

$\displaystyle
d(|{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}|^{2})={\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}d({\mathbf{G}}){\mathbf{w}}_{k}^{1}{\mathbf{w}}_{k}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{h}}_{k,\mathrm{LoS}}$
$\displaystyle+{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}{\mathbf{w}}_{k}^{1^{{\scriptscriptstyle\mathsf{H}}}}d({\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}){\mathbf{h}}_{k,\mathrm{LoS}}$
(12maf)
$\displaystyle=\tr\big({\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{A}}_{l}d({\bm{\Phi}}_{l}){\mathbf{C}}_{l}{\mathbf{w}}_{k}^{1}{\mathbf{w}}_{k}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{h}}_{k,\mathrm{LoS}}\big{missing})$
$\displaystyle+\tr({\mathbf{A}}_{l}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{h}}_{k,\mathrm{LoS}}{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{G}}{\mathbf{w}}_{k}^{1}{\mathbf{w}}_{k}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{C}}_{l}^{{\scriptscriptstyle\mathsf{H}}}d({\bm{\Phi}}_{l}^{{\scriptscriptstyle\mathsf{H}}})).$
(12mag)

In (12mag), we have substituted
$d({\mathbf{G}})={\mathbf{A}}_{l}d({\bm{\Phi}}_{l}){\mathbf{C}}_{l}$, where
${\mathbf{A}}_{l}={\bm{\Phi}}_{L}{\mathbf{W}}^{L}\cdots{\bm{\Phi}}_{l+1}{\mathbf{W}}^{l+1}$,
and
${\mathbf{C}}_{l}={\mathbf{W}}^{l}{\bm{\Phi}}_{l-1}{\mathbf{W}}^{l-1}\cdots{\bm{\Phi}}_{1}$.
Having obtained the differential, we have

$\displaystyle\nabla_{{\bm{\phi}}_{l}}D_{k}$
$\displaystyle=\frac{\partial}{\partial{\bm{\phi}}_{l}^{*}}D_{k}$ (12mah)
$\displaystyle=p_{k}\kappa_{k}\text{diag}\left(\right)({\mathbf{C}}_{l}^{*}{\mathbf{w}}_{k}^{1^{*}}{\mathbf{w}}_{k}^{1^{{\scriptscriptstyle\mathsf{T}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{T}}}{\mathbf{h}}_{k,\mathrm{LoS}}^{*}{\mathbf{h}}_{k,\mathrm{LoS}}^{{\scriptscriptstyle\mathsf{T}}}{\mathbf{A}}_{l}^{*})$

The first term in the denominator is written as

$\displaystyle
d\big{(}\tr\big({\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{R}}\big{missing})\big{)}=\tr\big({\mathbf{A}}_{l}d({\bm{\Phi}}_{l}){\mathbf{C}}_{l}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{R}}\big{missing})$
$\displaystyle+\tr\big({\mathbf{A}}_{l}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{R}}{\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{C}}_{l}^{{\scriptscriptstyle\mathsf{H}}}d({\bm{\Phi}}_{l}^{{\scriptscriptstyle\mathsf{H}}})\big{missing}).$
(12mai)

Thus, we have

$\displaystyle\\!\nabla_{{\bm{\phi}}_{l}}\\!\tr\big({\mathbf{G}}{\mathbf{w}}_{i}^{1}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{H}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{H}}}{\mathbf{R}}\big{missing})\\!=\\!\text{diag}\left(\right)({\mathbf{C}}_{l}^{*}{\mathbf{w}}_{i}^{1^{*}}{\mathbf{w}}_{i}^{1^{{\scriptscriptstyle\mathsf{T}}}}{\mathbf{G}}^{{\scriptscriptstyle\mathsf{T}}}{\mathbf{R}}{\mathbf{A}}_{l}^{*})$

The second term in the denominator is similar to the numerator. Hence, the
derivation is similar.

## References

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solutionfile


$\displaystyle=1_{A}(\omega)\sup\Big{\\{}x\in\mathbb{R}_{+}\;|\;\int_{0}^{\infty}\psi_{x}(\mathbb{P}(-1_{A}X>y\;|\;\mathscr{F}_{t})(\omega))\operatorname{d\\!}y$
$\displaystyle\qquad+\int^{0}_{-\infty}[\psi_{x}(\mathbb{P}(-1_{A}X>y\;|\;\mathscr{F}_{t})(\omega))-1]\operatorname{d\\!}y\leq
0\Big{\\}}$ $\displaystyle=1_{A}(\omega)\alpha^{\Psi}_{t}(1_{A}X)(\omega).$

Hence locality of $\alpha_{t}^{\Psi}$ is established.

Quasi-concavity. For any $t\in\mathcal{T}$, $X,Y\in L^{\infty}$ and
$\omega\in\Omega\backslash N^{X}$. Let $m>0$ be some
$\mathscr{F}_{t}$-measurable random variable such that
$\alpha^{\Psi}_{t}(X)(\omega)\geq m(\omega)$ and
$\alpha^{\Psi}_{t}\left(Y\right)(\omega)\geq m(\omega)$.
Fix one such $\omega$ and denote $m^{\star}=m(\omega)$. By (B.35), we have
that $\alpha^{\Psi}_{t}(X)(\omega)\geq m^{\star}$ and
$\alpha^{\Psi}_{t}\left(Y\right)(\omega)\geq m^{\star}$ imply that

$\rho_{t}^{\psi_{m(\omega)}}(X)(\omega)=\rho_{t}^{\psi_{m^{\star}}}(X)(\omega)\leq
0,$
$\rho_{t}^{\psi_{m(\omega)}}(Y)(\omega)=\rho_{t}^{\psi_{m^{\star}}}(Y)(\omega)\leq
0,$

respectively. Since sublinearity implies convexity, then by sublinearity of
$\rho^{\psi}_{t}$,

$\displaystyle\rho_{t}^{\psi_{m^{\star}}}\left(\lambda
X+(1-\lambda)Y\right)(\omega)\leq\lambda\rho_{t}^{\psi_{m^{\star}}}(X)(\omega)+(1-\lambda)\rho_{t}^{\psi_{m^{\star}}}(Y)(\omega)\leq
0.$ (B.36)

By (B.35) again, we show (B.36) is equivalent to
$\alpha^{\Psi}_{t}\left(\lambda X+(1-\lambda)Y\right)(\omega)\geq m^{\star}$.
Therefore, quasi-concavity holds.

Monotonicity. For any $t\in\mathcal{T}$, $X,Y\in L^{\infty}$ with $X\leq Y$
and $\omega\in\Omega\backslash N^{X}$. For any $\mathscr{F}_{t}$-measurable
random variable $m>0$, such that $\alpha_{t}^{\Psi}(X)(\omega)>m(\omega)$, we
have $\rho_{t}^{\psi_{m(\omega)}}(X)(\omega)\leq 0$. By monotonicity of
$\rho^{\psi}_{t}$,
$\rho_{t}^{\psi_{m(\omega)}}(Y)(\omega)\leq\rho_{t}^{\psi_{m(\omega)}}(X)(\omega)\leq
0$. Then, $\alpha_{t}^{\Psi}(Y)(\omega)\geq m(\omega)$, and hence
$\alpha_{t}^{\Psi}(Y)(\omega)\geq\alpha_{t}^{\Psi}(X)(\omega)$.

Scale Invariance. For any $t\in\mathcal{T}$, $X\in L^{\infty}$, $\lambda\in
L^{\infty}_{t}$ with $\lambda>0$, and $\omega\in\Omega\backslash N^{X}$, we
have that,

$\alpha_{t}^{\Psi}(\lambda
X)(\omega)=\sup\left\\{x\in\mathbb{R}_{+}\;|\;\rho_{t}^{\psi_{x}}(\lambda
X)(\omega)\leq 0\right\\}.$

In view of the positive homogeneity of $\rho^{\psi}_{t}$,

$\alpha_{t}^{\Psi}(\lambda
X)(\omega)=\sup\left\\{x\in\mathbb{R}_{+}\;|\;\lambda\rho_{t}^{\psi_{x}}(X)(\omega)\leq
0\right\\}.$

Since $\lambda>0$, then $\lambda\rho_{t}^{\psi_{x}}(X)\leq 0$ is equivalent to
$\rho_{t}^{\psi_{x}}(X)\leq 0$. Hence, $\alpha_{t}^{\Psi}(\lambda
X)=\alpha_{t}^{\Psi}(X)$.

The proof is complete.

∎

###  B.3 On conditional quantiles

The conditional versions of qualtiles are defined naturally using
‘probabilistic conditioning’ of the corresponding regular (or static) notions.
Same hold true for most properties. While morally this is true, the
difficulties are hidden in technical details related to measurably and well-
definiteness of these objects. Here we present some results related to that.
We also refer to [BCF18] for some relevant discussion on this topic.

######  Definition B.13.

For any $\alpha\in(0,1)$, the conditional upper and lower $\alpha$-quantile of
$X\in L^{\infty}$ with respect to $\sigma$-field $\mathscr{F}_{t}$ are defined
as,

$\displaystyle
q_{\alpha}^{+}(X\;|\;\mathscr{F}_{t})=\operatorname*{ess\,sup}\\{m\in
L^{\infty}_{t}\;|\;\mathbb{P}(X\leq m\;|\;\mathscr{F}_{t})\leq\alpha\\},$
$\displaystyle
q_{\alpha}^{-}(X\;|\;\mathscr{F}_{t})=\operatorname*{ess\,inf}\\{m\in
L^{\infty}_{t}\;|\;\mathbb{P}(X\leq m\;|\;\mathscr{F}_{t})\geq\alpha\\}.$

Below wWe will give other representations of the conditional quantile
functions, and we start with an auxiliary result.

######  Lemma B.14.

For any $X\in L^{\infty},\ t\in\mathcal{T},\ a\in L^{\infty}_{t}$, and for
fixed $\omega^{\prime}\in\Omega$,

$\displaystyle P(X\leq
a\;|\;\mathscr{F}_{t})(\omega^{\prime})=\mathbb{P}(X\leq
a(\omega^{\prime})\;|\;\mathscr{F}_{t})(\omega^{\prime}).$ (B.37)

###### Proof.

For fixed $\omega^{\prime}\in\Omega$,

$\displaystyle\mathbb{P}(X\leq a\;|\;\mathscr{F}_{t})(\omega^{\prime})$
$\displaystyle=\mathbbm{1}_{a=a(\omega^{\prime})}\mathbb{P}(X\leq
a\;|\;\mathscr{F}_{t})=\mathbbm{1}_{a=a(\omega^{\prime})}\mathbb{E}(\mathbbm{1}_{X\leq
a}\;|\;\mathscr{F}_{t})$
$\displaystyle=\mathbbm{1}_{a=a(\omega^{\prime})}\mathbb{E}(\mathbbm{1}_{a=a(\omega^{\prime})}\mathbbm{1}_{X\leq
a}\;|\;\mathscr{F}_{t})=\mathbbm{1}_{a=a(\omega^{\prime})}\mathbb{E}(\mathbbm{1}_{X\leq
a(\omega^{\prime})}\;|\;\mathscr{F}_{t})$ $\displaystyle=\mathbb{P}(X\leq
a(\omega^{\prime})\;|\;\mathscr{F}_{t})(\omega^{\prime}).$

∎

######  Lemma B.15.

For any $X\in L^{\infty}$, $t\in\mathcal{T}$, $\alpha\in(0,1)$ and
$\omega\in\Omega\backslash N^{X}$, the conditional lower and upper
$\alpha$-quantile admit the representation

$\displaystyle q_{\alpha}^{+}(X\;|\;\mathscr{F}_{t})(\omega)$
$\displaystyle=\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\},$ $\displaystyle
q_{\alpha}^{-}(X\;|\;\mathscr{F}_{t})(\omega)$
$\displaystyle=\inf\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq\alpha\\}.$

###### Proof.

We will prove the first identity, and the second identity about
$q_{\alpha}^{-}(X\;|\;\mathscr{F}_{t})$ can be proved similarly. For
simplicity, denote by $m^{*}:=q_{\alpha}^{+}(X\;|\;\mathscr{F}_{t})$ and
$x^{*}(\omega):=\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}$. We will show that the set
$\\{m^{*}\neq x^{*}\\}\cap(\Omega\setminus N^{X})$ is an empty set. We proceed
by splitting $\\{m^{*}\neq x^{*}\\}$ into two sets $\\{m^{*}<x^{*}\\}$ and
$\\{m^{*}>x^{*}\\}$.

First assume that $\\{m^{*}<x^{*}\\}\cap(\Omega\setminus N^{X})\neq\emptyset$.
Then, there exist $M_{1}\in\mathscr{F}_{t}$ and $a\in L^{\infty}_{t}$, such
that on non-empty set $M_{1}\backslash N^{X}$,

$m^{*}<a<x^{*}.$ (B.38)

Since $a<x^{*}$ on $M_{1}\backslash N^{X}$, we have for fixed
$\omega^{\prime}\in M_{1}\backslash N^{X}$,

$a(\omega^{\prime})<x^{*}(\omega^{\prime})=\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega^{\prime})\leq\alpha\\}.$

Thus, $\mathbb{P}(X\leq
a(\omega^{\prime})\;|\;\mathscr{F}_{t})(\omega^{\prime})\leq\alpha$, which by
by Lemma B.15 becomes, $\mathbb{P}(X\leq
a\;|\;\mathscr{F}_{t})(\omega^{\prime})\leq\alpha.$ Since $\omega^{\prime}$
can be chosen arbitrarily in $M_{1}\backslash N^{X}$, we have,
$\mathbb{P}(X\leq a\;|\;\mathscr{F}_{t})\mathbbm{1}_{M_{1}\backslash
N^{X}}\leq\alpha.$ By locality of conditional probability, $\mathbb{P}(X\leq
a\mathbbm{1}_{M_{1}\backslash N^{X}}\;|\;\mathscr{F}_{t})\leq\alpha,$ which by
the definition of $q_{\alpha}^{+}(X\;|\;\mathscr{F}_{t})$ implies that
$a\mathbbm{1}_{M_{1}\backslash N^{X}}\leq m^{*}$. This contradicts (B.38), and
thus $\\{m^{*}<x^{*}\\}=\emptyset$ on $\Omega\backslash N^{X}$.

Next we prove that $\\{m^{*}>x^{*}\\}\cap(\Omega\backslash N^{X})$ is also an
empty. Assume this is not true. Then, there exist $M_{2}\in\mathscr{F}_{t}$
and $b\in L^{\infty}_{t}$, such that on non-empty set $M_{2}\backslash N^{X}$,

$\displaystyle x^{*}<b<m^{*}.$ (B.39)

Since $x^{*}<b$ on $M_{2}\backslash N^{X}$, for fixed $\omega^{\prime}\in
M_{2}\backslash N^{X}$,

$\displaystyle
b(\omega^{\prime})>x^{*}(\omega^{\prime})=\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega^{\prime})\leq\alpha\\}.$

Thus, $\mathbb{P}(X\leq
b(\omega^{\prime})\;|\;\mathscr{F}_{t})(\omega^{\prime})>\alpha$, and by Lemma
B.15, $\mathbb{P}(X\leq b\;|\;\mathscr{F}_{t})(\omega^{\prime})>\alpha.$ Since
$\omega^{\prime}$ can be chosen arbitrarily in $M_{2}\backslash N^{X}$, we
have on $M_{2}\backslash N^{X}$, that $\mathbb{P}(X\leq
b\;|\;\mathscr{F}_{t})>\alpha$. By the definition of
$q_{\alpha}^{+}(X\;|\;\mathscr{F}_{t})$, we have that $b>m^{*}$ on
$M_{2}\backslash N^{X}$, which contradicts (B.39).

The proof is complete. ∎

As a consequence of Lemma B.15, one can show that (see also [FS04, Section
4.4] for unconditional case)

$\displaystyle q_{\alpha}^{+}(X\;|\;\mathscr{F}_{t})(\omega)$
$\displaystyle=\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}(X<x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}.$
$\displaystyle q_{\alpha}^{-}(X\;|\;\mathscr{F}_{t})(\omega)$
$\displaystyle=\inf\\{x\in\mathbb{R}\;|\;\mathbb{P}(X<x\;|\;\mathscr{F}_{t})(\omega)\geq\alpha\\}.$

For brefity, we show the first identity only. Let
$a=\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}(X<x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}$
and $b=\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}$. Then, since

$\left\\{x\in\mathbb{R}\;|\;\mathbb{P}(X<x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\right\\}\supset\left\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\right\\},$

we have $a\geq b$. To show that $a\leq b$, we proceed by contradiction. Assume
that $a>b$, then there exists $s_{0},s_{1}\in\mathbb{R}$, such that
$b<s_{0}<s_{1}<a$. The inequality $s_{1}<a$ implies that
$\mathbb{P}(X<s_{1}\;|\;\mathscr{F}_{t})(\omega)\leq\alpha$, and $b<s_{0}$
implies that $\mathbb{P}(X\leq s_{0}\;|\;\mathscr{F}_{t})(\omega)>\alpha$.
Consequently, we get

$\displaystyle\mathbb{P}(X<s_{1}\;|\;\mathscr{F}_{t})(\omega)\leq\alpha<\mathbb{P}(X\leq
s_{0}\;|\;\mathscr{F}_{t})(\omega).$ (B.40)

On the other hand, by [BCF18, (A.4) and (A.5)], since $s_{0}<s_{1}$, we have
that $\mathbb{P}(X\leq
s_{0}\;|\;\mathscr{F}_{t})(\omega)\leq\mathbb{P}(X<s_{1}\;|\;\mathscr{F}_{t})(\omega)\leq\alpha$,
which contradicts (B.40).

Finally, we present a result that relates lower conditional quantile to upper
conditional quantile.

######  Lemma B.16.

For any $X\in L^{\infty},\ \omega\in\Omega\backslash N^{X}$,

$\displaystyle
q^{+}_{\alpha}(X\;|\;\mathscr{F}_{t})(\omega)=-q^{-}_{1-\alpha}(-X\;|\;\mathscr{F}_{t})(\omega).$

###### Proof.

By Lemma B.15, for any $\omega\in\Omega\backslash N^{\prime}$,

$\displaystyle q^{+}_{\alpha}(X\;|\;\mathscr{F}_{t})(\omega)$
$\displaystyle=\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}(X<x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}=-\inf\\{-x\in\mathbb{R}\;|\;\mathbb{P}(X<x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}$
$\displaystyle=-\inf\\{x\in\mathbb{R}\;|\;\mathbb{P}(X<-x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}$
$\displaystyle=-\inf\\{x\in\mathbb{R}\;|\;\mathbb{P}(-X>x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}$
$\displaystyle=-\inf\\{x\in\mathbb{R}\;|\;1-\mathbb{P}(-X\leq
x\;|\;\mathscr{F}_{t})(\omega)\leq\alpha\\}$
$\displaystyle=-\inf\\{x\in\mathbb{R}\;|\;\mathbb{P}(-X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq 1-\alpha\\}$
$\displaystyle=-q^{-}_{1-\alpha}(-X\;|\;\mathscr{F}_{t})(\omega).$

∎

######  Lemma B.17.

For any $X\in L^{\infty}$, $t\in\mathcal{T}$ and $\omega\in\Omega\backslash
N^{X}$,

$\displaystyle\mathbb{E}[X\;|\;\mathscr{F}_{t}](\omega)=\int_{(0,1)}q^{-}_{1-z}(X\;|\;\mathscr{F}_{t})(\omega)\operatorname{d\\!}z.$

###### Proof.

By [Kal06, Theorem 5.4], fix two measurable spaces $S$ and $T$, a
$\sigma$-field $\mathscr{F}\subset\mathcal{A}$, and a random element $\xi$ in
$S$ such that $\mathbb{P}[\xi\in\cdot\;|\;\mathscr{F}]$ has a regular version
$\nu$. Further consider an $\mathscr{F}$-measurable random element $\eta$ in
$T$ and a measurable function $f$ on $S\times T$ with
$\mathbb{E}|f(\xi,\eta)|<\infty$. Then, for any $\omega\in\Omega\backslash
N^{\xi}$,

$\displaystyle\mathbb{E}[f(\xi,\eta)\;|\;\mathscr{F}](\omega)=\int\nu(\operatorname{d\\!}s)(\omega)f(s,\eta)(\omega).$
(B.41)

Note that in our settings, there is only one variable $X$ which plays the role
of $\xi$ in (B.41). Since $X\in L^{\infty}$, we only consider $X\geq 0$,
otherwise take $X^{\prime}=X+C$, where $C:=\operatorname*{ess\,sup}X$. Thus we
have,

$\displaystyle\mathbb{E}[X\;|\;\mathscr{F}_{t}](\omega)$
$\displaystyle=\int_{\mathbb{R}}xd\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)=\int_{[0,C]}x\operatorname{d\\!}\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)$ $\displaystyle=x\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)|^{C}_{0-}-\int_{[0,C]}\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\operatorname{d\\!}x$
$\displaystyle=C-\int_{[0,C]}\left[1-\mathbb{P}(X>x\;|\;\mathscr{F}_{t})(\omega)\right]\operatorname{d\\!}x=\int_{[0,C]}\mathbb{P}(X>x\;|\;\mathscr{F}_{t})(\omega)\operatorname{d\\!}x$
$\displaystyle=\int_{[0,C]}\int_{(0,1)}\mathbbm{1}_{z<\mathbb{P}(X>x\;|\;\mathscr{F}_{t})(\omega)}\operatorname{d\\!}z\operatorname{d\\!}x=\int_{(0,1)}\int_{[0,C]}\mathbbm{1}_{z<\mathbb{P}(X>x\;|\;\mathscr{F}_{t})(\omega)}\operatorname{d\\!}x\operatorname{d\\!}z$
$\displaystyle=\int_{(0,1)}\int_{[0,C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)<1-z}\operatorname{d\\!}x\operatorname{d\\!}z,$

Next, we will consider the continuous part and the discrete part of $X$
separately. For fixed $z\in(0,1)$, let
$x^{\prime}=\inf\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq 1-z\\}$.

If $X$ is continuous at $x^{\prime}$, then
$\mathbb{P}(X=x^{\prime}\;|\;\mathscr{F}_{t})(\omega)=0$, we have
$\mathbb{P}(X\leq x^{\prime}\;|\;\mathscr{F}_{t})(\omega)=1-z$, and

$\displaystyle\int_{[0,C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)<1-z}\operatorname{d\\!}x$
$\displaystyle=C-\int_{[0,C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq 1-z}\operatorname{d\\!}x$
$\displaystyle=C-\int_{[x^{\prime},C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq 1-z}\operatorname{d\\!}x$
$\displaystyle=C-\int_{[x^{\prime},C]}\mathbbm{1}\operatorname{d\\!}x=x^{\prime}.$

If $X$ is discrete at $x^{\prime}$, then
$\mathbb{P}(X=x^{\prime}\;|\;\mathscr{F}_{t})(\omega)>0$. For those
$z^{\prime}$ such that $\mathbb{P}(X\leq
x^{\prime}-\;|\;\mathscr{F}_{t})(\omega)<1-z^{\prime}\leq\mathbb{P}(X\leq
x^{\prime}\;|\;\mathscr{F}_{t})(\omega)$, we deduce

$\displaystyle\int_{[0,C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)<1-z}\operatorname{d\\!}x$
$\displaystyle=C-\int_{[0,C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq 1-z}\operatorname{d\\!}x$
$\displaystyle=C-\int_{[x^{\prime},C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq 1-z}\operatorname{d\\!}x$
$\displaystyle=C-\int_{[x^{\prime},C]}\mathbbm{1}dx=x^{\prime}.$

Thus for fixed $z\in(0,1)$, and any $X$, we have

$\int_{[0,C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)<1-z}\operatorname{d\\!}x=\inf\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq 1-z\\},$

and we continue

$\displaystyle\mathbb{E}[X\;|\;\mathscr{F}_{t}](\omega)$
$\displaystyle=\int_{(0,1)}\int_{[0,C]}\mathbbm{1}_{\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)<1-z}\operatorname{d\\!}x\operatorname{d\\!}z$
$\displaystyle=\int_{(0,1)}\inf\\{x\in\mathbb{R}\;|\;\mathbb{P}(X\leq
x\;|\;\mathscr{F}_{t})(\omega)\geq 1-z\\}\operatorname{d\\!}z$
$\displaystyle=\int_{(0,1)}q^{-}_{1-z}(X\;|\;\mathscr{F}_{t})(\omega)\operatorname{d\\!}z.$

∎

###  B.4 On $\operatorname{\mathrm{V}@\mathrm{R}}$ and
$\operatorname{\mathrm{AV}@\mathrm{R}}$

The conditional $\operatorname{\mathrm{V}@\mathrm{R}}$, similar to its static
counterpart, is defined in terms of conditional $\alpha$-quantile function.

######  Definition B.18.

For fixed $\alpha\in(0,1)$ and $X\in L^{\infty}$, conditional
$\operatorname{\mathrm{V}@\mathrm{R}}$ at level $\alpha$ with respect to
$\sigma$-field $\mathscr{F}_{t}$ is defined as,

$\displaystyle\operatorname{\mathrm{V}@\mathrm{R}}_{\alpha}(X\;|\;\mathscr{F}_{t}):=-q_{\alpha}^{+}(X\;|\;\mathscr{F}_{t})=\operatorname*{ess\,inf}\\{m\in
L^{\infty}_{t}\;|\;\mathbb{P}(X+m<0\;|\;\mathscr{F}_{t})\leq\alpha\\}.$ (B.42)

From financial point of view,
$\operatorname{\mathrm{V}@\mathrm{R}}_{\alpha}(X|\mathscr{F}_{t})$ can be
viewed as the smallest amount of capital, which, if added to the position $X$
at time $t$, will yield a secured position $X+m$ that encounters losses with a
(conditional) probability below the level $\alpha$. Clearly, the
$\operatorname{\mathrm{V}@\mathrm{R}}_{\alpha}$ does not capture the size or
the distribution of the losses beyond the $\alpha$-quantile. To overcome this,
the notion of average value at risk is introduced.

######  Definition B.19.

The conditional Average Value at Risk at level $\alpha\in(0,1]$ of a position
$X\in L^{\infty}$ is given by

$\displaystyle\operatorname{\mathrm{AV}@\mathrm{R}}_{\alpha}(X\;|\;\mathscr{F}_{t}):=\frac{1}{\alpha}\int_{(0,\alpha)}\operatorname{\mathrm{V}@\mathrm{R}}_{z}(X\;|\;\mathscr{F}_{t})dz=-\frac{1}{\alpha}\int_{(0,\alpha)}q_{z}^{+}(X\;|\;\mathscr{F}_{t})dz.$
(B.43)

Note that $q_{z}^{+}(X\;|\;\mathscr{F}_{t})$ is monotone increasing with
respect to $z$. Since monotonicity implies Borel measurability, $z\mapsto
q_{z}^{+}(X\;|\;\mathscr{F}_{t})$ is Borel measurable. Thus, the integral in
(B.43) is well-defined.

In the existing literature, the $\operatorname{\mathrm{AV}@\mathrm{R}}$
sometimes is defined through the so-called robust or dual representations; cf.
[BCF18] for the conditional case. Next results shows that these definitions
are equivalent.

######  Lemma B.20.

For any $X\in L^{\infty}$ and $\alpha\in(0,1]$,

$\operatorname{\mathrm{AV}@\mathrm{R}}_{\alpha}(X\;|\;\mathscr{F}_{t})=\operatorname*{ess\,sup}\\{\mathbb{E}[-XZ\;|\;\mathscr{F}_{t}]\;|\;Z\in\mathscr{F},0\leq
Z\leq 1/\alpha,\mathbb{E}(Z\;|\;\mathscr{F}_{t})=1\\}.$ (B.44)

Moreover, a maximizer $Z^{*}$ in the right hand side of (B.44) exists, and it
is given by

$Z_{\alpha}^{*}=\frac{1}{\alpha}\left(\mathbbm{1}_{X<q_{\alpha}^{\pm}(X\;|\;\mathscr{F}_{t})}+\varepsilon\mathbbm{1}_{X=q_{\alpha}^{\pm}(X\;|\;\mathscr{F}_{t})}\right),$
(B.45)

here

$\displaystyle\varepsilon=\left\\{\begin{array}[]{ll}0,&\mathbb{P}(X=q_{\alpha}^{\pm}(X\;|\;\mathscr{F}_{t})\;|\;\mathscr{F}_{t})=0,\\\
\frac{\alpha-\mathbb{P}(X<q_{\alpha}^{\pm}(X\;|\;\mathscr{F}_{t})\;|\;\mathscr{F}_{t})}{\mathbb{P}(X=q_{\alpha}^{\pm}(X\;|\;\mathscr{F}_{t})\;|\;\mathscr{F}_{t})},&\text{\
otherwise}.\end{array}\right.$

###### Proof.

In view of (B.43), it is enough to show that,

$-\frac{1}{\alpha}\int_{(0,\alpha)}q^{+}_{z}(X\;|\;\mathscr{F}_{t})(\omega)\operatorname{d\\!}z=\mathbb{E}[-XZ_{\alpha}^{*}\;|\;\mathscr{F}_{t}](\omega).$
(B.46)

Let
$L=-\frac{1}{\alpha}\int_{(0,\alpha)}q^{+}_{z}(X\;|\;\mathscr{F}_{t})(\omega)\operatorname{d\\!}z$.
Then,

$\begin{split}L&=\frac{1}{\alpha}\left[\int_{(0,\alpha)}q^{+}_{\alpha}(X\;|\;\mathscr{F}_{t})(\omega)-q^{+}_{z}(X\;|\;\mathscr{F}_{t})(\omega)\operatorname{d\\!}z\right]-q^{+}_{\alpha}(X\;|\;\mathscr{F}_{t})(\omega)\\\
&=\frac{1}{\alpha}\left[\int_{(0,1)}\left(q^{+}_{\alpha}\left(X\;|\;\mathscr{F}_{t}\right)(\omega)-q^{+}_{z}\left(X\;|\;\mathscr{F}_{t}\right)(\omega)\right)^{+}\operatorname{d\\!}z\right]-q^{+}_{\alpha}(X\;|\;\mathscr{F}_{t})(\omega).\end{split}$
(B.47)

Next we prove an intermediary equality

$\mathbb{E}\left[\left(q^{+}_{\alpha}\left(X\;|\;\mathcal{F}_{t}\right)-X\right)^{+}\;|\;\mathcal{F}_{t}\right](\omega)=\int_{(0,1)}\left(q^{+}_{\alpha}\left(X\;|\;\mathcal{F}_{t}\right)(\omega)-q^{+}_{z}\left(X\;|\;\mathcal{F}_{t}\right)(\omega)\right)^{+}\operatorname{d\\!}z.$
(B.48)

By Lemma B.17

$\begin{split}\mathbb{E}\big{[}\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X\right)^{+}&\;|\;\mathcal{F}_{t}\big{]}(\omega)=\int_{(0,1)}q^{-}_{1-z}\left(\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X\right)^{+}\;|\;\mathcal{F}_{t}\right)(\omega)\operatorname{d\\!}z\\\
&=\int_{(0,1)}\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}\left((q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X)^{+}<x\;|\;\mathcal{F}_{t}\right)(\omega)<1-z\\}\operatorname{d\\!}z.\end{split}$
(B.49)

Since

$\mathbb{P}(X^{+}<x)=\mathbb{P}(\max\\{0,X\\}<x)=\mathbb{P}(0<x,X<x)=\begin{cases}0,&x\geq
0,\\\ \mathbb{P}(X<x),&x>0,\end{cases}$

we have that

$\displaystyle\sup\\{x\in\mathbb{R}\;|\;\mathbb{P}\left(\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X\right)^{+}<x\;|\;\mathcal{F}_{t}\right)(\omega)<1-z\\}$
(B.50)
$\displaystyle\qquad\qquad=\sup\\{x>0\;|\;\mathbb{P}\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X<x\;|\;\mathcal{F}_{t}\right)(\omega)<1-z\\}.$
(B.51)

Then, we combine (B.49) with (B.50), and obtain

$\mathbb{E}\left[\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X\right)^{+}\;|\;\mathcal{F}_{t}\right](\omega)=\int_{(0,1)}\sup\\{x>0\;|\;\mathbb{P}\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X<x\;|\;\mathcal{F}_{t}\right)(\omega)<1-z\\}\operatorname{d\\!}z.$

Using this, to show (B.48), it is sufficient to show that for any $z\in(0,1)$,

$\sup\\{x>0\;|\;\mathbb{P}\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X<x\;|\;\mathcal{F}_{t}\right)(\omega)<1-z\\}\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-q^{+}_{z}(X\;|\;\mathcal{F}_{t})(\omega)\right)^{+},$
(B.52)

which we prove next. Denote by
$J:=\sup\\{x>0\;|\;\mathbb{P}\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X<x\;|\;\mathcal{F}_{t}\right)(\omega)<1-z\\}$.
Then,

$\displaystyle J$
$\displaystyle=\sup\\{x>0\;|\;\mathbb{P}\left(-X<-q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)+x\;|\;\mathcal{F}_{t}\right)(\omega)<1-z\\}$
$\displaystyle=\sup\\{x>0\;|\;\mathbb{P}\left(X>q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-x\;|\;\mathcal{F}_{t}\right)(\omega)<1-z\\}$
$\displaystyle=\sup\\{x>0\;|\;\mathbb{P}\left(X\leq
q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-x\;|\;\mathcal{F}_{t}\right)(\omega)>z\\}.$

Let $y:=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-x$, and we continue

$\displaystyle J$
$\displaystyle=\sup\\{q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-y>0\;|\;\mathbb{P}\left(X\leq
y\;|\;\mathcal{F}_{t}\right)(\omega)>z\\}$
$\displaystyle=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)+\sup\\{-y>-q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)\;|\;\mathbb{P}\left(X\leq
y\;|\;\mathcal{F}_{t}\right)(\omega)>z\\}$
$\displaystyle=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-\inf\\{y<q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)\;|\;\mathbb{P}\left(X\leq
y\;|\;\mathcal{F}_{t}\right)(\omega)>z\\}$
$\displaystyle=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-\min\left\\{q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega),\inf\left\\{y\in\mathbb{R}\;|\;\mathbb{P}\left(X\leq
y\;|\;\mathcal{F}_{t}\right)(\omega)>z\right\\}\right\\}$
$\displaystyle=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)+\max\left\\{-q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega),-\inf\left\\{y\in\mathbb{R}\;|\;\mathbb{P}\left(X\leq
y\;|\;\mathcal{F}_{t}\right)(\omega)>z\right\\}\right\\}$
$\displaystyle=\max\left\\{0,q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-q^{+}_{z}(X\;|\;\mathcal{F}_{t})(\omega)\right\\}$
$\displaystyle=\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)-q^{+}_{z}(X\;|\;\mathcal{F}_{t})(\omega)\right)^{+},$

and thus (B.52), is proved, and consequently (B.48) is established.

Then by (B.47) and (B.48), we get,

$\displaystyle L$
$\displaystyle=\frac{1}{\alpha}\mathbb{E}\left[\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X\right)^{+}\;|\;\mathcal{F}_{t}\right](\omega)-q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)$
$\displaystyle=\frac{1}{\alpha}\mathbb{E}\left[\left(q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})-X\right)\mathbbm{1}_{X<q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)-q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)$
$\displaystyle=\frac{1}{\alpha}q^{+}_{\alpha}\left(X\;|\;\mathcal{F}_{t}\right)(\omega)\mathbb{P}\left(X<q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\;|\;\mathcal{F}_{t}\right)(\omega)$
$\displaystyle\qquad\qquad+\frac{1}{\alpha}\mathbb{E}\left[-X\mathbbm{1}_{X<q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)-q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})(\omega)$
$\displaystyle=-\frac{1}{\alpha}q^{+}_{\alpha}\left(X\;|\;\mathcal{F}_{t}\right)(\omega)\left[\alpha-\mathbb{P}\left(X<q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\;|\;\mathcal{F}_{t}\right)(\omega)\right]$
$\displaystyle\qquad\qquad+\frac{1}{\alpha}\mathbb{E}\left[-X\mathbbm{1}_{X<q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)$
$\displaystyle=-\frac{1}{\alpha}q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})(\omega)\mathbb{E}\left[\frac{\alpha-\mathbb{P}(X<q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\;|\;\mathcal{F}_{t})}{\mathbb{P}(X=q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\;|\;\mathcal{F}_{t})}\mathbbm{1}_{X=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)$
$\displaystyle\qquad\qquad+\frac{1}{\alpha}\mathbb{E}\left[-X\mathbbm{1}_{X<q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)$
$\displaystyle=\frac{1}{\alpha}\mathbb{E}\left[-q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\frac{\alpha-\mathbb{P}(X<q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\;|\;\mathcal{F}_{t})}{\mathbb{P}(X=q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\;|\;\mathcal{F}_{t})}\mathbbm{1}_{X=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)$
$\displaystyle\qquad\qquad+\frac{1}{\alpha}\mathbb{E}\left[-X\mathbbm{1}_{X<q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)$
$\displaystyle=\frac{1}{\alpha}\mathbb{E}\left[-X\frac{\alpha-\mathbb{P}(X<q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\;|\;\mathcal{F}_{t})}{\mathbb{P}(X=q_{\alpha}^{+}(X\;|\;\mathcal{F}_{t})\;|\;\mathcal{F}_{t})}\mathbbm{1}_{X=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)$
$\displaystyle\qquad\qquad+\frac{1}{\alpha}\mathbb{E}\left[-X\mathbbm{1}_{X<q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)$
$\displaystyle=\frac{1}{\alpha}\mathbb{E}\left[-X\varepsilon\mathbbm{1}_{X=q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)+\frac{1}{\alpha}\mathbb{E}\left[-X\mathbbm{1}_{X<q^{+}_{\alpha}(X\;|\;\mathcal{F}_{t})}\;|\;\mathcal{F}_{t}\right](\omega)$
$\displaystyle=\mathbb{E}[-XZ_{\alpha}^{*}\;|\;\mathcal{F}_{t}](\omega),$

where $Z_{\alpha}^{*}$ is given as (B.45). This is exactly (B.46), and the
proof is complete. ∎