# Mean-field optimal control problem of SDDEs driven by fractional Brownian
motion

Soukaina Douissi , Astrid Hilbert , Nacira Agram  Laboratory LIBMA, Faculty
Semlalia, University Cadi Ayyad, Marrakech, Morocco. Email:
<EMAIL_ADDRESS>of Mathematics, Linnaeus University,
Sweden. Email<EMAIL_ADDRESS>of Mathematics, University of
Oslo, P.O. Box 1053 Blindern, N–0316 Oslo, Norway. Email<EMAIL_ADDRESS>
###### Abstract

We consider a mean-field optimal control problem for stochastic differential
equations with delay driven by fractional Brownian motion with Hurst parameter
greater than one half. Stochastic optimal control problems driven by
fractional Brownian motion can not be studied using classical methods, because
the fractional Brownian motion is neither a Markov pocess nor a semi-
martingale. However, using the fractional White noise calculus combined with
some special tools related to the differentiation for functions of measures,
we establish and proove necessary and sufficient stochastic maximum
principles. To illustrate our study, we consider two applications: we solve a
problem of optimal consumption from a cash flow with delay and a linear-
quadratique (LQ) problem with delay.

Keywords Mean-Field, Stochastic Delayed Differential Equations, Fractional
Brownian Motion, Stochastic Maximum Principles.

Mathematics Subject Classification 60G22. 60H07. 60H04. 93E20

##  1 Introduction

The interest for stochastic delayed differential equations is constantly
increasing. They are frequently used to model the evolution of systems with
past-dependence nature. Such systems usually appear in biology, engineering
and mathematical finance.
There is a rich litterature on stochastic optimal problems with delay. A lot
of authors studied both the case where the stochastic systems are driven by a
classical Brownian motion as well as where there is jumps, see,e.g., [1, 2, 3,
4].
Stochastic control problems driven by fractional Brownian motion (fBm) were
also studied by many authors, see,e.g., [5, 6, 7, 8]. However, compared with
the papers on stochastic control problems driven by the classical Brownian
motion, few has been done because classical methods to solve control problems
can not be used direclty, since the fractional Brownian motion is not a semi-
martingale and not a Markov process.
Mean-field problems have also attracted wide attention recently, due to their
several applications in physics, economics, finance and stochastic games.
Mean-field games were first studied by Lasry and Lions [9]. Buckdhan, Li and
Peng [10] studied a special mean-field games and introduced the so-called
mean-field backward stochastic differential equations. Later, Carmona and
Delarue [11] studied a class of mean-field forward-backward stochastic
differential equations and gave many applications.
In this paper, all the previous fields are combined to study the optimal
control problem of mean-field stochastic differential equations driven by
fractional Brownian motion. The dynamic of the controlled state process
depends on the state, the control, their laws but also on their values at
previous time instants.
The dynamics of this work are close to those in the paper of Qiuxi Wang, Feng
Chen and Fushan Huang [8]. In [8], the adjoint equation is an anticipated
backward stochastic diferential equation (ABSDE) driven by both a fBm and a
standard Brownian motion and the integral with respect to the fBm, is defined
in the Stratonovich sense.
This work is also inspired by the recent paper of Buckdhan Rainer and Shuai
Jing [6], if the system has a past-dependence feature. In [6] the dynamic of
the adjoint process is driven by a standard Brownian motion, here the
anticipated BSDE is driven by a fBM.
The approach used here is an extension of the work of Biagini, Hu, Øksendal
and Sulem [5] when the dynamic of the system is of delayed mean-field type.
In our paper, we establish and proove necessary and sufficient maximum
principles and we illustrate our study by solving two optimal control problems
: a mean-field optimal consumption problem from a cash flow with delay and a
linear-quadratique (LQ) problem with delay.
We present now more specifically the general problem we consider:

##  2 Statement of the problem

Let $B^{H}$ be a fractional Brownian motion on a filtred probability space
$(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$.
We consider a mean-field controlled stochastic delay equation of the form:

$\begin{array}[c]{lll}dX(t)&=&b(t,X(t),X(t-\delta),M(t),M(t-\delta),u(t))dt+\sigma(t,M(t),M(t-\delta))dB^{H}_{t},\text{
}t\in\left[0,T\right],\text{\ }\\\ X(t)&=&x_{0}(t);\text{ \
}t\in\left[-\delta,0\right],\end{array}$

where

$M(t):=\mathbb{P}_{X(t)},\text{ \ }M(t-\delta):=\mathbb{P}_{X(t-\delta)}$

and $T>0$, $\delta>0$ are given constants. Here

$b:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\times\mathcal{P}_{2}(\mathbb{R})\times\mathcal{U}\rightarrow\mathbb{R},$
$\sigma:\left[0,T\right]\times\mathcal{P}_{2}(\mathbb{R})\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R},$

are given functions such that, for all $t\in[0,T]$,
$b(.,t,x,\bar{x},m,\bar{m},u)$ is supposed to be $\mathcal{F}_{t}$\-
measurable for all $x,\bar{x}\in\mathbb{R}$, $u\in\mathcal{U}$,
$m,\bar{m}\in\mathcal{P}_{2}(\mathbb{R})$. The function $\sigma$ is assumed to
be deterministic such that its integral with respect to the fBm will be a
Wiener type integral. $\mathcal{P}_{2}(\mathbb{R})$ denotes the space of all
probability measures $m$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, such that
$\int_{\mathbb{R}}|x|^{2}m(dx)<\infty$.

The function $x_{0}$ is assumed to be continuous and deterministic. The set
$\mathcal{U}\subset\mathbb{R}$ consists of the admissible control values. The
information available to the controller is given by the filtration
$\mathbb{F}$ generated by the fBm $B^{H}$. The set of admissible controls
denoted by $\mathcal{A}_{\mathbb{F}}$ are the strategies available to the
controller, required to be $\mathcal{U}$-valued and $\mathbb{F}$-adapted
processes. Through the paper, we assume that $X$ exists and belongs to
$L^{2}(\Omega\times[0,T])$.

The performance functional is assumed to have the form

$\begin{array}[c]{lll}J(u)&=&\mathbb{E[}g(X(T),M(T))+{\textstyle\int_{0}^{T}}f(t,X(t),X(t-\delta),M(t),M(t-\delta),u(t))dt]\end{array},$

where
$f:\Omega\times\left[0,T\right]\times\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\times\mathcal{P}_{2}(\mathbb{R})\times\mathcal{U}\rightarrow\mathbb{R}$
and
$g:\Omega\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
are given functions, such that for all $t\in[0,T]$,
$f(.,t,x,\bar{x},m,\bar{m},u)$ is assumed to be $\mathcal{F}_{t}$-measurable
for all $x,\bar{x}\in\mathbb{R}$, $u\in\mathcal{U}$,
$m,\bar{m}\in\mathcal{P}_{2}(\mathbb{R})$. The function $g(.,x,m)$ is assumed
to be $\mathcal{F}_{T}$-measurable for all $x\in\mathbb{R}$,
$m\in\mathcal{P}_{2}(\mathbb{R})$.

We also assume the following integrability condition

$\begin{array}[c]{lll}\mathbb{E[}|g(X(T),M(T))|+{\textstyle\int_{0}^{T}}|f(t,X(t),X(t-\delta),M(t),M(t-\delta),u(t))|dt]<+\infty.\end{array},$

The functions $\sigma$, $b$, $f$ and $g$ are assumed to be continuously
differentiable with respect to $x,\bar{x},u$ with bounded derivatives and
admit Fréchet bounded derivatives with respect to $m,\bar{m}$.

The problem we consider in this paper is the following :

Problem: Find a control $u^{*}\in\mathcal{A}_{\mathbb{F}}$ such that

$J(u^{*})=\sup_{u\in\mathcal{A}_{\mathbb{F}}}J(u).$ (1)

any control $u^{*}\in\mathcal{A}_{\mathbb{F}}$ satisfying (1) is called an
optimal control.

##  3 Generalities

In this section we give some preliminaries concerning fractional Brownian
motion based on fractional White noise calculus and some generalities on
differentiability with respect to the measures. For a general introduction to
fractional White noise theory the reader may consult the books [12, 13].

###  3.1 Fractional Brownian motion

Let $T>0$ be a finite time horizon and $\Omega$ be the space
$\mathcal{S}^{{\prime}}([0,T])$ of tempered distributions on $[0,T]$, which is
the dual of the Schwartz space $\mathcal{S}([0,T])$ of rapidly decreasing
smooth functions on $[0,T]$.

For $1/2<H<1$. We put

$\varphi_{H}(t,s)=H(2H-1)|t-s|^{2H-2},\hskip 14.22636pts,t\in[0,T].$

If $\omega\in\mathcal{S}^{{\prime}}([0,T])$ and $f\in\mathcal{S}([0,T])$ , we
let $<\omega,f>=\omega(f)$ denote the action of $\omega$ applied to $f$, it
can be extended to $f:[0,T]\rightarrow\mathbb{R}$ such that

$\|f\|_{{H}}^{2}:={\textstyle\int_{0}^{T}}{\textstyle\int_{0}^{T}}f(s)f(t)\varphi_{H}(t,s)dsdt<+\infty.$

The space of all such functions $f$ is denoted by $\mathbb{L}_{H}^{2}([0,T])$.
The map $f\mapsto\exp(-1/2\|f\|_{{H}}^{2})$ with $f\in\mathcal{S}([0,T])$, is
positive definite on $\mathcal{S}([0,T])$, then by Bochner-Minlos theorem
there exists a probability measure $\mathbb{P}^{H}=\mathbb{P}$ on the Borel
subsets $\mathcal{B}(\Omega)$ such that

$\int_{\Omega}e^{i<\omega,f>}d\mathbb{P}(\omega)=e^{-1/2\|f\|_{{H}}^{2}}\hskip
28.45274pt\forall f\in\mathcal{S}([0,T])$ (2)

It follows from (2) that

$\mathbb{E}[<.,f>]=0\hskip 5.69046pt\text{and}\hskip
5.69046pt\mathbb{E}[<.,f>^{2}]=\|f\|_{{H}}^{2}$ (3)

where $\mathbb{E}$ denotes the expectation under the probability measure
$\mathbb{P}$. Hence, if we put

$B^{H}_{t}:=<\omega,\chi_{[0,t]}(.)>$

then using (3), $B^{H}$ is a fractional Brownian motion with Hurst parameter
$H$, that is a centred Gaussian process with covariance function

$\mathbb{E}[B^{H}_{t}B^{H}_{s}]=\frac{1}{2}(t^{2H}+s^{2H}-|t-s|^{2H}),\hskip
5.69046pts,t\in[0,T]$

From now on, we endow $\Omega$ with the natural filtration
$\mathbb{F}:=\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$ generated by $B^{H}$.

######  Lemma 3.1.

If $f,g$ belong to $\mathbb{L}_{H}^{2}([0,T])$, then the Wiener integrals
$\int_{0}^{T}f_{s}dB^{H}_{s}$ and $\int_{0}^{T}g_{s}dB^{H}_{s}$ are well
defined as zero mean Gaussian random variables with variances
$\|f\|_{{H}}^{2}$ and $\|g\|_{{H}}^{2}$ respectively and

$\mathbb{E}[\int_{0}^{T}f_{s}dB^{H}_{s}\int_{0}^{T}g_{s}dB^{H}_{s}]=\int_{0}^{T}\int_{0}^{T}f(s)g(t)\varphi_{H}(t,s)dtds$

We denote by $\tilde{\mathcal{S}}$ the set of all polynomial functions of
$B^{H}(\psi_{j})=\int^{T}_{0}\psi_{j}(t)dB^{H}(t)$. For an element
$F\in\tilde{\mathcal{S}}$, having the form

$F=g(B^{H}(\psi_{1}),\cdots,B^{H}(\psi_{n})),$

where $g$ is a polynomial function of $n$ variables, we define its Malliavin
derivative $D_{s}^{H}F$ by

$D^{H}_{s}F:=\sum_{i=1}^{n}\frac{\partial g}{\partial
x_{i}}(B^{H}(\psi_{1}),\cdots,B^{H}(\psi_{n}))\psi_{i}(s),\qquad 0\leq s\leq
T.$

We also introduce another derivative

$\mathbb{D}_{t}^{H}F:={\textstyle\int_{0}^{T}}D^{H}_{r}F\varphi_{H}(r,t)dr,$

Let $\mathcal{L}_{H}^{1,2}([0,T])$ be the set of processes
$G:[0,T]\times\Omega\rightarrow\mathbb{R}$ such that $\mathbb{D}_{s}^{H}G(s)$
exists for all $s\in[0,T]$ and

$\|G\|_{\mathcal{L}_{H}^{1,2}}^{2}:=\mathbb{E}[{\textstyle\int_{0}^{T}}{\textstyle\int_{0}^{T}}G(s)G(t)\varphi_{H}(t,s)dtds+({\textstyle\int_{0}^{T}}\mathbb{D}_{s}^{H}G(s)ds)^{2}]<\infty.$

We let $\int_{0}^{T}G(s)dB^{H}(s)$ denote the fractional Wick-Itô-Skorohod
(fWIS) integral of the process $G$ with respect to $B^{H}$. We recall its
construction: if $G$ belongs to the family $\mathbb{S}$ of step functions of
the form

$G(t,\omega)={\textstyle\sum_{i=1}^{N}}G_{i}(\omega)\chi_{[t_{i},t_{i+1}[}(t),$

where $0\leq t_{1}<t_{2}<...<t_{N+1}\leq T$, then the fWIS integral is defined
naturally as follows

${\textstyle\int_{0}^{T}}G(t,\omega)dB^{H}(t):={\textstyle\sum_{i=1}^{N}}G_{i}(\omega)\diamondsuit(B^{H}(t_{i+1})-B^{H}(t_{i})),$

where $\diamondsuit$ denotes the Wick product, see [12] for its definition.
For $G\in\mathbb{S}\cap\mathcal{L}_{H}^{1,2}([0,T])$, we have the isometry

$\mathbb{E}[({\textstyle\int_{0}^{T}}G(t)dB^{H}(t))^{2}]=\mathbb{E}[{\textstyle\int_{0}^{T}}{\textstyle\int_{0}^{T}}G(s)G(t)\varphi_{H}(t,s)dtds+({\textstyle\int_{0}^{T}}\mathbb{D}_{s}^{H}G(s)ds)^{2}],$

Using this we can extend the integral ${\textstyle\int_{0}^{T}}G(t)dB^{H}(t)$
to $\mathcal{L}_{H}^{1,2}([0,T])$. Note that if
$G_{1},G_{2}\in\mathcal{L}_{H}^{1,2}([0,T])$, we have by polarization

$\mathbb{E}[{\textstyle\int_{0}^{T}}G_{1}(t)dB^{H}(t){\textstyle\int_{0}^{T}}G_{2}(t)dB^{H}(t)]=\mathbb{E}[{\textstyle\int_{0}^{T}}G_{1}(s)G_{2}(t)\varphi_{H}(s,t)dsdt+{\textstyle\int_{0}^{T}}{\textstyle\int_{0}^{T}}\mathbb{D}_{s}^{H}G_{1}(s)\mathbb{D}_{t}^{H}G_{2}(t)dsdt]$

An important property of this integral is that

$\mathbb{E}[{\textstyle\int_{0}^{T}}G(t)dB^{H}(t)]=0;\hskip 14.22636pt\text{
for all}\hskip 14.22636ptG\in\mathcal{L}_{H}^{1,2}([0,T]).$

We will need the following integration by parts formula.

######  Proposition 3.1 (Integration by parts).

Let $X$ and $Y$ be two processes of the form

$dX(t)=F_{1}(t)dt+G_{1}(t)dB^{H}(t),\hskip 14.22636ptX(0)=x\in\mathbb{R},$

and

$dY(t)=F_{2}(t)dt+G_{2}(t)dB^{H}(t),\hskip 14.22636ptY(0)=y\in\mathbb{R},$

where $F_{1}:[0,T]\times\Omega\rightarrow\mathbb{R}$,
$F_{2}:[0,T]\times\Omega\rightarrow\mathbb{R}$,
$G_{1}:[0,T]\times\Omega\rightarrow\mathbb{R}$ and
$G_{2}:[0,T]\times\Omega\rightarrow\mathbb{R}$ are given processes such that
$G_{1}$, $G_{2}\in\mathcal{L}_{H}^{1,2}([0,T])$.

  1. 1.

Then, for $T>0$,

$\displaystyle\mathbb{E}[X(T)Y(T)]$
$\displaystyle=xy+\mathbb{E}[{\textstyle\int_{0}^{T}}X(s)dY(s)]+\mathbb{E}[{\textstyle\int_{0}^{T}}Y(s)dX(s)]$
$\displaystyle\hskip
8.5359pt+\mathbb{E}[{\textstyle\int_{0}^{T}}{\textstyle\int_{0}^{T}}G_{1}(t)G_{2}(s)\varphi_{H}(t,s)dsdt]+\mathbb{E}[{\textstyle\int_{0}^{T}\int_{0}^{T}}\mathbb{D}_{t}^{H}G_{1}(t)\mathbb{D}_{s}^{H}G_{2}(s)dsdt],$

provided that the first two integrals exist.

  2. 2.

In particular, if $G_{1}$ or $G_{2}$ is deterministic, then

$\mathbb{E}[X(T)Y(T)]=xy+\mathbb{E}[{\textstyle\int_{0}^{T}}X(s)dY(s)]+\mathbb{E}[{\textstyle\int_{0}^{T}}Y(s)dX(s)]+\mathbb{E}[{\textstyle\int_{0}^{T}}{\textstyle\int_{0}^{T}}G_{1}(t)G_{2}(s)\varphi_{H}(t,s)dsdt].$

###  3.2 Differentiability of Functions of Measures

Let $\mathcal{P}(\mathbb{R})$ be the space of all probability measures on
$(\mathbb{R},\mathcal{B}(\mathbb{R}))$. We denote by
$\mathcal{P}_{p}(\mathbb{R})$ the subspace of $\mathcal{P}(\mathbb{R})$ of
order $p$, which means

$\mathcal{P}_{p}(\mathbb{R})=\\{m\in\mathcal{P}(\mathbb{R}):\int_{\mathbb{R}}|x|^{p}m(dx)<+\infty\\}.$

  * $\bullet$

The Wasserstein metric :

On $\mathcal{P}_{p}(\mathbb{R})$, the Wasserstein metric of order $p$ is
defined by

$\displaystyle
W_{p}(m,m^{\prime})=\inf\bigg{\\{}\left(\int_{\mathbb{R}^{2}}|x-y|^{p}\rho(dx,dy)\right)^{\frac{1}{p}},$
$\displaystyle\ \rho\in\mathcal{P}_{p}(\mathbb{R}\times\mathbb{R})\ \hbox{such
that}$ $\displaystyle\rho(\cdot\times\mathbb{R})=m\ \hbox{and}\
\rho(\mathbb{R}\times\cdot)=m^{\prime}\bigg{\\}}.$

Notice that if $\xi$ and $\eta$ are two real $p$-integrable random variables
with laws $\mathbb{P}_{\xi}$ and $\mathbb{P}_{\eta}$, then we have
$W_{p}(\mathbb{P}_{\xi},\mathbb{P}_{\eta})\leq\left(\mathbb{E}[{|\xi-\eta|^{p}}]\right)^{\frac{1}{p}}$
since we can choose a special $\rho=\mathbb{P}_{(\xi,\eta)}$ in the above
definition.

  * $\bullet$

Diffentiability of functions of measures :

The notion of differentiability for functions of measures that we will use in
the paper, is the one introduced by Lions in his course at Collège de France
[14] and summarized by Cardaliaguet [15]. We also refer to Carmona and Delarue
[11].

It’s based on the lifting of fuctions
$m\in\mathcal{P}_{2}(\mathbb{R})\mapsto\sigma(m)$ into functions
$\tilde{\xi}\in
L^{2}(\tilde{\Omega};\mathbb{R})\mapsto\tilde{\sigma}(\tilde{\xi})$, over some
probability space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$,
by setting
$\tilde{\sigma}(\tilde{\xi}):=\sigma(\tilde{\mathbb{P}}_{\tilde{\xi}})$.

######  Definition 3.2.

A function $\sigma$ is said to be differentiable at
$m_{0}\in\mathcal{P}_{2}(\mathbb{R})$, if there exists a random variable
$\tilde{\xi}_{0}\in
L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ over some
probability space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$
with $\tilde{\mathbb{P}}_{\tilde{\xi}_{0}}=m_{0}$ such that
$\tilde{\sigma}:L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})\to\mathbb{R}$
is Fréchet differentiable at $\tilde{\xi}_{0}$.

We suppose for simplicity that
$\tilde{\sigma}:L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})\to\mathbb{R}$
is Fréchet differentiable. We denote its Fréchet derivative at
$\tilde{\xi}_{0}$ by $D\tilde{\sigma}(\tilde{\xi}_{0})$. Recall that
$D\tilde{\sigma}(\tilde{\xi}_{0}):L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})\to\mathbb{R}$
is a continuous linear mapping; i.e. $D\tilde{\sigma}(\tilde{\xi}_{0})\in
L(L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}}),\mathbb{R})$.
With the identification that
$L(L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}}),\mathbb{R})\equiv
L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ given by Riesz
representation theorem, $D\tilde{\sigma}(\tilde{\xi}_{0})$ is viewed as an
element of $L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$,
hence we can write

$\sigma(m)-\sigma(m_{0})=\tilde{\sigma}(\tilde{\xi})-\tilde{\sigma}(\tilde{\xi}_{0})=\tilde{\mathbb{E}}[(D\tilde{\sigma})({\tilde{\xi}}_{0}).(\tilde{\xi}-\tilde{\xi}_{0})]+o(\tilde{\mathbb{E}}[|\tilde{\xi}-\tilde{\xi}_{0}|^{2}]^{1/2}),\
\textrm{as}\ \tilde{\mathbb{E}}[|\tilde{\xi}-\tilde{\xi}_{0}|^{2}]^{1/2}\to
0.$

where $\tilde{\xi}$ is a random variable with law $m$. Moreover, according to
Cardaliaguet [15], there exists a Borel function
$h_{m_{0}}:\mathbb{R}\to\mathbb{R}$, such that
$D\tilde{\sigma}(\tilde{\xi}_{0})=h_{m_{0}}(\tilde{\xi}_{0})$,
$\tilde{\mathbb{P}}$-a.s. We define the derivative of $\sigma$ with respect to
the measure at $m_{0}$ by putting
$\partial_{m}\sigma(m_{0})(x):=h_{m_{0}}(x)$. Notice that
$\partial_{m}\sigma(m_{0})(x)$ is defined $m_{0}(dx)$-a.e. uniquely.
Therefore, the following differentiation formula is invariant by modification
of the space $\tilde{\Omega}$ where the random variables $\tilde{\xi}_{0}$ and
$\tilde{\xi}$ are defined, i.e.

$\sigma(m)-\sigma(m_{0})=\tilde{\mathbb{E}}[\partial_{m}\sigma(m_{0})(\tilde{\xi}_{0}).(\tilde{\xi}-\tilde{\xi}_{0})]+o(\tilde{\mathbb{E}}[|\tilde{\xi}-\tilde{\xi}_{0}|^{2}]^{1/2}),\
\textrm{as}\ \tilde{\mathbb{E}}[|\tilde{\xi}-\tilde{\xi}_{0}|^{2}]^{1/2}\to
0.$

whenever $\tilde{\xi}$ and $\tilde{\xi}_{0}$ are random variables with laws
$m$ and $m_{0}$ respectively.

  * $\bullet$

Joint concavity

We will need the joint concavity of a function on
$(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$. A differentiable function $b$
defined on $(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$ is concave, if for
every $(x^{\prime},m^{\prime})$ and
$(x,m)\in(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$, we have

$\displaystyle
b(x^{\prime},m^{\prime})-b(x,m)-\partial_{x}b(x,m)(x^{\prime}-x)-\tilde{{\mathbb{E}}}[\partial_{m}b(x,m)(\tilde{X})(\tilde{X}^{\prime}-\tilde{X})]\leq
0,$ (4)

whenever $\tilde{X},\tilde{X}^{\prime}\in
L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R})$ with
laws $m$ and $m^{\prime}$ respectively.

##  4 Necessary maximum principle

In this section, we establish a maximum principle of necessary type.

For this end, we assume that $\mathcal{U}$ is a closed convex set (and hence
$\mathcal{A}_{\mathbb{F}}$ is convex). Now for a given
$u^{\ast}\in\mathcal{A}_{\mathbb{F}}$ and an arbitrary but fixed control
$u\in\mathcal{A}_{\mathbb{F}}$, we define

$\begin{array}[c]{ll}u^{\theta}:=u^{\ast}+\theta\left(u-u^{\ast}\right),&\theta\in\left[0,1\right].\end{array}$

Note that, thanks to the convexity of $\mathcal{A}_{\mathbb{F}}$,
$u^{\theta}\in\mathcal{A}_{\mathbb{F}},\text{for all \hskip
2.84544pt}\theta\in\left[0,1\right]$. We denote by
$X^{\theta}:=X^{u^{\theta}}$ and by $X^{\ast}:=X^{u^{\ast}}$ the controlled
state processes corresponding to $u^{\theta}$ and $u^{\ast}$ respectively.

For $u^{\ast}\in\mathcal{A}_{\mathbb{F}}$ and the associated controlled state
process $X^{\ast}$, let $Y(t):=\frac{d}{d\theta}X^{\theta}(t)|_{\theta=0}$,
hence $Y$ satisfies the following SDDE :

$\begin{array}[c]{ll}dY(t)&=\\{\partial_{x}b^{*}(t)Y(t)+\partial_{\bar{x}}b^{*}(t)Y(t-\delta)+\mathbb{\tilde{E}}[\partial_{m}b^{*}(t)(\tilde{X}^{*}(t))\tilde{Y}(t)]\\\
&\hskip
22.76228pt+\mathbb{\tilde{E}}[\partial_{\bar{m}}b^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{Y}(t-\delta)]+\partial_{u}b^{*}(t)(u(t)-u^{\ast}(t))\\}dt\\\
&\hskip
36.98866pt+\\{\mathbb{\tilde{E}}[\partial_{m}\sigma(t)(\tilde{X}^{*}(t))\tilde{Y}(t)]+\mathbb{\tilde{E}}[\partial_{\bar{m}}\sigma^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{Y}(t-\delta)]\\}dB_{t}^{H},\text{
\ }t\in[0,T],\\\ \\\ Y(t)&=0;\hskip
14.22636ptt\in\left[-\delta,0\right]\text{.}\end{array}$ (5)

where we used the following notations:

$\begin{array}[c]{ll}\partial_{x}b^{*}(t)&:=\partial_{x}b(t,X^{\ast}(t),X^{\ast}(t-\delta),M^{\ast}(t),M^{\ast}(t-\delta),u^{\ast}(t)),\\\
\partial_{m}b^{*}(t)(.)&:=\partial_{m}b(t,X^{\ast}(t),X^{\ast}(t-\delta),M^{\ast}(t),M^{\ast}(t-\delta),u^{\ast}(t))(.),\\\
\partial_{m}\sigma^{*}(t)(.)&:=\partial_{m}\sigma(t,M^{\ast}(t),M^{\ast}(t-\delta))(.).\end{array}$

and $(\tilde{X},\tilde{Y},\tilde{u})$ is an independant copy of $(X,Y,u)$
defined on some probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ and
$\tilde{\mathbb{E}}$ denotes the expectation on
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$.

######  Remark 4.1.

From the definition of the tilde random variables and since $\sigma$ is
deterministic, we have

$\tilde{\mathbb{E}}[\partial_{m}\sigma^{\ast}(t)(\tilde{X}^{\ast}(t))\tilde{Y}^{\ast}(t)]=\mathbb{E}[\partial_{m}\sigma^{\ast}(t)({X}^{\ast}(t)){Y}^{\ast}(t)].$

Note that using the previous notations,
$\mathbb{\tilde{E}}[\partial_{m}b^{\ast}(t)(\tilde{X}^{\ast}(t))\tilde{Y}(t)]$
is a function of the random vector
$(X^{\ast}(t),X^{\ast}({t-\delta}),u^{\ast}(t))$ as it stands for
$\mathbb{\tilde{E}}[\partial_{m}b(t,x,\bar{x},M^{\ast}(t),M^{\ast}(t-\delta),u)(\tilde{X}^{*}(t))\tilde{Y}(t)]|_{x=X^{\ast}(t),\bar{x}=X^{\ast}(t-\delta),u=u^{\ast}(t)}$.

We assume that the derivative process $Y$ exists and belongs to
$L^{2}(\Omega\times[0,T])$ and that the function
$\psi_{\delta}^{*}:t\mapsto\mathbb{\tilde{E}}[\partial_{m}\sigma^{*}(t)(\tilde{X}^{*}(t))\tilde{Y}(t)]+\mathbb{\tilde{E}}[\partial_{\bar{m}}\sigma^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{Y}(t-\delta)]$
is in $\mathbb{L}_{H}^{2}([0,T])$, the integral with respect to the fBm is
therefore well defined in the Wiener sense.

Now if $u^{\ast}$ is an optimal control, we have $J(u^{\ast})\leq
J(u^{\theta})$, for all $\theta\in\left[0,1\right]$, i.e.,

$0\leq\underset{\theta\rightarrow
0}{\lim}\tfrac{J(u^{\theta})-J(u^{\ast})}{\theta}\text{.}$ (6)

with

$\displaystyle\underset{\theta\rightarrow
0}{\lim}\tfrac{1}{\theta}(J(u^{\theta})-J(u^{\ast}))$
$\displaystyle=\mathbb{E[}\partial_{x}g^{*}(T)Y(T)+\mathbb{\tilde{E}}[\partial_{m}g^{*}(T)(\tilde{X}^{*}(T))\tilde{Y}(T)]]$
$\displaystyle+\mathbb{E[}{\textstyle\int_{0}^{T}}\\{\partial_{x}f^{*}(t)Y(t)+\partial_{\bar{x}}f^{*}(t)Y(t-\delta)+\mathbb{\tilde{E}}[\partial_{m}f^{*}(t)(\tilde{X}^{*}(t))\tilde{Y}(t)]$
$\displaystyle+\mathbb{\tilde{E}}[\partial_{\bar{m}}f^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{Y}(t-\delta)]+\partial_{u}f^{*}(t)\left(u(t)-u^{\ast}(t)\right)\\}dt].$
(7)

where we have used the simplified notations :

$\begin{array}[c]{ll}\partial_{x}g^{*}(T)&:=\partial_{x}g(X^{\ast}(T),M^{\ast}(T)),\\\
\partial_{m}g^{*}(T)(.)&:=\partial_{m}g(X^{\ast}(T),M^{\ast}(T))(.),\\\
\partial_{x}f^{*}(t)&:=\partial_{x}f(t,X^{\ast}(t),X^{\ast}(t-\delta),M^{\ast}(t),M^{\ast}(t-\delta),u^{\ast}(t)),\\\
\partial_{{m}}f^{*}(t)(.)&:=\partial_{{m}}f(t,X^{\ast}(t),X^{\ast}(t-\delta),M^{\ast}(t),M^{\ast}(t-\delta),u^{\ast}(t))(.),\\\
\partial_{m}\sigma^{*}(t)(.)&:=\partial_{m}\sigma(t,M^{\ast}(t),M^{\ast}(t-\delta))(.).\end{array}$

In order to determine the adjoint backward equation associated to
$(u^{*},X^{*})$, we suppose that it has in general the following form

$\left\\{\begin{array}[c]{ll}dp^{*}(t)&=-\alpha(t)dt+q^{*}(t)dB_{t}^{H},\text{
\ }t\in\left[0,T\right],\\\
p^{*}(T)&=\partial_{x}g^{*}(T)+\mathbb{\tilde{E}}[\partial_{m}\tilde{g}^{*}(T)(X(T))].\end{array}\right.$
(8)

where $(p^{*},q^{*})$ is assumed to be in
$\mathcal{L}_{H}^{1,2}([0,T])\times\mathcal{L}_{H}^{1,2}([0,T])$, the integral
with respect to the fBm is a fractional Wick-Itô-Skorohod integral and
$\alpha$ is some $\mathbb{F}$-adapted process which we have to determine.

Applying the integration by parts formula of Proposition 3.1, to
$p^{*}\left(t\right)$ and $Y\left(t\right)$, we obtain

$\displaystyle\mathbb{E}[p^{*}(T)Y(T)]$
$\displaystyle=\mathbb{E}[\int_{0}^{T}p^{*}(t)dY(t)]+\mathbb{E}[\int_{0}^{T}Y(t)dp^{*}(t)]+\mathbb{E}[\int_{0}^{T}\int_{0}^{T}q^{*}(t)\psi_{\delta}^{*}(t)\varphi_{H}(s,t)dsdt]$
$\displaystyle=\mathbb{E}[\int_{0}^{T}p^{*}(t)\\{\partial_{x}b^{*}(t)Y(t)+\partial_{\bar{x}}b^{*}(t)Y(t-\delta)+\mathbb{\tilde{E}}[\partial_{m}b^{*}(t)(\tilde{X}^{*}(t))\tilde{Y}(t)]$
$\displaystyle\hskip
8.5359pt+\mathbb{\tilde{E}}[\partial_{\bar{m}}b^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{Y}(t-\delta)]+\partial_{u}b^{*}(t)(u(t)-u^{*}(t))\\}dt]-\mathbb{E}[\int_{0}^{T}Y(t)\alpha(t)dt]$
$\displaystyle\hskip
8.5359pt+\mathbb{E}[\int_{0}^{T}\int_{0}^{T}q^{*}(s)\psi^{*}_{\delta}(t)\varphi_{H}(s,t)dsdt]$

where we assumed that $Y(t)q^{*}(t)\in\mathcal{L}_{H}^{1,2}([0,T])$ and
$\psi^{*}_{\delta}(t)p^{*}(t)\in\mathcal{L}_{H}^{1,2}([0,T])$. By Fubini’s
theorem, remark (4.1), remplacing $\psi^{*}_{\delta}(t)$ by its value and by a
change of variables using the fact that $Y(t)=0$ for all $t\in[-\delta,0]$ and
, we get

$\displaystyle\mathbb{E}[p^{*}(T)Y(T)]$
$\displaystyle=\mathbb{E}[\int_{0}^{T}Y(t)\\{p^{*}(t)\partial_{x}b^{*}(t)+p^{*}(t+\delta)\partial_{\bar{x}}b^{*}(t+\delta)\chi_{[0,T-\delta]}(t)-\alpha(t)$
$\displaystyle\hskip
17.07182pt+\mathbb{\tilde{E}}[\tilde{p}^{*}(t)\partial_{m}\tilde{b}^{*}(t)(X^{*}(t))]+\mathbb{\tilde{E}}[\tilde{p}^{*}(t+\delta)\partial_{\bar{m}}\tilde{b}^{*}(t+\delta)(X^{*}(t))]\chi_{[0,T-\delta]}(t)$
$\displaystyle\hskip
17.07182pt+\int_{0}^{T}\\{\tilde{\mathbb{E}}[\tilde{q}^{*}(s)]\partial_{m}{\sigma}^{*}(t)(X^{*}(t))\varphi_{H}(t,s)$
$\displaystyle\hskip
17.07182pt+\tilde{\mathbb{E}}[\tilde{q}^{*}(s)]\partial_{\bar{m}}{{\sigma}}^{*}(t+\delta)(X^{*}(t))\chi_{[0,T-\delta]}(t)\varphi_{H}(t+\delta,s)\\}ds\\}dt]$
$\displaystyle\hskip
17.07182pt+\mathbb{E}[\int_{0}^{T}p^{*}(t)\partial_{u}b^{*}(t)(u(t)-u^{*}(t))dt].$
(9)

where $(\tilde{p}^{*},\tilde{q}^{*})$ is an independant copy of
$(p^{*},q^{*})$ defined on some probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$.

By substituting $\left(\ref{PY}\right)$ in $\left(\ref{J}\right)$ and using
the terminal value of the BSDE, we get

$\displaystyle 0$
$\displaystyle\leq\mathbb{E}[\int_{0}^{T}Y(t)\\{p^{*}(t)\partial_{x}b^{*}(t)+p^{*}(t+\delta)\partial_{\bar{x}}b^{*}(t+\delta)\chi_{[0,T-\delta]}(t)+\mathbb{\tilde{E}}[\tilde{p}^{*}(t)\partial_{m}\tilde{b}^{*}(t)(X^{*}({t}))]$
$\displaystyle\hskip
11.38092pt+\mathbb{\tilde{E}}[\tilde{p}^{*}(t+\delta)\partial_{\bar{m}}\tilde{b}^{*}(t+\delta)(X^{*}({t}))]\chi_{[0,T-\delta]}(t)-\alpha(t)+\int_{0}^{T}\\{\tilde{\mathbb{E}}[\tilde{q}^{*}(s)]\partial_{m}{{\sigma}}^{*}(t)(X^{*}(t))\varphi_{H}(t,s)$
$\displaystyle\hskip
11.38092pt+\tilde{\mathbb{E}}[\tilde{q}^{*}(s)]\partial_{\bar{m}}{{\sigma}}^{*}(t+\delta)(X^{*}(t))\chi_{[0,T-\delta]}(t)\varphi_{H}(t+\delta,s)\\}ds+\partial_{x}f^{*}(t)$
$\displaystyle\hskip
11.38092pt+\partial_{\bar{x}}f^{*}(t+\delta)\chi_{[0,T-\delta]}(t)+\mathbb{\tilde{E}}[\partial_{{m}}\tilde{f}^{*}(t)(X^{*}(t))]+\mathbb{\tilde{E}}[\partial_{\bar{m}}\tilde{f}^{*}(t+\delta)(X^{*}(t))]\chi_{[0,T-\delta]}(t)\\}dt]$
$\displaystyle\hskip
11.38092pt+\mathbb{E}[\int_{0}^{T}\\{p^{*}(t)\partial_{u}b^{*}(t)+\partial_{u}f^{*}(t)\\}(u(t)-u^{*}(t))dt].$

Letting the integrand which contains $Y(t)$ equal to zero, we get

$\displaystyle\alpha(t)$
$\displaystyle=p^{*}(t)\partial_{x}b^{*}(t)+p^{*}(t+\delta)\partial_{\bar{x}}b^{*}(t+\delta)\chi_{[0,T-\delta]}(t)+\mathbb{\tilde{E}}[\tilde{p}^{*}(t)\partial_{m}\tilde{b}^{*}(t)(X^{*}({t}))]$
$\displaystyle\hskip
11.38092pt+\mathbb{\tilde{E}}[\tilde{p}^{*}(t+\delta)\partial_{\bar{m}}\tilde{b}^{*}(t+\delta)(X^{*}({t}))]\chi_{[0,T-\delta]}(t)+\int_{0}^{T}\\{\tilde{\mathbb{E}}[\tilde{q}^{*}(s)]\partial_{m}{{\sigma}}^{*}(t)(X^{*}(t))\varphi_{H}(t,s)$
$\displaystyle\hskip
11.38092pt+\tilde{\mathbb{E}}[\tilde{q}^{*}(s)]\partial_{\bar{m}}{{\sigma}}^{*}(t+\delta)(X^{*}(t))\chi_{[0,T-\delta]}(t)\varphi_{H}(t+\delta,s)\\}ds+\partial_{x}f^{*}(t)+\partial_{\bar{x}}f^{*}(t+\delta)\chi_{[0,T-\delta]}(t)$
$\displaystyle\hskip
11.38092pt+\mathbb{\tilde{E}}[\partial_{m}\tilde{f}^{*}(t)(X^{*}(t))]+\mathbb{\tilde{E}}[\partial_{\bar{m}}\tilde{f}^{*}(t+\delta)(X^{*}(t))]\chi_{[0,T-\delta]}(t).$

where, for simplicity of notations, we have set :

$\begin{array}[c]{ll}\partial_{x}\tilde{b}^{*}(t)&:=\partial_{x}b(t,\tilde{X}^{\ast}(t),\tilde{X}^{\ast}(t-\delta),M^{\ast}(t),M^{\ast}(t-\delta),\tilde{u}^{\ast}(t)),\\\
\partial_{m}\tilde{b}^{*}(t)(.)&:=\partial_{m}b(t,\tilde{X}^{\ast}(t),\tilde{X}^{\ast}(t-\delta),M^{\ast}(t),M^{\ast}(t-\delta),\tilde{u}^{\ast}(t))(.),\\\
\partial_{m}\tilde{f}^{*}(t)(.)&:=\partial_{m}f(t,\tilde{X}^{\ast}(t),\tilde{X}^{\ast}(t-\delta),M^{\ast}(t),M^{\ast}(t-\delta),\tilde{u}^{\ast}(t))(.).\end{array}$

We define now the Hamiltonian associated to our problem by:

$H:\Omega\times\left[0,T\right]\times\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\times\mathcal{P}_{2}(\mathbb{R})\times\mathcal{U}\times\mathbb{R}\times\mathcal{R}\rightarrow\mathbb{R}$

by

$\begin{array}[c]{ll}H(t,x,\overline{x},m,\overline{m},u,p,q(.))&=f(t,x,\overline{x},m,\overline{m},u)+p\text{\
}b(t,x,\overline{x},m,\overline{m},u)+\sigma(t,m,\overline{m}){\textstyle\int_{0}^{T}}q(s)\varphi_{H}(s,t)ds.\end{array}$
(10)

where $\mathcal{R}$ is the set of continuous functions from $[0,T]$ into
$\mathbb{R}$.

For $u\in\mathcal{A}_{\mathbb{F}}$ with corresponding solution $X=X^{u}$,
define, whenever solutions exist, $(p,q):=(p^{u},q^{u})$, by the adjoint
equation, in terms of the Hamiltonian, as follows:

$\left\\{\begin{array}[c]{lll}dp(t)&=&-\\{\partial_{x}H(t)+\mathbb{E}[\partial_{\bar{x}}H(t+\delta)\chi_{[0,T-\delta]}(t)|\mathcal{F}_{t}]\\\
&+&\tilde{\mathbb{E}}[\partial_{m}\tilde{H}(t)(X(t))]+\mathbb{\tilde{E}}[\partial_{\bar{m}}\tilde{H}(t+\delta)(X(t))\chi_{[0,T-\delta]}(t)]\\}dt+q(t)dB^{H}(t);t\in\left[0,T\right],\\\
\\\
p(T)&=&\partial_{x}g(T)+\mathbb{\tilde{E}}[\partial_{m}\tilde{g}(T)({X}(T))]\end{array}\right.$
(11)

We assume that $(p,q)$ is in
$\mathcal{L}_{H}^{1,2}([0,T])\times\mathcal{L}_{H}^{1,2}([0,T])$, the integral
with respect to the fBm is understood in the fractional Wick-Itô-Skorohod
sense.

For simplicity of notations, we have put :

$\begin{array}[c]{ll}\partial_{x}H(t)&:=\partial_{x}H(t,X(t),X(t-\delta),M(t),M(t-\delta),u(t),p(t),q(.)),\\\
\partial_{m}\tilde{H}(t)(.)&:=\partial_{m}H(t,\tilde{X}(t),\tilde{X}(t-\delta),M(t),M(t-\delta),\tilde{u}(t),\tilde{p}(t),\tilde{q}(.))(.),\\\
\partial_{x}g(T)&:=\partial_{x}g(X(T),M(T)),\text{ \
}\partial_{m}\tilde{g}(T)(.)=\partial_{m}g(\tilde{X}(T),M(T))(.).\end{array}$

######  Remark 4.2.

By the definition of the Hamiltonian given above, the time advanced BSDE (11)
is expressed as a first part which appears to be linear and a second part with
coefficients that contain the laws or more precisely the joint distribution of
the solution processes. This type of Backward Stochastic Differential
Equations was never been studied before. However, when there is no mean-field
terms and no advance in time, several resolutions were proposed see for
instance [5], [7] and for a more general setting [16]. In the section devoted
to the applications, we suggest some dynamics where the mean-field term
appears mainly in the terminal cost functional, the resolution of the BSDE
(11) is in this case possible following a recursive procedure.

We establish in the following theorem the necessary condition of optimality.

######  Theorem 4.1 (Necessary condition of optimality).

We assume that the couple $(u^{\ast},X^{\ast})$ is optimal. Suppose that there
exists $p^{*}(t)$, $q^{*}(t)$, solutions of the adjoint equation
$\left(\ref{eq:2}\right)$ associated to the pair $(u^{\ast},X^{\ast})$ and
that $Y(t)q^{*}(t)\in\mathcal{L}_{H}^{1,2}([0,T])$ and
$\psi^{*}_{\delta}(t)p^{*}(t)\in\mathcal{L}_{H}^{1,2}([0,T])$. Then

$0\leq\mathbb{E[}\int_{0}^{T}\partial_{u}H^{\ast}(t)\left(u(t)-u^{\ast}(t)\right)dt].$

###### Proof.

Suppose that $u^{\ast}$ is an optimal control, then we have

$\begin{array}[c]{l}0\leq\text{ }\underset{\theta\rightarrow
0}{\lim}\tfrac{1}{\theta}(J(u^{\theta})-J(u^{\ast}))\\\
=\mathbb{E[}p^{*}(T)Y(T)]+\mathbb{E[}{\textstyle\int_{0}^{T}}\\{\partial_{x}f^{*}(t)Y(t)+\partial_{\bar{x}}f^{*}(t)Y(t-\delta)\\\
\hskip
8.5359pt+\mathbb{\tilde{E}}[\partial_{m}f^{*}(t)(\tilde{X}^{*}(t))\tilde{Y}(t)]+\mathbb{\tilde{E}}[\partial_{\bar{m}}f^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{Y}(t-\delta)]+\partial_{u}f^{*}(t)\left(u(t)-u^{\ast}(t)\right)\\}dt].\end{array}$
(12)

Applying again the integration by parts formula of Proposition 3.1 to
$p^{*}\left(t\right)$ and $Y\left(t\right)$ then using Fubini’s theorem,
remplacing $\psi^{*}_{\delta}(t)$ by its value, and using a change of variable
and the fact that $Y(t)=0$ for all $t\in[-\delta,0]$, we get

$\displaystyle\mathbb{E}[A_{1}]$
$\displaystyle=\mathbb{E}[\int_{0}^{T}p^{*}(t)dY(t)]+\mathbb{E}[\int_{0}^{T}Y(t)dp^{*}(t)]+\mathbb{E}[\int_{0}^{T}\int_{0}^{T}q^{*}(s)\psi^{*}_{\delta}(t)\varphi_{H}(t,s)dsdt]$
$\displaystyle=\mathbb{E}[\int_{0}^{T}Y(t)\\{p^{*}(t)\partial_{x}b^{*}(t)+p^{*}(t+\delta)\partial_{\bar{x}}b^{*}(t+\delta)\chi_{[0,T-\delta]}(t)+\mathbb{\tilde{E}}[\tilde{p}^{*}(t)\partial_{m}\tilde{b}^{*}(t)(X^{*}(t))]$
$\displaystyle\hskip
5.69046pt+\mathbb{\tilde{E}}[\tilde{p}^{*}(t+\delta)\partial_{\bar{m}}\tilde{b}^{*}(t+\delta)(X^{*}(t))]\chi_{[0,T-\delta]}(t)-\\{\partial_{x}H^{*}(t)+\mathbb{E}[\partial_{\bar{x}}H^{*}(t+\delta)\chi_{[0,T-\delta]}(t)|\mathcal{F}_{t}]$
$\displaystyle\hskip
5.69046pt+\tilde{\mathbb{E}}[\partial_{m}\tilde{H}^{*}(t)(X^{*}(t))]+\mathbb{E}[\mathbb{\tilde{E}}[\partial_{\bar{m}}\tilde{H}^{*}(t+\delta)(X^{*}(t))\chi_{[0,T-\delta]}(t)]|\mathcal{F}_{t}]\\}$
$\displaystyle\hskip
5.69046pt+\int_{0}^{T}\\{\tilde{\mathbb{E}}[\tilde{q}^{*}(s)]\partial_{m}{{\sigma}}^{*}(t)(X^{*}(t))\varphi_{H}(t,s)+\tilde{\mathbb{E}}[\tilde{q}^{*}(s)]\partial_{\bar{m}}{{\sigma}}^{*}(t+\delta)(X^{*}(t))\chi_{[0,T-\delta]}(t)\varphi_{H}(t+\delta,s)\\}ds\\}dt]$
$\displaystyle\hskip
5.69046pt+\mathbb{E}[\int_{0}^{T}p^{*}(t)\partial_{u}b^{*}(t)(u(t)-u^{*}(t))dt].]$
(13)

where $A_{1}:=\mathbb{E}[p^{*}(T)Y(T)]$, then by (12), (4) and applying the
definition of the Hamiltonian, we get the desired result, that is

$0\leq\mathbb{E[}{\textstyle\int_{0}^{T}}\partial_{u}H^{\ast}(t)\left(u(t)-u^{\ast}(t)\right)dt].$

∎

##  5 Sufficient maximum principle

In this section, we proove sufficient stochastic maximum principle.

######  Theorem 5.1 (Sufficient condition of optimality).

Let $u^{\ast}\in\mathcal{A}_{\mathbb{F}}$ with corresponding controlled state
$X^{\ast}$. Suppose that there exist $p^{*}(t)$, $q^{*}(t)$ solution of the
associated adjoint equation $\left(\ref{eq:2}\right)$. Assume the following:

  1. 1.

$X^{u}(t)q^{*}(t)\in\mathcal{L}_{H}^{1,2}([0,T])$,
$p^{*}(t)\sigma(t,M(t),M(t-\delta))\in\mathcal{L}_{H}^{1,2}([0,T])$ $\forall
u\in\mathcal{A}_{\mathbb{F}}$.

  2. 2.

(Concavity) The functions

$\begin{array}[c]{ll}(x,\bar{x},m,\bar{m},u)&\mapsto
H(t,x,\bar{x},m,\bar{m},u,p^{*}(t),q^{*}(.)),\\\ (x,m)&\mapsto
g(x,m)\text{,}\end{array}$

are concave for each $t\in[0,T]$ almost surely.

  3. 3.

(Maximum condition)

$H(t,X^{*}(t),X^{*}(t-\delta),M^{*}(t),M^{*}(t-\delta),u^{*}(t),p^{*}(t),q^{*}(.))=\hskip
56.9055pt\underset{u\in\mathcal{U}}{\text{
}\sup}H(t,X^{*}(t),X^{*}(t-\delta),M^{*}(t),M^{*}(t-\delta),u,p^{*}(t),q^{*}(.))$

for all $t\in[0,T]$ almost surely.

Then $(u^{\ast},X^{\ast})$ is an optimal couple for our problem.

###### Proof.

Let $u\in\mathcal{A}_{\mathbb{F}}$ be a generic admissible control, and
$X=X^{u}$ the corresponding controlled state process. By the definition of the
performance functional $J$ given by $\left(\ref{P}\right)$, we have

$\begin{array}[c]{lll}J(u)-J(u^{\ast})&=&A_{2}+A_{3},\end{array}$ (14)

where

$\displaystyle\begin{array}[c]{lll}A_{2}&:=&\mathbb{E[}{\textstyle\int_{0}^{T}}\\{f(t)-f^{\ast}(t)\\}dt],\end{array}$
$\displaystyle\begin{array}[c]{lll}A_{3}&:=&\mathbb{E[}g(T)-g^{\ast}(T)].\end{array}$

Applying the definition of the Hamiltonian $\left(\ref{ham}\right)$, we have

$\begin{array}[c]{ccc}A_{2}&=&\mathbb{E[}{\textstyle\int_{0}^{T}}\\{H(t)-H^{\ast}(t)-p^{*}(t){b}^{\prime}(t)-{\textstyle\int_{0}^{T}}q^{*}(s){\sigma}^{\prime}(t)\varphi_{H}(s,t)ds\\}dt]\end{array}$
(15)

where we used the following notations

$\begin{array}[c]{ll}b(t)&:=b(t,X(t),X(t-\delta),M(t),M(t-\delta),u(t)),\\\
b^{*}(t)&:=b(t,X^{*}(t),X^{*}(t-\delta),M^{*}(t),M^{*}(t-\delta),u^{*}(t)),\\\
f(t)&:=f(t,X(t),X(t-\delta),M(t),M(t-\delta),u(t)),\\\
f^{*}(t)&:=f(t,X^{*}(t),X^{*}(t-\delta),M^{*}(t),M^{*}(t-\delta),u^{*}(t)),\\\
g(T)&:=g(X(T),M(T)),\text{ \ }g^{*}(T):=g(X^{*}(T),M^{*}(T)),\\\
\sigma(t)&:=\sigma(t,M(t),M(t-\delta)),\text{ \
}\sigma^{*}(t):=\sigma(t,M^{*}(t),M^{*}(t-\delta)),\\\
H(t)&:=H(t,X(t),X(t-\delta),M(t),M(t-\delta),u(t),p^{*}(t),q^{*}(.)),\\\
H^{*}(t)&:=H(t,X^{*}(t),X^{*}(t-\delta),M^{*}(t),M^{*}(t-\delta),u^{*}(t),p^{*}(t),q^{*}(.)),\\\
{b}^{\prime}(t)&:=b(t)-b^{\ast}(t),\\\
{\sigma}^{\prime}(t)&:=\sigma(t)-\sigma^{\ast}(t)\\\
{X}^{\prime}(t)&:=X(t)-X^{\ast}(t)\end{array}$

Now using the concavity of $g$ and the terminal value of the BSDE (11)
associated to $(u^{*},X^{*})$, we get by Fubini’s theorem

$\displaystyle A_{3}$
$\displaystyle\leq\mathbb{E}[\partial_{x}g^{*}(T){X}^{\prime}(T)]+\mathbb{E}[\tilde{\mathbb{E}}[\partial_{m}g^{*}(T)(\tilde{X}(T))\tilde{X}^{\prime}(T)]]$
$\displaystyle=\mathbb{E}[(\partial_{x}g^{*}(T)+\tilde{\mathbb{E}}[\partial_{m}\tilde{g}^{*}(T)({X}(T))]){X}^{\prime}(T)]$
$\displaystyle=\mathbb{E}[p^{*}(T){X}^{\prime}(T)]$

Applying the integration by parts formula to $p^{*}(t)$ and $X^{\prime}(t)$,
we get

$\displaystyle\mathbb{E}[p^{*}(T)X^{\prime}(T)]$
$\displaystyle=\mathbb{E}[\int_{0}^{T}p^{*}(t)dX^{\prime}(t)]+\mathbb{E}[\int_{0}^{T}X^{\prime}(t)dp^{*}(t)]+\mathbb{E}[\int_{0}^{T}\int_{0}^{T}q^{*}(s)\sigma^{\prime}(t)\varphi_{H}(t,s)dsdt]$
$\displaystyle=\mathbb{E}[\int_{0}^{T}p^{*}(t)b^{\prime}(t)dt]-\mathbb{E}[\int_{0}^{T}X^{\prime}(t)\\{\partial_{x}{H}^{*}(t)+\partial_{\bar{x}}H^{*}(t+\delta)\chi_{[0,T-\delta]}(t)$
$\displaystyle\hskip
14.22636pt+\mathbb{\tilde{E}}[\partial_{m}\tilde{H}^{*}(t)({X}^{*}(t))]+\mathbb{\tilde{E}}[\partial_{\bar{m}}\tilde{H}^{*}(t)({X}^{*}(t))]\chi_{[0,T-\delta]}(t)\\}dt]$
$\displaystyle\hskip
14.22636pt+\mathbb{E}[\int_{0}^{T}\int_{0}^{T}q^{*}(s)\sigma^{\prime}(t)\varphi_{H}(t,s)dtds].$

Note that by the change of variables $r=t+\delta$, we have

$\mathbb{E}[\int_{0}^{T-\delta}{X}^{\prime}(t)\partial_{\bar{x}}H^{*}(t+\delta)dt]=\mathbb{E}[\int_{\delta}^{T}{X}^{\prime}(r-\delta)\partial_{\bar{x}}H^{*}(r)dr]=\mathbb{E}[\int_{0}^{T}{X}^{\prime}(r-\delta)\partial_{\bar{x}}H^{*}(r)dr]$

where we used that
$\mathbb{E}[\int_{0}^{\delta}{X}^{\prime}(r-\delta)\partial_{\bar{x}}H^{*}(r)dr]=\mathbb{E}[\int_{-\delta}^{0}{X}^{\prime}(u)\partial_{\bar{x}}H^{*}(u+\delta)du]=0$
since ${X}^{\prime}(u)=0$ for all $u\in[-\delta,0]$, because
$X^{*}(t)=X(t)=x_{0}(t)$ for all $t\in[-\delta,0]$.

Similarly, we get using the previous argument and by Fubini’s theorem

$\mathbb{E}[\int_{0}^{T}X^{\prime}(t)\mathbb{\tilde{E}}[\partial_{\bar{m}}\tilde{H}^{*}(t+\delta)(X^{*}(t))]\chi_{[0,T-\delta]}(t)dt]=\mathbb{E}[\int_{0}^{T}\mathbb{\tilde{E}}[\partial_{\bar{m}}{H}^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{X}^{\prime}(t-\delta)]dt]$

Hence, we have

$\displaystyle\mathbb{E}[p^{*}(T)X^{\prime}(T)]$
$\displaystyle=\mathbb{E}[\int_{0}^{T}p^{*}(t)b^{\prime}(t)dt]-\mathbb{E}[\int_{0}^{T}X^{\prime}(t)\partial_{x}H^{*}(t)dt]-\mathbb{E}[\int_{0}^{T}\partial_{\bar{x}}H^{*}(t)X^{\prime}(t-\delta)dt]$
$\displaystyle\hskip
14.22636pt-\mathbb{E}[\int_{0}^{T}\mathbb{\tilde{E}}[\partial_{m}H^{*}(t)(\tilde{X}^{*}(t))\tilde{X}^{\prime}(t)]dt]-\mathbb{E}[\int_{0}^{T}\mathbb{\tilde{E}}[\partial_{\bar{m}}H^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{X}^{\prime}(t-\delta)]dt]$
$\displaystyle\hskip
14.22636pt+\mathbb{E}[\int_{0}^{T}\int_{0}^{T}q^{*}(s)\sigma^{\prime}(t)\varphi_{H}(t,s)dtds].$
(16)

By (14), (15) and (5), we obtain

$\displaystyle J(u)-J(u^{*})$
$\displaystyle\leq\mathbb{E}[\int_{0}^{T}(H(t)-H^{*}(t))dt]-\mathbb{E}[\int_{0}^{T}\partial_{x}H^{*}(t){X}^{\prime}(t)dt]-\mathbb{E}[\int_{0}^{T}\partial_{\bar{x}}H^{*}(t){X}^{\prime}(t-\delta)dt]$
$\displaystyle\hskip
22.76228pt-\mathbb{E}[\int_{0}^{T}\mathbb{\tilde{E}}[\partial_{m}H^{*}(t)(\tilde{X}^{*}(t))\tilde{X}^{\prime}(t)]dt]-\mathbb{E}[\int_{0}^{T}\mathbb{\tilde{E}}[\partial_{\bar{m}}H^{*}(t)(\tilde{X}^{*}(t-\delta))\tilde{X}^{\prime}(t-\delta)]dt].$
$\displaystyle\leq 0.$

due to the concavity assumption on $H$ and because $u^{*}$ satisfies the
maximum condition, the first order derivative in $u^{*}$ vanishes. ∎

##  6 Applications

The main applications of mean-field dynamics that appear in the literature
rely mainly on a dependence upon the probability measures through functions of
scalar moments of the measures. More precisely, we assume that:

$b(t,x,\bar{x},m,\bar{m},u)=\hat{b}(t,x,\bar{x},(\psi_{1},m),(\psi_{2},\bar{m}),u),$
$\sigma(t,m,\bar{m})=\hat{\sigma}(t,(\phi_{1},m),(\phi_{2},\bar{m})),$
$f(t,x,\bar{x},m,\bar{m},u)=\hat{f}(t,x,\bar{x},(\gamma_{1},m),(\gamma_{2},\bar{m}),u),$
$g(x,m)=\hat{g}(x,(\gamma_{3},m)).$

for some scalar functions $\psi_{1}$, $\psi_{2}$, $\phi_{1}$, $\phi_{2}$,
$\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ with at most quadratic growth at
$\infty$. The functions $\hat{b}$, $\hat{f}$ are defined on
$[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathcal{U}$,
the function $\hat{\sigma}$ is defined on
$[0,T]\times\mathbb{R}\times\mathbb{R}$ and $\hat{g}$ is defined on
$\mathbb{R}\times\mathbb{R}$. The notation $(\psi,m)$ denotes the integral of
the function $\psi$ with respect to the probability measure $m$. The
Hamiltonian that we defined in the previous section takes now the following
form:

$\begin{array}[c]{ll}H(t,x,\overline{x},m,\overline{m},u,p,q(.))&=\hat{f}(t,x,\overline{x},(\gamma_{1},m),(\gamma_{2},\bar{m}),u)+p\text{\
}\hat{b}(t,x,\overline{x},(\psi_{1},m),(\psi_{2},\bar{m}),u)\\\ &\hskip
14.22636pt+\hat{\sigma}(t,(\phi_{1},m),(\phi_{2},\bar{m})){\textstyle\int_{0}^{T}}q(s)\varphi_{H}(s,t)ds.\end{array}$

According to the definition of the differentiability with respect to functions
of measures recalled in the preliminaries, the derivative of the Hamiltonian
with respect to the measure $m$ for instance, is computed as follows :

$\displaystyle\partial_{m}H(t,x,\overline{x},m,\overline{m},u,p,q(.))(x^{\prime})$
$\displaystyle=\partial_{x^{\prime}}\hat{f}(t,x,\bar{x},(\gamma_{1},m),(\gamma_{2},\bar{m}),u)\gamma_{1}^{\prime}(x^{\prime})$
$\displaystyle\hskip
22.76228pt+p\times\partial_{x^{\prime}}\hat{b}(t,x,\bar{x},(\psi_{1},m),(\psi_{2},\bar{m}),u)\psi_{1}^{\prime}(x^{\prime})$
$\displaystyle\hskip
22.76228pt+\partial_{x^{\prime}}\hat{\sigma}(t,(\phi_{1},m),(\phi_{2},\bar{m}))\phi_{1}^{\prime}(x^{\prime}){\textstyle\int_{0}^{T}}q(s)\varphi_{H}(s,t)ds.$

The terminal value of the adjoint BSDE(11) which is
$p(T)=\partial_{x}g(T)+\mathbb{\tilde{E}}[\partial_{m}\tilde{g}(T)({X}(T))]$,
can be written in terms of the derivatives of the function $\hat{g}$ as
follows:

$p(T)=\partial_{x}\hat{g}(X_{T},\mathbb{E}[\gamma_{3}(X_{T})])+\mathbb{\tilde{E}}[\partial_{x^{\prime}}\hat{g}(\tilde{X}_{T},\mathbb{E}[\gamma_{3}(X_{T}))]\gamma_{3}^{\prime}(X_{T})$

where $\tilde{X}_{T}$ is an independant copy of $X_{T}$. We study in the
following two applications that illustrate the previous results.

###  6.1 Optimal consumption from a cash flow with delay

We consider the problem of an optimal consumption with a cash flow with delay
$X:=X^{\rho}$ given by

$\left\\{\begin{array}[c]{lll}dX(t)&=[X(t-\delta)-\rho(t)]dt+\beta(t)dB^{H}(t),\text{
\ }t\in[0,T],\\\ X(t)&=x_{0}(t)\text{ \ }t\in[-\delta,0].\end{array}\right.$
(17)

where $\rho$ is the relative consumption rate (our control), $x_{0}$ a bounded
deterministic function, $\delta$ a strictly positive constant and $\beta$ is a
given deterministic function in $\mathbb{L}_{H}^{2}([0,T])$. The integral with
respect to the fBm is therefore a Wiener type integral.

The problem we consider is to find the consumption rate $\rho^{*}$ such that

$J(\rho^{*})=\sup_{\rho\in\mathcal{\bar{A}}_{\mathbb{F}}}J(\rho)$

where

$J(\rho)=\mathbb{E}[\int_{0}^{T}\log(\rho(t))dt+\xi_{1}\mathbb{E}[X(T)]],$

over the set $\mathcal{\bar{A}}_{\mathbb{F}}$ of admissible controls which are
$\mathbb{F}$-adapted processes with values in $\mathbb{R}^{*}_{+}$,
$\xi_{1}>0$ is a given bounded $\mathcal{F}_{T}$-measurable random variable
assumed to be in ${\mathcal{L}}_{H}^{1,2}([0,T])$, we also assume that $X$
exists and belongs to $L^{2}(\Omega\times[0,T])$.

Note that the running cost functional we consider in this example is the
function $\rho(t)\mapsto\log(\rho(t))$ which is a utility function. Moreover,
in order to control the fluctuations of the terminal time-value of the cash
flow $X^{\rho}_{T}$, we chose to introduce its mean in the terminal cost
functional. Therefore according to the notations used in the previous
paragraph, the terminal cost functional is of mean-field type, more precisely
it has the following form:

$g(X_{T},\mathbb{P}_{X_{T}})=g(\mathbb{P}_{X_{T}})=\xi_{1}\mathbb{E}[X_{T}]=\hat{g}_{\xi_{1}}((\textsc{Id},\mathbb{P}_{X_{T}}))$

where $\hat{g}_{\xi_{1}}(x^{\prime})=\xi_{1}x^{\prime}$, therefore
$\hat{g}^{\prime}_{\xi_{1}}(x^{\prime})=\xi_{1}$ a.s.

The Hamiltonian of this control problem is given by :

$H(t,x,\bar{x},\rho,p,q(.))=\log(\rho)+(\bar{x}-\rho)p+\beta(t)\int_{0}^{T}q(s)\varphi_{H}(s,t)ds,$

where $(p,q)$ is the solution of the associated adjoint BSDE

$\left\\{\begin{array}[c]{lll}dp(t)&=-\mathbb{E}[p(t+\delta)\chi_{[0,T-\delta]}(t)|\mathcal{F}_{t}]dt+q(t)dB^{H}(t),\text{
\ }t\in[0,T],\\\ p(T)&=\xi_{1}.\end{array}\right.$ (18)

Inspired by the resolution of the linear BSDE driven by a fractional Brownian
motion with Hurst parameter $H>1/2$ done in [5] and [16], we propose a
resolution of the anticipated BSDE (18) by solving a sequence of linear BSDEs
following this procedure :

Step 1  If $t\in[T-\delta,T]$, the previous BSDE takes the form

$\left\\{\begin{array}[c]{lll}dp(t)&=q(t)dB^{H}(t),\text{ \
}t\in[T-\delta,T],\\\ p(T)&=\xi_{1}.\end{array}\right.$

which has the solution

$p(t)=\hat{\mathbb{E}}[\xi_{1}|\mathcal{F}_{t}],\text{ \
}q(t)=\hat{\mathbb{E}}[D^{H}_{t}\xi_{1}|\mathcal{F}_{t}],\text{ \
}t\in[T-\delta,T]$

where $\hat{\mathbb{E}}$ is the quasi-conditional expectation, see [12] for
its definition.

Step 2 : If $t\in[T-2\delta,T-\delta]$ and $T-2\delta>0$, we obtain the BSDE

$\left\\{\begin{array}[c]{lll}dp(t)&=-\mathbb{E}[p(t+\delta)|\mathcal{F}_{t}]+q(t)dB^{H}(t),\text{
\ }t\in[T-2\delta,T-\delta],\\\ p(T-\delta)&\text{ \ }\text{known from step
1}.\end{array}\right.$

this BSDE has an expicit solution given by

$p(t)=\hat{\mathbb{E}}[p(T-\delta)+\int_{t}^{T}\theta(s)dt|\mathcal{F}_{t}],\text{
\ }q(t)=D^{H}_{t}p(t),\text{ \ }t\in[T-2\delta,T-\delta].$

where $\theta(t)=\mathbb{E}[p(t+\delta)|\mathcal{F}_{t}]$ and $p(t+\delta)$ is
known by step 1.

We continue like this by induction up to and including step n, where n is such
that $T-n\delta\leq 0<T-(n-1)\delta$ and we solve the corresponding BSDE on
the time interval $[0,T-(n-1)\delta]$ and we solve the corresponding BSDE on
the time interval $[0,T-(n-1)\delta]$.

Maximizing H with respect to $\rho$ gives the following first order condition
for an optimal consumption rate $\rho^{*}$ :

${\partial_{\rho}H^{*}(t)}=\frac{1}{\rho^{*}(t)}-p(t)=0,$

Then if

$p(t)>0\text{ \ for all }t\in[0,T].$ (19)

We get

$\rho^{*}(t)=\frac{1}{p(t)}\text{ \ for all }t\in[0,T].$ (20)

where $p$ satisfies the previous anticipated BSDE.

######  Theorem 6.1.

Let $(p,q)$ be the solution of the BSDE (18) and suppose that (19) holds. Then
any optimal consumption rate $\rho^{*}$ satisfies (20) and the corresponding
optimal wealth $X^{*}$ is given by equation (17).

###  6.2 Linear-Quadratique Problem with delay

We consider now a Linear-Quadratique (LQ) model for a controlled process
$X=X^{\alpha}$ given by the following delayed stochastic differential
equation:

$\left\\{\begin{array}[c]{lll}dX(t)&=[\beta_{1}(t)X(t-\delta)+\alpha(t)]dt+\beta_{2}(t)dB^{H}(t),\text{
\ }t\in[0,T],\\\ X(t)&=x_{0}(t)\text{ \ }t\in[-\delta,0].\end{array}\right.$
(21)

where $\delta>0$ is a given constant, $\beta_{1}$, $x_{0}$ are given bounded
deterministic functions, $\beta_{2}$ is a given deterministic function in
$\mathbb{L}_{H}^{2}([0,T])$. The integral with respect to the fBm is therefore
a Wiener type integral and $\alpha\in\mathcal{A}_{\mathbb{F}}$ is our control
process, the set $\mathcal{A}_{\mathbb{F}}$ are the admissible controls
assumed to be square integrable $\mathbb{F}$-adapted processes with real
values.

We want to minimize the expected value of $(X_{T}-\mathbb{E}[X_{T}])^{2}$
which is the variance of $X_{T}$ with a minimal average use of energy,
measured by the integral $\mathbb{E}[\int_{0}^{T}\alpha^{2}(t)dt]$, more
precisely, the performance functional is of the following form:

$J(\alpha)=-\frac{1}{2}(\textsc{Var}(X_{T})+\mathbb{E}[\int_{0}^{T}\alpha^{2}(t)dt]).$
(22)

Our goal is therefore to find the control process
$\alpha^{*}\in\mathcal{A}_{\mathbb{F}}$, such that

$J(\alpha^{*})=\sup_{\alpha\in\mathcal{A}_{\mathbb{F}}}J(\alpha)$ (23)

######  Remark 6.1.

Including the variance of the state process in the cost functional in order to
keep it small is a way to control its sensitivity with respect to the possible
variations of the random events. The form of this cost functional is inspired
from [20].

Note that the terminal cost functional of our problem has the following form:

$g(X_{T},\mathbb{P}_{X_{T}})=\hat{g}(X_{T},(\textsc{Id},\mathbb{P}_{X_{T}}))=-\frac{1}{2}(X_{T}-\mathbb{E}[X_{T}])^{2}$

where $\hat{g}(x,x^{\prime})=-\frac{1}{2}(x-x^{\prime})^{2}$, therefore
$\partial_{x}\hat{g}(x,x^{\prime})=-(x-x^{\prime})=-\partial_{x^{\prime}}\hat{g}(x,x^{\prime})$.
Therefore the terminal value of the solution of the adjoint BSDE is:

$\displaystyle p(T)$
$\displaystyle=\partial_{x}\hat{g}(X_{T},(\textsc{Id},\mathbb{P}_{X_{T}}))+\mathbb{\tilde{E}}[\partial_{x^{\prime}}\hat{g}(\tilde{X}_{T},(\textsc{Id},\mathbb{P}_{X_{T}}))]$
$\displaystyle=-(X_{T}-\mathbb{E}[X_{T}])+\tilde{\mathbb{E}}[\tilde{X}_{T}-\mathbb{E}[X_{T}]]$
$\displaystyle=-(X_{T}-\mathbb{E}[X_{T}])$

where we used the fact that $\tilde{X}$ and $X$ have the same distribution.

The Hamiltonian of our control problem takes the following form:

$H(t,x,\bar{x},\alpha,p,q(.))=-\frac{1}{2}\alpha^{2}+(\beta_{1}(t)\bar{x}+\alpha)p+\beta_{2}(t)\int_{0}^{T}q(s)\varphi_{H}(s,t)ds,$

where $(p,q)$ is the solution of the associated adjoint BSDE:

$\left\\{\begin{array}[c]{lll}dp(t)&=-\beta_{1}(t+\delta)\mathbb{E}[p(t+\delta)\chi_{[0,T-\delta]}(t)|\mathcal{F}_{t}]dt+q(t)dB^{H}(t),\text{
\ }t\in[0,T],\\\ p(T)&=\xi_{2}(T).\end{array}\right.$

where we put $\xi_{2}(T)=-(X_{T}-\mathbb{E}[X_{T}])$, we proceed now as we did
in the previous exemple by solving a sequence of linear BSDEs.

The function $\alpha\mapsto H(t,X(t),X(t-\delta),\alpha(t),p(t),q(.))$ is
maximal when

$\alpha(t)=\alpha^{*}(t)=p^{*}(t)$ (24)

where $p^{*}$ satisfies:

$\left\\{\begin{array}[c]{lll}dp^{*}(t)&=-\beta_{1}(t+\delta)\mathbb{E}[p^{*}(t+\delta)\chi_{[0,T-\delta]}(t)|\mathcal{F}_{t}]dt+q^{*}(t)dB^{H}(t),\text{
\ }t\in[0,T],\\\ p(T)&=\xi_{2}^{*}(T).\end{array}\right.$ (25)

where $\xi_{2}^{*}(T)=-(X_{T}^{*}-\mathbb{E}[X_{T}^{*}])$. Therefore, we have
proved the following theorem.

######  Theorem 6.2.

The optimal control $\alpha^{*}$ of the LQ problem (23) is given by (24),
where $(X^{*},p^{*},q^{*})$ solve the couple of systems (21) and (25) of
forward-backward stochastic differential equations.

## Acknowledgement

The first author is supported by the Erasmus + International Credit mobility
between Linnaeus University and Cadi Ayyad University for the academic year
2016-2017. The research of the third author is carried out with the support of
the Norwegian Research Council, within the research project Challenges in
Stochastic Control, Information and Applications (STOCONINF), project number
250768/F20.

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# Quantum Gates Between Distant Qubits via Spin-Independent Scattering

Leonardo Banchi1 Enrico Compagno1 Vladimir Korepin2 Sougato Bose1 1Department
of Physics and Astronomy, University College London, Gower St., London WC1E
6BT, UK  2 C. N. Yang Institute for Theoretical Physics, State University of
New York at Stony Brook, NY 11794-3840, USA

###### Abstract

We show how the spin independent scattering of two initially distant qubits,
say, in distinct traps or in remote sites of a lattice, can be used to
implement an entangling quantum gate between them. The scattering takes place
under 1D confinement for which we consider two different scenarios: a 1D wave-
guide and a tight-binding lattice. We consider models with a delta interaction
between two fermionic or two bosonic particles. A qubit is encoded in two
distinct spin (or other internal) states of each particle. Our scheme, by
providing a gate between qubits which are initially too far to interact
directly, provides an alternative to photonic mediators for the scaling of
quantum computers. Fundamentally, an interesting feature is that “identical
particles” (e.g., two atoms of the same species) and the 1D confinement, are
both necessary for the action of the gate. Effects of the departure from 1D
confinement, the degree of control required on the initial motional states of
particles, and how to achieve the same, are discussed, as well as the
feasibility of atomic implementations.

Recent progress in the control of the motion of neutral atoms in restricted
geometries such as traps Muldoon ; Schlosser , 1D optical lattices fukuhara ;
GreinerNew  and wave-guides waveguide  has been astounding. Naturally, the
question arises as to whether they can be used in a similar manner as photons
are used, i.e., as “flying qubits” for logic as well as for connecting well
separated registers in quantum information processing. Quantum logic between
flying qubits exploits their indistinguishability and assume them to be
mutually non-interacting – hence the names “linear optics” KLM  and “free
electron” Beenakker  quantum computation. In fact, for such an approach to be
viable one has to engineer circumstances so that the effect of the inter-qubit
interactions can be ignored Popescu . On the other hand, in the context of
photonic qubits, it is known that effective interactions, engineered using
atomic or other media, may enhance the efficacy of processing information
Duan-Kimble ; Angelakis ; Munro ; Tiecke ; Gorshkov . One is thereby motivated
to seek similarly efficient quantum information processing (QIP) with material
flying qubits which have the advantage of naturally interacting with each
other. Further motivation stems from the fact that for non-interacting mobile
fermions, additional “which-way” detection is necessary for quantum
computation Beenakker  and even for generating entanglement Bose-Home , which
are not necessarily easy. Thus, if interactions do exist between flying qubits
of a given species, one should aim to exploit these for QIP.

While it is known that both spin-dependent ciccarello  and spin-independent
lamata ; saraga  scattering can entangle, it is highly non-trivial to obtain a
useful quantum gate. The amplitudes of reflection and transmission in
scattering generally depend on the internal states of the particles involved
which makes it difficult to ensure that a unitary operation i.e., a quantum
gate acts exclusively on the limited logical (e.g. internal/spin) space that
encodes the qubits. Only recently, for non-identical (one static and one
mobile) particles, it has been shown that a quantum gate can be engineered
from a spin dependent scattering combined with an extra potential guillermo .
We will show here that one can accomplish a quantum gate merely from the spin
independent elastic scattering of two identical particles. This crucially
exploits quantum indistinguishability to label qubits by their momenta, as
well as the equality of the incoming pair and outgoing pair of momenta in one
dimension (1D). In our scheme the quantum gate is only dictated by the
Scattering matrix or S-matrix acting on the initial state of the two free
moving qubits. This is thus an example of minimal control QIP where nothing
other than the initial momenta of the qubits is controlled. Not only will it
enable QIP beyond the paradigm of linear optics with material flying qubits,
but also potentially connect well separated registers of static qubits. One
static qubit from each register should be out-coupled to momenta states in
matter wave-guides and made to scatter from each other. The resulting quantum
gate will connect separated quantum registers. This may be simpler than
interfacing static qubits with photons.

While quantum gates exploiting the mutual interactions of two material flying
qubits has not been considered yet, the corresponding situation for static
qubits has been widely studied (e.g., Refs.loss ; jaksch ; mandel ; deutsch1 ;
deutsch2 ; philips ; grangier ). However, these methods typically require a
precise control of the interaction time of the qubits or between them and a
mediating bus (e.g., Refs.cirac-zoller ; munro-spiller ; banchi ). On the
other hand, our method exploits a much lower control process, namely the
scattering of flying qubits. Still static qubits offer the natural candidate
for information storage. Motivated by this, and also by the high degree of
control reached in current optical lattice experiments fukuhara ; GreinerNew ,
as a second result of this Letter we consider a lattice implementation of our
gate. This experimental proposal is particularly compelling also because the
qubit can be made either static of mobile depending on the tunable potential
barrier on different lattice sites, thus avoiding to seek some mechanism to
couple static and mobile particles and allowing for both storage and
computation with the same physical setup.

Our study interfaces QIP and quantum indistinguishability with two other
areas, namely the Bethe-Ansatz exact solution of many-body models Korepin  and
the 1D confinement of atoms achieved in recent experiments Tonks ; chips1 ;
chips2 ; Aspect ; Druten-recent ; Druten ; article:Inguscio2013 ; CataniNew
Quantum gate between flying qubits:– A two qubit entangling gate is important
as it enables universal quantum computation when combined with arbitrary one
qubit rotations bremner . We consider the spin independent interaction to be a
contact interaction between point-like non-relativistic particles. For two two
spinless bosons on a line (1D) the Hamiltonian with a delta-function
interaction is Korepin

$H=-\frac{\partial^{2}}{\partial x_{1}^{2}}-\frac{\partial^{2}}{\partial
x_{2}^{2}}+2c\delta(x_{1}-x_{2}),$ (1)

where $x_{1}$ and $x_{2}$ are the coordinates of the two particles. The above
model is called the Lieb-Liniger model and has an interesting feature which we
will actively exploit. This is the fact that the momenta are individually
conserved during scattering. If the incoming particles have momenta $p_{1}$
and $p_{2}$, then the outgoing particles also have momenta $p_{1}$ and $p_{2}$
energy , as shown in Fig.1(a). Thus the scattering matrix is diagonal in the
basis of momenta pairs and, is, in fact, only a phase which accumulates on
scattering. The scattering matrix extracted from these wavefunctions is given,
for incident particles with momenta $p_{2}{>}p_{1}$, by Korepin
$S(p_{2},p_{1}){=}\frac{p_{2}-p_{1}-ic}{p_{2}-p_{1}+ic}.$ The phase
accumulated on scattering is ${-}i\ln S(p_{2},p_{1})$. Note that, as expected,
for non-interacting bosons ($c{\rightarrow}0$), their exchange causes no phase
change, while when $c{\rightarrow}\infty$ (impenetrable bosons equivalent to
free fermions) and have a ${-}1$ factor multiplying on exchange.

Figure 1: Part (a) shows the nature of 1D scattering of two identical
particles, labeled as $A$ and $B$ according to their momenta directions, where
some internal states encode the logical states of the qubit. Incoming
particles of momenta $p_{A}$ and $p_{B}$ imply outgoing particles of exactly
the same momenta. Their internal degrees of freedom on the other hand get
entangled after the collision. Part (b) and (c) shows two different physical
implementations of the scheme depicted in part (a). Part (b) deals with flying
qubits, where a momentum $p$ is obtained via suitably controlled local traps.
Part (c) considers an optical lattice implementation, where a higher barrier
on the left forces the qubit to travel to the right.

We consider the case of colliding particles having some internal degrees of
freedom in which a qubit can be encoded (Fig.1(a)). The collision is assumed
to have the form of a spin independent contact (delta) interaction of point-
like particles as in Eq.(1). We first consider bosons with two relevant states
$|{\uparrow}\rangle$ and $|{\downarrow}\rangle$ of some internal degree of
freedom (could be any two spin states of a spin-1 boson, for example). For
symmetric states of the internal degrees of freedom, the external degrees of
freedom also have to be symmetric and have the same scattering matrix as
spinless bosons. On the other hand, for antisymmetric spin states, the spatial
wave function of the two particles is fermionic so that the amplitude for
$x_{1}{=}x_{2}$ (the chance of a contact delta interaction) is zero implying
that they do not scatter from each other. The above observations lead to the
S-matrix (for $p_{2}{>}p_{1}$)

$S^{B}(p_{2},p_{1})=\frac{(p_{2}-p_{1})-ic\Pi_{12}}{p_{2}-p_{1}+ic},$ (2)

where $\Pi_{12}$ is the SWAP (permutation) operator acting on the internal
(spin) degree of freedom
($\Pi_{12}(|u\rangle_{1}|v\rangle_{2}){=}|v\rangle_{1}|u\rangle_{2}$, where
$|u\rangle_{1}$ and $|v\rangle_{2}$ are arbitrary spin states of the
particles, so that $\Pi_{12}$ is a $4{\times}4$ matrix). We also consider the
case where qubit states are spin states of a spin-1/2 particle (say, electrons
or fermionic atoms). This is the conventional encoding in many quantum
computation schemes. In this case the $S-$matrix was computed by C. N. Yang
Yang  to be (for $p_{2}{>}p_{1}$)

$S^{F}(p_{2},p_{1})=\frac{ic\Pi_{12}+p_{2}-p_{1}}{ic+p_{2}-p_{1}}.$ (3)

We consider a frame in which two qubits are moving towards each other so that
they eventually collide. Let us call the qubit with momentum towards the right
as qubit $A$, while the qubit with momentum towards the left is called qubit
$B$. Each qubit is in a definite momenta state, whose magnitudes are $p_{A}$
and $p_{B}$ respectively second . Thus $p_{2}{=}p_{A}$ and $p_{1}{=}{-}p_{B}$.
The evolution of the 4 possible qubit states due to the scattering is thereby
given by

$\displaystyle S^{B/F}|{\uparrow}\rangle_{A}|{\uparrow}\rangle_{B}$
$\displaystyle=e^{i\phi_{B/F}}|{\uparrow}\rangle_{A}|{\uparrow}\rangle_{B}$
$\displaystyle S^{B/F}|{\downarrow}\rangle_{A}|{\downarrow}\rangle_{B}$
$\displaystyle=e^{i\phi_{B/F}}|{\downarrow}\rangle_{A}|{\downarrow}\rangle_{B}$
$\displaystyle S^{B/F}|{\uparrow}\rangle_{A}|{\downarrow}\rangle_{B}$
$\displaystyle=\frac{p_{A{+}B}|{\uparrow}\rangle_{A}|{\downarrow}\rangle_{B}\mp
ic|{\downarrow}\rangle_{A}|{\uparrow}\rangle_{B}}{p_{A{+}B}+ic}$ (4)
$\displaystyle S^{B/F}|{\downarrow}\rangle_{A}|{\uparrow}\rangle_{B}$
$\displaystyle=\frac{p_{A{+}B}|{\downarrow}\rangle_{A}|{\uparrow}\rangle_{B}\mp
ic|{\uparrow}\rangle_{A}|{\downarrow}\rangle_{B}}{p_{A{+}B}+ic}$

where $p_{A{+}B}{=}p_{A}{+}p_{B}$,
$e^{i\phi_{B}}{=}\frac{p_{A{+}B}-ic}{p_{A{+}B}+ic}$ and $e^{i\phi_{F}}{=}1$.
Unless either $p_{A{+}B}$ or $c$ vanishes, the above is manifestly an
entangling gate, as is evident from the fact that the right hand sides of the
last two lines of Eq. (Quantum Gates Between Distant Qubits via Spin-
Independent Scattering) is an entangled state. This gate is most entangling
(i.e., the most useful in context of quantum computation, equivalent in
usefulness to the well known Controlled NOT or CNOT gate) when
$p_{A{+}B}{\approx}c$, as then the right hand sides of the last two lines of
Eq(Quantum Gates Between Distant Qubits via Spin-Independent Scattering)
correspond to maximally entangled states
$\frac{e^{-i\frac{\pi}{4}}}{\sqrt{2}}(|{\uparrow}\rangle_{A}|{\downarrow}\rangle_{B}{\mp}i|{\downarrow}\rangle_{A}|{\uparrow}\rangle_{B})$
and
$\frac{e^{-i\frac{\pi}{4}}}{\sqrt{2}}(|{\downarrow}\rangle_{A}|{\uparrow}\rangle_{B}{\mp}i|{\uparrow}\rangle_{A}|{\downarrow}\rangle_{B})$
respectively. The above gates would aid universal quantum computation by means
of scattering with both bosonic and fermionic qubits. The gates of
Eqs.(Quantum Gates Between Distant Qubits via Spin-Independent Scattering) are
easiest to exploit as the only other requirement, namely local rotations of
the qubit states are accomplishable by means of laser induced transitions
between different atomic internal levels or electronic spin rotations by
magnetic fields.

Error estimates:– The amplitudes in Eqs.(Quantum Gates Between Distant Qubits
via Spin-Independent Scattering) depend only on the ratio of $p_{A{+}B}/c$ and
thereby any spread $\delta p_{A{+}B}$ of the relative momenta of the incoming
particles only affects the amplitudes as $\delta p_{A{+}B}/c$. As a relevant
example we consider two Gaussian wavepackets in the internal state
$|{\uparrow}\rangle_{A}|{\downarrow}\rangle_{B}$ where $p_{A{+}B}$ is centered
around $c(1{+}\delta)$ with a relative variance $\eta c$. After the scattering
the generated entanglement between the internal degrees of freedom is measured
by the concurrence Concurrence  $C$. It is $C{=}2\Re[z]\Im[f(z)]$, being
$z{=}\frac{1-i(1+\delta)}{\sqrt{2}\eta}$ and $f(z){=}\sqrt{\pi}e^{z^{2}}{\rm
erfc}(z)$. From the asymptotic expansion Abramowitz
$zf(z){\approx}1{-}z^{-2}/2$ one obtains that $C$ slowly decays as a function
of $(\delta,\eta)$, and that the case $\delta{\gtrsim}0$ is less prone to
errors when $\eta$ increases. Errors can thereby be arbitrarily reduced in
principle by choosing particles with higher $c$. This is opposite to the usual
paradigm of gates based on “timed” interactions, where for a given timing
error $\delta t$, stronger interactions enhance the error (while weaker
interactions make gates both slower and susceptible to decoherence).

Implementation via flying qubits:– One of the most promising implementation of
our gates is with neutral bosonic/fermionic atoms. The delta function
interaction we use is, in fact, very realistic and realizable for neutral
atoms under strong 1D confinement Olshanii . 87Rb atoms have already been
strongly confined to 1D atomic waveguides leading to delta interactions Tonks
. For 87Rb, with a 3D scattering length $a\,{\simeq}50{\AA}$ an axial (for 1D)
confinement of $\omega_{\perp}{\simeq}100$kHz gives (using e.g. Refs.Olshanii
; Busch ) $c\,{\simeq}10^{6}$ m-1. Velocities of atoms in 1D waveguides (c.f.
atom lasers Aspect ) can be mm s-1, which translates to
$p_{A{+}B}{\simeq}10^{6}$ m-1 (in units of wavenumbers). Thereby,
$p_{A{+}B}{\approx}c$ for optimal gates is achievable in current technology
comment . Deviation from the 1D effective $\delta$-potential are expected when
the condition $p_{A{+}B}{\ll}\sqrt{\hbar\mu\omega_{\perp}}$ is not satisfied
($\mu$ being the atomic mass). In that case the scattering matrix has still
the form Eq.(Quantum Gates Between Distant Qubits via Spin-Independent
Scattering) where $c$ shows a (weak) dependence on $p_{A{+}B}$ Olshanii . The
optimal gate is then found by solving
$p_{A{+}B}{=}c(p_{A{+}B}){\simeq}c{-}\zeta_{3/2}(\mu\omega_{\perp}/\hbar)^{{-}3/2}(c\,p_{A{+}B}/4)^{2}$,
being $\zeta$ the Riemann zeta function. State independent waveguides for two
spin states has been met Druten-recent  in magnetic waveguides (trivially
possible in optical waveguides/hollow fibers). Our gate will be an extension
of recent collision experiments between different spin species Zwierlein  with
pairs of atoms at a time. Launching exactly two atoms towards each other in 1D
should be feasible with microtrap arrays Muldoon ; Schlosser ; Marray  or in
atom chips chips1 ; chips2  and is also a key assumption in many works Popescu
; Calarco . For example, our gates can be made with the technique of
Ref.Calarco  whereby atoms are trapped initially in potential dips inside a
larger well and let to roll towards each other in a harmonic potential to
acquire their momenta (note that our gate scheme is completely different from
Ref.Calarco , where the atomic motion is guided by internal states). The
initial position of the particles $x_{B}{=}{-}x_{A}{=}x_{0}$ can be tailored
so that their relative momentum has minimum variance $\Delta p_{A{+}B}$ when
the particles reach the collision point $(x{=}0)$. As
$\omega_{z}{\ll}\omega_{\perp}$ (being $\omega_{z}$ the frequency of the
longitudinal harmonic confinement) the collision does not feel the
longitudinal potential, so it is approximated by Eq.(Quantum Gates Between
Distant Qubits via Spin-Independent Scattering). As shown previously, the
generated entanglement is very high ($C{\simeq}1$) provided that
$\eta{\sim}\Delta p_{A{+}B}/p_{A{+}B}{\sim}\Delta x_{0}/x_{0}{\ll}1$. A
different approach (depicted is Fig. 1b) consists in suddenly moving the local
trapping potentials so that the particles in $A$ and $B$ move towards their
new potential minima. As in the previous case, wave-packets with well-defined
and tunable momenta are created by switching off the potential when they reach
the minima where their momentum uncertainties are minimal. Lastly, static
atomic qubits (c.f. grangier ; Hofmann ) in well separated traps may be Raman
outcoupled philips  imparting them momenta towards each other in a 1D
waveguide – this will link distinct registers.

Ballistic electrons inside carbon nano-tubes or semiconductor nanowires, or
edge states may be another implementation Owen ; Buttiker . As electrons are
typically screened in a solid, a contact interaction should describe their
scattering. Note that the Hubbard model with solely on site interactions
models a variety of electronic systems. In a low energy limit, in which the
electronic de Broglie wavelength is far larger than the lattice spacing of the
Hubbard model, the Hamiltonian reduces to that of Eq.(1).

Quantum gates between distant stationary qubits:– A discrete variant of the
system with a $\delta$-interaction is the Bose-Hubbard Hamiltonian
Book:Gaudin2014 ; Korepin :

$H{=}\sum\limits_{j,\alpha}\frac{J_{j}}{2}\left[a_{j,\alpha}^{\dagger}a_{j+1,\alpha}+{\rm
h.c.}\right]+\sum\limits_{j,\alpha,\beta}\\!\frac{U_{j}^{\alpha\beta}}{2}n_{j,\alpha}n_{j,\beta}~{},$
(5)

where $\alpha{=}\\{\uparrow,\downarrow\\}$ labels two internal degrees of
freedom of the particles. We call $N$ the length of the chain. In a lattice
setup, free-space evolution is replaced with particle hopping. Particle
collisions lead to a scattering matrix which, for uniform couplings
$J_{j}{=}J$, $U^{\alpha\beta}_{j}{=}U^{\alpha\beta}$, is given by Eq.(2) with
the substitutions Book:Gaudin2014 ; article:Krauth

$\displaystyle p_{j}$ $\displaystyle\rightarrow\sin p_{j},$ $\displaystyle c$
$\displaystyle\rightarrow U^{\alpha\beta}/J.$ (6)

We consider different initial internal degrees of freedom and set
$U_{j}^{\alpha\alpha}{=}0$. A maximally entangling gate is therefore realized
when $\sin p_{1}{-}\sin p_{2}{\approx}2U$, being
$U{=}U^{\uparrow\downarrow}/(2J)$. In particular, $p_{1}{=}\sin^{-1}U$ when
$p_{1}{=}{-}p_{2}$.

The Bose-Hubbard Hamiltonian (5) naturally models cold bosonic atoms in
optical lattice Book:Sanpera2012 . Owing to single atom addressing techniques
article:Weitenberg2011  ${}^{87}Rb$ atoms in different lattice sites can be
initialized in either two distinguishable hyperfine internal states
$|{\downarrow}\rangle{\equiv}|{F{=}1,m_{F}{=}{-}1}\rangle$ and
$|{\uparrow}\rangle{\equiv}|{F{=}2,m_{F}{=}{-}2}\rangle$. The coupling
constants $U_{j}^{\alpha\beta}$ depend on the strength $g_{\alpha\beta}$ of
the effective interaction between cold atoms Jaksch1998 . These parameters are
usually experimentally measured article:Egorov2013 ; article:Weiner1999  and
can be tuned by Feshbach resonances article:Marte2002 . Spin-exchange
collisions are highly suppressed due to the little difference (less than 5%)
between singlet and triplet scattering length of ${}^{87}Rb$ Book:Sanpera2012
. The one-dimensional regime is obtained by increasing the harmonic lattice
transverse confinement ($\omega_{\perp}/2\pi{\simeq}18$ kHz see
article:Inguscio2013 ; article:Bloch2005  for typical values). We obtain the
1D pseudo-potential coupling constants $g_{\alpha\beta}$ from the 3D measured
values article:Egorov2013  following Olshanii , respectively
$g_{\uparrow\uparrow}{=}1.14{\times}10^{-37}{\rm J\,m}$,
$g_{\uparrow\downarrow}{=}1.12{\times}10^{-37}{\rm J\,m}$,
$g_{\downarrow\downarrow}{=}1.09{\times}10^{-37}{\rm J\,m}$. The internal spin
state and the position of particles are detected by fluorescence microscopy
techniques article:Weitenberg2011 ; article:Bakr2009 . The parameters
$U_{j}^{\alpha\beta}$ and $J_{j}$ can be physically controlled in optical
lattice systems locally varying the depth of the optical potential
article:HubbardToolbox2004 . Arbitrary optical potential landscapes are
generated directly projecting a light pattern by using holographic masks or
micromirror device GreinerNew ; article:Bakr2009 . In particular,
$U^{\alpha\beta}{\simeq}\sqrt{2\pi}\left(g_{\alpha\beta}/\lambda\right)\left(V_{0}/E_{R}\right)^{1/4}$
and
$J_{j}\simeq\left(4/\sqrt{\pi}\right)E_{R}\left(V{/}E_{R}\right)^{3/4}\exp\left[-2\left(V{/}E_{R}\right)^{1{/}2}\right]$
where $\lambda$ is the laser wavelength, $V_{0}$ is the lattice depth and
$E_{R}$ is the recoil energy Jaksch1998 .

For flying qubits we considered a fixed $c$ and we tuned $p_{j}$ to obtain the
desired gate. In a lattice, on the other hand, $U_{\alpha\beta}$ can be
controlled precisely, while the creation of a wave-packet requires the control
and initialization of many-sites. This kind of control can be avoided by
initially placing two particles at the two distant boundaries of the lattice
(particle $A$ on the left and particle $B$ on the right) and locally tuning
the coupling $J_{0}$ between the boundaries and the rest of the chain
article:Banchi2011  (all the other couplings are uniform $J_{j}{=}J$). An
optimal choice of $J_{0}/J$ has a twofold effect: firstly it generates two
wave-packets whose momentum distribution is Lorentzian, narrow around
$p_{A}{=}{-}p_{B}{\simeq}{\pm}\pi/2$, respectively, and with a width dependent
on $J_{0}$; secondly it generates a quasi-dispersionless evolution, allowing
an almost perfect reconstruction of the wave-packets after the transmission
(occurring in a time ${\approx}N/J$) to the opposite end. In this scheme (Fig.
1c), the particles starts from opposite locations, interact close to the
center of the chain and then reach the opposite end where the wave-function is
again almost completely localized, allowing a proper particle addressing.
Since $p_{A/B}$ is fixed, a high amount of entanglement is generated when
$U{=}|\sin p_{A/B}|{=}1$. For 87Rb we found that the latter condition is
satisfied, e.g., when $V_{0}/E_{R}{\simeq}2.2$, giving also
$J/h{\simeq}240{\rm Hz}$.

In this scheme there are two error sources. The first is due to the
transmission quality, though it is above $85\%$ even for long chains
article:Banchi2011 . The second one is due to the finite width of the
Lorentzian momentum profile around $|p|{=}\pi/2$ which, in turn, yields
slightly different gates for different momentum components. To quantify the
amount of those errors we evaluate numerically the join probability amplitude
$A^{\alpha\beta}_{ij}(\tilde{t})$ to have respectively particle $A$ in sites
$i$ and particle $B$ in $j$ as function of the inter-particle interactions
$U$. The indices $\alpha,\beta$ refer to the initial internal state of
particles $A$ and $B$, $\tilde{t}$ is the transfer time, and the initial
condition is $A^{\alpha\beta}_{1N}(0){=}1$. We find that for distinguishable
particles
$A^{\uparrow\downarrow}_{1N}{/}A^{\uparrow\downarrow}_{N1}(\tilde{t}){=}{-}iU/U_{\rm
opt}$ in agreement with the prediction, apart from $U_{\rm opt}$ which models
finite size effects: e.g. $U_{\rm opt}{=}0.95$ ($U_{\rm opt}{=}0.97$) for
$N{=}25$ ($N{=}51$) and $U_{\rm opt}{\to}1$ for long chains. For
indistinguishable particles we obtain that
$A^{\alpha\alpha}_{11}/A^{\alpha\alpha}_{1N}(\tilde{t})$ is zero for
$\alpha{=}{\uparrow},{\downarrow}$ and for any value of
$U_{\uparrow\downarrow}$. Therefore, apart from a global damping factor due to
the non-perfect wave-packet reconstruction, the resulting transformation is in
agreement with the gate (Quantum Gates Between Distant Qubits via Spin-
Independent Scattering). The entanglement generation between the boundaries is
evaluated via the concurrence Concurrence
$C_{1N}{=}2|A^{\uparrow\downarrow}_{1N}A^{\uparrow\downarrow}_{N1}{}^{*}|$.
From the asymptotic analysis article:Banchi2011  we find that
$C_{1N}{=}f_{1N}^{4}\frac{2U/U_{\rm opt}}{(U/U_{\rm opt})^{2}+1}$ where
$f_{1N}$ is the transmission probability from site $1$ to site $N$ at the
transmission time. Surprisingly, $C_{1N}$ does not directly depend on the
width of the wave-packet in momentum space: its maximum value depends only on
the transfer quality $f_{1N}$. For example, for $N{=}25$, $f_{1N}{=}0.97$ so
$C^{\rm opt}_{1N}{=}0.88$; for $N{=}51$, $f_{1N}{=}0.95$ so $C^{\rm
opt}_{1N}{=}0.81$.

Concluding remarks: – We have described a mechanism for useful gates between
flying qubits and hopping qubits in a lattice using the low control method of
scattering. 1D matter waveguides, identical particles and a contact
interaction are key requirements for the gate. Exactly solvable models enable
the study Korepin , while experiments and results on 1D gases Tonks ; chips1 ;
chips2 ; Aspect ; Druten-recent ; Muldoon ; Schlosser ; article:Inguscio2013 ;
CataniNew  and optical lattices article:Weitenberg2011 ; fukuhara  suggest
implementations with flying and static atomic qubits. The scheme may both
enhance linear optics-like QIP and connect separated atomic qubit registers.

Acknowledgements: – VK acknowledges financial support by NSF Grant No.
DMS-1205422. SB, LB and EC are currently supported by the ERC grant
PACOMANEDIA.

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