# Existence of density for the solution of stochastic delay differential
equations with reflection driven by a fractional Brownian motion

M. Besalú, D. Márquez-Carreras, C. Rovira  Mireia Besalú, Dep. Genètica,
Microbiologia i Estadística, Facultat de Biologia, Universitat de Barcelona.
Diagonal, 645, 08028 Barcelona<EMAIL_ADDRESS>David Márquez-Carreras,
Facultat de Matemàtiques i Informàtica, Universitat de Barcelona. Gran Via de
les Corts Catalanes, 585, 08007 Barcelona<EMAIL_ADDRESS>Carles Rovira,
Facultat de Matemàtiques i Informàtica, Universitat de Barcelona. Gran Via de
les Corts Catalanes, 585, 08007 Barcelona<EMAIL_ADDRESS>
###### Abstract.

In this note we prove the existence of a density for the law of the solution
for 1-dimensional stochastic delay differential equations with normal
reflection. The equations are driven by a fractional Brownian motion with
Hurst parameter $H>1/2$. The stochastic integral with respect to the
fractional Brownian motion is a pathwise Riemann-Stieltjes integral.

D. Márquez-Carreras and C. Rovira are supported by the grant
PID2021-123733NB-I00 from SEIDI, Ministerio de Economia y Competividad.

##  1\. Introduction

There are some models affected by some type of noise where the dynamics are
related to propagation delay and some of them are naturally non-negative
quantities. For instance, applications as rated and prices in internet model,
concentrations of ions or proportions of a population that are infected (see
the references in [13]). It is then natural to consider stochastic
differential equations with delay and non-negativity constraints. In our paper
[1], we initiate their study when the noise is a fractional Brownian motion,
obtaining the existence and uniqueness of solution.

More precisely, consider a stochastic delay differential equation with
positivity constraints of the form:

$\displaystyle X_{t}$ $\displaystyle=$
$\displaystyle\eta_{0}+\int_{0}^{t}b(X_{s})ds+\int_{0}^{t}\sigma(X_{s-r})dW_{s}^{H}+Y_{t},\quad
t\in(0,T_{0}],$ $\displaystyle X_{t}$ $\displaystyle=$
$\displaystyle\eta_{t},\qquad t\in[-r,0].$ (1.1)

Here $r$ denotes a strictly positive time delay; $W^{H}$ is a fractional
Brownian motion with Hurst parameter $H>\frac{1}{2}$ defined in a complete
probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$;
$\eta:[-r,0]\rightarrow{\mathbb{R}}_{+}$ is a non negative smooth function,
with ${\mathbb{R}}_{+}=[0,+\infty)$ and $Y$ is a vector-valued non-decreasing
process which ensures that the non-negativity constraints on X are enforced.

Set

$Z_{t}=\eta_{0}+\int_{0}^{t}b(X_{s})ds+\int_{0}^{t}\sigma(X_{s-r})dW_{s}^{H},\qquad
t\in[0,T_{0}].$ (1.2)

It is known that we have an explicit formula for the regulator term $Y$ in
terms of $Z$:

$Y_{t}=\max_{s\in[0,t]}\left(Z_{s}\right)^{-},\qquad t\in[0,T_{0}],$

where $x^{-}=\max(0,-x)$. Then, using the called Skorohod’s mapping, the
solution of (1.1) satisfies

$X_{t}=\qquad\begin{cases}Z_{t}+Y_{t},\qquad t\in[0,T_{0}],\\\
\eta_{t},\qquad\quad t\in[-r,0].\end{cases}$

As $H>\frac{1}{2}$, the integral with respect to $W^{H}$ can be defined as a
pathwise Riemann-Stieltjes integral using the results given by Young [24].
Moreover, Zälhe [25] introduced a generalized Stieltjes integral using the
techniques of fractional calculus. In particular, she obtained a formula for
the Riemann-Stieltjes integral using fractional derivatives (see (2.2) below).
Using this formula, Nualart and Răşcanu [19] proved a general result on
existence, uniqueness and finite moments of the solution to a class of general
differential equations. In [1] we extended these results to equation (1.1).

Nualart and Sausserau [21] obtained the Malliavin differentiability and the
existence of density for the solutions to equations considered in [19]. Our
aim is to extend these results to our equations and to study the existence of
density of the solution to (1).

As in all the papers following the methods presented in [19], we will obtain
first deterministic results and then we will apply pathwise to the stochastic
equations.

Our main result states as follow:

######  Theorem 1.1.

Assume $b,\sigma\in{\mathcal{C}}^{2}_{b}$ and
$\eta\in{\mathcal{C}}^{1-\alpha}(-r,0)$. Set $X$ the solution of (1) and
assume that $\sigma>0$ on ${\mathbb{R}}^{+}$. Then, for any
$t_{0}\in(0,T_{0}]$, the law of $X_{t_{0}}$ restricted on $(0,+\infty)$ is
absolutely continuous.

Notice that we obtain the regularity restricted on $(0,+\infty)$ since we are
not able to prove that $P(X_{t_{0}}=0)=0$. It is the same situation that
appears in the study of SPDEs [6]. In our paper, the main difficult of the
problem lies in the lack of properties of the Skorohod map with respect to the
topologies used working with fractional Brownian motion.

Stochastic systems with delay have been studied deeply (see, as a basic
reference, [17]), but the literature about stochastic differential equations
with delay driven by a fractional Brownian motion is scarce. There are
references dealing with existence and uniqueness of solution [7, 16, 18, 23],
existence and regularity of the density [16] and convergence when the delay
goes to zero [8]. There has been little work on stochastic differential
equations with delay and non-negativity constraints driven by standard
Brownian motion. We can only refer the reader to [12], dedicated to study
numerical methods, and [13] where the authors obtain sufficient conditions for
existence and uniqueness of stationary solutions. The study of equations with
fractional Brownian motion and non negative constraints is also weak. The
existence of solution is given in [9] and the existence and uniqueness in
[10]. In [4] the authors deal with existence and uniqueness using rough path
techniques.

On the other hand, the study of the regularity of the solution to stochastic
differential equations with reflection is also scarce. We can refer the reader
to the initial papers dealing with diffusions with reflection [14] and [15]
and the work dedicated to stochastic partial differential equations [6] and
[20].

The structure of the paper is as follows. In the next Section, we give some
preliminaries about the Skorohod problem and fractional calculus. In Sections
3, 4, 5 and 6 we deal with the deterministic case. Section 3 is devoted to get
some comparison, existence and uniqueness theorems for deterministic
equations. In Section 4 we obtain the convergence results to solve the
deterministic Skorohod problem. The Fréchet differentiability of our
deterministic equations is considered in Section 5 and in Section 6 we put
together the deterministic results of the previous sections to obtain the
approximations that we will consider pathwise in our stochastic case. Finally,
Section 7 is devoted to apply all the deterministic results to the stochastic
case proving our main theorem.

##  2\. Preliminaries

###  2.1. Skorohod Problem

Let

${\mathcal{C}}_{+}({\mathbb{R}}_{+},{\mathbb{R}}):=\left\\{x\in{\mathcal{C}}({\mathbb{R}}_{+},{\mathbb{R}}):x(0)\in{\mathbb{R}}_{+}\right\\}.$

We can recall now the Skorokhod problem.

######  Definition 2.1.

Given a path $z\in{\mathcal{C}}_{+}({\mathbb{R}}_{+},{\mathbb{R}})$, we say
that a pair $(x,y)$ of functions in
${\mathcal{C}}_{+}({\mathbb{R}}_{+},{\mathbb{R}})$ solves the Skorokhod
problem for $z$ with reflection if

  1. (1)

$x_{t}=z_{t}+y_{t}$ for all $t\geq 0$ and $x_{t}\in{\mathbb{R}}_{+}$ for each
$t\geq 0$,

  2. (2)

$y(0)=0$ and $y$ is non-decreasing,

  3. (3)

${\displaystyle\int_{0}^{t}x_{s}dy_{s}=0}$ for all $t\geq 0$, so $y$ can
increase only when $x$ is at zero.

It is known that we have an explicit formula for $y$ in terms of $z$:

$y_{t}=\max_{s\in[0,t]}\left(z_{s}\right)^{-}.$

The path $z$ is called the reflector of $x$ and the path $y$ is called the
regulator of $x$. We use the Skorokhod mapping for constraining a continuous
real-valued function to be non-negative by means of reflection at the origin.
We will apply it to each path of $z$ defined by (1.2).

###  2.2. Fractional calculus

Let $\alpha\in\left(0,\frac{1}{2}\right)$ and $(a,b)\subseteq[-r,T]$. We
denote by $W_{1}^{\alpha}(a,b)$ the space of measurable functions
$f:[a,b]\rightarrow{\mathbb{R}}$ such that

$\left\|f\right\|_{\alpha,1(a,b)}:=\sup_{u\in[a,b]}\Delta_{t}^{\alpha}(f)=\sup_{u\in[a,b]}\left(\left|f(u)\right|+\int_{a}^{u}\frac{\left|f(u)-f(v)\right|}{(u-v)^{\alpha+1}}dv\right)<\infty.$

We also denote by $W_{2}^{1-\alpha}(0,T)$ the space of measurable functions
$g:[0,T]\rightarrow{\mathbb{R}}$ such that

$\left\|g\right\|_{1-\alpha,2}:=\sup_{0\leq u<v\leq
T}\left(\frac{\left|g(v)-g(u)\right|}{|v-u|^{1-\alpha}}+\int_{u}^{v}\frac{\left|g(y)-g(u)\right|}{(y-u)^{2-\alpha}}dy\right)<\infty.$

Finally, we define the supremum norm for functions
$f:[a,b]\rightarrow{\mathbb{R}}$ as

$\|f\|_{\infty(a,b)}=\sup_{u\in[a,b]}|f(u)|,$

the space of $\lambda$-Hölder continuous functions
$f:[a,b]\rightarrow{\mathbb{R}}$ denoted by ${\mathcal{C}}^{\lambda}(a,b)$ for
any $0<\lambda\leq 1$ with the norm

$\left\|f\right\|_{\lambda(a,b)}:=\left\|f\right\|_{\infty(a,b)}+\sup_{a\leq
u<v\leq b}\frac{\left|f(v)-f(u)\right|}{(v-u)^{\lambda}}<\infty,$

and for any integer $k\geq 1$ we denote by ${\mathcal{C}}^{k}_{b}$ the class
of real-valued functions on ${\mathbb{R}}$ which are $k$ times continuously
differentiable with bounded partial derivatives up to the $k$-th order.

Clearly, for any $\varepsilon>0$,

$C^{1-\alpha+\varepsilon}(0,T)\subset W^{1-\alpha}_{2}(0,T)\subset
C^{1-\alpha}(0,T).$ (2.1)

Moreover, as $\alpha\in(0,\frac{1}{2})$,

$C^{1-\alpha}(0,T)\subset W^{\alpha}_{1}(0,T).$

If $f\in C^{\lambda}(a,b)$ and $g\in C^{\mu}(a,b)$ with $\lambda+\mu>1$, it is
proved in [25] that the Riemman-Stieltjes integral $\int_{a}^{b}fdg$ exists
and it can be expressed as

$\int_{a}^{b}fdg=(-1)^{\alpha}\int_{a}^{b}D_{a+}^{\alpha}f(t)D_{b-}^{1-\alpha}g_{b-}(t)dt,$
(2.2)

where $g_{b-}(t)=g(t)-g(b),$ $1-\mu<\alpha<\lambda$, and the fractional
derivatives are defined as

$\displaystyle D_{a+}^{\alpha}f(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(1-\alpha)}\left(\frac{f(t)}{(t-a)^{\alpha}}+\alpha\int_{a}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds\right),$
$\displaystyle D_{b-}^{\alpha}f(t)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\alpha}}{\Gamma(1-\alpha)}\left(\frac{f(t)}{(b-t)^{\alpha}}+\alpha\int_{t}^{b}\frac{f(t)-f(s)}{(s-t)^{\alpha+1}}ds\right).$

We refer to [19] and [25] and the references therein for a detailed account
about this generalized integral and the fractional calculus.

Let $\Omega=C_{0}([0,T];{\mathbb{R}})$ be the Banach space of continuous
functions, null at time 0, equipped with the supremum norm. Let ${\rm P}$ be
the unique probability measure on $\Omega$ such that the canonical process
$\\{W^{H}_{t},t\in[0,T]\\}$ is an $1$-dimensional fractional Brownian motion
with Hurst parameter $H>\frac{1}{2}$.

We denote by $\mathcal{E}$ the space of step functions on $[0,T]$ with values
in ${\mathbb{R}}$. Let $\mathcal{H}$ be the Hilbert space defined as the
closure of $\mathcal{E}$ with respect to the scalar product

$\langle{\bf 1}_{[0,t]},{\bf 1}_{[0,s]}\rangle_{\mathcal{H}}=R_{H}(t,s),$

where

$R_{H}(t,s)=\int_{0}^{t\wedge s}K_{H}(t,r)K_{H}(s,r)dr,$

and $K_{H}(t,s)$ is the square integrable kernel defined by

$K_{H}(t,s)=c_{H}s^{1/2-H}\int_{s}^{t}(u-s)^{H-3/2}u^{H-1/2}du,\qquad\mbox{for}\;t>s,$
(2.3)

where $c_{H}=\sqrt{\frac{H(2H-1)}{\beta(2-2H,H-1/2)}}$ and $\beta$ denotes the
Beta function. And for $t\leq s$, we set $K_{H}(t,s)=0$.

The mapping ${\bf 1}_{[0,t]}\rightarrow W^{H}_{t}$ can be extended to an
isometry between $\mathcal{H}$ and the Gaussian space $\mathcal{H}_{1}$
associated to $W^{H}$. We denote this isometry by $\varphi\rightarrow
W^{H}(\varphi)$.

Consider the operator $K^{\ast}_{H}$ from $\mathcal{E}$ to
$L^{2}(0,T;{\mathbb{R}})$ defined by

$(K^{\ast}_{H}\varphi)(s)=\int_{s}^{T}\varphi(t)\partial_{t}K_{H}(t,s)dt.$

From (2.3), we get

$\partial_{t}K_{H}(t,s)=c_{H}\left(\frac{t}{s}\right)^{H-1/2}(t-s)^{H-3/2}.$

Notice that

$K^{\ast}_{H}({\bf 1}_{[0,t_{1}]},\ldots,{\bf
1}_{[0,t_{m}]})=(K_{H}(t_{1},\cdot),\ldots,K_{H}(t_{m},\cdot)).$

For any $\varphi,\psi\in\mathcal{E}$,

$\langle\varphi,\psi\rangle_{\mathcal{H}}=\langle
K^{\ast}_{H}\varphi,K^{\ast}_{H}\psi\rangle_{L^{2}(0,T;{\mathbb{R}}^{m})}={\rm
E}(W^{H}(\varphi)W^{H}(\psi))$

and $K^{\ast}_{H}$ provides an isometry between the Hilbert space
$\mathcal{H}$ and a closed subspace of $L^{2}(0,T;{\mathbb{R}})$.

Following [21], we consider the fractional version of the Cameron-Martin space
$\mathcal{H}_{H}:=\mathcal{K}_{H}(L^{2}(0,T;{\mathbb{R}}))$, where for $h\in
L^{2}(0,T;{\mathbb{R}})$,

$(\mathcal{K}_{H}h)(t):=\int_{0}^{t}K_{H}(t,s)h_{s}ds.$

We finally denote by
$\mathcal{R}_{H}=\mathcal{K}_{H}\circ\mathcal{K}_{H}^{\ast}:\mathcal{H}\rightarrow\mathcal{H}_{H}$
the operator

$\mathcal{R}_{H}\varphi=\int_{0}^{\cdot}K_{H}(\cdot,s)(\mathcal{K}^{\ast}_{H}h)(s)ds.$

We remark that for any $\varphi\in\mathcal{H}$, $\mathcal{R}_{H}\varphi$ is
Hölder continuous of order $H$. Therefore, for any $1-H<\alpha<1/2$,

$\mathcal{H}_{H}\subset C^{H}(0,T;{\mathbb{R}})\subset
W^{1-\alpha}_{2}(0,T;{\mathbb{R}}).$

Notice that $\mathcal{R}_{H}{\bf 1}_{[0,t]}=R_{H}(t,\cdot)$, and, as a
consequence, $\mathcal{H}_{H}$ is the Reproducing Kernel Hilbert Space
associated with the Gaussian process $W^{H}$. The injection
$\mathcal{R}_{H}:\mathcal{H}\rightarrow\Omega$ embeds $\mathcal{H}$ densely
into $\Omega$ and for any $\varphi\in\Omega^{\ast}\subset\mathcal{H}$,

${\rm
E}\left(e^{iW^{H}(\varphi)}\right)=\exp\left(-\frac{1}{2}\|\varphi\|^{2}_{\mathcal{H}}\right).$

As a consequence, $(\Omega,\mathcal{H},{\rm P})$ is an abstract Wiener space
in the sense of Gross.

##  3\. Comparison and existence and uniqueness theorems

Let $0<\alpha<\frac{1}{2}$. Consider now the following equation

$w_{t}=\eta_{0}+\zeta_{t}+\int_{0}^{t}b(w_{s})ds,\quad t\in(0,T],$ (3.1)

where $\zeta\in{\mathcal{C}}^{1-\alpha}(0,T)$ and $\eta_{0}>0$.

In this section we will prove the existence and uniqueness of solution to
equation (3.1) and a comparison result.

######  Lemma 3.1.

Assume that $b$ is a Lipschitz function and that
$\zeta\in{\mathcal{C}}^{1-\alpha}(0,T)$. Then (3.1) has a unique solution
$w\in{\mathcal{C}}^{1-\alpha}(0,T)$.

###### Proof.

The proof is a simple case of the proof of Theorem 5.1 in [21]. ∎

######  Lemma 3.2.

Let $b,\,\tilde{b}$ Lipschitz functions, such that $b\leq\tilde{b}$,
$\zeta\in{\mathcal{C}}^{1-\alpha}(0,T)$ and $t\in[0,T]$, then if

$\displaystyle w_{t}$ $\displaystyle=$
$\displaystyle\eta_{0}+\zeta_{t}+\int_{0}^{t}b(w_{s})ds,$
$\displaystyle\tilde{w}_{t}$ $\displaystyle=$
$\displaystyle\eta_{0}+\zeta_{t}+\int_{0}^{t}\tilde{b}(\tilde{w}_{s})ds,$

we have that $w_{t}\leq\tilde{w}_{t}$, for any $t\in[0,T]$.

###### Proof.

For $k>0$, consider the function

$\varphi_{k}(x)={\bf 1}_{\\{x\geq
0\\}}\int_{0}^{x}\int_{0}^{y}\rho_{k}(z)dzdy,$

where

$\rho_{k}(z)=\left\\{\begin{array}[]{ll}2kz,&\textrm{if}\;\;z\in\left[0,\frac{1}{k}\right],\\\\[5.0pt]
2\times{\bf
1}_{\\{z\geq\frac{1}{k}\\}},&\textrm{otherwise}.\end{array}\right.$

From [11, Theorem 5.1], we can check that
$\varphi_{k}\in{\mathcal{C}}^{2}_{b}$ and that their derivatives are bounded:

$0\leq\varphi^{\prime}_{k}(x)\leq
2x^{+},\;\textrm{and}\;0\leq\varphi^{\prime\prime}_{k}(x)\leq 2\times{\bf
1}_{\\{x\geq 0\\}},$

where $x^{+}=\max(0,x)$. We also have that $\varphi_{k}(x)\uparrow(x^{+})^{2}$
when $k\rightarrow 0$.

Then using this properties on $\varphi_{k}$ we have

$\displaystyle\varphi_{k}(w_{t}-\tilde{w}_{t})$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\varphi^{\prime}_{k}(w_{s}-\tilde{w}_{s})(b(w_{s})-\tilde{b}(\tilde{w}_{s}))ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\varphi^{\prime}_{k}(w_{s}-\tilde{w}_{s})(\tilde{b}(w_{s})-\tilde{b}(\tilde{w}_{s}))ds$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{t}(w_{s}-\tilde{w}_{s})^{+}|w_{s}-\tilde{w}_{s}|ds$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{t}\left((w_{s}-\tilde{w}_{s})^{+}\right)^{2}ds,$

where $C$ does not depend on $k$. Now, letting $k\rightarrow 0$ the last
inequality becomes

$\left((w_{t}-\tilde{w}_{t})^{+}\right)^{2}\leq
C\int_{0}^{t}\left((w_{s}-\tilde{w}_{s})^{+}\right)^{2}ds,$

and using the Gronwall’s inequality we obtain that
$\left((w_{t}-\tilde{w}_{t})^{+}\right)^{2}=0$ and from here the desired
result. ∎

##  4\. Convergence results

In this section, we consider the following equations

$\displaystyle x_{t}^{(\zeta)}$ $\displaystyle=$
$\displaystyle\eta_{0}+\zeta_{t}+\int_{0}^{t}b(x_{s}^{(\zeta)})ds+y_{t}^{(\zeta)},\,\qquad
t\in[0,T],$ (4.1)

where $y^{(\zeta)}$ is the regulator term of $x^{(\zeta)}$, and

$\displaystyle x_{t}^{(\zeta),\varepsilon}$ $\displaystyle=$
$\displaystyle\eta_{0}+\zeta_{t}+\int_{0}^{t}\left[b(x_{s}^{(\zeta),\varepsilon})+\frac{1}{\varepsilon}f(x_{s}^{{(\zeta),\varepsilon}})\right]ds,\,\qquad
t\in[0,T].$ (4.2)

We assume that $f$ is a ${\mathcal{C}}^{2}_{b}$ not increasing function such
that

$\begin{cases}f(x)=0,&\mbox{if}\;x\geq 0,\\\ 0<f(x)\leq
x^{-},&\mbox{if}\;x<0,\end{cases}$

with $x^{-}=\max(0,-x)$.

We will assume this definition of $f$ throughout the paper.

Following the same computations in [1], we can easily check that equation
(4.1) has a unique solution. The purpose of this section is to prove that
equation (4.2) has a unique solution and moreover, these solutions converge to
the solution of equation (4.1) when $\varepsilon$ goes to zero. The result
states as follows:

######  Theorem 4.1.

Assume that $b$ is a Lipschitz function,
$\zeta\in{\mathcal{C}}^{1-\alpha}(0,T)$. Then the solutions
$x^{(\zeta),\varepsilon}$ of (4.2) converge uniformly, as
$\varepsilon\rightarrow 0$, to $x^{(\zeta)}$, the unique solution of the
equation (4.1). Moreover, $x^{(\zeta)}\in{\mathcal{C}}^{1-\alpha}(0,T).$

The proof of this theorem follows easily from the following three propositions
in this section.

######  Proposition 4.2.

Under the hypothesis of Theorem 4.1, equation (4.2) has a unique solution
$x_{t}^{(\zeta),\varepsilon}\in{\mathcal{C}}^{1-\alpha}(0,T)$.

###### Proof.

If we consider $\tilde{b}(x)=b(x)+\frac{1}{\varepsilon}f(x)$, since $f$ is
Lipschitz we can apply Lemma 3.1 and we obtain the desired result. ∎

######  Proposition 4.3.

Under the hypothesis of Theorem 4.1, the solutions $x^{(\zeta),\varepsilon}$
of (4.2) converge uniformly to a continuous function $G$ on $[0,T]$ as
$\varepsilon\rightarrow 0$.

###### Proof.

Following the method used in [5], we will organize the proof in several steps.

Step 1. We consider the following equation for $t\in[0,T]$,

$v_{t}=\eta_{0}+\zeta_{t}+\int_{0}^{t}b(v_{s})ds.$ (4.3)

We want to prove that $v_{t}\leq x_{t}^{(\zeta),\varepsilon}$ for all
$\varepsilon>0$ and $t\in[0,T]$.

Using Lemma 3.1, equation (4.3) has a unique solution. Moreover, since
$b\leq\tilde{b}$ ($\tilde{b}(x)=b(x)+\frac{1}{\varepsilon}f(x)$ where
$\frac{1}{\varepsilon}f(x)\geq 0$), we can apply Lemma 3.2 and we obtain the
desired result. Step 2. Now we consider

$u_{t}=\eta_{0}+\zeta_{t}+\int_{0}^{t}b\left(u_{s}+\sup_{s^{\prime}\leq
s}u_{s^{\prime}}^{-}\right)ds.$ (4.4)

We also define $\tilde{u}_{t}=u_{t}+\sup_{s^{\prime}\leq t}u_{s^{\prime}}^{-}$
and $\phi_{t}=\sup_{s^{\prime}\leq t}u_{s^{\prime}}^{-}$ so we can write
$\tilde{u}_{t}=u_{t}+\phi_{t}$.

We observe that $\tilde{u}_{t}\geq 0$ and $\phi_{t}$ is an increasing
function. Following the same ideas in the proof of Lemma 3.1 we can prove that
equation (4.4) has a unique solution.

We want to prove that $x_{t}^{(\zeta),\varepsilon}\leq\tilde{u}_{t}$ for all
$\varepsilon>0$. We write
$\tilde{\theta}_{t}^{\varepsilon}=x_{t}^{(\zeta),\varepsilon}-\tilde{u}_{t}$.

We can use again the function $\varphi_{k}$ defined in the proof of Lemma 3.2
and we have

$\displaystyle\varphi_{k}(\tilde{\theta}_{t}^{\varepsilon})$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\varphi^{\prime}_{k}(\tilde{\theta}_{s}^{\varepsilon})\left(b(x_{s}^{(\zeta),\varepsilon})+\frac{1}{\varepsilon}f(x_{s}^{(\zeta),\varepsilon})-b(\tilde{u}_{s})\right)ds-\int_{0}^{t}\varphi^{\prime}_{k}(\tilde{\theta}_{s}^{\varepsilon})d\phi_{s}$
(4.5) $\displaystyle=$
$\displaystyle\int_{0}^{t}\varphi^{\prime}_{k}(\tilde{\theta}_{s}^{\varepsilon})\left(b(x_{s}^{(\zeta),\varepsilon})-b(\tilde{u}_{s})\right)ds+\frac{1}{\varepsilon}\int_{0}^{t}\varphi^{\prime}_{k}(\tilde{\theta}_{s}^{\varepsilon})f(x_{s}^{(\zeta),\varepsilon})ds-\int_{0}^{t}\varphi^{\prime}_{k}(\tilde{\theta}_{s}^{\varepsilon})d\phi_{s}$
$\displaystyle=$ $\displaystyle
T_{1}^{\varepsilon}(k)+T_{2}^{\varepsilon}(k)+T_{3}^{\varepsilon}(k).$

Now we observe that,

$0\leq
T_{2}^{\varepsilon}(k)\leq\frac{1}{\varepsilon}\int_{0}^{t}2(\tilde{\theta}_{s}^{\varepsilon})^{+}(x_{s}^{(\zeta),\varepsilon})^{-}ds=0.$

In fact, if $(\tilde{\theta}_{s}^{\varepsilon})^{+}>0$, then
$\tilde{\theta}_{s}^{\varepsilon}=x_{s}^{(\zeta),\varepsilon}-\tilde{u}_{s}>0$.
So, since we have proved that $\tilde{u}_{t}\geq 0$, it is necessary that
$x_{s}^{(\zeta),\varepsilon}>0$ and then
$(x_{s}^{(\zeta),\varepsilon})^{-}=0$.

On the other hand, $T_{3}^{\varepsilon}(k)\leq 0$, since
$\varphi^{\prime}_{k}(\tilde{\theta}_{s}^{\varepsilon})\geq 0$ and $\phi_{t}$
is an increasing process. Then,

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
T_{1}^{\varepsilon}(k)\leq\int_{0}^{t}2(\tilde{\theta}_{s}^{\varepsilon})^{+}|b(x_{s}^{(\zeta),\varepsilon})-b(\tilde{u}_{s})|ds$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{t}(\tilde{\theta}_{s}^{\varepsilon})^{+}|x_{s}^{(\zeta),\varepsilon}-\tilde{u}_{s}|ds=C\int_{0}^{t}(\tilde{\theta}_{s}^{\varepsilon})^{+}|\tilde{\theta}_{s}^{\varepsilon}|ds=C\int_{0}^{t}\left((\tilde{\theta}_{s}^{\varepsilon})^{+}\right)^{2}ds.$

Using the previous computations of $T_{1}^{\varepsilon}(k)$,
$T_{2}^{\varepsilon}(k)$ and $T_{3}^{\varepsilon}(k)$ on (4.5) and for
$k\rightarrow 0$ we have

$\left((\tilde{\theta}_{t}^{\varepsilon})^{+}\right)^{2}\leq
C\int_{0}^{t}\left((\tilde{\theta}_{s}^{\varepsilon})^{+}\right)^{2}ds.$

Now we have just to apply Gronwall’s lemma and we obtain that
$(\tilde{\theta}_{t}^{\varepsilon})^{+}=0$. So
$x_{t}^{(\zeta),\varepsilon}\leq\tilde{u}_{t}$ for all $\varepsilon$.

Step 3. Since we have seen that $v_{t}\leq
x_{t}^{(\zeta),\varepsilon}\leq\tilde{u}_{t}$ and $|\tilde{u}_{t}|\leq
2\sup_{s\leq t}|u_{s}|$ we can state

$|x_{t}^{(\zeta),\varepsilon}|\leq|v_{t}|+2\sup_{s\leq t}|u_{s}|.$

So,

$\sup_{\varepsilon}|x_{t}^{(\zeta),\varepsilon}|\leq|v_{t}|+2\sup_{s\leq
t}|u_{s}|.$

Step 4. We can use Lemma 3.2 to see that $x_{t}^{(\zeta),\varepsilon}$ is an
increasing sequence when $\varepsilon$ decreases. Using that we have proved
that it is bounded for all $\varepsilon>0$ we can define
$G_{t}:=\lim_{\varepsilon\rightarrow 0}x_{t}^{(\zeta),\varepsilon}$. Since the
sequence $\\{x^{(\zeta),\varepsilon}\\}_{\varepsilon}$ is increasing, the
converge is uniform in $[0,T]$ (see [20]). That is,
$\sup_{t}|x_{t}^{(\zeta),\varepsilon}-G_{t}|\rightarrow 0$ when
$\varepsilon\rightarrow 0$, and so $G$ is a continuous function. ∎

######  Proposition 4.4.

If $G$ is the function appearing in Proposition 4.3, then $G_{t}$ is the
solution of the equation (4.1) and $G\in{\mathcal{C}}^{1-\alpha}(0,T).$

###### Proof.

We will prove that $G$ is a solution of our equation. So, by uniqueness we
will obtain the desired result. Then, we have to check that we have a pair
$(G_{t},y_{t}^{(\zeta)})\in{\mathcal{C}}_{+}({\mathbb{R}}_{+},{\mathbb{R}})$
that solves the Skorohod problem with reflection for $z^{(\zeta)}$ a path in
${\mathcal{C}}_{+}({\mathbb{R}}_{+},{\mathbb{R}})$. By Definition 2.1 we have
to prove the following facts

  * (i)

$G_{t}=z_{t}^{(\zeta)}+y_{t}^{(\zeta)}$ and $G_{t}\geq 0$, for each $t\geq 0$.

  * (ii)

$y_{0}^{(\zeta)}=0$ and $y^{(\zeta)}$ is non-decreasing.

  * (iii)

For all $t\in[0,T]$, $\int_{0}^{t}G_{s}dy^{(\zeta)}_{s}=0$.

Step 1: Define, using monotone convergence theorem,

$\displaystyle y_{t}^{(\zeta)}$ $\displaystyle=$
$\displaystyle\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\int_{0}^{t}f\left(x_{s}^{(\zeta),\varepsilon}\right)ds,$
$\displaystyle z_{t}^{(\zeta)}$ $\displaystyle=$
$\displaystyle\eta_{0}+\zeta_{t}+\lim_{\varepsilon\rightarrow
0}\int_{0}^{t}b(x_{s}^{(\zeta),\varepsilon})ds=\eta_{0}+\zeta_{t}+\int_{0}^{t}b(G_{s})ds.$
(4.6)

So, it only remains to check that $G_{t}\geq 0$, $\forall t\in[0,T]$.

Since

$x_{t}^{(\zeta),\varepsilon}-\eta_{0}-\zeta_{t}-\int_{0}^{t}b(x_{s}^{(\zeta),\varepsilon})ds=\frac{1}{\varepsilon}\int_{0}^{t}f\left(x_{s}^{(\zeta),\varepsilon}\right)ds$

and

$\lim_{\varepsilon\rightarrow 0}\left[\varepsilon
x_{t}^{(\zeta),\varepsilon}-\varepsilon\eta_{0}-\varepsilon\zeta_{t}-\varepsilon\int_{0}^{t}b(x_{s}^{(\zeta),\varepsilon})ds\right]=0,\quad\forall\,t\in[0,T],$

we have

$\lim_{\varepsilon\rightarrow
0}\int_{0}^{t}f\left(x_{s}^{(\zeta),\varepsilon}\right)ds=0,\quad\forall\,t\in[0,T].$

The monotone convergence theorem implies that
$\int_{0}^{t}f\left(G_{s}\right)ds=0,$ $\forall\,t\in[0,T]$. Then,
$G_{s}^{-}=0$ and $G_{s}\geq 0$. Step 2: Obviously $y_{0}^{(\zeta)}=0$.
Moreover, $y^{(\zeta)}$ is non-decreasing since $f\geq 0.$

Step 3: With the definition of $y_{t}^{(\zeta)}$ in mind, we define the
following two measures on $[0,T]$

$\mu_{\varepsilon}(A)=\int_{A}\frac{1}{\varepsilon}f\left(x_{s}^{(\zeta),\varepsilon}\right)ds,\qquad\qquad\mu(A)=\int_{A}dy_{s}^{(\zeta)},\qquad\quad
A\in\mathcal{B}([0,T]).$

We have that $\mu_{\varepsilon}$ converges weakly to $\mu$, this means that

$\lim_{\varepsilon\downarrow
0}\mu_{\varepsilon}([0,t])=\mu([0,t]),\qquad\forall\,t\geq 0.$

Observe that if $\varepsilon\leq\varepsilon^{\prime}$, then
$x^{(\zeta),\varepsilon}_{t}\geq x^{(\zeta),\varepsilon^{\prime}}_{t}$ for all
$t\in[0,T]$, and $f(x^{(\zeta),\varepsilon}_{t})\leq
f(x^{(\zeta),\varepsilon^{\prime}}_{t}),$ for all $t\in[0,T]$. It implies that
${\rm supp}\ \mu_{\varepsilon}\subseteq\ {\rm
supp}\mu_{\varepsilon^{\prime}}$, and so, ${\rm supp}\ \mu\subseteq{\rm supp}\
\mu_{\varepsilon}$, for any $\varepsilon>0$.

So, if $s\in{\rm supp}\ \mu$, then $f(x^{(\zeta),\varepsilon}_{s})\geq 0$ and
it satisfies that $x^{(\zeta),\varepsilon}_{s}\leq 0$. So, we get, for any
$t\in[0,T]$,

$\int_{0}^{t}x^{(\zeta),\varepsilon}_{s}dy_{s}^{(\zeta)}=\int_{0}^{t}x^{(\zeta),\varepsilon}_{s}d\mu(s)\leq
0.$

Finally, by the monotone convergence theorem

$\int_{0}^{t}G_{s}dy_{s}^{(\zeta)}\leq 0.$

The condition $G\geq 0$ implies that

$\int_{0}^{t}G_{s}dy_{s}^{(\zeta)}=0.$

Step 4: To finish the proof let us check that
$G\in{\mathcal{C}}^{1-\alpha}(0,T).$ From expression (4.6) it is clear that
$z^{\zeta}$ belongs to ${\mathcal{C}}^{1-\alpha}(0,T)$ and, on the other hand,
in the proof of Proposition 4.2 in [1] we have checked that
$\|G\|_{1-\alpha(0,T)}\leq 2\|z^{(\zeta)}\|_{1-\alpha(0,T)}$. ∎

##  5\. Differentiability

The aim of this section is to study the Fréchet differentiability in the
direction $h\in W_{1}^{1-\alpha}(0,T)$ of the solution to equation

$\displaystyle
x_{t}^{g}=\eta_{0}+\int_{0}^{t}b(x_{s}^{g})ds+\int_{0}^{t}\sigma(\bar{x}^{g}_{s-r})dg_{s},\qquad
t\in[0,T],$ $\displaystyle x_{t}^{g}=\eta_{t},\qquad t\in[-r,0]$ (5.1)

where $\bar{x}^{g}$ is Fréchet differentiable. We follow the ideas presented
in [21]. The main result of this section is the following:

######  Theorem 5.1.

Let $\bar{x}$ be a map $g\mapsto\bar{x}^{g}$ from $W_{2}^{1-\alpha}(0,T)$ to
$W_{1}^{\alpha}(-r,T-r),$ continuously Fréchet differentiable in the direction
$h\in W_{2}^{1-\alpha}(0,T)$. Assume $b,\sigma\in{\mathcal{C}}^{2}_{b}$ and
set $x^{g}$ the solution to equation (5.1). Then the mapping

$g\in W_{2}^{1-\alpha}(0,T)\mapsto x^{g}\in W_{1}^{\alpha}(0,T)$

is Fréchet differentiable in the direction $h\in W_{2}^{1-\alpha}(0,T)$ and it
holds that

$D_{h}x_{t}^{g}=\int_{0}^{t}b^{\prime}(x_{s}^{g})D_{h}x^{g}_{s}ds+\int_{0}^{t}\sigma(\bar{x}^{g}_{s-r})dh_{s}+\int_{0}^{t}\sigma^{\prime}(\bar{x}^{g}_{s-r})D_{h}\bar{x}^{g}_{s-r}dg_{s}.$

###### Proof.

It follows easily combining Propositions 5.2 and 5.4. ∎

We begin studying the differentiability of the integral.

######  Proposition 5.2.

Let $x$ be a map $g\mapsto\bar{x}^{g}$ from $W_{2}^{1-\alpha}(-r,T-r)$ to
$W_{1}^{\alpha}(-r,T),$ continuously Fréchet differentiable in the direction
$h\in W_{2}^{1-\alpha}(0,T)$. Let $\sigma\in{\mathcal{C}}^{1}_{b}$. Then the
mapping

$F:W_{2}^{1-\alpha}(0,T)\mapsto W_{1}^{\alpha}(0,T),$

defined by

$F(g)_{\cdot}:=\int_{0}^{\cdot}\sigma(\bar{x}_{s-r}^{g})dg_{s}$

is Fréchet differentiable in the direction $h\in W_{2}^{1-\alpha}(0,T)$ with
directional derivative

$D_{h}F(g)_{t}=\int_{0}^{t}\sigma(\bar{x}^{g}_{s-r})dh_{s}+\int_{0}^{t}\sigma^{\prime}(\bar{x}^{g}_{s-r})D_{h}\bar{x}^{g}_{s-r}dg_{s}.$

###### Proof.

Let us consider the mapping

$H:W_{2}^{1-\alpha}(0,T)\times W_{1}^{\alpha}(-r,T-r)\mapsto
W_{1}^{\alpha}(0,T),$

defined by

$H(g,x):=\int_{0}^{\cdot}\sigma(x_{s-r})dg_{s}.$

Following the same computations as in Lemma 3 in [21] we can check that $H$ is
Fréchet differentiable with directional derivatives:

  * •

for any $h\in W_{2}^{1-\alpha}(0,T)$

$D_{1,h}H(g,x)_{t}=\int_{0}^{t}\sigma(x_{s-r})dh_{s},$

  * •

for any $w\in W_{1}^{\alpha}(-r,T-r)$

$D_{2,w}H(g,x)_{t}=\int_{0}^{t}\sigma^{\prime}(x_{s-r})w_{s-r}dg_{s}.$

Then, since $F(g)=H(g,\bar{x}^{g})$, using the chain rule we get that, for
$h\in W_{2}^{1-\alpha}(0,T)$

$\displaystyle D_{h}F(g)_{t}$ $\displaystyle=$ $\displaystyle
D_{1,h}H(g,\bar{x}^{g})_{t}+D_{2,D_{h}\bar{x}^{g}}H(g,\bar{x}^{g})_{t}$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\sigma(\bar{x}^{g}_{s-r})dh_{s}+\int_{0}^{t}\sigma^{\prime}(\bar{x}^{g}_{s-r})D_{h}\bar{x}^{g}_{s-r}dg_{s}.$

∎

We need a technical lemma before the last proposition.

######  Lemma 5.3.

Assume that $\eta_{0}\in{\mathbb{R}}$ and that $b$ belongs to $C^{2}_{b}$ and
that the mapping

$\zeta:W_{2}^{1-\alpha}(0,T)\mapsto W_{1}^{\alpha}(0,T)$

is Fréchet differentiable in the direction $h\in W_{2}^{1-\alpha}(0,T)$. Then
the mapping

$F:W_{2}^{1-\alpha}(0,T)\times W_{1}^{\alpha}(0,T)\rightarrow
W_{1}^{\alpha}(0,T)$

defined by

$(g,x)\mapsto F(g,x):=x-\eta_{0}-\int_{0}^{\cdot}b(x_{s})ds-\zeta^{g}_{\cdot}$

is Fréchet differentiable in the direction $h\in W_{2}^{1-\alpha}(0,T)$.
Moreover, for $h\in W_{2}^{1-\alpha}(0,T)$, $v\in W_{1}^{\alpha}(0,T)$ and
$(g,x)\in W_{2}^{1-\alpha}(0,T)\times W_{1}^{\alpha}(0,T)$, the Fréchet
derivatives with respect to $h$ and $v$ are given respectively by

$\displaystyle D_{1,h}F(g,x)_{t}$ $\displaystyle=-D_{h}\zeta_{t}^{g},$ (5.2)
$\displaystyle D_{2,v}F(g,x)_{t}$
$\displaystyle=v_{t}-\int_{0}^{t}b^{\prime}(x_{s})v_{s}ds.$ (5.3)

###### Proof.

For $(h,x)$ and $(\tilde{h},\tilde{x})$ in $W_{2}^{1-\alpha}(0,T)\times
W_{1}^{\alpha}(0,T)$ we have

$F(h,x)_{t}-F(\tilde{h},\tilde{x})_{t}=x_{t}-\tilde{x}_{t}-\int_{0}^{t}(b(x_{s})-b(\tilde{x}_{s}))ds-(\zeta_{t}^{h}-\zeta_{t}^{\tilde{h}}).$

Using [2, Proposition 2.2], we get that

$\begin{split}\bigg{\|}x-\tilde{x}-\int_{0}^{\cdot}(b(x_{s})-b(\tilde{x}_{s}))ds\bigg{\|}_{\alpha,1}&\leq
c_{\alpha,T}\|x-\tilde{x}\|_{\alpha,1}.\end{split}$

On the other hand, since $\zeta$ is differentiable, it will be continuous and
$|\zeta^{h}-\zeta^{\tilde{h}}|$ can be controlled. Therefore, $F$ is
continuous in both variables $(h,x)$.

We next show the Fréchet differentiability. Let $v,w\in W_{1}^{\alpha}(0,T)$.
By [2, Proposition 2.2], we have that

$\|D_{2,v}F(h,x)-D_{2,w}F(h,x)\|_{\alpha,1}\leq
c_{\alpha,T}\|v-w\|_{\alpha,1}.$

Thus, $D_{2,\cdot}F(h,x)$ is a bounded linear operator. Moreover,

$F(h,x+v)_{t}-F(h,x)_{t}-D_{2,v}F(h,x)_{t}=\int_{0}^{t}(b(x_{s})-b(x_{s}+v_{s})+b^{\prime}(x_{s})v_{s})ds.$

By the mean value theorem and [2, Proposition 2.2],

$\bigg{\|}\int_{0}^{\cdot}(b(x_{s})-b(x_{s}+v_{s})+b^{\prime}(x_{s})v_{s})ds\bigg{\|}_{\alpha,1}\leq
c_{\alpha,T}\|v\|_{\alpha,1}^{2}.$

This shows that (5.3) is the Fréchet derivative of $F(h,x)$ with respect to
$x$. The Fréchet differentiability with respect to $h$ and the derivative is
given easily by (5.2) from our hypothesis. ∎

######  Proposition 5.4.

Assume the hypotheses of Lemma 5.3. Let $x^{g}$ be the solution of

$x_{t}^{g}=\eta_{0}+\zeta_{t}^{g}+\int_{0}^{t}b(x_{s}^{g})ds,\quad t\in(0,T].$

Then the mapping

$g\in W_{2}^{1-\alpha}(0,T)\rightarrow x^{g}\in W_{1}^{\alpha}(0,T)$

is Fréchet differentiable in the direction $h$ and the derivative is given by

$D_{h}x_{t}^{g}=D_{h}\zeta^{g}_{t}+\int_{0}^{t}b^{\prime}(x_{s}^{g})D_{h}x_{s}^{g}ds.$

###### Proof.

Following the ideas of the proof of Proposition 4 in [21] and the result of
Lemma 5.3 we have

$\displaystyle D_{2,v}F(0,x^{g})_{t}$ $\displaystyle=$ $\displaystyle
v_{t}-\int_{0}^{t}b^{\prime}(x_{s}^{g})v_{s}ds.$ $\displaystyle Dx^{g}$
$\displaystyle=$ $\displaystyle-D_{2,\cdot}F(0,x^{g})^{-1}\circ
D_{1}F(0,x^{g}).$

Then, for any $h\in W_{2}^{1-\alpha}(0,T)$, $-D_{h}x^{g}$ is the unique
solution of the differential equation

$D_{1,h}F(0,x^{g})_{t}=-D_{2,D_{h}x^{g}}F(0,x^{g})_{t},$

that is

$-D_{h}\zeta^{g}_{t}=-D_{h}x^{g}_{t}+\int_{0}^{t}b^{\prime}(x_{s}^{g})D_{h}x^{g}_{s}ds.$

So, we obtain the desired derivative. ∎

We finish this section with an expression of the Fréchet derivative.

######  Proposition 5.5.

Assume the hypotheses of Theorem 5.1. Assume that the derivative in the
direction $h\in W_{1}^{1-\alpha}(0,T)$ is given by

$D_{h}\bar{x}_{t}^{g}=\int_{0}^{t}D_{s}\bar{x}_{t}^{g}dh_{s},$

with $D_{s}\bar{x}_{u}^{g}=0$ if $s>u$. Then the derivative of the solution of
equation (5.1) in the direction $h$ is given by

$D_{h}x_{t}^{g}=\int_{0}^{t}\Phi_{t}^{g}(s)dh_{s},$

and $\Phi_{t}^{g}(s)$ defined as

$\Phi_{t}^{g}(s)=\begin{cases}\displaystyle\int_{s}^{t}b^{\prime}(x_{u}^{g})\Phi_{u}^{g}(s)du+\sigma(\bar{x}^{g}_{s-r})+\int_{s}^{t}\sigma^{\prime}(\bar{x}^{g}_{u-r})D_{s}\bar{x}^{g}_{u-r}dg_{u},&\qquad\mbox{if}\;s\leq
t,\\\\[10.0pt] 0,&\qquad\mbox{if}\;s>t.\end{cases}$ (5.4)

###### Proof.

The proof follows similarly as the proof of [21, Proposition 4]. Indeed, for
$t\in[T]$

$\displaystyle\int_{0}^{t}\Phi^{g}_{t}(s)dh_{s}=\int_{0}^{t}\Big{(}\int_{s}^{t}b^{\prime}(x_{u}^{g})\Phi_{u}^{g}(s)du+\sigma(\bar{x}^{g}_{s-r})+\int_{s}^{t}\sigma^{\prime}(\bar{x}^{g}_{u-r})D_{s}\bar{x}^{g}_{u-r}dg_{u}\Big{)}dh_{s}$
$\displaystyle=\int_{0}^{t}b^{\prime}(x_{u}^{g})\Big{(}\int_{0}^{u}\Phi_{u}^{g}(s)dh_{s}\Big{)}du+\int_{0}^{t}\sigma(\bar{x}^{g}_{s-r})dh_{s}+\int_{0}^{t}\sigma^{\prime}(\bar{x}^{g}_{u-r})\Big{(}\int_{0}^{u-r}D_{s}\bar{x}^{g}_{u-r}dh_{s}\Big{)}dg_{u}$
$\displaystyle=\int_{0}^{t}b^{\prime}(x_{u}^{g})D_{h}x_{u}^{g}du+\int_{0}^{t}\sigma(\bar{x}^{g}_{s-r})dh_{s}+\int_{0}^{t}\sigma^{\prime}(\bar{x}^{g}_{u-r})D_{h}\bar{x}^{g}_{u-r}dg_{u},$

and $\Phi_{t}^{g}(s)=0$ if $s>t$. Notice that we have used that for $s>u-r$ we
know that $D_{s}\bar{x}_{u-r}^{g}=0$ ∎

##  6\. Definition of a sequence that converges to the deterministic solution

Our aim is to construct an approximated sequence in intervals of length $r$
that converges to the solution of the deterministic equation

$\displaystyle
x_{t}=\begin{cases}\displaystyle\eta_{0}+\int_{0}^{t}b(x_{s})ds+\int_{0}^{t}\sigma(x_{s-r})dg_{s}+y_{t},&\textrm{if}\;\;t\in\left[0,T\right],\\\
\eta_{t},&\textrm{if}\;\;t\in[-r,0].\end{cases}$

We will not write the dependence on $g$ of the solution of this equation in
order to simplify the notation. Let us recall that
$b_{\varepsilon}(x)=b(x)+\frac{1}{\varepsilon}f(x)$ for $f$ defined as in
Section 4 and assume throughout this section that
$b,\sigma\in{\mathcal{C}}^{2}_{b}$, $\eta\in{\mathcal{C}}^{1-\alpha}(-r,0)$
and $T=mr$.

We start with the interval $[-r,r]$. Since the map
$t\to\int_{0}^{t}\sigma(\eta_{s-r})dg_{s}$ belongs to
${\mathcal{C}}^{1-\alpha}(0,r)$, we have seen in Section 4 that the solutions
to the equations

$\displaystyle
x^{\varepsilon}_{t}=\begin{cases}\displaystyle\eta_{0}+\int_{0}^{t}b_{\varepsilon}(x_{s}^{\varepsilon})ds+\int_{0}^{t}\sigma(\eta_{s-r})dg_{s},&\textrm{if}\;\;t\in\left[0,r\right],\\\
\eta_{u},&\textrm{if}\;\;t\in[-r,0],\end{cases}$

converge uniformly to a continuous function $x$ belonging to
${\mathcal{C}}^{1-\alpha}(-r,r)$ such that

$\lim_{\varepsilon\to 0}\sup_{t\in[-r,r]}|x_{t}^{\varepsilon}-x_{t}|=0,$

and that satisfies the reflected equation

$\displaystyle
x_{t}=\begin{cases}\displaystyle\eta_{0}+\int_{0}^{t}b(x_{s})ds+\int_{0}^{t}\sigma(x_{s-r})dg_{s}+y_{t},&\textrm{if}\;\;t\in\left[0,r\right],\\\
\eta_{u},&\textrm{if}\;\;t\in[-r,0].\end{cases}$

Let us consider now the interval $[-r,2r]$ and the equation

$x^{\varepsilon}_{t}=\begin{cases}\displaystyle
x_{t}^{\varepsilon},&\textrm{if}\;\;t\in\left[-r,r\right],\\\ \displaystyle
x_{r}^{\varepsilon}+\int_{r}^{t}b_{\varepsilon}(x^{\varepsilon}_{s})ds+\int_{r}^{t}\sigma(x_{s-r})dg_{s},&\textrm{if}\;\;t\in\left[r,2r\right],\end{cases}$

that can be written

$x^{\varepsilon}_{t}=\eta_{0}+\int_{0}^{t}b_{\varepsilon}(x^{\varepsilon}_{s})ds+\int_{0}^{t}\sigma(x_{s-r})dg_{s},$

for $t\in[0,2r]$. Following the same arguments, there exists a continuous
function $x$ such that

$\lim_{\varepsilon\to 0}\sup_{t\in[-r,2r]}|x_{t}^{\varepsilon}-x_{t}|=0,$

and that satisfies the reflected equation

$\displaystyle
x_{t}=\begin{cases}\displaystyle\eta_{0}+\int_{0}^{t}b(x_{s})ds+\int_{0}^{t}\sigma(x_{s-r})dg_{s}+y_{t},&\textrm{if}\;\;t\in\left[0,2r\right],\\\
\eta_{u},&\textrm{if}\;\;t\in[-r,0].\end{cases}$

Notice that we have used the same notation $x$ since the solution is the
obvious extension of the function defined in $[-r,r].$

Repeating the same argument $m$ times (recall that $T=mr$), we will obtain a
sequence

$x^{\varepsilon}_{t}=\begin{cases}\displaystyle\eta_{t},&\textrm{if}\;\;t\in\left[-r,0\right]\\\
\displaystyle
x_{r}^{\varepsilon}+\int_{r}^{t}b_{\varepsilon}(x^{\varepsilon}_{s})ds+\int_{r}^{t}\sigma(x_{s-r})dg_{s},&\textrm{if}\;\;t\in\left[0,T\right],\end{cases}$

and a function $x$ belonging to ${\mathcal{C}}^{1-\alpha}(0,T)$ satisfying

$\lim_{\varepsilon\to 0}\sup_{t\in[-r,T]}|x_{t}^{\varepsilon}-x_{t}|=0,$

such that

$x_{t}=\eta_{0}+\int_{0}^{t}b(x_{s})ds+\int_{0}^{t}\sigma(x_{s-r})dg_{s}+y_{t},$

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

Now we have a sequence that converges to $x$ uniformly in $(-r,T]$. We want to
deal now with its Fréchet differentiability.

First, we deal again with the interval $[0,r]$. Clearly, from Theorem 5.1
yields that $x_{t}^{\varepsilon}$ are Fréchet differentiable in the directions
$h\in W_{2}^{1-\alpha}(0,r)$ and that its derivative is, for $t\in[0,r]$

$D_{h}x^{\varepsilon}_{t}=\int_{0}^{t}b^{\prime}_{\varepsilon}(x_{s}^{\varepsilon})D_{h}x_{s}^{\varepsilon}ds+\int_{0}^{t}\sigma(\eta_{s-r})dh_{s}.$

Moreover, from Proposition 5.5, the derivative in the direction $h$ is given
by

$D_{h}x_{t}^{\varepsilon}=\int_{0}^{t}\Phi_{t}^{g,\varepsilon}(s)dh_{s},$

for $t\in[0,r]$ and $\Phi_{t}^{\varepsilon}(s)$ defined as

$\Phi_{t}^{g,\varepsilon}(s)=\begin{cases}\displaystyle\int_{s}^{t}b^{\prime}_{\varepsilon}(x_{u}^{\varepsilon})\Phi_{u}^{g,\varepsilon}(s)du+\sigma(\eta_{s-r}),&\qquad\mbox{if}\;s\leq
t,\\\\[10.0pt] 0,&\qquad\mbox{if}\;s>t.\end{cases}$

Let us consider a the general case, that is, for $t\in[(l-1)r,lr]$. We assume
that $x_{t}$ is differentiable in the direction $h$. Then by Theorem 5.1 we
clearly have that $x_{t}^{\varepsilon}$ are Fréchet differentiable in the
direction $h$ and also we can yield its derivative.

$\displaystyle D_{h}x_{t}^{\varepsilon}$ $\displaystyle=$ $\displaystyle
D_{h}\int_{0}^{t}\sigma\left(x_{s-r}\right)dg_{s}+\int_{0}^{t}b^{\prime}_{\varepsilon}\left(x_{s}^{\varepsilon}\right)D_{h}x_{s}^{\varepsilon}ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}b^{\prime}_{\varepsilon}\left(x_{s}^{\varepsilon}\right)D_{h}x_{s}^{\varepsilon}ds+\int_{0}^{t}\sigma\left(x_{s-r}\right)dh_{s}+\int_{0}^{t}\sigma^{\prime}(x_{s-r})D_{h}x_{s-r}dg_{s}.$

By Proposition 5.5, this derivative in the direction $h$ can be written

$D_{h}x_{t}^{\varepsilon}=\int_{0}^{t}\Phi_{t}^{g,\varepsilon}(s)dh_{s},$

for $t\in[0,lr]$ and $\Phi_{t}^{\varepsilon}(s)$ defined as

$\Phi_{t}^{g,\varepsilon}(s)=\begin{cases}\displaystyle\int_{s}^{t}b^{\prime}_{\varepsilon}(x_{u}^{\varepsilon})\Phi_{u}^{g,\varepsilon}(s)du+\sigma(x_{s-r})+\int_{s}^{t}\sigma^{\prime}(x_{u-r})D_{s}x_{u-r}dg_{u},&\qquad\mbox{if}\;s\leq
t,\\\\[10.0pt] 0,&\qquad\mbox{if}\;s>t.\end{cases}$

##  7\. Stochastic case

In this section we apply the results obtained in the previous sections to the
stochastic delay differential equation with positivity constraints (1.1). We
can assume without loss of generality that $T_{0}=mr$.

Recall that $W^{H}=\\{W_{t}^{H},\,t\in[0,T_{0}]\\}$ is a one-dimensional
fractional Brownian motion with Hurst parameter $H>\frac{1}{2}$. That is, a
centered Gaussian process with covariance function

$\mathbf{E}(W_{t}^{H}W_{t}^{H})=R_{H}(t,s)=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right).$

Fix $\alpha\in(1-H,\frac{1}{2})$. As the trajectories of $W^{H}$ are
$(1-\alpha+\epsilon)$-Hölder continuous for all $\epsilon<H+\alpha-1$, by the
first inclusion in (2.1), we can apply the framework of the previous sections.
In particular, under the assumptions of Theorem 1.1, there exist the solutions
of equation (1.1) and all the other stochastic equations that appear in this
section.

We next proceed with the study of the Malliavin differentiability of the
solution. We begin applying the deterministic results of Theorem 5.1 to our
class of stochastic equations.

######  Theorem 7.1.

Let $\\{\bar{X}_{t},t\in[-r,kr]\\}$ be a stochastic process in
$W_{1}^{\alpha}(-r,kr),$ such that $\bar{X}$ is almost surely differentiable
in the directions of the Cameron-Martin space. Assume
$b,\sigma\in{\mathcal{C}}^{2}_{b}$ and set $\tilde{X}$ the solution to
equation

$\displaystyle\tilde{X}_{t}=\eta_{0}+\int_{0}^{t}b(\tilde{X}_{s})ds+\int_{0}^{t}\sigma(\bar{X}_{s-r})dW^{H}_{s},\qquad
t\in[0,r(k+1)],$ $\displaystyle\tilde{X}_{t}=\eta_{t},\qquad t\in[-r,0].$

Then, the solution $\tilde{X}$ is almost surely differentiable in the
directions of the Cameron-Martin space. Moreover, for any $t>0$, the
derivative satisfies

$D_{s}\tilde{X}_{t}=\int_{0}^{t}b^{\prime}(\tilde{X}_{u})D_{s}\tilde{X}_{u}du+\sigma(\bar{X}_{s-r})+\int_{0}^{t}\sigma^{\prime}(\bar{X}_{u-r})D_{s}\bar{X}_{u-r}dW^{H}_{u}$
(7.1)

if $s\leq t$ and $D_{s}\tilde{X}_{t}=0$ if $s>t$.

###### Proof.

By Proposition 5.5, the solution is Fréchet differentiable and for all
$\varphi\in\mathcal{H}$ the Fréchet derivative

$D_{\mathcal{R}_{H}\varphi}\tilde{X}_{t}^{i}=\frac{d}{d\epsilon}\tilde{X}_{t}^{i}(\omega+\epsilon\mathcal{R}_{H}\varphi)|_{\epsilon=0}$

exists, which proves the first statement of the theorem.

The derivative $D_{\mathcal{R}_{H}\varphi}\tilde{X}_{t}$ coincides with
$\langle D\tilde{X}_{t},\varphi\rangle_{\mathcal{H}},$ where $D$ is the usual
Malliavin derivative. Furthermore, for any $\varphi\in\mathcal{H}$, since we
can write

$D_{\mathcal{R}_{H}\varphi}\bar{X}_{t}=\langle
D\bar{X}_{t},\varphi\rangle_{\mathcal{H}},$

by Proposition 5.5, we get,

$\begin{split}D_{\mathcal{R}_{H}\varphi}\tilde{X}_{t}&=\int_{0}^{t}\Phi_{t}^{W^{H}}(s)d(\mathcal{R}_{H}\varphi)(s)\\\
&=\int_{0}^{t}\Phi_{t}^{W^{H}}(s)\left(\int_{0}^{s}\partial_{s}K_{H}(s,u)(\mathcal{K}_{H}^{\ast}\varphi)(u)du\right)ds\\\
&=\sum_{j=1}^{m}\int_{0}^{T}(\mathcal{K}_{H}^{\ast}\Phi_{t}^{W^{H}})(s)(\mathcal{K}_{H}^{\ast}\varphi)(s)ds\\\
&=\langle\Phi_{t}^{W^{H}},\varphi\rangle_{\mathcal{H}}\end{split}$

and equation (7.1) follows from (5.4). This concludes the proof. ∎

Now, we can deal with the Malliavin derivability of the solution to the
equation (1.1). First, we recall a technical lemma proved in [22].

######  Lemma 7.2.

Let $\\{F_{n},\,n\geq 1\\}$ be a sequence of random variables in
${\mathbb{D}}^{1,p}$ that converges to F in $L^{p}(\Omega)$ and such that

$\sup_{n}\mathbf{E}\left[\|DF_{n}\|^{p}_{\mathcal{H}}\right]<\infty.$

Then, $F$ belongs to ${\mathbb{D}}^{1,p}$, and the sequence of derivatives
$\\{DF_{n},\,n\geq 1\\}$ converges to $DF$ in the weak topology of
$L^{p}(\Omega,{\mathcal{H}})$.

Using the lemma above we can state the following result:

######  Proposition 7.3.

For any $t\in(0,T_{0}]$, $X_{t}\in{\mathbb{D}}^{1,2}$.

###### Proof.

Let us recall that we have for the deterministic case that
$x_{t}=\lim_{\varepsilon\to 0}x_{t}^{\varepsilon}$ and that
$x_{t}^{\varepsilon}$ is an increasing sequence (Propositions 4.3 and 4.4). So
$X_{t}^{\varepsilon}$ converges almost surely to $X_{t}$. Since
$X_{t}^{\varepsilon}$ is an increasing sequence and we know that
$E(X_{t}^{2})<\infty$, we get that $\\{X_{t}^{\varepsilon},\varepsilon>0\\}$
converges to $X_{t}$ in $L^{2}$.

We begin studying $t\in[0,r]$. Using Theorem 7.1, since the initial condition
$\eta$ is deterministic we get that, for any $\varepsilon>0$,
$X_{t}^{\varepsilon}$ belongs to ${\mathbb{D}}^{1,2}$ and its derivative
satisfies, for $s<t$

$D_{s}X_{t}^{\varepsilon}=\int_{s}^{t}b^{\prime}_{\varepsilon}(X_{u}^{\varepsilon})D_{s}X_{u}^{\varepsilon}du+\sigma(\eta_{s-r}).$

The solution of this equation can be written

$\displaystyle D_{s}X_{t}^{\varepsilon}$ $\displaystyle=$
$\displaystyle\sigma(\eta_{s-r})\exp\left(\int_{s}^{t}b^{\prime}_{\varepsilon}(X_{u}^{\varepsilon})du\right)$
$\displaystyle=$
$\displaystyle\sigma(\eta_{s-r})\exp\left(\int_{s}^{t}b^{\prime}(X_{u}^{\varepsilon})du\right)\exp\left(\frac{1}{\varepsilon}\int_{s}^{t}f^{\prime}(X_{u}^{\varepsilon})du\right).$

Using that

$\mathbf{E}\left[\|DX_{t}^{\varepsilon}\|^{2}_{\mathcal{H}}\right]\leq
C_{H}\int_{0}^{t}\mathbf{E}\left(|D_{u}X_{t}^{\varepsilon}|^{2}\right)du,$
(7.2)

since $f^{\prime}<0$ we get easily that

$\sup_{\varepsilon}\mathbf{E}\left[\|DX_{t}^{\varepsilon}\|^{2}_{\mathcal{H}}\right]\leq
C_{H}\int_{0}^{t}\sigma^{2}(\eta_{u-r})\exp\left(2(t-u)\|b^{\prime}\|_{\infty}\right)du<\infty.$

Applying Lemma 7.2, we get that for any
$t\in[0,r],X_{t}\in{\mathbb{D}}^{1,2}$.

We continue studying $t\in[0,2r]$. Using again Theorem 7.1 we get that each
$X_{t}^{\varepsilon}$ belongs to ${\mathbb{D}}^{1,2}$ and its derivative
satisfies, for $t\in[r,2r]$ and for $s<t$

$D_{s}X_{t}^{\varepsilon}=\int_{s}^{t}b^{\prime}_{\varepsilon}(X_{u}^{\varepsilon})D_{s}X_{u}^{\varepsilon}du+\sigma(X_{s-r})+\int_{s}^{t}\sigma^{\prime}(X_{u-r})D_{s}X_{u-r}dW_{u}.$

The solution of this equation can be written

$\displaystyle D_{s}X_{t}^{\varepsilon}$
$\displaystyle=\left(\sigma(X_{s-r})+\int_{s}^{t}\sigma^{\prime}(X_{u-r})D_{s}X_{u-r}dW_{u}\right)$
(7.3)
$\displaystyle\qquad\times\exp\left(\int_{s}^{t}b^{\prime}(X_{u}^{\varepsilon})du\Bigg{)}\exp\Bigg{(}\frac{1}{\varepsilon}\int_{s}^{t}f^{\prime}(X_{u}^{\varepsilon})du\right).$

Using again (7.2), since $f^{\prime}<0$ we get easily that

$\displaystyle\sup_{\varepsilon}\mathbf{E}\left[\|DX_{t}^{\varepsilon}\|^{2}_{\mathcal{H}}\right]\leq
2C_{H}\int_{0}^{t}\mathbf{E}\left(\sigma^{2}(X_{s-r})+\left|\int_{s}^{t}\sigma^{\prime}(X_{u-r})D_{s}X_{u-r}dW_{u}\right|^{2}\right)$
$\displaystyle\hskip
142.26378pt\times\exp\left(2(t-u)\|b^{\prime}\|_{\infty}\right)du<\infty.$

So, $X_{t}\in{\mathbb{D}}^{1,2}$.

Iterating this procedure $m$ times, we get that $X_{t}\in{\mathbb{D}}^{1,2}$
for any $t\in[0,T_{0}]$. Obviously, it implies that
$Y_{t}\in{\mathbb{D}}^{1,2}$ for any $t\in[0,T_{0}]$ and the we can write, for
$s<t$

$D_{s}X_{t}=\int_{s}^{t}b^{\prime}(X_{u})D_{s}X_{u}du+\sigma(X_{s-r})+\int_{s}^{t}\sigma^{\prime}(X_{u-r})D_{s}X_{u-r}dW_{u}+D_{s}Y_{t}.$

∎

######  Remark 7.4.

Notice that in the previous proof we have seen that

$\sup_{\varepsilon}\int_{0}^{t_{0}}\mathbf{E}\left(|D_{s}X_{t_{0}}^{\varepsilon}|^{2}\right)ds<\infty.$

So, there will exist a convergent sub-sequence that will converge in the weak
topology of $L^{2}(\Omega;L^{2}([0,T]))$. Moreover, by uniqueness, the limit
will be $DX_{t_{0}}.$

Now we can give the proof of the main result of this paper.

Proof of Theorem 1.1. We will prove that for any $a>0$, the restriction on
$[a,+\infty)$ of the law of $X_{t_{0}}$ is absolutely continuous. We have seen
in Proposition 7.3 that $X_{t_{0}}$ belongs to ${\mathbb{D}}^{1,2}$. Then,
using the classical result due to Bouleau and Hirsch (see [3] or [22, Theorem
2.1.2]) it is enough to check that on the set
$\Omega_{a}=\\{\omega,X_{t_{0}}(\omega)\geq a\\}$

$\displaystyle\int_{0}^{t_{0}}(D_{s}X_{t_{0}})^{2}ds>0,\qquad\mbox{ a.s on
}\Omega_{a}.$

Since

$\int_{0}^{t_{0}}|D_{s}X_{t_{0}}|ds<t_{0}^{\frac{1}{2}}\left(\int_{0}^{t_{0}}(D_{s}X_{t_{0}})^{2}ds\right)^{\frac{1}{2}}$

it suffices to check that

$\int_{0}^{t_{0}}|D_{s}X_{t_{0}}|ds>0,\qquad\mbox{ a.s on }\Omega_{a}.$

On the other hand, since $\sigma>0$, we have that $\sigma(X_{t_{0}-r})>0$ and
we can find $t_{1}(\omega)<t_{0}$ such that for $s\in[t_{1},t_{0}]$

$\sigma(X_{s-r})+\int_{s}^{t_{0}}\sigma^{\prime}(X_{u-r})D_{s}X_{u-r}dW_{u}>\frac{1}{2}\sigma(X_{t_{0}-r})>0.$

From expression (7.3), we get that for any $s,\,t_{1}(\omega)<s<t_{0}$ and for
all $\varepsilon>0$ it holds that $D_{s}X_{t_{0}}^{\varepsilon}>0$.

Passing to the limit, we obtain that for any $s,\,t_{1}(\omega)<s<t_{0}$ it
holds that $D_{s}X_{t_{0}}\geq 0$ and so, it will be enough to prove that

$\int_{t_{1}}^{t_{0}}D_{s}X_{t_{0}}ds>0,\qquad\mbox{ a.s on }\Omega_{a}.$

Fixed $a>0$ set $B_{s}=\\{\omega,X_{s}(\omega)>\frac{a}{2}\\}$ and
$\tau_{s}=\inf\\{t\geq s,X_{t}=\frac{a}{2}\\}$. We will assume that our path
$\omega$ is in $B_{t_{0}}.$ Then

$\int_{t_{1}}^{t_{0}}D_{s}X_{t_{0}}ds\geq\int_{t_{1}}^{t_{0}}D_{s}X_{t_{0}}{\bf
1}_{\\{\tau_{s}>t_{0}\\}}ds.$ (7.4)

From Remark 7.4 it yields that there exists a sub-sequence that will converge
in the weak topology, such that,

$\int_{t_{1}}^{t_{0}}D_{s}X_{t_{0}}{\bf
1}_{\\{\tau_{s}>t_{0}\\}}ds=\lim_{\varepsilon\to
0}\int_{t_{1}}^{t_{0}}D_{s}X_{t_{0}}^{\varepsilon}{\bf
1}_{\\{\tau_{s}>t_{0}\\}}ds,$ (7.5)

in the weak topology of $L^{2}(\Omega,L^{2}(0,T)).$

Set $F_{s,t}^{\varepsilon}$ the solution to the integral equation, for $s<t$

$F_{s,t}^{\varepsilon}=\sigma(X_{s-r})+\int_{s}^{t}\sigma^{\prime}(X_{u-r})D_{s}X_{u-r}dW_{u}+\int_{s}^{t}b^{\prime}(X_{u}^{\varepsilon})F_{s,u}^{\varepsilon}du.$

Consider a $s\in[t_{1},t_{0}]$ with $\tau_{s}>t_{0}$. For all $u\in[s,t_{0}]$
it holds that $X_{u}\geq\frac{a}{2}.$

By the uniform convergence of $X^{\varepsilon}$ to $X$, there exists a
$\varepsilon_{0}(\omega)$ such that for all $\varepsilon<\varepsilon_{0}$ it
holds that $X_{u}^{\varepsilon}>\frac{a}{4}$ for any $u\in[s,t_{0}]$. So, for
all $\varepsilon<\varepsilon_{0}$ it holds that
$f^{\prime}(X_{u}^{\varepsilon})=0$ for any $u\in[s,t_{0}]$. Then, for any
$\varepsilon<\varepsilon_{0}(\omega)$ and for all $u\in[s,t_{0}]$ we have by
uniqueness of the solution that

$F_{s,t_{0}}^{\varepsilon}=D_{s}X_{t_{0}}^{\varepsilon}.$ (7.6)

For $s<t$, let us introduce

$G_{s,t}=\sigma(X_{s-r})+\int_{s}^{t}\sigma^{\prime}(X_{u-r})D_{s}X_{u-r}dW_{u}+\int_{s}^{t}b^{\prime}(X_{u})G_{s,u}du.$

So,

$G_{s,t}=\left(\sigma(X_{s-r})+\int_{s}^{t}\sigma^{\prime}(X_{u-r})D_{s}X_{u-r}dW_{u}\right)\exp\left(\int_{s}^{t}b^{\prime}(X_{u})du\right),$

and it yields that $G_{s,t_{0}}>0$ for any $s\in[t_{1},t_{0}]$.

Notice that

$\displaystyle G_{s,t}-F_{s,t}^{\varepsilon}$ $\displaystyle=$
$\displaystyle\int_{s}^{t}b^{\prime}(X_{u})G_{s,u}du-\int_{s}^{t}b^{\prime}(X_{u}^{\varepsilon})F_{s,u}^{\varepsilon}du$
$\displaystyle=$
$\displaystyle\int_{s}^{t}(b^{\prime}(X_{u})-b^{\prime}(X_{u}^{\varepsilon}))G_{s,u}du+\int_{s}^{t}b^{\prime}(X_{u}^{\varepsilon})(G_{s,u}-F_{s,u}^{\varepsilon})du$

Then,

$G_{s,t_{0}}-F_{s,t_{0}}^{\varepsilon}=\left(\int_{s}^{t_{0}}(b^{\prime}(X_{u})-b^{\prime}(X_{u}^{\varepsilon}))G_{s,u}du\right)\exp\left(\int_{s}^{t_{0}}b^{\prime}(X_{u}^{\varepsilon})du\right).$

Using that $b^{\prime}$ is differentiable with bounded derivative we have that
for $s\in[t_{1},t_{0}]$

$|G_{s,t_{0}}-F_{s,t_{0}}^{\varepsilon}|\leq
K\|X-X^{\varepsilon}\|_{\infty}\left(\int_{s}^{t_{0}}|G_{s,u}|du\right).$

and so $F_{s,t_{0}}^{\varepsilon}$ converges a.s. to $G_{s,t_{0}}$.

Using a bounded convergence result we get

$\lim_{\varepsilon}\int_{t_{1}}^{t_{0}}F_{s,t_{0}}^{\varepsilon}{\bf
1}_{\\{\tau_{s}>t_{0}\\}}ds=\int_{t_{1}}^{t_{0}}G_{s,t_{0}}{\bf
1}_{\\{\tau_{s}>t_{0}\\}}ds>0,\qquad{\rm a.s.}$ (7.7)

Putting together (7.4), (7.5), (7.6) and (7.7) we obtain

$\int_{0}^{t_{0}}\left|D_{s}X_{t_{0}}\right|ds=\int_{0}^{t_{1}}\left|D_{s}X_{t_{0}}\right|ds+\int_{t_{1}}^{t_{0}}\left|D_{s}X_{t_{0}}\right|ds\geq\int_{t_{1}}^{t_{0}}\left|D_{s}X_{t_{0}}\right|ds\geq\int_{t_{1}}^{t_{0}}D_{s}X_{t_{0}}ds>0.$

$\square$

## References

  * [1] M. Besalú and C. Rovira: Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion. Bernoulli 18 (2012), 24-45.
  * [2] M. Besalú and C. Rovira: Stochastic Volterra equations driven by fractional Brownian motion with Hurst parameter $H>\frac{1}{2}$. Stochastics and Dynamics 12 (2012).
  * [3] B. Bouleau and F. Hirsch: Propriété d’absolute continuiée dans les espaces de Dirichlet el applications aux équations diférentielles stochastiques. Séminaire de Probabilités XX, Lectures Notes in Math 1204 (1986).
  * [4] A. Deya, M. Gubinelli, M. Hofmanová and S.Tindel: One-dimensional reflected rough differential equations. Stochastic Processes and their Applications  129 (2019), 3261-3281,
  * [5] C. Donati-Martin and E. Pardoux: White noise driven by SPDEs with reflection. Probab. Theory Related Fields 95 (1993), 1-24.
  * [6] C. Donati-Martin and E. Pardoux: EDPS Réfléchies et Calcul de Maliiavin. Bull. Sci. Math. 121 (1997), 405-422.
  * [7] M. Ferrante and C. Rovira: Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$. Bernoulli 12 (2006), 85-100.
  * [8] M. Ferrante and C. Rovira: Convergence of delay differential equations driven by fractional Brownian motion. Journal of Evolution Equations 10 (2010), 761–783.
  * [9] M. Ferrante and C. Rovira: Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion. J. Evol. Equ. 13 (2013), 617–632.
  * [10] A. Falkowski, L.Slomińsk: SDEs with constraints driven by semimartingales and processes with bounded p-variation. Stochastic Processes and their Applications 127 (2017), 3536-3557,
  * [11] I. Gyöngy: Existence and uniqueness results for semilinear stochastic partial differential equations. Stochastic Process. Appl. 73 (1998), no. 2, 271-299.
  * [12] H.J. Kusher: Numerical methods for controlled stochastic delay systems. Birkhäuser, Boston (2008).
  * [13] M.S. Kinnally, J. Williams : On Existence and Uniqueness of Stationary Distributions for Stochastic Delay Differential Equations with Non-Negativity Constraints. Electronic Journal of Probability 15 (2010), 409-451.
  * [14] D. Lépingle and C. Marois: Équations différentielles stochastiques multivoques unidimensionnelles. Séminaire de Probabilités, XXI, 520–533, Lecture Notes in Math., 1247, Springer, Berlin, 1987.
  * [15] D. Lépingle, D. Nualart and M. Sanz: Dérivation stochastique de diffusions réfléchies.  Ann. Inst. H. Poincaré Probab. Statist.25 (1989), 283–305.
  * [16] J. León and S. Tindel: Malliavin calculus for fractional delay equations. J. Theoret. Probab. 25 (2012), 854–889.
  * [17] S.-E. A. Mohammed: Stochastic differential systems with memory: theory, examples and applications. In Stochastic Analysis and Related Topics VI (L. Decreusefond, J. Gjerde, B. Oksendal and A.S. Üstünel, eds), Birkhäuser, Boston, 1-77 (1998).
  * [18] A. Neuenkirch, I. Nourdin, S. Tindel: Delay equations driven by rough paths. Electron. J. Probab. 13  (2008), 2031–2068.
  * [19] D. Nualart and A. Răşcanu: Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81.
  * [20] D. Nualart and E. Pardoux: White noise driven quasilinear SPDEs with reflection. Probab. Theory Related Fields 93 (1992) 77–89.
  * [21] D. Nualart and B. Saussereau: Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stochastic Process. Appl. 119 (2009) 391–409.
  * [22] D. Nualart: _The Malliavin Calculus and Related Topics._ Probability and its Applications. Springer-Verlag, 2nd
  * [23] S. Tindel, I. Torrecilla: Some differential systems driven by a fBm with Hurst parameter greater than 1/4. Stochastic analysis and related topics, 169–202, Springer Proc. Math. Stat., 22, Springer, Heidelberg, 2012.
  * [24] L.C. Young: An inequality of the Hölder type connected with Stieltjes integration. Acta Math. 67 (1936) 251–282.
  * [25] M. Zähle: Integration with respect to fractal functions and stochastic calculus I. Prob. Theory Relat. Fields 111 (1998) 333–374.


./images/

Real-time, low latency virtual DBA hypervisor for SLA-compliant multi-service
operations over shared Passive Optical Networks

Arijeet Ganguli, Frank Slyne and Marco Ruffini

CONNECT Center, School of Computer Science and Statistics, Trinity College
Dublin, Ireland

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a heuristic algorithm for a PON upstream scheduling hypervisor,
supporting low latency services with strict service-level agreement. The
algorithm achieves near-optimal performance while running in only 3.5 $\mu s$,
thus operating in real-time.

##  1 Introduction

Since the last decade, networks have started to migrate from closed,
monolithic architectures towards open and disaggregated systems. Software
Defined Networking (SDN) and Network Function Virtualization (NFV) have played
a fundamental role in providing open interfaces for programmability and
integration, adding flexibility in service development and provisioning, and
bringing down costs by commoditising much of the networking and computing
equipment. Passive Optical Networks (PONs) have experienced a similar
transition, with early implementations, like the SDN Enabled Broadband
Architecture (SEBA) [1], enabling virtualisation of control and management
planes. However, as PONs are being considered to offer services that go beyond
residential broadband, especially supporting 5G and future 6G base stations
[2], and Multi-access Edge Computing (MEC) [3], deep virtualisation, which
enables control of scheduling algorithms, becomes important to meet these new
performance targets. A prominent example is the Flexible Access System
Architecture (FASA) [4], which enables virtualisation and dynamic update of
scheduling algorithms, which can be matched to the performance required by
specific services. In [5] we proposed a virtual Dynamic Bandwidth Allocation
(vDBA) architecture for deep PON virtualisation, where independent Virtual
Network Operators (VNOs) are able to run multiple different upstream
scheduling algorithms (DBAs) in a shared PON. This enables both a multi-tenant
and multi-service approach to passive optical networks.

The approach consists on multiple VNOs running different schedulers in
parallel, each of them proposing a virtual Bandwidth Map (vBMap) (which
allocates upstream transmission slots to a group of Optical Network Units -
ONUs). The key element is the scheduling hypervisor (which we also call
merging engine), which collects all such virtual bandwidth maps to create a
single physical bandwidth map that is transmitted to all ONUs (thus providing
a solution that is compatible with typical PON standards).

The performance of the hypervisor determines the quality of service allocated
to each VNO. Since the virtual BMaps from the different VNOs are independent,
their proposed allocation will in general generate collisions (i.e., two
independent BMaps propose scheduling of upstream resource over the same time
sots). These needs to be resolved by the hypervisor, which has thus the
ability to decide which allocations have to be rejected or delayed, when
constructing the final bandwidth Map. An ideal hypervisor design is one that
can make decisions based on specific Service Level Agreements (SLAs), so that
it minimises the probability of breaching SLAs (which is key for supporting 5G
and future 6G services). The use of stateful algorithms, which take into
consideration the history of a service flow when making scheduling
prioritisation decisions, is preferred to stateless algorithms [6]. This is
because a stateful algorithm can prioritise flows depending on how close they
are to breaching their specific SLA target. It should be noticed that this
optimisation approach is not possible through classical queue management
systems, where priority is allocated through stateless decisions, packet by
packet. The drawback of stateful algorithms is that they require computation
times that can be incompatible with the short duration of PON frames, thus it
is difficult to use them in practice on real systems.

In this work we develop a heuristic stateful algorithm for the hypervisor that
provides close to optimal performance, while running in real time. The
algorithm is compared to an optimal MILP model to provide an upper bound on
performance. We also provide comparison with a stateless algorithm based on
simple packet priority rules as baseline reference for execution time. Our
results show that the proposed heuristic can provide near optimal performance,
with a run time of only 3.5$\mu s$, which is comparable to the run time of the
stateless algorithm.

##  2 SLA Stateful DBA Hypervisor

The aim of the stateful hypervisor is to minimise the probability of SLA
breach for each individual service (or flow). An SLA breach occurs when a
given flow accumulates a number of delayed upstream slots that is above its
target SLA threshold. For example for an SLA with maximum upstream packet
delay of 25 $\mu s$ with 99% compliance, every time an upstream slot is
delayed by more than 25 $\mu s$ with respect to the requested time slot in the
virtual BMap, we increment a counter. If the counter goes above the non-
compliance rate (in this case 100-99=1%), calculated over a frame (i.e., of
125 $\mu s$ duration), then we consider that an SLA breach has occurred. The
key idea of our real-time heuristic algorithm, reported in Fig. 2, is thus to
keep track of the history of the counter (through a flow-breach table shown in
Fig 2) for each flow, and prioritise the flows based on the number of previous
breaches. This allows to deal effectively with multiple flows even when they
have different SLA targets (i.e., target latency and compliance rate). The key
data structure is a flow-breach likelihood table that keeps track of how far
each traffic flow is from breaching its SLA (i.e., going above non-compliance
rate). With reference to Fig. 2, the heuristic first calculates the allocation
maxtime which is the latest time an allocation can be scheduled within its
latency target (code lines 1 to 3). Secondly, it starts allocating slots to
the various allocations according to the time assigned by their originating
virtual BMaps (code lines 5 to 6). Thirdly, it resolves collisions (lines 8 to
21) by allocating slots first in increasing order of non-compliance rate (line
18), then increasing order of their maxtimes (line 19) and finally increasing
order of their sizes (line 20). Lines 22 to 26 initialise the SLA table shown
in Fig. 2. Finally, the heuristic recalculates the non-compliance rate of all
the flows and updates the flow-breach table for scheduling of the allocations
in the next time frame (lines 27 to 32).

Fig. 1: Stateful DBA Hypervisor

Fig. 2: Pseudocode of the heuristic stateful algorithm

Our heuristic is compared to a Mixed Integer Linear Programming (MILP)
formulation, used as an upper bound for performance (but unable to run in
real-time). The MILP model is shown in (Fig. 6): equations (1)-(4) calculate,
respectively, the maximum delay of any given allocation to remain with the
target threshold, the status of packet level breaches of the allocations after
scheduling, the fraction of allocations in a flow that breached packet-level
latency and the flow-level SLA breach status after scheduling. The objective
(5) aims to minimize the overall flow-level SLA breaches defined in (4), by
optimally allocating slots to the various virtual BMaps requests. Equations
(6)-(8) puts constraints on the maximum possible size of the final merged
BMap, slot allocation uniqueness (non-overlap) and maximum and minimum slot
values.

##  3 Experiment and Performance Evaluation

We have carried out our experiments by feeding BMaps from different VNOs to
our real-time hypervisor, running on an AMD-Ryzen7 4K Series Processor. The
input allocation load on the shared PON was considered for 20%, 50% and 90% of
the total upstream capacity. For each of this allocation loads, we then varied
the percentage allocated to SLA-driven flows from 10% to 90% of the total load
(the remaining part is allocated to best effort flows). The comparative MILP
implementation was executed on a CPLEX solver from IBM ILOG, which is a high
performance solver for Linear Programming (LP), Mixed Integer Programming
(MIP) and Quadratic Programming (QP) problems.

The parameters for our experiment are as follows: we consider 5 VNOs (each one
generating a virtual BMap every frame) and two types of SLAs (type-1: 95%
compliance for latency target of 12.5 $\mu s$, type-2: 90% compliance for
latency target of 25 $\mu s$). Each bandwidth map has a uniformly distributed
set of allocations, and each individual burst allocation can be of 1.3KB,
4.7KB and 9.5KB, in average representing, respectively, about 1%, 3% and 6% of
the total frame size (the same ONU is also allowed to provide multiple burst
per frame). An empty time slot of around 0.1 $\mu s$, is introduced between
allocation to account for guard time between upstream transmissions. We also
report the performance of a stateless algorithm based on simple prioritisation
of one of the SLAs over the other, previously considered in [6]. The
experiment was run for 1000 time frames (each of 125 $\mu s$) and the average
run-time of the algorithm was recorded, together with the number of SLA
breaches as a function of the allocation load. The plots report the ability to
meet SLAs, depending on overall allocation load, percentage of load that is
SLA oriented, and size of burst allocation (shown with different colors in the
plots).

The key part of the results are the conditions where the SLA are not breached
(i.e., at the 100% value of the y axis). This information can be used by an
operator to understand how much load from SLA-driven 5G services and other
best effort services can be injected into a shared PON to achieve full SLA
compliance.

By comparing the plots in Fig. 6 and Fig. 6, we can see that our heuristic
provides performance close to optimal, across all load scenarios, and most
importantly provides similar compliance level to both types of SLAs, although
they have different latency and compliance targets. For example, at 90%
overall load (lower plots), the heuristic can maintain the same 100%
compliance as the MILP (for up to 20% of SLA-oriented flows in the system,
shown in the x axis). For the 50% load case (middle plots), we can see a
slightly sub-optimal behaviour as our heuristic can meet SLAs up to a 40%
value in the x axis, while the MILP can reach 50%. We can also notice that in
general the performance are lower for shorter burst, as this reduces the
available capacity due to an increasing number of guard intervals (this
reduction of performance for short bursts is typical in PON upstream
allocations). In addition, we can see that our heuristic performs much better
than the baseline stateless algorithm reported in Fig. 6, which, being based
on simple packet-by-packet prioritisation, is only able to satisfy flows with
one type of SLA (type-1), while it is totally non compliant for flows with SLA
type-2.

We have also carried out run time profiling of the algorithms using the
C-profiler gprof. This showed that every call to the hypervisor (i.e., for
each physical BMap) is completed in average in 3.52 $\mu s$, which is only a
small percentage (less than 3%) of the 125 $\mu s$ frame duration. It is also
close to the run time of the stateless algorithm, running in 2.72 $\mu s$.

Fig. 3: MILP Notations and Equations

Fig. 4: Stateless algorithm

Fig. 5: Stateful heuristic algorithm

Fig. 6: MILP implementation

##  4 Conclusions

In this work we presented a heuristic algorithm for a PON hypervisor, capable
of satisfying multiple different SLAs, when scheduling upstream capacity in a
multi-tenant PON infrastructure, thus supporting convergence of mobile and
residential services. We have also shown the maximum percentage of SLA-
oriented traffic that can be supported, for two sample SLA types, to satisfy
requirements for low latency services with no SLA breach. This for example
could support the Cooperative Transport Interface (CTI) over a shared PON, to
transport fronthaul eCPRI signals for 5G base stations (and beyond). A key
result is that the performance of our proposed heuristic is close to optimal
and it is capable of running in real-time, with a run time less than 3% of a
frame duration.

##  5 Acknowledgments

Financial support from Science Foundation Ireland grants 12/RC/2276_p2,
14/IA/2527 and 13/RC/2077_p2 is acknowledged.

## References

  * [1] Optical Networking Forum (ONF) SEBA reference architecture, https://opennetworking.org/seba/
  * [2] M. Wang et al. SDN-oriented Disaggregated Optical Access Node for Converged 5G Mobile and Residential Services. ECOC’21.
  * [3] G. Simon et al. MEC and Fixed Access Networks Synergies. OFC’21
  * [4] K. Asaka et al. Flexible access system architecture (FASA). OFC’18, paper Tu3L.7
  * [5] M. Ruffini et al. The Virtual DBA: Virtualizing Passive Optical Networks to Enable Multi-Service Operation in True Multi-Tenant Environments. JOCN, No.4, Vol.12, April 2020.
  * [6] F. Slyne et al. Stateful DBA Hypervisor Supporting SLAs with Low Latency High Availability in Shared PON. OFC’21.