# Row-finite systems of stochastic differential equations with dissipative
drift

Georgy Chargaziya
Mathematics Department, The University of York

Abstract

Motivated by studies of stochastic systems describing non-equilibrium dynamics
of (real-valued) spins of an infinite particle system in $\mathbb{R}^{n}$ we
consider a row-finite system of stochastic differential equations with
dissipative drift. The existence and uniqueness of infinite time solutions is
proved via finite volume approximation and a version of the Ovsjannikov
method.

###### Table of Contents

  1. 1 Introduction
  2. 2 Main Framework
    1. 2.1 General Notation
    2. 2.2 Scale and Ovsjannikov Map
    3. 2.3 Probability and Measure Spaces
    4. 2.4 $\mathcal{Z}$ spaces
    5. 2.5 Stochastic System
  3. 3 Auxiliary Results
  4. 4 Truncated Systems
  5. 5 One Dimesional Special Case
  6. 6 Existence and Uniqueness
  7. 7 Complementary Theory
    1. 7.1 Expectation, Measurability and Related Inequalities
    2. 7.2 Martingales and Wiener Process in $\mathbb{R}$
    3. 7.3 Deterministic Ovsjannikov Equation
  8. References

Acknowledgment
Much of this work is based on joint research with Alexei Daletskii. His advise
and guidance is gratefully acknowledged. I am grateful to Zdzislaw Brzezniak
for helpful and stimulating discussions.

##  1 Introduction

The study of properties of various physical phenomena has led to consideration
of systems of infinitely many coupled finite dimensional stochastic
differential equations. Such systems are known as lattice models with certain
conditions on the so-called “spin variables”, which are being modelled by the
SDEs. Term “stochastic dynamics” is also often used to describe such systems
in general and in particular SDEs that model the time dependence of spin
variables. Origins of this terminology can be found in [21] and additional
mathematical framework can be found, for example, in [23] and [22]. Questions
concerning existence and uniqueness of solutions of such systems have also
been studied in [25] and [24].
In recent decades studies of physical phenomena pertaining to non-crystalline
(amorphous) substances and ferrofluids and amorphous magnets has led to an
increased interest in studying countable systems of particles randomly
distributed in $\mathbb{R}^{d}$. Characterisation of each particle in such a
system by an internal real or vector valued “spin” parameter naturally leads
to the consideration of a lattice model based on a fixed configuration
$\gamma\subset\mathbb{R}^{d}$ of particle positions. Instances when
$\gamma\equiv\mathbb{Z}^{d}$ are well studied and have an extensive
literature, see for example [26, 28] and [27]. However, as described in [2]
there are instances when the configuration $\gamma$ of particle positions
doesn’t have a regular structure but instead lends itself as a locally finite
subset of $\mathbb{R}^{d}$ where the typical number of “neighbour variables”
of a particle located at $x\in\gamma$ is proportional to $\log|x|$ for large
$|x|$.
In [29] we saw an extension of work by [1]. This extension showed, under a
suitable choice of coefficients, how to construct a unique strong solution of
a stochastic differential equation, driven by a cylinder Wiener process, in a
separable Hilbert space

$d\xi(t)=F(\xi(t))dt+\Phi(\xi(t))dW(t),\ t\geq 0,$ (1.1)

using the method of Ovsjannikov. The end result was a strong solution that
takes values in an intersection of a suitably chosen scale of Hilbert spaces.
This general theory was subsequently used to extend the work of [2] [in a
sense of considering a stochastic version] by considering a lattice system on
a locally finite subset $\gamma\subset\mathbb{R}^{d}$ such that the spins
$q_{x}$ and $q_{y}$ are allowed to interact via a pair potential if the
distance between $x,y\in\gamma$ is no more than a fixed and positive
interaction radius $r$, that is, they are neighbours in the geometric graph
defined by $\gamma$ and $r$. Precisely speaking we considered a system

$\displaystyle d\xi_{x}(t)=\phi_{x}(\Xi(t))dt+\Psi_{x}(\Xi(t))dW_{x}(t),\
x\in\gamma,\ t\in[0,T],$ (1.2)

where $\phi_{x}$ and $\Psi_{x}$ were required to satisfy the so-called “finite
range ” and “uniform Lipschitz continuity” conditions and showed that system
(1.2) can be realised in a suitable scale of separable Hilbert spaces and
hence studied using the method of Ovsjannikov.
In this paper, we would like to further build upon results of [1, 2] and [17,
16] and consider a lattice system of the form

$\displaystyle
d\xi_{x}(t)=\Phi_{x}(\xi_{x}(t),\Xi(t))dt+\Psi_{x}(\xi_{x}(t),\Xi(t))dW_{x}(t),\
x\in\gamma,\ t\in[0,T],$ (1.3)

where $\Phi_{x}(a,b)\equiv V(a)+\phi_{x}(b)$, where $V$ is a real valued one
particle potential satisfying the dissipativity condition, and $\Psi_{x}$ is
Lipschitz. In our approach we will assume, as in [2], that configuration of
particles $\gamma\subset\mathbb{R}^{d}$ is a locally finite subset of
$\mathbb{R}^{d}$ distributed according to a Poisson or, more generally, Gibbs
measure with a superstable low regular interaction energy, so that for all
$x\in\gamma$ then number of particle in a certain compact vicinity of $x$ is
proportional to $\log|x|$ for large $|x|$.
Unfortunately, system (1.3) doesn’t lend itself for an immediate and
straightforward application of an Ovsjannikov method. Hence in this part we
opt for an approach that was used in [17] and consider a so-called sequence of
“finite volume approximations” of the system (1.3). Precisely speaking a
sequence of finite volume approximations is a sequence of solutions of
truncated systems of the following form

$\displaystyle\xi_{x,t}^{n}=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})ds+\int_{0}^{t}\Psi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})dW_{x}(s),$
$\displaystyle\quad\forall x\in\Lambda_{n}\land t\in[0,T],$ (1.4)
$\displaystyle\xi_{x,t}^{n}=\zeta_{x},$ $\displaystyle\quad\forall
x\not\in\Lambda_{n}\land t\in[0,T].$

where $\gamma\supset\Lambda_{n}\uparrow\gamma$ are finite. Using a comparison
Theorem 3.2, which builds upon the method of Ovsjannikov, we ultimately show
that the sequence of finite volume approximations converges to a unique strong
solution of the system (1.3) in a certain scale of Banach spaces.

##  2 Main Framework

###  2.1 General Notation

In our framework all vector spaces will be over $\mathbb{R}$ and the cardinal
number of any given set $A$ will always be denoted by $\\#A$. We now start
this subsection by introducing the following sets

$\displaystyle\mathbb{R}^{0}\coloneqq(0,\infty),\quad\mathbb{R}^{+}\coloneqq[0,\infty),\quad\mathbb{R}^{++}\coloneqq[1,\infty),\quad\mathbb{N}^{0}\coloneqq\mathbb{N}\cup\\{0\\},$
(2.1)

We also introduce constants $T,\ \rho,\ \underline{\mathfrak{a}},\
\overline{\mathfrak{a}}\in\mathbb{R}^{0}$, $d\in\mathbb{R}^{+}$ and a special
notation for the following closed intervals;

$\displaystyle\mathcal{A}\coloneqq[\underline{\mathfrak{a}},\overline{\mathfrak{a}}],$
(2.2) $\displaystyle\mathcal{T}\coloneqq[0,T].$ (2.3)

We let $\gamma$ be a locally finite subset of $\mathbb{R}^{d}$ and $|\cdot|,\
|\cdot|_{S}$ be respectively the Euclidean and supremum norm in
$\mathbb{R}^{d}$. Moreover we agree to use the following abbreviations;

$\displaystyle B(x,\rho)$ $\displaystyle\coloneqq\\{y\in\mathbb{R}^{d}\ |\
|x-y$ $\displaystyle|\ <\rho\\},$ (2.4) $\displaystyle\overline{B(x,\rho)}$
$\displaystyle\coloneqq\\{y\in\mathbb{R}^{d}\ |\ |x-y$ $\displaystyle|\
\leq\rho\\},$ $\displaystyle B_{x}$
$\displaystyle\coloneq\gamma\cap\overline{B(x,\rho)},$
$\displaystyle\\!\forall x\in\gamma,$ $\displaystyle n_{x}$
$\displaystyle\coloneq\\#B_{x},$ $\displaystyle\\!\forall x\in\gamma.$

######  Remark.

The fact that $\gamma$ is a locally finite subset of $\mathbb{R}^{d}$ means
that $\gamma\cap X$ is finite if $X\subset\mathbb{R}^{d}$ is compact and also
implies that $\gamma$ is a countable subset of $\mathbb{R}^{d}$.

Next, we fix in place a real valued function
$a:\mathbb{R}^{d}\to\mathbb{R}^{+}$ and make the following assumptions;

  1. (A)

$a(x)\leq\bar{a}\quad\quad\quad\quad$ for some constant
$\bar{a}\in\mathbb{R}^{+}$,

  2. (B)

$n_{x}\leq\mathcal{N}\log(1+|x|)$ for some constant
$\mathcal{N}\in\mathbb{R}^{+}$ and all $x\in\gamma$.

Given two vector spaes $A$ and $B$ let us now also introduce the following
notation

$\displaystyle A\prec B\iff\text{A is a subspace of B,}$ (2.5)

and agree that $A^{B}$ will be understood as the infinite cartesian product,
that is

$\displaystyle A^{B}=\bigtimes_{b\in B}A=\bigg{\\{}\\{z_{b}\\}_{b\in B}\
\bigg{|}\ z_{b}\in A\ \text{for all}\ b\in B\bigg{\\}}.$ (2.6)

Suppose now that
$\mathbf{X}\coloneqq\\{X_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}}$ is a
family of sets. We define for convenience and later use the following
notation;

$\displaystyle\mathbf{X}(\cup)\coloneqq\bigcup_{\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})}X_{\mathfrak{a}},$
(2.7)
$\displaystyle\mathbf{X}(\cap)\coloneqq\bigcap_{\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})}X_{\mathfrak{a}},$
(2.8)

###  2.2 Scale and Ovsjannikov Map

We now proceed to introduce several important definitions.

######  Definition 2.1.

A family
$\mathbf{X}\coloneqq\\{X_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}}$ of
Banach spaces is called a scale if for all $\alpha<\beta\in\mathcal{A}$ and
all $x\in X_{\alpha}$

  1. (1)

$X_{\alpha}\prec X_{\beta}$,

  2. (2)

$||x||_{X_{\beta}}\ \leq\ ||x||_{X_{\alpha}}$.

######  Definition 2.2.

Let $\mathbf{X^{1}}$ be a scale and
$\mathbf{X^{2}}\coloneqq\\{X^{2}_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}}$
be a family of Banach spaces. Then

$G:\mathbf{X^{1}}(\cup)\to X_{\overline{\mathfrak{a}}}^{2}$

is called an Ovsjannikov map of order $q$ and $L$ from $\mathbf{X^{1}}$ to
$\mathbf{X^{2}}$ if there exist $q,L\in\mathbb{R}^{0}$ such that for all
$\alpha<\beta\in\mathcal{A}$ and all $x,y\in X_{\alpha}$

  1. (1)

$G|_{X_{\alpha}^{1}}:X_{\alpha}^{1}\to X_{\beta}^{2}$,

  2. (2)

$||G(x)-G(y)||_{X_{\beta}^{2}}\leq\frac{L}{(\beta-\alpha)^{q}}||x-y||_{X_{\alpha}^{1}}$.

######  Definition 2.3.

Suppose $\mathbf{X^{1}}$ is a scale and
$\mathbf{X^{2}}\coloneqq\\{X^{2}_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}}$
is a family of Banach spaces.

$\displaystyle\mathcal{O}(\mathbf{X^{1}},\mathbf{X^{2}},L,q)$
$\displaystyle\coloneqq\\{\text{space of Ovsjannikov maps of order $q$ and $L$
from $\mathbf{X^{1}}$ to $\mathbf{X^{2}}$}\\},$ (2.9)
$\displaystyle\mathcal{O}(\mathbf{X^{1}},L,q)$
$\displaystyle\coloneqq\\{\text{space of Ovsjannikov maps of order $q$ and $L$
from $\mathbf{X^{1}}$ to $\mathbf{X^{1}}$}\\}.$ (2.10)

######  Definition 2.4.

For all $p\in\mathbb{R}^{++}$ and all $\mathfrak{a}\in\mathcal{A}$ let

$\displaystyle l^{p}_{\mathfrak{a}}$
$\displaystyle\coloneqq\bigg{\\{}z\in\mathbb{R}^{\gamma}\ \bigg{|}\
\|z\|_{l^{p}_{\mathfrak{a}}}\coloneqq\bigg{(}\sum_{x\in\gamma}e^{-\mathfrak{a}|x|}|z_{x}|^{p}\bigg{)}^{\frac{1}{p}}<\infty\bigg{\\}},$
(2.11) $\displaystyle\mathscr{L}^{p}$
$\displaystyle\coloneqq\\{l^{p}_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}},$
(2.12)

be, respectively, a normed linear space of weighted real sequences and a
family of such spaces.

######  Theorem 2.1.

Let $n\coloneqq\\{n_{x}\\}_{x\in\gamma}$. Then $n\in
l_{\underline{\mathfrak{a}}}^{1}$.

###### Proof.

Observe that assumption (B) implies that

$\displaystyle\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|n_{x}|\ \leq\
\mathcal{N}\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}\log(1+|x|)\ \leq\
\mathcal{N}\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|x|.$ (2.13)

Hence to conclude that proof we show that

$\displaystyle\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|x|\ <\infty.$
(2.14)

We now make a couple of preliminary observations and additional definitions.
Fix a suitable $k\in\mathbb{N}$ such that $\sqrt{d}\frac{1}{2^{k}}<\rho$ and
consider the following $k^{\text{th}}$ grid-partition or $\mathbb{R}^{d}$

$\displaystyle\mathcal{R}^{k}$
$\displaystyle\coloneqq\\{\mathcal{R}^{k}_{z}\\}_{z\in\mathbb{Z}^{d}},$ (2.15)
$\displaystyle\mathcal{R}^{k}_{z}$
$\displaystyle\coloneqq\bigg{\\{}x\in\mathbb{R}^{d}\ \bigg{|}\
\frac{z_{i}-1}{2^{k}}\leq x_{i}\leq\frac{z_{i}}{2^{k}}\bigg{\\}}.$ (2.16)

We shall call members of the family $\mathcal{R}^{k}$,
$k^{\text{th}}-$rectangles. Observe that for all $z\in\mathbb{Z}^{d}$

$\displaystyle Diam(\mathcal{R}^{k}_{z})\coloneqq\sup\\{|x-y|_{S}\ |\
x,y\in\mathcal{R}^{k}_{z}\\}=\frac{1}{2^{k}}.$ (2.17)

Now introduce the following sets

$\displaystyle I_{n}$ $\displaystyle\coloneqq\bigg{\\{}x\in\mathbb{R}^{d}\
\bigg{|}\ |x|_{S}\ \leq\frac{1}{2}n\bigg{\\}},\ $ $\displaystyle\forall
n\in\mathbb{N}^{0},$ (2.18) $\displaystyle J_{n}$ $\displaystyle\coloneqq
I_{n}-I_{n-1},$ $\displaystyle\forall n\in\mathbb{N}.$

Consider now the real function
$e^{-\underline{\mathfrak{a}}x}x:[0,\infty)\to\mathbb{R}$. We see that
$\frac{d}{dx}e^{-\underline{\mathfrak{a}}x}x=e^{-\underline{\mathfrak{a}}x}(1-\underline{\mathfrak{a}}x)$
and so it follows that $\frac{d}{dx}e^{-\underline{\mathfrak{a}}x}x<0$ if
$x>\frac{1}{\underline{\mathfrak{a}}}$. Therfore letting $m\in\mathbb{N}$ be
the smallest natural number such that
$\max\\{\frac{1}{\underline{\mathfrak{a}}},2\\}\leq m$ we see that
$e^{-\underline{\mathfrak{a}}x}x:(m,\infty)\to\mathbb{R}$ is a decreasing
function. Finally observe that the following statements are true

  1. (1)

$I_{1}$ contains exactly $2^{k+1}$ of $k^{\text{th}}-$rectangles.

  2. (2)

$J_{n}$ contains fewer then $n2^{k+2}$ of $k^{\text{th}}-$rectangles.

  3. (3)

For all $n\in\mathbb{N}$, if $x\in\gamma\cap J_{n}$ then $|x|\ \geq n-1$.

  4. (4)

Suppose that $n\in\mathbb{N}$ and $z\in\mathbb{Z}^{d}$. Consider
$x,y\in\gamma\cap\mathcal{R}^{k}_{z}\subset J_{n}$. It follows that

$\displaystyle|x-y|$ $\displaystyle\leq\sqrt{d}|x-y|_{S},$ (2.19)
$\displaystyle\leq\sqrt{d}Diam(\mathcal{R}^{k}_{z}),$ (2.20)
$\displaystyle\leq\sqrt{d}\frac{1}{2^{k}},$ (2.21) $\displaystyle\leq\rho.$
(2.22)

Hence we see that $y\in B_{x}$ and so from asumption B we see that

$\displaystyle\\#\gamma\cap\mathcal{R}^{k}_{z}$ $\displaystyle\leq n_{x},$
(2.23) $\displaystyle\leq\mathcal{N}\log(1+|x|),$ (2.24)
$\displaystyle\leq\mathcal{N}|x|,$ (2.25) $\displaystyle\leq\mathcal{N}n.$
(2.26)

Therefore we conclude that for all $n\in\mathbb{N}$, $\\#\gamma\cap
J_{n}\leq\mathcal{N}n^{2}2^{k+2}$.

Returning now to the inequality (2.14) we see that because $J_{m}$ is compact
and $\gamma$ is locally finite we can let

$\displaystyle B\coloneqq\sum_{x\in\gamma\cap
J_{m}}e^{-\underline{\mathfrak{a}}|x|}|x|,$ (2.27)

and observe that

$\displaystyle\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|x|\ $
$\displaystyle\leq B+\sum_{n\in\mathbb{N}\atop n>m}\sum_{x\in\gamma\cap
J_{n}}e^{-\underline{\mathfrak{a}}|x|}|x|,$ (2.28) $\displaystyle\leq
B+\mathcal{N}2^{k+2}\sum_{n\in\mathbb{N}\atop
n>m}e^{-\underline{\mathfrak{a}}(n-1)}(n-1)n^{2}.$ (2.29)

Hence letting $\mathcal{K}\coloneqq\frac{m-1}{m}$ we see that

$\displaystyle\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|x|\ \leq
B+\mathcal{N}2^{k+2}\sum_{n\in\mathbb{N}\atop
n>m}e^{-\mathcal{K}\underline{\mathfrak{a}}n}n^{3}.$ (2.30)

Now, one can show via a simple calculation involving the integral test (for
details see [19]) that the right hand side of the inequality (2.30) above is
finite hence the proof is complete. ∎

######  Remark.

From Theorem 2.1 above it is clear that
$\displaystyle\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}<\infty.$
(2.31) Moreover if $\|\cdot\|$ is any norm in $\mathbb{R}^{d}$ equivalent to
the Euclidean norm then, given a suitable modification of the asumption B,
Theorem 2.1 also implies that
$\displaystyle\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}\|x\|}\|x\|\
<\infty.$ (2.32)

######  Theorem 2.2.

Suppose that $p\in\mathbb{R}^{++}$. Then $\mathscr{L}^{p}$ is the scale.

###### Proof.

It is clear from the Definition 2.4 that $\mathscr{L}^{p}$ is a family of
normed linear spaces. Moreover conditions (1) and (2) of the Definition 2.1
follow immediately from the simple fact that if $\alpha<\beta\in\mathcal{A}$
then $e^{-\alpha}>e^{-\beta}$. Hence to conclude the proof we fix
$\mathfrak{a}\in\mathcal{A}$ and show that $l^{p}_{\mathfrak{a}}$ is a Banach
space.

Let us begin by assuming that $\\{z^{n}\\}_{n\in\mathbb{N}}$ is a Cauchy
sequence in $l^{p}_{\mathfrak{a}}$. Now fix an arbitrary $\epsilon>0$ and a
suitable constant $N_{\epsilon}\in\mathbb{N}$ such that for all
$n,m>N_{\epsilon}$ we have

$\displaystyle\bigg{(}\sum_{x\in\gamma}e^{-\mathfrak{a}|x|}|z^{n}_{x}-z^{m}_{x}|^{p}\bigg{)}^{\frac{1}{p}}<\epsilon.$
(2.33)

Because $\epsilon$ is arbitrary we see from inequality (2.33) above that for
all $x\in\gamma$ sequence $\\{z^{n}_{x}\\}_{n\in\mathbb{N}}$ is Cauchy in
$\mathbb{R}$. Hence, it follows that we can define a new sequence
$\mathbf{z}\coloneqq\\{\mathbf{z}_{x}\\}_{x\in\gamma}$ in
$\mathbb{R}^{\gamma}$ as follows

$\displaystyle\mathbf{z}_{x}\coloneqq\lim_{n\to\infty}z^{n}_{x},\ \forall
x\in\gamma.$ (2.34)

Now we complete the proof by showing that $\mathbf{z}\in l^{p}_{\mathfrak{a}}$
and $\overbrace{\lim_{n\to\infty}z^{n}}^{\text{in}\
l^{p}_{\mathfrak{a}}}=\mathbf{z}$. To begin, we fix an arbitrary finite subset
$A$ of $\gamma$. Now for all $n,m>N_{\epsilon}$ we see from inequality (2.33)
that

$\displaystyle\sum_{x\in
A}e^{-\mathfrak{a}|x|}|z^{n}_{x}-z^{m}_{x}|^{p}<\epsilon^{p}.$ (2.35)

Hence we see that for all $n>N_{\epsilon}$

$\displaystyle\lim_{m\to\infty}\sum_{x\in
A}e^{-\mathfrak{a}|x|}|z^{n}_{x}-z^{m}_{x}|^{p}$
$\displaystyle=\mathbin{{\color[rgb]{1,1,1}\sum_{x\in
A}e^{-\mathfrak{a}|x|}\lim_{m\to\infty}|z^{n}_{x}-z^{m}_{x}|^{p}}}$ (2.36)
$\displaystyle=\sum_{x\in
A}e^{-\mathfrak{a}|x|}|z^{n}_{x}-\lim_{m\to\infty}z^{m}_{x}|^{p}=$
$\displaystyle\\!\\!\\!\\!\\!\\!=\sum_{x\in
A}e^{-\mathfrak{a}|x|}|z^{n}_{x}-\mathbf{z}_{x}|^{p}\ \leq\ \epsilon^{p}.$

Since $A\subset\gamma$ is arbitrary we see from inequality (2.36) above that
for all $n>N_{\epsilon}$

$\displaystyle\sum_{x\in\gamma}e^{-\mathfrak{a}|x|}|z^{n}_{x}-\mathbf{z}_{x}|^{p}\leq\epsilon^{p}.$
(2.37)

Because $\epsilon$ is also arbitrary we conclude that
$\overbrace{\lim_{n\to\infty}z^{n}}^{\text{in}\
l^{p}_{\mathfrak{a}}}=\mathbf{z}$. Moreover we see that if $n>N_{\epsilon}$
then $z^{n}-\mathbf{z}\in l^{p}_{\mathfrak{a}}$. Since $l^{p}_{\mathfrak{a}}$
is a vector space we conclude that $\mathbf{z}\in l^{p}_{\mathfrak{a}}$ hence
the proof is complete. ∎

###  2.3 Probability and Measure Spaces

We shall now proceed to describe the probability space and also a couple of
important spaces of measurable maps and stochastic processes, that will become
important in the main body of this text.
Hence let us assume the following.

  1. (1)

Let us agree in the first palce that all probability and measure spaces in our
discussion are completed.

  2. (2)

Now we fix a filtered probability space

$\displaystyle\mathbf{P}\coloneqq(\Omega,\mathcal{F},\mathbb{P},\mathbb{F}),$
(2.38)

on which of our subsequent work will be based. Moreover:

    1. (a)

For all $t\in\mathcal{T}$ we let
$\mathbf{P}_{t}\coloneqq(\Omega,\mathcal{F}_{t},\mathbb{P})$.

    2. (b)

Filtration $\mathbb{F}\coloneqq\\{\mathcal{F}_{t}\\}_{t\in\mathcal{T}}$ is
assumed to be right continuous. That is for all $t\in\mathcal{T}$,

$\displaystyle\mathcal{F}_{t}=\bigcap_{n\in\mathbb{N}}\mathcal{F}_{t+\frac{1}{n}}.$
(2.39)

  3. (3)

We fix a measure space
$\mathbf{M}\coloneqq(\mathcal{T},\mathscr{B}(\mathcal{T}),\mu)$, where $\mu$
is a Lebesgue measure and $\mathscr{B}(\mathcal{T})$ is a Borel
$\sigma-$algebra.

  4. (4)

We now agree to work on a fixed product measure space

$\displaystyle\mathbf{M}\mathbf{P}\coloneqq(\overline{\Omega}\coloneqq\mathcal{T}\times\Omega,\overline{\mathcal{F}}\coloneqq\mathscr{B}(\mathcal{T})\times\mathcal{F},\overline{\mathbb{P}}\coloneqq\mu\times\mathbb{P}).$
(2.40)

  5. (5)

Given two measurable spaces $\mathbf{A}$ and $\mathbf{B}$ we denote by
$\mathcal{M}(\mathbf{A},\mathbf{B})$ the space of all measurable maps from
$\mathbf{A}$ to $\mathbf{B}$. In particular, the following measurable spaces
will be frequently mentioned

    1. (a)

$\mathbf{M}^{p}_{\mathfrak{a}}\coloneqq(l_{\mathfrak{a}}^{p},\mathscr{B}(l_{\mathfrak{a}}^{p}))$,

    2. (b)

$\mathbf{M}^{\mathbb{R}}\coloneqq(\mathbb{R},\mathscr{B}(\mathbb{R}))$,

Following definition fixes how we understand and denote stochastic processes
in this text.

######  Definition 2.5.

Let $Y$ be a normed linear space and $\mathbf{Y}\coloneqq(Y,\mathcal{B})$ be a
measurable space. Stochastic process is an element of
$\mathcal{M}(\mathbf{M}\mathbf{P},\mathbf{Y})$. In particular for all
$t\in\mathcal{T}$ and all $\omega\in\Omega$

$\mathcal{M}(\mathbf{P},\mathbf{Y})\ni\xi_{t}(\cdot):\Omega\to Y,$
$\mathcal{M}(\mathbf{M},\mathbf{Y})\ni\xi_{\cdot}(\omega):\mathcal{T}\to Y.$

For brevity we shall denote by $\mathcal{S}(\mathbf{Y})$ the set of all
stochastic processes from $\mathbf{MP}$ to $\mathbf{Y}$.

Following Bnach spaces will be frequently used.

######  Definition 2.6.

Let $\mathscr{X}\coloneqq(X,\mathcal{A},\eta)$ be a measure space, $Y$ be a
normed linear space, with norm denoted by $\|\cdot\|_{Y}$, and
$\mathscr{Y}\coloneqq(Y,\mathcal{B})$ be a measurable space. For all
$p\in\mathbb{R}^{++}$ we define the following Banach spaces.

$\displaystyle\mathcal{L}^{p}(\mathscr{X},\mathscr{Y})\coloneqq\left\\{f:X\to
Y\ \begin{tabular}[]{|l}\
$\|f\|_{\mathcal{L}^{p}(\mathscr{X},\mathscr{Y})}\coloneqq\left(\bigint_{\\!\\!\\!\\!X}\|f\|^{p}_{Y}d\eta\right)^{\frac{1}{p}}<\infty$.\\\
\ $f\in\mathcal{M}(\mathscr{X},\mathscr{Y})$.\end{tabular}\right\\}$ (2.43)

######  Remark.

As it is often done in the literature, we will not consider explicitly the
dependence of $\mathcal{L}^{p}(\cdot,\cdot)$ spaces on equivalence classes. We
will work directly with the Definition 2.6 and when necessary acknowledge any
issues arising from such dependence.

######  Theorem 2.3.

Suppose that $p\in\mathbb{R}^{++}$. Then
$\mathbf{L}\coloneqq\\{\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\mathfrak{a}})\\}_{\mathfrak{a}\in\mathcal{A}}$
is a scale.

###### Proof.

The fact that $\mathbf{L}$ is a family of Banach spaces is a standart result
from functional analysis, see [7] for example. Therefore it remains to verify
that conditions (1) and (2) of the Definition 2.1 hold. To this end let us
start by fixing $\alpha<\beta\in\mathcal{A}$ and
$f\in\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\alpha})$. By Definition 2.6
it follows that $f\in\mathcal{M}(\mathbf{P},\mathbf{M}^{p}_{\alpha})$. Because
$\mathscr{L}^{p}$ is the scale we conclude that
$f\in\mathcal{M}(\mathbf{P},\mathbf{M}^{p}_{\beta})$ and
$\|f\|_{l^{p}_{\beta}}^{p}\ \leq\ \|f\|_{l^{p}_{\alpha}}^{p}$. From Theorem
7.12 we see that

$\displaystyle\int_{\Omega}\|f\|_{l^{p}_{\beta}}^{p}d\mathbb{P}\ \leq\
\int_{\Omega}\|f\|_{l^{p}_{\alpha}}^{p}d\mathbb{P}.$ (2.44)

It follows that $f\in\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\beta})$ and
$\|f\|_{\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\beta})}\ \leq\
\|f\|_{\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\alpha})}$ hence the proof
is complete. ∎

######  Definition 2.7.

For all $p\in\mathbb{R}^{++}$ we introduce the following spaces of stochastic
processes.

$\displaystyle
L^{p}_{ad}\coloneqq\\{\xi\in\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})\
|\ \xi\ \text{is adapted to}\ \mathbb{F}\\}.$ (2.45)

######  Remark.

Suppose that $p\geq 2$ and $\xi\in L^{p}_{ad}$. Then $\xi\in L^{2}_{ad}$ by
Theorem 7.9 and by Fubini Theorem 7.15 we also see that
$\displaystyle\int_{0}^{T}\mathbb{E}\bigg{[}|\xi(t)|^{2}\bigg{]}dt<\infty.$
(2.46) This fact allows us to conclude that if $p\geq 2$ then every process in
$L^{p}_{ad}$ can be stochastically integrated with respect to the standard
Wiener proces. See [12] and section 7.2 for more details.

  1. (5)

Finally we fix a family of independent real valued Wiener processes
$W\coloneqq\\{W_{x}\\}_{x\in\gamma}$ on $\mathbf{M}\mathbf{P}$ and require our
filtration $\mathbb{F}\coloneqq\\{\mathcal{F}_{t}\\}_{t\in\mathcal{T}}$ to
satisfy the following standart properties

    1. (a)

For all $t\in\mathcal{T}$ and all $x\in\gamma$, $W_{x}(t)$ is
$\mathcal{F}_{t}$ measurable,

    2. (b)

For all $s\leq t\in\mathcal{T}$ and all $x\in\gamma$ $W_{x}(t)-W_{x}(s)$ is
independent of $\mathcal{F}_{s}$.

###  2.4 $\mathcal{Z}$ spaces

######  Definition 2.8.

For all $p\in\mathbb{R}^{++}$ and all $\mathfrak{a}\in\mathcal{A}$ let

$\displaystyle Z_{\mathfrak{a}}^{p}$
$\displaystyle\coloneqq\left\\{\xi\in\mathcal{S}(l_{\mathfrak{a}}^{p})\
\begin{tabular}[]{|l}\
$\|\xi\|_{Z^{p}_{\mathfrak{a}}}\coloneqq\bigg{(}\sup\bigg{\\{}\mathbb{E}\bigg{[}\|\xi\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}\
\bigg{|}\ t\in\mathcal{T}\bigg{\\}}\bigg{)}^{\frac{1}{p}}<\infty$.\\\ \
$\xi_{x}$ is adapted to $\mathbb{F}$ for all
$x\in\gamma$.\end{tabular}\right\\},$ (2.49) $\displaystyle\mathcal{Z}^{p}$
$\displaystyle\coloneqq\\{Z^{p}_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}},$
(2.50)

be, respectively, a normed linear space of $l_{\mathfrak{a}}^{p}$ valued
processes and a family of such spaces.

######  Remark.

Rigorously speaking $Z_{\mathfrak{a}}^{p}$ are a normed linear spaces only
after we partitioned them into equivalence classes. However as with
$\mathcal{L}^{p}(\cdot,\cdot)$ spaces we will not consider explicitly such
dependence at least untill the moment when we start addressing the question of
uniqueness of certain processes belonging to these spaces.

######  Theorem 2.4.

Let $p\in\mathbb{R}^{++}$, $\mathfrak{a}\in\mathcal{A}$ and suppose that
$\xi\in Z_{\mathfrak{a}}^{p}$. Then $\xi_{x}\in L^{p}_{ad}$ for all
$x\in\gamma$.

###### Proof.

From Definition 2.8 we see that to complete the proof we need to show that for
all $x\in\gamma$ we have
$\xi_{x}\in\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})$. Let us begin
by establishing that
$\xi_{x}\in\mathcal{M}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})$.
For each $x\in\gamma$ let us define maps

$\mathscr{I}^{x}:l_{\mathfrak{a}}^{p}\to
l_{\mathfrak{a}}^{p},\quad\mathscr{R}^{x}:\mathbb{R}\to
l_{\mathfrak{a}}^{p},\quad\xi|_{x}:\overline{\Omega}\to l_{\mathfrak{a}}^{p},$

in the following way.

$\displaystyle\mathscr{I}^{x}(\psi)_{y}$
$\displaystyle\coloneqq\begin{cases}\begin{tabular}[]{l|l}$\psi_{y}$&\
$y\in\gamma\land y=x$,\\\ $0$&\ $y\in\gamma\land
y\not=x$.\end{tabular}\end{cases}$ (2.51)
$\displaystyle\mathscr{R}^{x}(z)_{y}$
$\displaystyle\coloneqq\begin{cases}\begin{tabular}[]{l|l}$z$&\
$y\in\gamma\land y=x$,\\\ $0$&\ $y\in\gamma\land
y\not=x$.\end{tabular}\end{cases}$ (2.52) $\displaystyle\xi|_{x}$
$\displaystyle\coloneqq\mathscr{I}^{x}(\xi).$ (2.53)

Now observe that for each $x\in\gamma$ map $\mathscr{I}^{x}$ is continuous
which implies that
$\xi|_{x}\in\mathcal{M}(\mathbf{MP},\mathbf{M}^{p}_{\mathfrak{a}})$. Moreover
observe that each $x\in\gamma$ map $\mathscr{R}^{x}$ is continuous and
$\xi|_{x}=\mathscr{R}^{x}\circ\xi_{x}$. Consider now arbitary
$A\coloneqq[a,b]\subset\mathbb{R}$ and $x\in\gamma$. By continuity
$B\coloneqq\mathscr{R}^{x}([a,b])$ is compact and so
$B\in\mathscr{B}(l_{\mathfrak{a}}^{p})$. Since
$\xi|_{x}\in\mathcal{M}(\mathbf{MP},\mathbf{M}^{p}_{\mathfrak{a}})$ it follows
that $(\xi|_{x})^{-1}(B)\in\overline{\mathcal{F}}$. However

$(\xi|_{x})^{-1}(B)=(\xi_{x})^{-1}\circ(\mathscr{R}^{x})^{-1}(B)=(\xi_{x}^{-1})(A),$

which establishes that
$\xi_{x}\in\mathcal{M}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})$ for all
$x\in\gamma$.
Finally since for all $x\in\gamma$ we have $|\xi_{x}|\leq
e^{\frac{\mathfrak{a}}{p}|x|}\|\xi\|_{l^{p}_{\mathfrak{a}}}$ we may now
conclude using Theorem 7.12 that
$\xi_{x}\in\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})$ for all
$x\in\gamma$ and the proof is complete. ∎

######  Remark.

In simple terms, Theorem 2.4 above shows that, for all $p\in\mathbb{R}^{++}$
and $\mathfrak{a}\in\mathcal{A}$, component processes of each $\xi\in
Z_{\mathfrak{a}}^{p}$ can be stochastically integrated with respect to the
standard Wiener proces.

######  Theorem 2.5.

Let $p\in\mathbb{R}^{++}$ and $\mathfrak{a}\in\mathcal{A}$. Then
$Z^{p}_{\mathfrak{a}}$ is a Banach space.

###### Proof.

According to Definition 2.8 we need to show that $Z^{p}_{\mathfrak{a}}$ is
complete. Therfore let us start by assuming that
$\mathscr{X}\coloneqq\\{\xi^{n}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence in
$Z^{p}_{\mathfrak{a}}$. Moreover, let us also define
$\mathscr{X}_{t}\coloneqq\\{\xi^{n}_{t}\\}_{n\in\mathbb{N}}$ and observe now
the following

  1. (1)

From Definition 2.8 we see that $\mathscr{X}$ is a Cauchy sequence in
$\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{p}_{\mathfrak{a}})$. Hence let us
define $\overline{\xi}:\overline{\Omega}\to l^{p}_{\mathfrak{a}}$ in the
following way

$\displaystyle\overline{\xi}\coloneqq\overbrace{\bigg{[}\
\lim_{n\to\infty}\xi^{n}\ \bigg{]}}^{\text{in}\
\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{p}_{\mathfrak{a}})}\\!\\!.$ (2.54)

  2. (2)

From Definition 2.8 we see that for all $t\in\mathcal{T}$ sequence
$\mathscr{X}_{t}$ is Cauchy in
$\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\mathfrak{a}})$. Moreover we see
that

$\displaystyle\lim_{n,m\to\infty}\|\xi^{n}_{t}-\xi^{m}_{t}\|_{\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\mathfrak{a}})}\
=0\ \text{uniformly on }\mathcal{T}.$ (2.55)

We define $\overline{\overline{\xi}}:\overline{\Omega}\to
l^{p}_{\mathfrak{a}}$ in the following way

$\displaystyle\overline{\overline{\xi}}(t,\omega)\coloneqq\overbrace{\bigg{[}\lim_{n\to\infty}\xi^{n}_{t}\bigg{]}}^{\text{in}\
\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\mathfrak{a}})}(\omega),$ (2.56)

and conclude from a convergence result (2.55) above that

$\displaystyle\lim_{n\to\infty}\|\xi^{n}_{t}-\overline{\overline{\xi}}_{t}\|_{\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{p}_{\mathfrak{a}})}\
=0\ \text{uniformly on }\mathcal{T}.$ (2.57)

Now we make an important observation that underpins the rest of this proof.
From the fact that

$\displaystyle\lim_{n\to\infty}\|\xi^{n}_{t}-\overline{\xi}_{t}\|_{\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{p}_{\mathfrak{a}})}\
=0,$ (2.58)

We can coclude using Fubini Theorem 7.15 and Theorem 7.5 that there exist a
subsequence $\sigma$ such that

$\displaystyle\mu-a.s.,\
\mathbb{E}\bigg{[}\|\xi^{\sigma(n)}_{t}-\overline{\xi}_{t}\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}\to
0.$ (2.59)

Using Egoroff Theorem 7.6 and the fact that $\mu$ is a regular measure we find
a sequence $\\{A_{k}\\}_{k\in\mathbb{N}}$ of subsets of $\mathcal{T}$ such
that for all $k\in\mathbb{N}$

  1. (1)

$A_{k}\subset\mathcal{T}$ is compact and $\mu(A_{k})\leq\frac{1}{2^{k}}$,

  2. (2)

Moreover

$\displaystyle\mathbb{E}\bigg{[}\|\xi^{\sigma(n)}_{t}-\overline{\xi}_{t}\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}\to
0\ \text{uniformly on }\mathcal{T}-A_{k}.$ (2.60)

Let us now define a null set

$\displaystyle\widetilde{\mathcal{T}}\coloneqq\bigg{\\{}t\in\mathcal{T}\
\big{|}\ t\in\bigcap_{k\in\mathbb{N}}A_{k}\bigg{\\}},$ (2.61)

a sequence $\\{B_{k}\\}_{k\in\mathbb{N}}$ of subsets of $\mathcal{T}$ where
for all $k\in\mathbb{N}$

$\displaystyle B_{k}\coloneqq A_{k}-\widetilde{\mathcal{T}},$ (2.62)

and a map $\xi:\overline{\Omega}\to l^{p}_{\mathfrak{a}}$ in the following way

$\displaystyle\xi(t,\omega)$
$\displaystyle\coloneqq\begin{cases}\begin{tabular}[]{l|l}$\overline{\xi}(t,\omega)$&\
$\omega\in\Omega\land t\in(\mathcal{T}-\widetilde{\mathcal{T}})$,\\\
$\overline{\overline{\xi}}(t,\omega)$&\ $\omega\in\Omega\land
t\in\widetilde{\mathcal{T}}$.\end{tabular}\end{cases}$ (2.63)

We conclude immediately tha $\xi\in\mathcal{S}(l_{\mathfrak{a}}^{p})$ because
$\mathbf{MP}$ is complete, $\widetilde{\mathcal{T}}$ a null set and so
$\xi=\overline{\xi}$ almost everywhere on $\overline{\Omega}$. Now, to
conclude this proof it remains to show the following:

  1. I.

$\lim_{n\to\infty}\|\xi^{n}-\xi\|_{Z^{p}_{\mathfrak{a}}}\to 0$ as
$n\to\infty$,

  2. II.

$\xi_{x}$ is adapted to $\mathbb{F}$ for all $x\in\gamma$.

Proof of I.
Fix $k\in\mathbb{N}$ and using composite definition of $\xi$ together with
estimates (2.57) and (2.60) observe that

$\displaystyle\mathbb{E}\bigg{[}\|\xi^{\sigma(n)}_{t}-\xi_{t}\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}\to
0\ \text{uniformly on }(\mathcal{T}-B_{k})\cup\widetilde{\mathcal{T}}.$ (2.64)

Therefore we fix an arbitrary $\epsilon>0$ and a suitable
$N_{\epsilon}\in\mathbb{N}$ such that for all $n>N_{\epsilon}$ and all
$k\in\mathbb{N}$ we have

$\displaystyle\mathbb{E}\bigg{[}\|\xi^{\sigma(n)}_{t}-\xi_{t}\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}<\epsilon\
\text{uniformly on }(\mathcal{T}-B_{k})\cup\widetilde{\mathcal{T}}.$ (2.65)

Hence for all $n>N_{\epsilon}$ and all $k\in\mathbb{N}$ we see that

$\displaystyle\sup\bigg{\\{}\mathbb{E}\bigg{[}\|\xi^{\sigma(n)}_{t}-\xi_{t}\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}\
\bigg{|}\
t\in(\mathcal{T}-B_{k})\cup\widetilde{\mathcal{T}}\bigg{\\}}\leq\epsilon.$
(2.66)

Moreover, inequality (2.66) above shows that for all $n>N_{\epsilon}$

$\displaystyle\sup\bigg{\\{}\mathbb{E}\bigg{[}\|\xi^{\sigma(n)}_{t}-\xi_{t}\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}\
\bigg{|}\ t\in\mathcal{T}\bigg{\\}}\leq\epsilon,$ (2.67)

for otherwise there exists $\bar{n}>N_{\epsilon}$ and $\bar{t}\in\mathcal{T}$
such that

$\displaystyle\mathbb{E}\bigg{[}\|\xi^{\sigma(\bar{n})}_{\bar{t}}-\xi_{\bar{t}}\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}>\epsilon.$
(2.68)

Moreover $\bar{t}\not\in\widetilde{\mathcal{T}}$ for otherwise we will
contradict inequality (2.66). Therefore, by definition of
$\widetilde{\mathcal{T}}$ it follows that there exists $\bar{k}\in\mathbb{N}$
such that $t\not\in A_{\bar{k}}$ hence $t\not\in B_{\bar{k}}$ and so
$t\in(\mathcal{T}-B_{\bar{k}})$. Therefore from inequality (2.66) we get a
contradiction

$\displaystyle\mathbb{E}\bigg{[}\|\xi^{\sigma(\bar{n})}_{\bar{t}}-\xi_{\bar{t}}\|_{l^{p}_{\mathfrak{a}}}^{p}\bigg{]}\leq\epsilon,$
(2.69)

hence inequality (2.67) holds and we conclude that
$\lim_{n\to\infty}\|\xi^{\sigma(n)}-\xi\|_{Z^{p}_{\mathfrak{a}}}\to 0$ as
$n\to\infty$. Finally because $\mathscr{X}$ is a Cauchy sequence we conclude
that $\lim_{n\to\infty}\|\xi^{n}-\xi\|_{Z^{p}_{\mathfrak{a}}}\to 0$ as
$n\to\infty$.
Proof of II.
We start by fixing $x\in\gamma$ and $t\in\mathcal{T}$. Now from the previous
part (i.e. Proof of I) we can deduce that

$\displaystyle\lim_{n\to\infty}\|\xi^{n}_{x,t}-\xi_{x,t}\|_{\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{\mathbb{R}})}\
=0.$ (2.70)

Therefore using Theorem 7.5 we find a subsequence $\sigma$ such that
$\mathbb{P}-a.s.$ we have $\xi^{\sigma(n)}_{x,t}\to\xi_{x,t}$. Since for all
$n\in\mathbb{N}$ processes $\xi^{\sigma(n)}_{x,t}$ is $\mathcal{F}_{t}$
measurable we conclude by Theorem 7.7 that $\xi_{x,t}$ is $\mathcal{F}_{t}$
measurable and the proof is complete. ∎

######  Theorem 2.6.

Suppose that $p\in\mathbb{R}^{++}$. Then $\mathcal{Z}^{p}$ is the scale.

###### Proof.

From Theorem 2.5 we alredy know that $\mathcal{Z}^{p}$ is a family of Banach
spaces so to conclude the proof it only remains to show that conditions (1)
and (2) of the Definition 2.1 are satisfied.
Let us begin by fixing $\alpha<\beta\in\mathcal{A}$ and observing that
condition (1) of the Definition 2.1 can be verified by showing that
$Z^{p}_{\alpha}\subset Z^{p}_{\beta}$. To see that this is true we now fix
$\xi\in Z^{p}_{\alpha}$. Hence, $\xi\in\mathcal{S}(l^{p}_{\alpha})$ and
components of $\xi$ are adapted to $\mathbb{F}$ by Definition 2.8. Since
$\mathscr{L}^{p}$ is a scale we also see that
$\xi\in\mathcal{S}(l^{p}_{\beta})$. Moreover for all $t\in\mathcal{T}$ we
observe that
$\|\xi_{t}\|_{l_{\alpha}^{p}}\in\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{\mathbb{R}})$,
$\|\xi_{t}\|_{l_{\beta}^{p}}\in\mathcal{M}(\mathbf{P},\mathbf{M}^{\mathbb{R}})$
and $\|\xi_{t}\|_{l_{\beta}^{p}}\leq\|\xi_{t}\|_{l_{\alpha}^{p}}$. Now from
Theorem 7.12 we see that

$\displaystyle\mathbb{E}\bigg{[}\|\xi_{t}\|_{l_{\beta}^{p}}^{p}\bigg{]}\leq\mathbb{E}\bigg{[}\|\xi_{t}\|_{l_{\alpha}^{p}}^{p}\bigg{]},$
(2.71)

which establishes that $\|\xi\|_{Z_{\beta}^{p}}\ \leq\|\xi\|_{Z_{\alpha}^{p}}$
hence proving that both conditions (1) and (2) of the Definition 2.1 are
satisfied. ∎

###  2.5 Stochastic System

Throughout this section let us assume that $\mathbb{R}\ni p\geq 2$. We now
wish to introduce and study the following stochastic system, which we will
denote by $\mathscr{O}^{p}$.

$\displaystyle\xi_{x,t}$
$\displaystyle=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\xi_{x,s},\Xi_{s})ds+\int_{0}^{t}\Psi_{x}(\xi_{x,s},\Xi_{s})dW_{x}(s),\quad
x\in\gamma,\quad t\in\mathcal{T},$ ($\mathscr{O}^{p}$)

where:

  1. (1)

we assume that $\zeta\in l^{p}_{\underline{\mathfrak{a}}}$.

  2. (2)

we let $V$ in $C(\mathbb{R})$ and assume that for all $x\in\gamma$ maps
$\Phi_{x}:\mathbb{R}\times\mathbb{R}^{\gamma}\to\mathbb{R}$ are measurable
defined in the following way

$\displaystyle\Phi_{x}(q,\\{z_{y}\\}_{y\in\gamma})\coloneqq V(q)+\sum_{y\in
B_{x}}a(x-y)z_{y},$ (2.72)

for all $q\in\mathbb{R}$ and all
$\\{z_{y}\\}_{y\in\gamma}\in\mathbb{R}^{\gamma}$, where function $a$ was
defined previously in (A).

For all $x\in\gamma$ the following conditions are placed on maps $\Phi_{x}$.

  1. (C)

There exists $c\in\mathbb{R}^{0}$ and $\mathbb{R}^{++}\ni R\leq p$ such that
for all $q\in\mathbb{R}$ and all $x\in\gamma$

$\displaystyle|\Phi_{x}(q,0)|\leq c(1+|q|^{R})_{.}$ (2.73)

  2. (D)

There exists $b\in\mathbb{R}$ such that for all $q_{1},q_{2}\in\mathbb{R}$ and
all $x\in\gamma$

$\displaystyle(q_{1}-q_{2})(\Phi_{x}(q_{1},0)-\Phi_{x}(q_{2},0))\leq
b(q_{1}-q_{2})^{2}_{.}$ (2.74)

  1. (3)

For all $x\in\gamma$ we assume that maps
$\Psi_{x}:\mathbb{R}\times\mathbb{R}^{\gamma}\to\mathbb{R}$ are measurable.

Moreover for all $x\in\gamma$ the following conditions are also placed on maps
$\Psi_{x}$.

  1. (E)

There exists $M_{1},M_{2}\in\mathbb{R}$ such that for all
$q_{1},q_{2}\in\mathbb{R}$, $Z_{1},Z_{2}\in\mathbb{R}^{\gamma}$ and all
$x\in\gamma$

$\displaystyle|\Psi_{x}(q_{1},Z_{1})-\Psi_{x}(q_{2},Z_{2})|$
$\displaystyle\leq M_{1}|q_{1}-q_{2}|+M_{2}n_{x}\sum_{y\in
B_{x}}|z_{1,y}-z_{2,y}|,$ (2.75) $\displaystyle|\Psi_{x}(0,0)|$
$\displaystyle\leq c.$ (2.76)

######  Definition 2.9.

Suppose that $\mathbb{R}\ni p\geq 2$. A stochastic process $\Xi$ is called a
strong solution of the system ($\mathscr{O}^{p}$) if
$\Xi\in\mathcal{Z}^{p}(\cap)$ and for all $x\in\gamma$ and all
$t\in\mathcal{T}$ we have

$\displaystyle\xi_{x,t}$
$\displaystyle=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\xi_{x,s},\Xi_{s})ds+\int_{0}^{t}\Psi_{x}(\xi_{x,s},\Xi_{s})dW_{x}(s),\
\mathbb{P}-a.s.$ (2.77)

The main goal of this document is to show that for all $\mathbb{R}\ni p\geq 2$
the stochastic system ($\mathscr{O}^{p}$) admits a unique strong solution. We
now conclude this subsection with the following Lemma.

######  Lemma 2.10.

Suppose that $q_{1},q_{2}\in\mathbb{R}$ and let
$Z_{1},Z_{2}\in\mathbb{R}^{\gamma}$. Then for all $x\in\gamma$ we have

$\displaystyle|\Phi_{x}(q_{1},Z_{1})|$ $\displaystyle\leq
c(1+|q_{1}|^{R})+\tilde{a}_{x}\bigg{(}\sum_{y\in
B_{x}}z_{1,y}^{2}\bigg{)}^{\frac{1}{2}}_{,}$ (2.78)
$\displaystyle(q_{1}-q_{2})(\Phi_{x}(q_{1},Z_{1})-\Phi_{x}(q_{2},Z_{2}))$
$\displaystyle\leq(b+\frac{1}{2})(q_{1}-q_{2})^{2}+\frac{1}{2}\tilde{a}_{x}^{2}\sum_{y\in
B_{x}}(z_{1,y}-z_{2,y})^{2}_{,}$ (2.79)

where $\tilde{a}_{x}=\bigg{(}\sum_{y\in
B_{x}}a^{2}(x-y)\bigg{)}^{\frac{1}{2}}_{.}$

###### Proof.

First we prove inequality (2.78). We begin by considering the following chain
of calculations

$\displaystyle|\Phi_{x}(q_{1},Z_{1})|$
$\displaystyle=|\frac{1}{2}V(q_{1})-\sum_{y\in B_{x}}a(x-y)z_{1,y}|,$ (2.80)
$\displaystyle=|\Phi_{x}(q_{1},0)-\sum_{y\in B_{x}}a(x-y)z_{1,y}|,$ (2.81)
$\displaystyle\leq|\Phi_{x}(q_{1},0)|+|\sum_{y\in B_{x}}a(x-y)z_{1,y}|.$
(2.82)

Therefore using assumption (C) we see that

$\displaystyle|\Phi_{x}(q_{1},Z_{1})|$ $\displaystyle\leq
c(1+|q_{1}|^{R})+\sum_{y\in B_{x}}|a(x-y)z_{1,y}|,$ (2.83) $\displaystyle\leq
c(1+|q_{1}|^{R})+\bigg{(}\sum_{y\in
B_{x}}a^{2}(x-y)\bigg{)}^{\frac{1}{2}}\bigg{(}\sum_{y\in
B_{x}}z_{1,y}^{2}\bigg{)}^{\frac{1}{2}}_{.}$ (2.84)

Hence using the definition of $\tilde{a}_{x}$ above we see that

$\displaystyle|\Phi_{x}(q_{1},Z_{1})|\leq
c(1+|q_{1}|^{R})+\tilde{a}_{x}\bigg{(}\sum_{y\in
B_{x}}z_{1,y}^{2}\bigg{)}^{\frac{1}{2}}_{,}$ (2.85)

which establishes that inequality (2.78) is true. Now we show that inequality
(2.79) above is also true.
We start by observing from equations (2.81) and (2.72) that

$\displaystyle(q_{1}-q_{2})(\Phi_{x}(q_{1},Z_{1})-$
$\displaystyle\Phi_{x}(q_{2},Z_{2}))=$ (2.86)
$\displaystyle=(q_{1}-q_{2})(\Phi_{x}(q_{1},0)-\Phi_{x}(q_{2},0))+$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ \
+(q_{1}-q_{2})\sum_{y\in B_{x}}a(x-y)(z_{1,y}-z_{2,y}).$

Hence using assumption (D) we see that

$\displaystyle(q_{1}-q_{2})(\Phi_{x}(q_{1},Z_{1})-$
$\displaystyle\Phi_{x}(q_{2},Z_{2}))\leq$ (2.87) $\displaystyle\leq
b(q_{1}-q_{2})^{2}+\frac{1}{2}(q_{1}-q_{2})^{2}+\frac{1}{2}\bigg{(}\sum_{y\in
B_{x}}a(x-y)(z_{1,y}-z_{2,y})\bigg{)}^{2}_{,}$ (2.88)
$\displaystyle\leq(b+\frac{1}{2})(q_{1}-q_{2})^{2}+\frac{1}{2}\sum_{y\in
B_{x}}a^{2}(x-y)\sum_{y\in B_{x}}(z_{1,y}-z_{2,y})^{2}_{.}$ (2.89)

Finally using, once again, the definition of $\tilde{a}_{x}$ above we see that

$\displaystyle(q_{1}-q_{2})$
$\displaystyle(\Phi_{x}(q_{1},Z_{1})-\Phi_{x}(q_{2},Z_{2}))\leq(b+\frac{1}{2})(q_{1}-q_{2})^{2}+\frac{1}{2}\tilde{a}_{x}^{2}\sum_{y\in
B_{x}}(z_{1,y}-z_{2,y})^{2}_{,}$ (2.90)

and the proof is complete. ∎

##  3 Auxiliary Results

In this section we prove two results that will be used later on to show that
the stochastic system ($\mathscr{O}^{p}$) admits a unique strong solution.
Throughout this section let us assume that $\mathbb{R}\ni p\geq 2$.

######  Theorem 3.1.

Suppose that $\mathfrak{a}\in\mathcal{A}$ and let
$\Xi\coloneqq\\{\xi_{x}\\}_{x\in\gamma}$ be an element in
$Z^{p}_{\mathfrak{a}}$. Then for all $x\in\gamma$ we have
$\Phi_{x}(\xi_{x},\Xi)\in L^{1}_{ad}$ and $\Psi_{x}(\xi_{x},\Xi)\in
L^{2}_{ad}$.

###### Proof.

We combine Theorems 2.4 and 7.9 to conclude that for all $x\in\gamma$ we have

$\displaystyle\xi_{x}\in L^{p}_{ad}\subset L^{2}_{ad}\subset L^{1}_{ad}.$
(3.1)

Since composition of mesurable maps is measurable we conclude that
$x\in\gamma$ we have

$\displaystyle\Phi_{x}(\xi_{x},\Xi),\Psi_{x}(\xi_{x},\Xi)\in\mathcal{M}(\mathbf{MP},\mathbf{M}^{\mathbb{R}}),$
(3.2)

and adapted to $\mathbb{F}$. Now according to the definition (2.72) and the
assumption (C) we have for all $x\in\gamma$ the following inequality

$\displaystyle|\Phi_{x}(\xi_{x},\Xi)|\ \leq|c|(1+|\xi_{x}|^{R})+\sum_{y\in
B_{x}}a(x-y)|\xi_{y}|.$ (3.3)

Moreover, because $R\leq p$ we can use Theorem 7.12 to conclude that
$\Phi_{x}(\xi_{x},\Xi)\in L^{1}_{ad}$. Finally we combine Theorem 7.17 with
the assumption (E) to conclude that for all $x\in\gamma$ we have

$\displaystyle|\Psi_{x}(\xi_{x},\Xi)|^{2}\leq
4|\Psi_{x}(0,0)|^{2}+4M_{1}^{2}|\xi_{x}|^{2}+4M_{2}^{2}n_{x}^{3}\sum_{y\in
B_{x}}|\xi_{x}|^{2}.$ (3.4)

Once again applying Theorem 7.12 to the inequality (3.4) above we conclude
that $\Psi_{x}(\xi_{x},\Xi)\in L^{2}_{ad}$ hence the proof is complete. ∎

######  Theorem 3.2.

Let $L\coloneqq
4e^{\underline{\mathfrak{a}}\rho}C\mathcal{N}^{q+1}(1+\rho)^{\frac{1}{2}}$ and
let $Q\coloneqq\\{Q_{x,y}\\}_{x,y\in\gamma}$ be an infinite real matrix such
that for all $x,y\in\gamma$ we have

$\displaystyle x\not\in B_{y}\iff Q_{x,y}=0\iff y\not\in B_{x}.$ (3.5)

Moreover assume that for all $x,y\in\gamma$ there exist $C\in\mathbb{R}^{0}$
and $q\in\mathbb{R}^{++}$ such that

$\displaystyle|Q_{x,y}|\leq Cn_{x}^{q}.$ (3.6)

Then $Q\in\mathcal{O}(\mathscr{L}^{1},L,\frac{1}{2})$. That is, $Q$ is the
Ovsjannikov map of order $L$ and $\frac{1}{2}$ on $\mathscr{L}^{1}$.

###### Proof.

Consider arbitrary $\alpha<\beta\in\mathcal{A}$ and fix $z\in l_{\alpha}^{1}$.
We will complete this proof by showing that

$\displaystyle\|Qz\|_{\beta}\leq\frac{L}{(\beta-\alpha)^{\frac{1}{2}}}\|z\|_{\alpha},$
(3.7)

which establishes that $Q$ is a linear operator from $l_{\alpha}^{1}$ to
$l_{\beta}^{1}$ hence verifing conditions (1) and (2) of the Definition 2.2.
Consider now the following equation

$\displaystyle\|Qz\|_{\beta}$
$\displaystyle=\sum_{x\in\gamma}e^{-\beta|x|}\bigg{|}\sum_{y\in\gamma}Q_{x,y}z_{y}\bigg{|}.$
(3.8)

Moreover, for all $x\in\gamma$ we will make use of the following facts

$\displaystyle\mathbf{I}.\ $ $\displaystyle x\not\in B_{y}\lor y\not\in B_{x}$
$\displaystyle\implies Q_{x,y}=0.$ (3.9) $\displaystyle\mathbf{II}.\ $
$\displaystyle y\in B_{x}$ $\displaystyle\implies-|x|\leq-|y|+\rho.$
$\displaystyle\mathbf{III}.\ $ $\displaystyle x\in B_{y}$
$\displaystyle\implies\sqrt{|x|}\leq\sqrt{|y|}+\sqrt{\rho}.$

Now, using equation (3.8) together with the facts $\mathbf{I}$ and
$\mathbf{II}$ we see that

$\displaystyle\|Qz\|_{\beta}$
$\displaystyle\leq\sum_{x\in\gamma}\sum_{y\in\gamma}|Q_{x,y}|e^{-\beta|x|}|z_{y}|,$
(3.10) $\displaystyle\leq e^{\beta\rho}\sum_{x\in\gamma}\sum_{y\in
B_{x}}|Q_{x,y}|e^{-\beta|y|}|z_{y}|,$ (3.11) $\displaystyle\leq
e^{\beta\rho}\sum_{x\in\gamma}\sum_{y\in
B_{x}}|Q_{x,y}|e^{-(\beta-\alpha)|y|}e^{-\alpha|y|}|z_{y}|.$ (3.12)

Hence from inequality (3.12) we see that

$\displaystyle\|Qz\|_{\beta}$ $\displaystyle\leq
e^{\beta\rho}\sum_{x\in\gamma}\sum_{y\in\gamma}|Q_{x,y}|e^{-(\beta-\alpha)|y|}e^{-\alpha|y|}|z_{y}|,$
(3.13)
$\displaystyle=e^{\beta\rho}\sum_{y\in\gamma}\sum_{x\in\gamma}|Q_{x,y}|e^{-(\beta-\alpha)|y|}e^{-\alpha|y|}|z_{y}|,$
(3.14) $\displaystyle\leq e^{\overline{\mathfrak{a}}\rho}K\|z\|_{\alpha},$
(3.15)

where

$\displaystyle
K\coloneqq\sup\bigg{\\{}\sum_{x\in\gamma}|Q_{x,y}|e^{-(\beta-\alpha)|y|}\
\bigg{|}\ y\in\gamma\bigg{\\}}.$ (3.16)

We now estimate the value of supremum in the definition (3.16) above. Hence
using condition (3.6) together with the fact $\mathbf{I}$ we see that for all
$y\in\gamma$

$\displaystyle\sum_{x\in\gamma}|Q_{x,y}|e^{-(\beta-\alpha)|y|}$
$\displaystyle=\sum_{x\in B_{y}}|Q_{x,y}|e^{-(\beta-\alpha)|y|},$ (3.17)
$\displaystyle\leq C\sum_{x\in B_{y}}n_{x}^{q}e^{-(\beta-\alpha)|y|}.$ (3.18)

Using now assumption (B) together with the fact $\mathbf{III}$ we see that for
all $y\in\gamma$

$\displaystyle\sum_{x\in\gamma}|Q_{x,y}|e^{-(\beta-\alpha)|y|}$
$\displaystyle\leq C\sum_{x\in
B_{y}}\mathcal{N}^{q}|x|^{\frac{1}{2}}e^{-(\beta-\alpha)|y|},$ (3.19)
$\displaystyle\leq C\mathcal{N}^{q}\sum_{x\in
B_{y}}(|y|^{\frac{1}{2}}+\rho^{\frac{1}{2}})e^{-(\beta-\alpha)|y|},$ (3.20)
$\displaystyle\leq
C\mathcal{N}^{q}n_{y}(|y|^{\frac{1}{2}}+\rho^{\frac{1}{2}})e^{-(\beta-\alpha)|y|},$
(3.21) $\displaystyle\leq
C\mathcal{N}^{q+1}(|y|^{\frac{1}{4}}+|y|^{\frac{1}{2}}\rho^{\frac{1}{2}})e^{-(\beta-\alpha)|y|},$
(3.22) $\displaystyle\leq
B(|y|^{\frac{1}{4}}+|y|^{\frac{1}{2}})e^{-(\beta-\alpha)|y|},$ (3.23)

where $B\coloneqq C\mathcal{N}^{q+1}(1+\rho)^{\frac{1}{2}}$.
Now returning to equation (3.16) we see that

$\displaystyle K$ $\displaystyle\leq
B\sup\bigg{\\{}(|y|^{\frac{1}{4}}+|y|^{\frac{1}{2}})e^{-(\beta-\alpha)|y|}\
\bigg{|}\ y\in\gamma\bigg{\\}},$ (3.24) $\displaystyle\leq
B\sup\bigg{\\{}(h^{\frac{1}{4}}+h^{\frac{1}{2}})e^{-(\beta-\alpha)h}\
\bigg{|}\ h>0\bigg{\\}},$ (3.25) $\displaystyle\leq
4B\sup\bigg{\\{}h^{\frac{1}{2}}e^{-(\beta-\alpha)h}\ \bigg{|}\ h>0\bigg{\\}},$
(3.26) $\displaystyle\leq
4B\sup\bigg{\\{}\bigg{(}he^{-2(\beta-\alpha)h}\bigg{)}^{\frac{1}{2}}\
\bigg{|}\ h>0\bigg{\\}},$ (3.27) $\displaystyle\leq
4B\bigg{(}\sup\bigg{\\{}he^{-2(\beta-\alpha)h}\ \bigg{|}\
h>0\bigg{\\}}\bigg{)}^{\frac{1}{2}}.$ (3.28)

Now, we can deduce that function
$he^{-2(\beta-\alpha)h}:(0,\infty)\to\mathbb{R}$ attains its supremum when
$\frac{d}{dh}he^{-2(\beta-\alpha)h}=0$ that is when
$h=\frac{1}{2(\beta-\alpha)}$. Hence it follows from inequality (3.28) that

$\displaystyle K$
$\displaystyle\leq\frac{4B}{(\beta-\alpha)^{\frac{1}{2}}}\frac{1}{e\sqrt{2}}.$
(3.29)

Now, continuing from equation (3.15) we finally see that

$\displaystyle\|Qz\|_{\beta}$ $\displaystyle\leq
e^{\underline{\mathfrak{a}}\rho}K\|z\|_{\alpha},$ (3.30)
$\displaystyle\leq\frac{4e^{\underline{\mathfrak{a}}\rho}C\mathcal{N}^{q+1}(1+\rho)^{\frac{1}{2}}}{(\beta-\alpha)^{\frac{1}{2}}}\|z\|_{\alpha},$
(3.31)

hence the proof is complete. ∎

######  Remark.

In the following Theorem we will describe an equation of the form
$\displaystyle f(t)=z_{\underline{\mathfrak{a}}}+\int_{0}^{t}Q(f(s))ds,\
t\in\mathcal{T},$ (3.32) and rely on our work in subsection 7.3 to conclude,
with the choice $\mathbf{X}\equiv\mathscr{L}^{1}\quad\text{and}\quad F\equiv
Q,$ that equation (3.32) has a unique continuous solution, in the context of
Theorem (7.27).

######  Theorem 3.3 (Comparison Theorem).

Suppose $z_{\underline{\mathfrak{a}}}\in l^{1}_{\underline{\mathfrak{a}}}$,
$q<1$ and $Q\coloneqq\\{Q_{x,y}\\}_{x,y\in\gamma}$ is an element of
$\mathcal{O}(\mathscr{L}^{1},L,q)$. Moreover suppose that $Q_{x,y}\geq 0$ for
all $x,y\in\gamma$ and, in the context of Theorem (7.27), let $f$ be the
unique continuous solution of the integral equation

$\displaystyle f(t)=z_{\underline{\mathfrak{a}}}+\int_{0}^{t}Q(f(s))ds,\
t\in\mathcal{T}.$ (3.33)

Finally, suppose that $g:[0,T]\to l^{1}_{\underline{\mathfrak{a}}}$ is a
continuous map such that for all $x\in\gamma$

$\displaystyle g_{x}(t)\leq
z_{\underline{\mathfrak{a}},x}+\bigg{[}\int_{0}^{t}Q(g(s))ds\bigg{]}_{x},\
t\in\mathcal{T}.$ (3.34)

Then for all $t\in\mathcal{T}$ and all $x\in\gamma$

$\displaystyle g_{x}(t)\leq f_{x}(t).$ (3.35)

###### Proof.

For all $\mathfrak{a}\in\mathcal{A}$ let
$H_{\mathfrak{a}}=\mathcal{C}([0,T],l^{1}_{\mathfrak{a}})$ and define a family
$\mathbf{H}\coloneqq\\{H_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}}$. It
follows from subsection 7.3 that $\mathbf{H}$ is a scale. Moreover from
Theorem 7.23 we know that map $\mathcal{I}:\mathbf{H}(\cup)\to
H_{\overline{\mathfrak{a}}}$ defined for all $t\in[0,T]$ and all $\kappa\in
H_{\alpha}$ via formula

$\displaystyle\mathcal{I}(\kappa)(t)\coloneqq
z_{\underline{\mathfrak{a}}}+\int_{0}^{t}Q(\kappa(s))ds,$ (3.36)

is an Ovsjannikov map of order $TL$ and $q$ on $\mathbf{H}$. That is
$\mathcal{I}\in\mathcal{O}(\mathbf{H},TL,q)$.
Therefore, using Theorem 7.26, we see that if
$\underline{\mathfrak{a}}<\beta\in\mathcal{A}$ then the sequence
$\\{\mathcal{I}^{n}(g)\\}_{n\in\mathbb{N}}$ where

$\displaystyle\begin{rcases}\mathcal{I}^{1}(g)(t)&\coloneqq
z_{\underline{\mathfrak{a}}}+\int_{0}^{t}Q(g(s))ds,\\\ &\vdots\\\
\mathcal{I}^{n+1}(g)(t)&\coloneqq\mathcal{I}(\mathcal{I}^{n}(g))(t),\end{rcases}\
\forall\ t\in\mathcal{T}.$ (3.37)

is such that

$\displaystyle\overbrace{\bigg{[}\lim_{n\to\infty}\mathcal{I}^{n}(g)\bigg{]}}^{\text{in}\
\mathcal{C}([0,T],l^{1}_{\beta})}=f.$ (3.38)

Therefore it is also true that
$\lim_{n\to\infty}\mathcal{I}^{n}_{x}(g)(t)=f_{x}(t)$ for all $x\in\gamma$ and
all $t\in\mathcal{T}$. Hence to conclude the proof it is sufficient to fix
$x\in\gamma$ and $t\in\mathcal{T}$ and prove by induction that

$\displaystyle g_{x}(t)\leq\mathcal{I}^{n}_{x}(g)(t),\ \forall\
n\in\mathbb{N}.$ (3.39)

Case $n=1$ is satisfied by the initial assumption on $g$, so let us now assume
that the induction hypothesis (3.39) is true for some $n\geq 1$ and proceed by
considering the following chain of inequalities

$\displaystyle\mathcal{I}^{n+1}_{x}(g)(t)$
$\displaystyle=\mathcal{I}_{x}(\mathcal{I}^{n}(g))(t)_{,}$ (3.40)
$\displaystyle=z_{\underline{\mathfrak{a}},x}+\bigg{[}\int_{0}^{t}Q(\mathcal{I}^{n}(g)(s))ds\bigg{]}_{x,}$
(3.41)
$\displaystyle=z_{\underline{\mathfrak{a}},x}+\sum_{y\in\gamma}Q_{x,y}\int_{0}^{t}\mathcal{I}^{n}_{y}(g)(s)ds,$
(3.42) $\displaystyle\geq z_{\underline{\mathfrak{a}},x}+\sum_{y\in
p}Q_{x,y}\int_{0}^{t}g_{y}(s)ds,$ (3.43)
$\displaystyle=z_{\underline{\mathfrak{a}},x}+\bigg{[}\int_{0}^{t}Q(g(s))ds\bigg{]}_{x,}$
(3.44) $\displaystyle\geq g_{x}(t).$ (3.45)

Finally from inequalities (3.40) - (3.45) we conclude that ineauality (3.39)
holds hence the proof is complete. ∎

######  Corollary 3.4.

Suppose that $z_{\underline{\mathfrak{a}},x}\geq 0$ $x\in\gamma$. Moreover
assume that components of $g$ are non-negative functions, that is
$g_{x}(t)\geq 0$ for all $x\in\gamma$ and all $t\in\mathcal{T}$. Then for all
$\beta>\alpha\in\mathcal{A}$ there exists a constant
$K(\alpha,\beta)\in\mathbb{R}$ such that

$\displaystyle\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}g_{x}(t)\leq
K(\alpha,\beta)\sum_{x\in\gamma}e^{-\alpha|x|}z_{\underline{\mathfrak{a}},x}.$
(3.46)

###### Proof.

Using Theorem 3.3, we start by making an observation that for all $x\in\gamma$
and all $t\in\mathcal{T}$

$\displaystyle g_{x}(t)$ $\displaystyle\leq
z_{\underline{\mathfrak{a}},x}+\bigg{[}\int_{0}^{t}Q(g(s))ds\bigg{]}_{x,}$
(3.47) $\displaystyle\leq
z_{\underline{\mathfrak{a}},x}+\bigg{[}\int_{0}^{t}Q(f(s))ds\bigg{]}_{x.}$
(3.48)

Therefore we see that for all $x\in\gamma$

$\displaystyle\sup_{t\in\mathcal{T}}g_{x}(t)$ $\displaystyle\leq
z_{\underline{\mathfrak{a}},x}+\bigg{[}\int_{0}^{T}Q(f(s))ds\bigg{]}_{x,}$
(3.49) $\displaystyle=f_{x}(T).$ (3.50)

Hence it follows that

$\displaystyle\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}g_{x}(t)$
$\displaystyle\leq\sum_{x\in\gamma}e^{-\beta|x|}f_{x}(T),$ (3.51)
$\displaystyle\leq\|f(T)\|_{l_{\beta}^{1}}.$ (3.52)

Norm in the inequality (3.52) above can be estimated using Theorem 7.28 and
remark that proceeds it. In particular we get

$\displaystyle\|f(T)\|_{l_{\beta}^{1}}\leq\sum_{n=0}^{\infty}\frac{L^{n}T^{n}}{(\beta-\alpha)^{q}}\frac{n^{q}}{n!}\|z_{\underline{\mathfrak{a}}}\|_{l_{\alpha}^{1}}.$
(3.53)

Finally letting
$K(\alpha,\beta)=\sum_{n=0}^{\infty}\frac{L^{n}T^{n}}{(\beta-\alpha)^{q}}\frac{n^{q}}{n!}$
we see that

$\displaystyle\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}g_{x}(t)\leq
K(\alpha,\beta)\|z_{\underline{\mathfrak{a}}}\|_{l_{\alpha}^{1}},$ (3.54)

hence the proof is complete. ∎

##  4 Truncated Systems

Throughout this section let us assume that $\mathbb{R}\ni p\geq 2$.
We now start working with a sequence $\\{\Lambda_{n}\\}_{n\in\mathbb{N}}$ of
finite subsets of $\gamma$ such that $\Lambda_{n}\uparrow\gamma$ as
$n\to\infty$. Moreover for each $n\in\mathbb{N}$ we now wish to introduce and
study the following stochastic system, which we will denote by
$\mathscr{O}^{p}_{n}$.

$\displaystyle\xi_{x,t}^{n}=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})ds+\int_{0}^{t}\Psi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})dW_{x}(s),$
$\displaystyle\quad\forall x\in\Lambda_{n}\land t\in\mathcal{T},$
($\mathscr{O}^{p}_{n}$) $\displaystyle\xi_{x,t}^{n}=\zeta_{x},$
$\displaystyle\quad\forall x\not\in\Lambda_{n}\land t\in\mathcal{T}.$

In simple words, for each $n\in\mathbb{N}$ system ($\mathscr{O}^{p}_{n}$) is a
stoped/truncated version of the system ($\mathscr{O}^{p}$), which was
described in subsection 2.5.
In this section our goal is to prove two important results concerning systems
($\mathscr{O}^{p}_{n}$). In the subsequent sections these results will help us
to establish that system ($\mathscr{O}^{p}$) admits a unique strong solution.
We shall now rely on [14, 13] and state the next result without a proof.

######  Theorem 4.1.

For all $n\in\mathbb{N}$ and $\zeta\in l^{p}_{\underline{\mathfrak{a}}}$
system ($\mathscr{O}^{p}_{n}$) has a continuous solution $\Xi^{n}\in
Z_{\underline{\mathfrak{a}}}^{p}$.

######  Remark.

For all $n\in\mathbb{N}$, a term solution in the Theorem 4.1 above is to be
understood in the same sence as explained in the Definition 2.9 except we do
not require $\Xi^{n}$ to be a map from $\overline{\Omega}$ to
$\mathcal{Z}^{p}(\cap)$.

######  Remark.

Combining Theorems (4.1) and (3.1) with the Definition (7.6) we see that
$\xi_{x}^{n}$ in an Itô process for all $n\in\mathbb{N}$ and $x\in\gamma$.

In the next two sections of this document it will be shown that the sequence
$\\{\Xi^{n}\\}_{n\in\mathbb{N}}$ converges to the unique strong solution of
the system ($\mathscr{O}^{p}$). However before this can be achieved we need to
establish the following two theorems.

######  Theorem 4.2.

Suppose that $n\in\mathbb{N}$ and $\mathbb{R}\ni p\geq 2$. Moreover let
$\Xi^{n}$ be the process defined in the Theorem 4.1 and for all $x\in\gamma$
let $\xi^{n}_{x}$ be components of $\Xi^{n}$. Then for all
$\underline{\mathfrak{a}}<\alpha\in\mathcal{A}$ we have

$\displaystyle\sum_{x\in\gamma}e^{-\alpha|x|}\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}|^{p}\bigg{]}<\infty.$
(4.1)

###### Proof.

Let us start by recalling that

$\displaystyle\xi_{x,t}^{n}=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})ds+\int_{0}^{t}\Psi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})dW_{x}(s),$
$\displaystyle\quad\forall x\in\Lambda_{n}\land t\in\mathcal{T},$ (4.2)
$\displaystyle\xi_{x,t}^{n}=\zeta_{x},$ $\displaystyle\quad\forall
x\not\in\Lambda_{n}\land t\in\mathcal{T}.$

Hence using Itô Lemma 7.22 we see that if $x\in\Lambda_{n}$ then for all
$t\in\mathcal{T}$

$\displaystyle|\xi_{x,t}^{n}|^{p}=|\zeta_{x}|^{p}+\int_{0}^{t}p(\xi_{x,s}^{n})^{p-1}$
$\displaystyle\Phi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})ds+\mathbin{{\color[rgb]{1,1,1}\int_{0}^{t}\frac{(p-1)p}{2}}}$
(4.3)
$\displaystyle+\int_{0}^{t}\frac{(p-1)p}{2}(\xi_{x,s}^{n})^{p-2}(\Psi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n}))^{2}ds+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\int_{0}^{t}p(\xi_{x,s}^{n})^{p-1}\Psi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})dW_{x}(s).$

Now from assumptions (C), (D) and Lemma 2.10 we can deduce that for all
$t\in\mathcal{T}$

$\displaystyle(\xi_{x,t}^{n})^{p-1}\Phi_{x}(\xi_{x,t}^{n},\Xi_{t}^{n})$
$\displaystyle=(\xi_{x,t}^{n})^{p-2}(\xi_{x,t}^{n})\Phi_{x}(\xi_{x,t}^{n},\Xi_{t}^{n}),$
(4.4)
$\displaystyle\leq(\xi_{x,t}^{n})^{p-2}\bigg{[}(b+\frac{1}{2})|\xi_{x,t}^{n}|^{2}+\frac{1}{2}\tilde{a}_{x}^{2}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{2}+\xi_{x,t}^{n}\Phi_{x}(0,0)\bigg{]},$ (4.5)
$\displaystyle\leq(\xi_{x,t}^{n})^{p-2}\bigg{[}(b+1)|\xi_{x,t}^{n}|^{2}+\tilde{a}_{x}^{2}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{2}+c^{2}\bigg{]},$ (4.6)
$\displaystyle\leq(b+1)|\xi_{x,t}^{n}|^{p}+\tilde{a}_{x}^{2}|\xi_{x,t}^{n}|^{p-2}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{2}+|\xi_{x,t}^{n}|^{p-2}c^{2},$ (4.7)
$\displaystyle\leq(b+1)|\xi_{x,t}^{n}|^{p}+\tilde{a}_{x}^{2}n_{x}\max_{y\in
B_{x}}|\xi_{y,t}^{n}|^{p-2}\max_{y\in
B_{x}}|\xi_{y,t}^{n}|^{2}+|\xi_{x,t}^{n}|^{p-2}c^{2},$ (4.8)
$\displaystyle\leq(b+1)|\xi_{x,t}^{n}|^{p}+\tilde{a}_{x}^{2}n_{x}\max_{y\in
B_{x}}|\xi_{y,t}^{n}|^{p}+(1+|\xi_{x,t}^{n}|)^{p}c^{2}.$ (4.9)

Now using in addition Theorem 7.17 we see that for all $t\in\mathcal{T}$ we
have

$\displaystyle(\xi_{x,t}^{n})^{p-1}\Phi_{x}(\xi_{x,t}^{n},\Xi_{t}^{n})$
$\displaystyle\leq(b+1)|\xi_{x,t}^{n}|^{p}+\tilde{a}_{x}^{2}n_{x}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{p}+2^{p-1}c^{2}+2^{p-1}c^{2}|\xi_{x,t}^{n}|^{p},$
(4.10)
$\displaystyle\leq(b+1+2^{p-1}c^{2})|\xi_{x,t}^{n}|^{p}+\tilde{a}_{x}^{2}n_{x}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{p}+2^{p-1}c^{2},$ (4.11)
$\displaystyle\leq(b+1+2^{p-1}c^{2})|\xi_{x,t}^{n}|^{p}+\bar{a}^{2}n_{x}^{3}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{p}+2^{p-1}c^{2},$ (4.12)

Moreover from assumption (E) we know that for all $t\in\mathcal{T}$

$\displaystyle(\xi_{x,s}^{n})^{p-2}(\Psi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n}))^{2}$
$\displaystyle\leq(\xi_{x,s}^{n})^{p-2}\bigg{[}4M_{1}^{2}|\xi_{x,t}^{n}|^{2}+4M_{2}^{2}n_{x}^{2}\bigg{(}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|\bigg{)}^{2}+4|\Psi_{x}(0,0)|^{2}\bigg{]},$ (4.13)
$\displaystyle\leq(\xi_{x,s}^{n})^{p-2}\bigg{[}4M_{1}^{2}|\xi_{x,t}^{n}|^{2}+4M_{2}^{2}n_{x}^{3}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{2}+4c^{2}\bigg{]},$ (4.14) $\displaystyle\leq
4M_{1}^{2}|\xi_{x,t}^{n}|^{p}+4M_{2}^{2}n_{x}^{4}\max_{y\in
B_{x}}|\xi_{y,t}^{n}|^{p-2}\max_{y\in
B_{x}}|\xi_{y,t}^{n}|^{2}+4c^{2}|\xi_{x,s}^{n}|^{p-2},$ (4.15)
$\displaystyle\leq 4M_{1}^{2}|\xi_{x,t}^{n}|^{p}+4M_{2}^{2}n_{x}^{4}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{p}+4c^{2}2^{p-1}(1+|\xi_{x,s}^{n}|^{p}),$ (4.16)
$\displaystyle\leq(4M_{1}^{2}+4c^{2}2^{p-1})|\xi_{x,t}^{n}|^{p}+4M_{2}^{2}n_{x}^{4}\sum_{y\in
B_{x}}|\xi_{y,t}^{n}|^{p}+4c^{2}2^{p-1}.$ (4.17)

Now letting

$\displaystyle A_{1}\coloneqq(b+1+2^{p-1}c^{2}),$ (4.18) $\displaystyle
A_{2}\coloneqq(4M_{1}^{2}+4c^{2}2^{p-1}),$ (4.19) $\displaystyle
A_{3}\coloneqq(p\bar{a}^{2}+p^{2}4M_{2}^{2}),$ (4.20) $\displaystyle
A_{4}\coloneqq 5p^{2}2^{p}c^{2}T.$ (4.21)

we observe from inequalities (4.12) and (4.17) together with the system
($\mathscr{O}^{p}_{n}$) that for all $x\in\Lambda_{n}$ we have

$\displaystyle\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}\leq
p^{2}(A_{1}+A_{2})\int_{0}^{t}\mathbb{E}\bigg{[}|\xi_{x,s}^{n}|^{p}\bigg{]}ds+$
$\displaystyle+A_{3}n_{x}^{4}\sum_{y\in
B_{x}}\int_{0}^{t}\mathbb{E}\bigg{[}|\xi_{y,s}^{n}|^{p}\bigg{]}ds+A_{4},\
t\in\mathcal{T}.$ (4.22)

Now we fix an arbitrary $n\in\mathbb{N}$ and also define a measurable map
$\eta^{n}:\mathcal{T}\to l^{1}_{\underline{\mathfrak{a}}}$, that is a map
$\eta^{n}\in\mathcal{M}(\mathbf{M},\mathbf{M}^{p}_{\underline{\mathfrak{a}}})$,
via the following formula

$\displaystyle\eta_{x}^{n}(t)\coloneqq\max_{m\leq n}\
\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]},\ \forall t\in\mathcal{T}.$
(4.23)

Hence we deduce from the inequality (4.22) and from the system
($\mathscr{O}^{p}_{n}$) that for all $x\in\gamma$

$\displaystyle\eta_{x}^{n}(t)\leq\sum_{y\in\gamma}Q_{x,y}\int_{0}^{t}\eta_{y}^{n}(s)ds+A_{x},\
t\in\mathcal{T}.$ (4.24)

where

$\displaystyle Q_{x,y}=\begin{cases}p^{2}(A_{1}+A_{2})+A_{3}n_{x}^{4},&x=y,\\\
A_{3}n_{x}^{4},&0<|x-y|<\rho,\\\ 0,&|x-y|>\rho.\end{cases}$ (4.25)

and

$\displaystyle A_{x}=|\zeta_{x}|^{p}+A_{4}.$ (4.26)

Moreover the following facts can now also be deduced.

  1. (1)

$A\in l^{1}_{\underline{\mathfrak{a}}}$ as a result of Theorem 2.1 and the
choice $\zeta\in l^{p}_{\underline{\mathfrak{a}}}$.

  2. (2)

Using Theorem 7.8, we see that $\eta^{n}\in
C([0,T],l^{1}_{\underline{\mathfrak{a}}})$,

  3. (3)

From equation (4.25) we see that there exists a constant $C$ such that
$|Q_{x,y}|\leq Cn_{x}^{4}$. Therefore using Theorem 3.2 we conclude that there
exists some $L\in\mathbb{R}^{0}$ such that $Q$ is the Ovsjannikov operator of
order $L$ and $\frac{1}{2}$ on $\mathcal{L}^{1}$.

Now since $n\in\mathbb{N}$ was arbitrary, application of Theorem 3.3 and
Corrolary 3.4 to the inequality (4.24) tells us that for all $n\in\mathbb{N}$
we have

$\displaystyle\sum_{x\in\gamma}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\eta_{x}^{n}(t)\leq
K(\underline{\mathfrak{a}},\alpha)\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|A_{x}|.$
(4.27)

Hence we see that

$\displaystyle\sum_{x\in\gamma}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\max_{m\leq
n}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}\leq
K(\underline{\mathfrak{a}},\alpha)\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|A_{x}|.$
(4.28)

Therefore

$\displaystyle\sup_{n\in\mathbb{N}}\bigg{\\{}\sum_{x\in\gamma}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\max_{m\leq
n}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}\bigg{\\}}\leq
K(\underline{\mathfrak{a}},\alpha)\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|A_{x}|.$
(4.29)

######  Remark.

Consider now arbitrary $x\in\gamma$. It is clear that
$\sup_{n\in\mathbb{N}}\bigg{(}\max_{m\leq
n}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}\bigg{)}\leq\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}.$
Moreover for all $\epsilon>0$ there exists $k\in\mathbb{N}$ such that
$\displaystyle\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}-\epsilon$
$\displaystyle\leq\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{k}|^{p}\bigg{]},$
(4.30) $\displaystyle\leq\max_{m\leq
k}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]},$ (4.31)
$\displaystyle\leq\sup_{n\in\mathbb{N}}\bigg{(}\max_{m\leq
n}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}\bigg{)}.$
(4.32) Since $\epsilon$ is arbitrary It follows that
$\displaystyle\sup_{t\in\mathcal{T}}\sup_{n\in\mathbb{N}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}$
$\displaystyle=\sup_{n\in\mathbb{N}}\bigg{(}\max_{m\leq
n}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}\bigg{)},$
(4.33)
$\displaystyle=\sup_{n\in\mathbb{N}}\bigg{(}\sup_{t\in\mathcal{T}}\max_{m\leq
n}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}\bigg{)}.$ (4.34)

Remark above shows that if an arbitrary set $A\subset\gamma$ is finite then

$\displaystyle\sup_{n\in\mathbb{N}}\bigg{\\{}\sum_{x\in
A}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\max_{m\leq
n}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}\bigg{\\}}=\sum_{x\in
A}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\sup_{n\in\mathbb{N}}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}.$
(4.35)

Hence from inequality (4.29) we finally learn that

$\displaystyle\sum_{x\in\gamma}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\sup_{n\in\mathbb{N}}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}|^{p}\bigg{]}\leq
K(\underline{\mathfrak{a}},\alpha)\sum_{x\in\gamma}e^{-\underline{\mathfrak{a}}|x|}|A_{x}|,$
(4.36)

and the proof is complete. ∎

######  Theorem 4.3.

Suppose that $\mathbb{R}\ni p\geq 2$ and for all $n\in\mathbb{N}$ process
$\Xi^{n}$ is the solution of the truncated system ($\mathscr{O}^{p}_{n}$) as
defined in the Theorem 4.1. Then for all
$\underline{\mathfrak{a}}<\alpha\in\mathcal{A}$ sequence
$\\{\Xi^{n}\\}_{n\in\mathbb{N}}$ is Cauchy in $Z^{p}_{\alpha}.$

###### Proof.

Fix $n,m\in\mathbb{N}$ and define

$\displaystyle\bar{\Xi}^{n,m}\coloneqq\Xi^{n}-\Xi^{m}.$ (4.37)

In addition let us assume, without loss of generality, that
$\Lambda_{n}\subset\Lambda_{m}$. For all $x\in\gamma$ we shall now estimate
components $\bar{\xi}^{n,m}_{x}$ of $\bar{\Xi}^{n,m}$ by considering three
separate cases namely; $x\not\in\Lambda_{m}$, $x\in\Lambda_{n}$ and
$x\in\Lambda_{m}-\Lambda_{n}$.
First of all, from the definition of the system ($\mathscr{O}^{p}_{n}$) we see
that if $x\not\in\Lambda_{m}$ then we have

$\displaystyle\bar{\xi}^{n,m}_{x,t}=0,\ \forall t\in\mathcal{T}.$ (4.38)

Let us now define for all $x\in\gamma$ and all $t\in\mathcal{T}$ the following
processes

$\displaystyle\Phi_{x}^{n,m}(t)$
$\displaystyle\coloneqq\Phi_{x}(\xi^{n}_{x,t},\Xi^{n}_{t})-\Phi_{x}(\xi^{m}_{x,t},\Xi^{m}_{t}),$
(4.39) $\displaystyle\Psi_{x}^{n,m}(t)$
$\displaystyle\coloneqq\Psi_{x}(\xi^{n}_{x,t},\Xi^{n}_{t})-\Psi_{x}(\xi^{m}_{x,t},\Xi^{m}_{t}),$
(4.40)

and consider the situation when $x\in\Lambda_{n}$. In this case we have

$\displaystyle\bar{\xi}^{n,m}_{x,t}=\int_{0}^{t}\Phi_{x}^{n,m}(s)ds+\int_{0}^{t}\Psi_{x}^{n,m}(s)dW_{x}(s),\
t\in\mathcal{T}.$ (4.41)

Hence using Itô Lemma 7.22 we see that if $x\in\Lambda_{n}$ then for all
$t\in\mathcal{T}$

$\displaystyle|\bar{\xi}^{n,m}_{x,t}|^{p}=\int_{0}^{t}p(\bar{\xi}^{n,m}_{x,s})^{p-1}$
$\displaystyle\Phi_{x}^{n,m}(s)ds+\mathbin{{\color[rgb]{1,1,1}\int_{0}^{t}\frac{p(p-1)}{2}(\bar{\xi}^{n,m}_{x,s})^{p-2}(\Psi_{x}^{n,m}(s))^{2}ds}}$
(4.42)
$\displaystyle+\int_{0}^{t}\frac{p(p-1)}{2}(\bar{\xi}^{n,m}_{x,s})^{p-2}(\Psi_{x}^{n,m}(s))^{2}ds+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\int_{0}^{t}p(\bar{\xi}^{n,m}_{x,t})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s).$

Now, from Lemma 2.10 we can see that for all $t\in\mathcal{T}$ we have

$\displaystyle(\bar{\xi}^{n,m}_{x,t})^{p-1}\Phi_{x}^{n,m}(t)$
$\displaystyle=(\bar{\xi}^{n,m}_{x,t})^{p-2}\bar{\xi}^{n,m}_{x,t}\bigg{(}\Phi_{x}(\xi^{n}_{x,t},\Xi^{n}_{t})-\Phi_{x}(\xi^{m}_{x,t},\Xi^{m}_{t})\bigg{)},$
(4.43)
$\displaystyle\leq(\bar{\xi}^{n,m}_{x,t})^{p-2}\bigg{(}(b+\frac{1}{2})(\xi^{n}_{x,t}-\xi^{m}_{x,t})^{2}+\frac{1}{2}\tilde{a}_{x}^{2}\sum_{y\in
B_{x}}(\xi^{n}_{y,t}-\xi^{m}_{y,t})^{2}\bigg{)},$ (4.44)
$\displaystyle\leq(b+1)|\bar{\xi}^{n,m}_{x,t}|^{p}+\max_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p-2}\bigg{(}\tilde{a}_{x}^{2}n_{x}\max_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{2}\bigg{)},$ (4.45)
$\displaystyle\leq(b+1)|\bar{\xi}^{n,m}_{x,t}|^{p}+\tilde{a}_{x}^{2}n_{x}\max_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p},$ (4.46)
$\displaystyle\leq(b+1)|\bar{\xi}^{n,m}_{x,t}|^{p}+\tilde{a}_{x}^{2}n_{x}\sum_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p},$ (4.47)
$\displaystyle\leq(b+1)|\bar{\xi}^{n,m}_{x,t}|^{p}+\bar{a}^{2}n_{x}^{3}\sum_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p}.$ (4.48)

Moreover, using assumption (E) we can see that for all $t\in\mathcal{T}$ we
also have

$\displaystyle(\bar{\xi}^{n,m}_{x,t})^{p-2}(\Psi_{x}^{n,m}(t))^{2}$
$\displaystyle=(\bar{\xi}^{n,m}_{x,t})^{p-2}\bigg{(}\Phi_{x}(\xi^{n}_{x,t},\Xi^{n}_{t})-\Phi_{x}(\xi^{m}_{x,t},\Xi^{m}_{t})\bigg{)}^{2},$
(4.49)
$\displaystyle\leq(\bar{\xi}^{n,m}_{x,t})^{p-2}\bigg{(}2M_{1}^{2}(\xi^{n}_{x,t}-\xi^{m}_{x,t})^{2}+2M_{2}^{2}n_{x}^{3}\sum_{y\in
B_{x}}(\xi^{n}_{y,t}-\xi^{m}_{y,t})^{2}\bigg{)},$ (4.50) $\displaystyle\leq
2M_{1}^{2}|\bar{\xi}^{n,m}_{x,t}|^{p}+\max_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p-2}\bigg{(}2M_{2}^{2}n_{x}^{4}\max_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{2}\bigg{)},$ (4.51) $\displaystyle\leq
2M_{1}^{2}|\bar{\xi}^{n,m}_{x,t}|^{p}+2M_{2}^{2}n_{x}^{4}\max_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p},$ (4.52) $\displaystyle\leq
2M_{1}^{2}|\bar{\xi}^{n,m}_{x,t}|^{p}+2M_{2}^{2}n_{x}^{4}\sum_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p}.$ (4.53)

Therefore letting

$\displaystyle B_{1}$ $\displaystyle\coloneqq(b+1+2M_{1}^{2}),$ (4.54)
$\displaystyle B_{2}$ $\displaystyle\coloneqq(p\bar{a}^{2}+2p^{2}M_{2}^{2}),$
(4.55)

we can deduce from equation 4.42 that if $x\in\Lambda_{n}$ then

$\displaystyle\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{p}\bigg{]}\leq
p^{2}B_{1}\int_{0}^{t}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,s}|^{p}\bigg{]}ds+B_{2}n_{x}^{4}\sum_{y\in
B_{x}}\int_{0}^{t}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{y,s}|^{p}\bigg{]}ds,\
t\in\mathcal{T}.$ (4.56)

Finally, when $x\in\Lambda_{m}-\Lambda_{n}$ we see using Theorem 7.17 that for
all $t\in\mathcal{T}$

$\displaystyle|\bar{\xi}^{n,m}_{x,t}|^{p}$
$\displaystyle\leq(|\xi^{n}_{x,t}|+|\xi^{m}_{x,t}|)^{p},$ (4.57)
$\displaystyle\leq 2^{p-1}|\xi^{n}_{x,t}|^{p}+2^{p-1}|\xi^{m}_{x,t}|^{p}.$
(4.58)

Therefore, using Theorem 4.2 and equation (4.38), we see now that if
$x\in\Lambda_{m}-\Lambda_{n}$ then for all $t\in\mathcal{T}$ we have

$\displaystyle\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{2}\bigg{]}$
$\displaystyle\leq
2^{p}\sup_{n\in\mathbb{N}}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}|^{p}\bigg{]},$
(4.59) $\displaystyle\leq
2^{p}\mathbbm{1}_{\Lambda_{m}-\Lambda_{n}}(x)\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}|^{p}\bigg{]}.$
(4.60)

Therefore we can finally deduce, combining equations (4.38), (4.56) and
(4.60), that all $x\in\gamma$ and for all $t\in\mathcal{T}$ we have

$\begin{split}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{p}\bigg{]}\leq
p^{2}B_{1}\int_{0}^{t}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,s}|^{p}\bigg{]}&ds+\\\\[10.00002pt]
&\\!\\!\\!\\!\\!\\!\\!\\!+B_{2}n_{x}^{4}\sum_{y\in
B_{x}}\int_{0}^{t}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{y,s}|^{p}\bigg{]}ds+\\\\[10.00002pt]
&\quad\quad\quad\quad\quad\quad\quad+2^{p}\mathbbm{1}_{\Lambda_{m}-\Lambda_{n}}(x)\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}|^{p}\bigg{]}.\end{split}$
(4.61)

Now, as in the proof of Theorem 4.2, infinite system of inequalities (4.61)
can be rewritten in the following way.
Define, relying on the inequality (4.61) a measurable map
$\varrho^{n,m}:\mathcal{T}\to\mathbb{R}^{\gamma}$, that is a map
$\varrho^{n,m}\in\mathcal{M}(\mathbf{M},\mathbf{M}^{p}_{\underline{\mathfrak{a}}})$,
via the following formula

$\displaystyle\varrho_{x}^{n,m}(t)\coloneqq\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{p}\bigg{]},\
\forall t\in\mathcal{T},$ (4.62)

and deduce from inequalities (4.56) - (4.60) that

$\displaystyle\varrho_{x}^{n,m}(t)\leq\sum_{y\in\gamma}Q_{x,y}\int_{0}^{t}\varrho_{y}^{n,m}(s)ds+A_{x},\
t\in\mathcal{T},$ (4.63)

where

$\displaystyle Q_{x,y}=\begin{cases}p^{2}B_{1}+B_{2}n_{x}^{4},&x=y,\\\
B_{2}n_{x}^{4},&0<|x-y|<p,\\\ 0,&|x-y|>p.\end{cases}$ (4.64)

and

$\displaystyle
A_{x}=2^{p}\mathbbm{1}_{\Lambda_{m}-\Lambda_{n}}(x)\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}|^{p}\bigg{]}.$
(4.65)

Now, fixing $\underline{\mathfrak{a}}<\tilde{\alpha}<\alpha\in\mathcal{A}$ we
can also deduce the following facts.

  1. (1)

$A\in l^{1}_{\tilde{\alpha}}$ as a result of Theorem 4.2.

  2. (2)

Using Theorem 7.8, we see that $\varrho^{n,m}\in
C([0,T],l^{1}_{\tilde{\alpha}})$,

  3. (3)

From equation (4.25) we see that there exists a constant $D$ such that
$|Q_{x,y}|\leq Dn_{x}^{4}$. Therefore using Theorem 3.2 we conclude that there
exists some $L\in\mathbb{R}^{0}$ such that $Q$ is the Ovsjannikov operator of
order $L$ and $\frac{1}{2}$ on $\mathcal{L}^{1}$.

Therefore we can now use Theorem 3.3 and Corollary 3.4 to conclude that

$\displaystyle\sum_{x\in\gamma}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\varrho_{x}^{n,m}(t)\leq
K(\tilde{\alpha},\alpha)\sum_{x\in\gamma}e^{-\tilde{\alpha}|x|}|A_{x}|.$
(4.66)

From equation (4.66) and definition (2.8) we therefore see that we have the
following estimate

$\displaystyle\|\Xi^{n}-\Xi^{m}\|_{Z^{p}_{\alpha}}^{p}$
$\displaystyle=\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}||\Xi^{n}_{t}-\Xi^{m}_{t}||^{p}_{l^{p}_{\alpha}}\bigg{]},$
(4.67)
$\displaystyle=\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}\sum_{x\in\gamma}e^{-\alpha|x|}|\xi^{n}_{x,t}-\xi^{m}_{x,t}|^{p}\bigg{]},$
(4.68)
$\displaystyle\leq\sum_{x\in\gamma}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\varrho_{x}^{n,m}(t),$
(4.69) $\displaystyle\leq
K(\tilde{\alpha},\alpha)\sum_{x\in\gamma}e^{-\tilde{\alpha}|x|}|A_{x}|,$
(4.70) $\displaystyle\leq
K(\tilde{\alpha},\alpha)\sum_{x\in\gamma}e^{-\tilde{\alpha}|x|}2^{p}\mathbbm{1}_{\Lambda_{m}-\Lambda_{n}}(x)\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}|^{p}\bigg{]},$
(4.71) $\displaystyle\leq
2^{p}K(\tilde{\alpha},\alpha)\sum_{x\in\Lambda_{m}-\Lambda_{n}}e^{-\tilde{\alpha}|x|}\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}|^{p}\bigg{]}.$
(4.72)

Estimate above implies that the right hand side of equation (4.72) is the
remainder of a convergent series hence the proof is complete. ∎

##  5 One Dimesional Special Case

Suppose that $\mathbb{R}\ni p\geq 2$ and
$\underline{\mathfrak{a}}<\alpha\in\mathcal{A}$. For all $n\in\mathbb{N}$ let
$\Xi^{n}$ be a solution of the truncated system ($\mathscr{O}^{p}_{n}$) and
using Theorem 4.3 let $\\{\Xi^{n}\\}_{n\in\mathbb{N}}$ be a Cauchy sequence in
$Z^{p}_{\alpha}$. Since $Z^{p}_{\alpha}$ is a Banach space, by Theorem 2.5, we
now define the following process

$\displaystyle\overbrace{\ \Xi\coloneqq\lim_{n\to\infty}\Xi^{n}\ }^{\text{in}\
Z_{\alpha}^{p}}.$ (5.1)

Consider now an arbitrary $x\in\gamma$. The main goal of this section is to
prove that the following stochastic integral equation

$\displaystyle\eta_{x,t}$
$\displaystyle=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\eta_{x,s},\Xi_{s})ds+\int_{0}^{t}\Psi_{x}(\eta_{x,s},\Xi_{s})dW_{x}(s),\
t\in\mathcal{T},$ (5.2)

has a solution in $\mathcal{S}(\mathbf{M}^{\mathbb{R}})$. We begin our work by
proving an auxiliary result.

######  Theorem 5.1.

Suppose that $x\in\gamma$ and $\Xi$ is a process defined by (5.1). Then

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\xi_{x,t}|^{p}\bigg{]}<\infty.$
(5.3)

###### Proof.

We shall prove this theorem by showing that for all $\epsilon>0$ there exist
$N\in\mathbb{N}$ such that for all $n,m\geq N$ we have

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\xi^{n}_{x,t}-\xi^{m}_{x,t}|^{p}\bigg{]}<\epsilon,$
(5.4)

where for all $n\in\mathbb{N}$ processes $\xi^{n}_{x}$ are components of
$\Xi^{n}$.
Since $\Lambda_{n}\uparrow\gamma$ we begin by finding some
$\bar{N}\in\mathbb{N}$ such that $x\in\Lambda_{\bar{N}}$ and temporary fixing
some $n,m\geq\bar{N}$. Moreover let us assume, without loss of generality,
that $n<m$ so that $x\in\Lambda_{n}\subset\Lambda_{m}$ and we define

$\displaystyle\bar{\xi}_{x,t}^{n,m}\coloneqq\xi^{n}_{x,t}-\xi^{m}_{x,t},\
\forall t\in\mathcal{T}.$ (5.5)

Now we recal Theorem 4.3. In particular we are interested in using definitions
(4.39) - (4.40) and an equation (4.41).
Hence an application of Itô Lemma shows that for all $t\in\mathcal{T}$

$\displaystyle|\bar{\xi}^{n,m}_{x,t}|^{p}=\int_{0}^{t}p(\bar{\xi}^{n,m}_{x,s})^{p-1}$
$\displaystyle\Phi_{x}^{n,m}(s)ds+\mathbin{{\color[rgb]{1,1,1}\int_{0}^{t}\frac{p(p-1)}{2}(\bar{\xi}^{n,m}_{x,s})^{p-2}(\Psi_{x}^{n,m}(s))^{2}ds}}$
(5.6)
$\displaystyle+\int_{0}^{t}\frac{p(p-1)}{2}(\bar{\xi}^{n,m}_{x,s})^{p-2}(\Psi_{x}^{n,m}(s))^{2}ds+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\int_{0}^{t}p(\bar{\xi}^{n,m}_{x,t})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s).$

Therefore we see from equation (5.6) above that

$\displaystyle\sup_{t\in\mathcal{T}}|\bar{\xi}^{n,m}_{x,t}|^{p}=\int_{0}^{T}p(\bar{\xi}^{n,m}_{x,s})^{p-1}$
$\displaystyle\Phi_{x}^{n,m}(s)ds+\mathbin{{\color[rgb]{1,1,1}\int_{0}^{T}\frac{p(p-1)}{2}(\bar{\xi}^{n,m}_{x,s})^{p-2}(\Psi_{x}^{n,m}(s))^{2}ds}}$
(5.7)
$\displaystyle+\int_{0}^{T}\frac{p(p-1)}{2}(\bar{\xi}^{n,m}_{x,s})^{p-2}(\Psi_{x}^{n,m}(s))^{2}ds+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sup_{t\in\mathcal{T}}\int_{0}^{t}p(\bar{\xi}^{n,m}_{x,t})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s).$

Moreover from inequalities (4.48) and (4.53) we see that

$\displaystyle(\bar{\xi}^{n,m}_{x,t})^{p-1}\Phi_{x}^{n,m}(t)\leq(b+1)|\bar{\xi}^{n,m}_{x,t}|^{p}+\bar{a}^{2}n_{x}^{3}\sum_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p}..$ (5.8)

and

$\displaystyle(\bar{\xi}^{n,m}_{x,t})^{p-2}(\Psi_{x}^{n,m}(t))^{2}\leq
2M_{1}^{2}|\bar{\xi}^{n,m}_{x,t}|^{p}+2M_{2}^{2}n_{x}^{4}\sum_{y\in
B_{x}}|\bar{\xi}^{n,m}_{x,t}|^{p}..$ (5.9)

Hence from inequality (5.7) and inequalities (5.8) - (5.9) above we see by
letting

$\displaystyle C_{1}$ $\displaystyle\coloneqq p^{2}(b+1+2M_{1}^{2}),$ (5.10)
$\displaystyle C_{2}$ $\displaystyle\coloneqq
n_{x}^{4}(p\bar{a}^{2}+2p^{2}M_{2}^{2}),$ (5.11) $\displaystyle K$
$\displaystyle\coloneqq
C_{1}\int_{0}^{T}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,s}|^{p}\bigg{]}ds+C_{2}\sum_{y\in
B_{x}}\int_{0}^{T}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{y,s}|^{p}\bigg{]}ds.$
(5.12)

that we have the following inequality

$\begin{split}\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\bar{\xi}^{n,m}_{x,t}|^{p}\bigg{]}\leq
K+\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}\int_{0}^{t}p(\bar{\xi}^{n,m}_{x,s})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s)\bigg{]},\end{split}$
(5.13)

Now using results from subsection 7.2, in particular Burkholder-Davis-Gundy
inequality 7.21 and also using Jensen inequality 7.14 we see that the
following estimate on the stochastic term from the inequality (5.13) above
holds.

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}\int_{0}^{t}p(\bar{\xi}^{n,m}_{x,s})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s)\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}\bigg{(}p(\bar{\xi}^{n,m}_{x,s})^{p-1}\Psi_{x}^{n,m}(s)\bigg{)}^{2}ds\bigg{)}^{\frac{1}{2}}\bigg{]},$
(5.14)
$\displaystyle\leq\bigg{(}\mathbb{E}\bigg{[}\int_{0}^{t}\bigg{(}p(\bar{\xi}^{n,m}_{x,s})^{p-1}\Psi_{x}^{n,m}(s)\bigg{)}^{2}ds\bigg{]}\bigg{)}^{\frac{1}{2}}.$
(5.15)

To simplify inequality (5.15) we note that according to the (E) for all
$t\in\mathcal{T}$ the following estimate is true

$\displaystyle\bigg{(}(\bar{\xi}^{n,m}_{x,t})^{p-1}\Psi_{x}^{n,m}(t)\bigg{)}^{2}$
$\displaystyle=(\bar{\xi}^{n,m}_{x,t})^{2p-2}\bigg{(}M_{1}|\bar{\xi}^{n,m}_{x,t}|+M_{2}n_{x}\sum_{y\in
B_{x}}|\bar{\xi}^{n,m}_{y,t}|\bigg{)}^{2},$ (5.16)
$\displaystyle\leq(\bar{\xi}^{n,m}_{x,t})^{2p-2}\bigg{(}2M_{1}^{2}|\bar{\xi}^{n,m}_{x,t}|^{2}+2M_{2}^{2}n_{x}^{3}\sum_{y\in
B_{x}}|\bar{\xi}^{n,m}_{y,t}|^{2}\bigg{)},$ (5.17) $\displaystyle\leq
2M_{1}^{2}|\bar{\xi}^{n,m}_{x,t}|^{2p}+\max_{y\in
B_{x}}|\bar{\xi}^{n,m}_{y,t}|^{2p-2}\bigg{(}2M_{2}^{2}n_{x}^{4}\max_{y\in
B_{x}}|\bar{\xi}^{n,m}_{y,t}|^{2}\bigg{)},$ (5.18) $\displaystyle\leq
2M_{1}^{2}|\bar{\xi}^{n,m}_{x,t}|^{2p}+2M_{2}^{2}n_{x}^{4}\sum_{y\in
B_{x}}|\bar{\xi}^{n,m}_{y,t}|^{2p}.$ (5.19)

Now letting

$\displaystyle C_{3}$ $\displaystyle\coloneqq 2p^{2}M_{1}^{2}T,$ (5.20)
$\displaystyle C_{4}$ $\displaystyle\coloneqq 2p^{2}M_{2}^{2}n_{x}^{4}T,$
(5.21)

It follows now that inequality (5.15) can be written as follows

$\begin{split}\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}\int_{0}^{t}p(\bar{\xi}^{n,m}_{x,s})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s)\bigg{]}\leq
C_{3}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{2p}\bigg{]}+C_{4}\sum_{y\in
B_{x}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{2p}\bigg{]},\end{split}$
(5.22)

Therefore returning to the inequality (5.13) we see that

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\bar{\xi}^{n,m}_{x,t}|^{p}\bigg{]}\leq
TC_{1}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{p}\bigg{]}$
$\displaystyle+TC_{2}\sum_{y\in
B_{x}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{y,t}|^{p}\bigg{]}+$
(5.23)
$\displaystyle+C_{3}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{2p}\bigg{]}+C_{4}\sum_{y\in
B_{x}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}^{n,m}_{x,t}|^{2p}\bigg{]}.$

Since $B_{x}$ is finite we can now use Theorem 4.3 to conclude that, with a
suitable choice of $n,m\in\mathbb{N}$, the right hand side of the inequality
(5.23) above can be made arbitrary small hence the proof is complete. ∎

Relying on [14] we now state without proof the following result.

######  Theorem 5.2.

There exists $\tau\in\mathbb{R}$ such that the stochastic integral equation
(5.2) admits a unique local maximal solution
$\eta_{x}:[0,\tau]\times\Omega\to\mathbb{R}$.

Now using Theorem 5.2 we can establish the existence of a global solution.
Precisely speaking we have the following result.

######  Theorem 5.3.

There exists a solution $\eta_{x}:\overline{\Omega}\to\mathbb{R}$ of the
stochastic integral equation (5.2).

###### Proof.

Clearly if $\tau\geq\mathcal{T}$ then there is nothing to prove so let us
assume that $\tau<\mathcal{T}$. Now by Theorem 5.2 there exists a unique,
maximal local solution $\eta_{x}:[0,\tau]\times\Omega\to\mathbb{R}$ to the
stochastic equation (5.2). Hence to complete the proof we will show that this
solution is also a global one. That is we will establishing that almost surely
$\lim_{n\to\infty}\tau_{n}=\infty$, where $\tau_{n}$ is the first exit time of
the maximal local solution from the interval (-n, n) for all $n\in\mathbb{N}$.
Now, since $\eta$ is a local solution to (5.2) we conclude that for all
$n\in\mathbb{N}$ and all $t\in[0,\infty)$ we have

$\displaystyle\eta_{x,t\land\tau_{n}}$
$\displaystyle=\zeta_{x}+\int_{0}^{t\land\tau_{n}}\Phi_{x}(\eta_{x,s\land\tau_{n}},\Xi_{s\land\tau_{n}})ds+\int_{0}^{t\land\tau_{n}}\Psi_{x}(\eta_{x,s\land\tau_{n}},\Xi_{s\land\tau_{n}})dW_{x}(s).$
(5.24)

Hence using Itô Lemma 7.22 we see that for all $t\in[0,\infty)$ we have the
following

$\displaystyle|\eta_{x,t\land\tau_{n}}|^{p}=$
$\displaystyle\int_{0}^{t\land\tau_{n}}p(\eta_{x,s\land\tau_{n}})^{p-1}\Phi_{x}(\eta_{x,s\land\tau_{n}},\Xi_{s\land\tau_{n}})ds+$
(5.25)
$\displaystyle\quad\quad\quad\quad\quad\quad+\int_{0}^{t\land\tau_{n}}\frac{p(p-1)}{2}(\eta_{x,s\land\tau_{n}})^{p-2}(\Psi_{x}(\eta_{x,s\land\tau_{n}},\Xi_{s\land\tau_{n}}))^{2}ds+$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\int_{0}^{t\land\tau_{n}}p(\eta_{x,s\land\tau_{n}})^{p-1}\Psi_{x}(\eta_{x,s\land\tau_{n}},\Xi_{s\land\tau_{n}})dW_{x}(s).$

Now by letting

$\displaystyle\bar{\Phi}_{x}^{p}(\eta,t)\coloneqq(\eta_{x,t\land\tau_{n}})^{p-1}\Phi_{x}(\eta_{x,t\land\tau_{n}},\Xi_{t\land\tau_{n}}),\
\forall t\in[0,\infty),$ (5.26)

we conclude from inequalities (4.12) - (4.17) and definitions (4.18) - (4.21)
that for all $t\in[0,\infty)$

$\displaystyle\Phi_{x}^{p}(\eta,t)$ $\displaystyle\leq
A_{1}|\eta_{x,t\land\tau_{n}}|^{p}+\bar{a}^{2}n_{x}^{3}\sum_{y\in
B_{x}}|\xi_{y,t\land\tau_{n}}|^{p}+2^{p-1}c^{2},$ (5.27) $\displaystyle\leq
A_{1}|\eta_{x,t\land\tau_{n}}|^{p}+\bar{a}^{2}n_{x}^{3}\sum_{y\in
B_{x}}\sup_{t\in\mathcal{T}}|\xi_{y,t\land\tau_{n}}|^{p}+2^{p-1}c^{2},$ (5.28)
$\displaystyle\leq
A_{1}|\eta_{x,t\land\tau_{n}}|^{p}+\bar{a}^{2}n_{x}^{3}\sum_{y\in
B_{x}}\sup_{t\in\mathcal{T}}|\xi_{y,t}|^{p}+2^{p-1}c^{2},$ (5.29)

and

$\displaystyle(\eta_{x,s\land\tau_{n}})^{p-2}(\Psi_{x}(\eta_{x,s\land\tau_{n}},\Xi_{s\land\tau_{n}}))^{2}$
$\displaystyle\leq
A_{2}|\eta_{x,t\land\tau_{n}}|^{p}+4M_{2}^{2}n_{x}^{4}\sum_{y\in
B_{x}}|\xi_{y,t\land\tau_{n}}|^{p}+4c^{2}2^{p-1},$ (5.30) $\displaystyle\leq
A_{2}|\eta_{x,t\land\tau_{n}}|^{p}+4M_{2}^{2}n_{x}^{4}\sum_{y\in
B_{x}}\sup_{t\in\mathcal{T}}|\xi_{y,t}|^{p}+4c^{2}2^{p-1}.$ (5.31)

Threfore, combining inequalities (5.29) - (5.31) with an inequality (5.25) we
see that for all $t\in[0,\infty)$ we have

$\displaystyle\mathbb{E}\bigg{[}|\eta_{x,t\land\tau_{n}}|^{p}\bigg{]}$
$\displaystyle\leq
p^{2}(A_{1}+A_{2})\int_{0}^{t}\mathbb{E}\bigg{[}|\eta_{x,s\land\tau_{n}}|^{p}\bigg{]}ds+Tn^{4}_{x}A_{3}\sum_{y\in
B_{x}}\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\xi_{y,t}^{n}|^{p}\bigg{]}+A_{4},$
(5.32) $\displaystyle\leq
D\int_{0}^{t}\mathbb{E}\bigg{[}|\eta_{x,s\land\tau_{n}}|^{p}\bigg{]}ds+K(x).$
(5.33)

Where

$\displaystyle D$ $\displaystyle\coloneqq p^{2}(A_{1}+A_{2}),$ (5.34)
$\displaystyle K(x)$ $\displaystyle\coloneqq Tn^{4}_{x}A_{3}\sum_{y\in
B_{x}}\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\xi_{y,t}^{n}|^{p}\bigg{]}+A_{4}.$
(5.35)

Now using Gronwall’s inequality 7.13 together with the inequality (5.33) above
we see that for all $t\in[0,\infty)$ we have

$\displaystyle\mathbb{E}\bigg{[}|\eta_{x,t\land\tau_{n}}|^{p}\bigg{]}\leq
K(x)e^{Dt}.$ (5.36)

However using the definition of $\tau_{n}$ we see that for all
$n\in\mathbb{N}$ we have $|\eta_{x,\tau_{n}}|\geq n$. Moreover, because
$\mathbb{P}(\tau_{n}<t)=\mathbb{E}\bigg{[}\mathbbm{1}_{\\{\tau_{n}<t\\}}\bigg{]}$
we also see that for all $t\in[0,\infty)$

$\displaystyle n^{p}\mathbb{P}(\tau_{n}<t)$
$\displaystyle\leq\mathbb{E}\bigg{[}|\eta_{x,\tau_{n}}|^{p}\mathbbm{1}_{\\{\tau_{n}<t\\}}\bigg{]},$
(5.37)
$\displaystyle\leq\mathbb{E}\bigg{[}|\eta_{x,\tau_{n}}|^{p}\mathbbm{1}_{\\{\tau_{n}<t\\}}\bigg{]}+\mathbb{E}\bigg{[}|\eta_{x,\tau_{n}}|^{p}\mathbbm{1}_{\\{\tau_{n}\geq
t\\}}\bigg{]},$ (5.38)
$\displaystyle=\mathbb{E}\bigg{[}|\eta_{x,t\land\tau_{n}}|^{p}\mathbbm{1}_{\\{\tau_{n}<t\\}}\bigg{]}+\mathbb{E}\bigg{[}|\eta_{x,t\land\tau_{n}}|^{p}\mathbbm{1}_{\\{\tau_{n}\geq
t\\}}\bigg{]},$ (5.39)
$\displaystyle=\mathbb{E}\bigg{[}|\eta_{x,t\land\tau_{n}}|^{p}\bigg{]}.$
(5.40)

Therefore using inequalities (5.36) - (5.40) above we see that for all
$n\in\mathbb{N}$ and for all $t\in[0,\infty)$ we have

$\displaystyle\mathbb{P}(\tau_{n}<t)\leq\frac{1}{n^{p}}K(x)e^{Dt}.$ (5.41)

Hence for all $t\in[0,\infty)$ we have

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\tau_{n}<t)=0.$ (5.42)

Now convergence in probability and the fact that
$\\{\tau_{n}\\}_{n\in\mathbb{N}}$ is an increasing sequence impliy that almost
surely $\lim_{n\to\infty}\tau_{n}=\infty$ hence the proof is complete. ∎

##  6 Existence and Uniqueness

Throughout this section let us assume that $\mathbb{R}\ni p\geq 2$.
In this section we will learn that system ($\mathscr{O}^{p}$) admits a unique
strong solution. We shall start by showing existence.

######  Theorem 6.1.

Stochastic system ($\mathscr{O}^{p}$) admits a strong solution.

###### Proof.

Let us start by fixing some $\underline{\mathfrak{a}}<\alpha\in\mathcal{A}$.
Now, according to the Theorem 4.3 sequence $\\{\Xi^{n}\\}_{n\in\mathbb{N}}$
converges in $Z^{p}_{\alpha}$. Therefore, this proof can be completed by
letting

$\displaystyle\overbrace{\ \Xi\coloneqq\lim_{n\to\infty}\Xi^{n}\ }^{\text{in}\
Z_{\alpha}^{p}},$ (6.1)

and showing that $\Xi\equiv\\{\xi_{x}\\}_{x\in\gamma}$ is also a strong
solution of the system ($\mathscr{O}^{p}$). However because $\Xi$ in
$Z^{p}_{\alpha}$ we see from the Definition 2.9 that to complete the proof it
only remains to show that for all $x\in\gamma$ and all $t\in\mathcal{T}$ we
have

$\displaystyle\xi_{x,t}$
$\displaystyle=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\xi_{x,s},\Xi_{s})ds+\int_{0}^{t}\Psi_{x}(\xi_{x,s},\Xi_{s})dW_{x}(s),\quad\mathbb{P}-a.s.$
(6.2)

Using our work in the previous section 5, in particular using Theorem 5.3 we
begin by defining a family of processes
$H\coloneqq\\{\eta_{x}\\}_{x\in\gamma}$ such that for all $x\in\gamma$ and all
$t\in\mathcal{T}$ we have

$\displaystyle\eta_{x,t}$
$\displaystyle=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\eta_{x,s},\Xi_{s})ds+\int_{0}^{t}\Psi_{x}(\eta_{x,s},\Xi_{s})dW_{x}(s),\quad\mathbb{P}-a.s.$
(6.3)

Now, if $n\in\mathbb{N}$ then we also recall from the Theorem 4.1 and the
Definition, of the truncated system, ($\mathscr{O}^{p}_{n}$) that for all
$x\in\gamma$ and all $t\in\mathcal{T}$ we have

$\begin{rcases}\begin{aligned}
&\xi_{x,t}^{n}=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})ds+\int_{0}^{t}\Psi_{x}(\xi_{x,s}^{n},\Xi_{s}^{n})dW_{x}(s)&&\quad\forall
x\in\Lambda_{n}\\\ &\xi_{x,t}^{n}=\zeta_{x}&&\quad\forall
x\not\in\Lambda_{n}\end{aligned}\end{rcases},\ \mathbb{P}-a.s.$ (6.4)

Moreover convergence $\overbrace{\ \Xi=\lim_{n\to\infty}\Xi^{n}\ }^{\text{in}\
Z_{\alpha}^{p}}$ in particular implies that

$\displaystyle\adjustlimits{lim}_{n\to\infty}{sup}_{t\in\mathcal{T}}\mathbb{E}\bigg{[}\
\sum_{x\in\gamma}e^{-\alpha|x|}|\xi^{n}_{x,t}-\xi_{x,t}|^{p}\bigg{]}=0.$ (6.5)

Now, from equation (6.5) above and Theorem 7.9 it follows that for all
$x\in\Lambda_{n}$ and uniformly on $\mathcal{T}$ we have

$\displaystyle\lim_{n\to\infty}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}-\xi_{x,t}|\bigg{]}=0.$
(6.6)

Therefore, observing that $\Lambda_{n}\uparrow\gamma$ as $n\to\infty$ we see
that in order to establish the equation (6.2), and hence conclude the proof,
it remains to show that for all $x\in\gamma$ and uniformly on $\mathcal{T}$ we
have

$\displaystyle\lim_{n\to\infty}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}-\eta_{x,t}|\bigg{]}=0.$
(6.7)

######  Remark.

Indeed, this will show that for all $x\in\gamma$ and uniformly on
$\mathcal{T}$ we have
$\displaystyle\mathbb{E}\bigg{[}|\xi_{x,t}-\eta_{x,t}|\bigg{]}=0,$ (6.8)
which, because $\gamma$ is countable, will establish that for all $x\in\gamma$
and uniformly on $\mathcal{T}$ we have $\displaystyle\xi_{x,t}=\eta_{x,t},\
\mathbb{P}-a.s.,$ (6.9) hence establishing equation (6.2) and compleating the
proof.

Now, let us fix an arbitrary $n\in\mathbb{N}$ and define for all $x\in\gamma$
and all $t\in\mathcal{T}$ the following processes

$\displaystyle\Phi_{x}^{n}(t)$
$\displaystyle\coloneqq\Phi_{x}(\xi^{n}_{x,t},\Xi^{n}_{t})-\Phi_{x}(\eta_{x,t},\Xi_{t}),$
(6.10) $\displaystyle\Psi_{x}^{n}(t)$
$\displaystyle\coloneqq\Psi_{x}(\xi^{n}_{x,t},\Xi^{n}_{t})-\Psi_{x}(\eta_{x,t},\Xi_{t}),$
(6.11) $\displaystyle\mathscr{X}^{n}_{x,t}$
$\displaystyle\coloneqq\xi^{n}_{x,t}-\eta_{x,t}.$ (6.12)

Hence using Itô Lemma we begin observing that for all $x\in\gamma$ and all
$t\in\mathcal{T}$ we have

$\displaystyle|\mathscr{X}^{n}_{x,t}|^{p}=\int_{0}^{t}p(\mathscr{X}^{n}_{x,t})^{p-1}$
$\displaystyle\Phi_{x}^{n,m}(s)ds+\mathbin{{\color[rgb]{1,1,1}\int_{0}^{t}\frac{p(p-1)}{2}(\bar{\xi}^{n,m}_{x,s})^{p-2}(\Psi_{x}^{n,m}(s))^{2}ds}}$
(6.13)
$\displaystyle+\int_{0}^{t}\frac{p(p-1)}{2}(\mathscr{X}^{n}_{x,t})^{p-2}(\Psi_{x}^{n,m}(s))^{2}ds+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\int_{0}^{t}p(\mathscr{X}^{n}_{x,t})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s).$

Therefore, from inequalities (4.48) - (4.53) we can see that for all
$x\in\gamma$ and all $t\in\mathcal{T}$ we have

$\displaystyle(\xi^{n}_{x,t}-\eta_{x,t})^{p-1}\Phi_{x}^{n}(t)\leq(b+1)(\xi^{n}_{x,t}-\eta_{x,t})^{p}+\bar{a}^{2}n_{x}^{3}\sum_{y\in
B_{x}}(\xi^{n}_{y,t}-\xi_{y,t})^{p},$ (6.14)

and

$\displaystyle(\xi^{n}_{x,t}-\eta_{x,t})^{p-2}\bigg{(}\Psi_{x}^{n}(t)\bigg{)}^{2}\leq
2M_{1}^{2}(\xi^{n}_{x,t}-\eta_{x,t})^{p}+2M_{2}^{2}n_{x}^{4}\sum_{y\in
B_{x}}(\xi^{n}_{y,t}-\xi_{y,t})^{p}.$ (6.15)

Now, because $B_{x}$ is finite for all $x\in\gamma$ it is clear from equation
(6.5) that

$\displaystyle\mathbb{E}\bigg{[}\sum_{y\in
B_{x}}(\xi^{n}_{y,t}-\xi_{y,t})^{p}\bigg{]},$ (6.16)

can be made arbitrary small uniformly on $\mathcal{T}$ by taking
$n\in\mathbb{N}$ sufficiently large. Therefore from inequalities (6.14) and
(6.15) above we see that we can establish the following inequlities for all
$t\in\mathcal{T}$

$\displaystyle\mathbb{E}\bigg{[}(\xi^{n}_{x,t}-\eta_{x,t})^{p-1}\Phi_{x}^{n}(t)\bigg{]}\leq(b+1)\mathbb{E}\bigg{[}(\xi^{n}_{x,t}-\eta_{x,t})^{p}\bigg{]}+A_{x}^{n},$
(6.17)

and

$\displaystyle\mathbb{E}\bigg{[}(\xi^{n}_{x,t}-\eta_{x,t})^{p-2}\bigg{(}\Psi_{x}^{n}(t)\bigg{)}^{2}\bigg{]}\leq
2M_{1}^{2}\mathbb{E}\bigg{[}(\xi^{n}_{x,t}-\eta_{x,t})^{p}\bigg{]}+A_{x}^{n}.$
(6.18)

Where for all $x\in\gamma$ we have

$\displaystyle A_{x}^{n}\coloneqq\max\\{\bar{a}^{2}n_{x}^{3},\
2M_{2}^{2}n_{x}^{4}\\}\mathbb{E}\bigg{[}\sum_{y\in
B_{x}}(\xi^{n}_{y,t}-\xi_{y,t})^{p}\bigg{]}.$ (6.19)

Moreover $A_{x}^{n}\to 0$ uniformly on $\mathcal{T}$ as $n\to\infty$.
Therefore using inequalities (6.17) and (6.18) above we can conclude from
equation (6.13) that for all $x\in\gamma$ and all $t\in\mathcal{T}$ we have

$\displaystyle\mathbb{E}\bigg{[}|\xi^{n}_{x,t}-\eta_{x,t}|^{p}\bigg{]}\leq
C\int_{0}^{t}\mathbb{E}\bigg{[}|\xi^{n}_{x,s}-\eta_{x,s}|^{p}\bigg{]}ds+\bar{A}_{x}^{n},$
(6.20)

where

$\displaystyle C\coloneqq p^{2}(b+1+2M_{1}^{2}),$ (6.21)
$\displaystyle\bar{A}_{x}^{n}\coloneqq 2p^{2}TA_{x}^{n}.$ (6.22)

Finally using Gronwall inequality 7.13 we see that for all $x\in\gamma$ and
all $t\in\mathcal{T}$ we have

$\displaystyle\mathbb{E}\bigg{[}|\xi^{n}_{x,t}-\eta_{x,t}|^{p}\bigg{]}\leq
A_{x}^{n}e^{CT},$ (6.23)

which shows that for all $x\in\gamma$ and uniformly on $\mathcal{T}$

$\displaystyle\lim_{n\to\infty}\mathbb{E}\bigg{[}|\xi^{n}_{x,t}-\eta_{x,t}|^{p}\bigg{]}=0.$
(6.24)

Equation (6.7) now follows via application of Therorem 7.9 hence the proof is
complete. ∎

In the following theorem we now address uniqueness.

######  Theorem 6.2.

Suppose $\zeta\in l^{p}_{\underline{\mathfrak{a}}}$ and
$\underline{\mathfrak{a}}<\alpha\in\mathcal{A}$. Then stochastic system
($\mathscr{O}^{p}$) admits a unique strong solution $\Xi$ in $Z^{p}_{\alpha}$.

###### Proof.

For contradiction, using Theorem 6.1, suppose that $\Xi^{1}$ and $\Xi^{2}$ are
distinct strong solutions of the system ($\mathscr{O}^{p}$) in
$Z^{p}_{\alpha}$. Now let us define a map $\bar{\Xi}\in Z^{p}_{\alpha}$ via
the following formula

$\displaystyle\bar{\Xi}_{t}\coloneqq\Xi^{1}_{t}-\Xi^{2}_{t}.$ (6.25)

We see that almost surely we have

$\displaystyle\bar{\xi}_{x,t}=\int_{0}^{t}\Phi_{x}(\xi^{1}_{x,s},\Xi^{1}_{s})-\Phi_{x}(\xi^{2}_{x,s},\Xi^{2}_{s})ds+\int_{0}^{t}\Psi_{x}(\xi^{1}_{x,s},\Xi^{1}_{s})-\Psi_{x}(\xi^{2}_{x,s},\Xi^{2}_{s})dW_{x}(s).$
(6.26)

Now as in the proof of Theorem 4.3 we deduce, using Ito Lemma, that

$\displaystyle|\bar{\xi}_{x,t}|^{p}=\int_{0}^{t}p(\bar{\xi}_{x,s})^{p-1}\Phi_{x}^{1,2}(s)ds$
$\displaystyle+\int_{0}^{t}\frac{p(p-1)}{2}(\bar{\xi}_{x,s})^{p-2}(\Psi_{x}^{1,2}(s))^{2}ds\
+$ (6.27)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\int_{0}^{t}p(\bar{\xi}_{x,s})^{p-1}\Psi_{x}^{1,2}(s)dW_{x}(s),$

where we have chosen for all $t\in\mathcal{T}$ to let

$\displaystyle\Phi_{x}^{1,2}(t)$
$\displaystyle\coloneqq\Phi_{x}(\xi^{1}_{x,t},\Xi^{1}_{t})-\Phi_{x}(\xi^{2}_{x,t},\Xi^{2}_{t}),$
(6.28) $\displaystyle\Psi_{x}^{1,2}(t)$
$\displaystyle\coloneqq\Psi_{x}(\xi^{1}_{x,t},\Xi^{1}_{t})-\Psi_{x}(\xi^{2}_{x,t},\Xi^{2}_{t}).$
(6.29)

Therefore we see that

$\displaystyle\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}|^{p}\bigg{]}\leq
B_{1}(p,b,c,M_{1})\int_{0}^{t}\mathbb{E}\bigg{[}|\bar{\xi}_{x,s}|^{p}\bigg{]}ds+B_{2}(x,p,M_{2})\sum_{y\in
B_{x}}\int_{0}^{t}\mathbb{E}\bigg{[}|\bar{\xi}_{y,s}|^{p}\bigg{]}ds,$ (6.30)

where

$\displaystyle B_{1}(p,b,c,M_{1})\coloneqq pb+\frac{p}{2}+M_{1}^{2}p(p-1),$
(6.31) $\displaystyle B_{2}(x,p,M_{2})\coloneqq
p\tilde{a}^{2}_{x}+n_{x}^{4}(p+M_{2}^{2}p(p-1)).$ (6.32)

Let us now use inequality (6.30) to define a map
$\kappa:\mathcal{T}\to\mathbb{R}^{\gamma}$ via the following formula

$\displaystyle\kappa_{x}(t)\coloneqq\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}|^{p}\bigg{]},$
(6.33)

and deduce from inequality (6.30) that

$\displaystyle\kappa_{x}(t)\leq\sum_{y\in\gamma}Q_{x,y}\int_{0}^{t}\kappa_{y}(s)ds,$
(6.34)

where for all $x,y\in\gamma$ we have

$\displaystyle
Q_{x,y}=\begin{cases}B_{1}(p,b,c,M_{1})+B_{2}(x,p,M_{2}),&x=y,\\\
B_{2}(x,p,M_{2}),&0<|x-y|\leq p,\\\ 0,&|x-y|>p.\end{cases}$ (6.35)

Fixing $\underline{\mathfrak{a}}<\tilde{\alpha}\leq\alpha\in\mathcal{A}$ we
can now deduce the following facts.

  1. (1)

Using Theorem 7.8, we see that $\kappa\in C([0,T],l^{1}_{\tilde{\alpha}})$,

  2. (2)

From equation (4.25) we see that there exists a constant $C$ such that
$|Q_{x,y}|\leq Cn_{x}$. Therefore $Q$ is the Ovsiannikov operator on
$\mathcal{L}^{1}$.

Therefore we can now use Theorem 3.3 and Corollary 3.4 to conclude that

$\displaystyle\sum_{x\in\gamma}e^{-\alpha|x|}\sup_{t\in\mathcal{T}}\kappa_{x}(t)\leq
K(\tilde{\alpha},\alpha)\sum_{x\in\gamma}e^{-\tilde{\alpha}|x|}|A_{x}|,$
(6.36)

where $A_{x}$ is the zero sequence in $l^{1}_{\tilde{\alpha}}$. Therefore we
can establish that

$\displaystyle\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}\sum_{x\in\gamma}e^{-\alpha|x|}|\bar{\xi}_{x,t}|^{p}\bigg{]}=0.$
(6.37)

Hence

$\displaystyle||F^{1}-F^{2}||_{Z_{\alpha}^{p}}=0,$ (6.38)

and the proof is complete. ∎

##  7 Complementary Theory

Let us begin this section by recalling a couple of earlier statements and
definitions.
Throughout this section assume the following.

  1. (1)

We fix $T\in\mathbb{R}^{+}$, define $\mathcal{T}\coloneqq[0,T]$ and work on a
complete filtered probability space

$\displaystyle\mathbf{P}\coloneqq(\Omega,\mathcal{F},\mathbb{P},\mathbb{F}).$
(7.1)

    1. (a)

Completeness of $\mathbf{P}$ implies that for all $t\in\mathcal{T}$,
$\mathbf{P}_{t}\coloneqq(\Omega,\mathcal{F}_{t},\mathbb{P})$ is complete.

    2. (b)

Filtration $\mathbb{F}\coloneqq\\{\mathcal{F}_{t}\\}_{t\in\mathcal{T}}$ is
assumed to be right continuous. That is for all $t\in\mathcal{T}$,

$\displaystyle\mathcal{F}_{t}=\bigcap_{n\in\mathbb{N}}\mathcal{F}_{t+\frac{1}{n}}.$
(7.2)

  2. (2)

We fix a measure space
$\mathbf{M}\coloneqq(\mathcal{T},\mathscr{B}(\mathcal{T}),\mu)$, where $\mu$
is a Lebesgue measure and $\mathscr{B}(\mathcal{T})$ is a Borel
$\sigma-$algebra.

  3. (3)

We now agree to work on a fixed product measure space

$\displaystyle\mathbf{M}\mathbf{P}\coloneqq(\overline{\Omega}\coloneqq\mathcal{T}\times\Omega,\overline{\mathcal{F}}\coloneqq\mathscr{B}(\mathcal{T})\times\mathcal{F},\overline{\mathbb{P}}\coloneqq\mu\times\mathbb{P}).$
(7.3)

  4. (4)

Given two measurable spaces $\mathbf{A}$ and $\mathbf{B}$ we denote by
$\mathcal{M}(\mathbf{A},\mathbf{B})$ the space of all measurable maps from
$\mathbf{A}$ to $\mathbf{B}$. In particular, the following spaces will be
frequently mentioned

    1. (a)

$\mathbf{M}^{p}_{\mathfrak{a}}\coloneqq(l_{\mathfrak{a}}^{p},\mathscr{B}(l_{\mathfrak{a}}^{p}))$,
where $l_{\mathfrak{a}}^{p}$ was introduced by Definition 2.4.

    2. (b)

$\mathbf{M}^{\mathbb{R}}\coloneqq(\mathbb{R},\mathscr{B}(\mathbb{R}))$,

Here is how we will understand and denote stochastic processes in this
section.

######  Definition 7.1.

Let $Y$ be a normed linear space and $\mathbf{Y}\coloneqq(Y,\mathcal{B})$ be a
measurable space. Stochastic process is an element of
$\mathcal{M}(\mathbf{M}\mathbf{P},\mathbf{Y})$. In particular for all
$t\in\mathcal{T}$ and all $\omega\in\Omega$

$\mathcal{M}(\mathbf{P},\mathbf{Y})\ni\xi_{t}(\cdot):\Omega\to Y,$
$\mathcal{M}(\mathbf{M},\mathbf{Y})\ni\xi_{\cdot}(\omega):\mathcal{T}\to Y.$

For brevity we shall denote by $\mathcal{S}(\mathbf{Y})$ the set of all
stochastic processes.

Following Bnach spaces will be frequently used.

######  Definition 7.2.

Let $\mathscr{X}\coloneqq(X,\mathcal{A},\eta)$ be a measure space, $Y$ be a
normed linear space, with norm denoted by $\|\cdot\|_{Y}$, and
$\mathscr{Y}\coloneqq(Y,\mathcal{B})$ be a measurable space. For all
$p\in\mathbb{R}^{++}$ we define the following Banach spaces.

$\displaystyle\mathcal{L}^{p}(\mathscr{X},\mathscr{Y})\coloneqq\left\\{f:X\to
Y\ \begin{tabular}[]{|l}\
$\|f\|_{\mathcal{L}^{p}(\mathscr{X},\mathscr{Y})}\coloneqq\left(\bigint_{\\!\\!\\!\\!X}\|f\|^{p}_{Y}d\eta\right)^{\frac{1}{p}}<\infty$.\\\
\ $f\in\mathcal{M}(\mathscr{X},\mathscr{Y})$.\end{tabular}\right\\}$ (7.6)

Let us also make the following definitions

$\displaystyle S_{1}\coloneqq\bigg{\\{}K\subset\mathcal{T}\times\Omega\
\bigg{|}\ K=(s,t]\times A\text{ where }s<t\in\mathcal{T}\land
A\in\mathcal{F}_{s}\bigg{\\}},$ (7.7) $\displaystyle
S_{2}\coloneqq\bigg{\\{}K\subset\mathcal{T}\times\Omega\ \bigg{|}\
K=\\{0\\}\times A\text{ where }A\in\mathcal{F}_{0}\bigg{\\}},$ (7.8)
$\displaystyle\mathcal{P}\coloneqq\sigma(S_{1}\cup S_{2}),$ (7.9)
$\displaystyle\mathbb{L}\coloneqq\left\\{\xi\in\mathcal{S}(\mathbf{M}^{\mathbb{R}})\
\begin{tabular}[]{|l}\ $\text{trajectories of }\xi\text{ are left continuous,
almost surely}$.\\\ \ $\xi\ \text{is adapted to}\
\mathbb{F}$.\end{tabular}\right\\}.$ (7.12)

We note that $\mathcal{P}$ above is the smallest $\sigma-$algebra with respect
to which all elements of $\mathbb{L}$ are measurable.

  1. (5)

We now fix the following product measure space

$\displaystyle\overline{\mathbf{M}\mathbf{P}}\coloneqq(\overline{\Omega},\mathcal{P},\overline{\mathbb{P}}).$
(7.13)

###  7.1 Expectation, Measurability and Related Inequalities

Unless stated otherwise, information in this subsection is based on [8, 11].
For brevity and convenience in this subsection we will be working with the
following definition.

######  Definition 7.3.

Suppose $\mathbf{X}\coloneqq(X,\mathcal{A},\mu)$ is a measure spaces. For all
$p\in\mathbb{R}^{++}$ we make the following definition.

$\displaystyle\mathcal{L}^{p}\coloneqq\mathcal{L}^{p}(\mathbf{P},\mathbf{M}^{\mathbb{R}}),$
(7.14)
$\displaystyle\mathcal{L}^{p}(\mathbf{X})\coloneqq\mathcal{L}^{p}(\mathbf{X},\mathbf{M}^{\mathbb{R}}),$
(7.15) $\displaystyle\mathcal{L}^{p}_{+}\coloneqq\\{f\in\mathcal{L}^{p}|f\geq
0\ \text{almost surely}\\},$ (7.16)
$\displaystyle\mathcal{L}^{p}_{+}(\mathbf{X})\coloneqq\\{f\in\mathcal{L}^{p}(\mathbf{X})|f\geq
0\ \text{almost surely}\\}.$ (7.17)

######  Theorem 7.1 (Borel–Cantelli Theorem).

Let $\\{A_{i}\\}_{i\in\mathbb{N}}$ be a sequence of measurable subsets of
$\Omega$. Then

$\displaystyle\sum_{i=0}^{\infty}\mathbb{P}(A_{i})<\infty\implies\mathbb{P}\bigg{(}\bigcap_{j=0}^{\infty}\bigcup_{i=j}^{\infty}A_{i}\bigg{)}=0.$
(7.18)

######  Corollary 7.1.

Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be a sequence of real valued random
variables and let $X$ be another real valued random variable. For all
$\epsilon>0$ and all $i\in\mathbb{N}$ define also the followig measurable sets

$A_{i}(\epsilon)=\\{\omega\in\Omega\ |\
|X_{i}(\omega)-X(\omega)|\geq\epsilon\\}.$

If for all $\epsilon>0$

$\displaystyle\sum_{i=0}^{\infty}\mathbb{P}(A_{i}(\epsilon))<\infty,$ (7.19)

then $X_{i}\overset{a.s.}{\to}X$.

######  Corollary 7.2.

Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be a sequaence of real valued random
variables. For all $\epsilon>0$ and all $i\in\mathbb{N}$ define also the
followig measurable sets

$A_{i}(\epsilon)=\\{\omega\in\Omega\ |\
|X_{i+1}(\omega)-X_{i}(\omega)|\geq\epsilon\\}.$

If for all $\epsilon>0$

$\displaystyle\sum_{i=0}^{\infty}\mathbb{P}(A_{i}(\epsilon))<\infty,$ (7.20)

then almost surely $\\{X_{i}\\}_{i\in\mathbb{N}}$ is a Cauchy sequence.

######  Theorem 7.2 (Minkowski’s inequality).

Let $p\in[1,\infty)$ and also let $f,g\in\mathcal{L}^{p}$. Then
$f+g\in\mathcal{L}^{p}$ and

$\displaystyle\|f+g\|_{\mathcal{L}^{p}}\leq\|f\|_{\mathcal{L}^{p}}+\|g\|_{\mathcal{L}^{p}}.$
(7.21)

######  Theorem 7.3.

Let $p\in[1,\infty)$ and also let
$\\{f_{n}\\}_{n\in\mathbb{N}}\in\mathcal{L}^{p}$. In addition, suppose that
there exists $g\in\mathcal{L}^{p}$ such that $|f_{n}|<g$ for all
$n\in\mathbb{N}$ and almost surely $\lim_{n\to\infty}f_{n}(\omega)=f(\omega)$.
Then

$\displaystyle
f\in\mathcal{L}^{p}\quad\text{and}\quad\lim_{n\to\infty}\|f_{n}-f\|_{\mathcal{L}^{p}}\
\to\ 0.$ (7.22)

######  Theorem 7.4.

Let $p\in[1,\infty)$ and also let
$\\{f_{n}\\}_{n\in\mathbb{N}}\in\mathcal{L}^{p}$. In addition, let
$f\in\mathcal{L}^{p}$ and suppose that almost surely
$\lim_{n\to\infty}f_{n}(\omega)=f(\omega)$. Then

$\displaystyle\lim_{n\to\infty}\|f_{n}-f\|_{\mathcal{L}^{p}}\ \to\
0\quad\iff\quad\lim_{n\to\infty}\|f_{n}\|_{\mathcal{L}^{p}}\ \to\
\|f\|_{\mathcal{L}^{p}}.$ (7.23)

######  Theorem 7.5.

Let $\\{f_{n}\\}_{n\in\mathbb{N}}\in\mathcal{L}^{p}$ and
$f\in\mathcal{L}^{p}$. Suppose that

$\displaystyle\lim_{n\to\infty}\|f_{n}-f\|_{\mathcal{L}^{p}}\ \to\ 0.$ (7.24)

Then there exists a subsequence $\\{f_{\sigma(n)}\\}_{n\in\mathbb{N}}$ such
that almost surely $\lim_{n\to\infty}f_{\sigma(n)}(\omega)=f(\omega)$.

######  Remark.

Suppose that $\mathbf{X}\coloneqq(X,\mathcal{A},\mu)$ is any finite measure
space. Then Theorem 7.5 remains true if $\mathcal{L}^{p}$ is replaced by
$\mathcal{L}^{p}(\mathbf{X})$.

######  Theorem 7.6 (Egoroff Theorem).

Let $\\{f_{n}\\}_{n\in\mathbb{N}}$ and $f$ be measurable functions on a finite
measure space $\mathbf{X}\coloneqq(X,\mathcal{A},\mu)$. Suppose that
$\lim_{n\to\infty}f_{n}(x)=f(x)$ almost surely. Then given any $\delta>0$
there exists a measurable set $F$ such that $\mu(F)\leq\delta$ and
$\lim_{n\to\infty}f_{n}(x)=f(x)$ uniformly on $X-F$.

######  Theorem 7.7.

Let $\\{f_{n}\\}_{n\in\mathbb{N}}$ be a sequence of measurable functions on a
measure space $\mathbf{X}\coloneqq(X,\mathcal{A},\mu)$. Suppose that we have a
function $f:X\to\mathbb{R}$ such that $\lim_{n\to\infty}f_{n}(x)=f(x)$ almost
surely. Then $f$ is measurable.

######  Theorem 7.8.

Let $p\in\mathbb{R}^{++}$, $\widetilde{\Omega}\subset\Omega$ such that
$\mathbb{P}(\widetilde{\Omega})=1$ and also let
$f:\overline{\Omega}\to\mathbb{R}$. Assume that $f$ has the following
properties

  1. (1)

$f(t,\cdot)\in\mathcal{L}^{p}$ for all $t\in\mathcal{T},$

  2. (2)

$f(\cdot,\omega)$ is continuous almost surely,

  3. (3)

$|f(t,\omega)|<g(\omega)$ for all
$(t,\omega)\in\mathcal{T}\times\widetilde{\Omega}$ and some positive
$g\in\mathcal{L}^{p}.$

If we now define for all $t\in\mathcal{T}$ the following function

$\displaystyle h(t)\coloneqq\int_{\Omega}|f(t)|^{p}d\mathbb{P}.$ (7.25)

Then $h$ is continuous.

######  Theorem 7.9.

Let $1\leq q\leq p$ be some real numbers and also let $f\in\mathcal{L}^{p}$.
Then

$\displaystyle\mathcal{L}^{p}$ $\displaystyle\subset\mathcal{L}^{q},$ (7.26)
$\displaystyle\|f\|_{\mathcal{L}^{q}}$
$\displaystyle\leq\|f\|_{\mathcal{L}^{p}}.$ (7.27)

######  Remark.

Note that Theorem 7.9 remains true if $\mathcal{L}^{p}$ is replaced by
$\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})$.

######  Theorem 7.10.

Let $p\in[1,\infty)$ and also let
$\\{f_{n}\\}_{n\in\mathbb{N}}\in\mathcal{L}^{p}$. In addition, let
$f\in\mathcal{L}^{p}$. Then

  1. (1)

$\|f_{n}-f\|_{\mathcal{L}^{p}}\ \to\ 0\ \text{as}\
n\to\infty\quad\implies\quad f_{n}\xrightarrow{\mathbb{P}}f\ \text{as}\
n\to\infty$,

  2. (2)

$\\{f_{n}\\}_{n\in\mathbb{N}}\ \text{is Cauchy in}\
\mathcal{L}^{p}\quad\quad\implies\quad\\{f_{n}\\}_{n\in\mathbb{N}}\ \text{is
Cauchy in}\ \mathbb{P}$.

######  Theorem 7.11.

Let $\\{f_{n}\\}_{n\in\mathbb{N}}$ be a sequence of measurable functions from
$\Omega$ to $\mathbb{R}$ such that $\\{f_{n}\\}_{n\in\mathbb{N}}$ is Cauchy in
$\mathbb{P}$. Then there exists a measurable function $f:\Omega\to\mathbb{R}$
such that

$\displaystyle f_{n}\xrightarrow{\mathbb{P}}f\ \text{as}\ n\to\infty,$ (7.28)

and almost surely $f$ is unique.

######  Theorem 7.12.

Let $f:\Omega\to\mathbb{R}$ be a measurable function. If there exist
$g\in\mathcal{L}^{p}$ such that $|f|\leq g$ then $f\in\mathcal{L}^{p}$.

######  Remark.

Note that Theorem 7.12 remains true if $\mathcal{L}^{p}$ is replaced by
$\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})$.

######  Theorem 7.13 (Grönwall Inequality).

Suppose that $\alpha,\beta\in\mathbb{R}$ are constants and
$f\in\mathcal{L}^{1}(\mathbf{M},\mathbf{M}^{\mathbb{R}})$ satisfies the
following inequality

$\displaystyle f(t)\leq\alpha+\beta\int_{0}^{t}f(s)ds,\quad\forall
t\in\mathcal{T}.$ (7.29)

Then

$\displaystyle f(t)\leq\alpha e^{\beta t},\quad\forall t\in\mathcal{T}.$
(7.30)

######  Theorem 7.14 (Jensen Inequality).

Let $\Lambda:\mathbb{R}^{+}\to\mathbb{R}^{+}$ and
$V:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a concave and a convex function
respectively. Suppose that $w,u\in\mathcal{L}^{1}_{+}$ and
$uw\in\mathcal{L}^{1}$. Then $\Lambda(u)w\in\mathcal{L}^{1}$ and

$\displaystyle\frac{\int_{\Omega}\Lambda(u)wd\mathbb{P}}{\int_{\Omega}wd\mathbb{P}}$
$\displaystyle\leq\Lambda\bigg{(}\frac{\int_{\Omega}uwd\mathbb{P}}{\int_{\Omega}wd\mathbb{P}}\bigg{)},$
(7.31) $\displaystyle
V\bigg{(}\frac{\int_{\Omega}uwd\mathbb{P}}{\int_{\Omega}wd\mathbb{P}}\bigg{)}$
$\displaystyle\leq\frac{\int_{\Omega}V(u)wd\mathbb{P}}{\int_{\Omega}wd\mathbb{P}}.$
(7.32)

######  Theorem 7.15 (Fubini Theorem).

Let $\mathbf{X}\coloneqq(X,\mathcal{A},\mu)$ and
$\mathbf{Y}\coloneqq(Y,\mathcal{B},\eta)$ be two $\sigma-\text{finite}$
measure spaces. Let

$\displaystyle\mathbf{XY}\coloneqq(X\times
Y,\mathcal{A}\times\mathcal{B},\mu\times\eta)$ (7.33)

be a product measure space and let $u:X\times Y\to\mathbb{R}$ be
$\mathcal{A}\times\mathcal{B}$ measurable. If at least one of the following
integrals is finite

$\displaystyle\int_{X\times Y}|u|d(\mu\times\eta),\quad\int_{X}\int_{Y}|u|d\mu
d\eta,\quad\int_{Y}\int_{X}|u|d\eta d\mu,$ (7.34)

then all three integrals are finite, $u\in\mathcal{L}^{1}(\mathbf{XY})$ and

  1. (1)

$x\to u(x,y)\in\mathcal{L}^{1}(\mathbf{X})$, $\eta-\text{almost everywhere}$,

  2. (2)

$y\to u(x,y)\in\mathcal{L}^{1}(\mathbf{Y})$, $\mu-\text{almost everywhere}$,

  3. (3)

$y\to\int_{X}u(x,y)d\mu(x)\in\mathcal{L}^{1}(\mathbf{Y})$,

  4. (4)

$x\to\int_{Y}u(x,y)d\eta(y)\in\mathcal{L}^{1}(\mathbf{X})$,

  5. (5)

$\int_{X\times Y}|u|d(\mu\times\eta)=\int_{X}\int_{Y}|u|d\mu
d\eta=\int_{Y}\int_{X}|u|d\eta d\mu$.

######  Remark.

Note that Theorem 7.15 above remains true for Banach space valued maps. For
details see [20].

######  Remark.

It follows that if $f\in\mathcal{L}^{1}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})$
then by Theorem 7.15 function $\displaystyle
t\to\int_{\Omega}f(t)d\mathbb{P},$ (7.35) is $\mathscr{B}(\mathcal{T})$
measurable.

######  Theorem 7.16.

Let $X$ be a Banach space and let $Y$ be a dense subset of $X$. Moreover
suppose that $X$ is compact and $f:X\to\mathbb{R}$ is continuous. Then

$\displaystyle\sup\\{f(y)\ |\ y\in Y\\}=\sup\\{f(x)\ |\ x\in X\\}.$ (7.36)

######  Theorem 7.17.

For some $n\in\mathbb{N}$, suppose that $x_{k}\geq 0$ for all $1\leq k\leq n$
and $p\geq 1$. Then

$\displaystyle\bigg{(}\sum_{k=1}^{n}x_{k}\bigg{)}^{p}\leq
n^{p-1}\sum_{k=1}^{n}x_{k}^{p}.$ (7.37)

######  Theorem 7.18 (Young Inequality).

Suppose that $p,q\in(1,\infty)$ are such that $\frac{1}{p}+\frac{1}{q}=1$ and
$x,y\in\mathbb{R}^{+}$. Then

$\displaystyle xy\leq\frac{x^{p}}{p}+\frac{y^{q}}{q}.$ (7.38)

Moreover equality in (7.38) above occures if and only if $y=x^{p-1}$.

###  7.2 Martingales and Wiener Process in $\mathbb{R}$

In this section we work with a real valued Wiener process $W$ defined on
$\mathbf{M}\mathbf{P}$ and assume that a filtration
$\mathbb{F}\coloneqq\\{\mathcal{F}_{t}\\}_{t\in\mathcal{T}}$ is suitably
chosen so that the following properties are satisfied;

  1. (1)

For all $t\in\mathcal{T}$, $W(t)$ is $\mathcal{F}_{t}$ measurable,

  2. (2)

For all $s\leq t\in\mathcal{T}$, $W(t)-W(s)$ is independent of
$\mathcal{F}_{s}$.

Unless stated otherwise, information in this subsection is based on [12].

######  Definition 7.4.

For all $p\in\mathbb{R}^{++}$ we introduce the folowing spaces of stochastic
processes.

$\displaystyle
L^{p}_{ad}\coloneqq\\{\xi\in\mathcal{L}^{p}(\mathbf{MP},\mathbf{M}^{\mathbb{R}})\
|\ \xi\ \text{is adapted to}\ \mathbb{F}.\\},$ (7.39)

and a space

$\displaystyle\mathcal{M}_{\mathbb{F}}\coloneqq\left\\{\xi\in\mathcal{S}(\mathbf{M}^{\mathbb{R}})\
\begin{tabular}[]{|l}\
$\xi_{t}\in\mathcal{L}(\mathbf{P},\mathbf{M}^{\mathbb{R}}),\ \forall
t\in\mathcal{T}$.\\\ \ $\xi\ \text{is adapted to}\
\mathbb{F}$.\end{tabular}\right\\}$ (7.42)

######  Definition 7.5.

  1. (1)

$\xi\in\mathcal{M}_{\mathbb{F}}$ is called a martingale with respect to
$\mathbb{F}$ if almost surely for all $s\leq t\in\mathcal{T}$

$\displaystyle\mathbb{E}[\xi_{t}|\mathcal{F}_{s}]=\xi_{s}.$ (7.43)

  2. (2)

$\xi\in\mathcal{S}(\mathbf{M}^{\mathbb{R}})$ is called square integrable if
$\xi_{t}\in\mathcal{L}^{2}(\mathbf{P},\mathbf{M}^{\mathbb{R}}),\ \forall
t\in\mathcal{T}$.

  3. (3)

$\xi\in\mathcal{S}(\mathbf{M}^{\mathbb{R}})$ is called predictable if
$\xi\in\mathcal{M}(\overline{\mathbf{MP}},\mathbf{M}^{\mathbb{R}})$.

######  Theorem 7.19.

Let $\xi$ be a right continuous, square integrable martingale with left-hand
limits. Then there is a unique decomposition

$\displaystyle\xi^{2}_{t}=L_{t}+A_{t},\ \forall t\in\mathcal{T},$ (7.44)

where $L$ is a right continuous martingale with left-hand limits and $A$ is a
predictable, right continuous, and increasing process such that $A(0)=0$ and
$A_{t}\in\mathcal{L}(\mathbf{P},\mathbf{M}^{\mathbb{R}}),\ \forall
t\in\mathcal{T}$.

######  Remark.

Process $A$ found by Theorem 7.19 will be called a Meyer process in this text
and the following abbreviation will be used
$\displaystyle\langle\xi\rangle_{t}=A_{t},\ \forall t\in\mathcal{T}.$ (7.45)
Moreover, one can show that $\displaystyle\langle W\rangle_{t}=t,\ \forall
t\in\mathcal{T}.$ (7.46)

######  Theorem 7.20.

Suppose that $\xi\in L^{2}_{ad}$ and define a stochastic process $X$ in the
following way

$\displaystyle X_{t}\coloneqq\int_{0}^{t}\xi(s)dW(s).$ (7.47)

Then

  1. (A)

$X$ is a martingale with respect to $\mathbb{F}$ and trajectories of $X$ are
almost surely continuous.

  1. (B)

For all $t\in\mathcal{T}$

    1. (1)

$\mathbb{E}\bigg{[}\mathop{{\int}}_{\\!\\!\\!0}^{t}\xi(s)dW(s)\bigg{]}=0$,

    2. (2)

$\mathbb{E}\bigg{[}\bigg{|}\mathop{{\int}}_{\\!\\!\\!0}^{t}\xi(s)dW(s)\bigg{|}^{2}\bigg{]}=\mathop{{\int}}_{\\!\\!\\!0}^{t}\mathbb{E}\bigg{[}|\xi(s)|^{2}\bigg{]}ds$,

    3. (3)

$\langle X\rangle_{t}=\mathop{{\int}}_{\\!\\!\\!0}^{t}|\xi(s)|^{2}d\langle
W\rangle_{s}$.

Following theorem is a usefull result from [15].

######  Theorem 7.21 (Burkholder, Davis and Gundy Inequality).

Let $X$ be a continuous martingale. Then for all $t\in\mathcal{T}$ and all
$p\in(0,\infty)$

$\displaystyle\mathbb{E}\bigg{[}\sup\bigg{\\{}|X_{s}|^{p}\ \bigg{|}\ 0\leq
s\leq t\bigg{\\}}\bigg{]}=\mathbb{E}\bigg{[}\bigg{(}\langle
X\rangle_{t}\bigg{)}^{\frac{p}{2}}\bigg{]}.$ (7.48)

######  Definition 7.6.

Suppose that $f\in L^{2}_{ad}$, $g\in L^{1}_{ad}$ and let $\xi_{0}$ be a
$\mathcal{F}_{0}$ measurable random variable. An Itô process is a real valued
stochastic process $\xi$ satisfying

$\displaystyle\xi_{t}$
$\displaystyle=\xi_{0}+\int_{0}^{t}g(s)ds+\int_{0}^{t}f(s)dW(s),\forall
t\in\mathcal{T}.$ (7.49)

######  Theorem 7.22 (Itô Lemma).

Let $\xi$ be an Itô process satisfying equation (7.49) above and suppose that
$\theta:\mathbb{R}^{2}\to\mathbb{R}$ is a continuous function such that all
$\frac{\partial\theta}{\partial t}$, $\frac{\partial\theta}{\partial x}$ and
$\frac{\partial^{2}\theta}{\partial x^{2}}$ are continuous functions from
$\mathbb{R}^{2}$ to $\mathbb{R}$. Then $\theta\circ\xi$ is an Itô process
satisfying

$\displaystyle\theta(t,\xi_{t})=\theta(0,\xi_{0})+\int_{0}^{t}\mathcal{K}(s,\xi_{s})ds+\int_{0}^{t}\frac{\partial\theta}{\partial
x}(s,\xi_{s})f(s)dW(s),\ \forall t\in\mathcal{T},$ (7.50)

where

$\displaystyle\mathcal{K}(t,\xi_{t})\coloneqq\frac{\partial\theta}{\partial
t}(t,\xi_{t})+\frac{\partial\theta}{\partial
x}(t,\xi_{t})g(t)+\frac{1}{2}\frac{\partial^{2}\theta}{\partial
x^{2}}(t,\xi_{t})f^{2}(t),\ \forall t\in\mathcal{T}.$ (7.51)

###  7.3 Deterministic Ovsjannikov Equation

Unless stated otherwise, information in this subsection is based on [1, 2, 5,
6].
In this subsection we would like to address the problem of finding a unique
continuous infinite time solution f satisfying the following integral equation

$\displaystyle f(t)=x_{\underline{\mathfrak{a}}}+\int_{0}^{t}F(f(s))ds,$
(7.52)

where we let
$\mathbf{X}\coloneqq\\{X_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}}$ be a
suitable scale of Banach spaces, $x_{\underline{\mathfrak{a}}}\in
X_{\underline{\mathfrak{a}}}$ and $F\in\mathcal{O}(\mathbf{X},L,q)$ be an
Ovsjannikov map on $\mathbf{X}$. The main result of this appendix, that is
existence and uniqueness of $f$, is summarised in the Theorem 7.27 bellow.

We will now show how the proof of Theorem 7.27 can be obtained. We start with
a result that will be needed later on.

######  Lemma 7.1.

Supose that $A,B\in\mathbb{R}^{+}$, $p\in\mathbb{N}$ and
$q\in[0,\frac{1}{p})$. Then

$\displaystyle\sum_{n=0}^{\infty}\frac{\sqrt[p]{A^{n}\
}}{B^{qn}}\frac{n^{qn}}{\sqrt[p]{n!\ }}<\infty.$ (7.53)

###### Proof.

We will consider the following two cases separately.

  1. (1)

$q\in(0,\frac{1}{2})$.
Then by analyzing ratio of terms of series (7.53) we get

$\displaystyle\frac{\sqrt[p]{A^{n+1}}}{B^{q(n+1)}}\frac{(n+1)^{q(n+1)}}{\sqrt[p]{(n+1)!}}\bigg{/}\frac{\sqrt[p]{A^{n}}}{B^{qn}}\frac{n^{qn}}{\sqrt[p]{n!}}$
$\displaystyle=\frac{\sqrt[p]{A}}{B^{q}}(n+1)^{qn+q-\frac{1}{p}}\frac{1}{n^{qn}},$
(7.54)
$\displaystyle=\frac{\sqrt[p]{A}}{B^{q}}\frac{1}{(n+1)^{\frac{1}{p}-q}}\bigg{(}1+\frac{1}{n}\bigg{)}^{qn}_{.}$
(7.55)

Now since

$\displaystyle\lim_{n\to\infty}\frac{\sqrt[p]{A}}{B^{q}}\frac{1}{(n+1)^{\frac{1}{p}-q}}\bigg{(}1+\frac{1}{n}\bigg{)}^{qn}$
$\displaystyle=\frac{\sqrt[p]{A}}{B^{q}}\bigg{(}\lim_{n\to\infty}\frac{1}{(n+1)^{\frac{1}{p}-q}}\bigg{)}\bigg{(}\lim_{n\to\infty}\bigg{(}1+\frac{1}{n}\bigg{)}^{qn}\bigg{)},$
$\displaystyle=\frac{\sqrt[p]{A}}{B^{q}}(0)(e^{q}),$ $\displaystyle=0.$ (7.56)

Hence we conclude by ratio test that when $q\in(0,\frac{1}{2})$ series (7.53)
converges.

  2. (2)

$q=0$.
Then series (7.53) reduces to $\sum_{n=0}^{\infty}\sqrt[p]{\frac{A^{n}}{n!}\
}$. By ratio test, as shown above, it is clear that when $q=0$ series (7.53)
also converges hence the proof is complete.

∎

Continuing, we fix some $T\in\mathbb{R}^{+}$ and introduce a family
$\mathbf{Z}\coloneqq\\{Z_{\mathfrak{a}}\\}_{\mathfrak{a}\in\mathcal{A}}$ where
$Z_{\mathfrak{a}}$ is the classical space of continuous $X_{\mathfrak{a}}$
valued maps. That is for all $\mathfrak{a}\in\mathcal{A}$ we define

$\displaystyle Z_{\mathfrak{a}}\coloneqq\mathcal{C}([0,T],X_{\mathfrak{a}}).$
(7.57)

Now, for all $\alpha<\beta\in\mathcal{A}$ and $f\in Z_{\alpha}$ the following
consequences are immediate.

  1. (1)

$\mathbf{Z}$ is a family of Banach spaces,

  2. (2)

$Z_{\alpha}\prec Z_{\beta}$, (7.58)

  3. (3)

$||f||_{Z_{\beta}}\leq||f||_{Z_{\alpha}}$.

Therefore, from the list (2) above we can conclude, using the Definition 2.1,
that $\mathbf{Z}$ is the scale. Continuing, we let

$\displaystyle\overline{\mathbf{Z}}\coloneqq\bigcup_{\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})}Z_{\mathfrak{a}},$
(7.59)

and define a map $\mathcal{I}:\overline{\mathbf{Z}}\to
Z_{\overline{\mathfrak{a}}}$ by letting for all $t\in[0,T]$ and all
$f\in\overline{\mathbf{Z}}$

$\displaystyle\mathcal{I}(f)(t)\coloneqq
x_{\underline{\mathfrak{a}}}+\int_{0}^{t}F(f(s))ds.$ (7.60)

The following result can now be proved.

######  Theorem 7.23.

$\mathcal{I}\in\mathcal{O}(\mathbf{X},LT,q)$.

###### Proof.

Fix $\alpha<\beta\in\mathcal{A}$, $f,g\in Z_{\alpha}$ and $t\in[0,T]$. We now
check that the integral map $\mathcal{I}$ satisfies the Definition 2.2. We
begin by using the definition of Bochner integral and the fact that
$F\in\mathcal{O}(\mathbf{X},L,q)$ to conclude that
$\mathcal{I}|_{Z_{\alpha}}:Z_{\alpha}\to Z_{\beta}$. Moreover we see that

$\displaystyle||\mathcal{I}(f)(t)-\mathcal{I}(g)(t)||_{X_{\beta}}$
$\displaystyle\leq\int_{0}^{t}||F(f(s))-F(g(s))||_{X_{\beta}}ds,$ (7.61)
$\displaystyle\leq\frac{L}{(\beta-\alpha)^{q}}\int_{0}^{t}||f(s)-g(s)||_{X_{\alpha}}ds,$
(7.62)
$\displaystyle\leq\frac{L}{(\beta-\alpha)^{q}}\int_{0}^{t}||f-g||_{Z_{\alpha}}ds.$
(7.63)

Therefore we see that

$\displaystyle||\mathcal{I}(f)-\mathcal{I}(g)||_{Z_{\beta}}$
$\displaystyle\leq\frac{L}{(\beta-\alpha)^{q}}\int_{0}^{T}||f-g||_{Z_{\alpha}}ds,$
(7.64) $\displaystyle\leq\frac{LT}{(\beta-\alpha)^{q}}||f-g||_{Z_{\alpha}},$
(7.65)

hence the proof is complete. ∎

We now would like to define something called an itterated or a composite map.
That is for all $n\in\mathbb{N}$ we define

$\displaystyle\mathcal{I}^{n}\coloneqq\overbrace{\mathcal{I}\circ\mathcal{I}\circ\cdots\circ\mathcal{I}}^{n\text{\
times}},$ (7.66)

and let $\mathcal{T}^{0}$ be the identity map from
$Z_{\underline{\mathfrak{a}}}$ to $Z_{\underline{\mathfrak{a}}}$. Letting

$\displaystyle\underline{\mathbf{Z}}\coloneqq\bigcap_{\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})}Z_{\mathfrak{a}},$
(7.67)

our next result shows that for all $n\in\mathbb{N}$ the composite map
$\mathcal{I}^{n}$ is well defined.

######  Theorem 7.24.

For all $n\in\mathbb{N}^{0}$

$\displaystyle\mathcal{I}^{n}:Z_{\underline{\mathfrak{a}}}\to\underline{\mathbf{Z}}.$
(7.68)

###### Proof.

We prove this statement by induction. For $n=0$ the statement (7.68) is
trivially true because
$Z_{\underline{\mathfrak{a}}}\subset\underline{\mathbf{Z}}$. Now suppose that
induction hypothesis holds for some $n\geq 0$. Fix arbitrary
$\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})$ and
$p\in(\underline{\mathfrak{a}},\mathfrak{a})$. Observe that induction
hypothesis implies that $\mathcal{I}^{n}:Z_{\underline{\mathfrak{a}}}\to
Z_{p}$. However because $\mathcal{I}\in\mathcal{O}(\mathbf{X},LT,q)$ we know
that $\mathcal{I}|_{Z_{p}}:Z_{p}\to Z_{\mathfrak{a}}$ hence by composition
$\mathcal{I}\circ\mathcal{I}^{n}$ it follows that
$\mathcal{I}^{n+1}:Z_{\underline{\mathfrak{a}}}\to Z_{\mathfrak{a}}$ and since
$\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})$ is
arbitrary the proof is complete. ∎

######  Remark.

Observe that Theorem 7.24 shows that if $f\in Z_{\underline{\mathfrak{a}}}$
then the sequence $\\{\mathcal{I}^{n}(f)\\}_{n=0}^{\infty}$ belogs to
$Z_{\mathfrak{a}}$ for all
$\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})$.

Let us now, for a moment, consider some fixed $t_{0}\in[0,T]$,
$\alpha<\beta\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})$ and $f\in
Z_{\underline{\mathfrak{a}}}$. Moreover let us consider arbitrary
$n\in\mathbb{N}$ and a partition $\\{\psi_{i}\\}_{i=0}^{n}$ of
$[\alpha,\beta]$ into $n$ intervals of equal length. That is
$\psi_{0}=\alpha$, $\psi_{n}=\beta$ and $\psi_{i+1}-\psi_{i}=\frac{b-a}{n}$
for all $0\leq i\leq n$. Letting

$\displaystyle
K_{n}^{n+1}(t)=\mathcal{I}^{n}(f)(t_{0})-\mathcal{I}^{n+1}(f)(t_{0}),\ \forall
t\in[0,t_{0}],$ (7.69)

we see from Theorem 7.23 and 7.24 that

$\displaystyle||K_{n}^{n+1}(t_{0})||_{X_{\psi_{n}}}$
$\displaystyle\leq\frac{L}{(\psi_{n}-\psi_{n-1})^{q}}\int_{0}^{t_{0}}||K_{n-1}^{n}(t_{1})||_{X_{\psi_{n-1}}}dt_{1},$
$\displaystyle\leq\frac{L}{(\psi_{n}-\psi_{n-1})^{q}}\frac{L}{(\psi_{n-1}-\psi_{n-2})^{q}}\int_{0}^{t_{0}}\int_{0}^{t_{1}}||K_{n-2}^{n-1}(t_{2})||_{X_{\psi_{n-2}}}dt_{2}dt_{1},$
$\displaystyle\leq
L^{n}\bigg{(}\frac{\beta-\alpha}{n}\bigg{)}^{-qn}\int_{0}^{t_{0}}\int_{0}^{t_{1}}\cdots\int_{0}^{t_{n-1}}||K_{0}^{1}(t_{n})||_{X_{\psi_{0}}}dt_{n}dt_{n-1}\cdots
dt_{1},$ (7.70)
$\displaystyle\leq\frac{L^{n}}{(\beta-\alpha)^{qn}}n^{qn}||K_{0}^{1}||_{Z_{\psi_{0}}}\int_{0}^{t_{0}}\int_{0}^{t_{1}}\cdots\int_{0}^{t_{n-1}}\
dt_{n}dt_{n-1}\cdots dt_{1},$
$\displaystyle\leq\frac{L^{n}t_{0}^{n}}{(\beta-\alpha)^{qn}}\frac{n^{qn}}{n!}||K_{0}^{1}||_{Z_{\psi_{0}}}.$

Hence, defining recursively
$\mathcal{H}^{n}:\mathcal{C}([0,T],\mathbb{R})\to\mathcal{C}([0,T],\mathbb{R})$
for all $n\in\mathbb{N}^{0}$ via formula

$\displaystyle\mathcal{H}^{n}(t,f)\coloneqq\begin{cases}\begin{tabular}[]{l|l}$f(t)$&\
$t\in[0,T]\land n=0$,\\\ $\int_{0}^{t}f(s)ds$&\ $t\in[0,T]\land n=1$,\\\
$\int_{0}^{t}\mathcal{H}^{n-1}(s,f)ds$&\ $t\in[0,T]\land
n>1$.\end{tabular}\end{cases}$ (7.71)

we see from inequalities (7.70) that the following result can be formulateed
and proved.

######  Theorem 7.25.

Suppose $\alpha<\beta\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})$
and $f,g\in Z_{\underline{\mathfrak{a}}}$. Then for all $n\in\mathbb{N}$

$\displaystyle||\mathcal{I}^{n}(f)-\mathcal{I}^{n+1}(g)||_{Z_{\beta}}\leq\frac{L^{n}T^{n}}{(\beta-\alpha)^{qn}}\frac{n^{qn}}{n!}||f-\mathcal{I}(g)||_{Z_{\alpha}}.$
(7.72)

###### Proof.

Fixing $t\in[0,T]$ we prove by induction that

$\displaystyle||\mathcal{I}^{n}(f)(t)-\mathcal{I}^{n+1}(g)(t)||_{X_{\beta}}\leq\frac{L^{n}}{(\beta-\alpha)^{qn}}n^{qn}\mathcal{H}^{n}(t,||f-\mathcal{I}(g)||_{X_{\alpha}}),$
(7.73)

from where inequality (7.72) follows directly. Clearly case $n=1$ follows
immediately from the Theorem 7.23. Precisely spaking inequality (7.62) shows
that the induction hypothesis holds for $n=1$. Now, suppose that the induction
hypothesis holds for some $n\geq 1$. Chosing $\psi\in(\alpha,\beta)$ such that
$\beta-\psi=\frac{\beta-\alpha}{n+1}$ we see, using Theorem 7.23, that

$\displaystyle||\mathcal{I}^{n+1}(f)(t)-\mathcal{I}^{n+2}(g)(t)||_{X_{\beta}}\leq\frac{L}{(\beta-\psi)}\int_{0}^{t}||\mathcal{I}^{n}(f)(s)-\mathcal{I}^{n+1}(g)(s)||_{X_{\psi}}ds.$
(7.74)

Hence letting

$\displaystyle\mathbf{A}\coloneqq||f-\mathcal{I}(g)||_{X_{\alpha}},$ (7.75)

and applying the induction hypothesis we get

$\displaystyle||\mathcal{I}^{n+1}(f)(t)-\mathcal{I}^{n+2}(g)(t)||_{X_{\beta}}$
$\displaystyle\leq\frac{L}{(\beta-\psi)^{q}}\frac{L^{n}}{(\psi-\alpha)^{qn}}n^{qn}\int_{0}^{t}\mathcal{H}^{n}(s,\mathbf{A})ds,$
$\displaystyle\leq\frac{L^{n+1}}{(\beta-\psi)^{q}(\psi-\alpha)^{qn}}n^{qn}\mathcal{H}^{n+1}(t,\mathbf{A}),$
$\displaystyle\leq
L^{n+1}\bigg{(\frac{\beta-\alpha}{n+1}}\bigg{)}^{-q}\bigg{(\frac{n(\beta-\alpha)}{n+1}}\bigg{)}^{-qn}n^{qn}\mathcal{H}^{n+1}(t,\mathbf{A}),$
(7.76)
$\displaystyle\leq\frac{L^{n+1}}{(\beta-\alpha)^{q(n+1)}}\frac{(n+1)^{q(n+1)}}{n^{qn}}n^{qn}\mathcal{H}^{n+1}(t,\mathbf{A}),$
$\displaystyle\leq\frac{L^{n+1}}{(\beta-\alpha)^{q(n+1)}}(n+1)^{q(n+1)}\mathcal{H}^{n+1}(t,\mathbf{A}).$

Hence

$\displaystyle||\mathcal{I}^{n+1}(f)(t)-\mathcal{I}^{n+2}(g)(t)||_{X_{\beta}}\leq\frac{L^{n+1}}{(\beta-\alpha)^{q(n+1)}}(n+1)^{q(n+1)}\mathcal{H}^{n+1}(t,||f-\mathcal{I}(g)||_{X_{\alpha}}),$
(7.77)

and the proof is complete. ∎

######  Remark.

It is clear from the definition of the composite map $\mathcal{I}^{n}$ that
the Theorem 7.25 is trivially true for $n=0$. Moreover it is essential that
$\alpha\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})$ because it is
possible that $\mathcal{I}(f)$ does not belogn to
$Z_{\underline{\mathfrak{a}}}$.

Theorem 7.25 puts us in a position to prove the following.

######  Theorem 7.26.

Suppose that $q<1$ and $F\in\mathcal{O}(\mathbf{X},L,q)$. Then there exists a
unique element $\phi\in\underline{Z}$ such that $\mathcal{I}(\phi)=\phi$.
Moreover if
$\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})$ and $f\in
Z_{\underline{\mathfrak{a}}}$ then

$\displaystyle\overbrace{\ \lim_{n\to\infty}\mathcal{I}^{n}(f)\ }^{\text{in}\
Z_{\mathfrak{a}}}=\phi.$ (7.78)

###### Proof.

Fix $f\in Z_{\underline{\mathfrak{a}}}$ and
$\mathfrak{a}\in(\underline{\mathfrak{a}},\overline{\mathfrak{a}})$. Fix also
an arbitrary $\gamma\in(\underline{\mathfrak{a}},\mathfrak{a})$ and using
theorem 7.25 observe that for all $m\geq n\in\mathbb{N}$ we have