11institutetext: Department of Mathematics, University of Milan,
via Saldini 50, 10133 Milan Italy
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# A set–valued framework for birth–and–growth process

Giacomo Aletti  Enea G. Bongiorno  Vincenzo Capasso

###### Abstract

We propose a set–valued framework for the well–posedness of birth–and–growth
process. Our birth–and–growth model is rigorously defined as a suitable
combination, involving Minkowski sum and Aumann integral, of two very general
set–valued processes representing nucleation and growth respectively. The
simplicity of the used geometrical approach leads us to avoid problems arising
by an analytical definition of the front growth such as boundary regularities.
In this framework, growth is generally anisotropic and, according to a
mesoscale point of view, it is not local, i.e. for a fixed time instant,
growth is the same at each space point.

## Introduction

Nucleation and growth processes arise in several natural and technological
applications (cf. [8, 7] and the references therein) such as, for example,
solidification and phase–transition of materials, semiconductor crystal
growth, biomineralization, and DNA replication (cf., e.g., [15]).

A _birth–and–growth process_ is a RaCS family given by
$\Theta_{t}=\bigcup_{n:T_{n}\leq t}\Theta_{T_{n}}^{t}(X_{n})$, for
$t\in\mathbb{R}_{+}$, where $\Theta^{t}_{T_{n}}\left(X_{n}\right)$ is the RaCS
obtained as the evolution up to time $t>T_{n}$ of the germ born at (random)
time $T_{n}$ in (random) location $X_{n}$, according to some growth model.
An analytical approach is often used to model birth–and–growth process, in
particular it is assumed that the growth is driven according to a non–negative
normal velocity, i.e. for every instant $t$, a border point
$x\in\partial\Theta_{t}$ “grows” along the outward normal unit (e.g. [3, 22,
13, 4, 6, 5, 11]). Thus, growth is pointwise isotropic; i.e. given a point
belonging $\partial\Theta_{t}$, the growth rate is independently from outward
normal direction. Note that, the existence of the outward normal vector
imposes a regularity condition on $\partial\Theta_{t}$ and also on the
nucleation process (it cannot be a point process).

This paper is an attempt to offer an original alternative approach based on a
purely geometric stochastic point of view, in order to avoid regularity
assumptions describing birth–and–growth process. In particular, Minkowski sum
(already employed in [19] to describe self–similar growth for a single convex
germ) and Aumann integral are used here to derive a mathematical model of such
process. This model, that emphasizes the geometric growth without regularity
assumptions on $\partial\Theta_{t}$, is rigorously defined as a suitable
combination of two very general set–valued processes representing nucleation
$\left\\{B_{t}\right\\}_{t\in[t_{0},T]}$ and growth
$\left\\{G_{t}\right\\}_{t\in[t_{0},T]}$ respectively

$\begin{array}[]{rl}\Theta_{t}=&\left(\Theta_{t_{0}}\oplus\int_{t_{0}}^{t}G_{s}ds\right)\cup\bigcup_{s\in[t_{0},t]}dB_{s}\\\
d\Theta_{t}=&\oplus G_{t}dt\cup dB_{t}\qquad\textrm{ or
}\qquad\Theta_{t+dt}=(\Theta_{t}\oplus G_{t}dt)\cup dB_{t}.\end{array}$

Roughly speaking, increment $d\Theta_{t}$, during an infinitesimal time
interval $dt$, is an enlargement due to an infinitesimal Minkowski addend
$G_{t}dt$ followed by the union with the infinitesimal nucleation $dB_{t}$.
As a consequence of Minkowski sum definition, for every instant $t$, each
point $x\in\Theta_{t}$ (and then each point $x\in\partial\Theta_{t}$) grows up
by $G_{t}dt$ and no regularity border assumptions are required. Then we deal
with _not–local_ growth; i.e. growth is the same Minkowski addend for every
$x\in\Theta_{t}$. Nevertheless, under mesoscale hypothesis we can only
consider constant growth region as described, for example, in [6]. On the
other hand, growth is anisotropic whenever $G_{t}$ is not a ball.

The aim of this paper is to ensure the well–posedness of such a model and,
hence, to show that above “integral” and “differential” notations are
meaningful.
In view of well–posedness, in [1], the authors show how the model leads to
different and significant statistical results.

The article is organized as follows. Section 0.1 contains some assumptions
about (random) closed sets and their basilar properties. Model assumptions are
collected in Section 0.2 and integrability properties of growth process are
studied in Section 0.3. For the sake of simplicity, we present, in Section
0.4, main results of the paper (that imply well-posedness of the model),
whilst correspondent proofs are in Section 0.4.1. At the last, Section 0.5
proposes a discrete time point of view, also justifying integral and
differential notations.

##  0.1 Preliminary results

Let $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, $\mathbb{R}_{+}$ be the sets of
all non–negative integer, integer, real and non–negative real numbers
respectively. Let $\mathfrak{X}$, ${\mathfrak{X}^{*}}$, $B^{*}_{1}$ be a
Banach space, its dual space and the unit ball of the dual space centered in
the origin respectively. We shall consider

$\begin{array}[]{llcll}\mathfrak{P}^{\,0}(\mathfrak{X})&=\textrm{ the family
of all subsets of
}\mathfrak{X},&&\mathfrak{P}(\mathfrak{X})&=\mathfrak{P}^{\,0}(\mathfrak{X})\setminus\\{\emptyset\\}\\\
\mathbb{F}^{\,0}(\mathfrak{X})&=\textrm{ the family of all closed subsets of
}\mathfrak{X},&&\mathbb{F}(\mathfrak{X})&=\mathbb{F}^{\,0}(\mathfrak{X})\setminus\\{\emptyset\\}.\end{array}$

The suffixes $c$ and $b$ denote convexity and boundedness properties
respectively (e.g. $\mathbb{F}^{\,0}_{bc}(\mathfrak{X})$ denotes the family of
all closed, bounded and convex subsets of $\mathfrak{X}$).

For all $A,B\in\mathfrak{P}^{\,0}(\mathfrak{X})$ and
$\alpha\in\mathbb{R}_{+}$, let us define

$\begin{array}[]{rll}A+B=&\left\\{a+b:a\in A,\ b\in B\right\\}=\bigcup_{b\in
B}b+A,&\textrm{(Minkowski Sum)}\\\ \alpha\cdot A=&\alpha A=\left\\{\alpha
a:a\in A\right\\},&\textrm{(Scalar Product)}\end{array}$

By definition, $\forall A\in\mathfrak{P}^{\,0}(\mathfrak{X})$,
$\alpha\in\mathbb{R}_{+}$, we have $\emptyset+A=\emptyset=\alpha\emptyset$. It
is well known that $+$ is a commutative and associative operation with a
neutral element but $(\mathfrak{P}(\mathfrak{X}),+)$ is not a group (cf.
[20]). The following relations are useful in the sequel (see [21]): for all
$\forall A,B,C\in\mathfrak{P}(\mathfrak{X})$

$\begin{array}[]{c}(A\cup B)+C=(A+C)\cup(B+C)\\\ \textrm{if }B\subseteq
C,\quad A+B\subseteq A+C\end{array}$

In the following, we shall work with closed sets. In general, if
$A,B\in\mathbb{F}^{\,0}(\mathfrak{X})$ then $A+B$ does not belong to
$\mathbb{F}^{\,0}(\mathfrak{X})$ (e.g., in $\mathfrak{X}=\mathbb{R}$ let
$A=\left\\{n+1/n:n>1\right\\}$ and $B=\mathbb{Z}$, then
$\left\\{1/n=\left(n+1/n\right)+(-n)\right\\}\subset A+B$ and $1/n\downarrow
0$, but $0\not\in A+B$). In view of this fact, we define $A\oplus
B=\overline{A+B}$ where $\overline{(\cdot)}$ denotes the closure in
$\mathfrak{X}$.

For any $A,B\in\mathbb{F}(\mathfrak{X})$ the _Hausdorff distance_ (or
_metric_) is defined by

$\delta_{H}(A,B)=\max\left\\{\sup_{a\in A}\inf_{b\in
B}\left\|a-b\right\|_{\mathfrak{X}},\sup_{b\in B}\inf_{a\in
A}\left\|a-b\right\|_{\mathfrak{X}}\right\\}.$

For all $(x^{*},A)\in B^{*}_{1}\times\mathbb{F}(\mathfrak{X})$, the _support
function_ is defined by $s(x^{*},A)=\sup_{a\in A}x^{*}(a)$. It can be proved
(cf. [14, 2]) that for each $A,B\in\mathbb{F}_{bc}(\mathfrak{X})$,

$\delta_{H}(A,B)=\sup\left\\{\left|s(x^{*},A)-s(x^{*},B)\right|:x^{*}\in
B^{*}_{1}\right\\}.$ (1)

Let $(\Omega,\mathfrak{F})$ be a measurable space with $\mathfrak{F}$ complete
with respect to some $\sigma$-finite measure, let
$X:\Omega\to\mathfrak{P}^{\,0}(\mathfrak{X})$ be a set–valued map, and

$\displaystyle D(X)$
$\displaystyle=\left\\{\omega\in\Omega:X(\omega)\neq\emptyset\right\\}$ be the
domain of $X$ $\displaystyle X^{-1}(A)$
$\displaystyle=\left\\{\omega\in\Omega:X(\omega)\cap
A\neq\emptyset\right\\},\quad A\subset\mathfrak{X},$ be the inverse image of
$X$

Roughly speaking, $X^{-1}(A)$ is the set of all $\omega$ such that $X(\omega)$
hits set $A$.

Different definitions of measurability for set–valued functions are developed
over the years by several authors (cf. [17, 10, 2, 16] and reference therein).
Here, $X$ is _measurable_ if, for each $O$, open subset of $\mathfrak{X}$,
$X^{-1}(O)\in\mathfrak{F}$.

###### Proposition 0.1.1

_(See[17]) _ $X:\Omega\to\mathfrak{P}^{\,0}(\mathfrak{X})$ is a measurable
set–valued map if and only if $D(X)\in\mathfrak{F}$, and $\omega\mapsto
d(x,X(\omega))$ is a measurable function of $\omega\in D(X)$ for each
$x\in\mathfrak{X}$.

From now on, $\mathcal{U}[\Omega,\mathfrak{F},\mu;\mathbb{F}(\mathfrak{X})]$
($=\mathcal{U}[\Omega;\mathbb{F}(\mathfrak{X})]$ if the measure $\mu$ is
clear) denotes the family of $\mathbb{F}(\mathfrak{X})$–valued measurable maps
(analogous notation holds whenever $\mathbb{F}(\mathfrak{X})$ is replaced by
another family of subsets of $\mathfrak{X}$).

Let $(\Omega,\mathfrak{F},\mathbb{P})$ be a complete probability space and let
$X\in\mathcal{U}[\Omega,\mathfrak{F},\mathbb{P};\mathbb{F}(\mathfrak{X})]$,
then $X$ is a RaCS.
It can be proved (see [18]) that, if $X,X_{1},X_{2}$ are RaCS and if $\xi$ is
a measurable real–valued function, then $X_{1}\oplus X_{2}$, $X_{1}\ominus
X_{2}$, $\xi X$ and $(\textrm{Int}\ X)^{C}$ are RaCS. Moreover, if
$\left\\{X_{n}\right\\}_{n\in\mathbb{N}}$ is a sequence of RaCS then
$X=\overline{\bigcup_{n\in\mathbb{N}}X_{n}}$ is so.

Let $(\Omega,\mathfrak{F},\mu)$ be a finite measure space (although most of
the results are valid for $\sigma$-finite measures space). The _Aumann
integral_ of
$X\in\mathcal{U}[\Omega,\mathfrak{F},\mu;\mathbb{F}(\mathfrak{X})]$ is defined
by

$\int_{\Omega}Xd\mu=\left\\{\int_{\Omega}xd\mu:x\in S_{X}\right\\},$

where $S_{X}=\left\\{x\in L^{1}[\Omega;\mathfrak{X}]:x\in X\
\mu-\textrm{a.e.}\right\\}$ and $\int_{\Omega}xd\mu$ is the usual Bochner
integral in $L^{1}[\Omega;\mathfrak{X}]$. Moreover,
$\int_{A}Xd\mu=\left\\{\int_{A}xd\mu:x\in S_{X}\right\\}$ for
$A\in\mathfrak{F}$. If $\mu$ is a probability measure, we denote the Aumann
integral by $\mathbb{E}{X}=\int_{\Omega}Xd\mu$.

Let $X\in\mathcal{U}[\Omega,\mathfrak{F},\mu;\mathbb{F}(\mathfrak{X})]$, it is
_integrably bounded_ , and we shall write $X\in
L^{1}[\Omega,\mathfrak{F},\mu;\mathbb{F}(\mathfrak{X})]=L^{1}[\Omega;\mathbb{F}(\mathfrak{X})]$,
if $\left\|X\right\|_{h}\in L^{1}[\Omega,\mathfrak{F},\mu;\mathbb{R}]$.

##  0.2 Model assumptions

Let us consider

$\begin{array}[]{rl}\Theta_{t}=&\left(\Theta_{t_{0}}\oplus\int_{t_{0}}^{t}G_{s}ds\right)\cup\bigcup_{s\in[t_{0},t]}dB_{s}\\\
d\Theta_{t}=&\oplus G_{t}dt\cup dB_{t}\qquad\textrm{ or
}\qquad\Theta_{t+dt}=(\Theta_{t}\oplus G_{t}dt)\cup dB_{t}.\end{array}$ (2)

In fact, above equation is not a definition since, for example, problems arise
handling non–countable union of (random) closed sets. The well–posedness of
(2) and hence the existence of such a process are the main purpose of this
paper.
From now on, let us consider the following assumptions.

  * (A-0)

    * -

$(\mathfrak{X},\left\|\cdot\right\|_{\mathfrak{X}})$ is a reflexive Banach
space with separable dual space
$({\mathfrak{X}^{*}},\left\|\cdot\right\|_{{\mathfrak{X}^{*}}})$, (then,
$\mathfrak{X}$ is separable too, see [12, Lemma II.3.16 p. 65]).

    * -

$[t_{0},T]\subset\mathbb{R}$ is the _time observation interval_ (or _time
interval_),

    * -

$\left(\Omega,\mathfrak{F},\left\\{\mathfrak{F}_{t}\right\\}_{t\in[t_{0},T]},\mathbb{P}\right)$
is a filtered probability space, where the filtration
$\left\\{\mathfrak{F}_{t}\right\\}_{t\in[t_{0},T]}$ is assumed to have the
usual properties.

(_Nucleation Process_). $B=\left\\{B(\omega,t)=B_{t}:\omega\in\Omega,\
t\in[t_{0},T]\right\\}$ is a process with non–empty closed values, i.e.
$B:\Omega\times[t_{0},T]\to\mathbb{F}(\mathfrak{X})$ such that

  * (A-1)

$B(\cdot,t)\in\mathcal{U}[\Omega,\mathfrak{F}_{t},\mathbb{P};\mathbb{F}(\mathfrak{X})]$,
for every $t\in[t_{0},T]$, i.e. $B_{t}$ is an _adapted_ (to
$\left\\{\mathfrak{F}_{t}\right\\}_{t\in[t_{0},T]}$) process.

  * (A-2)

$B_{t}$ is increasing: for every $t,s\in[t_{0},T]$ with $s<t$, $B_{s}\subseteq
B_{t}$.

(_Growth Process_). $G=\left\\{G_{t}=G(\omega,t):\omega\in\Omega,\
t\in[t_{0},T]\right\\}$ is a process with non–empty closed values, i.e.
$G:\Omega\times[t_{0},T]\to\mathbb{F}(\mathfrak{X})$ such that

  * (A-3)

for every $\omega\in\Omega$ and $t\in[t_{0},T]$, $0\in G(\omega,t)$.

  * (A-4)

for every $\omega\in\Omega$ and $t\in[t_{0},T]$, $G(\omega,t)$ is convex, i.e.
$G:\Omega\times[t_{0},T]\to\mathbb{F}_{c}(\mathfrak{X})$.

  * (A-5)

there exists $K\in\mathbb{F}_{b}(\mathfrak{X})$ such that
$G(\omega,t)\subseteq K$ for every $t\in[t_{0},T]$ and $\omega\in\Omega$.

As a consequence, $G(\omega,t)\in\mathbb{F}_{b}(\mathfrak{X})$ and
$\left\|G(\omega,t)\right\|_{h}\leq\left\|K\right\|_{h}$,
$\forall(\omega,t)\in\Omega\times[t_{0},T]$.

In order to establish the well–posedness of integral $\int_{t_{0}}^{t}G_{s}ds$
in (2), let us consider a suitable hypothesis of measurability for $G$
(analogously to what is).
A $\mathbb{F}(\mathfrak{X})$–valued process
$G=\left\\{G_{t}\right\\}_{t\in[t_{0},T]}$ has _left continuous trajectories_
on $[t_{0},T]$ if, for every $\overline{t}\in[t_{0},T]$ with $t<\overline{t}$,

$\lim_{t\to\overline{t}}\delta_{H}\left(G(\omega,t),G(\omega,\overline{t})\right)=0,\qquad\textrm{a.s.}$

The $\sigma$-algebra on $\Omega\times[t_{0},T]$ generated by the processes
$\left\\{G_{t}\right\\}_{t\in[t_{0},T]}$ with left continuous trajectories on
$[t_{0},T]$, is called the _previsible_ (or _predictable_) $\sigma$-algebra
and it is denoted by $\mathcal{P}$.

###### Proposition 0.2.1

__The previsible $\sigma$-algebra is also generated by the collection of
random sets $A\times t_{0}$ where $A\in\mathfrak{F}_{t_{0}}$ and
$A\times(s,t]$ where $A\in\mathfrak{F}_{s}$ and $(s,t]\subset[t_{0},T]$.

Proof. Let the $\sigma$-algebra generated by the above collection of sets be
denoted by $\mathcal{P}^{\prime}$. We shall show
$\mathcal{P}=\mathcal{P}^{\prime}$. Let $G$ be a left continuous process and
let $\alpha=(T-t_{0})$, consider for $n\in\mathbb{N}$

$G_{n}(\omega,t)=\left\\{\begin{array}[]{ll}G(\omega,t_{0}),&t=t_{0}\\\ \\\
G\left(\omega,t_{0}+\frac{k\alpha}{2^{n}}\right),&\begin{array}[]{c}\left(t_{0}+\frac{k\alpha}{2^{n}}\right)<t\leq\left(t_{0}+\frac{(k+1)\alpha}{2^{n}}\right)\\\
k\in\left\\{0,\ldots,(2^{n}-1)\right\\}\end{array}\end{array}\right.$

It is clear that $G_{n}$ is $\mathcal{P}^{\prime}$-measurable, since $G$ is
adapted. As $G$ is left continuous, the above sequence of left-continuous
processes converges pointwise (with respect to $\delta_{H}$) to $G$ when $n$
tends to infinity, so $G$ is $\mathcal{P}^{\prime}$-measurable, thus
$\mathcal{P}\subseteq\mathcal{P}^{\prime}$.
Conversely consider $A\times(s,t]\in\mathcal{P}^{\prime}$ with
$(s,t]\subset[t_{0},T]$ and $A\in\mathfrak{F}_{s}$. Let
$b\in\mathfrak{X}\setminus\\{0\\}$ and $G$ be the process

$G(\omega,v)=\left\\{\begin{array}[]{ll}b,&v\in(s,t],\ \omega\in A\\\
0,&\textrm{otherwise}\end{array}\right.$

this function is adapted and left continuous, hence
$\mathcal{P}^{\prime}\subseteq\mathcal{P}$. $\blacksquare$

Then let us consider the following assumption.

  * (A-6)

$G$ is $\mathcal{P}$-measurable.

##  0.3 Growth process properties

Theorem 0.3.2 is the main result in this section. It shows that
$\omega\mapsto\int_{a}^{b}G(\omega,\tau)d\tau$ is a RaCS with non–empty
bounded convex values. This is the first step in order to obtain
well–posedness of (2).

###### Proposition 0.3.1

__Suppose (A-3), …, (A-6), and let $\mu_{\lambda}$ be the
Lebesgue measure on $[t_{0},T]$, then

  * •

$G(\omega,\cdot)\in\mathcal{U}\left[[t_{0},T],\mathcal{B}_{[t_{0},T]},\mu_{\lambda};\mathbb{F}_{bc}(\mathfrak{X})\right]$
for every $\omega\in\Omega$.

  * •

$G(\cdot,t)\in\mathcal{U}[\Omega,\widetilde{\mathfrak{F}}_{t^{-}},\mathbb{P};\mathbb{F}_{bc}(\mathfrak{X})]$
for each $t\in[t_{0},T]$, where $\widetilde{\mathfrak{F}}_{t^{-}}$ is the so
called _history $\sigma$-algebra_ i.e.
$\widetilde{\mathfrak{F}}_{t^{-}}=\sigma\left(\mathfrak{F}_{s}:0\leq
s<t\right)\subseteq\mathfrak{F}$.

  * •

$G\in\
L^{1}[[t_{0},T],\mathcal{B}_{[t_{0},T]},\mu_{\lambda};\mathbb{F}_{bc}(\mathfrak{X})]\cap
L^{1}[\Omega,\mathfrak{F},\mathbb{P};\mathbb{F}_{bc}(\mathfrak{X})]$

Proof. Assumptions (A-3) and (A-4) imply that $G$ is
non–empty and convex. Measurability and integrability properties are
consequence of (A-6) and (A-5) respectively.
$\blacksquare$

###### Theorem 0.3.2

__Suppose (A-3), …, (A-6). For every $a,b\in[t_{0},T]$,
the integral $\int_{a}^{b}G(\omega,\tau)d\tau$ is non–empty and the set–valued
map

$\begin{array}[]{rccl}G_{a,b}:&\Omega&\to&\mathfrak{P}(\mathfrak{X})\\\
&\omega&\mapsto&\int_{a}^{b}G(\omega,\tau)d\tau\end{array}$

is measurable. Moreover, $G_{a,b}$ is a non–empty, bounded convex RaCS.

In order to prove Theorem 0.3.2, consider following properties for real
processes.
A real–valued process $X=\left\\{X_{t}\right\\}_{t\in[t_{0},T]}$ is
_predictable_ with respect to filtration
$\left\\{\mathfrak{F}_{t}\right\\}_{t\in\mathbb{R}_{+}}$, if it is measurable
with respect to the _predictable $\sigma$-algebra_ $\mathcal{P}_{\mathbb{R}}$,
i.e. the $\sigma$-algebra generated by the collection of random sets
$A\times\left\\{0\right\\}$ where $A\in\mathfrak{F}_{0}$ and $A\times(s,t]$
where $A\in\mathfrak{F}_{s}$.

###### Proposition 0.3.3

_(See[9, Propositions 2.30, 2.32 and 2.41]) _Let
$X=\left\\{X_{t}\right\\}_{t\in[t_{0},T]}$ be a predictable real–valued
process, then $X$ is
$(\mathfrak{F}\otimes\mathcal{B}_{[t_{0},T]},\mathcal{B}_{\mathbb{R}})$-measurable.
Further, for every $\omega\in\Omega$, the trajectory
$X(\omega,\cdot):[t_{0},T]\to\mathbb{R}$ is
$(\mathcal{B}_{[t_{0},T]},\mathcal{B}_{\mathbb{R}})$-measurable .

###### Lemma 0.3.4

__Let $x^{*}$ be an element of the unit ball in the dual space $B^{*}_{1}$,
then $G\mapsto s(x^{*},G)$ is a measurable map.

Proof. By definition $s(x^{*},G)=\sup\left\\{x^{*}(g):g\in G\right\\}$. Since
$\mathfrak{X}$ is separable (A-0), there exists
$\left\\{g_{n}\right\\}_{n\in\mathbb{N}}\subset G$ such that
$G=\overline{\left\\{g_{n}\right\\}}$. Then, for every $x^{*}\in B^{*}_{1}$ we
have

$s(x^{*},G)=\sup_{g\in G}x^{*}(g)=\sup_{n\in\mathbb{N}}x^{*}(g_{n}).$

Since $x^{*}$ is a continuous map then, $s(x^{*},\cdot)$ is measurable.
$\blacksquare$

Proof of Theorem 0.3.2.  At first, we prove that $G_{a,b}$ is a measurable
map. From Proposition 0.3.1, integral
$G_{a,b}=\int_{a}^{b}G(\omega,\tau)d\tau$ is well defined for all
$\omega\in\Omega$. Assumption (A-3) implies $0\in
G_{a,b}(\omega)\neq\emptyset$ for every $\omega\in\Omega$. Hence, the domain
of $G_{a,b}$ is the whole $\Omega$ for all $a,b\in[t_{0},T]$

$D\left(G_{a,b}\right)=\left\\{\omega\in\Omega:G_{a,b}\neq\emptyset\right\\}=\Omega\in\mathfrak{F}.$

Thus, by Proposition 0.1.1 and for a fixed couple $a,b\in[t_{0},T]$, $G_{a,b}$
is (weakly) measurable if and only if, for every $x\in\mathfrak{X}$, the map

$\omega\mapsto
d\left(x,\int_{a}^{b}G(\omega,\tau)d\tau\right)=\delta_{H}\left(x,\int_{a}^{b}G(\omega,\tau)d\tau\right)$
(3)

is measurable. Equation (1) guarantees that (3) is measurable if and only if,
for every $x\in\mathfrak{X}$, the map

$\omega\mapsto\sup_{x^{*}\in
B^{*}_{1}}\left|s(x^{*},x)-s\left(x^{*},\int_{a}^{b}G(\omega,\tau)d\tau\right)\right|$

is measurable. The above expression can be computed on a countable family
dense in $B^{*}_{1}$ (note that such family exists since ${\mathfrak{X}^{*}}$
is assumed separable (A-0)):

$\omega\mapsto\sup_{n\in\mathbb{N}}\left|s(x^{*}_{i},x)-s\left(x^{*}_{i},\int_{a}^{b}G(\omega,\tau)d\tau\right)\right|.$

It can be proved ([18, Theorem 2.1.12 p. 46]) that

$s\left(x^{*},\int_{a}^{b}G(\omega,\tau)d\tau\right)=\int_{a}^{b}s\left(x^{*},G(\omega,\tau)\right)d\tau,\qquad\forall
x^{*}\in B^{*}_{1}$

and therefore, since $s(x^{*}_{i},x)$ is a constant, $G_{a,b}$ is measurable
if, for every $x^{*}\in\left\\{x_{i}^{*}\right\\}_{i\in\mathbb{N}}$, the
following map

$\begin{array}[]{ccl}(\Omega,\mathfrak{F})&\to&(\mathbb{R},\mathcal{B}_{\mathbb{R}})\\\
\omega&\mapsto&\int_{a}^{b}s\left(x^{*},G(\omega,\tau)\right)d\tau\end{array}$
(4)

is measurable. Note that $s(x^{*},G(\cdot,\cdot))$, as a map from
$\Omega\times[t_{0},T]$ to $\mathbb{R}$, is predictable since it is the
composition of a predictable map (A-6) with a measurable one (see
Lemma 0.3.4):

$\begin{array}[]{rccccc}s\left(x^{*},G(\cdot,\cdot)\right):&(\Omega\times[t_{0},T],\mathcal{P})&\to&(\mathbb{F}(\mathfrak{X}),\sigma_{f})&\to&(\mathbb{R},\mathcal{B}_{\mathbb{R}})\\\
&(\omega,t)&\mapsto&G(\omega,t)&\mapsto&s\left(x^{*},G(\omega,t)\right)\end{array}$

thus, by Proposition 0.3.3, it is a $\mathcal{P}$-measurable map and hence (4)
is a measurable map.

In view of the first part, it remains to prove that $G_{a,b}$ is a bounded
convex set for a.e. $\omega\in\Omega$. Since $\mathfrak{X}$ is reflexive
(A-0), by Proposition 0.3.1 we have that $G_{a,b}$ is closed ([18,
Theorem 2.2.3]). Further, $G_{a,b}$ is also convex (see [18, Theorem 2.1.5 and
Corollary 2.1.6]).
To conclude the proof, it is sufficient to show that $G_{a,b}$ is included in
a bounded set:

$\displaystyle\int_{a}^{b}G(\omega,\tau)d\tau$ $\displaystyle=$
$\displaystyle\left\\{\int_{a}^{b}g(\omega,\tau)d\tau:g(\omega,\cdot)\in
G(\omega,\cdot)\subseteq K\right\\}$ $\displaystyle\subseteq$
$\displaystyle\left\\{\int_{a}^{b}kd\tau:k\in K\right\\}=\left\\{(b-a)k:k\in
K\right\\}=(b-a)K.$

$\blacksquare$

##  0.4 Geometric Random Process

For the sake of simplicity, let us present the main results which proofs will
be given in Section 0.4.1.
Let us assume conditions from (A-0) to (A-6). For every
$t\in[t_{0},T]\subset\mathbb{R}$, $n\in\mathbb{N}$ and
$\Pi=\left(t_{i}\right)_{i=0}^{n}$ partition of $[t_{0},t]$, let us define

$\displaystyle s_{\Pi}(t)=$
$\displaystyle\left(B_{t_{0}}\oplus\int_{t_{0}}^{t}G(\tau)d\tau\right)\cup\bigcup_{i=1}^{n}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{t}G(\tau)d\tau\right)$ (5) $\displaystyle
S_{\Pi}(t)=$
$\displaystyle\left(B_{t_{0}}\oplus\int_{t_{0}}^{t}G(\tau)d\tau\right)\cup\bigcup_{i=1}^{n}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i-1}}^{t}G(\tau)d\tau\right)$ (6)

where $\Delta B_{t_{i}}=B_{t_{i}}\setminus B_{t_{i-1}}^{o}$ ($B_{t_{i-1}}^{o}$
denotes the interior set of $B_{t_{i-1}}$) and where the integral is in the
Aumann sense with respect to the Lebesgue measure $d\tau=d\mu_{\lambda}$. We
write $s_{\Pi}$ and $S_{\Pi}$ instead of $s_{\Pi}(t)$ and $S_{\Pi}(t)$ when
the dependence on $t$ is clear.

Proposition 0.4.1 guarantees that both $s_{\Pi}$ and $S_{\Pi}$ are well
defined RaCS, further, Proposition 0.4.3 shows $s_{\Pi}\subseteq S_{\Pi}$ as a
consequence of different time intervals integration: if the time interval
integration of $G$ increases then the integral of $G$ does not decrease with
respect to set-inclusion (Lemma 0.4.2). Proposition 0.4.4 means that
$\left\\{s_{\Pi}\right\\}$ ($\left\\{S_{\Pi}\right\\}$) increases (decreases)
whenever a refinement of $\Pi$ is considered. At the same time, Proposition
0.4.5 implies that $s_{\Pi}$ and $S_{\Pi}$ become closer each other (in the
Hausdorff distance sense) when partition $\Pi$ becomes finer. The “limit” is
independent on the choice of the refinement as consequence of Proposition
0.4.6.
Corollary 0.4.7 means that, given any
$\left\\{\Pi_{j}\right\\}_{j\in\mathbb{N}}$ refinement sequence of
$[t_{0},t]$, the random closed sets $s_{\Pi_{j}}$ and $S_{\Pi_{j}}$ play the
same role that lower sums and upper sums have in classical analysis when we
define the Riemann integral. In fact, if $\Theta_{t}$ denotes their limit
value (see (7)), $s_{\Pi_{j}}$ and $S_{\Pi_{j}}$ are a lower and an upper
approximation of $\Theta_{t}$ respectively. Note that, as a consequence of
monotonicity of $s_{\Pi_{j}}$ and $S_{\Pi_{j}}$, we avoid problems that may
arise considering uncountable unions in integral expression in (2).

###### Proposition 0.4.1

__Let $\Pi$ be a partition of $[t_{0},t]$. Both $s_{\Pi}$ and $S_{\Pi}$,
defined in (5) and (6), are RaCS.

###### Lemma 0.4.2

__Let $X\in L^{1}[I,\mathfrak{F},\mu_{\lambda};\mathbb{F}(\mathfrak{X})]$,
where $I$ is a bounded interval of $\mathbb{R}$, such that $0\in X$
$\mu_{\lambda}$-almost everywhere on $I$ and let $I_{1},I_{2}$ be two other
intervals of $\mathbb{R}$ with $I_{1}\subset I_{2}\subset I$. Then

$\int_{I_{1}}X(\tau)d\tau\subseteq\int_{I_{2}}X(\tau)d\tau.$

###### Proposition 0.4.3

__Let $\Pi$ be a partition of $[t_{0},t]$. Then $s_{\Pi}\subseteq S_{\Pi}$
almost surely.

###### Proposition 0.4.4

__Let $\Pi$ and $\Pi^{\prime}$ be two partitions of $[t_{0},t]$ such that
$\Pi^{\prime}$ is a refinement of $\Pi$. Then, almost surely,
$s_{\Pi}\subseteq s_{\Pi^{\prime}}$ and $S_{\Pi^{\prime}}\subseteq S_{\Pi}$.

###### Proposition 0.4.5

__Let $\left\\{\Pi_{j}\right\\}_{j\in\mathbb{N}}$ be a refinement sequence of
$[t_{0},t]$ (i.e. $\left|\Pi_{j}\right|\to 0$ if $j\to\infty$). Then, almost
surely, $\lim_{j\to\infty}\delta_{H}\left(s_{\Pi_{j}},S_{\Pi_{j}}\right)=0$.

###### Proposition 0.4.6

__Let $\left\\{\Pi_{j}\right\\}_{j\in\mathbb{N}}$ and
$\left\\{\Pi_{l}^{\prime}\right\\}_{l\in\mathbb{N}}$ be two distinct
refinement sequences of $[t_{0},t]$, then, almost surely,

$\lim_{\textrm{\tiny$\begin{array}[]{c}j\rightarrow\infty\\\
l\rightarrow\infty\end{array}$}}\delta_{H}\left(s_{\Pi_{j}},s_{\Pi^{\prime}_{l}}\right)=0\qquad\textrm{and}\qquad\lim_{\textrm{\tiny$\begin{array}[]{c}j\rightarrow\infty\\\
l\rightarrow\infty\end{array}$}}\delta_{H}\left(S_{\Pi_{j}},S_{\Pi^{\prime}_{l}}\right)=0.$

###### Corollary 0.4.7

__For every $\left\\{\Pi_{j}\right\\}_{j\in\mathbb{N}}$ refinement sequence of
$[t_{0},t]$, the following limits exist

$\overline{\left(\bigcup_{j\in\mathbb{N}}s_{\Pi_{j}}\right)},\
\overline{\left(\lim_{j\rightarrow\infty}s_{\Pi_{j}}\right)},\
\lim_{j\rightarrow\infty}S_{\Pi_{j}},\ \bigcap_{j\in\mathbb{N}}S_{\Pi_{j}},$
(7)

and they are equals almost surely. The convergences is taken with respect to
the Hausdorff distance.

We are now ready to define the continuous time stochastic process.

###### Definition 0.4.8

__Assume (A-0), …, (A-6). For every $t\in[t_{0},T]$, let
$\left\\{\Pi_{j}\right\\}_{j\in\mathbb{N}}$ be a refinement sequence of the
time interval $[t_{0},t]$ and let $\Theta_{t}$ be the RaCS defined by

$\overline{\left(\bigcup_{j\in\mathbb{N}}s_{\Pi_{j}}(t)\right)}=\overline{\left(\lim_{j\rightarrow\infty}s_{\Pi_{j}}(t)\right)}=\Theta_{t}=\lim_{j\rightarrow\infty}S_{\Pi_{j}}(t)=\bigcap_{j\in\mathbb{N}}S_{\Pi_{j}}(t),$

then, the family $\Theta=\left\\{\Theta_{t}:t\in[t_{0},T]\right\\}$ is called
_geometric random process G-RaP_ (on $[t_{0},T]$).

###### Theorem 0.4.9

__Let $\Theta$ be a G-RaP on $[t_{0},T]$, then $\Theta$ is a non-decreasing
process with respect to the set inclusion, i.e.

$\mathbb{P}\left(\Theta_{s}\subseteq\Theta_{t},\ \forall t_{0}\leq s<t\leq
T\right)=1.$

Moreover, $\Theta$ is adapted with respect to filtration
$\left\\{\mathfrak{F}_{t}\right\\}_{t\in[t_{0},T]}$.

###### Remark 0.4.10

__We want to point out that, assumptions we considered on
$\left\\{B_{t}\right\\}$ and $\left\\{G_{t}\right\\}$ are so general, that a
wide family of classical random sets and evolution processes can be described
(for example, Boolean model is a birth–and–growth process with “null growth”).

###  0.4.1 Proofs of Propositions in Section 0.4

Proof of Proposition 0.4.1.  For every $i\in\left\\{0,\ldots,n\right\\}$,
$\int_{t_{i-1}}^{t}G(\tau)d\tau$ is a RaCS (Theorem 0.3.2). Thus,
measurability Assumption (A-1) on $B$ guarantees that, for every
$t_{i}\in\Pi$, $B_{t_{i}}$, $\Delta B_{t_{i}}$, $\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{t}G(\tau)d\tau\right)$, and hence $s_{\Pi}$ and
$S_{\Pi}$ are RaCS. $\blacksquare$

Proof of Lemma 0.4.2.  Let $y\in\left(\int_{I_{1}}X(\tau)d\tau\right)$, then
there exists $x\in S_{X}$, for which
$y=\left(\int_{I_{1}}x(\tau)d\tau\right)$. Let us define on $I_{2}(\supset
I_{1})$

$x^{\prime}(\tau)=\left\\{\begin{array}[]{ll}x(\tau),&\tau\in I_{1}\\\
0,&\tau\in I_{2}\setminus I_{1}\end{array}\right.$

then $x^{\prime}\in S_{X}$ and
$y=\left(\int_{I_{2}}x^{\prime}(\tau)d\tau\right)\in\left(\int_{I_{2}}X(\tau)d\tau\right)$.
$\blacksquare$

Proof of Proposition 0.4.3.  Thesis is a consequence of Lemma 0.4.2 and
Minkowski addition properties, in fact
$\left(\int_{t_{i-1}}^{t}G(\tau)d\tau\right)\subseteq\left(\int_{t_{i}}^{t}G(\tau)d\tau\right)$
implies $s_{\Pi}\subseteq S_{\Pi}$. $\blacksquare$

Proof of Proposition 0.4.4.  Let $\Pi^{\prime}$ be a refinement of partition
$\Pi$ of $[t_{0},t]$, i.e. $\Pi\subset\Pi^{\prime}$. We prove that
$s_{\Pi}\subseteq s_{\Pi^{\prime}}$ ($S_{\Pi^{\prime}}\subseteq S_{\Pi}$ is
analogous). It is sufficient to show the thesis only for
$\Pi^{\prime}=\Pi\cup\left\\{\overline{t}\right\\}$ where
$\Pi=\left\\{t_{0},\ldots,t_{n}\right\\}$ with $t_{0}<\ldots<t_{n}=t$ and
$\overline{t}\in(t_{0},t)$. Let $i\in\left\\{0,\ldots,(n-1)\right\\}$ be such
that $t_{i}\leq\overline{t}\leq t_{i+1}$ then

$\displaystyle s_{\Pi}$ $\displaystyle=$
$\displaystyle\left(B_{t_{0}}\oplus\int_{t_{0}}^{t}G(\tau)d\tau\right)\cup\bigcup_{\textrm{\tiny$\begin{array}[]{c}j=1\\\
j\neq i+1\end{array}$}}^{n}\left(\Delta
B_{t_{j}}\oplus\int_{t_{j}}^{t}G(\tau)d\tau\right)\cup$
$\displaystyle\left[\left(B_{t_{i+1}}\setminus
B_{t_{i}}^{o}\right)\oplus\int_{t_{i+1}}^{t}G(\tau)d\tau\right]$

and

$\displaystyle s_{\Pi^{\prime}}$ $\displaystyle=$
$\displaystyle\left(B_{t_{0}}\oplus\int_{t_{0}}^{t}G(\tau)d\tau\right)\cup\bigcup_{\textrm{\tiny$\begin{array}[]{c}j=1\\\
j\neq i+1\end{array}$}}^{n}\left(\Delta
B_{t_{j}}\oplus\int_{t_{j}}^{t}G(\tau)d\tau\right)\cup$
$\displaystyle\left[\left(B_{\overline{t}}\setminus
B_{t_{i}}^{o}\right)\oplus\int_{\overline{t}}^{t}G(\tau)d\tau\right]\cup\left[\left(B_{t_{i+1}}\setminus
B_{\overline{t}}^{o}\right)\oplus\int_{t_{i+1}}^{t}G(\tau)d\tau\right]$

Definitely, in order to prove that $s_{\Pi}\subseteq s_{\Pi^{\prime}}$ we have
to prove that

$\displaystyle\left\\{\left[\left(B_{\overline{t}}\setminus
B_{t_{i}}^{o}\right)\oplus\int_{\overline{t}}^{t}G(\tau)d\tau\right]\cup\left[\left(B_{t_{i+1}}\setminus
B_{\overline{t}}^{o}\right)\oplus\int_{t_{i+1}}^{t}G(\tau)d\tau\right]\right\\}$
$\displaystyle\supseteq\left[\left(B_{t_{i+1}}\setminus
B_{t_{i}}^{o}\right)\oplus\int_{t_{i+1}}^{t}G(\tau)d\tau\right].$

This inclusion is a consequence of
$\left(\int_{\overline{t}}^{t}G(\tau)d\tau\right)\supseteq\left(\int_{t_{i+1}}^{t}G(\tau)d\tau\right)$
(Lemma 0.4.2) and of the Minkowski distribution property. $\blacksquare$

Proof of Proposition 0.4.5.  Let $\Pi_{j}=\left(t_{i}\right)_{i=0}^{n}$ be the
$j$-partition of the refinement sequence
$\left\\{\Pi_{j}\right\\}_{j\in\mathbb{N}}$, then

$\delta_{H}\left(s_{\Pi_{j}},S_{\Pi_{j}}\right)=\max\left\\{\sup_{x\in
s_{\Pi_{j}}}d(x,S_{\Pi_{j}}),\sup_{y\in S_{\Pi_{j}}}d(y,s_{\Pi_{j}})\right\\}$

where $d(x,S_{\Pi_{j}})=\inf_{y\in
S_{\Pi_{j}}}\left\|x-y\right\|_{\mathfrak{X}}$. By Proposition 0.4.3,
$s_{\Pi_{j}}\subseteq S_{\Pi_{j}}$ then

$\sup_{x\in s_{\Pi_{j}}}d(x,S_{\Pi_{j}})=0$

and hence we have to prove that, whenever $j\to\infty$ (i.e.
$\left|\Pi_{j}\right|\to 0$),

$\delta_{H}\left(s_{\Pi_{j}},S_{\Pi_{j}}\right)=\sup_{y\in
S_{\Pi_{j}}}d(y,s_{\Pi_{j}})=\sup_{y\in S_{\Pi_{j}}}\inf_{x\in
s_{\Pi_{j}}}\left\|x-y\right\|_{\mathfrak{X}}\longrightarrow 0.$

For every $\omega\in\Omega$, let $y$ be any element of $S_{\Pi_{j}}(\omega)$,
then we distinguish two cases:

(1) if
$y\in\left(B_{t_{0}}(\omega)\oplus\int_{t_{0}}^{t}G(\omega,\tau)d\tau\right)$,
then it is also an element of $s_{\Pi_{j}}(\omega)$, and hence
$d\left(s_{\Pi_{j}}(\omega),y\right)=0$.

(2) if
$y\not\in\left(B_{t_{0}}(\omega)\oplus\int_{t_{0}}^{t}G(\omega,\tau)d\tau\right)$,
then there exist $j\in\left\\{1,\ldots,n\right\\}$ such that

$y\in\left(\Delta
B_{t_{j}}(\omega)\oplus\int_{t_{j-1}}^{t}G(\omega,\tau)d\tau\right).$

By definition of $\oplus$, for every $\omega\in\Omega$, there exist

$\left\\{y_{m}\right\\}_{m\in\mathbb{N}}\subseteq\left(\Delta
B_{t_{j}}(\omega)+\int_{t_{j-1}}^{t}G(\omega,\tau)d\tau\right),$

such that $\lim_{m\to\infty}y_{m}=y$. Then, for every $\omega\in\Omega$, there
exist $h_{m}\in\Delta B_{t_{j}}(\omega)$ and
$g_{m}\in\left(\int_{t_{j-1}}^{t}G(\omega,\tau)d\tau\right)$ such that
$y_{m}=(h_{m}+g_{m})$ and hence

$y=\lim_{m\to\infty}(h_{m}+g_{m})=\lim_{m\to\infty}y_{m}$

where the convergence is in the Banach norm, then let
$\overline{m}\in\mathbb{N}$ be such that
$\left\|y-y_{m}\right\|_{\mathfrak{X}}<\left|\Pi_{j}\right|$, for every
$m>\overline{m}$.

Note that, for every $\omega\in\Omega$ and $m\in\mathbb{N}$, by Aumann
integral definition, there exists a selection $\widehat{g_{m}}(\cdot)$ of
$G(\omega,\cdot)$ (i.e. $\widehat{g_{m}}(t)\in G(\omega,t)$
$\mu_{\lambda}$-a.e.) such that

$g_{m}=\int_{t_{j-1}}^{t}\widehat{g_{m}}(\tau)d\tau\qquad\textrm{ and }\qquad
y_{m}={h_{m}+\int_{t_{j-1}}^{t}\widehat{g_{m}}(\tau)d\tau}.$

For every $\omega\in\Omega$, let us consider

$x_{m}=h_{m}+\int_{t_{j}}^{t}\widehat{g_{m}}(\tau)d\tau$

then $x_{m}\in s_{\Pi_{j}}(\omega)$ for all $m\in\mathbb{N}$. Moreover, the
following chain of inequalities hold, for all $m>\overline{m}$ and
$\omega\in\Omega$,

$\displaystyle\inf_{x^{\prime}\in
s_{\Pi_{j}}}\left\|x^{\prime}-y\right\|_{\mathfrak{X}}$ $\displaystyle\leq$
$\displaystyle\left\|x_{m}-y\right\|_{\mathfrak{X}}\leq\left\|x_{m}-y_{m}\right\|_{\mathfrak{X}}+\left\|y_{m}-y\right\|_{\mathfrak{X}}$
$\displaystyle\leq$
$\displaystyle\left\|\int_{t_{j-1}}^{t_{j}}\widehat{g_{m}}(\tau)d\tau\right\|_{\mathfrak{X}}+\left|\Pi_{j}\right|\leq\int_{t_{j-1}}^{t_{j}}\left\|\widehat{g_{m}}(\tau)\right\|_{\mathfrak{X}}d\tau+\left|\Pi_{j}\right|$
$\displaystyle\leq$
$\displaystyle\int_{t_{j-1}}^{t_{j}}\left\|G(\tau)\right\|_{h}d\tau+\left|\Pi_{j}\right|\leq\left|t_{j}-t_{j-1}\right|\left\|K\right\|_{h}+\left|\Pi_{j}\right|$
$\displaystyle\leq$
$\displaystyle\left|\Pi_{j}\right|\left(\left\|K\right\|_{h}+1\right)\stackrel{{\scriptstyle
j\to\infty}}{{\longrightarrow}}0$

since $\left\|K\right\|_{h}$ is a positive constant. By the arbitrariness of
$y\in S_{\Pi_{j}}(\omega)$ we obtain the thesis. $\blacksquare$

Proof of Proposition 0.4.6.  Let $\Pi_{j}$ and $\Pi_{l}^{\prime}$ be two
partitions of the two distinct refinement sequences
$\left\\{\Pi_{j}\right\\}_{j\in\mathbb{N}}$ and
$\left\\{\Pi_{l}^{\prime}\right\\}_{l\in\mathbb{N}}$ of $[t_{0},t]$. Let
$\Pi^{\prime\prime}=\Pi_{j}\cup\Pi_{l}^{\prime}$ be the refinement of both
$\Pi_{j}$ and $\Pi_{l}^{\prime}$. Then Proposition 0.4.4 and Proposition 0.4.3
imply that $s_{\Pi_{j}}\subseteq s_{\Pi^{\prime\prime}}\subseteq
S_{\Pi^{\prime\prime}}\subseteq S_{\Pi_{l}^{\prime}}$. Therefore
$s_{\Pi_{j}}\subseteq S_{\Pi_{l}^{\prime}}$ for every $j,l\in\mathbb{N}$. Then

$\overline{\left(\bigcup_{j\in\mathbb{N}}s_{\Pi_{j}}\right)}\subseteq\bigcap_{l\in\mathbb{N}}S_{\Pi_{l}^{\prime}}.$

Analogously

$\overline{\left(\bigcup_{l\in\mathbb{N}}s_{\Pi_{l}^{\prime}}\right)}\subseteq\bigcap_{j\in\mathbb{N}}S_{\Pi_{j}}.$

Proposition 0.4.5 concludes the proof. $\blacksquare$

In order to prove Theorem 0.4.9, let us consider the following Lemma.

###### Lemma 0.4.11

__Let $s,t\in[t_{0},T]$ with $t_{0}<s<t$ and let $\Pi^{s}$ and $\Pi^{t}$ be
two partition of $[t_{0},s]$ and $[t_{0},t]$ respectively, such that
$\Pi^{s}\subset\Pi^{t}$. Then

$s_{\Pi^{s}}(s)\subseteq s_{\Pi^{t}}(t)\qquad\textrm{ and }\qquad
S_{\Pi^{s}}(s)\subseteq S_{\Pi^{t}}(t).$

Proof. The proofs of the two inclusions are similar. Let us prove that
$s_{\Pi^{s}}(s)\subseteq s_{\Pi^{t}}(t)$.
Since $\Pi^{s}\subset\Pi^{t}$, then $\Pi^{s}=\left(t_{i}\right)_{i=0}^{n}$ and
$\Pi^{t}=\Pi^{s}\cup\left(t_{i}\right)_{i=n+1}^{n+m}$ with $m\in\mathbb{N}$.
By Lemma 0.4.2, we have that

$\displaystyle s_{\Pi^{s}}(s)$ $\displaystyle=$
$\displaystyle\left(B_{t_{0}}\oplus\int_{t_{0}}^{s}G(\tau)d\tau\right)\cup\bigcup_{i=1}^{n}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{s}G(\tau)d\tau\right)$ $\displaystyle\subseteq$
$\displaystyle\left(B_{t_{0}}\oplus\int_{t_{0}}^{t}G(\tau)d\tau\right)\cup\bigcup_{i=1}^{n}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{t}G(\tau)d\tau\right)$ $\displaystyle\subseteq$
$\displaystyle\left(B_{t_{0}}\oplus\int_{t_{0}}^{t}G(\tau)d\tau\right)\cup\bigcup_{i=1}^{n}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{t}G(\tau)d\tau\right)$
$\displaystyle\cup\bigcup_{i=n+1}^{n+m}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{t}G(\tau)d\tau\right)$

i.e. $s_{\Pi^{s}}(s)\subseteq s_{\Pi^{t}}(t)$. $\blacksquare$

Proof of Theorem 0.4.9.  For every $s,t\in[t_{0},T]$ with $s<t$, let
$\left\\{\Pi^{s}_{i}\right\\}_{i\in\mathbb{N}}$ and
$\left\\{\Pi^{t}_{i}\right\\}_{i\in\mathbb{N}}$ be two refinement sequences of
$[t_{0},s]$ and $[t_{0},t]$ respectively, such that
$\Pi^{s}_{i}\subset\Pi^{t}_{i}$ for every $i\in\mathbb{N}$. Then, by Lemma
0.4.11, $S_{\Pi^{s}_{i}}\subseteq S_{\Pi^{t}_{i}}$. Now, as $i$ tends to
infinity, we obtain

$\Theta_{s}=\bigcap_{i\rightarrow\infty}S_{\Pi^{s}_{i}}\subseteq\bigcap_{i\rightarrow\infty}S_{\Pi^{t}_{i}}=\Theta_{t}.$

For the second part, note that Theorem 0.3.2 still holds replacing
$\mathfrak{F}_{t}$ instead of $\mathfrak{F}$, so that for every
$s\in[t_{0},T]$, the family
$\left\\{\int_{s}^{t}G(\omega,\tau)d\tau\right\\}_{t\in[s,T]}$ is an adapted
process to the filtration $\left\\{\mathfrak{F}_{t}\right\\}_{t\in[t_{0},T]}$.
This fact together with Assumption (A-1) guarantees that
$\left\\{S_{\Pi}\right\\}_{t\in[s,T]}$ is adapted for every partition $\Pi$ of
$[s,T]$ and hence $\Theta$ is adapted too. $\blacksquare$

##  0.5 Discrete time case and infinitesimal notations

Let us consider $\Theta_{s}$ and $\Theta_{t}$ with $s<t$. Let
$\left\\{\Pi^{s}_{j}\right\\}_{j\in\mathbb{N}}$ and
$\left\\{\Pi^{t}_{j}\right\\}_{j\in\mathbb{N}}$ be two refinement sequences of
$[t_{0},s]$ and $[t_{0},t]$ respectively, such that
$\Pi^{s}_{j}\subset\Pi^{t}_{j}$ for every $j\in\mathbb{N}$ (i.e.
$\Pi_{j}^{s}=\left(t_{i}\right)_{i=0}^{n}$ and
$\Pi_{j}^{t}=\Pi_{j}^{s}\cup\left(t_{i}\right)_{i=n+1}^{n+m}$ with
$n,m\in\mathbb{N}$). It is easy to compute

$s_{\Pi^{t}_{j}}=\left(s_{\Pi^{s}_{j}}\oplus\int_{s}^{t}G(\tau)d\tau\right)\cup\bigcup_{i=n+1}^{n+m}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{t}G(\tau)d\tau\right).$

Then, by Definition 0.4.8, whenever $\left|\Pi^{t}_{j}\right|\to 0$, we obtain

$\Theta_{t}=\left(\Theta_{s}\oplus\int_{s}^{t}G(\tau)d\tau\right)\cup\lim_{\left|\Pi^{t}_{j}\right|\rightarrow
0}\bigcup_{i=n+1}^{n+m}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{t}G(\tau)d\tau\right).$ (10)

The following notations

$G_{k}=\int_{s}^{t}G(\tau)d\tau\qquad\textrm{and}\qquad
B_{k}=\lim_{\left|\Pi^{t}_{j}\right|\rightarrow
0}\bigcup_{i=n+1}^{n+m}\left(\Delta
B_{t_{i}}\oplus\int_{t_{i}}^{t}G(\tau)d\tau\right)$

lead us to the set-valued discrete time stochastic process

$\Theta_{k}=\left\\{\begin{array}[]{ll}(\Theta_{k-1}\oplus G_{k})\cup
B_{k},&k\geq 1,\\\ B_{0},&k=0.\end{array}\right.$

In view of this, we are able to justify infinitesimal notations introduced in
(2). In particular, from Equation (10), whenever
$\left|\Pi^{t}_{j}\right|\rightarrow 0$, we obtain

$\Theta_{t}=\left(B_{t_{0}}\oplus\int_{t_{0}}^{t}G(\tau)d\tau\right)\cup\bigcup_{s=t_{0}}^{t}\left(dB_{s}\oplus\int_{s}^{t}G(\tau)d\tau\right),\qquad
t\in[t_{0},T].$

Moreover, with a little abuse of this infinitesimal notation, we get two
differential formulations

$d\Theta_{t}=\oplus G_{t}dt\cup
dB_{t}\qquad\textrm{and}\qquad\Theta_{t+dt}=(\Theta_{t}\oplus G_{t}dt)\cup
dB_{t}.$

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