\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 86, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/86\hfil Random dynamical systems]
{Random dynamical systems on time scales}

\author[C. Lungan , V. Lupulescu \hfil EJDE-2012/86\hfilneg]
{Cristina Lungan, Vasile Lupulescu}  % in alphabetical order

\address{Cristina Lungan \newline
Gheorghe Tatarascu School of Targu Jiu, 23 August 47, Romania}
\email{crisslong@yahoo.com}

\address{Vasile Lupulescu \newline
Constantin Brancusi University of Targu Jiu, Republicii 1, Romania}
\email{lupulescu\_v@yahoo.com}

\thanks{Submitted January 12, 2012. Published May 31, 2012.}
\subjclass[2000]{34N05, 37H10, 26E70}
\keywords{Differential equation; random variable; time scale}

\begin{abstract}
The purpose of this paper is to prove the existence and uniqueness of
solution for random dynamic systems on time scales.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

The theory of dynamic systems on time scales allows us to study both
continuous and discrete dynamic systems simultaneously. Since Hilger's
initial work \cite{hil-88} there has been significant growth in the theory
of dynamic systems on time scales, covering a variety of different
qualitative aspects. We refer to the books \cite{boh,boh2}, and the papers
\cite{agra,agra2,til,til1}. In recent years, some authors studied stochastic
differential equations on time scales \cite{bohs,gro,san}. The main
theoretical and practical aspects of probability theory and stochastic
differential equations can be found in books \cite{ch,ol}. The organization
of this paper is as follows. Section 2 presents a few definitions and
concepts of time scales. Also, the notion of stochastic process on a time
scale is introduced. In Section 3 we prove the existence and uniqueness of
solution for the random dynamic systems on time scales.

\subsection*{Preliminaries}

 By a \emph{time scale} $\mathbb{T}$ we mean any
closed subset of $\mathbb{R}$. Then $\mathbb{T}$ is a complete metric space
with the metric defined by $d(t,s):=|t-s|$ for $t,s\in \mathbb{T}$. Since a
time scale $\mathbb{T}$ is not connected in generally, we need the concept
of jump operators. The \emph{forward} \emph{jump operator} $\sigma :
\mathbb{T}\to \mathbb{T}$ is defined by $\sigma (t):=\inf \{s\in
\mathbb{T}:s>t\}$, while the \emph{backward jump operator} $\rho :\mathbb{T
}\to \mathbb{T}$ is defined by $\rho (t):=\sup \{s\in \mathbb{T}
:s<t\}$. In this definition we put $\inf \emptyset =\sup \mathbb{T}$ and 
$\sup \emptyset =\inf \mathbb{T}$.\ The \emph{graininess function} $\mu :
\mathbb{T}\to [ 0,\infty )$ is defined by $\mu (t):=\sigma
(t)-t$. If $\sigma (t)>t,$ we say $t$ is a \emph{right-scattered point},
while if $\rho (t)<t,$ we say $t$ is a \emph{left-scattered point}. Points
that are right-scattered and left-scattered at the same time will be called
\emph{isolated points}. A point $t\in \mathbb{T}$ such that $t<\sup
\mathbb{T}$ and $\sigma (t)=t$, is called a \emph{right-dense point}. A
point $t\in \mathbb{T}$ such that $t>\inf \mathbb{T}$ and $\rho (t)=t$, is
called a \emph{left-dense point}. Points that are right-dense and
left-dense at the same time will be called \emph{dense points}. The set
$\mathbb{T}^{\kappa }$ is defined to be $\mathbb{T}^{\kappa }=\mathbb{T}
\setminus \{m\}$ if $\mathbb{T}$ has a left-scattered maximum $m$,
otherwise $\mathbb{T}^{\kappa }=\mathbb{T}$.\ Given a time scale interval
 $[a,b]_{\mathbb{T}}:=\{t\in \mathbb{T}:a\leq t\leq b\}$, then $[a,b]_{\mathbb{
T}}^{\kappa }$ denoted the interval $[a,b]_{\mathbb{T}}$ if $a<\rho (b)=b$
and denote the interval $[a,b)_{\mathbb{T}}$ if $a<\rho (b)<b$. In fact, $
[a,b)_{\mathbb{T}}=[a,\rho (b)]_{\mathbb{T}}$. Also, for $a\in \mathbb{T}$,
we define $[a,\infty )_{\mathbb{T}}=[a,\infty )\cap \mathbb{T}$. If $\mathbb{
T}$\ is a bounded time scale, then $\mathbb{T}$\ can be identified with 
$[\inf \mathbb{T},\sup \mathbb{T}]_{\mathbb{T}}$.

If $t_0\in \mathbb{T}$ and $\delta >0$, then we define the following
neighborhoods of $t_0$: $U_{\mathbb{T}}(t_0,\delta ):=(t_0-\delta
,t_0+\delta )\cap \mathbb{T}$, $U_{\mathbb{T}}^{+}(t_0,\delta
):=[t_0,t_0+\delta )\cap \mathbb{T}$, and $U_{\mathbb{T}
}^{-}(t_0,\delta ):=(t_0-\delta ,t_0]\cap \mathbb{T}$.

\begin{definition}[\cite{boh}] \label{def1} \rm
 A function $f:\mathbb{T}\to \mathbb{R}$ is called \emph{regulated} if its right-sided
limits exist (finite) at all right-dense points in $\mathbb{T}$, and its
left-sided limits exist (finite) at all left-dense points in $\mathbb{T}$. A
function $f:\mathbb{T}\to \mathbb{R}$ is called \emph{rd-continuous
} if it is continuous at all right-dense points in $\mathbb{T}$ and its
left-sided limits exist (finite) at all left-dense points in $\mathbb{T}$.
\end{definition}

Obviously, a continuous function is rd-continuous, and a rd-continuous
function is regulated (\cite[Theorem 1.60]{boh}).

\begin{definition} \label{def2}\rm
 A function $f:[a,b]_{\mathbb{T}}\times
\mathbb{R}\to \mathbb{R}$ is called Hilger continuous if $f$ is
continuous at each point $(t,x)$ where $t$ is right-dense, and the limits
\[
\lim_{(s,y)\to (t^{-},x)} f(s,y)\quad\text{and}\quad \lim_{y\to x} f(t,y)
\]
both exist and are finite at each point $(t,x)$ where $t$ is
left-dense.
\end{definition}

\begin{definition}[\cite{boh}] \label{def3} \rm
 Let $f:\mathbb{T}\to \mathbb{R}$ and $t\in \mathbb{T}^{\kappa }$.
 Let $f^{\Delta }(t)\in \mathbb{R}$ (provided it exists) with the property
that for every $\varepsilon >0$,
there exists $\delta >0$ such that
\begin{equation}
| f(\sigma (t))-f(s)-f^{\Delta }(t)[\sigma (t)-s]| \leq
\varepsilon | \sigma (t)-s|  \label{a-delta}
\end{equation}
for all $s\in U_{\mathbb{T}}(t,\delta )$. We call $f^{\Delta }(t)$ the
\emph{delta} (or \emph{Hilger}) derivative ($\Delta $-derivative for
short) of $f$ at $t$. Moreover, we say that $f$ is delta differentiable ($
\Delta $-differentiable for short) on $\mathbb{T}^{\kappa }$ provided $f(t)$
exists for all $t\in \mathbb{T}^{\kappa }$.
\end{definition}

The following result will be very useful.

\begin{proposition}[{\cite[Theorem 1.16]{boh}}] \label{prop1}
Assume that $f:\mathbb{T}\to \mathbb{R}$ and $t\in \mathbb{T}^{\kappa }$.
\begin{itemize}
\item[(i)]  If $f$ is $\Delta$-differentiable
at $t$, then $f$ is continuous at $t$.


\item[(ii)] If $f$ is continuous at $t$ and $t$ is right-scattered, then $f$
 is $\Delta $-differentiable at $t$ with
\[
f^{\Delta }(t)=\frac{f(\sigma (t))-f(t)}{\sigma (t)-t}.
\]

\item[(iii)]  If $f$  is $\Delta $-differentiable at $t$ and 
$t$ is right-dense then
\[
f^{\Delta }(t)=\lim_{s\to t} \frac{f(t)-f(s)}{t-s}.
\]

\item[(iv)] If $f$ is $\Delta $-differentiable
at $t$,  then $f(\sigma (t))=f(t)+\mu (t)f^{\Delta }(t)$.
\end{itemize}
\end{proposition}

It is known \cite{guse1} that for every $\delta >0$ there exists at least
one partition $P:a=t_0<t_1<\dots <t_n=b$ of $[a,b)_{\mathbb{T}}$ such
that for each $i\in \{1,2,\dots,n\}$ either $t_i-t_{i-1}\leq \delta $ or $
t_i-t_{i-1}>\delta $ and $\rho (t_i)=t_{i-1}$. For given $\delta >0$ we
denote by $\mathcal{P}([a,b)_{\mathbb{T}},\delta )$ the set of all
partitions $P:a=t_0<t_1<\dots<t_n=b$ that possess the above property.

Let $f:\mathbb{T}\to \mathbb{R}$ be a bounded function on
 $[a,b)_{\mathbb{T}}$, and let $P:a=t_0<t_1<\dots<t_n=b$ be a partition of
 $[a,b)_{\mathbb{T}}$. In each interval $[t_{i-1},t_i)_{\mathbb{T}}$,where
 $ 1\leq i\leq n$, we choose an arbitrary point $\xi _i$ and form the sum
\[
S=\sum_{i=1}^n(t_i-t_{i-1})f(\xi _i).
\]
We call $S$ a Riemann $\Delta $-sum of $f$ corresponding to the partition
$P$.

\begin{definition}[\cite{guse}] \label{def4} \rm
 We say that $f$ is Riemann $\Delta $-integrable from $a$ to $b$ 
(or on $[a,b)_{\mathbb{T}}$) if there
exists a number $I$ with the following property: for each $\varepsilon >0$
there exists $\delta >0$ such that $|S-I|<\varepsilon $ for every Riemann 
$\Delta $-sum $S$ of $f$ corresponding to a partition 
$P\in \mathcal{P}([a,b)_{\mathbb{T}},\delta )$ independent of the way in
 which we choose $\xi_i\in [ t_{i-1},t_i)_{\mathbb{T}}$,
 $i=1,2,\dots,n$. It is easily
seen that such a number $I$ is unique. The number $I$ is the Riemann 
$\Delta$-integral of $f$ from $a$ to $b$, and we will denote it by 
$\int_a^{b}f(t)\Delta t$.
\end{definition}

\begin{proposition}[{\cite[Theorem 5.8]{guse}}] \label{prop2}
A bounded function $f:[a,b)_{\mathbb{T}}\to \mathbb{R}$
 is Riemann $\Delta $-integrable on $[a,b)_{\mathbb{T}}$
if and only if the set of all right-dense points of
 $[a,b)_{\mathbb{T}}$ at which $f$ is discontinuous is a set of 
$\Delta$-measure zero. 
\end{proposition} 

It is no difficult to see that every regulated function on a compact
interval is bounded (see \cite[Theorem 1.65]{boh}). Then we get that every
regulated function $f:[a,b]_{\mathbb{T}}\to \mathbb{R}$, is Riemann 
$\Delta $-integrable from $a$ to $b$.

\begin{proposition}[{\cite[Theorem 5.8]{hil-90}}] \label{prop3} 
Assume that $a,b\in \mathbb{T}$, $a<b$  and $f:\mathbb{T}
\to \mathbb{R}$ is rd-continuous. Then the integral
has the following properties.
\begin{itemize}
\item[(i)] If $\mathbb{T}=\mathbb{R}$, then 
$\int_a^{b}f(t)\Delta t=\int_a^{b}f(t)dt$, where the integral on
the right-hand side is the Riemann integral.

\item[(ii)] If $\mathbb{T}$ consists of
isolated points, then
\[
\int_a^{b}f(t)\Delta t=\sum_{t\in [ a,b)_{\mathbb{T}}}\mu (t)f(t).
\]
\end{itemize}
\end{proposition}

If $f,g:\mathbb{T}\to \mathbb{R}$ are Riemann $\Delta $-integrable
on $[a,b)_{\mathbb{T}}$, then $\lambda f$, $f+g$ and 
$| f| $ are are Riemann $\Delta $-integrable on $[a,b)_{\mathbb{T}}$,
and the following properties are true \cite{boh}:
\begin{equation} 
\begin{gathered}
\int_a^{b}(\lambda f)(t)\Delta t=\lambda
\int_a^{b}f(t)\Delta t,\quad \lambda \in \mathbb{R}, \\
\int_a^{b}(f+g)(t)\Delta t=\int_a^{b}f(t)\Delta
t+\int_a^{b}g(t)\Delta t, \\
\int_a^{b}f(t)\Delta t=-\int_{b}^{a}f(t)\Delta t\\
| \int_a^{b}f(t)\Delta t| \leq
\int_a^{b}| f(t)| \Delta t, \\
\int_a^{b}f(t)\Delta t=\int_a^{c}f(t)\Delta
t+\int_{c}^{b}f(t)\Delta t,\quad a<c<b,
\end{gathered}
\label{integral-01}
\end{equation}

\begin{definition}[\cite{boh}] \label{def5}
 A function $g:\mathbb{T} \to \mathbb{R}$ is called a $\Delta $-antiderivative
 of $f: \mathbb{T}\to \mathbb{R}$ if $g^{\Delta }(t)=f(t)$ for all $t\in
\mathbb{T}^{\kappa }$.
\end{definition}
One can show that each rd-continuous function has a $\Delta $-antiderivative
\cite[Theorem 1.74]{boh}.

\begin{proposition}[{\cite[Theorem 4.1]{guse}}] \label{prop4}
Let $f:\mathbb{T}\to \mathbb{R}$ be Riemann $\Delta $-integrable function on 
$[a,b)_{\mathbb{T}}$. If $f$ has a $\Delta $-antiderivative 
$g:[a,b]_{\mathbb{T}}\to \mathbb{R}$, then 
$\int_a^{b}f(t)\Delta t=g(b)-g(a)$.
In particular, $\int_t^{\sigma (t)}f(s)\Delta s=\mu (t)f(t)$ for
all $t\in [ a,b)_{\mathbb{T}}$ (see \cite[Theorem 1.75]{boh})
\end{proposition}

\begin{proposition}[{\cite[Theorem 4.3]{guse}}] \label{prop5} 
Let $f:\mathbb{T}\to \mathbb{R}$ be a function which is Riemann 
$\Delta $-integrable from $a$  to $b$. For
$t\in [ a,b]_{\mathbb{T}}$, let $g(t)=\int_a^{t}f(t)\Delta t$.
Then $g$ is continuous on $[a,b]_{\mathbb{T}}$.
Further, let $t_0\in [ a,b)_{\mathbb{T}}$ and let
$f$  be arbitrary at $t_0$ if $t_0$ is
right-scattered, and let $f$ be continuous at $t_0$ if
$t_0$ is right-dense. Then $g$ is $\Delta $-differentiable at
$t_0$ and $g^{\Delta }(t_0)=f(t_0)$.
\end{proposition}

\begin{lemma} \label{lem1}
Let $g:\mathbb{R}\to \mathbb{R}$
 be a continuous and nondecreasing function. If $s,t\in \mathbb{T}$
 with $s\leq t$, then
\[
\int_{s}^{t}g(\tau )\Delta \tau \leq \int_{s}^{t}g(\tau )d\tau .
\]
\end{lemma}

\subsection*{Stochastic process on time scales}
 Denote by $\mathcal{B}$ the $\sigma $-algebra of all Borel subsets of $\mathbb{R}$. 
Let $(\Omega , \mathcal{F},P)$ be a complete probability measure space.
 A function $X(\cdot ):\Omega \to \mathbb{R}$ is called a random variable if $X$ is a
measurable function from $(\Omega ,\mathcal{F})$ into $(\mathbb{R},\mathcal{B
})$; that is, $X^{-1}(B):=\{\omega \in \Omega ;X(\omega )\in B\}\in \mathcal{
F}$ for all $B\in \mathcal{B}$. A time scale stochastic process is a
function $X(\cdot ,\cdot ):[a,b]_{\mathbb{T}}\times \Omega \to
\mathbb{R}$ such that $X(t,\cdot ):\Omega \to \mathbb{R}$ is a
random variable for each $t\in \mathbb{T}$. For each point $\omega \in
\Omega $, the function on $\mathbb{T}$ given by $t\mapsto X(t,\omega )$ is
will be called a trajectory (or a sample path) of the time scale stochastic
process $X(\cdot ,\cdot )$ corresponding to $\omega $. A time scale
stochastic process $X(\cdot ,\cdot )$ is said to be regulated
(rd-continuous, continuous)\ if there exists $\Omega _0\subset \Omega $
with $P(\Omega _0)=1$ and such that the trajectory $t\mapsto X(t,\omega )$
is a regulated (rd-continuous, continuous) function on $[a,b]_{\mathbb{T}}$
for each $\omega \in \Omega _0$. Let $X(\cdot )$ and $Y(\cdot )$ be two
random variables. If there exists $\Omega _0\subset \Omega $ with $
P(\Omega _0)=1$ and such that $X(\omega )=Y(\omega )$ for all $\omega \in
\Omega _0$, then we will write $X(\omega )=_PY(\omega )$. Similarly for
the inequalities. Let $X(\cdot ,\cdot )$ and $Y(\cdot ,\cdot )$ be two time
scale stochastic processes. If there exists $\Omega _0\subset \Omega $
with $P(\Omega _0)=1$ and such that for each $\omega \in \Omega _0$ we
have $X(t,\omega )=Y(t,\omega )$ for all $t\in [ a,b]_{\mathbb{T}}$,
then we will write $X(t,\omega )=_PY(t,\omega )$, $t\in [ a,b]_{
\mathbb{T}}$. Similarly for the inequalities. 

\begin{lemma} \label{lem2}
 Let $X(\cdot ,\cdot ):[a,b]_{\mathbb{T}
}\times \Omega \to \mathbb{R}$ be a time scale stochastic
process. If there exists $\Omega _0\subset \Omega $ with
 $P(\Omega _0)=1$ such that the function $t\mapsto X(t,\omega )$
 is Riemann $\Delta $-integrable on $[a,b)_{\mathbb{T}}$
 for every $\omega \in \Omega _0$, then the
function $Y(\cdot ,\cdot ):[a,b]_{\mathbb{T}}\times \Omega \to \mathbb{R}$
 given by
\[
Y(t,\omega )=\int_a^{t}X(s,\omega )\Delta s,\quad t\in [ a,b]_{\mathbb{T}}
\]
is a continuous time scale stochastic process.
\end{lemma}

\begin{proof}
 From Proposition \ref{prop5}, it follows that the function 
$t\mapsto \int_a^{t}X(s,\omega )\Delta s$ is continuous for each 
$\omega \in \Omega _0$. Since the Riemann $\Delta $-integral is a limit of the
finite sum $S(\omega) =\sum_{i=1}^n(t_i-t_{i-1})X(\xi_i,\omega )$ of measurable 
functions, we have that $\omega \mapsto \int_a^{t}X(s,\omega )\Delta s$ is a
 measurable function. Therefore, $Y(\cdot ,\cdot )$ is a continuous time 
scale stochastic process. 
\end{proof}

\section{Random initial value problem on time scales}

In the following, consider an initial value problem of the form
\begin{equation}
\begin{gathered}
X^{\Delta }(t,\omega )=_Pf(t,X(t,\omega ),\omega ), \quad
 t\in [a,b]_{\mathbb{T}}^{\kappa } \\
X(a,\omega )=_PX_0(\omega ),
\end{gathered}  \label{ivp}
\end{equation}
where $X_0:\Omega \to \mathbb{R}$ is a random variable and 
$f:[a,b]_{\mathbb{T}}^{\kappa }\times \mathbb{R}\times \Omega \to
\mathbb{R}$ satisfies the following assumptions:
\begin{itemize}
\item[(H1)] $f(t,x,\cdot ):\Omega \to \mathbb{R}$ is a
random variable for all $(t,x)\in [ a,b]_{\mathbb{T}}^{\kappa }\times
\mathbb{R}$,

\item[(H2)] with $P.1$, the function $f(\cdot ,\cdot ,\omega
):[a,b]_{\mathbb{T}}^{\kappa }\times \mathbb{R}\to \mathbb{R}$ is a
Hilger continuous function at every point $(t,x)\in [ a,b]_{\mathbb{T}
}^{\kappa }\times \mathbb{R}$; that is, there exists $\Omega _0\subset
\Omega $ with $P(\Omega _0)=1$ and such that for each $\omega \in \Omega
_0$, the function $(t,x)\mapsto f(t,x,\omega )$ is Hilger
continuous.
\end{itemize}

\begin{definition} \label{def6} \rm
 By a solution of \eqref{ivp} we mean a time
scale stochastic process $X(\cdot ,\cdot ):[a,b]_{\mathbb{T}}^{\kappa
}\times \Omega \to \mathbb{R}$ that satisfies conditions in
\eqref{ivp}. A solution $X(\cdot ,\cdot )$ is unique if $X(t,\omega
)=_PY(t,\omega )$, $t\in [ a,b]_{\mathbb{T}}^{\kappa }$ for any time
scale stochastic process $Y(\cdot ,\cdot ):[a,b]_{\mathbb{T}}^{\kappa
}\times \Omega \to \mathbb{R}$ which is a solution of \eqref{ivp}.
\end{definition}

Obviously, if there exists $\Omega _0\subset \Omega $ with 
$P(\Omega_0)=1$ and such that for each $\omega \in \Omega _0$ we have
 $|X(t,\omega )-Y(t,\omega )|=0$ for all $t\in [ a,b]_{\mathbb{T}}$,
then $X(t,\omega )=_PY(t,\omega )$, $t\in [ a,b]_{\mathbb{T}
}^{\kappa }$; that is, if $|X(t,\omega )-Y(t,\omega )|=_P0$ for all $t\in
[ a,b]_{\mathbb{T}}^{\kappa }$, then $X(t,\omega )=_PY(t,\omega )$, $
t\in [ a,b]_{\mathbb{T}}^{\kappa }$.

\begin{remark} \label{rmk1}\rm
 We can consider the random differential
equation \eqref{ivp} as a family (with respect to parameter $\omega $) of
deterministic differential equations, namely
\begin{equation}
\begin{gathered}
X^{\Delta }(t,\omega )=f(t,X(t,\omega ),\omega ), \quad t\in [ a,b]_{
\mathbb{T}}^{\kappa } \\
X(a,\omega )=X_0(\omega ).
\end{gathered}  \label{divp}
\end{equation}
\end{remark}

Generally, is not correct to solve each problem \eqref{divp} to obtain the
solutions of \eqref{ivp}. Let us give two examples.

\begin{example} \label{examp1}\rm
Let $(\Omega ,\mathcal{F},P)$ be a complete
probability measure space. Consider an initial value problem of the form
\begin{equation}
\begin{gathered}
X^{\Delta }(t,\omega )=K(\omega )X^{2}(t,\omega ),\quad 
t\in [0,\infty )_{\mathbb{R}} \\
X(0,\omega )=1,
\end{gathered}  \label{ex1}
\end{equation}
where $K:\Omega \to (0,\infty )$ is a random variable. It is easy to
see that, for each $\omega \in \Omega $,
 $X(t,\omega )=\frac{1}{1-K(\omega )t}$ is a solution of \eqref{ex1} 
on the interval $[0,1/K(\omega )]$. Since
for each $a\geq 0$ we have that $P(1/K(\omega )>a)<1$, it follows that not
all solutions $X(\cdot ,\omega )$ are well defined on some common interval 
$[0,a)$.
\end{example}

\begin{example} \label{examp2} \rm
 Let $(\Omega ,\mathcal{F},P)$ be a complete
probability measure space and let $\Omega _0\notin \mathcal{F}$. It is
easy to check that, for each $\omega \in \Omega $, the function
 $X(\cdot,\cdot ):[0,1]_{\mathbb{R}}\times \Omega \to \mathbb{R}$, given by
\[
X(t,\omega )=\begin{cases}
0 &\text{if } \omega \in \Omega _0 \\
t^{3/2} &\text{if }\omega \in \Omega \setminus \Omega _0,
\end{cases}
\]
is a solution of the initial-value problem
\begin{gather*}
X^{\Delta }(t,\omega )=\frac{3}{2}X(t,\omega ), \quad
t\in [ 0,\infty)_{\mathbb{R}} \\
X(0,\omega )=0.
\end{gather*}
But $X(\cdot ,\cdot )$ is not a stochastic process. Indeed, we have that
\[
\{\omega \in \Omega ;X(1,\omega )\in [ -\frac{1}{2},\frac{1}{2}
]\}=\Omega _0\notin \mathcal{F},
\]
that is, $\omega \mapsto X(1,\omega )$ is not a measurable function.
\end{example}

Using  Propositions \ref{prop4} and \ref{prop5} and \cite[Lemma 2.3]{til1}, it is easy to
prove the following result.

\begin{lemma} \label{lem3}.
 A time scale stochastic process 
$X(\cdot ,\cdot ):[a,b]_{\mathbb{T}}^{\kappa }\times \Omega \to
\mathbb{R}$ is the solution of the problem \eqref{ivp} if and only
if $X(\cdot ,\cdot )$ is a continuous time scale stochastic
process and it satisfies the following random integral equation
\begin{equation}
X(t,\omega )=_PX_0(\omega )+\int_a^{t}f(s,X(s,\omega ),\omega )\Delta s
, t\in [ a,b]_{\mathbb{T}}.  \label{inteq}
\end{equation}
\end{lemma}

The following results is known as Gronwall's inequality on time scale and
will be used in this paper.

\begin{lemma}[{\cite[Lemma 3.1]{til}}] \label{lem4}
Let an rd-continuous time scale stochastic processes 
$X(\cdot ,\cdot ),Y(\cdot,\cdot ):[a,b]_{\mathbb{T}}^{\kappa }\times
 \Omega \to \mathbb{R}_{+} $ be such that
\[
X(t,\omega )\leq _PY(t,\omega )+\int_a^{t}q(s)X(s,\omega )\Delta s,
\quad t\in [ a,b]_{\mathbb{T}},
\]
where $1+\mu (t)q(t)\neq 0$, for all
$t\in [ a,b]_{\mathbb{T}}$. Then we have
\[
X(t,\omega )\leq _PY(t,\omega )+e_{q}(t,a)\int_a^{t}q(s)Y(s,\omega )
\frac{1}{e_{q}(\sigma (s),a)}\Delta s, \quad t\in [ a,b]_{\mathbb{T}}.
\]
\end{lemma}

\begin{theorem} \label{thm1}
Let $f:[a,b]_{\mathbb{T}}^{\kappa}\times \mathbb{R}\times \Omega \to \mathbb{R}$ 
satisfy {\rm (H1)--(H2)} and assume that there exists a rd-continuous time 
scale stochastic process $L(\cdot ,\cdot ):[a,b]_{\mathbb{T}}^{\kappa }\times \Omega
\to \mathbb{R}$ such that
\begin{equation}
|f(t,x,\omega )-f(t,y,\omega )|\leq L(t,\omega )|x-y|  \label{Lip}
\end{equation}
for every $t\in [ a,b]_{\mathbb{T}}^{\kappa }$ and
every $x,y\in \mathbb{R}$ with $P.1$. Let 
$X_0:\Omega \to \mathbb{R}$ a random variable such that
\begin{equation}
|f(t,X_0(\omega ),\omega )|\leq _PM, \quad t\in [ a,b]_{\mathbb{T}
}^{\kappa },  \label{M}
\end{equation}
where $M>0$ is a constant. Then  problem \eqref{ivp}
has a unique solution.
\end{theorem}

\begin{proof}. 
To prove the theorem we apply the method of
successive approximations (see \cite{til}). For this, we define a sequence
of functions $X_n(\cdot ,\cdot ):[a,b]_{\mathbb{T}}^{\kappa }\times \Omega
\to \mathbb{R}$, $n\in \mathbb{N}$, as follows:
\begin{equation}
\begin{gathered}
X_0(t,\omega )=X_0(\omega ) \\
X_n(t,\omega )=X_0(\omega )+\int_a^{t}f(s,X_{n-1}(s,\omega
),\omega )\Delta s, \quad  n\geq 1,
\end{gathered}\label{succ}
\end{equation}
for every $t\in [ a,b]_{\mathbb{T}}^{\kappa }$ and every 
$\omega \in \Omega $. First, using \eqref{M} and the Lemma \ref{lem1}, we observe that
\begin{align*}
|X_1(t,\omega )-X_0(t,\omega )|
&\leq \big|\int_a^{t}f(s,X_0(\omega ),\omega )\Delta s\big|
 \leq \int_a^{t}| f(s,X_0(\omega ),\omega )| \Delta s \\
&\leq \int_a^{t}| f(s,X_0(\omega ),\omega )| ds\leq _PM(t-a) \\
&\leq  M(b-a),\quad t\in [ a,b]_{\mathbb{T}}.
\end{align*}
We prove by induction that for each integer $n\geq 2$ the following estimate
holds
\begin{equation}
|X_n(t,\omega )-X_{n-1}(t,\omega )|\leq _PM\widetilde{L}(\omega )\frac{
(t-a)^n}{n!}\leq M\widetilde{L}(\omega )\frac{(b-a)^n}{n!},\text{ }t\in
[ a,b]_{\mathbb{T}},  \label{ind}
\end{equation}
where $\widetilde{L}(\omega )=\sup_{[a,b]_{\mathbb{T}}} L(t,\omega
)$. Suppose that \eqref{ind} holds for $n=k\geq 2$. Then, using \eqref{Lip},
\eqref{M} and Lemma \ref{lem1}, we obtain
\begin{align*}
|X_{k+1}(t,\omega )-X_{k}(t,\omega )|
&\leq \int_a^{t}|f(s,X_{k}(s,\omega ),\omega )-f(s,X_{k-1}(s,\omega
),\omega )|\Delta s \\
&\leq _P\widetilde{L}(\omega )\int_a^{t}|X_{k}(s,\omega
)-X_{k-1}(s,\omega )|\Delta s\\
&\leq _P\widetilde{L}(\omega )\frac{M}{k!}
\int_a^{t}(s-a)^{k}\Delta s \\
&\leq \widetilde{L}(\omega )\frac{M}{k!}\int_a^{t}(s-a)^{k}ds\\
&=M \widetilde{L}(\omega )\frac{(t-a)^{k+1}}{(k+1)!}\\
&\leq M\widetilde{L}(\omega ) \frac{(b-a)^{k+1}}{(k+1)!},\quad 
t\in [ a,b]_{\mathbb{T}}.
\end{align*}
Thus, \eqref{ind} is true for $n=k+1$ and so \eqref{ind} holds for all 
$n\geq 2$. Further, we show that for every $n\in \mathbb{N}$ the functions 
$X_n(\cdot ,\omega ):[a,b]_{\mathbb{T}}\to \mathbb{R}$ are
continuous with $P.1$. Let $\varepsilon >0$ and
 $t,s\in [ a,b]_{\mathbb{T}}$ be such that $|t-s|<\varepsilon /M$. We have
\begin{align*}
|X_1(t,\omega )-X_1(s,\omega )|
&=| \int_a^{t}f(\tau,X_0(\omega ),\omega )\Delta \tau -\int_a^{s}f(\tau
,X_0(\omega ),\omega )\Delta \tau |  \\
&=| \int_{s}^{t}f(\tau ,X_0(\omega ),\omega )\Delta \tau | \\
&\leq \int_{s}^{t}| f(\tau ,X_0(\omega),\omega )| \Delta \tau \\
&\leq \int_{s}^{t}| f(\tau ,X_0(\omega ),\omega )| d\tau  \\
&\leq _PM|t-s|<\varepsilon
\end{align*}
and so $t\mapsto X_1(t,\omega )$ is continuous with $P.1$. Since for each 
$n\geq 2$
\begin{align*}
&|X_n(t,\omega )-X_n(s,\omega )|\\
&=| \int_a^{t}f(\tau ,X_{n-1}(\tau ,\omega ),\omega )\Delta \tau -\int_a^{s}f(\tau
,X_{n-1}(\tau ,\omega ),\omega )\Delta \tau |  \\
& \leq \int_{s}^{t}| f(\tau ,X_{n-1}(\tau ,\omega ),\omega )| \Delta \tau \\
&\leq \int_{s}^{t}| f(\tau ,X_0(\omega ),\omega )| \Delta \tau 
 +\int_{s}^{t}| f(\tau ,X_{n-1}(\tau ,\omega ),\omega )
 -f(\tau ,X_0(\omega ),\omega )| \Delta \tau \\
&\leq \int_{s}^{t}| f(\tau ,X_0(\omega ),\omega )|\Delta \tau \\
&\quad +\sum_{k=1}^{n-1}\int_{s}^{t}| f(\tau ,X_{k}(\tau
,\omega ),\omega )-f(\tau ,X_{k-1}(\tau ,\omega ),\omega )| \Delta \tau 
\end{align*}
then, by induction, we obtain
\[
|X_n(t,\omega )-X_n(s,\omega )|\leq _PM(
1+\sum_{k=1}^{n-1}\frac{\widetilde{L}(\omega )^{k-1}(b-a)^{k}}{k!}
) |t-s|\to 0
\]
 as $s\to t$ with $P.1$. Therefore, for every $n\in \mathbb{N}$ the
function $X_n(\cdot ,\omega ):[a,b]_{\mathbb{T}}\times \Omega \to \mathbb{R}$ is
continuous with $P.1$. Now, using Lemma \ref{lem3} and \eqref{succ},
we deduce that the functions $X_n(t,\cdot ):\Omega \to \mathbb{R}$ are
measurable. Consequently, it follows that for every $n\in \mathbb{N}$ the
function $X_n(\cdot ,\cdot ):[a,b]_{\mathbb{T}}\times \Omega \to
\mathbb{R}$ is a time scale stochastic process.

Further, we shall show that the sequence $(X_n(t,\cdot ))_{n\in \mathbb{N}}
$ is uniformly convergent with $P.1$. Denote
\[
Y_n(t,\omega )=|X_{n+1}(t,\omega )-X_n(t,\omega )|\text{, \ }n\in
\mathbb{N}.
\]
Since
\[
Y_n(t,\omega )-Y_n(s,\omega )\leq _P\widetilde{L}(\omega
)\int_{s}^{t}| X_n(\tau ,\omega )-X_{n-1}(\tau ,\omega
)| \Delta \tau
\]
then, reasoning as above, we deduce that the functions $t\mapsto
Y_n(t,\omega )$ are continuous with $P.1$. Now, using \eqref{ind}, we
obtain
\[
\underset{t\in [ a,b]_{\mathbb{T}}}{\sup }| X_n(t,\omega
)-X_{m}(t,\omega )| \leq \sum_{k=m}^{n-1}\underset{t\in
[ a,b]_{\mathbb{T}}}{\sup }Y_{k}(t,\omega )\leq
_PM\sum_{k=m}^{n-1}\frac{\widetilde{L}(\omega )^{k}(b-a)^{k+1}}{
(k+1)!}
\]
for all $n>m>0$. Since the series $\sum_{n=1}^{\infty }\widetilde{L}
(\omega )^{n-1}(b-a)^n/n!$ converges with $P.1$, then for each $
\varepsilon >0$ there exists $n_0\in \mathbb{N}$ such that
\begin{equation}
\underset{t\in [ a,b]_{\mathbb{T}}}{\sup }| X_n(t,\omega
)-X_{m}(t,\omega )| \leq _P\varepsilon \text{ \ \ for all }
n,m\geq n_0.  \label{a}
\end{equation}
Hence, since $([a,b]_{\mathbb{T}},|\cdot |)$ is a complete metric space, it
follows that there exists $\Omega _0\subset \Omega $ such that $P(\Omega
_0)=1$ and for every $\omega \in \Omega _0$ the sequence $(X_n(t,\cdot
))_{n\in \mathbb{N}}$ is uniformly convergent. For $\omega \in \Omega _0$
denote $\widetilde{X}(t,\omega )=\underset{n\to \infty }{\lim }
X_n(t,\omega )$. Next, we define the function $X(\cdot ,\cdot ):[a,b]_{
\mathbb{T}}\times \Omega \to \mathbb{R}$ as follows: $X(\cdot
,\omega )=\widetilde{X}(\cdot ,\omega )$ if $\omega \in \Omega _0$, and $
X(\cdot ,\omega )$ as an arbitrary function if $\omega \in \Omega \setminus
\Omega _0$. Obviously, $X(\cdot ,\omega )$ is continuos with $P.1$. Since,
by Lemma \ref{lem2} and \eqref{succ}, the functions $\omega \to X_n(\cdot
,\omega )$ are measurable and $X(t,\omega )=\underset{n\to \infty }{
\lim }X_n(t,\omega )$ for every $t\in [ a,b]_{\mathbb{T}}$\ with $P.1
$, we deduce that $\omega \to X(t,\omega )$ is measurable for every $
t\in [ a,b]_{\mathbb{T}}$. Therefore, $X(\cdot ,\cdot ):[a,b]_{\mathbb{
T}}\times \Omega \to \mathbb{R}$ is a continuous time scale
stochastic process. We show that $X(\cdot ,\cdot )$ satisfies the random
integral equation \eqref{inteq}. For each $n\in \mathbb{N}$ we put $
G_n(t,\omega )=f(t,X_n(t,\omega ),\omega )$, $t\in [ a,b]_{\mathbb{
T}}$, $\omega \in \Omega $. Then $G_n(t,\omega )$ is rd-continuous time
scale stochastic process, and we have that
\[
\sup_{t\in [ a,b]_{\mathbb{T}}} | G_n(t,\omega
)-G_{m}(t,\omega )| \leq _P\widetilde{L}(\omega )
\sup_{t\in [ a,b]_{\mathbb{T}}} | X_n(t,\omega )-X_{m}(t,\omega
)| , \quad t\in [ a,b]_{\mathbb{T}}
\]
for all $n,m\geq n_0$. Using \eqref{a} we infer that the sequence
$(G_n(\cdot ,\omega ))_{n\in \mathbb{N}}$ is uniformly convergent with $P.1$.
 If we take $m\to \infty $, then for each $\varepsilon >0$ there
exists $n_0\in \mathbb{N}$ such that for every $n\geq n_0$ we have
\[
\sup_{t\in [ a,b]_{\mathbb{T}}} | G_n(t,\omega
)-f(t,X(t,\omega ),\omega )|
\leq _P\widetilde{L}(\omega )
\sup_{t\in [ a,b]_{\mathbb{T}}} | X_n(t,\omega
)-X(t,\omega )| \text{, \ }t\in [ a,b]_{\mathbb{T}}
\]
and so $\lim_{n\to \infty } | G_n(t,\omega
)-f(t,X(t,\omega ),\omega )| =0$ for all
$t\in [ a,b]_{\mathbb{T}}$\ with $P.1$. Also, it easy to see that
\[
\sup_{t\in [ a,b]_{\mathbb{T}}} |\int_a^{t}G_n(s,\omega )\Delta
s-\int_a^{t}f(s,X(s,\omega ),\omega )\Delta s|
\leq _P \widetilde{L}(\omega )\int_a^{t}| X_n(s,\omega
)-X(s,\omega )| \Delta s.
\]
Since the sequence $X(t,\omega )=\lim_{n\to \infty } X_n(t,\omega )$
uniformly with $P.1$, then it follows that
\[
\lim_{n\to \infty } |\int_a^{t}G_n(s,\omega )\Delta
s-\int_a^{t}f(s,X(s,\omega ),\omega )\Delta s| =0
\quad \forall t\in [ a,b]_{\mathbb{T}}\; \text{with }P.1.
\]
Now, we have
\begin{align*}
&\sup_{t\in [ a,b]_{\mathbb{T}}} | X(t,\omega
)-(X_0(\omega )+\int_a^{t}f(s,X(s,\omega ),\omega )\Delta s) | \\
&\leq \sup_{t\in [ a,b]_{\mathbb{T}}} | X(t,\omega )-X_n(t,\omega )|  \\
&\quad +\underset{t\in [ a,b]_{\mathbb{T}}}{\sup }| X_n(t,\omega
)-(X_0(\omega )+\int_a^{t}f(s,X_{n-1}(s,\omega ),\omega
)\Delta s) |  \\
&\quad +\sup_{t\in [ a,b]_{\mathbb{T}}} |
\int_a^{t}f(s,X_{n-1}(s,\omega ),\omega )\Delta
s-\int_a^{t}f(s,X(s,\omega ),\omega )\Delta s| .
\end{align*}
Using the two previous convergence
\[
\big| X(t,\omega )-\big(X_0(\omega
)+\int_a^{t}f(s,X(s,\omega ),\omega )\Delta s\big) \big| =0
\text{ for all }t\in [ a,b]_{\mathbb{T}}\ \text{with }P.1;
\]
that is, $X(\cdot ,\cdot )$ satisfies the random integral equation
\eqref{inteq}. Then, by Lemma \ref{lem3}, it follows that
 $X(\cdot ,\cdot )$ is the
solution of  \eqref{ivp}.

 Finally, we show the uniqueness of the
solution. For this, we assume that 
$X(\cdot ,\cdot ),Y(\cdot ,\cdot ):[a,b]_{\mathbb{T}}\times \Omega \to \mathbb{R}$
 are two solutions of \eqref{inteq}. Since
\[
|X(t,\omega )-Y(t,\omega )|\leq _P\int_a^{t}\widetilde{L}(\omega
)|X(s,\omega )-Y(s,\omega )|ds, \quad t\in [ a,b]_{\mathbb{T}},
\]
from Lemma \ref{lem4}, it follows that $|X(t,\omega )-Y(t,\omega )|\leq _P0$,
$t\in [ a,b]_{\mathbb{T}}$ and so, the proof is complete.
\end{proof}

Let $\mathbb{T}$ be an upper unbounded time scale. Then under suitable
conditions we can extend the notion of the solution of (2.1) from 
$[a,b]_{\mathbb{T}}^{\kappa }$ to 
$[a,\infty )_{\mathbb{T}}:=[a,\infty )\cap \mathbb{T}$, if we define $f$ 
on $[a,\infty )_{\mathbb{T}}\times \mathbb{R}\times
\Omega $ and show that the solution exists on each $[a,b]_{\mathbb{T}}$
where $b\in (a,\infty )_{\mathbb{T}}$, $a<\rho (b)$.

\begin{theorem} \label{thm2}
Assume that $f:[a,\infty )_{\mathbb{T}}\times
\mathbb{R}\times \Omega \to \mathbb{R}$ satisfies the
assumptions of  Theorem \ref{thm1} on each interval 
$[a,b]_{\mathbb{T}}$ with $b\in (a,\infty )_{\mathbb{T}}$,
 $a<\rho (b)$. If there is a constant 
$M>0$ such that $| f(t,x,\omega )|\leq _PM$ for all
$(t,x)\in [ a,b)_{\mathbb{T}}\times \mathbb{R}$, 
then the problem \eqref{ivp} has a unique
solution on $[a,\infty )_{\mathbb{T}}$.
\end{theorem}

\begin{proof} 
Let $X(t,\cdot )$ be the solution of \eqref{ivp}
which exists on $[a,b)_{\mathbb{T}}$ with $b\in (a,\infty )_{\mathbb{T}}$, 
$ a<\rho (b)$, and the value of $b$ cannot be increased. First, we observe
that $b$ is a left-scattered point, then $\rho (b)\in (a,b)_{\mathbb{T}}$
and the solution $X(t,\cdot )$ exists on $[a,\rho (b)]_{\mathbb{T}}$. But
then the solution $X(t,\cdot )$ exists also on $[a,b]_{\mathbb{T}}$, namely
by putting
\begin{align*}
X(b,\omega ) &=_P X(\rho (b),\omega )+\mu (b)X^{\Delta }(\rho (b),\omega )
\\
&=_P X(\rho (b),\omega )+\mu (b)f(\rho (b),X(\rho (b),\omega ),\omega ).
\end{align*}
If $b$ is a left-dense point, then their neighborhoods contain infinitely
many points to the left of $b$. Then, for any $t,s\in (a,b)_{\mathbb{T}}$
such that $s<t$, we have
\[
| X(t,\omega )-X(s,\omega )| \leq \int_{s}^{t}|
f(\tau ,X(\tau ,\omega ),\omega )| \Delta \tau \leq
_P M| t-s| .
\]
Taking limit as $s,t\to b^{-}$ and using Cauchy criterion for
convergence, it follows $\lim_{t\to b^{-}} X(t,\omega )$
exists and is finite with $P.1$. Further, we define $X_{b}(\omega )=_P
\lim_{t\to b^{-}} X(t,\omega )$ and consider the initial
value problem
\begin{gather*}
X^{\Delta }(t,\omega )=_Pf(\tau ,X(\tau ,\omega ),\omega ), \quad
t\in [ b,b_1]_{\mathbb{T}}, \quad b_1>\sigma (b), \\
X(b,\omega )=_PX_{b}(\omega ).
\end{gather*}
By Theorem \ref{thm1}, one gets that $X(t,\omega )$ can be continued beyond $b$,
contradicting our assumptions. Hence every solution $X(t,\omega )$ of
e\ref{ivp} exists on $[a,\infty )_{\mathbb{T}}$ and the proof is
complete.
\end{proof}

\section{Random linear systems on time scales}

Let $a:\Omega \to \mathbb{R}$ be a positively regressive random
variable; that is, $1+\mu (t)a(\omega )>0$ for all $t\in \mathbb{T}$ and
 $\omega \in \Omega $. Then, by Lemma \ref{lem2}, the function 
$(t,\omega )\mapsto e_{a(\omega )}(t,t_0)$ defined by
\[
e_{a(\omega )}(t,t_0)=_P\Big(\int_{t_0}^{t}\frac{\log (1+\mu (\tau
)a(\omega ))}{\mu (\tau )}\Delta \tau\Big) , \quad t_0,t\in \mathbb{T},
\]
is a continuous time scale stochastic process. For each fixed
$\omega \in \Omega $, the sample path $t\mapsto e_{a(\omega )}(t,t_0)$ is the
exponential function on time scales (see \cite{boh}). It easy to check that
the stochastic process $(t,\omega )\mapsto e_{a(\omega )}(t,t_0)$ is a
solution of the initial value problem (for deterministic case, see
 \cite[Theorem 2.33]{boh})
\begin{equation}
\begin{gathered}
X^{\Delta }(t,\omega )=_Pa(\omega )X(t,\omega ), \quad
t\in [t_0,b]_{\mathbb{T}}^{\kappa } \\
X(t_0,\omega )=_P1.
\end{gathered}  \label{exp}
\end{equation}
If $a:\Omega \to \mathbb{R}$ is bounded with $P.1$ then, by the
Theorems \ref{thm1}] and \ref{thm2}, it follows that \eqref{exp} has a unique solution on
 $[t_0,\infty )_{\mathbb{T}}$.

Further, consider the following nonhomogeneous initial value problem
\begin{equation}
\begin{gathered}
X^{\Delta }(t,\omega )=_Pa(\omega )X(t,\omega )+h(t,\omega ), \quad
t\in [ t_0,b]_{\mathbb{T}}^{\kappa } \\
X(t_0,\omega )=_PX_0(\omega ),
\end{gathered}  \label{ivp-l}
\end{equation}
where $a:\Omega \to \mathbb{R}$ is a positively regressive random
variable, $X_0:\Omega \to \mathbb{R}$ is a bounded random
variable, and $h(,\cdot ,):[a,b]_{\mathbb{T}}^{\kappa }\times \Omega
\to \mathbb{R}$ is a rd-continuous time scale stochastic
process.

\begin{theorem} \label{thm3}
Suppose that $a:\Omega \to \mathbb{R}$
 is a positively regressive and bounded random variable, 
$X_0:\Omega \to \mathbb{R}$  is a bounded random variable,
and $h(,\cdot ,):[t_0,\infty )_{\mathbb{T}}\times \Omega \to
\mathbb{R}$  is a rd-continuous time scale stochastic process. If
there is a constant $\nu >0$ such that $| h(t,\omega
)| \leq _P\nu $  for all
$t\in [ t_0,b)_{ \mathbb{T}}$ with $b\in (t_0,\infty )_{\mathbb{T}}$,
 $t_0<\rho (b)$, then the initial-value problem \eqref{ivp-l} has a unique
solution on $[t_0,\infty )_{\mathbb{T}}$.
\end{theorem}

\begin{proof}
 First, we observe that we put $f(t,x,\omega ):=a(\omega
)x+h(t,\omega )$, then $f$ satisfies the conditions ($H_1$) and ($H_{2}$).
Moreover,
\[
| f(t,x,\omega )-f(t,y,\omega )| \leq _P|a(\omega )| | x-y|
\]
for every $t\in [ t_0,\infty )_{\mathbb{T}}$ and every $x,y\in
\mathbb{R}$. Therefore, by the Theorem \ref{thm1}, it follows that \eqref{ivp-l} has
a unique solution on $[t_0,b]_{\mathbb{T}}^{\kappa }$. Further, let
$X(t,\cdot )$ be the solution of \eqref{ivp-l} which exists on
$[t_0,b)_{\mathbb{T}}$ with $b\in (t_0,\infty )_{\mathbb{T}}$, $t_0<\rho (b)$.
Also, let $N>0$ be such that $| a(\omega )| \leq _PN$.
Then we have
\begin{gather*}
| X(t,\omega )|
 \leq | X(t_0,\omega )| +\int_{t_0}^{t}| a(\omega )X(s,\omega )|
\Delta s+\int_{t_0}^{t}| h(s,\omega )| \Delta s\leq _P
\\
1+\nu (t-t_0)+N\int_{t_0}^{t}| X(s,\omega )| \Delta s.
\end{gather*}
Then, by the \cite[Corollary 6.8]{boh}, it follows that
\[
| X(t,\omega )| \leq _P(1+\frac{\nu }{N})e_{N}(t,t_0)-
\frac{\nu }{N}\leq (1+\frac{\nu }{N})e_{N}(b,t_0).
\]
Hence $| f(t,X(t,\omega ),\omega )| \leq _PM:=\nu +(1+
\frac{\nu }{N})e_{N}(b,t_0)$. Proceeding as in the proof of the
Theorem \ref{thm2}
it follows that the unique solution of \eqref{ivp-l} exists on
$[t_0,\infty )_{\mathbb{T}}$.
\end{proof}

\begin{theorem}[Variation of Constants] \label{thm4}
A continuous time scale stochastic process 
$X(\cdot ,\cdot ):[t_0,\infty )_{\mathbb{T}}\times
\Omega \to \mathbb{R}$ is a solution of the initial-value
problem \eqref{ivp-l} if and only if
\[
X(t,\omega )=_Pe_{a(\omega )}(t,t_0)X_0(\omega
)+\int_{t_0}^{t}e_{a(\omega )}(t,\sigma (s))h(s,\omega )\Delta s,
t\in [ t_0,\infty )_{\mathbb{T}}.
\]
\end{theorem}

\begin{proof} 
Multiplying  $X^{\Delta }(t,\omega)=_Pa(\omega )X(t,\omega )+h(t,\omega )$ 
by $e_{a(\omega )}(t_0,\sigma(t))$, we obtain that
\[
X^{\Delta }(t,\omega )e_{a(\omega )}(t_0,\sigma (t))-a(\omega )X(t,\omega
)e_{a(\omega )}(t_0,\sigma (t))=_Ph(t,\omega )e_{a(\omega
)}(t_0,\sigma (t));
\]
that is,
\[
[ X(t,\omega )e_{a(\omega )}(t_0,t)]^{\Delta }=_Ph(t,\omega
)e_{a(\omega )}(t_0,\sigma (t)).
\]
Integrating both sides of the last equality from $t_0$ to $t$, it follows
that
\[
X(t,\omega )e_{a(\omega )}(t_0,t)-X(t_0,\omega )e_{a(\omega
)}(t_0,t_0)=_P\int_{t_0}^{t}e_{a(\omega )}(t_0,\sigma
(s))h(s,\omega )\Delta s.
\]
Multiplying the last equality by $e_{a(\omega )}(t,t_0)$, we obtain
\eqref{ivp-l}.
\end{proof}

\begin{corollary} \label{coro1}
Let $X_0:\Omega \to \mathbb{R}$ be a bounded random variable. 
If the positively regressive random variable
$a:\Omega \to \mathbb{R}$ is bounded with $P.1$, then the unique solution 
of the initial-value problem
\begin{gather*}
X^{\Delta }(t,\omega )=_Pa(\omega )X(t,\omega ), \quad 
t\in [t_0,\infty )_{\mathbb{T}} \\
X(t_0,\omega )=_PX_0(\omega )
\end{gather*}
is given by
\[
X(t,\omega )=_Pe_{a(\omega )}(t,t_0)X_0(\omega ), t\in [t_0,\infty )_{\mathbb{T}}.
\]
\end{corollary}

\begin{remark} \label{rmk2} \rm
Let $X_0:\Omega \to \mathbb{R}$ be a bounded random variable. 
If the positively regressive random variable $a:\Omega \to \mathbb{R}$ 
is bounded with $P.1$, then the unique
solution of the  initial-value problem
\begin{gather*}
X^{\Delta }(t,\omega )=_P-a(\omega )X^{\sigma }(t,\omega ), \quad 
t\in[ t_0,\infty )_{\mathbb{T}} \\
X(t_0,\omega )=_PX_0(\omega )
\end{gather*}
is given by
\[
X(t,\omega )=_Pe_{\ominus a(\omega )}(t,t_0)X_0(\omega ), t\in
[ t_0,\infty )_{\mathbb{T}},
\]
where $\ominus a(\omega )=-\frac{a(\omega )}{1+\mu (t)a(\omega )}$
(see \cite{boh}) and $X^{\sigma }(t,\omega )=X(\sigma (t),\omega )$.
Indeed, we have (see \cite{boh})
\begin{align*}
X^{\Delta }(t,\omega )
&=_P\Big(\frac{1}{e_{\ominus a(\omega )}(t,t_0)
}\Big) ^{\Delta }X_0(\omega )=_P-\frac{a(\omega )}{e_{a(\omega
)}(\sigma (t),t_0)}X_0(\omega ) \\
&=_P-a(\omega )e_{\ominus a(\omega )}(\sigma (t),t_0)X_0(\omega
)=_P-a(\omega )X^{\sigma }(t,\omega ).
\end{align*}
\end{remark}

\begin{theorem}[Variation of Constants] \label{thm5} 
Suppose that $a:\Omega \to \mathbb{R}$ is a positively regressive and bounded
random variable, $X_0:\Omega \to \mathbb{R}$ is a
bounded random variable, and 
$h(,\cdot ,):[t_0,\infty )_{\mathbb{T}}\times \Omega \to \mathbb{R}$
is a rd-continuous time scale stochastic process. If there is a constant 
$\nu >0$ such that $| h(t,\omega )| \leq _P\nu $ for all 
$ t\in [ t_0,b)_{\mathbb{T}}$ with $b\in (t_0,\infty )_{\mathbb{T}}$, 
$t_0<\rho (b)$, then the initial-value problem 
\begin{equation}
\begin{gathered}
X^{\Delta }(t,\omega )=_P-a(\omega )X^{\sigma }(t,\omega )+h(t,\omega ),\quad
t\in [ t_0,\infty )_{\mathbb{T}} \\
X(t_0,\omega )=_PX_0(\omega ),
\end{gathered}   \label{aivp}
\end{equation}
has a unique solution on  $[t_0,\infty )_{\mathbb{T}}$ given by
\begin{equation}
X(t,\omega )=_Pe_{\ominus a(\omega )}(t,t_0)X_0(\omega
)+\int_{t_0}^{t}e_{\ominus a(\omega )}(t,s)h(s,\omega )\Delta s,
t\in [ t_0,\infty )_{\mathbb{T}}.  \label{aivp-s}
\end{equation}
\end{theorem}

\begin{proof} 
Multiplying the both sides of the equation in \eqref{aivp}
by $e_{a(\omega )}(t,t_0)$. Then we have
\begin{align*}
(e_{a(\omega )}(t,t_0)X(t,\omega )) ^{\Delta }
&=_Pe_{a(\omega )}(t,t_0)X^{\Delta }(t,\omega )+a(\omega )e_{a(\omega
)}(t,t_0)X^{\sigma }(t,\omega ) \\
&=_Pe_{a(\omega )}(t,t_0)[X^{\Delta }(t,\omega )+a(\omega )X^{\sigma
}(t,\omega )] \\
&=_Pe_{a(\omega )}(t,t_0)h(t,\omega ).
\end{align*}
Next, we integrate both sides from $t_0$ to $t$ and we infer that
\[
e_{a(\omega )}(t,t_0)X(t,\omega )-e_{a(\omega
)}(t_0,t_0)X(t_0,\omega )=_P\int_{t_0}^{t}e_{a(\omega
)}(s,t_0)h(s,\omega )\Delta s;
\]
that is,
\[
e_{a(\omega )}(t,t_0)X(t,\omega )=_PX_0(\omega
)+\int_{t_0}^{t}e_{a(\omega )}(s,t_0)h(s,\omega )\Delta s.
\]
Since
$$
e_{a(\omega )}(t_0,t)=\frac{1}{e_{a(\omega )}(t,t_0)}=e_{\ominus
a(\omega )}(t,t_0), e_{a(\omega )}(t_0,t)e_{a(\omega )}(t,t_0)=1
$$
(see \cite[Theorem 2.36]{boh}), then multiplying the both sides of the last
equality by $e_{a(\omega )}(t_0,t)$, we obtain \eqref{aivp-s}.
\end{proof}

\begin{example} \label{examp3} \rm
 Let us consider $\Omega =(0,1)$, $\mathcal{F}$ the $\sigma $-algebra of all
 Borel subsets of $\Omega $, $P$ the Lebesgue measure
on $\Omega $, and the following initial-value problem
\begin{equation}
\begin{gathered}
X^{\Delta }(t,\omega )=_P\omega X(t,\omega )+e_{\omega }(t,0), \quad 
t\in [ 0,\infty )_{\mathbb{T}} \\
X(0,\omega )=_P1-\omega .
\end{gathered}   \label{ex-3}
\end{equation}
Then, by the Theorems \ref{thm2} and \ref{thm3}, the initial value problem \eqref{ex-3} has a
unique solution on $[0,\infty )_{\mathbb{T}}$, given by
\[
X(t,\omega )=_P(1-\omega )e_{\omega }(t,0)+\int_0^{t}e_{\omega
}(t,\sigma (s))e_{\omega }(s,0)\Delta s;
\]
that is,
\[
X(t,\omega )=_Pe_{\omega }(t,0)\Big[ 1-\omega +\int_0^{t}\frac{1}{1+\mu
(s)\omega }\Delta s\Big] , \quad t\in [ 0,\infty )_{\mathbb{T}}.
\]
\end{example}

Next, consider two particular cases.

If $\mathbb{T}=\mathbb{R}$, then $\mu (t)=0$ for all $t\in \mathbb{N}$, and 
$e_{\omega }(t,0)=e^{\omega t}$. Moreover, in this case we have
\[
\int_0^{t}\frac{1}{1+\mu (s)\omega }\Delta s=\int_0^{t}ds=t.
\]
It follows that the initial-value problem
\begin{gather*}
X^{\Delta }(t,\omega )=_P\omega X(t,\omega )+e^{\omega t}, \quad
t\in [ 0,\infty ) \\
X(0,\omega )=_P1-\omega ,
\end{gather*}
has the solution $X(t,\omega )=(1-\omega +t)e^{\omega t}$,
 $t\in [ 0,\infty )$.

If $\mathbb{T}=\mathbb{N}$, then $\mu (n)=1$ for all $n\in \mathbb{N}$, and 
$e_{\omega }(n,0)=(1+\omega )^n$. Moreover, in this case we have
\[
\int_0^{t}\frac{1}{1+\mu (s)\omega }\Delta s=\sum_{s\in [ 0,n)}\frac{
1}{1+\omega }=\frac{n}{1+\omega }.
\]
It follows that the difference initial-value problem
\begin{gather*}
X_{n+1}(\omega )=_P(1+\omega )X_n(\omega )+(1+\omega )^n, \quad n\in
\mathbb{N} \\
X_0(\omega )=_P1-\omega ,
\end{gather*}
has the solution $X_n(\omega )=(1-\omega +\frac{n}{1+\omega })(1+\omega
)^n$,  $n\in \mathbb{N}$.

\begin{example} \label{examp4} \rm
 Let us consider $\Omega =(0,1)$, $\mathcal{F}$ the 
$\sigma $-algebra of all Borel subsets of $\Omega $, $P$ the Lebesgue measure
on $\Omega $, and the  initial-value problem
\begin{equation}
\begin{gathered}
X^{\Delta }(t,\omega )=_P-\omega X^{\sigma }(t,\omega )+e_{\ominus \omega
}(t,t_0),\quad  t\in [ t_0,\infty )_{\mathbb{T}} \\
X(t_0,\omega )=_P1-\omega .
\end{gathered}  \label{ex-4}
\end{equation}
The initial-value problem \eqref{ex-4} has a unique solution on 
$[t_0,\infty )_{\mathbb{T}}$, given by
\[
X(t,\omega )=_P(1-\omega )e_{\ominus \omega
}(t,t_0)+\int_0^{t}e_{\ominus \omega }(t,s)e_{\ominus \omega
}(s,t_0)\Delta s;
\]
that is,
\[
X(t,\omega )=_P(1-\omega -t_0+t) e_{\ominus \omega }(t,t_0)
\text{, \ }t\in [ t_0,\infty )_{\mathbb{T}}.
\]
If $\mathbb{T}=h\mathbb{N}$ with $h>0$, then $\mu (t)=h$ for all
$t\in h \mathbb{N}$, and $e_{\ominus \omega }(t,0)=(1+\omega h)^{-t/h}$.
It follows that the $h$-difference initial-value problem
\begin{gather*}
X_{t+h}(\omega )=_P\frac{1}{1+\omega h}X_t(\omega )+h(1+\omega
h)^{-t/h-1}, \quad t\in h\mathbb{N} \\
X_0(\omega )=_P1-\omega ,
\end{gather*}
has the unique solution
$X_t(\omega )=_P(1-\omega +t) (1+\omega h)^{-t/h}$,  $t\in h\mathbb{N}$.

If $\mathbb{T}=2^{\mathbb{N}}$, then $\mu (t)=t$ for all
 $t\in 2^{\mathbb{N}}$, and 
$e_{\ominus \omega }(t,0)=\prod_{s\in [ 0,t)}(1+\omega s)^{-1}$. 
It follows that the $2$-difference initial value problem
\begin{gather*}
X_t(\omega )=_P(1+\omega t)X_{2t}(\omega )-t\prod_{s\in [
1,t)}(1+\omega s)^{-1},\quad t\in 2^{\mathbb{N}} \\
X_1(\omega )=_P1-\omega ,
\end{gather*}
has the unique solution $X_t(\omega )=_P(1-\omega +t)
\prod_{s\in [ 1,t)}(1+\omega s)^{-1}$,  $t\in 2^{\mathbb{N}}$.
\end{example}

\begin{thebibliography}{00}

\bibitem{agra}  R. P. Agarwal, M. Bohner;
\emph{Basic calculus on time scales and some of its applications}, 
Results Math. 35(1999) 3--22.

\bibitem{agra2}  R. P. Agarwal, M. Bohner, D. O'Regan, A. Peterson;
\emph{Dynamic equations on time scales: a survey}, J. Comput. Appl. Math.
141(1--2) (2002) 1--26.

\bibitem{boh}  M. Bohner, A. Peterson;
\emph{Dynamic Equations on Time Scales: An Introduction with Applications}, 
Birkh\"{a}user, Boston, 2001.

\bibitem{boh2}  M. Bohner, A. Peterson;
\emph{Advances in Dynamic Equations on Time Scales}, Birkh\"{a}user, Boston, 2003.

\bibitem{bohs}  M. Bohner, S. Sanyal;
\emph{The Stochastic Dynamic Exponential and Geometric Brownian Motion on 
Isolated Time Scales},
Communications in Mathematical Analysis 8(3)(2010) 120-135.

\bibitem{ch}  K. L. Chung;
\emph{Elementary Probability Theory with
Stochastic Processes}, Springer, 1975.

\bibitem{gro}  D. Grow, S. Sanyal;
\emph{Brownian Motion indexed by a Time Scale}, 
Stochastic Analysis and Applications. Accepted, November 2010.

\bibitem{guse}  G. Sh. Guseinov;
\emph{Integration on time scales}, J.
Math. Anal. Appl. 285(2003) 107--127.

\bibitem{guse1}  G. Sh. Guseinov, B. Kaymakcalan;
\emph{Basics of Riemann delta and nabla integration on time scales}, 
J. Difference Equations Appl. 8(2002) 1001--1017.

\bibitem{hil-88}  S. Hilger;
\emph{Ein Ma\ss kettenkalk\"{u}l mit
Anwendung auf Zentrumsmannigfaltigkeiten}, Ph.D. Thesis, Universit\"{a}t W
\"{u}rzburg, 1988.

\bibitem{hil-90}  S. Hilger;
\emph{Analysis on measure chains-a unified approach to continuous 
and discrete calculus}, Results Math. 18(1990) 18--56.

\bibitem{ol}  B. K. \O ksendal;
\emph{Stochastic Differential
Equations: An Introduction with Applications}, 4th ed., Springer, 1995.

\bibitem{san}  S. Sanyal;
\emph{Stochastic Dynamic Equations}. PhD
dissertation, Missouri University of Science and Technology, 2008.

\bibitem{til}  C. C. Tisdell, A. H. Zaidi;
\emph{Successive approximations to solutions of dynamic equations on time scales},
Communications on Applied Nonlinear Analysis 16(1)(2009) 61--87.

\bibitem{til1}  C. C. Tisdell, A. H. Zaidi;
\emph{Basic qualitative and quantitative results for solutions to nonlinear,
 dynamic equations on time scales with an application to economic modelling},
 Nonlinear Analysis, 68(11)(2008) 3504--3524.

\end{thebibliography}

\end{document}