\documentclass[letterpaper,10pt]{report}
\newcommand{\R}{\ensuremath{\mathbf{R}}}
\newcommand{\N}{\ensuremath{\mathbf{N}}}
\newcommand{\B}{\ensuremath{\mathcal{B}}}
\newcommand{\Borel}{\ensuremath{\mathcal{R}}}
\renewcommand{\L}{\ensuremath{\mathcal{L}}}
\usepackage{amsmath}
% Title Page
\title{Basic Stochastic Processes}
\author{Alex Gittens}


\begin{document}
\maketitle

\begin{abstract}
probability spaces, random variables, distribution functions; stochastic processes, marginal functions, extension theorems, existence theorems, sample functions, directly given stoch processes, Kolmogorov construction, equivalent processes; classes of stoch processes (indepent increments, stationarity, martingales, markov processes, finite dimensional markov processes); examples of stochastic processes
\end{abstract}

\section{ Probability spaces and random variables}

A probability space $(\Omega, \B, P)$ is a normalized measure space, i.e. $P(\Omega) = 1$. A random variable $X$ is a measurable function from $(\Omega, \B, P)$. The probability distribution of a random variable $X$ with values in $(E, \mathcal{E})$ is the measure on $\mathcal{E}$ defined by $P^X(B) = P(X^{-1}(B))$; the space $(E, \mathcal{E}, P^X)$ is a probability space. Note that several random variables may have the same probability distribution. 

Every random variable $X$ with values in $(\R, \Borel)$ (the Borel sets) has an associated distribution function $F^X : \R \rightarrow [0,1]$ defined by
\begin{equation*} 
	F^X(x) = P^X((-\infty,x]) = P\{X \leq x\}.
\end{equation*}
Note that the distribution function defines a probability measure on the algebra of rectangles which generates $\B$, so can be extended to a probability measure $\hat{P}$ on $\B$. Since this extension is unique, $\hat{P} = P^X$; that is, the distribution function of $X$ determines the probability distribution of $X$.

\section{Stochastic processes}
A stochastic process is a parameterized family of real or complex-valued random variables 
\begin{equation*}
	\{X(t, \cdot); t\in T\}
\end{equation*}
defined on a common probability space $(\Omega, \B, P)$, where the set of parameterization $T$ is a subset of $\R$. The state space $\{S, \L\}$ of the process consists of $S$, the set in which the process takes values, and $\L$, a sigma algebra on $S$. In the case that $T = \N_+ = \{0, 1, \ldots\}$, the process is called a discrete parameter stochastic process; if $T$ is an interval, it is called a continuous parameter stochastic process.

Let $\{X(t, \cdot); t \in T\}$ be a real-valued stochastic process and $\{t_1, \ldots, t_n\} \subset T$ with $t_1 < t_2 < \ldots < t_n$, then 
\begin{equation*}
 F_{t_1, \ldots, t_n}(x_1, \ldots, x_n) = P\{X(t_1, \cdot) \leq x_1, \ldots, X(t_n, \cdot) \leq x_n\}
\end{equation*}
is a finite-dimensional marginal distribution function of the process $\{X(t, \cdot); t\in T\}$. For each fixed $t$, $X(t, \cdot)$ is a random variable; when $\omega \in \Omega$ is fixed, $X(\cdot, \omega)$ is called a sample function. In most applications, only sample functions are observed; e.g., if we modeled the closing price of the stock exchange as a stochastic process, the historical data provides a sample function of the process.

A stochastic process $\{X(t, \cdot); t \in T\}$ is said to be given if its marginal distribution functions \( F_{t_1, \ldots, t_n} \) are given for each $\{t_1, \ldots, t_n\} \subset T$. Marginal distribution functions must be consistent:
\begin{enumerate}
 \item 
  \begin{equation*} F_{t_1, \ldots, t_n}(x_1, \ldots, x_n) = F_{t_{k_1}, t_{k_2}, \ldots, t_{k_n}}(x_{k_1}, \ldots, x_{k_n}) \end{equation*}
  for any permutation $k_1, \ldots, k_n$ of $1, \ldots, n$.
 \item For any $1 \leq k \leq n$,
  \begin{equation*} F_{t_1, \ldots, t_k} = F_{t_1, \ldots, t_n}(x_1, \ldots, x_k, \infty, \ldots, \infty)   \end{equation*}
\end{enumerate}

Let $\{\xi(t); t \in T\}$ be a stochastic process on $\{\Omega, \B, P\}$ with state space $\{\R, \Borel\}$, then the process determines a consistent family of marginal distributions. But is it true that a consistent family of marginal distributions uniquely determines a real-valued stochastic process? That is, can we justify calling a stochastic process given if we know a set of marginal distributions? We can directly construct such a stochastic process, using as $\Omega$ the set of coordinate mappings $\R^T$; that is, each $\omega \in \R^T$ is a mapping
\begin{equation*}
\omega : T \rightarrow \R,
\end{equation*}
and taking $\B$, the $\sigma$-algebra on $\Omega$, to be the $\sigma$-algebra generated by the finite-dimensional Borel cylinders,
\begin{equation*}
 \B = \sigma( \{\omega: \omega(t_1) \in B_1, \ldots, \omega(t_n) \in B_n\} ), \: B_1,\ldots,B_n \in \Borel
\end{equation*}

Let $\{X(t, \cdot); t \in T\}$ be the set of coordinate mappings, so $X(t, \omega) = \omega(t)$; then each $X(t, \cdot)$ is measurable. Finally, note that because of their consistency, the marginal distributions define a probability measure on the algebra of Borel cylinders, which generates the ...

\section{Equivalent Stochastic Processes}

There are three ways in which stochastic processes can be said to be equivalent, in increasing strength: these are stochastic equivalence in the wide sense, stochastic equivalence (also, just 'equivalence'), and indistinguishablility.

Let $\{X(t, \cdot); t \in T\}$ and $\{Y(t, \cdot); t \in T\}$ be two stochastic processes defined on a common probability space $(\Omega, \B, P)$ and taking values in the same state space $(S, \L)$. If, for each $n =1, 2, \ldots$
\begin{equation}
 P{X(t_1, \cot
\end{equation}

\end{document}