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\def\N{{\mathbb N}}        % positive integers
\def\Q{{\mathbb Q}}        % rationals
\def\Z{{\mathbb Z}}        % integers
\def\R{{\mathbb R}}        % reals
\def\Rn{{\R^{n}}}          % product of n copies of reals
\def\P{{\mathbb P}}        % probability
\def\E{{\mathbb E}}        % expectation
\def\1{{\mathbf 1}}        % indicator
\def\U{{\mathbb U}}        % potential measure
\def\var{{\mathop{\mathbf Var}}}    % variance


\def\borel{{\cal B}}    % borel sigmal field
\def\G{{\cal G}}        % some sigma field
\def\F{{\cal F}}        % some sigma field
\def\H{{\cal H}}        % some sigma field

\def\L{{\mathbf L}}     % L, as in L^2
\def\I{{\mathbf I}}

\def\ascv{\stackrel{\scriptscriptstyle a.s.}{\longrightarrow}}     % almost sure convergnece
\def\pcv{\stackrel{\scriptscriptstyle \P}{\longrightarrow}}        % convergnece in P
\def\ltcv{\stackrel{\scriptscriptstyle\L^2}{\longrightarrow}}      % L2 convergnece
\def\lpcv{\stackrel{\scriptscriptstyle\L^p}{\longrightarrow}}      % Lp convergnece

\def\ci{\perp\!\!\!\perp}  % conditional independence

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    \hbox to 6.28in { {\bf Stat205A:~Probability~Theory~(Fall 2002) \hfill Lecture: 26} }
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\begin{document}
\lecture{Processes with Independent Increments}{Jim Pitman}
  {Jonathan Weare {\tt weare@math.berkeley.edu }}{26} %

\section{Poisson Processes and Brownian Motion}
Let $(F_t)_{t \geq 0}$ be a filtration.  Usually, but not necessarily, 
  $F_t = \sigma(X_s, 0\leq s\leq t)$.

\begin{definition}
A real valued process $X_t$, $t\geq 0$, is an $F_t$-{\bf Brownian Motion} 
 ($BM$) if 
\begin{itemize}
\item[0)] {$X_t$ is $F_t$-measurable.}
\item[1)] {The mapping $t\rightarrow X_t(w)$ is continuous for almost every 
  $w$.}
\item[2)] {For $s,t \ge 0$ 
the increment $X_{t+s} - X_s$ is normally distributed with mean 
  0 and variance t.}
\item[3)] {$X_{t+s}-X_s$ is independent of $F_s$.}
\end{itemize}
\end{definition}

\begin{definition}
A real valued process $X_t$, $t\geq 0$, is an $F_t$-{\bf Poisson Process} 
 with rate $\lambda$ or $PP(\lambda)$ if 
\begin{itemize}
\item[0)] {$X_t$ is $F_t$-measurable.}
\item[1)] {The mapping $t\rightarrow X_t(w)$ is increasing, right continuous, 
  and takes nonnegative integer values.}
\item[2)] {For $s,t \ge 0$ 
The increment, $X_{t+s} - X_s$, is a Poisson random variable with 
  parameter $ \lambda t$.}
\item[3)] {$X_{t+s}-X_s$ is independent of $F_s$.}
\end{itemize}
\end{definition}

\begin{theorem}
{\em (L\'evy)}
A real-valued process $X_t, t \ge 0$ with $X_0 = 0$ and continuous paths is an $F_t$-Brownian Motion if and only 
  if
\begin{itemize}
\item[1)] {$X_t$ is an $F_t$-martingale.}
\item[2)] {$X_t^2-t$ is an $F_t$-martingale.}
\end{itemize}
\end{theorem}

\begin{theorem}
{\em (Watanabe)}
A process $X_t, t \ge 0$ with $X_0 = 0$ and increasing, right continuous step
function paths with all jumps of size 1 is an $F_t$-Poisson Process with rate
$\lambda$ if and only if $X_t- \lambda t$ is an $F_t$-martingale.
\end{theorem}

The proofs of the above two theorems require stochastic calculus and 
are not given here. 

\section{Construction of a Poisson Process}

Let $T_0 = 0$ and for $r = 1,2, \ldots$ let
$$T_r\:= \textrm{time of\ } r^{th} \textrm{\ jump\ }$$
and let $$N_t = \sum_{r=1}^\infty 1_{(T_r\leq t)} 
\textrm{\ and\ } 
  W_r = T_r-T_{r-1}$$.

\begin{theorem}
Assuming $(N_t)_{t\geq 0}$ is a simple counting process, it is a $PP(\lambda)$
  process if and only if $W_1,W_2,W_3,\dots$ are $i.i.d.$ with
  $$P(W_r>t)=e^{-\lambda t}$$.
\end{theorem}

\begin{proof}
See Durrett, Sec 2.6.
\end{proof}

\begin{definition}
For any $r>0$, a random variable, $\Gamma_r$, has {\bf gamma$(r,\lambda)$}
  distribution if 
  $$P(\Gamma_r \in dt) = \frac{1}{\Gamma(r)} t^{r-1} \lambda^r e^{- \lambda t}dt$$
\end{definition}
where $\Gamma(r)$ is a constant of normalization, called the {\em gamma
function}.

\begin{claim}
Fore $r = 1,2, \ldots$, the time $T_r$ of the $r$th point in a 
$PP(\lambda)$ has gamma$(r,\lambda)$ distribution.
\end{claim}

\begin{proof}
Using independence of increments of the Poisson process
\begin{eqnarray*}
P(T_r\in dt) &=& P(r-1 \textrm{\ arrivals in\ } (0,t) 
  \textrm{\ and 1 arrival in\ } dt)\\
&=& P(r-1 \textrm{\ arrivals in\ } (0,t))P(1 \textrm{\ arrival in\ } dt)\\
&=& e^{-\lambda t}\frac{(\lambda t)^{r-1}}{(r-1)!} \, \lambda dt
\end{eqnarray*}
\end{proof}

Note that for each fixed $\lambda >0$ the family of gamma$(r,\lambda)$ 
distributions forms a convolution semigroup, $i.e.$, if 
  $F_r(t)=P(\Gamma _r \leq t)$ is the $c.d.f.$ of $\Gamma_r$ then
  $$F_r\ast F_s = F_{r+s}$$
For $r,s=0,1,2,\dots$ this is obvious from the Poisson Process interpretation.
That this is also true for all real $r,s>0$ can be shown by 
computation.

\section{Compound Poisson Process}
Compound Poisson Process are frequently used to model losses in the insurance
  industry.  Let $J_1,J_2,J_3,\dots$ be $i.i.d.$ random variables with some
  $c.d.f.$ $F(x)=P(J_i\leq x)$, and independent of a Poisson process
  $(N_t)_{t\geq 0}$.  The 
  $J_i$ are now the jumps in our process.  They could be interpreted, for
  example, as the losses associated with a sequence of automobile accidents.
  
 Let $$X_t = \sum_{i=1}^{N_t} J_i = \textrm{ loss accrued up to time t}$$

Notice that $(X_t)_{t\geq 0}$ has stationary independent increments.  We can
compute the characteristic function for $X_t$ as follows
\begin{eqnarray*}
\phi_{X_t}(\theta) = E[ e^{i\theta X_t} ] &=& 
  E[ e^{i\theta (J_1+J_2+\dots +J_{N_t})} ] \\
&=& \sum_{n=0}^\infty P(N_t=n) E[ e^{i\theta J_1} ]^n \\
&=& \sum_{n=0}^\infty e^{-\lambda t}\frac{(\lambda t)^n}{n!}
  E[ e^{i\theta J_1} ]^n \\
&=& e^{-\lambda t} exp(\lambda tE[ e^{i\theta J_1} ] ) \\
&=& exp(\lambda t E[e^{i\theta J_1}-1])
\end{eqnarray*}

For $t=1$ and letting $F(dx) = P(J_1\in dx)$ we have
\begin{eqnarray*}
\phi_{X_1}(\theta) &=& exp(\lambda \int (e^{i\theta x }-1) F(dx) ) \\
&=& exp(\int (e^{i\theta x}-1) L(dx) )
\end{eqnarray*}
where $L:=\lambda F$ is a positive measure on $\mathbb{R}$ with total mass
  $\lambda$.

If for any Borel set $A$ we define
$$N(t,A) = \sum_{i=1}^{N_t} 1_{(J_i\in A)} 
  = \textrm{\ number of\ } J_i \textrm{\ in A up to time t}$$
then $N(t,A)$ is Poisson random variable with parameter $\lambda tF(A)$.
  Also, $N(t,A)$ and $N(t,A^c)$ are independent.  In fact if
  $A_1,A_2,A_3,\dots,A_n$ are disjoint Borel subsets then 
  $(N(t,A_i))_1^n$ are independent Poisson random variables with
  parameters $\lambda t F(A_i)$.




\end{document}
