\documentclass[twoside]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{epsfig} \usepackage{floatflt} \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 % the header/lecture structure\cdotsborrowed from alistair sinclairs cs270 format \newcommand{\lecture}[4]{ \pagestyle{myheadings} \thispagestyle{plain} \newpage \setcounter{page}{1} \setcounter{lecnum}{#4} \noindent \begin{center} \framebox{ \vbox{\vspace{2mm} \hbox to 6.28in { {\bf Stat205A:~Probability~Theory~(Fall 2002) \hfill Lecture: 26} } \vspace{6mm} \hbox to 6.28in { {\Large \hfill #1 \hfill} } \vspace{6mm} \hbox to 6.28in { {\it Lecturer: #2 \hfill Scribe: #3} } \vspace{2mm}} } \end{center} \markboth{#1}{#1} \vspace*{4mm} } %\hbox to 5.90in { \hfill {\it Scribe: #3} } % number everything using the lecture number counter % are referring to an example, theorem, exercise, etc. \newcounter{lecnum} \renewcommand{\thepage}{\thelecnum-\arabic{page}} \renewcommand{\thesection}{\thelecnum.\arabic{section}} \renewcommand{\theequation}{\thelecnum.\arabic{equation}} \renewcommand{\thefigure}{\thelecnum.\arabic{figure}} \renewcommand{\thetable}{\thelecnum.\arabic{table}} \newtheorem{theorem}{Theorem}[lecnum] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{claim}[theorem]{Claim} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}{Definition}[lecnum] \newenvironment{proof}{{\bf Proof:}}{\hfill\rule{2mm}{2mm}} \newenvironment{proofsketch}{{\bf Proof Sketch:}}{\hfill\rule{2mm}{2mm}} \newcommand{\fig}[3]{ \vspace{#2} \begin{center} Figure \thelecnum.#1:~#3 \end{center} } \newtheorem{exercise}{Exercise}[lecnum] \newtheorem{example}{Example}[lecnum] %some things well never understand.\cdotsalas \setlength{\oddsidemargin}{0.25 in} \setlength{\evensidemargin}{-0.25 in} \setlength{\topmargin}{-0.6 in} \setlength{\textwidth}{6.0 in} \setlength{\textheight}{8.5 in} \setlength{\headsep}{0.75 in} \setlength{\parindent}{0 in} \setlength{\parskip}{0.1 in} \advance\topmargin by 0.5in %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}