\documentclass[12pt]{article}

\usepackage{hyperref}
\hypersetup{
  linkcolor  = blue,
  citecolor  = green,
  urlcolor   = orange,
  colorlinks = true,
}

\usepackage{amscd}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amstext}
\usepackage{amsthm}
\usepackage{bbold}
\usepackage{bm}
\usepackage{colonequals}
\usepackage[dvips,letterpaper,margin=1in]{geometry}
\let\digamma\relax
\usepackage{tikz}

\definecolor{clightgray}{rgb}{0.6,0.6,0.6}
\newcommand{\hint}[1]{\textcolor{clightgray}{(\textsc{\textbf{Hint:}} #1)}}

\theoremstyle{definition}
\newtheorem{outproblem}{Problem}[]
\theoremstyle{plain}

\renewcommand{\AA}{\mathbb{A}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\DD}{\mathbb{D}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\renewcommand{\SS}{\mathbb{S}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\ZZ}{\mathbb{Z}}
\DeclareSymbolFont{bbold}{U}{bbold}{m}{n}
\DeclareSymbolFontAlphabet{\mathbbold}{bbold}
\newcommand{\One}{\mathbbold{1}}

\newcommand{\RP}{\RR\mathrm{P}}
\newcommand{\CP}{\CC\mathrm{P}}

\newcommand{\ba}{\bm a}
\newcommand{\bb}{\bm b}
\newcommand{\bc}{\bm c}
\newcommand{\bd}{\bm d}
\newcommand{\be}{\bm e}
\newcommand{\bg}{\bm g}
\newcommand{\bh}{\bm h}
\newcommand{\bi}{\bm i}
\newcommand{\bj}{\bm j}
\newcommand{\bk}{\bm k}
\newcommand{\bbm}{\bm m}
\newcommand{\bp}{\bm p}
\newcommand{\bq}{\bm q}
\newcommand{\br}{\bm r}
\newcommand{\bs}{\bm s}
\newcommand{\bt}{\bm t}
\newcommand{\bu}{\bm u}
\newcommand{\bv}{\bm v}
\newcommand{\bw}{\bm w}
\newcommand{\bx}{\bm x}
\newcommand{\by}{\bm y}
\newcommand{\bz}{\bm z}

\newcommand{\bA}{\bm A}
\newcommand{\bB}{\bm B}
\newcommand{\bC}{\bm C}
\newcommand{\bD}{\bm D}
\newcommand{\bE}{\bm E}
\newcommand{\bF}{\bm F}
\newcommand{\bG}{\bm G}
\newcommand{\bH}{\bm H}
\newcommand{\bI}{\bm I}
\newcommand{\bJ}{\bm J}
\newcommand{\bL}{\bm L}
\newcommand{\bM}{\bm M}
\newcommand{\bN}{\bm N}
\newcommand{\bP}{\bm P}
\newcommand{\bQ}{\bm Q}
\newcommand{\bR}{\bm R}
\newcommand{\bS}{\bm S}
\newcommand{\bT}{\bm T}
\newcommand{\bU}{\bm U}
\newcommand{\bV}{\bm V}
\newcommand{\bW}{\bm W}
\newcommand{\bX}{\bm X}
\newcommand{\bY}{\bm Y}
\newcommand{\bZ}{\bm Z}

\newcommand{\zero}{\bm{0}}
\newcommand{\one}{\bm{1}}

\newcommand{\balpha}{\boldsymbol{\alpha}}
\newcommand{\bbeta}{\boldsymbol{\beta}}
\newcommand{\bdelta}{\boldsymbol{\delta}}
\newcommand{\bepsilon}{\boldsymbol{\eps}}
\newcommand{\blambda}{\boldsymbol{\lambda}}
\newcommand{\bomega}{\boldsymbol{\omega}}
\newcommand{\bpi}{\boldsymbol{\pi}}
\newcommand{\bmmu}{\boldsymbol{\mu}}
\newcommand{\bmu}{\bm\mu}
\newcommand{\bnu}{\boldsymbol{\nu}}
\newcommand{\bphi}{\boldsymbol{\phi}}
\newcommand{\bvphi}{\boldsymbol{\varphi}}
\newcommand{\bpsi}{\boldsymbol{\psi}}
\newcommand{\bsigma}{\boldsymbol{\sigma}}
\newcommand{\btheta}{\boldsymbol{\theta}}
\newcommand{\bzeta}{\boldsymbol{\zeta}}
\newcommand{\bxi}{\boldsymbol{\xi}}

\newcommand{\bDelta}{\boldsymbol{\Delta}}
\newcommand{\bLambda}{\boldsymbol{\Lambda}}
\newcommand{\bOmega}{\boldsymbol{\Omega}}
\newcommand{\bPi}{\boldsymbol{\Pi}}
\newcommand{\bPhi}{\boldsymbol{\Phi}}
\newcommand{\bPsi}{\boldsymbol{\Psi}}
\newcommand{\bSigma}{\boldsymbol{\Sigma}}
\newcommand{\bTheta}{\boldsymbol{\Theta}}
\newcommand{\bXi}{\boldsymbol{\Xi}}

\newcommand{\sA}{\mathcal{A}}
\newcommand{\sB}{\mathcal{B}}
\newcommand{\sC}{\mathcal{C}}
\newcommand{\sD}{\mathcal{D}}
\newcommand{\sE}{\mathcal{E}}
\newcommand{\sF}{\mathcal{F}}
\newcommand{\sG}{\mathcal{G}}
\newcommand{\sH}{\mathcal{H}}
\newcommand{\sI}{\mathcal{I}}
\newcommand{\sJ}{\mathcal{J}}
\newcommand{\sK}{\mathcal{K}}
\newcommand{\sL}{\mathcal{L}}
\newcommand{\sM}{\mathcal{M}}
\newcommand{\sN}{\mathcal{N}}
\newcommand{\sO}{\mathcal{O}}
\newcommand{\sP}{\mathcal{P}}
\newcommand{\sQ}{\mathcal{Q}}
\newcommand{\sR}{\mathcal{R}}
\newcommand{\sS}{\mathcal{S}}
\newcommand{\sT}{\mathcal{T}}
\newcommand{\sU}{\mathcal{U}}
\newcommand{\sV}{\mathcal{V}}
\newcommand{\sX}{\mathcal{X}}
\newcommand{\sY}{\mathcal{Y}}

\newcommand{\fB}{\mathscr{B}}
\newcommand{\fC}{\mathscr{C}}
\newcommand{\fE}{\mathscr{E}}
\newcommand{\fH}{\mathscr{H}}
\newcommand{\fI}{\mathscr{I}}
\newcommand{\fP}{\mathscr{P}}
\newcommand{\fS}{\mathscr{S}}
\newcommand{\fV}{\mathscr{V}}

\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fq}{\mathfrak{q}}
\newcommand{\fg}{\mathfrak{g}}
\newcommand{\fh}{\mathfrak{h}}
\newcommand{\fa}{\mathfrak{a}}
\newcommand{\fb}{\mathfrak{b}}
\newcommand{\fc}{\mathfrak{c}}
\newcommand{\fm}{\mathfrak{m}}
\renewcommand{\sl}{\mathfrak{sl}}
\newcommand{\so}{\mathfrak{so}}
\newcommand{\gl}{\mathfrak{gl}}
\renewcommand{\sp}{\mathfrak{sp}}

\newcommand{\Mod}[1]{\ (\mathrm{mod}\ #1)}

\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\softmax}{soft\,max}
\DeclareMathOperator{\Gram}{Gram}
\DeclareMathOperator{\GOE}{GOE}
\DeclareMathOperator{\sym}{sym}
\DeclareMathOperator{\Part}{Part}
\DeclareMathOperator{\poly}{poly}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\img}{img}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\per}{per}
\DeclareMathOperator{\Frac}{Frac}
\DeclareMathOperator{\Irr}{Irr}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Der}{Der}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Rad}{Rad}
\DeclareMathOperator{\Unif}{Unif}
\DeclareMathOperator{\Ad}{Ad}
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\diam}{diam}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\supp}{supp}
\renewcommand\Re{\operatorname{Re}}
\renewcommand\Im{\operatorname{Im}}
\newcommand{\Px}{\mathop{\mathbb{P}}}
\newcommand{\Ex}{\mathop{\mathbb{E}}}
\newcommand{\Varx}{\mathop{\mathsf{Var}}}

\renewcommand{\bar}{\overline}
\newcommand{\what}{\widehat}
\newcommand{\wtil}{\widetilde}
\newcommand{\lvec}{\overrightarrow}
\newcommand{\ra}{\rangle}
\newcommand{\la}{\langle}
\renewcommand{\emptyset}{\varnothing}
\newcommand{\veq}{\mathrel{\rotatebox{90}{$=$}}}

\newcommand{\Law}{\mathsf{Law}}
\newcommand{\Exp}{\mathsf{Exp}}



\title{Assignment 5}
\author{Probability Theory II \\ (EN.553.721, Spring 2025)}
\date{Assigned: April 17, 2025 \,\,\,\,\, Due: 11:59pm EST, April 29, 2025}

\begin{document}

\maketitle

\noindent
\textbf{Solve any three out of the four problems.} If you solve more, we will grade the first four solutions you include. Each problem is worth an equal amount towards your grade.

\vspace{1em}

\noindent
Submit solutions in \LaTeX. Write in complete sentences. Include and justify all steps of your arguments, but avoid writing excessive explanation that is not contributing to your solution.

\vspace{1em}

\noindent
Keep in mind the late submission policy: you may use a total of five late days for homework submissions over the course of the semester without penalty. If you need an extension beyond these, \emph{you must ask me 48 hours before the due date of the homework} and have an excellent reason. After you have used up these late days, further late assignments will be penalized by 20\% per day they are late.

\vspace{1.5em}

\begin{outproblem}[Large deviations for Brownian motion]
    In this problem, you will derive the asymptotics of the very small probability that Brownian motion lies very close to the $x$-axis for a long time.
    Let $B(t)$ be a Brownian motion.
    Show that there is a constant $c > 0$ such that, for all $\epsilon > 0$ sufficiently small,
    \[ \PP\left[\sup_{t \in [0, 1]} |B(t)| \leq \epsilon\right] \leq \exp\left(-\frac{c}{\epsilon^2}\right). \]
    You do not need to prove it, but it turns out that this is the correct behavior, up to the precise value of $c$.

    \vspace{1em}

    \noindent
    \hint{Consider a grid of $n$ points $t_i$ of the interval $[0, 1]$. Show that, if $|B(t)| \leq \epsilon$ for all $t \in [0, 1]$, then $|B(t_i) - B(t_{i - 1})|$ cannot be too large for any $i$. Estimate the probability that this happens and optimize over $n$.}
\end{outproblem}

\vspace{1em}

\begin{outproblem}[Brownian bridge]
    Let $B(t)$ be a Brownian motion.
    Define $R(t) \colonequals B(t) - tB(1)$.
    Note that $R(0) = R(1) = 0$.
    Intuitively, think of $R(t)$ as a Brownian motion ``pinned'' to take value 0 at time 1.
    \begin{enumerate}
    \item Show that $R(t)$ is a Gaussian process (i.e., that all of its finite distributions are Gaussian random vectors).
        Compute its mean function $\mu(t) = \EE[R(t)]$ and covariance kernel $K(s, t) = \Cov[R(s), R(t)]$.
    \item Let $X_1, X_2, \dots$ be i.i.d.\ real-valued random variables that have distribution function $F(t) \colonequals \PP[X_i \leq t]$.
        You may assume that $F$ is strictly increasing.
        Note that $F$ is a function $F: \RR \to [0, 1]$.
        Let $F_n: \RR \to [0, 1]$ be the empirical distribution function of $n$ i.i.d.\ samples:
        \[ F_n(t) \colonequals \frac{\#\{i \in \{1, \dots, n\}: X_i \leq t\}}{n}. \]
        This is a random function, and thus we may view it as a continuous-time stochastic process.
        Define the associated normalized process
        \[ \what{F}_n(t) \colonequals \sqrt{n} \cdot (F_n(t) - F(t)). \]
        Define a final process $X(t) \colonequals R(F(t))$, a ``time change'' of $R(t)$.
        Show that $(\what{F}_n(t))_{t \in \RR}$ converges in finite distributions to $(X(t))_{t \in \RR}$.

        \hint{Show that a vector of the form $(\what{F}_n(t_1), \dots, \what{F}_n(t_k))$ may be viewed as a normalized sum of $n$ i.i.d.\ random vectors having some covariance matrix $\bm\Sigma \in \RR^{k \times k}$. Use a multidimensional central limit theorem (you may look this up and use it without proof) on this sum. and compare $\bm\Sigma$ with your computation of $K(s, t)$ above.}
    \end{enumerate}
\end{outproblem}

\vspace{1em}

\begin{outproblem}[Ornstein-Uhlenbeck process]
    Let $B$ be a Brownian motion.
    Define the process $U(t)$ over all $t \in \RR$ by
    \[ U(t) \colonequals e^{-t} B(e^{2t}). \]
    \begin{enumerate}
    \item Show that $U(t)$ is a Gaussian process.
        Compute its mean function $\mu(t) = \EE[U(t)]$ and covariance kernel $K(s, t) = \Cov[U(s), U(t)]$.
        Conclude that $(U(t))_{t \in \RR}$ has the same law (i.e., the same finite distributions) as $(U(c + t))_{t \in \RR}$ for any $c \in \RR$.
    \item For any $t \geq s$, compute $\EE[U(t) \mid U(s)]$ as a function (almost surely) of $U(s)$.
        Conclude that $U(t)$ is not a continuous-time martingale, in the sense that the value of the above is not $U(s)$.

        \hint{Reduce this to a matter of conditioning one Gaussian variable on another. Consult Problem 5 of Homework 2.}
    \item Derive a symmetry of the law of $U(t)$ from the time inversion symmetry of Brownian motion. Check that your conclusion is compatible with the calculation in Part 1.
    \item Consider the random variable $X = \int_0^t U(s)\,ds$ for some $t \geq 0$.
        (This should not be mysterious---it is just an integral of a continuous function, where the function happens to be random.)
        Compute the mean and variance of $X$ as functions of $t$.
    \end{enumerate}
\end{outproblem}


\vspace{1em}

\begin{outproblem}[A first stochastic integral]
    Let $B(t)$ be a Brownian motion.
    \begin{enumerate}
    \item Show that, almost surely,
        \[ \lim_{n \to \infty} \sum_{j = 1}^{2^n} \left|B\left(\frac{j}{2^n}\right) - B\left(\frac{j - 1}{2^n}\right)\right| = \infty. \]
    \item Show that, almost surely,
        \[ \lim_{n \to \infty} \sum_{j = 1}^{2^n} \left|B\left(\frac{j}{2^n}\right) - B\left(\frac{j - 1}{2^n}\right)\right|^2 = 1. \]
    \item Let $F: [0, 1] \to \RR$ be a smooth function.
        For each $n \geq 0$, define
        \[ I_n \colonequals \sum_{j = 1}^{2^n}2^n \cdot \left(F\left(\frac{j}{2^n}\right) - F\left(\frac{j - 1}{2^n}\right)\right)\cdot \left(B\left(\frac{j}{2^n}\right) - B\left(\frac{j - 1}{2^n}\right)\right). \]
        Show that the $I_n$ converge in $L^2$ to some (random) limit.
        Intuitively, you should view this limit as $\int_0^1 F^{\prime} dB$, a stochastic Riemann-Stieltjes integral.

        \hint{Show that the $I_n$ form a martingale and use a suitable convergence theorem.}
    \end{enumerate}
\end{outproblem}

\end{document}
