## MA5311. Take home exam due 03/20

Solve Problems 1,2,8,9,10,11 in Milnor’s book. (Extra credit for problem 6)

## MA5161. Take home exam. Due 03/20

Exercise 1.

Let $\alpha, \varepsilon, c >0$. Let $(X_t)_{t\in [0,1]}$ be a continuous Gaussian process such that for $s,t \in [0,1]$,

$\mathbb{E} \left( \| X_t - X_s \|^{\alpha} \right) \leq c \mid t-s \mid^{1+\varepsilon},$
Show that for every $\gamma \in [0, \varepsilon/\alpha)$, there is a positive random variable $\eta$ such that $\mathbb{E}(\eta^p)<\infty$, for every $p \ge 1$ and such that for every $s,t \in [0,1]$, $\|X_t-X_s\| \le \eta |t-s|^{\gamma}, \quad a.s.$ \textbf{Hint:} You may use without proof the Garsia-Rodemich-Rumsey inequality: Let $p \ge 1$ and $\alpha >p^{-1}$, then there exists a constant $C_{\alpha,p} >0$ such that for any continuous function $f$ on $[0,T]$, and for all $t,s \in [0,T]$ one has:
$\|f(t)-f(s)\|^p \le C_{\alpha,p} |t-s|^{\alpha p-1} \int_0^T \int_0^T \frac{ \|f(x)-f(y)\|^p}{ |x-y|^{\alpha p+1}} dx dy.$

Exercise.(Non-canonical representation of Brownian motion)

• Show that for $t \ge 0$, the Riemann integral $\int_0^t \frac{B_s}{s} ds$ almost surely exists.
• Show that the process $\left( B_t-\int_0^t \frac{B_s}{s}ds\right)_{t \ge 0}$ is a standard Brownian motion.

Exercise. [Non-differentiability of the Brownian paths]
1) Show that if $f: (0,1) \to \mathbb{R}$ is differentiable at $t \in (0,1)$, then there exist an interval $(t-\delta,t+\delta)$ and a constant $C>0$ such that for $s \in (t-\delta,t+\delta)$,
$| f(t)-f(s) | \le C |t-s|.$
2)  For $n \ge 1$, let
$M_n = \min_{1 \le k \le n} \{ \max \{| B_{k/n}-B_{(k-1)/n}|, | B_{(k+1)/n}-B_{k/n}|, | B_{(k+2)/n}-B_{k+1/n}| \} \}.$
Show that $\lim_{n \to +\infty} \mathbb{P}( \exists C>0, nM_n \le C)=0$.

3)  Deduce that
$\mathbb{P} \left( \exists t \in (0,1), B \text{ is differentiable at } t\right)=0.$

Exercise.[Fractional Brownian motion] Let $0. 1) Show that for $s \in \mathbb{R}$, the function $f_s(t)=(|t-s|^{H-\frac{1}{2}} - \mathbf{1}_{(-\infty, 0]}(t)|t|^{H-\frac{1}{2}}) \mathbf{1}_{(-\infty, s]}(t)$ is square integrable on $\mathbb{R}$. 2)  Deduce that $R(s,t)=\frac{1}{2} \left( s^{2H} +t^{2H} -|t-s|^{2H} \right), \quad s,t \ge 0$ is a covariance function. 3) A continuous and centered Gaussian process with covariance function $R$ is called a fractional Brownian motion with parameter $H$. Show that such process exists and study its Holder sample path regularity. 4) Let $(B_t)_{t \ge 0}$ be a fractional Brownian motion with parameter $H$. Show that for any $h \geq 0$, the process $(B_{t+h} - B_h)_{t \ge 0}$ is a fractional Brownian motion. 5) Show that for every $c >0$, the process $(B_{ct})_{t \geq 0}$ has the same law as the process $(c^H B_t)_{t \geq 0}$

Exercise. (Brownian bridge)
Let $T>0$ and $x \in \mathbb{R}$.

1. Show that the process
$X_t =\frac{t}{T}x +B_t -\frac{t}{T}B_T, \quad, 0 \le t \le T,$
is a Gaussian process. Compute its mean function and its covariance function.
2.  Show that $(X_t)_{0 \le t \le T}$ is a Brownian motion conditioned to be $x$ at time $T$, that is for every $0\le t_1 \le \cdots \le t_n , and $A_1,\cdots,A_n$ Borel sets of $\mathbb{R}$,
$\mathbb{P}( X_{t_1} \in A_1, \cdots, X_{t_n} \in A_n) = \mathbb{P}( B_{t_1} \in A_1, \cdots, B_{t_n} \in A_n | B_T =x).$
3. Let $(\alpha_n)_{n \ge 0}, (\beta_n)_{n \ge 1}$ be two independent sequences of i.i.d. Gaussian random variables with mean 0 and variance 1. By using the Fourier series decomposition of the process $B_t -t B_1$, show that the random series
$X_t=t \alpha_0 +\sqrt{2} \sum_{n=1}^{+\infty} \left( \frac{\alpha_n}{2\pi n} (\cos (2\pi nt)-1)+\frac{\beta_n}{2\pi n} \sin (2\pi nt)\right)$
is a Brownian motion on $[0,1]$.

## MA5311. Non orientable manifolds

Here are some videos to visualize non orientability.

## HW4 MA5161. Due February 24

Exercise. Let $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ be a filtered probability space that satisfies the usual conditions. We denote

$\mathcal{F}_{\infty}=\sigma \left( \mathcal{F}_t , t \ge 0 \right)$

and for $t\ge 0$, $\mathbb{P}_{/\mathcal{F}_t}$ is the restriction of $\mathbb{P}$ to $\mathcal{F}_t$. Let $\mathbb{Q}$ be a probability measure on $\mathcal{F}_{\infty}$ such that for every $t \ge 0$,

$\mathbb{Q}_{/\mathcal{F}_t} \ll \mathbb{P}_{/\mathcal{F}_t}.$

• Show that there exists a right continuous and left limited martingale $(D_t)_{t \ge 0}$ such that for every $t \ge 0$,$D_t=\frac{d\mathbb{Q}_{/\mathcal{F}_t}}{d\mathbb{P}_{/\mathcal{F}_t}},\text{ }\mathbb{P}-a.s.$
• Show that the following properties are equivalent:
1) $\mathbb{Q}_{/\mathcal{F}_\infty} \ll \mathbb{P}_{/\mathcal{F}_\infty}$;
2) The martingale $(D_t)_{t \ge 0}$ is uniformly integrable;
3) $(D_t)_{t \ge 0}$ converges in $L^1$;
4) $(D_t)_{t \ge 0}$ almost surely converges to an integrable and $\mathcal{F}_\infty$ measurable random variable $D$ such that $D_t =\mathbb{E}(D\mid \mathcal{F}_t), \quad t \ge 0.$

## HW4 MA5311. Due February 24

Exercise 1. Let $M$ be a smooth manifold and $V: C^\infty (M,R) \to C^\infty (M,R)$ be a linear operator such that for every smooth functions $f,g: M \to R$, $V(fg)=fVg+gVf$. Show that there exists a vector field $U$ on $M$ such that for every smooth function $g$, $Vg(x)=dg_x (U(x))$.

Exercise 2. Let $B_n$ be the open unit ball in $R^n$. Let $y$ in $B_n$. Show that there exists a smooth vector field on $R^n$, such that $e^V(0)=y$ and $V(x)=0$ if $x$ is not in $B_n$.

## HW3 MA5311. Due February 15

Exercise. Let $X \subset \mathbf{R}^k$ be a subset homeomorphic to the closed ball $B_n \subset \mathbf{R}^n$.  Show that if $f: X \to X$ is continuous, then there exists $x \in X$ such that $f(x)=x$.

Exercise. Let $X$ be a one-dimensional compact manifold with boundary. Show that $X$ is diffeomorphic to a finite union of segments and circles (You may use the appendix in Milnor’s lecture notes).

## HW3 MA5161. Due February 15

Exercise. (First hitting time of a closed set by a continuous stochastic process)
Let $(X_t)_{t \ge 0}$ be a continuous process adapted to a filtration $(\mathcal{F}_t)_{t \ge 0}$. Let

$T=\inf \{ t \ge 0, X_t \in F \},$

where $F$ is a closed subset of $\mathbb{R}$. Show that $T$ is a stopping time of the filtration $(\mathcal{F}_t)_{t\ge 0}$.

Exercise. (Closed martingale)
Let $(\mathcal{F}_t)_{t \ge 0}$ be a filtration defined on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and let $X$ be an integrable and $\mathcal{F}$-measurable random variable. Show that the process $\left( \mathbb{E}(X\mid \mathcal{F}_t) \right)_{t \ge 0}$ is a martingale with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$.

Exercise. Let $(\mathcal{F}_t)_{t \ge 0}$ be a filtration defined on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and let $(M_t)_{t \ge 0}$ be a submartingale with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$. Show that the function $t \rightarrow \mathbb{E} (M_t)$ is non-decreasing.

Exercise. Let $(\mathcal{F}_t)_{t \ge 0}$ be a filtration defined on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and let $(M_t)_{t \ge 0}$ be a martingale with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$. Let now $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a convex function such that for $t \ge 0$, $\mathbb{E} \left( \mid \psi(M_t) \mid \right) < + \infty$. Show that the process $(\psi(M_t))_{t \ge 0}$ is a submartingale.