## 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 $\phi: C^\infty (M,R) \to C^\infty (M,R)$ 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 $g$, $Vg(x)=df_x (U(x))$.

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.

## HW2 MA5161. Due February 3

Exercise 1. Let $m:\mathbb{R}_{\ge 0} \rightarrow \mathbb{R}$ and let $R: \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \rightarrow \mathbb{R}$ be a symmetric and positive function. Show that there exists a probability space $\left( \Omega , \mathcal{F}, \mathbb{P} \right)$ and a Gaussian process $(X_t)_{t \ge 0}$ defined on it, whose mean function is $m$ and whose covariance function is $R$.

Exercise 2. Let $(X_t)_{t \ge 0}$ be a continuous process adapted to a filtration $\mathcal{F}_t$. Show that $(X_t)_{t \ge 0}$ is progressively measurable.

## HW2 MA5311: Due February 3

Exercise 1. Let $g : \mathbf{R}^2 \to \mathbf{R}^4$, $(u,v) \to (\cos u , \sin u, \cos v, \sin v)$. Show that $g( \mathbf{R}^2)$ is a 2-dimensional smooth manifold homeomorphic to the torus $\mathbf{S}^1 \times \mathbf{S}^1$.

Exercise 2. Let $h_+: \mathbf{S}^2-N \to \mathbf{C}$ be the stereographic projection from the north pole $N$, and $h_-$ be the stereographic projection from the south pole $S$.

1. Show that for $z \neq 0$, $h_+ h_-^{-1} =\frac{1}{\bar z}$.
2. Show that if $P$ is a non constant polynomial, the map $f=h_+^{-1} P h_+$, $f(N)=N$ is smooth.
3. More generally, if $Q: \mathbf{C} \to \mathbf{C}$ is smooth, find a condition on $Q$ so that $f=h_+^{-1} Q h_+$, $f(N)=N$ is smooth.
4. By following Milnor’s argument in the proof of the fundamental theorem of algebra, find sufficient conditions so that a smooth map $Q: \mathbf{C} \to \mathbf{C}$ is onto.

Exercise 3.  By using Sard’s theorem, prove that the set of regular values of a smooth map $f : M \to N$ is dense in $N$.

## MA5161. HW 1 due Wednesday 1/25

Exercise 1. Show that the  $\sigma$-algebra $\mathcal{T}(\mathbb{R}_{\ge 0},\mathbb{R}^d)$ is also generated by the following families:

• $\{ f \in \mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d ), f(t_1) \in B_1,...,f(t_n) \in B_n \}$where $t_1,...,t_n \in \mathbb{R}_{\ge 0}$ and where $B_1,...,B_n$ are Borel sets in $\mathbb{R}^d$.
• $\{ f \in \mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d), (f(t_1),...,f(t_n)) \in B \}$where $t_1,...,t_n \in \mathbb{R}_{\ge 0}$ and where $B$ is a Borel set in $(\mathbb{R}^{d})^{\otimes n}$.

Exercise 2.  Show that the following sets are in $\mathcal{B} ([0,1],\mathbb{R})$:

• $\{ f \in \mathcal{C}([0,1], \mathbb{R}), \sup_{t\in [0,1]} f(t) <1 \}$
• $\{ f \in \mathcal{C}([0,1], \mathbb{R}), \exists t\in [0,1] f(t) =0 \}$