MA5311. Take home exam due 03/20

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

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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<H<1. 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 <T, 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].
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MA5311. Non orientable manifolds

Here are some videos to visualize non orientability.

 

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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.

 

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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.

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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).

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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.

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