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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