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 $[0,1]$.
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