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Continuous Time Processes - Exercises

Exercise 1

define the variations of \(f\) as the function

\[S_t=\sup_{\Pi=\{0=t_0\leq t_1\leq \cdots \leq t_n=t\}} \sum_{k=1}^n \left|f_{t_k}-f_{t_{k-1}}\right|, \quad t \in [0,\infty[\]

and we say that \(f\) has bounded variations if \(S_t<\infty\) for every \(t\). 1. show that if \(f\) has bounded variations, then

$$S_t-f_t\quad \text{and}\quad S_t+f_t$$

are both increasing functions.
  1. Show that if \(t\mapsto f_t\) is Lipschitz continuous, then \(f\) has bounded variations;

  2. Show that if \(t \mapsto f_t\) is continuous and has bounded variations, then for every sequence \(\Pi^n = \{0=t_0<t_1, \ldots, t_n=T\}\) of subdivisions of \([0, T]\) such that \(|\Pi^n| := \sup |t_i - t_{i-1}|\to 0\) then it holds

    \[\limsup_{n} \sum_{k=1}^n \left| f_{t_k} - f_{t_{k-1}} \right|^2 = 0\]

Exercise 2

\((\Omega, \mathcal{F},P)\), and \(\mathbb{F}\) the filtration generated by \(B\). Show that 1. \(B\) is a martingale and deduce that \(P[\{\omega\colon t\mapsto B_t(\omega) \text{ has bounded variations }\}]=0\). 2. Show that \(B\) and \(B^2_t-t\) are martingales. 3. \(\exp(\sigma B_t-\sigma^2 t/2)\) is a martingale for every \(\sigma>0\); 4. \(1/\sigma^2 B_{\sigma t}\) is a Brownian motion; 5. For fixed \(s\), \(B_{t+s}-B_s\) is a Brownian motion.

Exercise 3

is nowhere differentiable. Let \(B\) a Brownian motion considered on the interval \([0,1]\). Given a constant \(C>0\), for every \(n\), define

$$ \begin{aligned} A_n & = \left{ \omega \colon |B_t(\omega)-B_s(\omega)|\leq C |t-s| \text{ for some }0\leq s\leq 1\text{ and }|t-s|\leq 3/n \right}\ X_{k,n} & = \max_{j\in {0,1,2}} \left| B_{(k+j)/n}-B_{(k+j-1)/n} \right|\ C_n & = \bigcup_{1\leq k\leq n-2}\left{ X_{k,n}\leq 5C/n \right}

\end{aligned} $$

  1. Show that \(P[C_n]\leq \tilde{C} n n^{-3/2}=\tilde{C}n^{-1/2}\) for some constant \(\tilde{C}\).
  2. Show that \(A_n\subseteq C_n\) and deduce that \(B\) is nowhere Lipschitz, hence nowhere differentiable, on \([0,1]\).
  3. This argumentation with three steps \(j=0,1,2\) allows to show (why?) that \(B\) is nowhere H\u00f6lder-continuous on \([0,1]\) for every \(\gamma > 1/2+1/3=5/6\). Using \(k\) steps instead of just three, show that \(B\) is nowhere H\u00f6lder-continuous on \([0,1]\) for every \(\gamma > 1/2+1/k\).
  4. Deduce that \(B\) is nowhere H\u00f6lder continuous on \([0,1]\) for every \(\gamma>1/2\).

Exercise 4

increasing sequence of independent random variable \(T_k:\Omega \to [0,\infty[\), all exponentially distributed with parameter \(\lambda>0\), that is, \(dP_{T_k}=\lambda e^{-\lambda t} dt\) for every \(k\). We define the discrete process \(S\) as

\[S_0=0 \quad\text{and}\quad S_n=\sum_{k=1}^n T_k\]

which somehow model the number of persons arriving into a queue. We finally define the continuous time process

\[N_t=\max\left\{ n\in \mathbb{N}\colon S_n\leq t \right\}, \quad 0\leq t <\infty\]

representing the number of persons in the queue at time \(t\) and define \(\mathcal{F}_t=\sigma\left( N_s\colon s\leq t \right)\). Show that 1. Show that for \(0\leq s\leq t\), it holds[^1]

$$P\left[ S_{N_s+1}>t|\mathcal{F}_s \right]=e^{-\lambda(t-s)}$$

  1. (Difficult, bonus) Show that for \(s\leq t\), \(N_t-N_s\) is a Poisson distributed random variable with parameter \(\lambda(t-s)\) independent of \(\mathcal{F}_s\)[^2] that is

    \[E\left[ 1_A P\left[ N_t-N_s\leq k |\mathcal{F}_s \right] \right]=P\left[ A \right]\sum_{j=0}^k e^{-\lambda(t-s)}\frac{(\lambda(t-s))^j}{j!}\]

    for every \(A \in \mathcal{F}_s\).

  2. Show that the compensated Poisson process

    \[M_t:=N_t-\lambda t, \quad 0\leq t<\infty\]

    is a Martingale.

  3. Show that for any \(c>0\), it holds

    $$ \begin{aligned} \limsup_{t\to \infty} P\left[ \sup_{s\leq t} M_s \geq c\sqrt{\lambda t} \right] & \leq \frac{1}{c\sqrt{2\pi}}\ \liminf_{t \to \infty} P\left[ \inf_{s\leq t} M_s \leq -c\sqrt{\lambda t} \right] & \leq \frac{1}{c\sqrt{2\pi}}\ E\left[ \sup_{s\leq u\leq t}\left(\frac{M_u}{u}\right)^2 \right]&\leq \frac{4t\lambda}{s^2}

    \end{aligned} $$

    the latter inequality being for every \(0<s<t\).[^3]

Exercise 5

some probability space \((\Omega, \mathcal{F}, P)\). We define the Brownian bridge

\[B_t = W_t - t W_1 \quad \text{for }0\leq t\leq 1\]
  1. Show that \(B\) is a Gaussian process, that is, \((B_{t_1}, B_{t_2}, \ldots, B_{t_n})\) has a multivariate distribution for every \(0\leq t_1 < \ldots < t_n\leq 1\). Compute its mean vector and covariance matrix.
  2. Show that \(B\) does not have independent increments.
  3. Option: Draw sample path of the Brownian bridge on a computer.