Chapter 3

Suppose that \(\left\\{S_{n}\right\\}_{n \geq 0}\) is a symmetric simple random walk under \(\mathbb{P} .\) Show that $$ \mathbb{P}\left[\frac{S_{[n t]}}{\sqrt{n}} \leq x\right] \rightarrow \int_{-\infty}^{x} \frac{1}{\sqrt{2 \pi t}} \exp \left(-\frac{y^{2}}{2 t}\right) d y $$ as \(n \rightarrow \infty\) where \([n t]\) is the integer part of \(n t\).

Let \(Z\) be normally distributed with mean zero and variance one under the measure \(\mathbb{P} .\) What is the distribution of \(\sqrt{t} Z ?\) Is the process \(X_{t}=\sqrt{t} Z\) a Brownian motion?

Suppose that \(W_{t}\) and \(\tilde{W}_{t}\) are independent Brownian motions under the measure \(\mathbb{P}\) and let \(\rho \in[-1,1]\) be a constant. Is the process \(X_{t}=\rho W_{t}+\sqrt{1-\rho^{2}} \tilde{W}_{t}\) a Brownian motion?

Let \(\left\\{W_{t}\right\\}_{t \geq 0}\) be standard Brownian motion under the measure \(\mathbb{P}\). Which of the following are P-Brownian motions? (a) \(\left\\{-W_{t}\right\\}_{t \geq 0}\) (b) \(\left\\{c W_{t / c^{2}}\right\\}_{t \geq 0}\), where \(c\) is a constant, (c) \(\left\\{\sqrt{t} W_{1}\right\\}_{t \geq 0}\) (d) \(\left\\{W_{2 x}-W_{t}\right\\}_{t \geq 0}\) Justify your answers.

Suppose that \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma^{2}\). Calculate $$ \mathbb{E}\left[e^{\theta X}\right] $$ and hence evaluate \(\mathbb{E}\left[X^{4}\right]\).

Prove that if \(\left\\{W_{t}\right\\}_{t \geq 0}\) is standard Brownian motion under \(\mathbb{P}\) then, for \(x>0\) $$ \mathbb{P}\left[W_{t} \geq x\right] \equiv \int_{x}^{\infty} \frac{1}{\sqrt{2 \pi t}} e^{-y^{2} / 2 t} d y \leq \frac{\sqrt{t}}{x \sqrt{2 \pi}} e^{-x^{2} / 2 t} $$

Brownian motion is not going to be adequate as a stock market model. First, it has constant mean, whereas the stock of a company usually grows at some rate, if only due to inflation. Moreover, it may be too 'noisy' (that is the variance of the increments may be bigger than those observed for the stock) or not noisy enough. We can scale to change the 'noisiness' and we can artificially introduce a drift, but this still won't be a good model. Here is one reason why. Suppose that \(\left\\{W_{t}\right\\}_{t \geq 0}\) is standard Brownian motion under \(\mathbb{P} .\) Define a new process \(\left\\{S_{t}\right\\}_{I \geq 0}\) by \(S_{t}=\mu t+\sigma W_{t}\) where \(\sigma>0\) and \(\mu \in \mathbb{R}\) are constants. Show that for all values of \(\sigma>0, \mu \in \mathbb{R}\) and \(T>0\) there is a positive probability that \(S_{T}\) is negative.

Suppose that \(\left\\{W_{t}\right\\}_{t \geq 0}\) is standard Brownian motion. Prove that conditional on \(W_{t_{1}}=\) \(x_{1}\), the probability density function of \(W_{t_{1} / 2}\) is $$ \sqrt{\frac{2}{\pi t_{1}}} \exp \left(-\frac{1}{2}\left(\frac{\left(x-\frac{1}{2} x_{1}\right)^{2}}{t_{1} / 4}\right)\right) $$

Let \(\left\\{W_{t}\right\\}_{t \geq 0}\) be standard Brownian motion under \(\mathbb{P} .\) Let \(T_{a}\) be the 'hitting time of level \(a\), that is $$ T_{a}=\inf \left\\{t \geq 0: W_{t}=a\right\\} $$ Then we proved in Proposition \(3.4 .9\) that $$ \mathbb{E}\left[\exp \left(-\theta T_{a}\right)\right]=\exp (-a \sqrt{2 \theta}) $$ Use this result to calculate (a) \(\mathbb{E}\left[T_{a}\right]\), (b) \(\mathbb{P}\left[T_{a}<\infty\right]\).

Let \(\left\\{W_{t}\right\\}_{t \geq 0}\) denote standard Brownian motion under \(\mathbb{P}\) and define \(\left\\{M_{t}\right\\}_{r \geq 0}\) by $$ M_{t}=\max _{0 \leq s \leq t} W_{s} $$ Suppose that \(x \geq a .\) Calculate (a) \(\mathbb{P}\left[M_{t} \geq a, W_{t} \geq x\right]\), (b) \(\mathbb{P}\left[M_{t} \geq a, W_{t} \leq x\right]\).

Let \(\left\\{W_{t}\right\\}_{t \geq 0}\) be standard Brownian motion under the measure \(\mathbb{P}\) and let \(\left\\{\mathcal{F}_{I}\right\\}_{t \geq 0}\) denote its natural filtration. Which of the following are \(\left(\mathbb{P},\left\\{\mathcal{F}_{t}\right\\}_{t \geq 0}\right)\)-martingales? (a) \(\exp \left(\sigma W_{t}\right)\), (b) \(c W_{t / c^{2}}\), where \(c\) is a constant, (c) \(t W_{t}-\int_{0}^{t} W_{s} d s\)

Let \(\left\\{\mathcal{F}_{t}\right\\}_{0 \leq t \leq T}\) denote the natural filtration associated to a standard \(\mathbb{P}\)-Brownian motion, \(\left\\{W_{t}\right\\}_{0 \leq t \leq T}\). The result of Lemma 3.4.6.3 can be rewritten as $$ \mathbb{E}\left[\exp \left(\sigma W_{t}-\frac{1}{2} \sigma^{2} t\right) ; A\right]=\exp \left(\sigma W_{s}-\frac{1}{2} \sigma^{2} s\right) \mathbf{1}_{A}, \quad \text { for all } A \in \mathcal{F}_{S} $$ Use differentiation under the integral sign to provide another proof that \(\left\\{W_{t}^{2}-t\right\\}_{t \geq 0}\) is a \(\left(\mathbb{P},\left\\{\mathcal{F}_{t}\right\\}_{t \geq 0}\right)\)-martingale and show that the following are also \(\left(\mathbb{P},\left\\{\mathcal{F}_{t}\right\\}_{t \geq 0}\right)\) martingales: (a) \(W_{t}^{3}-3 t W_{t}\) (b) \(W_{t}^{4}-6 t W_{t}^{2}+3 t^{2}\)

Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space. Suppose that the real random variable \(T: \Omega \rightarrow\) \(\mathbb{R}\) is uniformly distributed on \([0,1]\) under the measure \(\mathbb{P}\). Define \(\left\\{X_{t}\right\\}_{t \geq 0}\) by $$ X_{t}(\omega)= \begin{cases}1, & T(\omega)=t \\ 0, & T(\omega) \neq t\end{cases} $$ Check that \(\left\\{X_{t}\right\\}_{t \geq 0}\) is a P-martingale with respect to its own filtration. [Hint: Conditional expectation is only unique to within a random variable that is almost surely zero.] Show that \(T\) is a stopping time for which the Optional Stopping Theorem fails.