## Math489/889 Stochastic Processes and Advanced Mathematical Finance Homework 7

Due Wed, October 20, 2010
1. Suppose $X$ is a continuous random variable with mean and variance both equal to $20$. What can be said about $ℙ\left[0\le X\le 40\right]$?
1. Look up the distribution of a Poisson random variable with parameter $\lambda$, state it and use that to calculate $ℙ\left[X\ge 1\right]$, and $ℙ\left[X\ge 2\right]$ where $X$ is a Poisson random variable with parameter $1$.
2. Given that the m.g.f. ${\varphi }_{X}\left(t\right)$ of a Poisson random variable with parameter $\lambda$ is ${e}^{\lambda \left({e}^{t}-1\right)}$, show that the sum of independent Poisson random variables ${X}_{1}$ with parameter ${\lambda }_{1}$ and ${X}_{2}$ with parameter ${\lambda }_{2}$ is again Poisson with parameter ${\lambda }_{1}+{\lambda }_{2}$.
3. Using the fact that the sum of two independent Poisson random variables with means ${\lambda }_{1}$ and ${\lambda }_{2}$ is again Poisson with mean ${\lambda }_{1}+{\lambda }_{2}$ find the exact probability that $ℙ\left[{X}_{1}+\cdots +{X}_{10}>15\right]$ where each ${X}_{i}$ is a Poisson random variable with parameter $1$.
4. Use the Markov Inequality to get a bound on $ℙ\left[{X}_{1}+\cdots +{X}_{10}>15\right]$ where each ${X}_{i}$ is a Poisson random variable with parameter $1$.
2. A first simple assumption is that the daily change of a company’s stock on the stock market is a random variable with mean $0$ and variance ${\sigma }^{2}$. That is, if ${S}_{n}$ represents the price of the stock on day $n$ with ${S}_{0}$ given, then
${S}_{n}={S}_{n-1}+{X}_{n},n\ge 1$

where ${X}_{1},{X}_{2},\dots$ are independent, identically distributed continuous random variables with mean $0$ and variance ${\sigma }^{2}$. (Note that this is an additive assumption about the change in a stock price. In the binomial tree models, we assumed that a stock’s price changes by a multiplicative factor up or down. We will have more to say about these two distinct models later.) Suppose that a stock’s price today is $100$. If ${\sigma }^{2}=1$, what can you say about the probability that after $10$ days, the stock’s price will be between $95$ and $105$ on the tenth day?

3. Find the moment generating function ${\varphi }_{X}\left(t\right)=E\left[exp\left(tX\right)\right]$ of the random variable $X$ which takes values $1$ with probability $1∕2$ and $-1$ with probability $1∕2$. Show directly (that is, without using Taylor polynomial approximations) that ${\varphi }_{X}{\left(t∕\sqrt{n}\right)}^{n}\to exp\left({t}^{2}∕2\right)$. (Hint: Use L’Hopital’s Theorem to evaluate the limit, after taking logarithms of both sides.)