Stochastic Processes and Financial Mathematics
(part two)
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11.5 Exercises on Chapter 11
In all the following questions, \(B_t\) denotes standard Brownian motion.
On Brownian motion
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11.1 Consider the process \(C_t=\mu t+ \sigma B_t\), for \(t\geq 0\), where \(\mu \in \R \) and \(\sigma >0\) are deterministic
constants.
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(a) Find the mean and variance of \(C_t\).
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(b) Let \(0\leq u\leq t\). What is the distribution of \(C_t-C_u\)?
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(c) Is \(C_t\) a random continuous function?
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(d) Is \(C_t\) a Brownian motion?
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11.2 Let \(0\leq u\leq t\). Use the properties of Brownian motion to show that \(\cov (B_t,B_u)=u\).
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11.3 Let \(u\geq 0\) and \(t\geq 0\). Show that \(\E [B_t\|\mc {F}_u]=B_{\min (u,t)}\).
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11.4
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(a) Show that \(\E [B_t^n]=t(n-1)\E [B_t^{n-2}]\) for all \(n\geq 2\). (Hint: Integrate by parts!)
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(b) Deduce that \(\E [B_t^2]=t\) and \(\var (B_t^2)=2t^2\).
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(c) Write down \(\E [B_t^n]\) for any \(n\in \N \).
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(d) Show that \(B_t^n\in L^1\) for all \(n\in \N \).
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11.5 Let \(Z\sim N(\mu ,\sigma ^2)\). Show that \(\E [e^Z]=\exp (\mu +\frac 12\sigma ^2)\). (Hint: Complete the
square!)
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11.6 Show that the following processes are martingales.
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11.7 Fix \(t>0\) and for each \(n\in \N \) let \((t_k)_{k=0}^n\) be such that \(0=t_0<t_1<\ldots <t_n=t\)
and \(\max _k|t_{k+1}-t_k|\to 0\) as \(n\to \infty \). (For example: \(t_k=\frac {kt}{n}\)).
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(a) Show that \(\sum \limits _{k=0}^{n-1} t_{k+1}-t_k=t\) and \(\sum \limits _{k=0}^{n-1} B_{t_{k+1}}-B_{t_k}=B_t\).
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(b) Show that \(\sum \limits _{k=0}^{n-1} (t_{k+1}-t_k)^2\to 0\) as \(n\to \infty \).
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(c) Set \(S_n=\sum \limits _{k=0}^{n-1} (B_{t_{k+1}}-B_{t_k})^2\). Show that \(\E [S_n]=t\) and that \(\var (S_n)\to 0\)
as \(n\to \infty \).
Challenge Questions
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11.8
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(a) Let \(y\geq 1\). Show that
\[\P [B_t\geq y]\leq \sqrt {\frac {t}{2\pi }}e^{-\frac {y^2}{2t}}.\]
Let \(\alpha >\frac 12\). Deduce that \(\P [B_t\geq t^\alpha ]\to 0\) as \(t\to \infty \). What about \(\alpha =\frac 12\)?
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(b) Let \(y\geq 0\). Show that
\[\P [B_t\geq y]\leq \frac {t}{\sqrt {2\pi }} \frac {1}{y}e^{-\frac {y^2}{2t}}.\]
Deduce that \(B_t\to 0\) in probability as \(t\searrow 0\).