Stochastic Processes and Financial Mathematics
(part one)
2.5 Exercises on Chapter 2
On probability spaces
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2.1 Consider the experiment of throwing two dice, then recording the uppermost faces of both dice. Write down a suitable sample space \(\Omega \) and suggest an appropriate \(\sigma \)-field \(\mc {F}\).
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2.2 Let \(\Omega =\{1,2,3\}\). Let
\(\seteqnumber{0}{2.}{7}\)\begin{align*} \mc {F}&=\{\emptyset , \{1\}, \{2,3\}, \{1,2,3\}\},\\ \mc {F}'&=\{\emptyset , \{2\}, \{1,3\}, \{1,2,3\}\}. \end{align*}
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(a) Show that \(\mc {F}\) and \(\mc {F}'\) are both \(\sigma \)-fields.
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(b) Show that \(\mc {F}\cup \mc {F}'\) is not a \(\sigma \)-field, but that \(\mc {F}\cap \mc {F}'\) is a \(\sigma \)-field.
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2.3 Let \(\Omega =\{H,T\}^\N \) be the probability space from Section 2.3, corresponding to an infinite sequence of independent fair coin tosses \((X_n)_{n=1}^\infty \).
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(a) Fix \(m\in \N \). Show that the probability that that random sequence \(X_1,X_2,\ldots ,\) contains precisely \(m\) heads is zero.
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(b) Deduce that, almost surely, the sequence \(X_1,X_2,\ldots \) contains infinitely many heads and infinitely many tails.
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2.4 Let \(\Omega =\{1,2,3,4,5,6\}\), corresponding to one roll of a die. In each of the following cases describe, in words, the information contained within the given \(\sigma \)-field.
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(a) \(\F _1=\big \{\emptyset ,\{1,2,3,4,5\},\{6\},\Omega \big \}\).
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(b) \(\F _2=\sigma \big (\{1\},\{2\},\{3\},\{4,5,6\}\big )\).
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(c) \(\F _3=\big \{\emptyset ,\{1,2\},\{3,4\},\{5,6\}, \{1,2,3,4\}, \{3,4,5,6\}, \{1,2,5,6\}, \Omega \big \}\).
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On random variables
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2.5 Let \(\Omega =\{1,2,3,4,5\}\) and set
\(\seteqnumber{0}{2.}{7}\)\begin{align*} \G _1&=\sigma \big (\{1,5\},\{2,4\},\{3\}\big )\\ \G _2&=\big \{\emptyset ,\{1,2\},\{3,4,5\},\Omega \big \}. \end{align*}
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(a) Define \(X_1:\Omega \to \R \) by \(X_1(\omega )=(\omega -3)^2\). Show that \(X_1\in m\G _1\) and \(X_1\notin m\G _2\).
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(b) Give an example of a function \(X_2:\Omega \to \R \) such that \(X_2\in m\G _2\) and \(X_2\notin m\G _1\).
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2.6 Let \(\Omega =\{HH,HT,TH,TT\}\), representing two coin tosses. Define \(X\) to be the total number of heads shown. Write down all the events in \(\sigma (X)\).
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2.7 Let \(X\) be a random variable. Explain why \(\frac {X}{X^2+1}\) and \(\sin (X)\) are also random variables.
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2.8 Let \(X\) be a random variable with the probability density function \(f:\R \to \R \) given by
\[f(x)= \begin {cases} 2x^{-3} & \text { if }x\in [1,\infty ),\\ 0 & \text { otherwise.} \end {cases} \]
Show that \(X\in L^1\) but \(X\notin L^2\).
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2.9 Let \(1\leq p\leq q<\infty \).
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(a) Generalize (2.7) to show that \(\E [|X|^p]\leq \E [|X|^q]+1\).
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(b) Deduce that if \(X\in L^q\) then \(X\in L^p\).
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2.10 Let \(a>0\) and let \(X\) be a random variable such that \(X\geq 0\). Show that \(\P [X\geq a]\leq \frac {1}{a}\E [X].\)