Stochastic Processes and Financial Mathematics
(part one)
8.6 Exercises on Chapter 8 \(\msconly \)
On the dominated convergence theorem
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8.1 Let \(U\) be a random variable that takes values in \((1,\infty )\). Define \(X_n=U^{-n}\). Show that \(\E [X_n]\to 0\).
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8.2 Let \(X\) be a random variable in \(L^1\) and set
\[X_n=X\1_{\{|X|\leq n\}}= \begin {cases} X & \text { if }|X|\leq n, \\ 0 & \text { otherwise.} \end {cases} \]
Show that \(\E [X_n]\to \E [X]\).
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8.3 Let \(Z\) be a random variable taking values in \([1,\infty )\) and for \(n\in \N \) define
\(\seteqnumber{0}{8.}{2}\)\begin{equation} \label {eq:Xn} X_n=Z\1_{\{Z\in [n,n+1)\}}= \begin{cases} Z & \text { if }Z\in [n,n+1)\\ 0 & \text { otherwise.} \end {cases} \end{equation}
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(a) Suppose that \(Z\in L^1\). Use the dominated convergence theorem to show that \(\E [X_n]\to 0\) as \(n\to \infty \).
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(b) Suppose, instead, that \(Z\) is a continuous random variable with probability density function
\[f(x)=\begin {cases} x^{-2} & \text { if }x\geq 1,\\ 0 & \text { otherwise}. \end {cases} \]
and define \(X_n\) using (8.3). Show that \(Z\) is not in \(L^1\), but that \(\E [X_n]\to 0\).
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(c) Comment on what part (b) tells us about the dominated convergence theorem.
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On stopping times and optional stopping
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8.4 Let \((\mc {F}_n)\) be a filtration. Check that \(T\) is a stopping time if and only if \(\{T=n\}\) is \(\mc {F}_n\) measurable for all \(n\).
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8.5 Let \((\mc {F}_n)\) be a filtration and let \((S_n)\) be an adapted process, with values in \(\Z \) and initial value \(S_0=0\). Show that \(R=\inf \{n\geq 1\-S_n=0\}\) is a stopping time.
Note: \(R\) is known as the first return time of \((S_n)\) to \(0\).
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8.6 Let \(S\) and \(T\) be stopping times with respect to the same filtration. Show that \(S+T\) is also a stopping time.
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8.7 Let \((\mc {F}_n)\) be a filtration. Let \(S\) and \(T\) be stopping times with \(S\leq T\). Show that \(\mc {F}_S\sw \mc {F}_T\).
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8.8 Recall the Pólya urn process from Section 4.2. Recall that \(B_n\) is the number of black balls in the urn at time \(n\), and that \(M_n=\frac {B_n}{n+2}\) is the fraction of black balls in the urn at time \(n\). Let \(T\) be the first time at which a black ball is drawn from the urn.
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(a) Show that \(\P [T\geq n]=\frac {1}{n}\).
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(b) Use the optional stopping theorem to show that \(\E [M_T]=\frac 12\) and \(\E [\frac {1}{T+2}]=\frac 14\).
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8.9 This question is a continuation of exercise 7.8 – recall the urn process from that exercise.
Let \(T\) be the first time at which the urn contains only balls of one colour, and suppose that initially the urn contains \(r\) red balls and \(b\) black balls. Recall that \(M_n\) denotes the fraction of red balls in the urn at time \(n\).
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(a) Deduce that \(\P [T<\infty ]=1\), and that \(\P [M_T=0\text { or }M_T=1]=1\).
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(b) Show that \(\E [M_T]=\frac {r}{r+b}\) and hence deduce that \(\P [M_T=1]=\frac {r}{r+b}\).
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8.10 Let \(m\in \N \) and \(m\geq 2\). At time \(n=0\), an urn contains \(2m\) balls, of which \(m\) are red and \(m\) are blue. At each time \(n=1,\ldots ,2m\) we draw a single ball from the urn; we discard this ball and do not replace it. We continue until the urn is empty. Therefore, at time \(n\in \{0,\ldots ,2m\}\) the urn contains \(2m-n\) balls.
Let \(N_n\) denote the number of red balls remaining in the urn at time \(n\). For \(n=0,\ldots ,2m-1\) let
\[P_n=\frac {N_n}{2m-n}\]
be the fraction of red balls remaining after time \(n\).
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(a) Show that \(P_n\) is a martingale, with respect to a natural filtration that you should specify.
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(b) [Challenge question] Let \(T\) be the first time at which we draw a red ball. Note that a \((T+1)^{st}\) ball will be drawn, because the urn initially contains at least two red balls. Show that the probability that the \((T+1)^{st}\) ball is red is \(\frac {1}{2}\).
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