Stochastic Processes and Financial Mathematics
(part one)
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3.4 Exercises on Chapter 3
On conditional expectation and martingales
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3.1 Let \((X_n)\) be a sequence of independent identically distributed random variables, such that \(\P [X_i=1]=\frac 12\) and
\(\P [X_i=-1]=\frac 12\). Let
\[S_n=\sum \limits _{i=1}^nX_i.\]
Use the properties of conditional expectation to find \(\E [S_2\|\sigma (X_1)]\) and \(\E [S_2^2\|\sigma (X_1)]\) in terms of \(X_1\), \(X_2\) and their expected values.
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3.2 Let \((X_n)\) be a sequence of independent random variables such that \(\P [X_n=2]=\frac 13\) and \(\P
[X_n=-1]=\frac 23\). Set \(\mc {F}_n=\sigma (X_i\-i\leq n)\). Show that \(S_n=\sum _{i=1}^nX_i\) is an \(\mc {F}_n\) martingale.
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3.3 Check that Examples 3.3.8 and 3.3.9 are
martingales.
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3.4 Let \((M_t)\) be a stochastic process that is both and submartingale and a supermartingale. Show that \((M_t)\) is a
martingale.
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3.5
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(a) Let \((M_n)\) be an \(\mc {F}_n\) martingale. Show that, for all \(0\leq n\leq m\), \(\E [M_m\|\mc {F}_n]=M_n\).
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(b) Guess and state (without proof) the analogous result to (a) for submartingales.
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3.6 Let \((M_n)\) be a \(\mc {F}_n\) martingale and suppose \(M_n\in L^2\) for all \(n\). Show that
\(\seteqnumber{0}{3.}{3}\)
\begin{equation}
\label {eq:doob_decomp_pre} \E [M_{n+1}^2|\mc {F}_n]=M_n^2+\E [(M_{n+1}-M_n)^2|\mc {F}_n]
\end{equation}
and deduce that \((M_n^2)\) is a submartingale.
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3.7 Let \(X_0,X_1,\ldots \) be a sequence of \(L^1\) random variables. Let \(\mc {F}_n\) be their generated filtration and
suppose that \(\E [X_{n+1}|\mc {F}_n]=aX_n+bX_{n-1}\) for all \(n\in \N \), where \(a,b>0\) and \(a+b=1\).
Find a value of \(\alpha \in \R \) (in terms of \(a,b\)) for which \(S_n=\alpha X_n+X_{n-1}\) is an \(\mc {F}_n\) martingale.
Challenge questions