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4.1 Let \(S_n=\sum _{i=1}^n X_i\) be the symmetric random walk from Section 4.1 and let \(Z_n=e^{S_n}\). Show that \((Z_n)\) is a submartingale and that
\[M_n=\l (\frac {2}{e+\frac {1}{e}}\r )^nZ_n\]
is a martingale.
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4.2 Let \(S_n=\sum _{i=1}^n X_i\) be the asymmetric random walk from Section 4.1, where \(\P [X_i=1]=p\), \(\P [X_i=-1]=q\) and with \(p>q\) and \(p+q=1\). Show that \((S_n)\) is a submartingale and that
\[M_n=\l (\frac {q}{p}\r )^{S_n}\]
is a martingale.
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4.3 Let \((X_i)\) be a sequence of identically distributed random variables with common distribution
\[X_i= \begin {cases} a & \text { with probability }p_a\\ -b & \text { with probability }p_b=1-p_a. \end {cases} \]
where \(0\leq a,b\). Let \(S_n=\sum _{i=1}^n X_i\). Under what conditions on \(a,b,p_a,p_b\) is \((S_n)\) a martingale?
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4.4 Let \(S_n=\sum _{i=1}^n X_i\) be the symmetric random walk from Section 4.1. Show that \(S_n^2\) is a submartingale and that \(M_n=S_n^2-n\) is a martingale.
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4.5 Let \((X_i)\) be an i.i.d. sequence of random variables such that \(\P [X_i=1]=\P [X_i=-1]=\frac 12\). Define
a stochastic process \(S_n\) by setting \(S_0=1\) and
\[S_{n+1}= \begin {cases} S_n+X_{n+1} & \text { if }S_n>0,\\ 1 & \text { if }S_n=0. \end {cases} \]
That is, \(S_n\) behaves like a symmetric random walk but, whenever it becomes zero, on the next time step it is ‘reflected’ back to \(1\). Let
\[L_n=\sum _{i=0}^{n-1}\1\{S_i=0\}\]
be the number of time steps, before time \(n\), at which \(S_n\) is zero. Show that
\[\E [S_{n+1}\|\mc {F}_n]=S_n+\1\{S_n=0\}\]
and hence show that \(S_n-L_n\) is a martingale.
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4.6 Consider an urn that may contain balls of three colours: red, blue and green. Initially the urn contains one ball of each
colour. Then, at each step of time \(n=1,2,\ldots \) we draw a ball from the urn. We place the drawn ball back into the urn and add an additional ball of the same colour.
Let \((M_n)\) be the proportion of balls that are red. Show that \((M_n)\) is a martingale.
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4.7 Let \(S_n=\sum _{i=1}^n X_i\) be the symmetric random walk from Section 4.1. State, with proof, which of the following processes are martingales:
\(\seteqnumber{0}{4.}{6}\)
\begin{equation*}
\text {(i) }S_n^2+n \hspace {3pc}\text {(ii) }S_n^2+S_n-n \hspace {3pc}\text {(iii) }\frac {S_n}{n}
\end{equation*}
Which of the above are submartingales?