last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part one)

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9.6 Exercises on Chapter 9 \(\msconly \)

In all questions below \((S_n)\) denotes a random walk started at the origin, but which random walk varies according to the question.

On one dimensional random walks
  • 9.1 Let \((S_n)\) be the simple asymmetric random walk, as in Section 9.1. Let \(a<0<b\) be integers and define the hitting times \(T_a=\inf \{n\in \N \-S_n=a\}\), \(T_b=\inf \{n\in \N \-S_n=b\}\) and \(T=T_a\wedge T_b\).

    • (a) Show that \(\E [S_T]=(p-q)\E [T]\).

    • (b) Calculate \(\E [S_T]\) directly using (9.5) and (9.6) and hence calculate \(\E [T]\).

  • 9.2 This question applies some of the techniques from Section 9.1 to the symmetric case.

    Let \((S_n)\) denote the simple symmetric random walk and let \(T_k\) be the hitting time of \(k\in \Z \). Let \(a<0<b\) be integers and let \(T=T_a\wedge T_b\).

    • (a) Explain carefully why both

      \begin{align*} 1 &= \P [T=T_a]+\P [T=T_b] \\ 0 &= a\P [T=T_a] + b\P [T=T_b]. \end{align*} Hint: Recall that \((S_n)\) is a martingale.

    • (b) Solve these equations to find explicit formulae for \(\P [T=T_a]\) and \(\P [T=T_b]\) in terms of \(a\) and \(b\).

    • (c) Show that \(\E [T]=-ab\).

      Hint: Can you think of a useful martingale?

  • 9.3 Let \((X_i)\) be a sequence of independent, identically distributed random variables with \(\P [X_i=2]=\frac 13\) and \(\P [X_i=-1]=\frac 23\). Set

    \[S_n=\sum _{i=1}^n X_i\]

    and define the stopping time \(R=\inf \{n\geq 1\-S_n=0\}\).

    • (a) Modify the argument in Exercise 7.11 to calculate \(\P [S_n=0]\) explicitly. Hence show that \(\P [S_{3n}=0] \sim \frac {\sqrt {3}}{2\sqrt {\pi n}}\) as \(n\to \infty \).

    • (b) Explain how to modify the proof of Lemma 9.3.1 to deduce that \(\P [R<\infty ]=1\).

  • 9.4 Let \(p\in (\frac 35,1]\). Let \((X_i)\) be a sequence of independent, identically distributed random variables with \(\P [X_i=1]=p\) and \(\P [X_i=-1]=\P [X_i=-2]=\frac {1-p}{2}\). Set

    \[S_n=\sum _{i=1}^n X_i\]

    and let \(T_1=\inf \{n\geq 1\-S_n=1\}\).

    • (a) Explain how to modify the argument in Lemma 9.4.1 to show that \(S_n\stackrel {a.s.}{\to }\infty \) as \(n\to \infty \). Hence show that \(\P [T_1<\infty ]=1\).

    • (b) Modify the argument in Lemma 9.4.3 to calculate \(\E [T_1]\).

  • 9.5 For the simple symmetric random walk, in Lemma 9.3.6 we showed that \(\P [T_1=2n-1]\sim \frac {1}{2n\sqrt {\pi n}}\). Use this fact to give a second proof (alongside that of Lemma 9.3.4) that \(\E [T_1]=\infty \).

  • 9.6 Let \((S_n)\) denote the simple symmetric random walk and let \(T_m=\inf \{n\geq 0\-S_n=m\}\) be the first hitting time of \(m\in \Z \). Let

    \[M^{(\theta )}_n = \frac {e^{\theta S_n}}{(\cosh \theta )^n}\]

    where \(\theta \in \R \).

    • (a) Show that \(M^{(\theta )}_n\) is a martingale.

    • (b) Check that none of the conditions (a)-(c) of the optional stopping theorem apply to the martingale \((M^{(\theta )}_n)\) at the stopping time \(T_m\).

    • (c) [Challenge question] Show that

      \[\E \l [\frac {1}{(\cosh \theta )^{T}}\r ]=\frac {1}{\cosh (m\theta )}\]

      where \(T=T_m\wedge T_{-m}\). You should start by applying the optional stopping theorem to a suitable martingale.

On random walks in two and three dimensions \(\offsyl \)
  • 9.7 Let \((S_n)\) denote the two dimensional simple symmetric random walk, as defined in Section 9.5. Prove that, almost surely, for each \(z\in \Z ^2\) there are infinitely many \(n\in \N \) such that \(S_n=z\).

    Hint: You can re-use some of the ideas from proof of Theorem 9.3.3.

  • 9.8 Let \((S_n)\) denote the three dimensional simple symmetric random walk, as defined in Section 9.5. Let \(G=\sum _{n=0}^\infty \1_{\{S_{2n}=0\}}\) denote the total number of visits to the origin. Let \(R=\min \{n=1,2,\ldots \-S_{n}=0\}\) and \(L=\max \{n=0,1,2,\ldots \-S_n=0\}\) denote, respectively, the first return time and the last visiting time of \((S_n)\) to the origin.

    • (a) Explain why \(\P [L<\infty ]=1\), as a consequence of Lemma 9.5.2.

    • (b) Is \(L\) is a stopping time? Give a brief reason for your answer.

    • (c) Show that \(\P [L=2n]=\P [S_{2n}=0]\P [R=\infty ]\) and hence prove that \(\E [G]=\frac {1}{1-\P [R<\infty ]}.\)

    • (d) We already came close to deducing this exact formula for \(\E [G]\), more than once, within the current chapter. Can you see where?

  • 9.9 Let \((S_n)\) denote the three dimensional simple symmetric random walk, as defined in Section 9.5. Prove that \(|S_n|\stackrel {a.s.}{\to }\infty \) as \(n\to \infty \).