Stochastic Processes and Financial Mathematics
(part two)
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12.5 Exercises on Chapter 12
In all the following questions, \(B_t\) denotes a Brownian motion and \(\mc {F}_t\) denotes its generated filtration.
On Ito integration
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12.1 Using (12.8), find \(\int _v^t 1\,dB_u\), where
\(0\leq v\leq t\).
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12.2 Show that the process \(e^{B_t}\) is in \(\mc {H}^2\). (Hint: Use (11.2).)
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12.3
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(a) Let \(Z\sim N(0,1)\). Show that the expectation of \(e^{\frac {Z^2}{2}}\) is infinite.
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(b) Give an example of a continuous, adapted, stochastic process that is not in \(\mc {H}^2\).
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12.4 Let \(X_t\) be an Ito process satisfying
\[X_t=2+\int _0^t t+B_u^2\,du+\int _0^t B_u^2\,dB_u.\]
Find \(\E [X_t]\).
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12.5 Which of the following stochastic processes are Ito processes?
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12.6 Let \(V_t\) be the stochastic process given by
\[V_t=e^{-kt}v+\sigma e^{-kt}\int _0^t e^{ku}\,dB_u\]
where \(k,\sigma ,v>0\) are deterministic constants. Find the mean and variance of \(V_t\).
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12.7 Suppose that \(\mu >0\) is a deterministic constant and that \(\sigma _t\in \mc {H}^2\). Let \(X_t\) be given by
\[X_t=\int _0^t\mu \,du+\int _0^t\sigma _u\,dB_u.\]
Show that \(X_t\) is a submartingale.
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12.8
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(a) Give an example of a stochastic process \(F_t\) such that \(\int _0^t \E [F_s]\,ds=\E [\int _0^t F_s\,dB_s]\).
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(b) Give an example of a stochastic process \(F_t\) such that \(\int _0^t \E [F_s]\,ds\neq \E [\int _0^t F_s\,dB_s]\).
Challenge Questions