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Math 635 -- Spring 2012\\[10pt]{\textbf{Homework 5\\[15pt]}}
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\noindent{\textbf {Due:  March 26, 2012}, beginning of the class. Late homework will \textbf{not} be accepted. }\\





\begin{enumerate}
\item \textbf{(Exercise 6.1)} Use the It\^o isometry to calculate the variances of
\[
\int_0^t|B_s|^{1/2} dBs \quad \textup{and} \quad \int_0^t(B_s+s)^2 dB_s
\]


\item \textbf{(Exercise 6.2)} The integrals 
\[
I_1=\int_0^t B_s ds\quad \textup{and} \quad I_2=\int_0^t B_s^2 ds
\]
are not stochastic integrals although they are random variables. In these cases we just integrate a (random) continuous function the usual (traditional) way. Find the mean and variance of $I_1$ and $I_2$.



\item Use the definition of the It\^o integral to prove that
\[
\int_0^t B_s^2 dB_s=\frac13 B_t^3-\int_0^t B_s ds.
\]



\end{enumerate}


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