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\noindent Math 236A, Fall 2012. 

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\centerline{\bf Homework Assignment 5}

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\noindent {\bf Due:} Nov.~26, 2012

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\noindent 1. Let an It\^o process $X_t$ be given by $dX_t=a\,dt+b\,dB_t$. Show that 
this decomposition is unique, that is, if also $dX_t=\alpha\,dt+\beta\,dB_t$, 
then $P(\alpha =a, \beta=b$ for almost all $t)$=1. 


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\noindent 2. Assume that $h_1,\dots, h_n$ are {\it deterministic\/} real functions
which are orthonormal in $L^2[0,t]$.  Find the distribution of the vector 
$(\int_0^t h_1\,dB_s, \dots, \int_0^t h_n\,dB_s)$. 
 
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\noindent 3. Find the stochastic integral representation of the following random 
variables (for a fixed $T$): $B_{T/2}B_T$, $\left(\int_0^Ts\,dB_s\right)^2$, 
$\int_0^T sB_s\,ds$. You may (but are not required to) use Malliavin calculus. 


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