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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.~16, 2012

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\noindent 1. 

\noindent (a) Assume that $f=f(t,x)$ is a $\cC^{1,2}(\bR^+\times \bR)$
such that $\partial_t f+\frac 12\partial_{xx}f=0$. Show that 
$f(t,B_t)$ is a local martingale. 

\noindent (b) For $f$ as in (a), 
let $g(t,x)=\int_0^x f(t,z)\, dz-\frac 12\int_0^t\partial_xf(s,0)\,ds$. 
Show that $g(t,B_t)$ is also a local martingale. 

\noindent (c) Let $h_n$ be the Hermite polynomial of degree $n$. (The first four are 
$1$, $x$, $x^2-1$, $x^3-3x$.) Let $f_n(t,x)=t^{n/2}h_n(x/\sqrt t)$ for $n\ge 0$. 
Show that $f_n(t,B_t)$ is a martingale for every $n$. 

\noindent (d) Define the processes $X^n_t$ recursively by $X^0_t\equiv 1$ and 
$X^{n+1}_t=\int_0^tX^n_s\, ds$. Find the connection between $X^n_t$ and the martingales 
from (c). 


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\noindent 2. Assume that $B^1_t$ and $B^2_t$ are independent Brownian 
motions. Consider $X_t=\int_0^tB^1_s\, dB^1_s$ and $X_t=\int_0^tB^2_s\, dB^1_s$. 
Then the two bracket processes $\langle X\rangle_t$ and $\langle Y\rangle_t$
are equal in distribution, but it is not even true that $X_t$ and $Y_t$ are 
equal in distribution for a fixed $t>0$. Prove this. 

 
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\noindent 3. What should be the quadratic variation process for 
a process given by $dX_t=a\,dt+b_1\,dB^1_t+b_2\,dB^2_t$?
Give the reasoning for your formula although a complete proof is not 
necessary. In particular, compute the quadratic variation process in the case
$dX_t=B^1_tB^2_t\,dB^1_t+|B^1_t|\,dB^2_t$.


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