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\noindent Math 236A, Fall 2012. 

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\centerline{\bf Homework Assignment 7}

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\noindent {\bf Due:} Dec.~10, 2012, noon. Please leave the assignment in my mailbox 
or slide it under the door of my office. {\it This assignment covers the entire 
course and you are required to make a credible attempt at every problem\/}. 

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\noindent 1. Let $\tau$ be a stopping time, with  $0\leq \tau \leq 1$, for the 
Brownian motion $B_t$ and $I=\int_0^1 B_t\,dt$.


\noindent
(a) Show (carefully!) that $E(B_tB_\tau)=E(\tau\wedge t)$ for each $t\in [0,1]$.  

\noindent
(b) Find a formula for
$$
E[(I-B_\tau)^2],
$$
in terms of $E\tau$, $E\tau^2$.

\noindent
(c) Find a stopping time $\tau \in [0,1]$ that minimizes the expression in (a). 
 
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\noindent 2. 
Assume that $h:[0,1]\to \bR$ is an absolutely continuous (deterministic) function with
$L^2$ derivative. 

\noindent (a) Show that for any $\epsilon>0$, $P(|B_t|<\epsilon$ for all $t\in [0,1])>0$. 

\noindent (b) Assume that $h:[0,1]\to \bR$ is an absolutely continuous (deterministic) function with an $L^2$ derivative, and with $h(0)=0$. Show that  
for any $\epsilon>0$, $P(|B_t-h(t)|<\epsilon$ for all $t\in [0,1])>0$. 

\noindent (c) Let $h$ be as in (b). Show that, with probability 1, 
there exists an integer $n$ so that $|B_n|<\epsilon$ and 
$|B_{t+n}-B_n-h|<\epsilon$  for all $t\in [0,1]$. 

\noindent (d) ``A two-dimensional Brownian motion
will eventually write your name near the origin.'' Formulate this precisely 
(with all necessary assumptions) and explain how it follows from (c). 

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\noindent 3. Consider the two processes
$$
X_t=\int_0^t \cos(s)\, dB_s^1
\qquad {\rm and } \qquad
Y_t=\int_0^t \sin(s)\, dB_s^2.
$$

\noindent
(a) Assume first that $B^1_t=B^2_t=B_t$ is the same Brownian motion. 
For which times $t$ are the random variables $X_t$ and $Y_t$ independent?
 
\noindent 
(b) Assume now that $B^1_t$ and $B^2_t$ are independent Brownian 
motions. Compute the bracket $\langle Z\rangle_t$ of the process $Z_t=X_t+Y_t$. 
What can you say about the distribution of the process $Z_t$?

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\noindent 4. 
Find the unique solution to the SDE
$$
dX_t=a\,dt+\sigma X_t\,dB_t
$$
with $X_0=0$. Here, $a$ and $\sigma$ are fixed constants.

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\noindent 5. Consider logistic growth with multiplicative noise,
that is, the SDE
$$
dX_t=(\lambda X_t-X_t^2) \,dt+\sigma X_t\,dB_t.
$$
Here, $\lambda>0$ and $\sigma>0$ are fixed constants. Find the unique solution 
for any deterministic $X_0=x_0>0$ explicitly, by deriving the equation 
for $Y_t=1/X_t$. What can you say about the behavior of $X_t$ as $t\to\infty$?  

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\noindent 6. In the Black-Scholes model, consider the {\it All or Nothing\/} 
option that pays one unit of cash at time $T$ provided the stock
exceeds the value $K$ some time during the interval $[0,T]$; otherwise 
it pays 0. That is, its value at time $T$ is
$$
X=1_{\{\max_{t\in[0,T]}S_t\ge K\}}.
$$
Find the arbitrage price of $X$ at time 0.

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