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{\bf Stochastic analysis, spring  2013, Exercises-14, 16.05.2013 }




\begin{enumerate}

\item

Let $(B_t: t\ge 0)$ a Brownian motion which generates the filtration $\F=( {\mathcal F}_t^B )$.  Compute the Ito representation
 of the random variables
\begin{enumerate}
\item
 \begin{eqnarray*}   
   \int_0^T B_t dt \end{eqnarray*}
\item 
  \begin{eqnarray*} 
 \exp\biggl( \int_0^T h(t) dB_t\biggr)
\end{eqnarray*}
 where $h(t)\in L^2([0,T],dt)$ is deterministic. 
\item $\sin(B_T)$ and $\cos(B_T)$. Hint: recall that 
\begin{eqnarray*} 
\exp( i B_t + t/2 )= \cos(B_t)\exp(t/2) + i \sin(B_t) \exp(t/2)
\end{eqnarray*} 
is a complex valued martingale.
\end{enumerate}
Recall also the formula

\begin{align*}
 f(B_T)=E_P\bigl( f(B_T)\bigr)
+
\int_0^T 
E_P\bigl ( f'( B_T ) \big\vert {\mathcal  F}^B_s\bigr) dB_s = 
E_P\bigl( f(B_T)  \bigr) + \int_0^T  E_P\biggl( f( B_T )
\frac{ B_T-B_t}{T-t} 
 \bigg\vert {\mathcal F}^B_s \biggr) dB_s 
\end{align*}
Integration by parts is also useful.

\item Prove the following version of Gronwall's lemma:
Let $a_t , b_t  : \R^+ \longrightarrow \R^+ $  non-decreasing functions with $a_0=0$.  If
\begin{align*}
 0\le  x_t \le  b_t+ \int_0^t x_t  da_t    \forall  t \ge 0
\end{align*}
then 
\begin{align*}
  x_t \le  b_t \exp( a_t)
\end{align*}
Hint: to start you can assume first that these are continuous functions.

{\bf Note } The following is also true when $a_t$ is continuous non-decreasing and $b\in \R$ is constant,
\begin{align*}
x_t \le b + \int_0^t x_s da_s  \Longrightarrow   x_t \le b + \int_0^t \exp( a_t-a_s)  b ds 
\end{align*}










\item
\begin{itemize}

\item  Write the following  Ito stochastic integral  as a  Stratonovich integral plus a process of finite variation
\begin{align}
 \int_0^t  \cos( B_s + W_s ) dB_s 
\end{align}
\item Write th   semimartingale decomposition of the following Stratonovich integrals 
\begin{align*} 
 \int_0^t  \exp( B_s + W_s) \circ dW_s
\end{align*}
where $B_t$ and $W_t$ are independent Brownian motions.
\end{itemize}


\item Solve the  linear   Ito stochastic differential equation
\begin{align*}
 X_t^x = x  + B_t + \int_0^t  \frac{ y-X_{s}^x }{T-s}ds  , \quad  t\in [0,T]
 \end{align*}
where $B_t$ is a Brownian motion.

Write and solve the linear Ito  stochastic differential equation
 for the derivative with respect to the initial value $x$:

\begin{align*}
   \dot X_t = \frac{\partial }{\partial x} X_t^x
\end{align*}

\item Let $X_t(
\omega)\in \R$ a solution of the stochastic differential equation

\begin{align*}
  X_t^x= x + \int_0^t  b( s, X_s^x) ds + \int_0^t \sigma(s,X_s^x) dB_s  
\end{align*}

If $f(x)$ is a smooth test function with bounded derivatives,
\begin{align*}
 f(X_t) - f(x)- \int_0^t ( {\mathcal L}_s f)  (X_s)ds 
\end{align*}
where 
\begin{align*}
( {\mathcal L}_s f )(x)  = b(s,x) \frac {d f}{ dx} (x)  + \frac 1 2 \sigma(s,x)^2 \frac {d^2 f}{ dx^²} (x)  
\end{align*}

\begin{itemize}

\item
Assume that $X_t$  has density with respect to Lebesgue measure $p(y;s,x)$ at every $s>0$.
Find  the partial differential equation satisfied by the density as follows:

For any smooth test function with compact support $f(x)$ write Ito formula for $f(X_t)$,
take expectation and use Fubini to interchange the order of integration with respect to
time and probability. Then write the expectations as Lebesgue integrals with  densities $p(y;s,x)$.

\item
Write also the  partial differential equation for the density of the derivative 
\begin{align*}
   \dot X_t = \frac{\partial }{\partial x} X_t^x
\end{align*}
assuming that the density exists.
\end{itemize}


\end{enumerate}



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