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\begin{document}{\bf UH Malliavin Calculus, Fall 2016, Exercises 7 (30.11, 7.12 2016) }
%Dario.Gasbarra@@helsinki.fi , room B314,
\begin{enumerate} \item  
 
 Consider the Black and Scholes stochastic differential equation
 \begin{align*}
   dS_t = S_t (  b( S_t) dt + a(S_t) dW_t )
 \end{align*}
and assume that $b$ and $a$ are $C^1$ functions. Compute the Malliavin derivative of $D_u S_t$ and
Write the stochastic differential equation for the Malliavin derivative $D_u S_t$, where $u$ is fixed and $t\in [0,T]$.



\item  Let $\xi_i $, $i\le n$ i.i.d. standard Gaussian random variables, and let $A$ be a $n\times n$ matrix with $\det(A)\ne 0$.


Consider the random variable
\begin{align*}
  F_i = \sum_{j=1}^n  A_{ij} (\xi_j^2 -1 ) =  \sum_{j=1}^n  A_{ij} h_2(\xi_j ) , \quad , i=1,\dots,n.
\end{align*}
where $h_2(x)$ is the 2nd Hermite polynomial.
\begin{enumerate}
\item 
Write the random $n\times n$ matrix $\Gamma$  with 
\begin{align*}
  \Gamma_{i,j} =  \langle DF_i, DF_j \rangle_H
\end{align*}
where you can take as Hilbert space $H=\R^n$  or 
\begin{align*}
H=\ell^2= \bigl \{ a=(a_i: i \in \N): \; \parallel a \parallel_{\ell^2}^2 =\sum_{i} a_i^2 < \infty \bigr \}
\end{align*}
\item 
Prove that $\Gamma$ is invertible.
\item 
Use the integration by part formula of Malliavin calculus to show that:

 for smooth $g: \R^n \to \R$ and denoting $\partial_i = \frac{\partial }{\partial x_i}$, by


\begin{align*}
 E\bigl(  Y \partial_i g(F)  \bigr ) =  E\bigl( g(F) \Psi_i^1(Y,F) \bigr)
\end{align*}
where 
\begin{align*}
 \Psi_i^1(Y,F)= \delta\biggl( Y \sum_{j=1}^n \Gamma_{i,j}^{-1} DF_j \biggr)
\end{align*}
see
Baudoin book page 216, see also Nualart, Proposition 2.1.4. p 100 in Nualart book (we have done this in the lectures for $Y$=1).
and by induction show
\begin{align*}
 E\bigl(  Y  \partial_{i_1,\dots,i_m}^{m} g(F)  \bigr ) =  E\bigl( g(F) \Psi_{i_1,\dots,i_m}^m(Y,F) \bigr)
\end{align*}
where
\begin{align*}
 \Psi_{i_1,\dots,i_m}^m(Y,F)=\delta\biggl( \Psi_{i_1,\dots,i_{m-1}}^{m-1}(Y,F) \sum_{j=1}^n \Gamma_{i,j}^{-1} DF_j \biggr)
\end{align*}
\item 
Apply this formula to compute an expession for
 the joint density of the vector $F=(F_1,\dots,F_n)$
with respect to the Lebesgue measure in $\R^n$.
\begin{align*}
&  p_F( x_1,\dots, x_n)= \frac{\partial ^n} {\partial_{x_1},\dots, \partial_{x_n}} P\bigl( X_1\le x_1,\dots ,X_n \le x_n \bigr)
& \\ &  = \frac{\partial^n}  {\partial_{x_1},\dots, \partial_{x_n}} E\bigl( {\bf 1}( F_1 \le x_1 )\times \dots\times   {\bf 1}( F_n \le x_n )   \bigr) 
&\end{align*}
To keep the calculation short  you can assume that $n=2$. 
\item
Use the result to plan a Monte Carlo experiment computing an approximation of the density $p_F(x)$.


\end{enumerate}
\end{enumerate}
\end{document}




















 
 
\end{enumerate}



\end{document} 




