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\begin{document}{\bf UH Malliavin Calculus, Fall 2016, Exercises 4 (5,12 October 2016) }
\begin{enumerate}



\item 


Show that the Malliavin derivative defines for simple random variable
\begin{align*}
 F= f( X(h_1),\dots, X(h_n) ) \in {\mathcal S}
\end{align*}
as
\begin{align*}
 DF= \sum_{i=1}^n \frac{\partial }{\partial x_i} f( X(h_1),\dots, X(h_n) ) h_i    \in L^2( \Omega,H)
\end{align*}
does not depend on the representation of $F$, 

Note that the representation of $F\in {\mathcal S}$ is not unique, it could be written
as 
\begin{align*}
 F= \varphi( X(\eta_1),\dots, X(\eta_m) )\end{align*}
for some other $m$ $\eta_i$ and $\varphi$.

\item  Let $\{ \xi_n : n \in \N \}$ be a sequence of Gaussian random variables,  $\xi_n \sim{\mathcal N}( \mu_n,\sigma_n)$
such that $\xi_n \stackrel{law}{\to} \xi$ with respect to the convergence in distribution,
meaning that the characteristic function converges pointwise
\begin{align*}
 \varphi_{\xi_n}(t):= E\bigl( \exp( i t \xi_n) \bigr)= \exp\bigl( i t \mu_n - t^2 \sigma_n^2 /2 \bigr ) \longrightarrow 
 \varphi_{\xi}(t) = E\bigl( \exp( i t \xi)         \quad \forall t
\end{align*}
where $i=\sqrt{(-1)}$ is the imaginary unit. Show that  $\xi$ is either Gaussian  or deterministic.

Hint: show that $\mu_n$ and $\sigma_n$ have limits.



\item 

For $g\in H$ we consider the shifted process  $(X+g)(h)=X(h)+\langle g , h \rangle_H$ , $\forall h \in H$,
which are jointly Gaussian, with mean $\langle g , h \rangle_H$ and covariance $E( (X+g)(h) (X+g)(h^') )= E( X(h) X(h^') )=\langle h,h' \rangle_H$.

We consider the shift operator $T_g : {\mathcal S} \longrightarrow {\mathcal S}$
such that 
\begin{align*}
 F = f\bigl( X(h_1), \dots ,X(h_n) \bigr) \mapsto  T_g F= f\bigl( X(h_1) + \langle g, h_1 \rangle_H, \dots  ,X(h_n) + \langle g, h_n \rangle_H \bigr)
\end{align*}

Show there is a probability measure $P_g\sim P$  (the shifted Gaussian measure) such  that if $F\in {\mathcal S}$
\begin{align*}
E_P( T_g F )=E_{P_g} (  F ) = E_P( F Z ), \quad Z=\frac{ dP_g}{dP } = \exp\bigl( X(g)- \langle g, g\rangle_H/2 \bigr) 
\end{align*}
Hint. Show that $Z>0$ $P$ a.s. and it is a Radon Nikodym derivative with $E_P(Z)=1$.


\item  Let $\{  e_i : i \in \N\}$ a complete orthonormal system of the separable Hilbert space, 
$\{ \xi_i(\omega) : i \in \N\}$ a sequence of i.i.d. standard Gaussian variables defines on $(\Omega,{\mathcal  F},P)$,
and let  
\begin{align*}
  X(h) =  \sum_{i=1}^{\infty} \langle h, e_i\rangle_H \xi_i(\omega) 
\end{align*}
be the isonormal Gaussian process.
Show that the shifted isonormal process has the representation
\begin{align*}
 (X+g)(h)\sum_{i=1}^{\infty}  \biggl( \langle h, e_i\rangle_H \xi_i(\omega)  +   \langle h, g\rangle_H  \biggr)
\end{align*}
 as a limit in $L^²(\Omega, \R)$ 





\item 

Show that $T_g$  is closed in $L^2(\Omega,{\mathcal F},P)$, and it has an unique extension $T_g : \mbox{Dom}(T_g) \to L^2(\Omega)$.

We consider now a fix direction $g$,
show for $F\in {\mathcal S}$ the {\it directional derivative } exists
\begin{align*}
D_g F:= \lim_{\varepsilon \to 0} \frac 1 {\varepsilon } ( T_{\varepsilon g} F  - F ) = \langle DF , g\rangle
\end{align*}
where we take the limit in $L^2(\Omega,\R)$. Show that
$D_g$ is a closed operator which is extended uniquely to an operator defined on
a closed subspace $\mbox{Dom}(D_g) \supseteq D^{1,2}$.


\item Compute the divergence  integrals with respect to the Brownian motion $W$, 
where  $H= L^2( [0,T], dt )$, and the isonormal Gaussian process is
\begin{align*}
 W( h) = \int_0^t h(s) W(ds)
\end{align*}
is the Wiener integral


\begin{enumerate}\begin{multicols}{3}
\item    $\int W(t) \delta W(t)$  \item   $\int W(t)^2 \delta W(t)$ \columnbreak 
\item  $\int W(t_0)^2 \delta W(t)$  \mbox{ with fixed $t_0$ }  \columnbreak 
\item  $\int_0^T \exp( W(T) ) \delta W(t)$
\end{multicols}
\end{enumerate} 


























\end{enumerate}
\end{document}