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\begin{document}{\bf UH Malliavin Calculus, Fall 2016, Exercises 3 (28.9 2016) }

 Let $N(\omega) \sim {\mathcal N}(0,1)$ is a standard Gaussian random variable, with probability density
\begin{align*}
  \phi(x) =  \frac 1 {\sqrt{2\pi} }  \exp\biggl( - \frac {x^2} 2 \biggr), \quad x \in \R
\end{align*}
and $\gamma(dx)= \phi(x)dx$ is the standard Gaussian distribution on $\R$.
\begin{enumerate}
\item  We denote by $H_n(x)= \bigl(\partial^{*n} 1 \bigr)(x)$ the $n$-th unnormalized Hermite polynomial.
Show that
\begin{align*}
  H_p(x) H_q(x)  = \sum_{r=0}^{\min\{p ,q \} } r! \binom{p}{r}\binom{q}{r}  H_{p+q-2r} (x)
\end{align*}
Hint: we have seen that for $f\in L^2(\gamma)$   $\bigl \{  h_r(x)=H_r(x)/\sqrt{r!} : \; r\in \N \}$
is a complete orthonormal system in $L^2(\gamma)$ and  for $f\in L^2(\gamma)$ 
\begin{align*}
f(x) = \sum_{r=0}^{\infty} \frac{ E( f(x) H_r(x) )  }{ r! } H_r(x)
\end{align*}
where the series is convergent in $L^2(\gamma)$.

Use the expansion with $f(x)=H_p(x)H_q(x)$.

\item Show {\it Rodrigues formula }
\begin{align*}
 H_p(x) = e^{x^2/2} (-1)^p   \frac {d^p}{ dx^p} e^{-x^2/2}
\end{align*}


\item Show that
\begin{align*}
 g(x,t):=\exp\bigl(  t x - t^2/2  \bigr) = \sum_{r=0}^{\infty} \frac{ t^r} {r! } H_r(x)
\end{align*}
$g(x,t)$ is the  {\it generating function } of the Hermite polynomials.
Hint: $g(x,t)=\exp( x^2) \exp( -(t-x)^2/2 )$


\item Show that
\begin{align*}
 \exp\bigl(  t x - {t^2\lambda^2}/2  \bigr) = 
 \sum_{r=0}^{\infty} \frac{ t^2 \lambda^2 } {r! } H_r\bigl(x/\lambda \bigr)
\end{align*}







\item Let $f\in {\mathcal S}$, $x\in \R$, and let $\bigl(P_t:t \in [0,+\infty]\bigr)$ be
the Ornstein Uhlenbeck semigroup. Show that
\begin{align*}
 \int_0^{\infty} e^{-2t} P_t f^{''}(x) dt - x \int_0^{\infty} e^{-t} P_t f^'(x) dt =f(x) -E\bigl( f(N) \bigr)
\end{align*}
Hint:   $\frac d {dt } P_t = L P_t$

\item 
 On the Wiener space, compute the Malliavin derivative $DF\in H=L^2( [0,T],dt)$ 
% \begin{multicols}{2}
\begin{enumerate}
\item $F= W_T^n$   $\quad n \in \N$,
\item $F= \exp( W_T)$ 
%\columnbreak
\item $F= \int_0^T W_s ds$
\item $F= f\bigl( \int_0^T g(s) dW_s)$, where $g\in L^2( [0,T],dt)$ is deterministic with $\int_0^T g(s)^2ds =1$
 and $f \in D^{1,2}(\R \gamma)$ with S obolev derivative $\partial f \in L^2(\gamma)$.
 \item $F= \exp( W_T)$ 
\end{enumerate}
%\end{multicols}
 \item On the Wiener space, let $g \in L^2([0,T],dt)$ deterministic
 write the chaos expansion of 
 \begin{align*}
    \exp( \int_0^T g_s dW_s)
 \end{align*}
 \end{enumerate}
\end{document}


\item On the Wiener space, let $X_t(\omega) $ be a  jointly measurable process process such that for each $t$ $X_t$
has Malliavin derivative $D X_t$  (which is understood as $D_s X_t$, for each $t$  $D_{\cdot } X_t \in L^2([0,T],dt)$.

Show that 
the  Malliavin derivative is linear, it commutes with the Lebesgue integral:
integral $\int_0^t X_t dt $ has Malliavin derivative
\begin{align*}
D \int_0^T  X_t dt = \int_0^T  D X_t dt
\end{align*} 
Hint: approximate
 

