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\begin{document}
{\bf  HY  Malliavin calcules, home exam, spring 2014  }
\\

Solve at least half of the questions for each problem.
 
\begin{enumerate}
\item 
Let $G(\omega) \sim {\mathcal N}(0,1)$ be a standard gaussian random varibale with
cumulative distribution function  $\Phi(x)= P( G \le x)$ and density
 $\phi(x)=\Phi'(x)$.
Recall that the standard gaussian density satisfies the ordinary differential equation
\begin{align*} 
\phi'(x)=-x \phi(x) \; 
\end{align*}
 

Consider the Gaussian integration by parts formula:  let 
\begin{align*}f(t)=f(0)+\int_0^t \partial f(s) ds 
\end{align*} with $E_P\bigl( \partial f(G) )$. When $f(G)\in L^1(P)$, by lemma 1.1.1. in the Nourdin Peccati book it follows that
$E_P( \partial f (G)  ) = E_P( f(G) G )$.

Show that
\begin{align*}
i) & \quad\quad E_P\bigl(   G {\bf 1} ( G \le t)   \bigr)=%\frac 1{\sqrt{2\pi} } \int_{-\infty}^t x \exp\biggl( - \frac{x^2} 2 \biggr) dx 
  - \phi(t)
& \\ ii) &\quad\quad
E_P\bigl( G^2 {\bf 1}( G\le t )  \bigr) =  \Phi(t)-t\phi(t)
&\end{align*}
Note: The derivative
 $\frac {\partial }{\partial x } {\bf 1}( x \le t )= - \delta_t(x)$, is not a function but a distribution. In the analysis language,
$\delta_t(x)$ denotes
the Dirac delta function, a  point mass at the location $t$.







\item  In the Wiener space, let $F(\omega)$ a random variable with Malliavin derivatives of all orders
 $D^n F \in L^2( \Omega, H^{\otimes n} )$ $\forall n \in \N$, $H$ denotes the Cameron Martin space of Brownian motion.
Prove (or check the proof from the literature) the Stroock formula
\begin{align*}
 F(\omega)= E_P(F) + \sum_{n=0}^{\infty} I_n( f_n), \quad     f_n(t_1,\dots,t_n) = \frac 1 {n!} E_P\bigl(  D^n_{t_1,\dots,t_n} F  \bigr)
\end{align*}
where $f_n(t_1,\dots,t_n)$ is symmetric $I_n(f_n)$ is an iterated Ito integral with respect to $W_t$. 

Hint: use the chaos expantion to  compute the $n$-fold Malliavin derivative of $F$  and take expectation.

\item
 Compute the Malliavin derivative, the 
Ito Clarck representation and the chaos expansion with respect to the Brownian motion $W_t$
 of the following variables:
\begin{eqnarray*} &&
 i ) \quad  F(\omega)=\int_0^T \biggl( \int_0^{t_2} cos(t_1+t_2) dW_{t_1} 
\biggr) dW_{t_2}   \\ &&
ii)\quad  F(\omega)= W^3(T)   \\ &&
iii)\quad   F(\omega)= \exp\biggl( \int_0^T g(s)dW_s\biggr) \quad \mbox{   where $g \in L^2([0,T],dt) $ , deterministic , }
\\ &&
iv) \quad  F(\omega)=\sin( W(T)) ,      \\ &&
v) \quad   F(\omega)=\cos( W(T)) ,  \\ &&
 vi) \quad  F(\omega)=\sinh( W(T)) ,      \\ &&
vii) \quad   F(\omega)=\cosh( W(T)) , \\ &&
viii) \quad  F(\omega)= \int_0^T W(s) ds 
      \end{eqnarray*}
{\bf Hints } For the martingale representation use the Ito Clarck Ocone formula,
it is  also possible to find the  martingale representation by computing  first the conditional
expectation and taking the  differential with Ito formula.

 When the functional has not polynomial form, the chaos  expansion will have infinitely many terms.
 use Hermite polynomials or the Stroock formula.

You can also use the generating function  
\begin{eqnarray*}
 F(x,t)= \exp(xt - \frac 1 2 t^2 ) = \sum_{n=0}^{\infty} t^n H_n(x)
\end{eqnarray*}
 together with 
\begin{eqnarray*} &&
 \sin( x )=\frac 1 {2i} ( \exp( i x ) -\exp(-ix) )  , \quad \cos( x )=\frac 1 {2} ( \exp( i x ) +\exp(-ix) )   ,\\ &&
\sinh( x )=\frac 1 2 ( \exp(  x ) -\exp(-x) )  , \quad \cosh( x )=\frac 1 2 ( \exp(  x ) +\exp(-x) )  ,
\end{eqnarray*}
 where  $i= \sqrt{(-1)  }$.

\item Let $\delta(u)=\int\limits_0^T u_s \delta W_s$  denote
the Skorokhod integral while 
$\int_0^T  u_s dW_s$ denotes the Ito integral when $u_s$ is adapted
to the Brownian filtration. 

Compute the following divergence (Skorohod) integrals:
\begin{eqnarray*} 
i)  &&  \delta(u) =\int_0^T \exp(W_T) W_s^2 \delta W_s  \\
ii) &&  \delta(u)=
\int_0^T \exp(W_T-W_t) W_t  \delta W_t \\
iii)&& \delta(u)= \int_0^T  W_{t_0}^3  \delta W_t , \quad  t_0\in [0,T].
\end{eqnarray*} 
In all these cases check also by a direct calculation that $E_P(\delta(u) )=0$ as it should be.

{\bf Hints}  For example , to compute 
\begin{align} \label{skorokhod example}
 \delta(u)= \int_0^T W_T^2 W_s^2  \delta W_s
\end{align}
one possibility is to 
 use the formula $ \delta( G u) = G \delta(u)- \langle DG, u \rangle_H $ 
where $G$ is a real valued random variable and $u$ is $H$-valued,
toghether with the fact that the Skorokhod integral $\delta(u)$ 
is just an Ito integral  when the integrand $u$ is adapted:
\begin{eqnarray*}  &&
\int_0^T W_T^2 W_s^2  \delta W_s =
 W_T^2\int_0^T  W_s^2 dW_s
 -\int_0^T D_s( W_T^2) W_s^2 ds =
 W_T^2\int_0^T  W_s^2 dW_s
 -\int_0^T 2  W_T (D_s W_T)  W_s^2 ds \\ && =
 W_T^2\int_0^T  W_s^2 dW_s
 -2 W_T\int_0^T   W_s^2 ds  
= \frac 1 3 W_T^5 - \frac 1 3 W_T^2 \int_0^t W_t dt
 - 2 W_T  \int_0^T   W_s^2 ds  
\end{eqnarray*}
where we used Ito formula. 
\item Compute   the Malliavin derivatives 
$D_t( \delta(u) )$ for $u$ as in 3.i), 3.ii), 3.iii) 
\\
We can either take directly the  Malliavin derivative, or use
the formula (1.57) in Nualart book:
\begin{eqnarray*}
D_t (\delta(u))=u_t + \int_0^T D_t u_s \delta W_s
\end{eqnarray*}

Note also that if $u$ is adapted, the Skorokhod integral is an
Ito integral and we have 
\begin{eqnarray*}
 D_t \biggl( \int_0^T u_s dW_s\biggr)= u_t + \int_0^T D_t u_s ds =
u_t + \int_t^T D_t u_s ds =
\end{eqnarray*}


{\bf Hints} For the example (\ref{skorokhod example}) at page  2, we obtain
\begin{eqnarray*}
 && 
D_t \delta(u) =  W_T^2 W_t^2 + \int_0^T D_t (W_T^2 W_s) \delta W_s 
=  W_T^2 W_t^2 + \int_0^T  ( 2 W_T W_s + W_T^2{\bf 1} ( t\le s))  \delta W_s \\ &&
=  W_T^2 W_t^2 + 
 2 W_T \int_0^T W_s dW_s + W_T^2 \int_t^T dW_s 
 - 2 \int_0^T W_s ds - 2 \int_t^T W_T ds   \\ && 
 =  W_T^2 W_t^2 + 
 2 W_T \int_0^T W_s dW_s + W_T^2 ( W_T-W_t ) 
 - 2 \int_0^T W_s ds - 2  W_T (T-t)
\end{eqnarray*}
Let us compute this result by taking directly the Malliavin derivative
of $\delta(u)$:
\begin{eqnarray*} &&
D_t( \delta(u))= D_t \biggl( W_T^2\int_0^T  W_s^2 dW_s
 -2 W_T\int_0^T   W_s^2 ds   \biggr)= \\ &&
2 W_T \int_0^T  W_s^2 dW_s
+ W_T^2  W_t^2 + 2 W_T^2\int_t^T  W_s   dW_s
- 2  \int_0^T   W_s^2 ds  -4 W_T \int_t^T    W_s  ds 
\end{eqnarray*}

\end{enumerate}
\end{document}


