

\documentclass[a4paper,12pt,finnish]{article}
\usepackage{hyperref}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{unicode}
\usepackage{multicol}
%\usepackage[geometry]{ifsym}
\usepackage[finnish]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{verbatim}
\usepackage{multicol}
%\usepackage{html,makeidx}
\usepackage{eurosym}
\def\eu{\mbox{\euro}}
\def\R{\mathbb{R}}
\def\Q{\mathbb{Q}}
\def\N{\mathbb{N}}
\def\E{\mathbb{E}}
\def\Z{\mathbb{Z}}
\def\P{\mathbb{P}}
\def\T{\mathbb{T}}
\def\F{\mathbb{F}}
\def\FF{ {\mathcal F} }
\def\D{\mathbb{D}}
\def\indep{\perp\!\!\perp}
%\def\mathcal{\cal}
\def\ind{\mathrel{\hbox{\rlap{\hbox to 7.5pt{\hrulefill}}\raise6.6pt\hbox{\eka\char'167}}}}
\newtheorem{thm}{Teoreema}[section]
\newtheorem{rmk}{Huomautus}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{propo}{Lause}[section]
\newtheorem{coro}{Seuraus}[section]
\newtheorem{assu}{Oletus}[section]
\newtheorem{har}{Tehtävä}[section]
\newtheorem{ex}{Esimerkki}[section]
\newtheorem{define}{Määritelmä}[section]
\numberwithin{equation}{section}

\def\Tau{{\mathcal T}}


\begin{document}{\bf UH Malliavin Calculus, Fall 2016, Exercises 6 (16 November 2016) }
%Dario.Gasbarra@@helsinki.fi , room B314,
\begin{enumerate} \item   \begin{enumerate} \item 
Let $f_1 \ne \dots \ne f_n \in H$. 
Find $m=m(n)$, coefficents $c_i \in \R$ and  $g_1,\dots, g_m\in H$ such that
\begin{align*}
  f_1 \widetilde \otimes f_2  \widetilde \otimes \dots  \widetilde \otimes f_n = \sum_{i=1}^m   c_i  g_i^{\otimes n}    \in H^{\widetilde\otimes n}
\end{align*}


\item 
Write the iterated integral 
$I_n( f_1 \otimes f_2 \otimes \dots \otimes  f_n)$
as linear combination of $h_n\bigl( X(\varphi_i ) \bigr)$ where $\varphi_i\in H$ and $\parallel \varphi_i \parallel_H=1$, and $h_n= \partial^{*n} 1$
is the unnormalized $n$-th Hermite polynomial.
\end{enumerate}


Hint: when $n=2$, $f_1,f_2\in H$ we have the polarization identity
\begin{align*}
 4 \;  f_1\widetilde \otimes f_2=  (f_1+f_2)\otimes (f_1+f_2) - (f_1-f_2)\otimes (f_1-f_2)   
\end{align*}
so that
\begin{align*} &
 4\; I_2 ( f_1\otimes f_2) = & \\ &    \parallel f_1 + f_2 \parallel_H^n h_n\bigl (  X(f_1+f_2)/  \parallel f_1 + f_2 \parallel_H \bigr)
  - \parallel f_1 - f_2 \parallel_H^n h_n\bigl (  X(f_1-f_2)/  \parallel f_1 - f_2 \parallel_H \bigr)
&\end{align*}
You can try first  with $n=3$  before looking at the general case.









\item 
We recall  Stroock formula: if $F\in D^{n,2}$ for all $n$,
i.e. $F$  has infinitely
many Malliavin differentiable infinitely many times in $L^2$, it is possible to write the chaos expansion as 
\begin{eqnarray*} 
 F(\omega)=E(F)+ \sum_{n=1}^{\infty} I_n(f_n) 
 =E(F)+ \sum_{n=1}^{\infty} \frac{ 1 }{n!} I_n ( E( D^n F) ) 
\end{eqnarray*}  
Note that $ D^n_{t_1,\dots,t_n} F(\omega)$ is a 
symmetric random function of  $n$  arguments.
This can be useful to compute  chaos expansions.
\\


{\bf Notation :  } We denote by $\delta(u)=\int u_s \delta W_s$ 
the Skorokhod integral while 
$\int u_s dW_s$ is 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 W_T^2 W_s^2  \delta W_s
\\  
ii)  &&  \delta(u) =\int_0^T \exp(W_T) W_s^2 \delta W_s  \\
iii) &&  \delta(u)=
\int_0^T \exp(W_T-W_t) W_t  \delta W_t \\
%iv)&& \delta(u)= \int_0^T  W_{t_0}^3  \delta W_t , \quad  t_0\in [0,T].
\end{eqnarray*} 

Hint:  use Stroock formula to compute the chaos expansion of the integrand, and then use the linearity
of $\delta$.
      Also recall that
      \begin{align*}
         \delta(  F u ) = F \delta u - \langle u, DF \rangle_H
      \end{align*}
when $F \in \D^{1,2}$ and $u\in L^2( \Omega; H)$.

\item   
Compute   the Malliavin derivatives 
$D_t( \delta(u) )$ for the divergence integrands  in 2.i), 2.ii), 2.iii) 
\\
We can either take directly the  Malliavin derivative, or use
the formula
\begin{align*}
 D \delta(u) =  u + \delta(D_u)
\end{align*}
which means
\begin{eqnarray*}
D_t (\delta(u))=u_t + \int_0^T D_t u_s \delta W_s
\end{eqnarray*}


ii) $F(\omega)=\sin(W_T)$ , 
iv) $F( \omega)= (W_T+T) \exp(-W(T)- \frac 1 2 T) $

Note  that $Z_t:=\exp(-W(t)- \frac 1 2 t) = \frac{ dQ_t }{dP_t} $ 
is the Radon-Nykodim derivative which appears in Girsanov
theorem. 
Under the measure $Q$  the process  $\tilde W_t:=W_t + t$ 
is a standard Brownian motion, while under
 the measure $P$ it is a Brownian motion with drift coefficient 1.
\item 
\begin{enumerate}\item 
Let  $A\in {\mathcal F}^X= \sigma( X(h): h\in H )$,
and $F(\omega)={\bf 1}_A(\omega)$.

Show that $F$ is Malliavin differentiable if and only if $P(A)=0$
or $P(A)=1$.

Hint:  Assume that the Malliavin derivative  $DF$ exists, take Malliavin derivative of
both sides in the identity  $F=F^2$
and find a contradiction. 


\item 

Find the Ito-Clarck representation of the digital 
option 

$$F=1(W_T > 0),$$ 

where $W_t$ is Brownian motion.

 Hint: Although we don't have  enough smoothness of the functional,
use formally the Clarck Ocone formula,
where we compute the Malliavin derivative by chain rule
obtaining a Dirac delta function,
 write the
conditional expectation explicitely and use
the gaussian integration by parts formula, and show
that the final result still makes sense (this is 
the hardest part).
\end{enumerate} 


{\bf Exercise 6 } The local time process $L^0_t$
of the Brownian motion at $0$ is defined
as
\begin{eqnarray*} 
 L_t^0 =  |W_t| - \int\limits_0^t \mbox{sign} (W_s) dW_s 
\end{eqnarray*}
where $\mbox{sign}(x)=2\; {\bf 1}( x \ge  0)-1 $.
The local time  $L^0_t$ is interpreted as the time the Brownian motion $W_t$
spends at the value $0$ within the time interval $[0,t]$.

\begin{enumerate}
\item 
Is $L_t^0$  Malliavin differentiable ?
\item  
Find the  Ito Clarck representation of  $L^0_T$.
Hint: as in Exercise 5, compute formally the Malliavin derivative,
take conditional expection and use Ito Clarck Ocone formula.
Explain why the obtained formula is correct.
\end{enumerate}
\end{enumerate}


\end{document} 




