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\input{settings_beamer.tex}
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%% Theorem - like Environments
\newtheorem{framework}[theorem]{Framework}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{question}[theorem]{Question}
\newtheorem{answer}[theorem]{Answer}
\newtheorem{price}[theorem]{The Price}
\newtheorem{conditions}[theorem]{Conditions on $\Jumpp$}


%% Data for front page and header / footer 
\author{Marc Corluy}
\title[Rates of Convergence in CLT for Markov Chains]{Rates of Convergence in the Central Limit Theorem for Markov Chains}
\date{July 31, 2008}
\def\shortinstitute{UConn}

\begin{document}


\frame{\titlepage}

\section{Framework}
\subsection{Introduction}

\begin{frame}
\frametitle{Motivation}    
Most non-deterministic phenomena are modeled by continuous processes,
often Brownian motion related. \\
\ \\
Some phenomena do not allow modeling with continuous processes.
\begin{itemize}
  \item Financial Mathematics : Occurrence of War, Pestilence et al.
  \item Quantum Mechanics : Discontinuity is inherent in its basic premises.
\end{itemize}
\ \\ \ \\
Solution : Use Processes with Jumps in your modeling. \\
\ \\
Caveat: In order to have good modeling, you might be forced to use processes where the
magnitude of the jumps is not bounded.

\end{frame}

\subsection{The Markov Chain}

\begin{frame}

\frametitle{The Markov Chain}    

\begin{framework}
\begin{enumerate}                
  \item Let $Y_n$ be a symmetric Markov chain on $\ZZ^d$.
  \item $Y_n$ can have unbounded range. i.e. for every $K$, there exists
        $x$ and $y$ s.t. $\| x - y \| > K$ and $\IP (Y_{n+1} = y \ |\ Y_n = x) > 0$.
  \item $C_{xy}$ be the conductances between $x$ and $y$; the transition
        probabilities for the Markov Chain $Y_n$ are defined by
        \begin{equation}
          p(x,y) = \IP^x (Y_1 = y) = \frac{C_{xy}}{\sum_z C_{xz}}
        \end{equation}
        where every $C_{xy} \in [0,\infty)$.
\end{enumerate}
\end{framework}

\end{frame}


\begin{frame}

\begin{example}[Space Homogeneous Case]
\begin{equation}
     \IP (Y_{n+1} = y \ |\ Y_n = x) = \frac{c}{\| y-x \|^{d+\gamma}}
\end{equation}
with $\gamma \in (0,2)$ and $c>0$ a normalization factor.
\end{example}

\begin{example}[Grid in $\ZZ^2$]
  \begin{tabular}{ll}   
  \begin{minipage}[t]{35mm}
    \begin{figure}[t]
    \centering
    \includegraphics[width=0.9\textwidth]{grid1.pdf}
    \end{figure}
  \end{minipage}
  &   
  \begin{minipage}[t]{80mm}
  \ \\
  \vskip -5mm
  In this grid \\
  $~~~~~\sum\limits_{y \in \ZZ^2} C_{ay} = 1+2+3+2+4 = 12$ \\
  and therefore \\
  $p(a,b) = \frac{1}{12}$, $p(a,c) = \frac{2}{12}$, $p(a,d) = \frac{3}{12}$, \\
  $p(a,e) = \frac{2}{12}$, and $p(a,f) = \frac{4}{12}$.
  \end{minipage}
\end{tabular}
\end{example}
\end{frame}

\begin{frame}
\frametitle{Conditions on the Conductances}              
\begin{enumerate}[({A}1)]
\setcounter{enumi}{-1}
  \item $\forall x,y \in \ZZ^d : C_{xy} = C_{yx}$
  \item $\forall x \in \ZZ^d, \exists c_1, c_2 > 0 \hbox{ s.t. } 
                       c_1 \leq \sum\limits_{y \in \ZZ^d} C_{xy} \leq c_2$
  \item There exist $M_o \geq 1, \delta > 0$ s.t. for any $x$, $y$ with $\| x-y \| = 1$,
        there exist $N>2$ and $x_1, \ldots, x_N$ in $B(x,M_o)$ s.t. $x_1 = x$ and $x_N = y$ and
        $C_{x_i x_{i+1}} \geq \delta$ for $i=1,\ldots,N-1$.\\
        Example in $\ZZ^2$ :
        \vskip -5mm
        \begin{figure}[t]
          \centering
          \includegraphics[width=0.35\textwidth]{grid2.pdf}
        \end{figure}
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Conditions on the Conductances}              
\begin{enumerate}[({A}1)]
\setcounter{enumi}{2}
  \item There exists a decreasing function
        \begin{equation}
          \Phi : \IN \to \IR^+ \hbox{\ with\ } \sum_{i=1}^{\infty} i^{d+1} \Phi(i) < \infty
        \end{equation}
        s.t. for all $x$, $y$ : $C_{xy} \leq \Phi(\| x-y \|)$. \\
        \vskip 2mm
        Note : (A3) states that the $C_{xy}$ satisfy a uniform finite second
        moment condition: \\
        \vskip -4mm
        {\relsize{-4}
        \begin{eqnarray}               
          \sum_{y \in \ZZ^d} \| x-y \|^2 C_{xy}
          &\leq& \sum_{y \in \ZZ^d} \| x-y \|^2 \Phi(\| x-y \|) \cr
          &\leq& \sum_{i=0}^{\infty} ~ \sum_{i < \| x-y \| \leq i+1} \| x-y \|^2 \Phi(\| x-y \|) \cr
          &\leq& \sum_{i=0}^{\infty} (i+1)^2 \Phi(i) \sum_{i < \| x-y \| \leq i+1} 1 \cr
          &\leq& c_3 \sum_{i=0}^{\infty} (i+1)^2 \Phi(i) (i+1)^{d-1} \cr
          &<& \infty
        \end{eqnarray}
        }
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Conditions on the Conductances}              
\begin{enumerate}[({A}1)]
\setcounter{enumi}{3}
  \item Define $A^{(h)} : h\ZZ^d \to \IR$ as
        \begin{equation}
           \left[ A^{(h)} (x) \right]_{ij} = \frac{1}{2} \sum_{z \in \ZZ^d} z_i ~ z_j ~ \Cxhz .
        \end{equation}
        Note that $x \in h\ZZ^d$, so $\Cxhz$ is a conductance on the grid $\ZZ^d$. Let there be a Borel measurable
        function $A: \IR^d \to M_{d \times d}$ such that $A$ is symmetric and uniformly elliptic where the map
        $x \to A(x)$ is continuous, and
        $A^{(h)}$ needs to converge to $A$ such that
        for some $c>0$ and $\beta > 0$,
        \begin{equation}
           \| A - A^{(h)} \|_{\infty} \leq c h^\beta .
        \end{equation}
  \item Let $C_{xy}$ be H\"older continuous with exponent $\alpha$. More precisely, there exist $c>0$ and $\alpha>0$ such for all $x$ and $y$, 
        \begin{equation}
           | C_{xy_1} - C_{xy_2} | \leq c~ \| y_1 - y_2 \|^\alpha .
        \end{equation}
\end{enumerate}
\end{frame}

\subsection{The Rescaled Markov Chain}

\begin{frame}[shrink=1]
\begin{definition}
The rescaled Markov Chain $X_t^{(h)}$ is defined as
\begin{equation}
  X_t^{(h)} := h Y_{\frac{t}{h^2}}        
\end{equation}
Note that the state space of this process is $h \ZZ^d$.
\end{definition}

\begin{notation}
Let $\ccC ([0,t_o],\IR^d)$ denote the collection of all continuous paths from $[0,t_o]$ to $\IR^d$.
The space $\ccD ([0,t_o],\IR^d)$ is the collection of all cadlag functions, i.e. all paths that are
continuous on the right and have left limits.
\end{notation}

\end{frame}

\begin{frame}
\begin{theorem}[Bass and Kumagai, 2008]          
Suppose that $(A0)$ to $(A3)$ hold and let $A^{(h)}$ converge to $A$ on compact sets. Also, put
$[x]_h := h\left(\left[\frac{1}{h}x_1\right],\ldots,\left[\frac{1}{h}x_d\right]\right)$.
Then the following holds:
\begin{enumerate}
  \item For each $x$ and for each $t_o$ the $\IP^{[x]_h}\hbox{-law}$ of
        $\{X_t^{(h)} \}_{0 \leq t \leq t_o}$ converges weakly with respect to the topology of the
        space $\ccD([0,t_o],\IR^d)$. The limit gives full measure to $\ccC([0,t_o],\IR^d)$.
  \item If $X_t$ is the canonical process on $\ccC([0,t_o],\IR^d)$
        and $\IP^x$ is the weak limit of the $\IP^{[x]_h}\hbox{-laws}$ of $X_t^{(h)}$,
        then the process $\{ X_t, \IP^x \}$ has continuous paths and is the
        symmetric process corresponding to the Dirichlet form
        \begin{equation}
          \cE_A (f,f) = \int_{\IR^d} \left< \nabla f(x)\ |\ A(x)\nabla f(x) \right> dx
        \end{equation}
\end{enumerate}
\end{theorem}
\end{frame}

\section{Rate of Convergence}

\begin{frame}
\begin{question}
A natural question to ask is how fast this convergence occurs. In other words, estimate
\begin{equation}\label{E:principal_inequality}
  | \IE \phi(X_t^{(h)}) - \IE \phi(X_t) | \leq F (h,t) ~ \| \phi \|
\end{equation}
for some function $F$.
\end{question}  
\end{frame}

\subsection{General Outline}

\begin{frame}
\begin{answer}
In general, for any Markov process $Y_t$, there is a semigroup $\{\cP_t\}_t$ such that $\cP_t \phi(x) = \IE \phi(Y_t^x)$
for any $\phi$ in the domain of $\cP_t$. Thus, finding $F$ for (\ref{E:principal_inequality}) transforms
into finding $F$ for
\begin{equation}\label{E:principal_inequality2}
  \|  P_t \phi - P_t^h \phi \| \leq F(h,t) ~ \| \phi \|
\end{equation}
where $P_t \phi(x) = [e^{tL}\phi](x)$ and $P_t^h \phi(x) = [e^{tL^h}\phi](x)$ with
\begin{eqnarray}
  L\phi(x) &=& \nabla \cdot A(x) \nabla ~ \phi (x) \cr
  L^h \phi(x) &=& \frac{1}{h^2} ~ \sum\limits_{z \in \ZZ^d} (\phi(x+hz)-\phi(x)) ~ \Cxhz
\end{eqnarray}
\end{answer}
\end{frame}

\begin{frame}
\begin{theorem}[Duhamel's Formula]
Put $P_t = e^{tL}$ and $P_t^h = e^{tL^h}$. We can then perform the following calculation:
\begin{eqnarray}
  P_t \phi- P_t^h \phi
  &=& \int_0^t \frac{d}{ds} \left( P_t P_{t-s}^h \phi\right) ~ds \\
  &=& \int_0^t \left( \frac{dP_s}{ds} \circ P_{t-s}^h \phi + P_s \circ \frac{dP_{t-s}^h}{ds} \phi\right)~ds\\ \label{E:Duhamel}
  &=& \int_0^t \left( L P_s P_{t-s}^h \phi- P_s L^h P_{t-s}^h \phi\right) ~ds \\
  &=& \int_0^t P_s \circ ( L - L^h ) \circ P_{t-s}^h \phi ~ds
\end{eqnarray}
The end result is usually referred to as Duhamel's formula.
\end{theorem}
\end{frame}

\begin{frame}
\begin{problem}
$L \circ P_{t-s}^h \phi$ will unavoidably involve terms of the form \\
\begin{center}
\Lightning \ \ $\nabla P_t^h \phi(x)$ \ \ \Lightning
\end{center}
\end{problem}

\begin{solution}
We will have to introduce smooth versions of $L$ and $L^h$. So,                                             
\begin{equation}
  \tilde L^h f(x) ~=~ \frac{1}{h^2} ~ \sum\limits_{z \in \ZZ^d} (f(x+hz)-f(x)) ~ \tCxhz
\end{equation}
where
\begin{equation}
  \tilde C^\varepsilon_{x,y} = \int_{\IR^d} \eta_\varepsilon (w) C_{x,y-w} ~dw ,
\end{equation}
and $L\phi(x) = \nabla \cdot \tilde A(x) \nabla ~ \phi (x)$ where
$  \tilde a_{ij}(x) = \int_{\IR^d} \eta_\varepsilon (w) a_{ij}(x) ~dw $.
\end{solution}
\end{frame}

\begin{frame}
\begin{price}
We will have to find $F(h,t)$ for
\begin{equation}
  \|  P_t \phi - \tilde P_t \phi \| + \|  \tilde P_t \phi - \tilde P_t^h \phi \| + \|  \tilde P_t^h \phi - P_t^h \phi \| ~\leq~ F(h,t) ~ \| \phi \| ,
\end{equation}
and each part has to be bound separately.
\end{price}

\begin{theorem}
If (A1) to (A4) are satisfied,
  \begin{equation}
     \| P_t^h - \tilde P_t^h \|_2 ~\leq~ c(t) \varepsilon^\alpha   
  \end{equation}
\end{theorem}

\begin{theorem}[Chen, Qian, Hu, and Zheng, 1998]
If (A1) to (A4) are satisfied,
  \begin{equation}
     \| P_t - \tilde P_t \|_2 ~\leq~ c(t) \varepsilon^\alpha   
  \end{equation}
\end{theorem}

\end{frame}

\subsection{Estimates on $\| \tilde P^h_t \phi - \tilde P_t \phi \|$ }

\begin{frame}
\frametitle{Norms} 
We introduce the following norm:
\begin{equation}
  \| \phi \|_{C_k} := \max_{1 \leq j \leq k} \sup_{x \in \IR^d}
                        \left|~ \frac{\partial^j \phi}{\partial x_{i_1} \ldots x_{i_j}}(x) ~\right| .
\end{equation}
We also introduce
\begin{equation}
\|b'\|_\infty = \max_{1 \leq i,j=1 \leq d} \sup_{x \in \IR^d} \left|\pdd{b^i}{x_j}\right| ,
\end{equation}
and 
\begin{equation}
\|b''\|_\infty = \max_{1 \leq i,j,k=1 \leq d} \sup_{x \in \IR^d} \left|\pddtwo{b^i}{x_j}{x_k}\right| ,
\end{equation}
as convenient shorthands, and we set
\begin{equation}
  \| Y_t \|_{\Delta} := \max_{1 \leq j \leq d} \sup_{x \in \IR^d} \left|~ Y_t^{x,(j)} (\omega) ~\right|.
\end{equation}
\end{frame}

\begin{frame}
\begin{remark}[Flows]              
  In our estimates we will repeatedly encounter first and second order derivatives of functions of the form
  $f(x) = \IE \phi(Y_t^x)$. A symbolic use of the chain rule would lead to
  \begin{equation}
    \pdd{}{x} \IE ~ \phi(Y_t^x) ~=~ \IE ~ \phi'(Y_t^x) ~ DY_t^x.
  \end{equation}
  As it turns out, $DY_t^x$ is a process that indeed behaves like a derivative in the sense that
  \begin{equation}
    Y_t^y - Y_t^x ~=~ \int_x^y DY_t^\xi ~ d\xi
  \end{equation}
  A rigorous treatment of so defined Derivatives for Ito Diffusions is widely available in the literature.
  We will revisit the topic for Integrators with Jumps later.                   
\end{remark}
\end{frame}

\begin{frame}
\begin{theorem}
If (A1) to (A5) are satisfied, then
\begin{eqnarray*}\label{E:Pbound1}
  && \hskip -10mm \| \tilde P^h_t \phi - \tilde P_t \phi \|_\infty \cr
  &\leq& c_1 ~ \varepsilon^{-1} ~ h^\beta ~ \|\phi\|_{C_2} \sumid \int_0^t \IE~ \|~ D_i \tilde X_s ~\|_\Delta ~ds \cr
  && ~+~ c_2 ~ h^\beta ~ \|\phi\|_{C_1} \sumijd \int_0^t \IE~ \|~ D^2_{ij} \tilde X_s ~\|_\Delta ~ds \cr
  && ~+~ c_3 ~ h^\beta ~ \|\phi\|_{C_2} \sumijd \int_0^t
               \max_{1 \leq k,l \leq d} \sup_{x \in \IR^d} \IE~ \left| D_i \tilde X_s^{x,(k)} D_j \tilde X_s^{x,(l)} \right| ds
\end{eqnarray*}
\end{theorem}
\end{frame}

\begin{frame}
\begin{theorem}
If $\tX_t^x$ is a weak solution of
\begin{equation}
  \tX_t^x = x + \intt b(\tX_s^x) ~ds + \intt \sigma (\tX_s^x) dW_s
\end{equation}
then,
\begin{equation}
  \IE \| D_k \tX_t \|_\Delta ~\leq~ c_o ~ e^{~\displaystyle (c_1 \|b'\|_\infty^2 + c_2 \|\sigma'\|_\infty^2)~t},
\end{equation}
and
\begin{eqnarray}
  \IE \| D_{km}^2 \tX_t \|_\Delta
    &\leq& \sqrt{c_{ob} \| b''\|_\infty^2 + c_{o\sigma} \| \sigma'' \|_\infty^2} ~ \sqrt{t} ~ \cr
    && \quad\quad e^{~\displaystyle (c_{1b} \|b'\|_\infty^2 + c_{1\sigma} \|\sigma'\|_\infty^2)~t} .
\end{eqnarray}
\end{theorem}
\end{frame}

\begin{frame}
\begin{theorem}
If (A1) to (A5) are satisfied, then
\begin{eqnarray}
  \| \tilde P_t^h \phi - \tilde P_t \phi \|_\infty
  &\leq& \left( \frac{c_1}{\varepsilon}t ~+~ \frac{c_2}{\varepsilon^2}\sqrt{\frac{c_{0b}}{\varepsilon^2}+1}~ 
         t^\frac{3}{2} ~+~ c_3 t\right) \cr
      && \quad\quad \cdot ~ \exp\left( \frac{c_4}{\varepsilon^2} \Big(\frac{c_5}{\varepsilon^2}+1\Big) t \right)
         \cdot ~ h^\beta ~ \|\phi\|_{C_2} \phantom{XXX}
\end{eqnarray}
\end{theorem}

\begin{lemma}
  If $\cP_t$ and $\cQ_t$ are Markov semigroups, then, for any $\phi$ that is in both domains,
  \begin{equation}
    \| \cP_t \phi - \cQ_t \phi \|_2 ~\leq~ \sqrt{2 ~ \|~ \cP_t \phi - \cQ_t \phi ~\|_\infty ~ \|\phi\|_1} .
  \end{equation}
  \phantom{q}
\end{lemma}

\end{frame}


\subsection{Bringing it All Together}

\begin{frame}
\begin{theorem}
  If (A1) to (A5) are satisfied, then
  \begin{equation}
    \| P_t \phi - P_t^h \phi \|_2 \leq c_1 ~ h^{\frac{\alpha\beta}{2\alpha+2}} \e^{\Big(c_2 h^{4-\frac{2\beta}{\alpha+2}} \Big)}
                                           ~\cdot~ \max\left\{\|\phi\|_2, \frac{\|\phi\|_{C_2} + \|\phi\|_1 }{2} \right\}
  \end{equation}
\end{theorem}

\begin{remark}
Note that for $h<1$, which is the only important part for convergence,
\begin{equation}
    \exp\Big(c_2 h^{4-\frac{2\beta}{\alpha+2}} \Big) \leq c'_2 ,
\end{equation}
as $0< \alpha, \beta <1$. So for $h<1$, we have
\begin{equation}
    \| P_t \phi - P_t^h \phi \|_2 \leq c ~ h^{\frac{\alpha\beta}{2\alpha+2}}
                                           ~\cdot~ \max\left\{ \|\phi\|_2, \frac{\|\phi\|_{C_2} + \|\phi\|_1 }{2} \right\} .
\end{equation}
\end{remark}

\end{frame}

\section{Derivatives of Flows}
\subsection{Jump Processes as Integrator}

\begin{frame}
\frametitle{Jump Processes as Integrator}
  Set
\begin{equation}\label{E:sdejumpp}
  X_t^x = x + \inttS \Jumpp (X_{\smin}^x,z) (\mu - \nu)(ds,dz)
\end{equation}
where $S$ is the state space, $\Jumpp$ a measurable function from $\IR \times S$ to $\IR$,
$\mu$ is a Poisson point process, and $\nu(A) = t \lambda(A)$ where $\lambda$ is the Lebesgue measure
on $\IR^d$. We can formally write
\begin{equation}
  D X_t^x =
   1 + \inttS \Jumpp' (X_{\smin}^x,z) D X_{\smin}^x (\mu - \nu)(ds,dz)
\end{equation}
where
\begin{equation}\label{E:sdeflow_thm3}
  X_t^x - X_t^y = \int_y^x D X_t^u du .
\end{equation}
This calculation is made rigorous within the framework of flows.
\end{frame}

\begin{frame}
\begin{definition}[Stochastic Flow]
The flow associated with a stochastic differential equation 
is the family of stochastic variables defined by
\vskip -7mm
\begin{equation}
  \Phi_X ~=~ \left\{~ X_t^x ~\big|~ x \in \IR^d, t \in [0,+\infty) , X_t^x \hbox{ solves the SDE} ~\right\} .
\end{equation}
\end{definition}
\begin{conditions}
\begin{enumerate}[(G1)]
  \item $\displaystyle\int_S | \Jumpp (x,z) - \Jumpp (y,z) |^{2p} \lambda(dz) \leq C |x-y|^{2p}$
  \item $\displaystyle\int_S | \Jumpp' (x,z) |^{2p} \lambda(dz) \leq C $
  \item $\displaystyle\int_S | \Jumpp' (x,z) - \Jumpp' (y,z) |^{2p} \lambda(dz) \leq C |x-y|^{2p}$
  \item $\displaystyle\int_S | \Jumpp'' (x,z) |^{2p} \lambda(dz) \leq C $
\end{enumerate}
\end{conditions}
\end{frame}

\subsection{First Order Derivatives}

\begin{frame}
\begin{theorem}
If $\Jumpp$ satisfies (G1), then there exists a version of $X_t^x$
that is jointly cadlag in $t$ and continuous in $x$.
\end{theorem}
\begin{theorem}
If $\Jumpp$ satisfies (G1) and (G2), then the SDE
\begin{equation}\label{E:diffsde}
  Y_t^x = 1 + \inttS \Jumpp' (X_{\smin}^x,z) Y_{\smin}^x (\mu -\nu) (ds,dz)
\end{equation}
has a unique solution $DX^x_t$ which has moments of all orders and there exists
a version of $DX_t^x$ that is jointly cadlag in $t$ and continuous in $x$.
\end{theorem}
\end{frame}
\begin{frame}
\begin{theorem}
If $\Jumpp$ satisfies (G1) to (G4), then for all $x$ and $y$,
\begin{equation}
  X_t^x - X_t^y = \int\limits_y^x DX^\xi_t d\xi .
\end{equation}
\end{theorem}
\end{frame}

\subsection{Higher Order Derivatives}

\begin{frame}
\frametitle{Fa\`a di Bruno's Formula}
For all $n>0$,
\begin{equation}
  \frac{d^n}{dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x)) B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right) ,
\end{equation}
where the Bell polynomial is defined as follows:
\begin{eqnarray}
  && \hskip -10mm B_{n,k}(x_1,x_2,\dots,x_{n-k+1}) \cr
  &&= \sum \frac{n!}{j_1!j_2! \cdots j_{n-k+1}!}
      \left( \frac{x_1}{1!} \right)^{j_1}
      \left( \frac{x_2}{2!} \right)^{j_2}
      \cdots \left( \frac{x_{n-k+1}}{(n-k+1)!} \right)^{j_{n-k+1}} .
\end{eqnarray}
The sum extends over all sequences $j_1, \ldots ,j_n$ of non-negative integers such that
$\sum_{i=1}^n j_i = k$ and $\sum_{i=1}^n i j_i = n$. \\
\end{frame}

\begin{frame}
\begin{definition}
We define the higher order derivatives ($n \geq 2$) of $X_t^x$, as
\begin{eqnarray*}
  && \hskip -10mm D^n X_t^x  \cr
  && \hskip -10mm = \inttS \sum_{k=1}^n \Jumpp^{(k)}(X_{\smin}^x,z) B_{n,k}(D^1 X_{\smin}^x,\ldots,D^{n-k+1} X_{\smin}^x) (\mu - \nu)(ds,dz)
\end{eqnarray*}
\end{definition}
\begin{conditions}
\begin{enumerate}[\ \ \ $(G1^{(n)})$]
  \item $\displaystyle\int_S | \Jumpp^{(n)} (x,z) - \Jumpp^{(n)} (y,z) |^{2p} \lambda(dz) \leq C |x-y|^{2p}$
  \item $\displaystyle\int_S | \Jumpp^{(n)} (x,z) |^{2p} \lambda(dz) \leq C $
\end{enumerate}
\end{conditions}


\end{frame}

\begin{frame}
\begin{theorem}\label{T:flow4}           
If $\Jumpp$ satisfies $(G1^{(1)})$ to $(G1^{(n)})$, and $(G2^{(1)})$ to $(G2^{(n)})$, then the SDE
\begin{eqnarray*}
  Y_t^x &=& \inttS \Jumpp' (X_{\smin}^x,z) Y_{\smin}^x ~ (\mu - \nu)(ds,dz) \cr
        && \hskip -15mm + \inttS \sum_{k=2}^{n} \Jumpp^{(k)}(X_{\smin}^x,z)~ B_{n,k}(D^1 X_{\smin}^x,\ldots,D^{n-k+1} X_{\smin}^x)
                    ~ (\mu - \nu)(ds,dz) \phantom{XXXX}
\end{eqnarray*}
has a unique solution $D^n X_t^x$ which has moments of all orders and there exists a version of $D^n X_t^x$
that is jointly cadlag in $t$ and continuous in $x$.
\end{theorem}
\end{frame}

\begin{frame}
\begin{theorem}\label{T:flow5}               
If $\Jumpp$ satisfies $(G1^{(1)})$ to $(G1^{(n+1)})$ and $(G2^{(1)})$ to $(G2^{(n+1)})$, then for all $x$ and $y$,
\begin{equation}
  D^{n-1} X_t^x - D^{n-1} X_t^y = \int\limits_y^x D^n X^\xi_t d\xi .
\end{equation}
\end{theorem}
\end{frame}

\subsection{Taylor Expansion}

\begin{frame}
\begin{theorem}[Taylor Expansion]\label{T:taylor_theorem}               
For any $n>0$, we have the Taylor expansion
\vskip -2mm
\begin{equation}\label{E:taylor_expansion}
  X_t^x = \sum_{k=0}^{n-1} \frac{(x-x_o)^k}{k!} D^k X_t^{x_o} ~+~ R_n (t;x,x_o) ,
\end{equation}
\vskip -2mm
where
\vskip -2mm
\begin{equation}\label{E:taylor_rest_term}
  R_n (t;x,x_o) ~=~ \int\limits_{x_o}^x\!\!\!\int\limits_{x_o}^{x_1}\!\!\cdots\!\!\int\limits_{x_o}^{x_{n-1}}
                     ~D^n X_t^{x_n}~ dx_n \ldots dx_1 dx_o .
\end{equation}
\vskip -2mm
Furthermore, for almost every path $\omega$, there exists an $\eta(\omega) \in [x_o \wedge x, x_o \vee x]$ such that
\vskip -2mm
\begin{equation}\label{E:taylor_rest_term_MVT}
  R_n (t;x,x_o) ~=~ D^n X_t^\eta ~ \frac{(x-x_o)^n}{n!} .
\end{equation}
\end{theorem}
\end{frame}

\begin{frame}
\begin{remark}[Generalization]
Let $X_t$ be defined as
\begin{equation}\label{E:b_sigma_Jumpp}
  X_t^x ~=~ x + \intt b(X^x_\smin) ds + \intt \sigma(X^x_\smin) dW_s + \inttS \Jumpp(x^x_\smin,z)(\mu-n)(ds,dz) ,
\end{equation}
and let $b$, $\sigma$ and $\Gamma$ satisfy
\begin{enumerate}[\ \ \ $(\Phi1^{(k)})$]
  \item $\displaystyle\int_S | \Phi^{(k)} (x,z) - \Phi^{(k)} (y,z) |^{2p} \lambda(dz) \leq C |x-y|^{2p}$
  \item $\displaystyle\int_S | \Phi^{(k)} (x,z) |^{2p} \lambda(dz) \leq C $
\end{enumerate}
for every $0 \leq k \leq N+1$ where $N$ is the highest order derivative wanted, then for every $0 < l \leq N$, the
previous theorems are also valid.
\end{remark}

\end{frame}

\section{Bibliography}

\begin{frame}[shrink=1]
\frametitle{References}

\bibliographystyle{amsplain}

\begin{thebibliography}{[88]}

  \bibitem{bass2}
     Richard F. Bass,
     \emph{Diffusions and Elliptic Operators},
     Probability and its Applications,
     ISBN 0-387-98315-5,
     Springer Verlag, 1998

  \bibitem{bass_kumagai}
     Richard F. Bass and Takashi Kumagai,
     \emph{Symmetric Markov Chains on $\ZZ^d$ with Unbounded Range},
     Transactions of the American Mathematical Society,
     Vol. 30, Nr 4, 2041-2075 (2008),
     AMS, 2008

  \bibitem{chen_qian_hu_zheng}
     Zhen-Qing Chen, Zhongmin Qian, Yaozhong Hu and Weian Zheng,
     \emph{Stability and Approximations of Symmetric Diffusion Semigroups and Kernels},
     Journal of Functional Analysis, 152, 225-280 (1998),
     Academic Press, 1998

   \bibitem{lepeltier_marchal}
     Jean-Pierre Lepeltier and Bernard Marchal,
     \emph{Probl\`eme des martingales et \'equations diff\'erentielles stochastiques associ\'ees
     \`a un opérateur in\'egro-diff\'erentiel},
     Annales de l' Institut Henri Poincar\'e (B) Vol. 12, 43-103, (1976)

   \bibitem{skorokhod}
     A.V. Skorokhod,
     \emph{Studies in the Theory of Random Processes},
     ISBN 9-780-48664240-6,
     Reading, Mass., Addison-Wesley Pub. Co., 1965

\end{thebibliography}

\bibliography{ref}

\end{frame}



\end{document}