ConstruccionLS.tex

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\begin{frame}[label=Extension]
	\frametitle{Extensi\'on al caso vectorial}    
	\begin{overlayarea}{\textwidth}{.1\textheight}
		\vspace*{-.25cm}
		\begin{empheq}[box={\Garybox[EDE Vectorial]}]{equation*}
			dy(t) = f(y(t)) dt + g(y(t)) dW_t, \quad Y_0=y_0.  \tag{EDE}
		\end{empheq}
	\end{overlayarea}
	%
	\tcbset{
		enhanced,
		colback=black!5!white,
		boxrule=0.4pt,
		colframe=black!65!black,
		fonttitle=\bfseries
	}
	
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		\begin{columns}
			\column{.4\textwidth}
				\only<2-3>{
					\vspace*{1.0cm}
					\begin{tcolorbox}[drop lifted shadow, title=Deriva]
						\begin{align*}
							f&: \R^d \to \R^d, \\  
							f&= 
							\left(
								f^{(1)}, \dots, f^{(d)}
							\right),			
						\end{align*}					
					\end{tcolorbox}
				}
				
			%
			\column{.4\textwidth}
				\only<3>{
					\begin{tcolorbox}[drop lifted shadow, title=Difusi\'on]
						\begin{align*}
							g&: \R^d \to \R^{d \times m},\\
							g&= 
								\left(
									g^{(i,j)}
								\right)_{
									\substack{
										i \in \{1,\dots, d\}\\
										j \in \{1,\dots, m\}
									}
								}  \\
							W &= \left(
							W^{(1)}, \dots, W^{(m)}				
							\right)					
						\end{align*}
					\end{tcolorbox}
				}
		\end{columns}
		%
		
		
		\only<4->{
			\begin{tcolorbox}[title=Hip\'otesis]
				\begin{itemize}		
				 \item[forma:]<4->
					\textcolor{black}{
						\textcolor<4>{red}{
							$f^{(j)}(x) = a_j(x) x^{(j)} + b_j (x^{(-j)})$,
						}\qquad
						\textcolor<5>{red}{
						$a_j,b_j \in \calC^{1}(\R^d)$ 
						}
					}
				 \item[(EU-1)]<6-> 
					\structure{Lipschitz Local} 
					\qquad
					\textcolor{black}{
						\only<6->{
							\textcolor<6-9>{cyan}{
								$\forall R>0$,   
							}
						}
						\only<7->{
							\textcolor<7-9>{cyan}{
								\  $\exists L_f=L_f(R)>0$
							}
						}
						%\vspace*{-0.4cm}
						\only<8->{
							\textcolor<8-9>{red}{
								$
									|f(u)-f(v)|^2
									\leq L_{f}|u-v|^2 
								$								
							}		
						}
						\only<9->{
							\qquad 
							\textcolor<9>{cyan}{
								$\forall \ u,v \in \R^d,  |u|\vee|v|\leq R$
							}						
						}
					}
					%\vspace*{.4cm}
				\item[(EU-2)]<10->
					\textcolor{black}{
						\structure{Lipschitz Global}				
						\only<10->{
							\qquad
							\textcolor<10-12>{cyan}{
								$\exists L_g>0$
							} 
						}
						\only<11->{
							\\
							\textcolor<11-12>{red}{
								$|g(u)-g(v)|^2 \leq L_{g}|u-v|^2,$
							}
						}
						\only<12->{
							\qquad
							\textcolor<12>{cyan}{
								$ \forall u,v \in \R^d.$
							 } 
						}
					}
				\item[(EU-3)]<13->
					\textcolor{black}{
						\structure{Monoton\'ia}
						\only<14->{ 
							\textcolor<14-16>{cyan}{
								$\exists$ $\alpha$, $\beta>0$
							}
						}
						\only<15>{
						\\
							\textcolor<15>{red}{
								$
									\innerprod{u}{f(u)} +\frac{1}{2}|g(u)|^2
									\leq \alpha +\beta |u|^2, 
								$
							}							
						}						
						\only<16->{
							\\								
							\textcolor<16-17>{red}{
								$
									\innerprod{u}{f(u)}+\frac{p-1}{2}|g(u)|^2 
									\leq \alpha + \beta |u|^2,
								$
							}						
						}
						\only<17->{
							\qquad
							\textcolor<17>{cyan}{
								$\forall u \in \R^d.$
							}					
						}
					}		
				\end{itemize}
			\only<18->{			
				\tcblower				
				$
					\Rightarrow
				$
				\hyperlink{ExistenciaMao}{				
					\beamerbutton{				
						$
						\exists ! \ y(t)				
						$
					}
				}			
			}		
		\end{tcolorbox}
	}
	\end{overlayarea}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[label=Construccion]
	\frametitle{Construcci\'on}    
	\begin{overlayarea}{\textwidth}{.1\textheight}
		\vspace*{-.3cm}
		\begin{empheq}[box=\shadowbox]{align*}
			dy(t) &= f(y(t)) dt + g(y(t)) dW_t, \quad   
			f^{(j)}(x) = a_j(x)  x^{(j)} + b_j (x^{(-j)})
		\end{empheq}
	\end{overlayarea}
	\begin{overlayarea}{\textwidth}{.4\textheight}
		\begin{columns}
			\column{.6\textwidth}
				\begin{tcolorbox}[
					enhanced,
					title=\only<1-2>{
						$
							f(y(t)) 
							\approx 
							\varphi_{f}(
								y(
								t_{\eta_{+}(t)}))
						$
					}
					\only<3->{
						$
							%\varphi_{f}(Y_k^{\star})=
								\left(
									\varphi_{f^{(1)}}(Y_k^{\star}),
										\ldots,
									\varphi_{f^{(d)}}(Y_k^{\star})
								\right)
						$
					},
					coltitle=green!25!black,	
					attach boxed title to top center={yshift*=-3mm},
					boxed title style={colframe=green!75!black,colback=yellow!50!green}
					%boxed title style={size=small,colback=green!40}
				]
					\only<1>{
						\vspace*{-.3cm}
						\begin{align*}
							\eta(t) &:=
								k,\  t\in [t_k, t_{k+1}), \quad k\geq 0,\\
							\eta_{+}(t) &:= 
								k+1,\  t\in [t_k, t_{k+1}), \quad k\geq 0
						\end{align*}
					}
					\vspace*{-0.5cm}
					\only<1-2>{
						\begin{align*}
							\varphi_{f}(y(t_{\eta_{+}(t)})) &=
								\frac{y(t_{\eta_+(t)})-y(t_{\eta (t)})}
								{ 
									\int_{
										\textcolor<2>{red}{
											y(t_{\eta(t)})
										}
									}^{y(t_{\eta_+(t)})}
										\frac{du}
										{
											\textcolor<2>{red}{
												a(y(t_{\eta(t)}))
											}u
											+b
										} 
								}				
						\end{align*}
					}
					\only<3-4>{
						\vspace*{.25cm}
						$
							a_{j,k} =a_j
								\left(
									Y^{(1)}_{k},
									\ldots, Y^{(d)}_{k}
								\right)
						$,
						\\
						$
							b_{j,k} =
							b_{j}
							\left(
								Y^{(-j)}_k
							\right)		
						$\\
					}
					\only<4->{
						\vspace*{.25cm}
						$
							\varphi_{f^{(j)}}\left(Y_k^{\star}\right)
							=
							\frac{Y_{k}^{\star(j)}-Y_{k}^{(j)}}
							{
								\int_{Y_{k}^{(j)}}^{Y_{k}^{\star(j)}}
								\frac{du}
								{
									a_{j,k} u
									+b_{j,k}
								}	
							}	
						$
					}
				\end{tcolorbox}
			\column{.5\textwidth}
				\vspace*{-1cm}
				\only<5->{
					\begin{align*}
						Y_k^{\star} &= Y_k + h \varphi_f(Y^{\star}_k),\\
						Y_{k+1}	&= Y_k^{\star} + g(Y_k^{\star})\Delta W_k,
					\end{align*}
				}

		\end{columns}
	\end{overlayarea}
		\begin{columns}
			\column{.5\textwidth}
				\begin{overlayarea}{\textwidth}{.5\textheight}
					\only<6->{
						\textcolor{orange}{Hip\'otesis:}
						$\forall x\in \R^d$
						\begin{enumerate}[({A}-1)]
							\item 
							$
							\exists \ L_a
							$,
							$
							a_{j}(x) \leq L_a
							$
							\item 
							$
							|b_j(x^{(-j)})|^2 \leq L_{b}(1+|x|^2)
							$
							\item Condiciones
							 \hyperlink{ZerosConditions}{\structure{ceros}} 
							 de 
							 $a_j(\cdot)$
						\end{enumerate}
					}
				\end{overlayarea}
			\column{.57\textwidth}
				\begin{overlayarea}{\textwidth}{.5\textheight}
					\only<6-11>{
						\begin{Teorema}
							\only<6>{
								Sea $u\in\mathbb{R}^d$ 
								\begin{align*}
										v &= u + h \varphi_f(v), \\
										v &= A^{(1)}(h,u)u +A^{(2)}(h,u) b(u).
								\end{align*}
								\hyperlink{MatrixFunctions}{\beamergotobutton{Def}}
							}
							\only<7->{
								$v = A^{(1)}(h,u)u +A^{(2)}(h,u) b(u)$
								\\
								$F_h(u) = v$,
									\   
								$\varphi_{f_h}(u) =\varphi_{f}(F_h(u))$,
								\\  
								$g_h(u) = g(F_h(u))$,\\
							}
							\only<8-11>{
								\textcolor<8>{orange}{
									$\Rightarrow$ son local Lipschitz\\
								}
							}
							\only<9-11>{
								\textcolor<9>{orange}{
									$\Rightarrow$
									$
										|\varphi_{f_h}(u)|\leq L_{\Phi} |f(u)|.	
									$
								}
							}
							\only<10-11>{
								\\
								\textcolor<10>{orange}{
									$\Rightarrow$
									$
										\innerprod{\varphi_{f_h}(u)}{u}
										\vee |g_h(u)|^2 \leq \alpha^* + \beta^* |u|^2 
									$
								}
							}
						\end{Teorema}
					}
					\only<12->{
						\begin{tcolorbox}[title= M\'etodo Explícito]
							\begin{align*}
								&Y_k^{\star} = A^{(1)}(h,Y_k)Y_k + A^{(2)}(h,Y_k)b(Yk),\\
								&Y_{k+1}	= Y_k^{\star} + g(Y_k^{\star})\Delta W_k,
							\end{align*}
						\end{tcolorbox}
					}
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		\end{columns}
\end{frame}