%
%  slide-sanjuan.tex 2007.6.11===>slide-iceland.tex 2006.6.28 in style
%              ===> Golden-poli-DP.tex  2006.9.6 in content (impotant!!)
%  slide-sanjuan  2007.6.11 is based upon Golden-poli-DP.tex  2006.9.6
%   However, many changes of Golden-poli-DP.tex should be added
%   on slide-sanjuan.tex!! (2007.6.11)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%          Golden-poli-DP.tex  2006.9.6===> 
%    Golden optimal policy in calculus of variation and dynamic programming 
%          (Springer's tex)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  slide-18-G-op-pol.tex 2006.3.18 ===> slide-G.ineq.tex, bunh18-G-op-pol.tex
%                                   and qu-sq.tex   2004.1.26 ==>
%%%%%%%
%   slide-18-G-op-pol.tex �Ŕ��\
%   slide-18-G-op-pol.tex 2006.3.18 ===> slide-G.ineq.tex, bunh18-G-op-pol.tex
%   bunh18-G-op-pol.tex 2005.10.22 ===>  bunh17-Gol-ineq.tex         
%   bunh18-G-op-pol.tex  ===> see also gold.opt-policy.tex      
%  1. The Golden optimal policy in dynamic programming
%  2. Optimal policy is golden in dynamic programming 
%  �u���ҁF ��{����i���j�E���c����                   
%  ���w��A �Q�O�O�U�N�R���Q�U���i���j�` �Q�X���i���j������
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%----------------------------------------------------------------------------
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%  2006�N3��26���i��)--29��(���j
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%  �Hpp. ???57--58.
%-----------------------------------------------------------------------------
%%%%%%    see ???.tex 200???
%\item %\bibitem{ma44} 
%��{ ����$^{*}$�E���c �����A 
%Optimal policy is Golden in dynamic programming�A
%�Q�O�O�U�N�R���Q�U���i���j�` �Q�X���i���j������E���H�D
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%      slide-G.ineq.tex  2005.9.4  (===> slide-IFORS05.tex  2005.6.4)
%                                         
%%%%%%    ���{���w�� �Q�O�O�T�N�H�G���ȉ�  ���v���w���ȉ�  �u���A�u�X�g���N�g
%%%%%%    2005�N9��19���i��)--22��(�؁j  pp.???55--56.
%%%%%%    bunh17-Gol-ineq.tex        The Golden inequalities
%%%%%%    �u���ҁF ��{����               
%%%%%%    ���w��  �Q�O�O�T�N�X���P�X���i���j�` �Q�Q���i�؁j���R��
%%%%%%    ���\    �Q�O�O�T�N�X���Q�P���i���F�R���ځj�̒��P�ԁFAM9:20--9:35��
%%%%%%        �ߑO����1�Ԗ�15���ԁi�S�̂Ōv��֌W��4�u���̑�1���ځj
%%%%%%    bunh17-Gol-ineq.tex  <===  bunh17-Gol-ineq-bypro.tex 2005.3.30 ===>
%%%%%%    see bunh17-Gol-ineq.tex
%%%%%%    \item %\bibitem{ma43} 
%%%%%%    ��{ ����A 
%%%%%%    The Golden inequalities�A
%%%%%%    �Q�O�O�T�N�X���P�X���i���j�` �Q�Q���i�؁j���R��D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      slide-IFORS05.tex  2005.6.4  (===> slide-NACA05.tex  2005.6.4)
%                                         and slide-IWIF.tex  2005.6.4
%
%  �i�T�j  slide-IFORS05.tex �� slide-NACA05.tex �� slide-IWIF.tex �������
%   see slide-IWIF.tex, dypricingIWIF1.tex
%   dypricingIWIF1.tex is a 95$%$ transliteration of dyval.tex 
%   except for random expiration date 
%   Hilton Hawaiian Village Beach Resort & Spa
%   dypaci.tex, dypricingIWIF1.tex ���瑁�}��
%   Pacific Option��p��tex.file���쐬����I�I
%  \item %\bibitem{tk48} 
%  S.~Iwamoto, 
%  Dynamic Pricing of Pacific Option, 
%The Seventeenth Triennial Conference of the International Federation of Operati%onal Research Societies (IFORS2005), hosted by INFORMS, 11 - 15 July, 2005 (Als%o Chair of Session \lq\lq Nonstochastic and Nonadditive Dynamic Programming").
%In session Nonstochastic and Nonadditive Dynamic Programming, cluster Dynamic P%rogramming, by invitation of Seiichi Iwamoto.
%
%  �i�U�j  2004.12.12  IWIF�u�� �����{����
%      slide-IWIF.tex  2004.12.3===> 
%   1. slide-Nobel.tex  2004.11.29===>
%      (see  slide-tu-iw.tex  2004.11.15===>slide-turu.tex)
%   2. dypricingIWIF1.tex
%      A dynamic pricing of exotic options : dypricingIWIW1.tex          
%    2004.10.4=> see dyval.tex 2001.2.23 and see also pachinko.tex (Latex 2e)
%    dyval.tex  Dynamic valuations of options 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      slide-NACA05.tex  2005.6.4  (===> slide-GoldenRIMS.tex  2005.2.1)
%
%  S.~Iwamoto, The Golden Optimum Solution in Quadratic Programming, 
%  The Fourth International Conference on 
%                 Nonlinear Analysis and Convex Analysis (NACA05), 
%  Okinawa Convension Center, Okinawa, JAPAN, July 31-July 4, 2005.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      slide-GoldenRIMS.tex  2005.2.1  (===> slide-Nobel.tex  2004.11.29)
%   ��{ ����A 
%   �����œK���̌����o�΁A
%   �����W��u�o�ς̐�����́v�A�Q�O�O�T�N�Q���P�P���i���j�`�Q���P�R���i���j�A
%   �����فB
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    slide-GoldenRIMS.tex  2005.2.1
%     ==> see  golden-opti-cro-RIMS.tex   2005.1.12
%          ==> see  golden-optimum.tex   2004.9.11 
%                ==> see golden-ququ.tex   2004.8.31  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      see also  golden-opti-cross.tex   2004.12.30
%           and  golden-opti-graph.tex   2004.12.23 
%      golden-opti-cross.tex   2004.12.30 ==>
%        Cross Dual on The Golden Optimum Solutions : �����œK���̌����o��
%      golden-opti-graph.tex   2004.12.23 ==>
%          golden-optimum.tex   2004.9.11 ==>
%                ==> see golden-ququ.tex   2004.8.31  ==> 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          golden-optimum.tex   2004.9.11 ==>
%                ==> see golden-ququ.tex   2004.8.31  ==> 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      see ===> slide-Nobel.tex  2004.11.29===>
%      (see  slide-tu-iw.tex  2004.11.15===>slide-turu.tex
%      �u2004�N�m�[�x���o�ϊw�܂̒��̐��w�v2004.12.3  BIC ���ʍu��
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%
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%\usepackage{txfonts} % Times�n�t�H���g�i�ŋ߂̃W���[�i���͂���j
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%\newtheorem{thm}{�藝}%[section]%
%\newtheorem{cor}{Corollary}%[section]%
%\newtheorem{lem}{Lemma}%[section]%
\newtheorem{lem}{Lem.}%[section]%
%\newtheorem{lem}{���}%[section]%
\newtheorem{df}{Def.}%[section]%
%\newtheorem{df}{Definition}%[section]%
\newtheorem{ex}{Example}%
\newtheorem{pr}{Problem}
%
%Institute for Operations Research and the Management Sciences (INFORMS) Interna%tional Puerto Rico 2007, Westin Rio Mar Beach Resort \& Spa, San Juan, Puerto R%ico, July 8-11, 2007. %%(talk AM10:00--11:30, Sunday July 8)
%
\title{SLIDE\\ INFORMS International Puerto Rico 2007\\
AM10:00--11:30, Sunday July 8 %
\special{dviout `keep y}
\special{dviout -e=0 -y=XGAP !AN0N5NP!w!p;!bdf}
}
\author{S. Iwamoto~~Kyushu Univ. \hspace{5mm}$\&$\hspace{5mm}M. Yasuda~~Chiba Univ.}

\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%       �ȉ� �{�ԗp    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
%\pagecolor{blackboard}  % �w�i�F
%\thispagestyle{empty}
\,~\,
\vspace{-40mm}
\begin{center}
\mbox{}
\vfill
{\LARGE \colorbox{red}{\WHITE{The Golden Optimal}}}\\
{\LARGE \colorbox{red}{\WHITE{Trajectories}}}\\
{\LARGE \colorbox{red}{\WHITE{in Continuous }}}\\
{\LARGE \colorbox{red}{\WHITE{Dynamic Optimization}}}
\\[1cm]
%%% The Golden Optimal Trajectories in Continuous Dynamic Optimization
%%%  \colorbox{red}{\WHITE{2004�N�m�[�x���o�ϊw�܂̒��̐��w}}
{\Large Seiichi~Iwamoto~\&~Masami~Yasuda}\\[0.7cm]
{\Large Faculty of Economics, Kyushu University\\[5mm] Faculty of Science, Chiba University}
\end{center}
%\vfill
%�� ���`�b�N�ɂ��Ă݂܂���\\
%\textcolor{red}{�ԐF}�E\textcolor{blue}{�F}�E\textcolor{yellow}{���F}�E
%\textcolor{green}{�ΐF}�E\textcolor{cyan}{���F}�E\textcolor{magenta}{�[�g�F}
\end{slide}
%%%%%
%  slide-sanjuan.tex 2007.6.11===>slide-iceland.tex 2006.6.28 in style
%              ===> Golden-poli-DP.tex  2006.9.6 in content (impotant!!)
%%%%%% slide-IFORS05.tex  2005.6.4  (===> slide-NACA05.tex  2005.6.4)
%   slide-IFORS05.tex ��slide-NACA05.tex �������
%   see slide-IWIF.tex, dypricingIWIF1.tex
%   dypricingIWIF1.tex is a 95$%$ transliteration of dyval.tex 
%   except for random expiration date 
%   Hilton Hawaiian Village Beach Resort & Spa
%   dypaci.tex, dypricingIWIF1.tex ���瑁�}��
%   Pacific Option��p��tex.file���쐬����I�I
%\item %\bibitem{tk48} 
%S.~Iwamoto, 
%Dynamic Pricing of Pacific Option, 
%The Seventeenth Triennial Conference of the International Federation of Operati%onal Research Societies (IFORS2005), hosted by INFORMS, 11 - 15 July, 2005 (Als%o Chair of Session \lq\lq Nonstochastic and Nonadditive Dynamic Programming").
%
%Dynamic pricing of Pacific option
%In session Nonstochastic and Nonadditive Dynamic Programming, cluster Dynamic P%rogramming, by invitation of Seiichi Iwamoto.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\colorbox{yellow}{{\color{black}�����œK}}
%%%%%%%%%%%%%%%%%%   �ȉ��̌��^�� IWIF Melbourne �p  %%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
%\pagecolor{blackboard}  % �w�i�F
%\thispagestyle{empty}
\,~\,
\vspace{-40mm}
\begin{center}
\mbox{}
\vfill
\colorbox{red}{\LARGE \YELLOW{{\bf The Golden Optimal}}}\\[1cm]
\colorbox{red}{\LARGE \YELLOW{{\bf Trajectories}}}\\[1cm]
\colorbox{red}{\Huge \YELLOW{{\bf in Continuous }}}\\[1cm]
\colorbox{red}{\Huge \YELLOW{{\bf Dynamic Optimization}}}\\[1cm]
%%  {\LARGE \colorbox{red}{\WHITE{The Golden inequalities}}}\\[1cm]
%%%  \colorbox{red}{\WHITE{A Dynamic Pricing of Exotic Options}}
{\Large Seiichi Iwamoto \& Masami Yasuda}\\[1cm]
{\Large Kyushu University \& Chiba University}\\[5mm]
{\Large Japan}
\end{center}
\vfill
%�� ���`�b�N�ɂ��Ă݂܂���\\
%\textcolor{red}{�ԐF}�E
%\textcolor{blue}{�F}�E
%\textcolor{yellow}{���F}�E
%\textcolor{green}{�ΐF}�E
%\textcolor{cyan}{���F}�E
%\textcolor{magenta}{�[�g�F}
\end{slide}
%%%%%%%%%%%%%%%    Memo   %%%%%%%%%%%
%A Dynamic Pricing of Exotic Options \\[2.78mm]
% : dypricingIWIF1.tex
%}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\,~
\vskip -35mm
\LARGE 
Is optimal policy \colorbox{red}{\Large \YELLOW{{\bf Golden}}}?

Continuous dynamic programming (DP) problems 

1. \textcolor{red}{Control process} 

2. \textcolor{blue}{Allocation process}

%\colorbox{red}{\LARGE \YELLOW{{\bf �u�����v�s����}}}
\end{slide}
%%%%%%%%%%%%%%%%  �T. CONTROL PROCESS   %%%%%%%%%%%%%%%%%%
\begin{slide}
\LARGE 
\textcolor{red}{�T. CONTROL PROCESSES} \\[6.56mm]
%
Min 
$$
\textcolor{red}{\int_{0}^{\infty} \hspace{-1.23mm}\left(x^{2} + u^{2} \right)dt 
}~~~\mbox{and}~~~
 \textcolor{blue}{\int_{0}^{\infty} \hspace{-1.23mm}\left(\dot{x}^{2} + u^{2} \right)dt}
$$ 
under 
$$ \dot{x} = bx + u,~~x(0) = c. $$
\end{slide}
%%%%%%%%%%%%%%%%  �U. ALLOCATION PROCESS   %%%%%%%%%%%%%%%%%%
\begin{slide}
\LARGE 
\textcolor{blue}{�U. ALLOCATION PROCESSES} \\[6.56mm]
%
Max 
$$
\textcolor{red}{\int_{0}^{\infty} \hspace{-1.23mm}e^{-\rho t}(\sqrt{x\,} + \sqrt{u \,} )dt 
}~~~\mbox{and}~~~
 \textcolor{blue}{\int_{0}^{\infty} \hspace{-1.23mm}e^{-\rho t}\sqrt{u\,}dt}
$$ 
under 
$$ \dot{x} = bx - u,~~x(0) = c. $$
\end{slide}
%%%%%%%%%%%%%%%%  �V. APPROACHES   %%%%%%%%%%%%%%%%%%
\begin{slide}
\LARGE 
\textcolor{black}{�V. APPROACHES} \\[4.56mm]
%
\textcolor{blue}{(i) one-variable optimization method} \\[1.56mm]
\textcolor{black}{(ii) calculus of variation~~(Euler Eq.)} \\[1.56mm]
\textcolor{red}{(iii) dynamic programming~~(Bellman Eq.)}.\\[6.56mm]
(i) and (ii) $\Longrightarrow$ \colorbox{red}{\YELLOW{{\bf Golden}}} trajectory\\[1.56mm]
(iii) $\Longrightarrow$ \colorbox{red}{\YELLOW{{\bf Golden}}} policy.
\end{slide}
%%%%%%%%%%%%%%%%  �W. Golden number   %%%%%%%%%%%%%%%%%%
\begin{slide}
\LARGE 
\begin{center}
\colorbox{red}{\LARGE \YELLOW{{\bf The Golden number}}} \\[2.45mm]
\BLUE{$ \phi = \dfrac{1 + \sqrt{5}}{2} \, \thickapprox \, 1.61803 $}
\end{center} 
\vskip -7.56mm
\Large
is the positive sol. to ~\,
\RED{$ x^{2} - x - 1 \dql 0. $}
\vskip -9.56mm
%
Solutions: \RED{$ \phi $} and \BLUE{{\it conjugate}  $ \overline{\phi} := 1 - \phi $}. %Note that 
\RED{$$    \phi + \overline{\phi} \dql 1,~~~~~\phi \ndot \overline{\phi} \dql -1. $$}
% 
 $$\BLUE{\phi^{2} \dql 1+ \phi, ~~~~~\overline{\phi}^{2} \dql 2 - \phi}$$
$$\BLUE{\phi^{2} + \overline{\phi}^{2} \dql 3,~~~~~\phi^{2} \ndot \overline{\phi}^{2} \dql 1.}$$ 
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Golden trajectory %%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\LARGE 
\begin{df} A d'ble ft $ x: [0, \infty) \to R^{1} $ is called \colorbox{red}{\YELLOW{{\bf Golden}}} iff either
$$   
\dfrac{\,\dot{x}(t)\,}{\,x(t) \,} \dql \RED{\log(\phi - 1)}~~\mbox{or}~~\dfrac{\,\dot{x}(t)\,}{\,x(t) \,} \dql \BLUE{\log(2 - \phi)}. $$
\end{df}

\begin{lem} A \colorbox{red}{\YELLOW{{\bf Golden ft $ x$}}}
%Golden ft $ x$ 
is either 
$$   
 \RED{x(t) \dql x(0)e^{-t \log \phi}}~~\mbox{or}~~\BLUE{x(t) \dql x(0)e^{-t \log (1 + \phi)}}. $$
\end{lem}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%  �����O���̃O���t gol-traj-1.tex  %%%%%%%%%%%%%%%%%%%%
\begin{slide}
\begin{center}
Fig.1~~\colorbox{red}{\YELLOW{{\bf Golden}}} trajectories $ \RED{x = c(\phi - 1)^{t}}~~~c = 1, 2, 3 $ 
\end{center}
\LARGE
\begin{center}
\vskip -32mm
%%%%%%%%%%%%%%%%%%%%%
\input{gol-traj-1.tex}
%%%%%%%%%%%%%%%%%%%%%%
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%  �����O���̃O���t gol-traj-2.tex  %%%%%%%%%%%%%%%%%%
\begin{slide}
\begin{center}
Fig.2~~\colorbox{red}{\YELLOW{{\bf Golden}}} trajectories $ \BLUE{x = c(2 - \phi)^{t}}~~~c = 1, 2, 3 $ 
\end{center}
\Huge
\begin{center}
\vskip -12mm
%%%%%%%%%%%%%%%%%%%%%
\input{gol-traj-2.tex}
%%%%%%%%%%%%%%%%%%%%%%
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%  Real Measurement %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Large 
In Fig. 1
$$
\RED{e^{-t \log \phi}} = e^{t \log (\phi - 1)} = \phi^{-t} =  \RED{(\phi - 1)^{t}}. $$
%where
$$ \log \phi \,\thickapprox \, 0.481,~~~\RED{\phi - 1 \dql \phi^{-1} \,\thickapprox \, 0.618} $$ 
%$$ \phi - 1 \dql \phi^{-1} \,\thickapprox \, 0.618033988,~~~~~\log \phi \,\thickapprox \, 0.481211825 $$ 
In Fig. 2
$$
 \BLUE{e^{-t \log(1 + \phi)}} = e^{t \log(2 - \phi)} = (1 + \phi)^{-t} = \BLUE{(2 - \phi)^{t}}.$$
%where
$$ \log(1 + \phi) \,\thickapprox \, 0.962,~~~ \BLUE{2 - \phi  \dql (1 + \phi)^{-1} \,\thickapprox \, 0.382} $$
%$$ 2 - \phi  \dql (1 + \phi)^{-1} \,\thickapprox \, 0.381966011,~~~~~\log(1 + \phi) \,\thickapprox \, 0.96242365  $$
\end{slide}
%%%%%%%%%%%%%%%%%%%        dynamics    %%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
Consider a controlled linear dynamics with real parameter $ b $: %as follows. %\cite{belc}. 
\begin{equation} 
 \RED{\dot{x} \dql b x + u ~~\mbox{on}~~[0, \infty)}  \label{eqn:dg1}
\end{equation}
where $ u : R^{1} \to R^{1} $ is called {\it \BLUE{control function}} or {\it \BLUE{control}}. If $ u(x) = px~ (\mbox{resp.}~ px + q) $, the control is called {\it\RED{ proportional} (\mbox{resp.} \RED{linear})}, where $ p,\, q $ are real constants. A differentiable solution $ x $ to (\ref{eqn:dg1}) is called \BLUE{{\it trajectory}}. 

%\clearpage
\begin{df} A \RED{proportional control $ u $} on dynamics (\ref{eqn:dg1}) is called \colorbox{red}{\YELLOW{{\bf Golden}}} iff it generates a \colorbox{red}{\YELLOW{{\bf Golden trajectory}}}. 
\end{df}
%Thus the definition of Golden trajectory depends on a dynamics. 

\begin{lem} A proportional control $ u = px $ on (\ref{eqn:dg1}) is \colorbox{red}{\YELLOW{{\bf Golden}}} iff 
$$  
 \RED{p \dql -b + \log (\phi - 1)}~~~\mbox{or}~~~\BLUE{p \dql -b + \log (2 - \phi)}. $$
\end{lem}
\end{slide}
%%%%%%%%%%%%%%%%  First CONTROL PROCESS   %%%%%%%%%%%%%%%%%%
\begin{slide}
\large 
Consider the {\it \RED{first}} control process with $b\,(-\infty < b < \infty) $ : 
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm minimize} \hspace{0.4cm} 
                  \int_{0}^{\infty} \hspace{-1.23mm}\left(\RED{x^{2}} + u^{2} \right)dt   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  \dot{x} = bx + u    \nonumber \\[-2.96mm]
{\rm C}(c)& & \hspace*{5.2cm}  ~~~0 \le t < \infty   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   u \in C^{1}   \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x(0) = c.    \nonumber  
\end{eqnarray}
%
%
Let $ v(c) $ be the minimum value. Then $ v(0) = 0. $ 
The value function $ v = v(x) $ satisfies the BE %\cite{bell,belc,belm,iwae} : 
\begin{equation} 
 \min_{-\infty < u < \infty}\left[x^{2} + u^{2} + v^{\prime}(x)(bx + u) \right] \dql 0,~v(0) = 0. \label{eqn:cb6} 
\end{equation}
\end{slide}
%%%%%%%%%%%%%%%%  First CONTROL PROCESS  Page 2   %%%%%%%%%%%%%%%%%%
\begin{slide}
\large 
%%%%      Begin of Lemma      %%%%%%%%%%%%%%%%
\begin{lem} \label{lem:be8} A pair of \RED{proportional control} and \BLUE{quadratic value function} 
\begin{eqnarray*}
& & \RED{\hat{u}(x) = -kx}, ~~ \BLUE{v(x) = kx^{2}}, ~~~ (\, k \dql b + \sqrt{b^{2} + 1\,}\,)  
\end{eqnarray*}
%where $$ k \dql b + \sqrt{b^{2} + 1\,} $$
satisfies the BE.  
%\begin{eqnarray*}
%& &\hspace{-5mm} \min_{-\infty < u < \infty}\left[x^{2} + u^{2} + v^{\prime}(x)%(bx + u) \right]  %\\
% \dql x^{2} + \hat{u}^{2}(x) + v^{\prime}(x)(bx + \hat{u}(x)) \\
%\dql 0.
%\end{eqnarray*}
\end{lem}
%%%%%     End of Lemma     %%%%%%%%%%%%%%%%

Then the dynamical system 
$$ \dot{x} = -\sqrt{b^{2} + 1\,} x, ~~~x(0) = c  $$
yields a trajectory 
$$   \RED{\hat{x}(t) \dql c e^{-\sqrt{b^{2} + 1\,}t}}.   $$
\end{slide}
%%%%%%%%%%%%%%%%%%%%%  First CONTROL PROCESS  Page 2+  %%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Large 
The proportional policy $ \hat{u} $ generates an optimal trajectory $ \hat{x} $, which is \BLUE{not Golden}. Thus the control process $ {\rm C}(c)$ \RED{does not admit an Golden optimal policy}. However, when $ b = \dfrac{1}{\, 2\,} $, the coefficient $ k $ reduces to the \colorbox{red}{\YELLOW{{\bf Golden ratio}}} $ \RED{k = \phi}. $  
\end{slide}
%%%%%%%%%%%%%%%%%%%%%  Second CONTROL PROCESS   %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\large 
Consider the {\it \BLUE{second}} control process with $b\,(-\infty < b < \infty) $ : 
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm minimize} \hspace{0.4cm} 
                  \int_{0}^{\infty} \hspace{-1.23mm}\left(\BLUE{\dot{x}^{2}} + u^{2} \right)dt   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  \dot{x} = bx + u    \nonumber \\[-2.96mm]
{\rm C}^{\prime}(c)& & \hspace*{5.2cm}  ~~~0 \le t < \infty   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   u \in C^{1}   \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x(0) = c.    \nonumber  
\end{eqnarray}
%
%
Let $ v(c) $ be the minimum value. Then
%Then the value function $ v = v(x) $ satisfies the BE with an initial condition : 
\begin{equation}
 \min_{-\infty < u < \infty}\left[(bx + u)^{2} + u^{2} + v^{\prime}(x)(bx + u) \right] \dql 0,~v(0) = 0.    \label{eqn:cgg1} 
\end{equation}
%
\end{slide}
%%%%%%%%%%%%%%%%%%%%%  Second CONTROL PROCESS   Page 2    %%%%%%%%%%%%%%%
\begin{slide}
%%%%%%     Begin of Lemma      %%%%%%%%%%%%%%%%
\begin{lem} \label{lem:be5} A pair of \RED{proportional control} and \BLUE{quadratic value function} 
\begin{eqnarray*}
& & \RED{\hat{u}(x) = px}, ~~~\BLUE{v(x) = kx^{2}}
\end{eqnarray*}
where 
$$ p \dql - \dfrac{1}{\, 2\,}(2 + \sqrt{2\,})b, ~~k \dql (1 + \sqrt{2\,})b 
$$
satisfies the BE. 
%\begin{eqnarray*}
%& &\hspace{-5mm} \min_{-\infty < u < \infty}\left[(bx + u)^{2} + u^{2} + v^{\pr%ime}(x)(bx + u) \right]. % \\
%\eql (bx + \hat{u}(x))^{2} + \hat{u}^{2}(x) + v^{\prime}(x)(bx + \hat{u}(x)).\
%\dql 0
%\end{eqnarray*}
\end{lem}
%%%  End of Lemma     %%%%%%%%%%%%%%%%
Then the dynamical system 
$$ \RED{\dot{x} = - \dfrac{\, b \,}{\, \sqrt{2\,}\,} x, \,~x(0) = c}  $$
yields an optimal trajectory 
$$   \BLUE{\hat{x}(t) \dql c e^{-(b/\sqrt{2})t}}.   $$
\end{slide}
%%%%%%%%%%%%%%%%%%%%%  Second CONTROL PROCESS   Page 3    %%%%%%%%%%%%%%%
\begin{slide}
\large
\begin{itemize}
\item Case {\rm (i)}~ When $ \RED{b \dql \sqrt{2\,}\log(1 + \phi) \,\thickapprox \, 1.361} $, the op. prop'l pol. $ \RED{\hat{u}(x) \dql -(1 + \sqrt{2\,}) \log(1 + \phi)\,x \,\thickapprox \, -2.323\,x} $ is \colorbox{red}{\YELLOW{{\bf Golden}}}. It generates the \colorbox{red}{\YELLOW{{\bf Golden trajectory}}}
$$\, \RED{\hat{x}(t) = c e^{-t \log (1 + \phi)}}. $$

\item Case {\rm (ii)}~ When $ \BLUE{b \dql \sqrt{2\,}\log \phi \,\thickapprox \, 0.681} $, the optimal prop'l policy $ \BLUE{\hat{u}(x) \dql -(1 + \sqrt{2\,})\log \phi \,x \,\thickapprox \, -1.162\,x} $ is \colorbox{red}{\YELLOW{{\bf Golden}}}. It generates the \colorbox{red}{\YELLOW{{\bf Golden trajectory}}} 
$$\, \BLUE{\hat{x}(t) = c e^{-t \log \phi}}. $$
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%  �U. ALLOCATION PROCESS   %%%%%%%%%%%%%%%%%%
\begin{slide}
\LARGE 
\textcolor{blue}{�U. ALLOCATION PROCESSES} \\[6.56mm]
%
Max 
$$
\textcolor{red}{\int_{0}^{\infty} \hspace{-1.23mm}e^{-\rho t}(\sqrt{x\,} + \sqrt{u \,} )dt 
}~~~\mbox{and}~~~
 \textcolor{blue}{\int_{0}^{\infty} \hspace{-1.23mm}e^{-\rho t}\sqrt{u\,}dt}
$$ 
under 
$$ \dot{x} = bx - u,~~x(0) = c. $$
\end{slide}
%%%%%%%%%%%%%%%%  First Allocation PROCESS   %%%%%%%%%%%%%%%%%%
\begin{slide}
\large 
Consider a \RED{general} control process %with $b\,(-\infty < b < \infty) $ : 
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm Maximize} \hspace{0.4cm} 
                  \int_{0}^{\infty} \hspace{-1.23mm}e^{-\rho t}f(x, u)dt   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  \dot{x} = g(x, u)    \nonumber \\[-2.96mm]
{\rm A}(c)& & \hspace*{6.2cm}  ~~~0 \le t < \infty   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   x \in C^{1},~ u \in U(x)   \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x(0) = c.    \nonumber  
\end{eqnarray}
%
%
Let $ v(c) $ be the maximum value. 
Then the value function $ v = v(x) $ satisfies BE : 
\begin{equation}
 \RED{\rho v(x) \dql \Max_{u \in U(x)}\left[f(x, u) + v^{\prime}(x)g(x, u) \right]}. \label{eqn:db1} 
\end{equation}
\end{slide}
%%%%%%%%%%%%%%%%  First Allocation PROCESS  Page 2   %%%%%%%%%%%%%%%%%%
\begin{slide} 
\large 
Let's solve the {\it \RED{first}} allocation process with a discouted total square-root criterion :
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm Maximize} \hspace{0.4cm} 
                  \int_{0}^{\infty} \hspace{-1.23mm}e^{-\rho t}(\RED{\sqrt{x\,}} + \sqrt{u \,} )dt   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  \dot{x} = bx - u    \nonumber \\[-2.96mm]
{\rm A}(c)& & \hspace*{6.2cm}  ~~~0 \le t < \infty   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   x \in C^{1} ,~ u \ge 0   \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x(0) = c.    \nonumber  
\end{eqnarray}
%
%
%Let us solve the Bellman equation: 
Then BE
\begin{equation}
 \rho v(x) \dql \Max_{u \ge 0}\left[\sqrt{x\,} + \sqrt{u\,} + v^{\prime}(x)(bx - u) \right]   \label{eqn:dc4} 
\end{equation}
\end{slide}
%%%%%%%%%%%%%%%%  First Allocation PROCESS  Page 3   %%%%%%%%%%%%%%%%%%
\begin{slide} 
\large 
%
has 
$$ v(x) = k \sqrt{x\,}~ (k > 0),~~u(x) = \dfrac{\,1\,}{\,k^{2}\,} $$ 
where ~$
  k \dql \dfrac{1}{\,\sqrt{1 + 2 \rho - b \,} - 1\,}.$ \
This %together with (\ref{eqn:gk1}) and {\rm(iii)} 
yields 
\begin{equation}
  \RED{\dot{x} \dql -ax,~~~~x(0) = c}. \nonumber   %  \label{eqn:gh3} 
\end{equation}
where 
$  \BLUE{a \dql 2(1 + \rho - b - \sqrt{1 + 2 \rho - b \,})}. $                 
We have the \RED{optimal trajectory} 
 $$ \RED{x^{*}(t) \dql c e^{-at}}  $$
and an \BLUE{optimal control} 
 $$  \BLUE{u^{*}(x) \dql (a + b)x}.  $$
\end{slide}
%%%%%%%%%%%%%%%%  First Allocation PROCESS  Page 4   %%%%%%%%%%%%%%%%%%
\begin{slide} 
%\large 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsubsection{Case $ b = 1$}
\bf{Case $ b = 1.$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Under the condition $ \rho \ge 2$, we have 
the \RED{prop'l policy} $ \RED{u^{*}(x) = (1 + 2 \rho - 2 \sqrt{2 \rho\,})\,x}. $ It generates an \BLUE{optimal trajectory} $ \BLUE{x^{*}(t) \dql c e^{-at}}  $, where $  a \dql 2(\rho - \sqrt{2 \rho \,}). $ 
%
\begin{itemize}
\item {\rm (i)}~~ When $ \RED{\rho \dql \dfrac{1}{\,2\,}(1 + \sqrt{1 + \log(1 + \phi) }\,)^{2}  \,\thickapprox \, 2.882} $, the optimal trajectory $ x^{*} $ is \colorbox{red}{\YELLOW{{\bf Golden}}} $ x_{1}^{*}(t) \dql c e^{-t\log(1 + \phi)}  $. This is generated by a \colorbox{red}{\YELLOW{{\bf Gol. opt. pol.}}} $ u_{1}^{*}(x) = (1 + \log(1 + \phi))\,x \,\thickapprox \, 1.962\,x $. 
%Thus we have the \colorbox{red}{\YELLOW{{\bf Golden}}} \RED{optimal policy} $ u_{1}^{*}.$
%
\item  {\rm (ii)}~~ When $ \BLUE{\rho \dql \dfrac{1}{\,2\,}(1 + \sqrt{1 + \log \phi }\,)^{2}  \,\thickapprox \,  2.458} $, the optimal trajectory $ x^{*} $ is \colorbox{red}{\YELLOW{{\bf Golden}}} $ x_{2}^{*}(t) \dql c e^{-t\log \phi}  $. This is generated by a \colorbox{red}{\YELLOW{{\bf Gol. opt. pol.}}} $ u_{2}^{*}(x) = (1 + \log \phi)\,x \,\thickapprox \, 1.481\,x $. 
%Thus $ u_{2}^{*}$ is Golden optimal. 
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%  First Allocation PROCESS  Page 4   %%%%%%%%%%%%%%%%%%
\begin{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsubsection{Case $ b = 0$}
\bf{Case $ b = 0.$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Under the condition $ \rho \ge 0$, we have 
the \BLUE{prop'l policy} $ \BLUE{u^{*}(x) = ax} $. It generates an \RED{optimal trajectory} $ \RED{x^{*}(t) \dql c e^{-at}}  $, where $  a \dql (\sqrt{1 + 2 \rho \,} - 1)^{2}. $ 
%
\begin{itemize}
\item {\rm (i)}~~When $ \BLUE{\rho \dql \sqrt{\log(1 + \phi)\,} + \dfrac{1}{\,2\,}\log(1 + \phi)  \,\thickapprox \, 1.462} $, the optimal trajectory $ \BLUE{x^{*}} $ is \colorbox{red}{\YELLOW{{\bf Golden}}} $ x_{1}^{*}(t) \dql c e^{-t\log(1 + \phi)}  $. This is generated by a \colorbox{red}{\YELLOW{{\bf Golden optimal policy}}} $ u_{1}^{*}(x) = x \log(1 + \phi) \,\thickapprox \, 0.962\,x $.  
%Thus we have the Golden optimal policy $ u_{1}^{*}$. 
%
\item {\rm (ii)}~~ When $ \RED{\rho \dql \sqrt{\log \phi\,} + \dfrac{1}{\,2\,}\log \phi  \,\thickapprox \,  0.934} $, the optimal trajectory $ x^{*} $ is \colorbox{red}{\YELLOW{{\bf Golden}}} $ x_{2}^{*}(t) \dql c e^{-t\log \phi}  $. This is generated by a \colorbox{red}{\YELLOW{{\bf Golden optimal policy}}} $ u_{2}^{*}(x) = x \log \phi \,\thickapprox \, 0.481\,x $. 
%Thus $ u_{2}^{*}$ is Golden optimal. 
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%  Second Allocation PROCESS   %%%%%%%%%%%%%%%%%%
\begin{slide} 
\large 
Let's solve the {\it \BLUE{second}} 
allocation process : 
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm Maximize} \hspace{0.4cm} 
                  \int_{0}^{\infty} \hspace{-1.23mm}e^{-\rho t}\BLUE{\sqrt{u\,}}dt   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  \dot{x} = bx - u    \nonumber \\[-2.96mm]
{\rm A}^{\prime}(c)& & \hspace*{6.2cm}  ~~~0 \le t < \infty   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   x \in C^{1},~ u \ge 0   \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x(0) = c.    \nonumber  
\end{eqnarray}
%
%
%Let us solve the Bellman equation: 
Then BE
\begin{equation}
 \rho v(x) \dql \Max_{u \ge 0}\left[\sqrt{u\,} + v^{\prime}(x)(bx - u) \right]  \label{eqn:kb4} 
\end{equation}
%%%%%%%%%%%%%%%%  Second Allocation PROCESS  Page 2   %%%%%%%%%%%%%%%%%%
\begin{slide} 
\large 
%
has 
$$ v(x) = k \sqrt{x\,}~ (k > 0),~~u(x) = \dfrac{\,1\,}{\,k^{2}\,} $$ 
where ~$
  k \dql \dfrac{1}{\,\sqrt{ 2 \rho - b \,} \,}.$ \
This %together with (\ref{eqn:gk1}) and {\rm(iii)} 
yields 
\begin{equation}
  \RED{\dot{x} \dql -ax,~~~~x(0) = c}. \nonumber   %  \label{eqn:gh3} 
\end{equation}
where 
$  \BLUE{a \dql 2(\rho - b)}. $                 
We have the \RED{optimal trajectory} 
 $$ \RED{x^{*}(t) \dql c e^{-at}}  $$
and an \BLUE{optimal control} 
 $$  \BLUE{u^{*}(x) \dql (b + a)x}.  $$
\end{slide}
\end{slide}
%%%%%%%%%%%%%%%%  Second Allocation PROCESS  Page 3   %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\begin{slide} 
\large 
%
%\subsubsection{Two cases}
\bf{Two cases:}
%%%%%%%%%%%%%%%%%%%%%%
%Let us specify two cases where the allocation process 
$ {\rm A}^{\prime}(c) $ has a \colorbox{red}{\YELLOW{{\bf Golden optimal policy}}} for $ b \ge 0.$ \\[2.56mm]
%
%\begin{itemize}
%\item 
\RED{Case {\rm (i)}}~~ When $\RED{ \rho \dql b + \log \phi} $, the optimal trajectory $ x^{*} $ is Golden $ \RED{x_{1}^{*}(t) \dql c e^{-t\log(1 + \phi)}}  $. This is generated by a Golden $ \RED{u_{1}^{*}(x) = (b + 2 \log \phi)\,x} $.  Thus we have the Golden optimal policy $ u_{1}^{*}.$ \\[2.56mm]
%
%\item  
\BLUE{Case {\rm (ii)}}~~ When $ \BLUE{\rho \dql b + \dfrac{1}{\,2\,}\log \phi} $, the optimal trajectory $ x^{*} $ is Golden $ \BLUE{x_{2}^{*}(t) \dql c e^{-t\log \phi}}  $. This is generated by a Golden $ \BLUE{u_{2}^{*}(x) = (b +  \log \phi)\,x} $.  Thus $ u_{2}^{*}$ is Golden optimal.
%\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage

\vskip 12mm

%
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