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%  price97sl.tex for 1997 IFIP 22 July 1997 (TALK) 
%  post talk corrections 01 august 1997
%  18th IFIP TC7 Conference on System Modelling and Optimization
% Session:  invited session on Stochastic Control and Filtering
% Organizer and Session Chair: George Yin, Math, Wayne State
%      Detroit, Michigan
%      F. B. Hanson*, University of Illinois at Chicago
%      D Ryan*, McKendree College
%  list of confirmed speakers for our session:
% 2pm  Floyd Hanson, U Illinois, Opt Ctrl under Res. & Price Uncert.
% 2:30 Jose Luis Menaldi, Wayne State U, Quasi-var Ineq and Hybrid Ctrl
% 3:30 Andrew Heunis, U Waterloo, Strong Approx for Recur Stoch Alg
% 4:00 Gangaram S. Ladde, UTA, Sing Pert Approach to Lg Scale Regulator Probs
% 4:30 Qing Zhang, U Georgia, Nonl Filt with Switch Diffs & SmObsNois
%  Cancel? Pao-Liu Chow, Wayne State University
% former source: acc97sl.tex
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\newcommand{\eq}{\begin{eqnarray}}
\newcommand{\nq}{\end{eqnarray}}
\newcommand{\equnn}{\begin{eqnarray*}}
\newcommand{\nqunn}{\end{eqnarray*}}
\newcommand{\grad}{\nabla}
\newcommand{\gradX}{\nabla_x}
\newcommand{\gradXT}{\nabla_x^T}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Mean}{\raisebox{-1ex}[1em][1.5em]{${\displaystyle\rm E}\atop{\scriptstyle\bf\{dZ,dQ,dW\}}$}}
\newcommand{\eps}{\varepsilon}
\newcommand{\UL}{\underline}
\newcommand{\PI}{{\bf\Pi}}
\newcommand{\vS}{{\bf{v}}^{\displaystyle *}}
\newcommand{\uS}{{\bf{u}}^{\displaystyle *}}
\newcommand{\SST}{S^{\displaystyle *}}
\newcommand{\HS}{^{\displaystyle *}}
\newcommand{\VE}{\mbox{\it VE}}
\newcommand{\SE}{\mbox{\it SE}}
\newcommand{\ZVE}{\mbox{\it ZVE}}
\newcommand{\QVE}{\mbox{\it QVE}}
\newcommand{\URE}{\mbox{\it URE}}
\newcommand{\UE}{\mbox{\it UE}}
\newcommand{\DVX}{\mbox{\it DVX}}
\newcommand{\DVXE}{\mbox{\it DVXE}}
\newcommand{\DVYE}{\mbox{\it DVYE}}
\newcommand{\DDVXE}{\mbox{\it DDVXE}}
\newcommand{\DDVYE}{\mbox{\it DDVYE}}
\newcommand{\VM}{\mbox{\it VM}}
\newcommand{\UM}{\mbox{\it UM}}
\newcommand{\UR}{\mbox{\it UR}}
\newcommand{\URM}{\mbox{\it URM}}
\newcommand{\SM}{\mbox{\it SM}}
\newcommand{\ZVM}{\mbox{\it ZVM}}
\newcommand{\QVM}{\mbox{\it QVM}}
\newcommand{\DVXM}{\mbox{\it DVXM}}
\newcommand{\DVYM}{\mbox{\it DVYM}}
\newcommand{\DDVXM}{\mbox{\it DDVXM}}
\newcommand{\DDVYM}{\mbox{\it DDVYM}}
\newcommand{\HB}{{\bf{H}}}
\newcommand{\RB}{{\bf{R}}}
\newcommand{\uB}{{\bf{u}}}
\newcommand{\UB}{{\bf{U}}}
\newcommand{\vB}{{\bf{v}}}
\newcommand{\xB}{{\bf{x}}}
\newcommand{\XB}{{\bf{X}}}
\newcommand{\yB}{{\bf{y}}}
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\newcommand{\vBO}{\overline{\bf{v}}}
\newcommand{\xiB}{\vec{\overline{\xi}}}%no boldmath large enough
\newcommand{\XBO}{\overline{\bf{X}}}
\newcommand{\reg}{{\small\rm reg}}
\newcommand{\BCEN}{\begin{center}}
\newcommand{\ECEN}{\end{center}}
\newcommand{\BDES}{\begin{description}}
\newcommand{\EDES}{\end{description}}
\renewcommand{\theenumi}{\bf\arabic{enumi}}
\setlength{\itemsep}{4ex}
\begin{document}
%
% Slide 0.1: Title Page
%
\thispagestyle{empty}
\BCEN
{\LARGE\bf
Optimal Control Under Resource\\*[0.1in] and Price Uncertainty
}\footnotetext[1]{\large
This work was supported by
the National Science Foundation Computational Mathematics Program
under grants DMS-93-01107 and DMS-96-26692.}
\\*[2in]
{\Large\bf Floyd B. Hanson}\\*[0.2in]
{\large\bf Laboratory for Advanced Computing\\
Department of Math., Stat., and Computer Science\\
University of Illinois at Chicago\\
E-Mail:~~ hanson@uic.edu\\
Web-Page:~~ {\it http://www.math.uic.edu/$\widetilde{~~}$hanson/}}\\*[0.25in]
{\large and}\\*[0.2in]
{\Large\bf Dennis Ryan}\\*[0.2in]
{\large\bf Division of Science and Mathematics\\
McKendree College\\
Lebanon, IL 62254, USA\\
Email:  dryan@a1.mckendree.edu}\\*[0.1in]
\ECEN
%
\newpage
%
% Slide 0.2: OverView Page
%
~\\*[0.2in]
\BCEN\UL{\LARGE\bf Overview}\ECEN
~\\*[0.1in]
{\Large%BeginSize
\begin{enumerate}
%\renewcommand{\theenumi}{\bf\Roman{enumi}}
\setlength{\itemsep}{8ex}
\item {\bf Noninflationary, Deterministic Model.}
\item {\bf Inflationary, Stochastic Control Model.}
\item {\bf Numerical Approximations.}
\item {\bf Numerical Results.}
\item {\bf Conclusions.}
\end{enumerate}
}%EndLargeSize
%
\newpage
%
% Slide 0.3: Abstract Page
%
\BCEN\UL{\LARGE\bf Outline of Abstract}\ECEN
~\\*[0.1in]
{\LARGE\bf%BeginSize
\begin{itemize}
\setlength{\itemsep}{8ex}
\item Optimal Control of Stochastic Resource\\in Continuous Time.  
\item Model Effects of Large Random Price\\Fluctuations.
\item Influence of Continuous growth\\and Jump Stochastic Noise.
\item Computational Stochastic Dynamic\\Programming.
\item Pronounced Effect of Inflationary Prices\\on Optimal Return.
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 1.1:  Noninflationary, Deterministic Model:  Introduction.
%
\BCEN\UL{\LARGE\bf Part 1.~~ 
Noninflationary, Deterministic Model:}\\ {\LARGE\bf Introduction.}\ECEN
~\\
\UL{\Large\bf 1.1.~~ Ordinary Differential Equation (ODE):}
\\*[1ex]
{\Large%BeginSize
\begin{itemize}
\setlength{\itemsep}{2ex}
\item {\bf Nonlinear (Logistic) Dynamics:}
\equnn
d\XB(s)&=&\left[r_1 \XB(s)(1-\XB(s)/K)-\HB(s)\right]\, ds, \\
&~&0 < t < s < T.
\nqunn
\item {\bf Initial Conditions:}
$\XB(0) = \xB_0 ~;~ \qquad 0 < t  < T$~
\item {\bf State Variable (Resource Size):}
$\XB(t) ~=~[X_i(t)]_{1\times 1} ~;$
\item {\bf BiLinear Control-State Dynamics Assumption\\
(for Resource Harvesting):}\\
$$\HB(t) ~=~ q \UB(t) \XB(t) ~;$$
where $q=$ Efficiency (Catchability) Coefficient;
\item {\bf Control Variable (Harvesting Effort):}\\
$$\UB(t) ~=~[U_i\{\XB(t),t\}]_{1 \times 1} ~, \qquad
U_{\min}\le \UB(t)\le U_{\max} < \infty ~;$$
\item {\bf Growth Parameters:}\\$r_1=$ Resource Intrinsic Growth Rate;\\
$K=$ Environment Carrying (Saturation) Capacity.
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 1.2:  Quadratic Performance Index
%
\UL{\Large\bf 1.2.~~ Quadratic Performance Index:}
\\*[1ex]
{\Large%BeginSize
\equnn
V(\XB,\UB,t)
= \int_t^{T}  e^{-\delta(s-t)}\left[pq\UB(s)\XB(s)-c(\UB(s))\right]\,ds ~,
\nqunn
where
\begin{itemize}
\setlength{\itemsep}{4ex}
\item  $V(\xB,\uB,t)=$ {\bf Current Value of Future Resources}\\
(i.e., $\exp(\delta t)$ times Present Value);
\item  $T=$ {\bf Time Horizon}~ ($T \geq t$);
\item  $\delta=$ {\bf Nominal Discount Rate} (NOT adjusted for inflation);
\item  $p=$ {\bf Price of Resource per Unit Harvest Rate};
\item  $c(\uB) = c_1 \uB +c_2 \uB^2 =$ {\bf Quadratic Costs}\\
(Assume Increasing, Convex Quadratic Costs: $c_1 >0$\\ and $c_2 >0$);
\item  {\bf Instantaneous Net Return:}\\
$\RB(\xB,\uB) = p q \uB \xB - c(\uB)$~. 
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 1.3:  Deterministic Dynamic Programming
%
\UL{\Large\bf 1.3.~~ Deterministic Dynamic Programming:}
\\*[1ex]
{\Large%BeginSize
\begin{itemize} 
\setlength{\itemsep}{2ex}
\item  {\bf Optimization Goal $=$ Maximize Total Return:}
$$\vS(\xB,t)=V(\xB,\uS,t)
=\max_u\left[V(\xB,\uB,t)\right] ~;$$
\item {\bf PDE of Deterministic Dynamic Programming:}
\equnn
\vB_t\HS(\xB,t)+r_1\xB(1-\xB/K)\vB_x\HS(\xB,t)
-\delta \vS(\xB,t)+S\HS(\xB,t) =0;  
\nqunn
\item {\bf Control Switching Term:}
\equnn
S\HS(\xB,t) = \max_u\left[\left(p - \vB_x\HS(\xB,t)\right)
q \uB \xB -c_1 \uB -c_2 \uB^2\right];
\nqunn
\item {\bf Regular (Unconstrained) Control:}
\equnn
\uB_R(\xB,t)=\frac{(p - \vB_x\HS(\xB,t))q\xB-c_1}{2\cdot c_2} ~, 
\qquad c_2 >0;
\nqunn
\item {\bf Optimal (Constrained) Control:}
\equnn
\uS(\xB,t) =  \left\{\begin{array}{ll}
U_{\max}, &U_{\max} \le \uB_R(\xB,t)\\
\uB_R(\xB,t), &U_{\min} \le \uB_R(\xB,t) \le U_{\max} \\    
U_{\min}, &\uB_R(\xB,t) \le U_{\min} 
\end{array}\right. ;
\nqunn
\item {\bf Final Boundary Condition:} $\qquad \vS(\xB,T) = 0$;
\item {\bf Extinction Natural Boundary Condition:}
\equnn
\vS (0,t) = -\frac{(c_1+c_2U_{\min})U_{\min}}{\delta}
\left(1-e^{-\delta(T-t)}\right) ~, \qquad \delta >0.
\nqunn
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 2.1:  SDE Formulation.
%
\BCEN\UL{\LARGE\bf Part 2.~~ Inflationary, Stochastic Control Model}\ECEN
~\\
\UL{\Large\bf 2.1.~~ Stochastic Dynamics Equation (SDE (1)):}
\\*[1ex]
{\Large%BeginSize
\begin{itemize}
\setlength{\itemsep}{3ex}
\item {\bf Nonlinear Dynamics with Gaussian (G)\\ and Poisson (Z) Noise:}
\equnn
d\XB(s) &=& [r_1 \XB(s)(1-\XB(s)/K)-\HB(s)]\,ds\\*[1ex]
&+&\sigma_1 \XB(s)\, dW_1(s) +\XB(s)\sum_{j=1}^n a_j\,dZ_j(s,f_j) ~,\\*[1ex]
\XB(t) &=& x ~,
\nqunn
\item {\bf Initial Conditions:}
$\XB(0) = \xB_0 ~,~~t_0 < t < s < T$ ~;
\item {\bf Gaussian (Wiener) Noise\\(Zero Mean and Normalized):}
$$E[dW_1(t)]=0 ~, \qquad  Var[dW_1(t)]= dt \qquad \sigma_1\leq 0~;$$
\item {\bf Poisson (Jump) Noise:}
$$E[dZ_j(t,f_j)]=f_j dt ~, \qquad  Var[dZ_j(t,f_j)]=f_j dt ~, \qquad 
1\leq j \leq n ~,$$
where $f_j=$ Jump Rate and $a_j=$ Jump Amplitude Coefficient ($-1<a_j$);
\item {\bf Independent (Uncorrelated) Processes Assumption:}
\equnn
\mbox{CoVar}[dW_1(t),~dZ_j(t,f_j)]=0 ~, \\*[0.5ex]
\mbox{CoVar}[dZ_j(t,f_j),~dZ_{j'}(t,f_{j'})]=\delta_{j,j'}f_j dt ~;
\nqunn
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 2.2:  Inflationary Factor Formulation
%
\UL{\Large\bf 2.2.~~ Inflationary Factor Model:}
\\*[0.5ex]
{\Large%BeginSize
\begin{itemize}
\setlength{\itemsep}{1.5ex}
\item {\bf Nonlinear Supply--Demand Model Relation:}
\equnn
{\bf P}(t)=\left(\frac{p_0}\HB(t)+p_1\right) \YB(t),
\nqunn
where
\begin{description}
\item * ${\bf P}(t)\cdot \HB(t)=$ Gross Return on Harvest;
\item * $p_0=$ Supply--Demand Price Coefficient;
\item * $p_1=$ Constant Price per Unit Harvest;
\item * $\YB(t)=$ Fluctuating Inflationary Factor;
\end{description}
\item {\bf Linear Fluctuating Inflationary Factor SDE (2):}
\equnn
d\YB(s)&=&r_2 \YB(s)\,ds+\sigma_2 \YB(s)\,dW_2 (s)+ 
\YB(s)\sum_{j=1}^m b_j \,dQ_j(s;g_j)~,
\nqunn
\begin{description}
\item * $\YB(t) = \yB$;
\item * $r_2=$ Annual Rate of Inflation without Fluctuations;
\item * $g_j=$ {\it j}th component of Inflationary Jump Rate;
\item * $b_j=$ {\it j}th component of Jump Amplitude Coefficient;
\end{description}
\item {\bf Inflationary Gaussian (Wiener) Noise:}
{\normalsize
$$E[dW_2(t)]=0 ~, \qquad  Var[dW_2(t)]= dt \qquad \sigma_2\leq 0~;$$}
\item {\bf Inflationary Poisson (Jump) Noise:}
{\normalsize
$$E[dQ_j(t,g_j)]=g_j dt ~, \qquad  Var[dQ_j(t,g_j)]=g_j dt ~, \qquad
1\leq j \leq m ~;$$}
\item {\bf Independent (Uncorrelated) Processes Assumption:}
{\normalsize
\equnn
\mbox{CoVar}[dW_2(t),~dQ_j(t,g_j)]=0 ~,\qquad
\mbox{CoVar}[dQ_j(t,g_j),~dQ_{j'}(t,g_{j'})]=\delta_{j,j'}g_j dt ~;
\nqunn}
\end{itemize}
}%EndLargeSize
%
\newpage
%
% Slide 2.2.1: Figure 1.
%
\BCEN\UL{\Large\bf Figure 1.}\ECEN
%
~\\\hspace*{0.2in}
\parbox[t]{6in}{\large\bf Pacific halibut prices in USdollars per kilogram for 
each year from 1935 to 1985 (Raw Data: IPHC 1984 and 1985 Annual Reports).}
%
%\begin{figure}[htbp]
\BCEN
~\\*[10pt]
\hspace*{-60pt}
\epsfysize400pt
\epsffile{Fig1YPBig.eps}
\ECEN
%\end{figure}  
%
\newpage
%
% Slide 2.2.2: Figure 2.
%
\BCEN\UL{\Large\bf Figure 2.}\ECEN
%
~\\\hspace*{0.2in}
\parbox[t]{6in}{\large\bf U.S.-Canadian catch in millions of kilograms for 
each year from 1935 to 1985 (Raw Data: IPHC 1984 and 1985 Annual Reports).}
%
%\begin{figure}[htbp]
\BCEN
~\\*[10pt]
\hspace*{-60pt}
\epsfysize400pt
\epsffile{Fig2YHBig.eps}
\ECEN
%\end{figure}
%
\newpage
%
% Slide 2.2.3: Figure 3.
%
\BCEN\UL{\Large\bf Figure 3.}\ECEN
%
~\\\hspace*{0.2in}
\parbox[t]{6in}{\large\bf Pacific halibut price in USdollars per kilogram 
versus catch in millions of kilograms for years from 1935 to 1985.
Linear regression for price times catch as a
function of catch from 1980 to 1985 displayed as smooth hyperbolic price curve.
(Raw Data: IPHC 1984 and 1985 Annual Reports).}
%
%\begin{figure}[htbp]
\BCEN
~\\*[10pt]
\hspace*{-60pt}
\epsfysize400pt
\epsffile{Fig3HPFitBig.eps}
\ECEN
%\end{figure}
%
\newpage
%
%  Slide 2.3:  Mean Quadratic Performance Index
%
\UL{\Large\bf 2.3.~~ Mean Quadratic Performance Index:}
\\*[1ex]
{\Large%BeginSize
\equnn
\overline{V}(\xB,\yB,\uB,t)&=&\Mean\left[\int_t^T e^{-\delta(s-t)}\left[(p_0+p_1
q\UB(s)\XB(s))\YB(s)\right.\right.
\\*[1em]
&-&\left.c(\UB(s))\right]\,ds\mid \XB(t)=\xB, \; 
\YB(t)=\yB, \; \UB(t)=\uB\biggr],
\nqunn
where
\begin{itemize}
\setlength{\itemsep}{6ex}
\item  $\overline{V}(\xB,\yB,\uB,t)=$ 
{\bf Expected Current Value of\\Future Resources} 
(i.e., $\exp(\delta t)$ times Present Value);
\item  $\{\xB,\yB\}=$ {\bf 2-Dim State of Inflationary
Stochastic\\ Dynamics };
\item  $T=$ {\bf Time Horizon}~ ($T \geq t$);
\item  $\delta=$ {\bf Nominal Discount Rate} (NOT adjusted for inflation);
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 2.4:  Stochastic Dynamic Programming
%
\UL{\Large\bf 2.4.~~ Stochastic Dynamic Programming:}
\\*[1ex]
{\Large%BeginSize
\begin{itemize} 
\setlength{\itemsep}{6ex}
\item  {\bf Optimization Goal $=$ Maximize Total Return:}
$$\vS(\xB,\yB,t)=\overline{V}(\xB,\yB,\uS,t)
=\max_u\left[\overline{V}(\xB,\yB,\uB,t)\right] ~;$$
\item {\bf PDE of Stochastic Dynamic Programming:}
\equnn
0&=&\vB_t\HS(\xB,\yB,t)+r_1\xB(1-\xB/K)\vB_x\HS(\xB,\yB,t)
-\delta \vS(\xB,\yB,t)
\\*[1ex]
&+& \frac{\sigma_1^2 \xB^2}{2}\vB_{xx}\HS
+ \sum_j f_j\left[\vS\left((1+a_j)\xB,\yB,t\right)
-\vS(\xB,\yB,t)\right] 
\\*[1ex]
&+& r_2\yB\vS_y + \frac{\sigma_2^2 \yB^2}{2}\vB_{yy}\HS
+ \sum_j g_j\left[\vS(\xB,(1+b_j)\yB,t)- \vS(\xB,\yB,t)\right]
\\*[1ex]
&+& S\HS(\xB,\yB,t),  
\nqunn
by General It\^{o} Chain Rule;
\item {\bf Control Switching Term:}
\equnn
S\HS(\xB,\yB,t) = \max_u\left[p_0\yB+\left(p_1\yB 
- \vB_x\HS(\xB,\yB,t)\right)
q \uB \xB -c_1 \uB-c_2 \uB^2\right];
\nqunn
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 2.4.1:  More Stochastic Dynamic Programming
%
\UL{\Large\bf 2.4.1.~~ More Stochastic Dynamic Programming:}
\\*[1ex]
{\Large%BeginSize
\begin{itemize}
\setlength{\itemsep}{8ex}
\item {\bf Regular (Unconstrained) Control:}
\equnn
\uB_R(\xB,\yB,t)
=\frac{(p_1\yB-\vB_x\HS(\xB,\yB,t))q\xB-c_1}{2 c_2} ~, 
\quad c_2 >0;
\nqunn
\item {\bf Optimal (Constrained) Control:}
\equnn
\uS(\xB,\yB,t) =  \left\{\begin{array}{ll}
U_{\max}, &U_{\max} \le \uB_R(\xB,\yB,t)\\
\uB_R(\xB,\yB,t), 
&U_{\min} \le \uB_R(\xB,\yB,t) \le U_{\max} \\    
U_{\min}, &\uB_R(\xB,\yB,t) \le U_{\min} 
\end{array}\right. ;
\nqunn
\item {\bf Final Boundary Condition:} $\qquad \vS(\xB,\yB,T) = 0$;
\item {\bf Extinction Natural Boundary Condition*:}
\equnn
\vS(0,0,t) = -\frac{(c_1+c_2U_{\min})U_{\min}}{\delta}
\left(1-e^{-\delta(T-t)}\right) ~, \qquad \delta >0.
\nqunn
*  {\it see Kushner and Dupuis (1992) for proper handling of
stochastic reflecting boundary conditions.}
\end{itemize}
}%EndLargeSize
%
\newpage
%
% Slide 3.1: Numerical Approximations.
%
\BCEN\UL{\LARGE\bf Part 3.~~ Numerical Approximations}\ECEN
~\\
\BCEN\UL{\Large\bf 3.1~~ Basic Hybrid Numerical Procedures.}\ECEN
%
~\\*[0.1in]
{\Large\bf%BeginSize
\begin{itemize}
\setlength{\itemsep}{8ex}
\item {\bf Extrapolated, Predictor-Corrector for Nonlinear\\Iteration.}
\item {\bf Crank-Nicolson Implicit for 2nd Order in Time and State.}
\item {\bf Modifications for Poisson Functional Terms.}
\item {\bf Modifications for Optimization in Switching Term.}
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 3.2:  Numerical Discretizations
%
\BCEN\UL{\Large\bf 3.2.~~ Numerical Discretizations:}\ECEN
%
~\\*[0.1in]
{\Large%BeginSize
\begin{itemize}
\setlength{\itemsep}{4ex}
\item {\bf State$_1$:}~ $X_i\equiv (i-1)\Delta x$, $i=1, \cdots, N_x$,~
$\Delta x\equiv K/(N_x-1)$; 
\item {\bf State$_2$:}~ $Y_j\equiv (j-1)\Delta y$, $j=1, \cdots, N_y$,~
$\Delta y\equiv e^{r_2T}/(N_y-1)$; 
\item {\bf Time:}~ $T_k\equiv T-(k-1)\Delta t$, $k=1, \cdots, N_t$,~
$\Delta t\equiv T/(N_t-1)$;
\item {\bf Optimal Expected Value:}~ $\vS(x_i,y_j,t_k) \longrightarrow 
V_{i,j,k}$;
\item $\vB_x\HS(X_i,Y_j,T_k) \longrightarrow \mbox{\it DVX}_{i,j,k}
\equiv 0.5(V_{i+1,j,k}-V_{i-1,j,k})/\Delta x$;
\item $\vB_y\HS(X_i,Y_j,T_k) \longrightarrow \mbox{\it DVY}_{i,j,k}
\equiv 0.5(V_{i,j+1,k}-V_{i,j-1,k})/\Delta y$;
\item $\vB_{xx}\HS(X_i,Y_j,T_k) \longrightarrow \mbox{\it DDVX}_{i,j,k}
\equiv (V_{i+1,j,k}-2V_{i,j,k}+V_{i-1,j,k})/(\Delta x)^2$;
\item $\vB_{yy}\HS(X_i,Y_j,T_k) \longrightarrow \mbox{\it DDVY}_{i,j,k}
\equiv (V_{i,j+1,k}-2V_{i,j,k}+V_{i,j-1,k})/(\Delta y)^2$; 
\item $\vB_t\HS(X_i,Y_j,T_{k+0.5}) \longrightarrow \mbox{\it DVT}_{i,j,k}
\equiv -(V_{i,j,k+1}-V_{i,j,k})/\Delta t$;
\end{itemize}
with Error: $O(\Delta x)^2+O(\Delta y)^2+O(\Delta t/2)^2$;
}%EndLargeSize
%
\newpage
%
%  Slide 3.2.1:  More Numerical Discretizations
%
\BCEN\UL{\Large\bf 3.2.1.~~ More Numerical Discretizations:}\ECEN
%
~\\*[0.1in]
{\Large%BeginSize
\begin{itemize}
\setlength{\itemsep}{8ex}
\item {\bf X-Poisson Term:}~ 
$\vS((1+a_l)X_i,Y_j,T_k) \longrightarrow \mbox{\it ZV}_{i,j,k,l}$ 
\\by 2nd order accurate interpolation between nearest nodes;
\item {\bf Y-Poisson Term:}~ 
$\vS(X_i,(1+b_l)Y_j,T_k) \longrightarrow \mbox{\it QV}_{i,j,k,l}$  
\\by 2nd order accurate interpolation between nearest nodes;
\item {\bf Regular Control:}~ 
\\$\uB_R(X_i,Y_j,T_k) \longrightarrow
\UR_{i,j,k}\equiv (p_1Y_j-\DVX_{i,j,k}\cdot q\cdot X_i-c_1)/(2c_2)$;
\item {\bf Optimal Control:}~ 
\\$\uS(X_i,Y_j,T_k) \longrightarrow
U_{i,j,k}\equiv$ same as exact composite expression;
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 3.3:  Computational Stochastic Dynamic Programming
%
\BCEN\UL{\Large\bf 3.3.~~ Computational Stochastic Dynamic Programming:}\ECEN
%
~\\*[-1ex]
{\Large%BeginSize
{\bf For $k+1=2$ to $N_t$ while $i=1$ to $N_x$ \& $j=1$ to $N_y$:}
\begin{itemize}
\setlength{\itemsep}{6ex}
\item {\bf Accelerating Extrapolating Start:}
$$
\VE_{i,j,k}\equiv 0.5(3V^{(c,*)}_{i,j,k}-V^{(c,*)}_{i,j,k-1})
\simeq V_{i,j,k+0.5}, 
\quad \mbox{if} \quad k\leq 2,$$
which are used to get components 
$\DVXE$, $\DVYE$, $\DDVXE$, $\DDVYE$, $\ZVE$, $\QVE$, $\URE$, $UE$ \& $\SE$,
and where $V^{(c,*)}_{i,j,k})$ is the final correction from step $k$;
\item {\bf Extrapolated-Predictor Step:}
\equnn
V_{i,j,k+1}^{(p)}
&=&V_{i,j,k}^{(c,*)} +\Delta t\left[r_1X_i(1-X_i/K)\DVXE_{i,j,k} \right.
\\
&+& \frac{1}{2}\sigma_1^2X_i^2\DDVXE_{i,j,k}
- \delta \VE_{i,j,k}
\\
&+&\Sigma_l f_l (\ZVE_{i,j,k,l}-\VE_{i,j,k})
\\
&+& r_2Y_j\DVYE_{i,j,k} 
+\frac{1}{2}\sigma_2^2Y_j^2\DDVYE_{i,j,k}
\\
&+& \left. \Sigma_l g_l (\QVE_{i,j,k,l}-\VE_{i,j,k}) + \SE_{i,j,k}\right],
\nqunn
\item {\bf Predictor Evaluation (Crank-Nicolson Midpoint):}
$$\VM_{i,j,k}^{(p)}\equiv 0.5(V^{(c,*)}_{i,j,k}+V^{(p)}_{i,j,k+1})
\simeq V_{i,j,k+0.5},$$
which are used to get predicted components of
$\DVXM$, $\DVYM$, $\DDVXM$, $\DDVYM$, $\ZVM$, $\QVM$, $\URM$, $UM$ \& $\SM$;
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 3.3.1:  More Computational Stochastic Dynamic Programming
%
\BCEN\UL{\Large\bf 3.3.1~~ More Computational Dynamic Programming:}\ECEN
%
~\\*[-6ex]
{\Large%BeginSize
\begin{itemize}
\setlength{\itemsep}{8ex}
\item {\bf $(L+1)$st Corrector Step:}
\equnn
V_{i,j,k+1}^{(c,L+1)}
&=&V_{i,j,k}^{(c,*)}+\Delta t\left[r_1x_i(1-x_i/K) \DVXM_{i,j,k}^{(c,L)}\right.
\\
&+&\frac{1}{2}\sigma_1^2x_i^2\DDVXM_{i,j,k}^{(c,L)}
-\delta \VM_{i,j,k}^{(c,L)}
\\
&+&\Sigma_l f_l\left(\ZVM_{i,j,k,l}^{(c,L)}-\VM_{i,j,k}^{(c,L)}\right)
\\
&+&r_2y_j\DVYM_{i,j,k}^{(c,L)} 
+\frac{1}{2}\sigma_2^2y_j^2\DDVYM_{i,j,k}^{(c,L)}
\\
&+& \left. \Sigma_l g_l\left(\QVM_{i,j,k,l}^{(c,L)}-\VM_{i,j,k}^{(c,L)}\right) 
+ \SM_{i,j,k}^{(c,L)}\right],
\nqunn
for $L+1=1$ to $L^*$, where $\VM_{i,j,k}^{(c,0)}=\VM_{i,j,k}^{(p)}$;
\item {\bf Corrector Evaluation:}
$$\VM_{i,j,k}^{(c,L)}=0.5(V_{i,j,k}^{(c,*)}+V_{i,j,k+1}^{(c,L)}),$$
which are used to get corrected components of
$\DVXM$, $\DVYM$, $\DDVXM$, $\DDVYM$, $\ZVM$, $\QVM$, $\URM$, $UM$ \& $\SM$;
\end{itemize}
}%EndLargeSize
%
\newpage
%
%  Slide 3.3.2:  More Computational Stochastic Dynamic Programming
%
\BCEN\UL{\Large\bf 3.3.2~~ More Computational Dynamic Programming:}\ECEN
%
~\\*[-6ex]
{\Large%BeginSize
\begin{itemize}
\setlength{\itemsep}{8ex}
\item {\bf Corrector Relative Stopping Criterion:}
$$
|V_{i,j,k+1}^{(c,L+1)}-V_{i,j,k+1}^{(c,L)}|<\varepsilon|V_{i,j,k+1}^{(c,L)}|
$$ 
for all $\{i,j\}$ at fixed $k+1$ and some relative tolerance $\varepsilon>0$ 
with $L+1 = L_k\HS$ and $V_{i,j,k}^{(c,*)}=V_{i,j,k}^{(c,L_k^*)}$. 

\item {\bf Mean Temporal-Spatial Mesh Corrector\\Convergence Condition:}
$$\Delta t<\frac{1}{2}
\frac{1}{\sqrt{(\overline{2A/(\Delta\xi)^2})^2+(\overline{B/\Delta\xi})^2}},$$
where for example $\overline{B/\Delta\xi}=0.5(B_x/\Delta x+B_y/\Delta y)$
represents some mean reciprocal of state meshes weighted by respective
linear comparison coefficients $B_x$ and $B_y$.  This condition is a
combined Parabolic-Hyperbolic (CFL) Mesh Ratio Condition.
\end{itemize}
}%EndLargeSize
%
\newpage
%
% Slide 4.1: Figure 4.
%
\BCEN\UL{\LARGE\bf Part 4.~~ Numerical Results}\ECEN
~\\
\BCEN\UL{\Large\bf Figure 4.}\ECEN
%
~\\\hspace*{0.2in}
\parbox[t]{6in}{\large\bf Optimal current value, $V^*(K,y,t)$, in millions of 
USdollars versus scaled price factor, $y\cdot\exp(-r_2\cdot T)$, 
with time parameter $t = 0.0$, 2.0, 4.0, 6.0, 8.0, 10.0 for each curve ordered
from top to bottom, respectively, 
and with population size fixed at carrying capacity $x = K$.
}
%
%\begin{figure}[htbp]
\BCEN
~\\*[10pt]
\hspace*{-60pt}
\epsfysize400pt
\epsffile{Fig4YVTBig.eps}
\ECEN
%\end{figure}
%
\newpage
%
% Slide 4.2: Figure 5.
%
\BCEN\UL{\Large\bf Figure 5.}\ECEN
%
~\\\hspace*{0.2in}
\parbox[t]{6in}{\large\bf Optimal feedback effort, $q\cdot E^*/r_1(K,y,t)$, 
in dimensionless form 
versus scaled price inflation factor, $y\cdot\exp(-r_2\cdot T)$,
with time parameter covering $t = 0.0$, 2.0, 4.0, 6.0, 8.0, 10.0 for each curve
closely spaced from bottom to top, respectively,
and with population size fixed at carrying capacity $x = K$.
}
%
%\begin{figure}[htbp]
\BCEN
~\\*[10pt]
\hspace*{-60pt}
\epsfysize400pt
\epsffile{Fig5YETBig.eps}
\ECEN
%\end{figure}
%
\newpage
%
% Slide 4.6: Figure 6.
%
\BCEN\UL{\Large\bf Figure 6.}\ECEN
%
~\\\hspace*{0.2in}
\parbox[t]{6in}{\large\bf Sensitivity of optimal current value, $V^*(K,y,0)$, 
to inflation price factor rate $r_2$, with curves parameterized by 
scaled inflation price factor, $y\cdot\exp(-r_2\cdot T)$, 
ranging from 1.0 at top to 0.2 at bottom in steps of 0.2,
with time fixed at initial value $t=0.0$, 
and with population size fixed carrying capacity $x = K$.
}
%
%\begin{figure}[htbp]
\BCEN
~\\*[10pt]
\hspace*{-60pt}
\epsfysize400pt
\epsffile{Fig6R2VYBig.eps}
\ECEN
%\end{figure}
%
\newpage
%
% Slide 5: Conclusions Page
%
\BCEN\UL{\LARGE\bf Part 5.~~ Conclusions}\ECEN
~\\*[2ex]
%
{\Large\bf%BeginSize
\begin{itemize}
\setlength{\itemsep}{6ex}
\item Examined Effects of Random Price Fluctuations\\
on Optimal Policy and Optimal Return.
\item Successfully Applied Computational Stochastic\\Dynamic Programming.
\item Random Price Jumps Strongly Affect\\Optimal Return.
\item Random Price Jumps have Less Impact on\\Optimal Policy.
\item Random Price Jumps needed as Serious\\
Consideration as Hazardous Environments and other Environmental Effects.
\end{itemize}
}%EndLargeSize
% 
\end{document}