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\begin{document}
\begin{center}\textbf{{\Large 
Computational Methods for Portfolio and Consumption 
\\*[0.6ex] 
Policy Optimization in Log-Normal Diffusion, 
\\*[1ex] 
Log-Uniform Jump Environments.
}\\*[2ex]
\parbox[c]{2.6in}{\centering Floyd B. Hanson\\
{\small Laboratory for Advanced Computing\\
University of Illinois at Chicago\\
851 Morgan St.; M/C 249\\
Chicago, IL  60607-7045, USA\\
hanson@math.uic.edu}}
and 
\parbox[c]{2.6in}{\centering J. J. Westman\\
{\small Department of Mathematics\\
University of California \\
Box 951555\\
Los Angeles, CA  90095-1555, USA\\
jwestman@math.ucla.edu}}%endauthors
}
\end{center}

\begin{abstract}
%
Computational methods for a jump-diffusion portfolio optimization
application using a log-uniform jump distribution are considered.
In contrast to the usual geometric Brownian motion problem based
upon two parameters, mean appreciation and diffusive volatility,
the jump-diffusion model will have at least five, since jump process
needs at least a rate, a mean and a variance, depending on the
jump-amplitude distribution.  As the number number of parameters
increases, the computational complexity of the problem of determining
the parameter set of the underlying model becomes greater.  In a
companion stochastic parameter estimation paper, real market data,
here a decade of log-returns for Standard and Poor's 500 index
closings, is used to fit the jump-diffusion parameters, with
constraints based on matching the data mean and variance to keep
the unconstrained parameter space to 3 dimensions.  A weighted
least squares method has been used.  The jump-diffusion theoretical
distribution and weights has been derived.  In this computational
paper, the computational features of a new multidimensional,
derivative-less global search method used in the companion paper
are discussed.  The main part of this paper is to discuss the
computational solution of an optimal portfolio and consumption
finance application with these more realistic parameter results.
The constant relative risk aversion (CRRA) canonical model is used
to reduce the high dimensionality of the PDE of stochastic dynamic
programming problem to something more reasonable.  Many computational
issues arise due to the jump process part of the model, since
several jump integrals arise which are not present in the pure
diffusion with drift model.  The log-uniformly distributed jumps
allow a wider range of portfolio policies than does previous work
with normally distributed jumps.
%
\end{abstract}
%

%
\section{Introduction}
\setcounter{equation}{0}
In portfolio optimization, large scale computations enter in at
least two ways.  The first is the estimation of realistic financial
parameters from appropriate large scale financial market data, such
that the parameter estimation does not suffer from over-fitting
problems.  The parameter estimation is facilitated by a robust,
global, multi-dimensional, optimization tool, which is under
development, and does not require derivatives.  This computational
tool is based upon the one-dimensional golden section search method.
The stochastic theory for this way is treated in the companion paper
\cite{MTNS02FMT}.

The second way large scale computations enter is in the stochastic
optimal control problem using computational dynamic programming
for general Markov processes in continuous time to compute the
portfolio and consumption of wealth optimization.  The jump process
leads to higher computational complexity since it adds global
dependence in the form of jump integrals to the local dependence
of the partial derivatives introduced by the diffusion process.
The computational complexity of PDE of dynamic programming already
suffers from the curse of dimensionality caused by the discretization
of the PDE state space.  Much of the computational complexity and 
large scale computations can
be reduced by employing canonical models such as Constant Relative
Risk-Aversion (CRRA) power utilities, for the utilities of terminal
wealth and instantaneous consumption.  The jump integral computations
can be systematically handled by our generalization of Gaussian
quadrature for general statistical distributions.

As an application to demonstrate these computational methods, we
treat a stochastic optimal control problem, constrained by the
stochastic dynamics of wealth and the investment objective is to
maximize the conditional, expected discounted utilities of terminal
wealth and instantaneous consumption.

%
\section{Optimal Portfolio and Consumption Problem}
%
Let S(t) be the price of a stock or mutual stock fund at time $t$ 
that satisfies the jump--diffusion stochastic differential equation 
(SDE),
%
\eq
\label{SDE}
dS(t) = S(t)\left[\mu_d dt + \sigma_d dZ(t) + J(Q)dP(t) \right]\;, 
~~ S(0) = S_0, ~~ S(t) > 0, 
\nq
%
where $\mu_d$ is the mean appreciation return rate, $\sigma_d$ is
the diffusive volatility, $dZ(t)$ is a one--dimensional mean--zero
differential diffusion process with variance $dt$, $J(Q)$ is a jump
amplitude depending on a random variable $Q$ with log--return mean
$\mu_j$ and variance $\sigma_j^2$, and $dP(t)$ is a standard
differential Poisson process with jump rate $\lambda$ with common
mean and variance of $\lambda {dt}$.  Here, we will assume that
the jump--diffusion parameters $\mu_d$, $\sigma_d$, $\mu_j$,
$\sigma_j$ and $\lambda$ are constants.  The stochastic processes
$dZ(t)$ and $dP(t)$ are Markov and pairwise independent.  The jump
amplitude process $J(Q)$, given a Poisson jump in time, is also
independently distributed.

Equation~(\ref{SDE}) can be transformed to the more convenient 
log-return form by an application of the stochastic calculus chain 
rule to find the logarithmic differential yielding,
%
\eq
\label{DLNS}
d[\ln(S(t))] &=& \mu_{ld}dt + \sigma_d dZ(t) + \ln(1+J(Q))dP(t)\;,
\nq
%
where the log--diffusion drift $\mu_{ld} \equiv \mu_d-\sigma_d^2/2$
corrects the diffusion drift by the diffusion coefficient.  Here, 
the random mark variable is chosen as the log-return jump amplitude,
i.e., $Q = \ln(1+J(Q))$, uniformly distributed with density
$\phi_Q(q) = \phi^{(u)}(q;Q_a,Q_b) = 1/(Q_b -Q_a)$ on $[Q_a,Q_b]$ where
$Q_a < 0 < Q_b$.  In a prior paper \cite{ACC02FM}, a normal mark
distribution was used with a fair amount of success, but the uniform
distribution appears to be more realistic since it has finite support.
Eq.~ (\ref{DLNS}) was used in our stochastic 
companion paper \cite{MTNS02FMT} to facilitate the parameter estimation.

In addition to a stock, the portfolio is hedged with a bond, which  is 
assumed to satisfy a deterministic exponential process 
%
\eq
\label{Bond}
dB(t) = r B(t) dt\;, ~~ B(0) = B_0\;.
\nq
%
with the bond price continuously compounded at a fixed rate of
interest, $r$.  Let $U_0(t)$ be the fraction of the instantaneous
change in the portfolio due to changes in the bond investment and 
$U_1(t)$ be the fraction due to changes in the stock investment,
such that $U_0(t) + U_1(t) = 1$.

The portfolio wealth process at time $t$ changes due to changes in 
the portfolio fraction depending on the relative change in portfolio 
prices less instantaneous consumption of wealth:
%
\eq
\label{Wealth}
dW(t) &=& 
W(t)\left[r dt + U_1(t)\left\{(\mu_d - r)dt \right.\right.
%\\\nonumber&&
+ \left.\left.\sigma_d dZ(t) + J(Q)dP(t) \right\}\right]
- C(t) dt\;,
\nq
%
where $C(t)$ is the instantaneous rate of consumption, assumed to
be non--negative as well as constrained relative to wealth, i.e.,
$0\leq C(t)\leq C_{\max}^{(0)}W(t)$ given $C_{\max}^{(0)}$, and 
$U_0(t)=1-U_1(t)$ has been eliminated by bond-stock fraction 
conservation.

The investor's objective is to maximize the conditional, expected 
current value of the discounted utility $\UC_f(w)$ of 
terminal wealth at the end of the investment terminal time $T$ and 
the discounted utility of instantaneous consumption, $\UC(C(t))$,
i.e.,
%
\eq
\label{Objective}%(
v^*(t,w)&=&\max_{\{u,c\}[t,T)}\left[E\left[e^{-\beta(T-t)}\UC_f(W(T))
%\right.\right.
%\nonumber\\&&
+
%\left.\left.
\left.\int_{t}^T e^{-\beta(\tau-t)}\UC(C(\tau))d\tau
\right|{\cal C}(t)\right]\right]\;,
\nq%]
%
conditioned on the state--control set ${\cal C}(t) = 
\{W(t)=w,U_1(t)=u,C(t)=c\}$, where ~$0\leq t < T$,~ $0 \leq c \leq
C_{\max}^{(0)}w$~ for non--negative consumption feasibility with
maximal relative limits $C_{\max}^{(0)}$, $w \geq 0$ for 
non--negative wealth
feasibility, and $\beta > 0$ is a fixed discount rate.  Thus, the
instantaneous consumption $c=C(t)$ and stock portfolio fraction
$u=U_1(t)$ serve as control variables, while the wealth $w=W(t)$
is the state variable.  
%Bellman's Principle of Optimality has the form,
%%
%\eq
%\label{PrinOpt}%((
%v^*(t,w)&=&\max_{\{u,c\}[t,t+dt)}\left[E_{[t,t+dt)}\left[\UC(c)dt
%\right.\right.
%\nonumber\\
%&&+\left.\left.(1-\beta dt)v^*(t+dt,w+dW(t))\right]\right]\;,
%\nq%]]
%%
%conditioned on set ${\cal C}$,
%for sufficiently small $dt$ when $0\leq t \leq T$. 
The objective (\ref{Objective}) is subject to the terminal wealth
condition $v^*(T,w)=\UC_f(w)$ and zero wealth absorbing boundary
condition to avoid the possibility of arbitrage \cite{Merton90},
%
\eq\label{ABC}
v^*(t,0^+)= \UC_f(0) e^{-\beta(T-t)} +\UC(0)(1-e^{-\beta(T-t)})/\beta
\nq
%
and assuming that the consumption must be zero when the wealth is zero.

Assuming the  $v^*(t,w)$ is continuously differentiable in $t$ 
and twice continuously differentiable in $w$ (see \cite{ACC02FM} for
more details), then the stochastic dynamic programming equation 
for Poisson jump versions follows from an application of the
principle of optimality and the stochastic calculus chain rule to the 
\eq
\label{SDPE} 
0 &=& v^*_t(t,w) - \beta v^*(t,w) + \UC(c^*)
%\nonumber\\&&
+\left[(r + (\mu_d-r)u^*)w-c^*\right] v^*_w(t,w)
\nonumber\\&& 
+\half \sigma_d^2 (u^*)^2 w^2 v^*_{ww}(t,w)
%\\\nonumber&& 
+\frac{\lambda}{Q_b-Q_a}\int_{Q_a}^{Q_b}\left[v^*(t,(1+J(q)u^*)w)\right.
%\\\nonumber&&
-\left.v^*(t,w)\right] dq\;,
\nq
%
where $u^* = u^*(t,w)\in [0,1]$ and $c^* = c^*(t,w) \in
[0,C_{\max}^{(0)}w]$ are the optimal controls if they exist, while
$v^*_w(t,w)$ and $v^*_{ww}(t,w)$ are the partial derivatives with
respect to wealth $w$ when $0\leq t < T$.

Non--negativity of wealth and the finite mark domain $[Q_a, Q_b]$
imply an additional consistency condition for the control, since
$(1+J(q)u^*)w$ is a wealth argument, $w \geq 0$ and $Q_a < 0 <
Q_b$, then $1+J(q)u \geq 0$ and consequently
%
\eq
\label{Uminmax}
U_{\min} \equiv -1/(\exp(Q_b)-1) \leq u 
\leq +1/(1-\exp(Q_a)) \equiv U_{\max},
\nq
%
defines the $u$ control domain for optimal objective (\ref{Objective}).
This result is in stark contrast to the $[0,1]$ control domain
restriction found in \cite{ACC02FM} in the case of a normally
distributed marks due to the infinite domain of the normal
distribution.  Here, recalling that the instantaneous bond fraction
is $u_0 = 1 - u$, since $U_{\min} < 0$ then $U_{\min} < u < 0$ and
$u_0 > 1$ mean that short-selling of stocks is permitted, while
since $U_{\max} > 1$ then $1 < u < U_{\max}$ and $u_0 < 0$ mean
that borrowing from bonds is permitted.

The utilities will be taken to be Constant Relative Risk--Aversion
(CRRA) power utilities \cite[Chapter 4-6]{Merton90} with the same
power for wealth and consumption:
%
\eq\label{UTIL}
\UC(x) = \UC_f(x) = x^{\gamma}/\gamma\;, ~~ x \geq 0, ~~ 0 <\gamma < 1\;. 
\nq
These power utilities for this optimal consumption and portfolio
problem lead to a canonical reduction in computational complexity
for the stochastic dynamic programming PDE problem to a simpler
ODE problem.  The optimal utility value function has a solution
separable in the wealth state variable and time,
%
\eq\label{Pow}
v^*(t,w) = \UC_f(w) v_0(t)\;,
\nq
%
where the wealth dependence is given explicitly and the time function
is to be determined.  Since $\UC_f(0^+) = \UC(0^+) = 0$ from (\ref{UTIL}), 
the absorbing boundary (\ref{ABC}), i.e., $v^*(t,0^+)$, is automatically 
satisfied.

Further, the regular (unconstrained) consumption control is a linear 
function of the wealth,
%
\eq\label{PowRegC}
c_{\reg}(t,w) \equiv w\cdot c_{\reg}^{(0)}(t) = w /v_0^{1/(1-\gamma)}(t)\;.
\nq
%
The regular stock fraction reduces to an implicitly defined 
wealth and time independent (essential for separability) control, 
$u_{\reg}(t,w)=u_{\reg}^{(0)}$,
%
\eq\label{PowRegU}
u_{\reg}^{(0)} 
&=& G(u_{\reg}^{(0)})
\equiv
\frac{1}{(1-\gamma)\sigma_d^2}
\left[
\mu_d-r +
\lambda I_1(u_{\reg}^{(0)})
\right]\!,
\\\nonumber
I_1(u) &\equiv& \frac{1}{Q_b-Q_a}\int_{Q_a}^{Q_b}
J(q) \left(1+J(q)u\right)^{\gamma-1}
dq,
\nq
%
where the uniform mark density on $[Q_a,Q_b]$ has been used.  Since
(\ref{PowRegU}) only defines $u_{\reg}^{(0)}$ implicitly in fixed
point form, $u_{\reg}^{(0)}$ must be found by iteration and a good
choice is Newton's method \cite{ACC02FM}, a fast and accurate fixed
point method.  The integrals are efficiently approximated by a
3--point Gauss--Statistics quadrature \cite{IJC00,ACC02FM} (a
general Gaussian quadrature that, with a standard log--uniform jump
density , is the Gauss--Legendre quadrature, but on $[0,1]$ with
different nodes $\{(5-\sqrt{15})/10,5/10,(5+\sqrt{15})/10\}$ and
weights $\{5/18,8/18,5/18\}$, having fifth degree polynomial
precision).  The optimal controls, when there are constraints, 
are given in the form: 
%
$c^*(t,w)/w =  c^*_0(t) = \max[\min[c_{\reg}^{(0)}(t),C_{\max}^{(0)}],0]$,
%
provided $w > 0$, and 
%
$u^*= \max[\min[u_{\reg}^{(0)},U_{\max}],U_{\min}]$,
%
independent of $w$ and $t$ along with $u_{\reg}^{(0)}$.
%

Substitution of the separable power solution (\ref{Pow})
and the regular controls in (\ref{PowRegC}-\ref{PowRegU}) 
into the stochastic dynamic programming equation (\ref{SDPE}),
leads to an ODE,
\eq\label{PowSDPE}
0=v_0^{\prime}(t) 
+ (1-\gamma)\left(g_1(u^*) v_0(t) 
+ g_2(t) v_{0}^{\frac{\gamma}{\gamma-1}}(t)\right)\;,
\nq
\eq
\nonumber
g_1(u)&\equiv&
\frac{1}{1-\gamma}\left[-\beta +\gamma \left(r+u(\mu_d-r)\right)
%\right.\\*[0.5ex]\nonumber &~& \left.
- \frac{\gamma(1-\gamma)}{2}\sigma_d^2 u^2
+\lambda (I_2(u)-1)\right] 
\\*[1ex]
\label{G2}
g_2(t) &\equiv& \frac{1}{1-\gamma}
\left[ \left(\frac{c^*_0(t)}{c_{\reg}^{(0)}(t)}\right)^{\gamma} 
-\gamma\left(\frac{c^*_0(t)}{c_{\reg}^{(0)}(t)}\right)\right],
\\*[0.5ex]
\nonumber
I_2(u)&\equiv& 
\frac{1}{Q_b-Q_a}\int_{Q_a}^{Q_b} (1+J(q)u)^{\gamma}dq\;,
\nq
%
for $0\leq t < T$.  The coupling of $v_0(t)$ to the time dependent
part of the consumption term $c_{\reg}^{(0)}(t)$ in $g_2(t)$
(\ref{G2}), and the relationship of $c_{\reg}^{(0)}(t)$ to $v_0(t)$
in (\ref{PowRegC}), means that the ODE (\ref{PowSDPE}) is actually
highly nonlinear and thus (\ref{PowSDPE}) is only of Bernoulli type
implicitly.  The implicit Bernoulli equation (\ref{PowSDPE}) can
be formerly transformed to a linear differential equation by using
%
$\theta(t) = v_0^{1/(1-\gamma)}(t)$,
% 
to obtain,
%
$0=\theta^{\prime}(t) + g_1(u^*)\theta(t) + g_2(t)$,
%
whose general solution can be inverse transformed to the particular
solution for the separated time function implicitly given by
%
\eq\label{LODEansT}
v_0(t)&=& \theta^{1-\gamma}(t) =
\left[e^{-g_1(u^*)(T-t)}\left(1
%\right.\right.\\\nonumber && ~~~ \left.\left.
+ \int_t^T g_2(\tau) e^{g_1(u^*)(T-\tau)} d\tau 
\right)\right]^{1-\gamma},
\nq
%
using the final condition $v_0(T)=1$.  Hence, both $v_0(t)$ and
$c_{\reg}^{(0)}(t)$ must be found by computational iteration (see
\cite{ACC02FM} for more details).  Assembling  solution for the
optimal value function is $v^*(t,w) = \UC_f(w)v_0(t)$, requires
only multiplication by the utility of wealth.

%
\section{Computational Finance Results}
%
In the companion stochastic parameter estimation paper \cite{MTNS02FMT}, 
the authors deduced
the following asymptotic result for the log-return in the form of a
log-normal diffusion, log-uniform jump process in the case when
the return-time $\Delta{t}$ is not an infinitesimal:

\begin{corollary}
%
As $\Delta{t} \rightarrow 0^+$, the log-uniform jump, log-normal diffusion 
density can be asymptotically approximated as
%
\eq
\label{AsDLNSDensity}
\phi_{\Delta\ln(S(t))}(x) &\sim& \phi\jdd(x)
\\\nonumber ~&\equiv&
(1-\lambda \Delta{t})
\phi^{(n)}(x;\mu_{ld}\Delta{t},\sigma_d^2\Delta{t})
+ \lambda \Delta{t}\frac{\Phi^{(n)}(x-Q_b,x-Q_a;\mu_{ld}\Delta{t},
\sigma_d^2\Delta{t})}{Q_b-Q_a}
\;,
\nq
neglecting $O((\Delta{t})^2)$.
%
\end{corollary}

Here $\phi^{(n)}(x;\mu_{ld}\Delta{t},\sigma_d^2\Delta{t})$ is the
normal density with mean $\mu_{ld}\Delta{t}$ and variance
$\sigma_d^2\Delta{t}$, while
$\Phi^{(n)}(x,y;\mu_{ld}\Delta{t},\sigma_d^2\Delta{t})$ is the
corresponding normal distribution on $[x,y]$.  In \cite{MTNS02FMT},
the histogram of the theoretical density (\ref{AsDLNSDensity}) was
fit to the histogram of empirical market data, namely the log
returns of the daily closings of the \SPI, $\Delta[\ln(SP_i)] \equiv
\ln(SP_{i+1}) - \ln(SP_i)$ for $i=1:2521$ values from 1992 to 2001.
This fitting was by the weighted least squares method in which two
of five jump-diffusion parameters were eliminated by matching the
theoretical and empirical mean $M_1 = \E[\Delta\ln(S(t))]$ and
variance $M_2 = \Var[\Delta\ln(S(t))]$.  

The minima was determined by our general multi-dimensional search
method \textit{Golden Super Finder (GSF)} \cite{GSF02}, that is a
generalization of the one-dimensional Golden Section Search (GSS)
method.  GSF is obviously slow due to the computational intensity,
but convenient for global optimization of complicated functions on
powerful workstations.  The GSF method has many modification over
the usual Golden Section Search method: (1) it is multi-dimensional,
(2) for $N$ dimensions or variables there are $4^N$ nodes using four
nodes per dimension (two golden interior nodes plus two endpoints),
(3) all nodes are tested for the current minimum, (4) if the current
minimum occurs at a purely golden interior point GSF proceeds with
a golden contraction like GSS, but if current minimum is at an end
point of any dimension then GSF shifts the hypercube golden template
by two nodes in that dimension in search of a better minimum, and
(5) a user can specify a bounding hypercube domain in which the
GSF hypercube search cannot leave, e.g., preserving non-negativity
of a variance parameter.

The final results for the jump-diffusion coefficients are
%
\eq
\mu_d &\simeq& 0.06386\;, ~~ \sigma_d^2 \simeq  0.005513\;,
%\nonumber\\
~~ \mu_j \simeq 0.0007624\;, ~~ \sigma_j^2 \simeq  0.0003679\;, 
~~ \lambda \simeq 55.46\;,
\nq
%
Here, the average time between trading days $\Delta{t} \simeq
0.003967$ was used since it was consistent with the assumption of
small $O((\Delta{t})^2)$ assumed in (\ref{AsDLNSDensity}).  Additional
economic rate parameters that will be used are the average rate
for Moody AAA bonds of $r \simeq 7.384\%$ for data in the period
1999-2001 \cite{Moody}, and a corresponding discount rate $\beta
\simeq 6.884\%$, 50 basis point smaller than the bond rate as is
typical with the Federal Market Rates.  Other parameters are $\gamma
= 0.50$ common terminal wealth and instant consumption CRRA utility
powers, $C_{\max}^{(0)} = 0.75$ upper bound on consumption relative
to wealth, and $T = 1$ trading year terminal time.

Fast and accurate approximations are very important in financial
engineering computations, so the computations were coded in MATLAB\TM
\cite{MATLAB} due to its facility for developing rapid prototype
solutions.

In Figure~\ref{Fig-vstw}, the numerical approximation to the optimal,
expected utility $v^*(t,w)$ is shown versus wealth $w$ in dollars
and $t$ in trading years.  When viewed for fixed time $t$, $v^*(t,w)$
follows the CRRA power utility template in wealth $w$, whereas for
fixed wealth $w$, $v^*(t,w)$ exhibits the dependence on the separated
time function $v_0(t)$ in time $t$.  In the finite difference
representation, the wealth $w$-intervals have been transformed into
constant intervals in the utility power $w^{\gamma}$ since as a
function of $w$ the utility is not differentiable as $w \rightarrow
0^+$.  Typically, the investor, given the terminal value $v*(T,w)
= \UC_f(w)$, is interested in the starting value $v^*(0,w)$ as as
function of wealth, but since the problem here is autonomous dynamic
programming also generates answers for lesser investment periods
$T_0 < T$ for which $v^*(T-T_0,w)$ would be the starting value.
The numerical result for the constant optimal stock fraction control
is $u^*(t,w) \simeq 3.271$, the same as the regular stock fraction
control $u_{\reg}(t,w) \simeq 3.271$ which is well within the
control domain $[U_{\min}, U_{\max}] \simeq [-28.93, +31.31]$ given
the estimated bounds on the marks, $Q_a \leq q \leq Q_b$.
%
\begin{figure}[ht]
\begin{center}{\includegraphics[width=3.25in]{mtns02fmbvstw.eps}}
\caption{\small
Optimal, expected utility numerical results $v^*(t,w)$ 
versus time $t$ and wealth $w$ for the CRRA power utility model.
\label{Fig-vstw}
}
\end{center}
\end{figure}
%

In Figure~\ref{Fig-cstw}, the computational approximation of the 
optimal consumption policy or control $c^*(t,w)$
is displayed versus the time $t$ in trading years and the wealth $w$
in dollars using the CRRA power utility model.  Recall that 
$c^*(t,w)$ is linear in the wealth $w$, but inversely proportional
to the square of the separated optimal value time function $v_0(t)$ to the 
power $1/(1-\gamma) = 2.00$ here when $\gamma = 0.5$. Hence,
lines constant in time are straight lines, while the dependence
in time $t$ for fixed wealth $w$ in $[0,100]$ is proportional to the 
reciprocal square of $v_0(t)$, i.e., $v_0^{-2}(t)$.

%
\begin{figure}[ht]
\begin{center}{\includegraphics[width=3.25in]{mtns02fmbcstw.eps}}
\caption{\small
Optimal consumption policy numerical results $c^*(t,w)$     
versus time $t$ and wealth $w$ for the CRRA power utility model.
\label{Fig-cstw}
}
\end{center}
\end{figure}
%

%
\section{Conclusions}
%
The log--normal diffusion, log--uniform jump
distribution has been demonstrated
on the canonical optimal portfolio and consumption control problem.
The log--uniform jump distribution has significant benefits over
the log--normal jump distribution used in our prior paper \cite{ACC02FM}
in that the stock fraction is not severely constrained on $[0,1]$ due
to the finite domain of the uniform distribution, allowing for
borrowing and short-selling, thus more realism.  This uniform distribution
is demonstrated on the optimal portfolio and consumption policy application,
yielding optimal stock fraction, consumption and expected discounted 
utility value.

Computational techniques are presented for handling the iterations
for implicitly defined solutions such as the optimal stock fraction
policy $u^*$ and the coupled optimal value separated time function
$v_0(t)$ and the optimal consumption policy $c^*$.  Also, the
Gauss--Statistics quadrature for handling the log--uniform jump
amplitude integral has been used, but this technique is also useful
for other jump distribution by using the appropriate standardized
distribution. The features multi-dimensional optimizer Golden Super Finder
\cite{GSF02} was used in a companion parameter estimation paper
\cite{MTNS02FMT} have also been discussed.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{.1in}
%
\noindent \textbf{Acknowledgement}:   Work supported in part by the
National Science Foundation Computational Program Mathematics Grant 
DMS--99--73231.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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