Applied Mathematics, 2011, 2, 912-913
doi:10.4236/am.2011.27123 Published Online July 2011 (http://www.SciRP.org/journal/am)
An Alternative Method of Stochastic Optimization: The
Portfolio Model
Moawia Alghalith
University of the West India, Saint Augustine, Trinidad and Tobago
Received May 19, 2011; revised May 26, 2011; accep te d Ma y 29, 2011
Abstract
We provide a new simple approach to stochastic dynamic optimization. In doing so, we derive the existing
(standard) results using a far simpler technique than the duality and the variational methods.
Keywords: Stochastic Optimization, Investment, Portfolio
1. Introduction
Previous studies in stochastic optimization relied on the
duality approach and/or variational techniques such as
using the Feynman Kac formula and the Hamilton-
Jacobi-Bellman partial differential equations. Examples
include [1-3], among many others.
In this paper, we offer a new simple approach to
stochastic dynamic optimization. That is, we prove the
previous results using a simpler method than the duality
or the Hamilton-Jacobi-Bellman partial differential
equations methods. We apply our method to the standard
investment model. Our approach is based on dividing the
time horizon into sub-horizons and applying Stein’s
lemma.
2. The Portfolio M odel
We use the standard investment model (see, for example,
[3], among many others). Similar to previous models, we
consider a risky asset and a risk-free asset. The risk-free
asset price process is given by where
is the rate of return.
0=
T
rds
t
Se

,


2
b
rCR
The dynamics of the risky asset price are given by

d=dd ,
ss s
SS s W   (1)
where

and

are the deterministic rate of return
and the volatility, respectively, and
s
W is a standard
Brownian motion.
The wealth process is given by


ππ
=πdπd,
TT
Tssssss
tt
s
X
xrXr sW   

(2)
where
x
is the initial wealth,

is the risky πstsT
portfolio process with . The trading strategy
2
πd<
T
s
t
Es



s
x
  is admissible (that is, ).
π0
s
X
The investor’s objective is to maximize the expected
utility of the terminal wealth



π
,=Sup= π,
Tt
t
Vtx EUXEU
 





(3)
where


.V is the (smooth) value function,


.U is
continuous, bounded and strictly concave utility function,
and is the filtration.

We rewrite (2) as






 
ππ
π
π
=ππ
πdπd
πdπd;
<<,,, ,, .
Tuuuuuuuu
TT
s
ssss ss
tt
TT
s
s
ssss ss
tt
XxrX rW
rXr sW
rXr sW
tututT tTutT
s





 
 
 
 


(4)
Substituting the above equation into (3) and dif-
ferentiating with respect to (and setting the
derivative equal to zero) yields
πu





..
uut uut
rEUEU W 
 




=0. (5)
By Stein’s lemma



 

.=, .
=..
utuut
uu t
EUWCovX WEU
EU 







 




(6)

M. ALGHALITH 913
Substituting this into (5) yields
 

 

2 2
..
π==
. .
uut uux
uuxx
ut
rEU rV
V EU

 

 






.
(7)
This solution can be generalized to any point on time s
 

2
.
π=
.
ssx
ssxx
rV
V



.
(8)
This is exactly the solution obtained by the previous
literature, but its derivation is far simpler. Furthermore,
this approach can be applied to many other stochastic
models.
3. References
[1] M. Alghalith, “A New Stochastic Factor Model: General
Explicit Solutions,” Applied Mathematics Letters, Vol. 22,
No. 12, 2009, pp. 1852-1854.
doi:10.1016/j.aml.2009.07.011
[2] W. Fleming, “Some Optimal Investment, Production and
Consumption Models,” Contemporary Mathematics: Ma-
thematics of Finance AMS-IMS-SIAM Proceedings, 2004,
pp. 115-124.
[3] F. Focardi and F. Fabozzi, “The Mathematics of Financial
Modeling and Investment Management,” Wiley, Hoboken,
2004.