Applied Mathematics, 2011, 2, 912-913

doi:10.4236/am.2011.27123 Published Online July 2011 (http://www.SciRP.org/journal/am)

Copyright © 2011 SciRes. AM

An Alternative Method of Stochastic Optimization: The

Portfolio Model

Moawia Alghalith

University of the West India, Saint Augustine, Trinidad and Tobago

E-mail: malghalith@gmail.com

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

W is a standard

Brownian motion.

The wealth process is given by

ππ

=πdπd,

TT

Tssssss

tt

s

xrXr sW

(2)

where

is the initial wealth,

is the risky πstsT

portfolio process with . The trading strategy

2

πd<

T

s

t

Es

s

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

ssss ss

tt

TT

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)