Advances in Pure Mathematics, 2011, 1, 81-83
doi:10.4236/apm.2011.13018 Published Online May 2011 (http://www.scirp.org/journal/apm)
Portfolio Optimization without the
Self-Financing Assumption
Moawia Alghalith
Department of Economics, University of the West Indies, St. Augustine, Trinidad and Tobago
Received January 13, 2011; revised February 28, 2011; accepted March 5, 2011
Abstract
In this paper, we relax the assumption of a self-financing strategy in the dynamic investment models. In so
doing we provide smooth solutions and constrained viscosity solutions.
Keywords: Portfolio, Investment, Stochastic, Viscosity Solutions, Self Financing
1. Introduction
The literature on dynamic portfolio optimization is vast.
However, previous literature on dynamic investment
relied on the assumption of a self-financing strategy; that
is, the investor cannot add or withdraw funds during the
trading horizon. Examples include [1], [2], [3] and [4]
among many others. However, this assumption is
somewhat restrictive and sometimes unrealistic.
Moreover, even with the assumption of a self-financ-
ing strategy, the previous literature usually provided
explicit solutions under the assumption of a logarithmic
or power utility function. Therefore, the assumption of a
self- financing strategy did not offer a significant
simplification of the solutions. Therefore, the
self-financing assumption needs to be relaxed.
Consequently, the goal of this paper is to relax the
assumption of self-financing strategies. In this paper, we
show that the assumption of a self-financing strategy can
be relaxed without a significant complication of the
optimal solutions. In so doing, we present a
stochastic-fac- tor incomplete-markets investment model
and provide both smooth solutions and constrained
viscosity solutions.
2. The Model
We consider an investment model, which includes a
risky asset, a risk-free asset and a random external
economic factor (see, for example, [5]). We use a
three-dimensional standard Brownian motion

1s 2s
,,WW

3s ,stsT
WF on the probability space

Ω,,
s
F
P, where

0
s
s
T
F is the augmentation of filtration. The risk-free
asset price process is

d
0,
T
s
t
rY s
Se

 where


s
rY


2
b
CRä is the rate of return and
s
Y is the stochastic
economic factor.
The dynamics of the risky asset price are given by




11
ddd,
ssss s
SSμYsσYW (1)
where


s
μY and


1
s
σY are the rate of return and the
volatility, respectively. The economic factor process is
given by


22
ddd,,
ss sst
YbYsσYWYy

 (2)
where


2
s
σY is its volatility and



1
s
bYC Rä .
The amount of money added to or withdrawn from the
investment at time
s
is denoted by Φ,
s
and its
dynamics are given by


33
dΦdd,
s
sss
aYs σYW (3)
where


3
s
σY is its volatility and



1.
ss
aY bYCRä
Thus the wealth process is given by
 
  


33
11
dd
d
d,
TT
Ts ss
tt
T
sss ss
t
T
ss s
t
XxaYsσYW
rY XμYrYπ
s
σYW



 





(4)
where
x
is the initial wealth,

,
ss
tsT
πF is the port-
folio process with

22
1d
T
ss
t
EσYπs

.
The investor’s objective is to maximize the expected

M. ALGHALITH 82
utility of the terminal wealth


,, sup,
t
Tt
VtxyEuX F



 (5)
where
()
.V is the value function,

.u is a
differentiable, bounded and concave utility function.
Under regularity conditions, the value function is
differentiable and thus satisfies the Hamiltonian-Jacobi-
Bellman PDE






 
  

 
  
22
22323 3
22
11313
12 12
11
22
sup
1
2
0,
t
txy
yyxy xx
tx
ttxx
txy
VryxayVbyV
σyV ρσ yσyV σ
y
V
yryV
yyyV
yyV


 
 
  

 








,, ,VTxyux (6)
where ij
ρ
is the correlation coefficient between the
Brownian motions. Hence, the optimal solution is
    

 
12 12
2
1
1
13 13.
x
xy
t
xx
μyryVρσ yσyV
πσyV
ρσ yσy





(7)
Similar to the previous literature, an explicit solution can
be obtained for specific forms of utility such as a
logarithmic utility function.
3. Viscosity Solutions
We can apply the constrained viscosity solutions to (6),
given the HJB is degenerate elliptic and monotone
increasing in V (see, for example, [6]).
Consider this HJB
 

 
 
,, ,0,Ω,
,Ω,
xxx
HxVxV xVxx
Vxgx x


ä
ä
(8)
where Ω is a bounded open set.
Definition 1 A continuous function

Vx is a
viscosity subsolution of (6) if







,,,0, ,
,
H
xV xPXPDV x
XJVx x




ä
滗 (9)
A continuous function

Vx is a viscosity
supersolution of (8) if



 

,,,0, ,
,,
H
xV xPXPDV x
XJVxx




ä
滗 (10)
where
  
,
:limsup0 ,
yx
VyVxPy x
DV xPyx



 









(11)
  
,
: liminf0,
yx
VyVxPy x
DV xPyx














(12)
are the super-differential and sub-differential,
respectively; and





 

2,:
1
,,
2
lim sup
0,
yx
JVx PX
Vy VxPyxXyxyx
yx




 


(13)





 

2
1
2
,:
,,
lim inf
0,
yx
JVx PX
VyVxPy xXy xy x
yx




 


(14)
are the superject and subject, respectively. A function


Vx is a viscosity solution if it is both a viscosity
subsolution and a viscosity supersolution.
Proposition 1


Vx is the unique constrained
viscosity solution of (6).
Proof Let


ΩVC

ä and let

s
V and


iV be
the upper and lower semicontinuous envelopes of V,
respectively, where






12
sup :,
s
Vuxuuu






12
inf :,iVu xuuu
where u1 and u2 are sub-solution and super-solution,
respectively.
Thus




ΩsV USCä and



ΩiV LSCä are a
viscosity subsolution and supersolution, respectively. At
the boundary we have




,Vx sViV
by the comparison principle


in Ω.sV iV
By definition




s
ViV and

M. ALGHALITH 83
thus

in ΩVx sViV
is the unique viscosity solution.
4. 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] N. Castaneda-Leyva and D. Hernandez-Hernandez, “Op-
timal Consumption-Investment Problems in Incomplete
Markets with Random Coefficients,” Proceedings of the
44th IEEE Conference on Decision and Control, and the
European Control Conference 2005, Sevilla, 12-15
December 2005, pp. 6650-6655.
[3] J. Cvitanic and F. Zapatero, “Introduction to the
Economics and Mathematics of Financial Markets,” MIT
Press, Cambridge, 2004.
[4] W. Fleming, “Some Optimal Investment, Production and
Consumption Models,” Contemporary Mathematics, Vol.
351, 2004, pp. 115-123.
[5] F. Focardi and F. Fabozzi, “The Mathematics of Financial
Modeling and Investment Management,” Wiley E-Series,
2004.
[6] F. Minani, “Hausdorff Continuous Viscosity Solutions to
Hamilton-Jacobi Equations and their Numerical Analy-
sis,” Unpublished Ph.D. Thesis, University of Pretoria,
Pretoria, 2007.