Applied Mathematics, 2011, 2, 433-435
doi:10.4236/am.2011.24054 Published Online April 2011 (http://www.SciRP.org/journal/am)
General Closed-Form Solutions to the Dynamic
Optimization Problem in Incomplete Markets
Moawia Alghalith
Economics Department, University of the West Indies (UWI), St. Augustine, Trinidad and Tobago
Received January 5, 2011; revised February 23, 2011; accepted February 25, 2011
Abstract
In this paper, we provide general closed-form solutions to the incomplete-market random-coefficient dy-
namic optimization problem without the restrictive assumption of exponential or HARA utility function.
Moreover, we explicitly express the optimal portfolio as a function of the optimal consumption and show the
impact of optimal consumption on the optimal portfolio.
Keywords: Stochastic, Incomplete Markets, Investment
1. Introduction
Dynamic optimization has been used extensively in the
economic and financial literature. Examples include in-
complete markets, stochastic volatility and random coef-
ficients models. The contemporary literatu re usually ado-
pts random coefficient models (the parameters of the mo-
del are dependent on a random external economic factor)
or non-tradable assets models. Examples include Bayrak-
tar and Young [1], Bayr aktar and Ludkovsky [2], Algha-
lith [3], Focardi and Fabozzi [4], and Flemin g [5].
In order to derive explicit solution to the optimization
problem, the previou s studies relied exclus ively on expo-
nential or HARA utility functions. Th is assumption is re-
strictive and sometimes unrealistic, since other common
and more appropriate functional forms exist.
In this paper, we relax the exponential or HARA util-
ity assumption. In doing so, we derive general closed
form solutions to the random-coefficient incomplete-
market dynamic optimization problem without imposing
restrictions on the functional form of u tility. Furthermore,
we explicitly derive a functional relationship between th e
optimal portfolio and optimal consumption and show the
impact of consumption on the optimal portfolio.
In Section 2, we present the theoretical model and the
results. In Section 3, we draw conclusions.
2. The Model
We consider a standard investment-consumption model,
which includes a risky asset, a risk-free asset and a ran-
dom external economic factor (see, for example, Fleming
[5] for background information on the investment-con-
sumption model). This implies a two-dimensional stan-
dard Brownian motion




12
,,
ss s
tsT
WW   on the pro-
bability space


,,
sP, where

stsT
, is the aug-
mentation of filtration. The risk-free asset price process
is

d
0
T
s
t
rY s
Se

, where

 
2
sb
rYC R is the rate of
return and
s
Y is the economic factor.
The risky asset price process is given by




1
ddd,
stss s
S SYtYW  
where


t
Y  and


t
Y  are the rate of return and the
volatility, respectively. The economic factor process is
given by


2
12
ddd1d,,
ss sst
YbYt WWYy 


where 1


is the correlation factor between the two
Brownian motions and

 
1
s
bYC R with bounded
derivative.
The wealth process is given by
  


,, 1
dd,
TT
cc
s
Tsssssssss
tt
bYCXxrY XYrYcsYW

   



M. ALGHALITH 434
where
x
is the initial wealth,

,tsT
ts


 is the port-
folio process and

,tsT
ts
c  is the consumption proc-
ess, with 2d
T
s
t
s  , d
T
s
t
cs and 0c. The trad-
ing strategy
 
,,
ss
cxy   is admissible (that is,
,0 c
s
X

).
The investor’s objective is to maximize the expected
utility of wealth and consumption



,
12
,
,,d| ,
tt
T
c
Tst
ct
VtxySupEu Xu cs










where


.V is the value function,


.u is a differenti-
able, bounded and strictly concave utility function.
The value function satisfies the Hamiltonian-Jacobi-
Bellman PDE
  
22 2
,
11 0,
22
t
txyyytxxttxtxyt
c
Vry xVgyVVSupyVyycVyVuc



  




,, ,VTxy ux
where
 


1
tttt
YYYrY  

. Differentiating
the above equation with respect πt and ct, respectively
and rearranging, we obtain the opti mal solu tions
 
1,
x
xy
t
xx
y
VV
yV


 
 (1)

2. tx
uc V


Lemma. We can obtain an exact fixed-coeffiecient
Taylor expansion of


2t
uc

.
Proof. Consider the following Taylor expansion
around a



 
 
222
2
2
1
,
2
tt
tt
ucuauac a
uac aRc


 

where


t
R
c is the remainder. Our objective is to mi-
nimize


t
Rc






2
222 2
1
min .
2
t
ttt
cRxucuauacauac a
















The solution yields
    
222
ˆˆ ˆ0,
tt t
Rcucuauaca

 
and thus
   
222
ˆˆ
.
tt
ucuauac a

 
Now since ˆt
c depends on the value of a, choose a
specific value of aa such that ˆtt
cc

; hence





222 . tt
ucuauac a
 
 
□
The above equation can be rewritten as

212
,
tt
ucb bc
 
 (2)
where i
b is a constant. Using the same procedure we
obtain the following exact expansion of

.
x
V
 
12
34 5
.()()
.
x xxxxy
VV V xVy
bbxby

   
  (3)
Since


2., tx
uc V

 we obtain

64 5 2
.
t
cbbxbyb

Substituting (2) and (3) into (1) yields
 

12 5
1
4
.
t
t
yb bcb
yb




 
 (4)
This is a general explicit formula that holds for any
utility function. Moreover, this formula allows us to de-
termine the impact of consumption on the portfolio. To
show this
 



2 11
2
4
0
t
xx
t
ua
b
yy yy
bV c
 





 

by the concavity of u.
This approach is empirically v ery convenien t since the
parameters in (4) b1, b2, b4 and b5 can be easily estimated
using a nonlinear regression based on historical data.
3. Conclusion
This approach is superior to the existing approaches in
several ways. First, we can obtain closed-form solutions
without the assumption of exponential or HARA utility
function. Secondly, we can easily obtain comparative
statics results. For example, there is a trade-off between
consumption and investment. Moreover, our approach
can be easily utilized by future empirical studies.
4. References
[1] E. Bayraktar and V. Young, “Optimal Investment Strate-

M. ALGHALITH 435
gy to Minimize Occupation Time,” Annals of Operations
Research, Vol. 176, No. 1, 2010, pp. 389-408.
doi:10.1007/s10479-008-0467-2
[2] E. Bayraktar and M. Ludkovski, “Inventory Management
with Partially Observed Nonstationary Demand,” Annals
of Operations Research, Vol. 176, No. 1, 2010, pp. 7-39.
doi:10.1007/s10479-009-0513-8
[3] 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
[4] F. Focardi and F. Fabozzi , “The Mathematics of Financial
Modeling and Investment Management,” Wiley E-Series,
2004.
[5] W. Fleming, “Some Optimal Investment, Production and
Consumption Models,” Contemporary Mathematics, Vol.
351, 2004, pp 115-124.