•^LT UBRARfES - D€WEY
m'^
®
y'
OBVEY
HD28
.M414
no.
529^5
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
OPTIMAL CONSUMPTION AND PORTFOLIO RULES
WITH LOCAL SUBSTITUTION
by
Ayman Hindy
Stanford University
and
Chi-fu Huang
Massachusetts Institute of Technology
WP#3273-91-EFA
May 1991
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
OPTIMAL CONSUMPTION AND PORTFOLIO RULES
WITH LOCAL SUBSTITUTION
by
Ayman Hindy
Stanford University
and
Chi-fu Huang
Massachusetts Institute of Technology
WP#3273-91-EFA
May 1991
Optimal Consumption and Portfolio Policies
with an Infinite Horizon: Existence and Convergence'*
Chi-fu Huang^ and Henri Pages*
March 198S
Revised, January 1991
To appear
in Annals of Applied Probability
Abstract
We
provide sufficient conditions for the existence of a solution to a consumption and portfolio
problem
in
continuous time under uncertainty with an infinite horizon.
When
the price processes
optimal policies can be computed by solving a linear partial
also provide conditions under which the solution to an infinite horizon
for securities are diffusion processes,
differential equation.
problem
is
We
the limit of the solutions to finite horizon problems
infinity.
'The authors would like to thank conversations with Robert Merton.
'Sloan School of Management, Mas.saciiusetts Institute of Technology
•Bank of France, Centre de Recherche
when the horizon increases to
[Mix
LIBRARIES
JUN 2 8 1991
\
I
•-*
1
Summary
Introduction and
Introduction and
1
1
Summary
Recent advances in the understanding of dynamic asset markets have made available a
consumption and portfolio decision of an individual
tools to analyze the optimal
set of
new
in continuous time
under uncertainty; see Cox and Huang (1989, 1990), Karatzas, Lehoczky, and Shreve (1987), He and
Pearson (1989), Pages (1988), and Pliska (1986). There are several attractive features of these new
tools. First, the existence
can be analyzed with
set
problem of an optimal consumption and portfolio policy
much
an individual
for
ease while the admissible policies do not take their values in a compact
and the consumption must obey a positive constraint. Second, when
securities prices follow
a diffusion process, the optimal policies can be computed by solving a linear partial differential
equation in contrast to a highly nonlinear Bellman equation in dynamic programming. Third, in
some
be computed directly by evaluating some
situations, optimal policies can even
The aforementioned
economies with a
papers, however, address the optimal consumption and portfolio problem in
finite horizon.
The purpose
be brought to bear on the same problem
that, except for
some important
in
of this paper
is
to
economies with an
show how
same
this set of
infinite horizon.
technical departures, the existence
policy can be analyzed similarly with the
We
integrals.
new
tools can
Our conclusions
are
and computation of an optimal
attractive features as in finite horizon economies.
also study the convergence of optimal policies in finite horizon economies to those in infinite
horizon economies.
policies.
We
show that pointwise convergence always occurs
But we do not know whether
portfolio policies converge in a certain
this occurs for
norm
optimal portfolio
for
optimal consumption
policies.
However, optimal
involving taking expectations. Thus,
portfolio policies for finite horizon economies do converge at all pointwise, they
if
the optimal
must converge to
is
very
(1989, 1990) extensively, which the reader
may
the optimal policy in the infinite horizon economy. Note that this type of convergence result
difficult to get
The
by using dynamic programming.
analysis in this paper utilizes
want to consult.
Cox and Huang
Two other papers complement our study.
First,
Merton (1989) uses the Cox-Huang
technique to solve, in infinite horizon, the optimal consumption and portfolio policy in closed-form
for a class utility functions
when
asset prices
foUow a geometric Brownian motion. Second, Foldes
(1989), using a different but related technique, also analyzes the optimal consumption
problem with an
infinite horizon in
and
portfolio
a stochastic environment more general than ours. He however,
does not give explicit conditions for existence nor does he study convergence properties of optimal
policies.
The
rest of this
paper
is
and computation of optimal
organized as follows. Section 2 formulates the model. The existence
policies are analyzed in Section 3
and Section
4, respectively.
Section 5
2
Formulation
2
some commonly used
gives closed-form solutions for
utility functions.
Section 6 gives results on
convergence and Section 7 contains concluding remarks.
Formulation
2
We
will use
we
an iV-dimensional Brownian motion to model the evolution of exogenous uncertainty. Thus
take the "state space"
Q,
to be the space of continuous functions from [0,oo) to 3?^ equipped
The
with the topology of uniform convergence on compact subintervals of [0,oo).
distinguishable events
is
the Borel sigma-field of
the agent to be considered
under
P
is
an
=
Vw G
u){t),
standard Brownian motion^, where
iV- dimensional
oo)
denoted by
T
collection of
and the probability
It is
belief of
well-known that
the coordinate process
continuous function
[0,
Q.
the Wiener measure on {0,,J^) denoted by P.
w{u,t)
is
We
consider a securities market under uncertainty in continuous time with an infinite horizon.
u
£
Q..
Since a state of nature
and one learns the true
state of nature
is
fi,
ij{t) is
the value at time
t
of the Jf^-valued
a complete realization of tt; on the time interval
by observing
w
over time,
we model
the intertemporal
resolution of uncertainty by an increasing and right-continuous family of sub-sigma-field of
filtration
F =
{w{s);0 <
{J^t\i
<
s
t}.^
G 3?+}, where ft =
One can
n,>(.?^°
verify that
T
and T°
— ^txiTu
is
.7^
the smallest sigma-field generated by
that
is,
all
the distinguishable events are
generated by sample paths of w. As a standard Brownian motion starts from zero at time
with probability one, Tq contains only subsets of
All the processes to appear
T
i
=
that are probability one or zero.
wiU be adapted processes.
progressively measurable; see Stroock
or a
In our setup, adapted processes are
and Varadhan (1979,
exercises 1.5.6
and
1.5.11).
Since
progressively measurable processes are naturally adapted, thus the set of adapted processes
equivalent to the set of progressively measurable processes.
(progressive) process that has right-continuous paths
one so that f;[X(5)|.7^(]
on Ti.
= X{i)
for all 5
>
(,
For any integrable random variable
that X{t)
=
martingale
X
under
and has continuous paths with
where
i;[-|.Ft] is
Y
(1),.^,
on
A
the expectation under
P
P
P
here
a
probability
conditional
P), there exists a (P)-martingale
E[Y\Tt], P-a.s.; see Jacod and Shiryaev (1987, remark 1.37).
is
is
X
so
All the conditional
expectations to appear wiU be martingales.
'a standard Brownian motion is a Brownian motion that starts at zero with probability one.
this paper we will use weak relations. For example, positive means nonnegative,
^Throughout
nondecreasing,
etc.
increasing
means
Formulation
2
Remark
3
In the above setup, the probability space {il,J^, P)
1
This
does not satisfy the usual conditions*.
and Huang (1989, 1990). This departure
follow
not complete'^ and the filtration
F
a departure from the earlier literature such as
Cox
important for our purpose; otherwise. Proposition
1 to
is
not valid.
is
We
is
is
will use the following notation: if
i
is
a matrix, then
\x\
=
ftrace(a;x^)j
where ^ denotes
,
"transpose."
There are
n
=
0, 1, 2,
.
A'^
.
.
,
+
iV.
1
securities traded continuously
Security n
^
is
risky
and
is
on the
infinite
time horizon [0,oo) indexed by
represented by a process of right-continuous and
bounded variation sample paths Dn with Dn{t) representing cumulative dividends paid by
n from time
=
let S{t)
to time
Denote the ex-dividend price of security n
t.
{Si{t),...,SN{t))'^ and Dit)
=
(I)i(f),.
.
.
,D;v(0)^-
ex-dividends, assume without loss of generality that Dn{0)
also that
S+
D
is s-n
=
/
As these
n
for all
b
and
<t,
=
t
by 5„(<) and
securities are traded
1,2,..
.
,N.
We
assume
iV-vector process:
S{t)+D{t) = SiO)+ f bis)ds+ f ais)dw{s),
Jo
Jo
where
at time
security
respectively, are iV-vector
and
N
te^+,
(1)
x iV-matrix predictable processes^
satisfying, for
all n,
T
"\b{s)\ds
<
oo,
P-a.5., V<>0,
(2)
(T{s)\^ds
<
oo,
P-a.5., Vt>0,
(3)
i:
for
some sequence of optional times (T„) with T^
t
oo P-a.s., and the second integral of (1)
is
a
stochastic integral.^
We
assume that a{t)
riskless. Its price at
is a.s.
time
of
B{t),
t,
full
is
rank for
all t.
The
zero-th security, called the bond,
is
locally
expj/J r{s)ds}. One can view this security as a bank account
that pays an instantaneous interest rate r{t) at time
t.
So $1 invested at time
grows to B{t)
at
any subset of a probability zero set is an element of ^.
^, = A,>,^j for all t g S?+; (ii) it is
complete: the probability space (Q,^, P) is complete and /o contains all the P-measure zero sets.
'a process is predictable if it is measurable with respect to the predictable sigma-field, which is the sigma-field
'a
*A
probability space (fl,/", P)
filtration
F
is
complete
if
satisfies the usuaJ conditions if (i) it is right-continuous:
adapted processes with continuous paths.
depends on the completion of a probability space and a filtration
satisfying the usuai conditions; see, for example, Liptser and Shiryayev (1977, chapter 4). The definition of a stochastic
integral here is based on Jacod and Shiryaev (1987, chapter 1) and has all the usual properties. In particular, the
stochastic integral of a continuous local martingale is a continuous local martingale and Ito's formula is valid, where
we recall that a process
is a local martingale under P if there exists a sequence of optional times T„ t oo. P-a.s.,
generated by
all
'The usual
definition of a stochastic integral
X
so that the process
{X{T„ A
<);(
€ ??+}
is
a (uniformly integrable) martingale for
all
T„.
2
4
Formulation
time
For
t.
B
to be well-defined,
we assume
Now
initial
in c,
is
Wo >
0.
possibly
risky securities
utility of
P-
oo,
U.S.,
>
Vf
0.
(4)
consider an agent with a time-additive utility function for consumption, u{c^t) and an
wealth
and
<
\r{s)\ds
/
that the interest rate process r satisfies
Assume throughout
that u{c,
unbounded from below
at c
=
t) is
0.
continuous in
This agent wants to manage a portfolio of the
and the bond, and withdraw funds out of the
consumption over time. Our task here
is
concave and increasing
(c,<),
maximize
portfolio to
on the
to find conditions
his
expected
utility function
and on
the price processes to guarantee the existence of a solution to the agent's problem.
We
Ito's
refer to the price
system as the
A'^-vector of
normalized prices defined by S*{t)
=
S{t)/ B{t).
formula implies that
i'
*•'"
+
i'
W)'"^'''
The process on the
=
''°'
left-hand side
is
^ - »'•
"""bw"'' "' + i' §w ^'"'"-
^
called the gains process
and here
""*
expressed in units of the
it is
bond. Putting
Git)
= S'{t)+ f-^dDis),
Jo B{s)
one sees that the difference G{t) — 5(0) represents the accumulated capital gains and accumulated
=
dividends on risky assets in units of the bond, where we note that since 5(0)
A
and
trading strategy
6^{t) are the
investor at time
t
is
a iV
number
-|-
1-vector predictable process (a,^^)
of shares of the
The
before trading.
receipt of dividends
bond and
=
{a,{6^
,.
.
1,
.
5'*(0)
bond
5(0).
,0^)), where a{t)
of risky asset n, respectively,
investor's wealth in units of the
=
owned by the
at time
t
after the
is
Wit) =
For now, a consumption plan c
is
ait)
+
dit)'^iS'it)
a process that
respect to the product measure generated by
the consumption rate at time
t.
A
P
is
+ ADit)/Bit)).
positive (except
on a
set that is negligible
and Lebesgue measure on
trading strategy
is
with
5f+) with c(t) denoting
said to finance the consumption plan c
if
the
following intertemporal budget constraint holds:
+ Wit)
f 4r\ds
B(S)
Jo
= WiO) +
r m'^dGit),
P-
a.s.,
vt e JJ+,
(5)
Jo
provided that the stochastic integral on the right-hand side
well-defined.
is
that, in units of the bond, the value of the portfolio at time
t
is
equal to
accumulated capital gains or losses and minus accumulated withdrawals
for
This equation says
its initial
value plus
consumption. For the
Formulation
2
5
we introduce the
stochastic integral of (5) to be well-defined,
class
C{G)
of trading strategies {a, 9),
of which satisfies
the
'
9{s)^a{s)
ds
i
for a
<
P-
oo,
a.s.
Bis)
sequence of optional times Tn
1 oo, P-a.s.
For the agent's problem to be well-posed, however,
necessary that something cannot be
it is
created from nothing through trading using reasonable trading strategies. This
is
the subject to
which we now turn.
Up
to
strategies
now nothing was
we have
defined
said about the existence of arbitrage opportunities.
it is
well
known
In fact with the
that such arbitrage opportunities do exist, even in finite
and Kreps (1979). Were
time; see the doubling strategy of Harrison
and portfolio choice problem would not be well-posed. To
this the case, the
consumption
rule out arbitrage opportunities,
we
will
impose a regularity conditions on the parameters of the price system and a natural institutional
constraint that wealth cannot be strictly negative. This constraiint has been ansdyzed by Dybvig
and Huang (1989) and Harrison and Pliska (1981).
We make
the following assumption throughout our analysis:
Assumption
that \k(u,
t)\
1
<
Let K{t)
K
for all
This assumption
is
=
-a{t)~^{b{t)
t,
P-a.s.
m
1, it is
There exists a positive constant
r{t)S{t)).
Now
= exp
P
define a martingale under
y^ K{sfdw{s) -
easily verified that £^[^(t)]
i
=
K
<
oo such
models originally considered by Samuelson
in particular satisfied in the
(1965) and Merton (1971).
By Assumption
-
that
jf* \Kis)\^
is
ds^
almost surely strictly positive:
te^+.
Thus
1.
VA6jr„
Qt{A)= h{u,t)P{du)
Ja
is
The family
a probability measure equivalent to P.
"consistent" in that
Proposition
1
Q, equals Qt on
P
on Tt for
G{t)
is
where oo >
There exists a probability
to the restriction of
and thus
Tt,
all
t
Q
6 5?+
.
5
of probability measures
>
0.
=
We
[0,oo)}
is
have the following result.
that its restriction
Moreover, under
= S{0)+ [ ^dw'is),
Jo i>{s)
standard Brownian motion under Q.
>
on {^,T) such
a local martingale^ under Q, where w'{t)
See footnote 6 foi a definition
f
{Qut G
on Tt
is
equivalent
Q
te[0,<x>),
w{t)
-
jQK,{s)ds
is
an
N -dimensional
2
6
Formulation
Define a consistent family Qt on {Q,Tt) as above. Stroock (1987,
Proof.
there exists a unique
!Ft,
we have the
theorem
(see, e.g.,
Since
On
first
P
and
Q
Q
on
{il,J^) so that Qll't
assertion.
lemma
are equivalent on
the other hand,
P
and
Q may
for all
f
G
§?+• Since
Q
is
4.2)
shows that
equivalent to
P
on
and Girsanov's
assertion follows from the definition of k
The second
Stroock (1987,
= Qt
lemma
I
4.3)).
any
(fi, J^t) for
finite
t,
they are said to be locally equivalent.
be mutually singular on the
cr-field .F, as
shown
in the following
lemma.
Lemma
The measures
1
By
Proof.
the
such that P(^oo
=
Q
and
P
are mutually singular if
Radon-Nikodym theorem,
oo)
=
Q{^oo
<
oo)
=
P
and
Q
and
Q
\K{t)\'^
|y
if
and only
dt
<
if /q°°
The
a.lmost surely statements
use Almost surely with respect to both
and
Q
2 Suppose {a, 9) G C{G).
and
Proof.
is
the
From
same whether
Then
it is
the definition of
G
P
Q
and
1,
\
oo, Q-a.s.
variable ^oo
^*
=
lemma above
P and Q
horizon case. However, one can
in restriction to
J^t?
for all
an
(F,P) or
to
f
G
still
3?+.
ltd process
under both
P
(F,Q).
one has
there exists
K
<
hand
oo such that
side
|k|
is
<
well defined under P.
K
.
By
h{t)-r{t)S{t)
9{tY-
^(t)T -(*)«(')
B{t)
'^'^'^^'Kit)
Bit)
On
the definition of
have thus
B{t)
I
can no longer be applied indifferently
finite
relative to
fact
'^» Q-a.s., as desired.
the integral f^ d{s)^ dG{s) is
computed
necessary and
Q-a.s.,
,
constant, so by the
!F
is
it
shows that in
[1979]
|«(0P
on
Since {a,0) G C{G), the stochastic integral on the right
other hand by Assumption
oo
is
with respect to either probability, as they can be in the
Lemma
random
to be mutually singular,
Samuelson or Merton, the process «
are mutually singular.
=
|/c(<)p(ii
VAGJ^;
oo}),
But theorem 8.19 of Jacod
0.
are mutually singular
In the models of
P
For
7.1).
{^oo<oo} =
and so
J^
and
Jacod (1979, theorem
sufficient that
if
there exists a extended -valued
QiA) = E[UlA] + QiAn{^ =
see, e.g.,
and only
/c,
the
we
2
Formulation
and so
jb{t)-rit)S{t)
<
0{t)
5(0
Thus the Lebesgue
integral
and
is finite,
/q
9{s)^ dG{s)
assertion follows from Stroock (1987, 111.4.3).
We
straint
are ready to
is
in force.
first recall
B{t)
is
an Ito process under P. The rest of the
I
show that there are no arbitrage opportunities when the positive wealth con-
Our proof is a
direct generalization of
Dybvig and Huang (1989, theorem
2).
We
the following usual definition of an arbitrage opportunity.
Definition
1
An
consumption plan
We
e{t)^ait)
K
arbitrage opportunity
a strategy {a,0) G jO{G) with
is
W{0) =
that finances a
and nonzero.
c that is positive
need a technical lemma to proceed:
Lemma
3 Let c be a consumption plan. Then
c{t)
E'
f
dt
B{t)
Jo
r
= E
Jo
dt
B{t)
where E' denotes expectation under Q.
Proof.
For any
finite
t.
'
E'
see DeUacherie
on any
l
c{s)
ds
and Meyer (1982, VI. 57). Given that
under either
finite subinterval of [0,oo)
the above relation are positive and increase in
theorem we have the assertion.
Proposition 2 Let
^i^) ^
Proof.
0(
(5)
<ۤ?+. Then P-almost
1
are locally equivalent, c
positive
and Q. Thus the integrands on both
sides of
letting
t
—- oo, by
monotone convergence
=
and with
surely, c is identically zero.
we have
r -Er\
B(s)
Proposition
Q
is
Thus
t.
and
I
'^^
+ ^(0 = w{0) +
and the
fact that (a, 6)
a positive local martingale under
f e{tfdG{t),
v< e »+.
Jo
Jo
By
P
P
ds
B{s)
consumption plan financed by {a,0) G C{G) with W{0)
c be a
P-a.s. for all
From
I
Bis)
Q
since
G C{G), the right-hand side of the above relation
by the positive wealth constraint the left-hand
side
is
is
2
8
Formulation
positive.
It is
known
that a positive local martingale
is
a supermartingale; see,
e.g.,
lemma
1
of
Dybvig and Huang (1989).
By
c is
the fact that c
is
a positive process under
a positive process under
Q
E'
[Jo
on any
P
and
P
and
finite subinterval [0,t].
Bit/']
Q
are locally equivalent,
This implies that
we know
2
Formulation
where Ua+{a,t) denotes the right-hand partial derivative ofu(a,t) with respect
Note that the
to its first
argument.^
hypothesis of Assumption 2 implies impatience, while relation (6) requires
first
that impatience to be sufficiently strong. Relation (7) will later be used to show that the space of
admissible consumption plans to be specified
those a's so that u{a,
t)
>
Remark
-
t;(c)e~*'' with 6t
2 Ifu{c,t)
Pick some a
>
and
for all
invariant with respect to the choices of a
is
t.
define a finite
\a{A)
=
>
for
c is
>
t
0,
VA
\u{a,t)\ dt,
j
Lebesgue measure since by impatience
a consumption plan
all
P
L^{Q X
|u(a,f)|
>
admissible
VM
where
Assumption
satisfies
2.
Ua-
Note that X^
except possibly for one
t.
is
equivalent to
We
Fix p £ [l,oo).
if
of admissible consumption plans
lfi+,'PM,i'a),
it
G B{^+),
and \a by
<
r\c{t)nu{a,t)\dt
Jo
Then the space
then
measure by
and denote the product measure generated by
will say that
among
00.
the positive orthant of the space L^{i^a)
is
denotes the progressive sigma-field.^
We
will
=
denote the positive
orthant of L^iva) by X^(/^a)-
The
following
lemma shows
respect to choices of a
Lemma
t,
4 For any
=
Lli,.,)
Proof.
E
>
first
invariant with
a and
a'
such that u{a,t) >
and
u{a',t)
>
for
all
Ll{u,.).
We
\c{t)\P\u{a\t)\dt
Jo
The
is
0.
strictly positive scalars,
Let c G L^it^a)-
/
that the space of admissible consumption plans
J
can write
<E Uo/
term on the right-hand
the second term
foo
/
Jo
is finite,
then
c
+E
\c{t)\^\u{a,t)\dt
side of the inequality
G L^{i^a')-
Assume
1
r
<
\c{t)\''\uia',t)-uia,t)\dt
E
is
first
\cit)\P\uia',t)
/
-
uia,t)\dt
.
L^o
J
finite
that
by assumption.
a'
>
a.
We
If
we can show that
have
^oo
/
\c{t)fua+ia,t){a-a)dt
L^o
J
<
r
{a'-a)E
\c{t)\^K,\ui<^,t)\dt
<
00,
Jo
Right-hand derivatives of a concave function always exist.
'Note that a process is progressively measurable if and only
sigma- field.
if it is
measurable with respect to the progressive
Existence of
3
where the
first
An Optimal
10
Policy
inequality follows from the concavity of u
Next suppose that
a'
<
a.
Arguments
and the second inequality
follows from (7).
above prove the assertion by noting
identical to those
that
\u{a',t)
-
u{a,t)\
< u+{a',t){a -
Similar arguments prove that c G
Now we
a')
L^'ii^a')
< Kau{a',t){a -
a')
a').
i
implies c G L^ii^a)-
are ready to specify completely the agent's problem.
< Kau{a,t){a -
He wants
to solve the following
program:
E[f^
sup(a,e)e/:(G)
W(t)>0
W{0) = Q{0)B(0) + e{0)'^SiO)<WQ,
s.t.
c
is
is
will say that there exists
and
finite
We
c
G L^{i>a).
a solution to the program (8)
if
the value of the program, val (Wo),
attained by a consumption plan financed by an admissible trading strategy that
the positive wealth constraint.
satisfies
3
is
(8)
financed by (a,^)
and
We
u{c{t),t)dt]
Existence of
An
Optimal Policy
provide in this section sufficient conditions for the existence of a solution to (8).
follows
Cox and Huang
variational
(1990).
We
problem whose solution
first
is
Our technique
transform the dynamic maximization problem into a static
well-understood.
The
solution of the static problem
is
then
implemented with a dynamic trading strategy uncovered by a martingale representation theorem.
Consider the following static variational problem:
SUPc€LP^(.,)
s-t.
We
first
E[!^u{c{t),t)dt]
E[l-^-^dt
show that the dynamic program
(9)
< Wo.
(8) is equivalent to the static variational
program
of (9).
Proposition 3
Proof.
c is a feasible
consumption plan in (8)
Let c G L\{i>a) be financed by (a,^) G
if
C{G)
and only
if it is
one in
(9).
satisfying the positive wealth constraint
and with W(0) < Wq. From the proof of Proposition 2 we know that
=
f ^^ds^W{t)
B(sj
Jo
W{Q)-V
f e{t)'dG{t),
Jo
VOO,
P-a.s.
3
is
Existence of
An Optima]
Policy
11
a supermartingale under Q. Thus
E'
Because W(t)
>
and
c(t)
>
f
C(5)
< W{0)
iw/'^'^^'i
Vi G 5?+.
under P, by the local equivalence of
rt
E'
c{s)
P
and Q, we know
<W{0) V<G»+,
ds
r
Jo B{s)
and
r 3h
c(0
E'
where the monotone convergence theorem
Lemma
dt
is
used for the second inequality.
It
then follows from
3 that
dt
< W{0) < Wq.
B{t )
Jo
Thus
< W{0),
B{t )
Jo
c is feasible in (9).
Conversely, let c G L^ii'a) be feasible in (9).
Lemma
c(0
<Wo.
dt
'
[Jo
3 implies that
B{t J
Thus
f ||*.m^,e).
Since
all
theorem
P-local martingales have the representation property relative to
III. 4. 33),
and since
Q
and
P
w
(see
Jacod and Shiryaev,
are locally equivalent, aU Q-local martingales also have the
representation property relative to w' (op.
cit.,
theorem
111.5.24).
process p and a sequence of optional times (T„) with limTn
T
Q-
/"|P(^)|- ds < oo,
Hence there
exists
oo Q-a.s. so that for
all
an
A'^-vector
T^,
a.s.,
Jo
and
E'
Let 0{t)'^
=
r wM
B{s)
= ^"
[f
wM
B{3)
i'"*^'""--*^'
+
^
"-
a.s.
'
B{t)pit)^(Tit)-\
w(,) = E-
ds\Tt
[j;
>
Vt
P-
a.s.,
B{s)
and
ait)
= W{t) -
e{t)'^{S'{t)
+ AD{t)/B{t)).
(10)
3
An Optimal
Existence of
Since c
>
W{t) +
/*
is,
We
(a, 6) finances
c.
Given
1
Finally,
first
1.
for almost
b
all
t,
and
Q we
a.s.,
j oo,
P-a.s.
have
teU+.
and only
if it is
denote the value of
c is feasible in (8).
I
3.
a solution to (9).
(9).
Note that since
and
(8)
(9) are equivalent,
(9).
provide conditions under which val [Wq)
Proposition 4 Suppose
P
W{0) < Wq. Thus
easily seen that
it is
we can then concentrate on
will also use T;a/(Wo) to
We
are locally equivalent, T„
Jo
c is a solution to (8) if
this corollary
P
r ^(5)'^dG(5), P -
+ e{OfS{0) +
a(0)
record an immediate corollary of Proposition
Corollary
we
=
and
local equivalence of
rs(s)
Jo
That
4^d5
Q
and P-a.s. Since
by construction and the
{a, 6) G 'C(G). Also
Hence
>0,Q
under P, W{t)
12
Policy
is finite.
that
u{c,t) is
unbounded from above
e (0,1) such that for some a
>
in c
and
there exists Pi
>
0, /32
>
and
0,
u{c,t)<\u{a,t)\{Pi
+ :^^{c'-'>-l)),
a.e.t;
(11)
and
(e/5)-^Ka,oiei''/*(^a).
2.
and
that, if u{c, t) is
unbounded from below
at the origin
on a
(12)
set
A
of
t
with strictly positive
Lehesgue measure,
Then val (Wo)
Proof.
with
b
6
is finite.
We first
show that va/(Wo) <
(0, 1), there exists
Relation (12) ensures that
oo.
When
a solution c to (9) only
c
budget constraint and yield a
G
^^.(^'a)-
finite
the utility function
if
there exists 7
>
is |u(a,t)|(/3i
so that
For c to actually be a solution,
expected
utility.
For the former
that
c(t)m
dt
Jo
B{t)
<
00,
+ i^(c^"''-l))
it is
it
must
also satisfy the
necessary and sufficient
An Optimal
3
Existence of
for
then 7 can be chosen to exhaust the
Policy
13
initial
wealth. Putting p
=
p/(l -b)
> p and l/p+
1/g
=
1,
Holder's inequality implies
cimt)
[i:
J_
dt
Bit)
)"'-r(t)"*w°'')i-^'--<"
02,
E
.
f^^^>^
"
^ar^r(fS)
where the
last inequality follows (12)
utility is finite.
\u{a,t)\^^ d\a{t)
and Assumption
J
7
<
(Wo) < 00 by
Next we take two
Suppose
cases.
on a
set
A
of
t.
first
(13), k
is finite.
to verify that the expected
Bit)
.
00.
is
bounded from below
Otherwise, suppose that u
m
L
Let
=E
is
for a.e.
t.
Then
unbounded from below
dt
A Bit)
Thus
c(u;,0=
is
i-Wo <
that u(c,<)
is finite.
k
By
00,
(11).
val{Wo) > -00 and thus val (Wq)
at the origin
we have
fE
Uo
P2
=
c{t)'-'\uia,t)\dt
/
va/
2. Finally,
<
|u(o,!t)\dt
For this we note that
.Jo
Thus
/
Wo.
-^lnxA(w,0>0
^u,t
a feasible consumption plan. Thus
valiWo) >
I
uiWo/k,t)dt > -00,
where the inequality follows from Assumption
I
>
1,
then u(c,() < Wia,t)\Pi. Thus the expected utility
our remaining task
is
to give conditions under which val
Note that,
less
2.
in (11),
if 6
is
always strictly
than +00.
Now
Cox and Huang (1990) by
(Wq)
is
attained.
We
will utilize
rewriting (9) as follows:
suPceLP(./a)
E
s.t.
E
r
(14)
B^r©yi dUt)] < Wo.
3
Existence of
An Optimal
14
Policy
program and the space of
In (14), both the objective functional of the maximization
objects are defined on a measurable space
the framework of
Here
is
Theorem
Proof.
our
Cox and Huang
first
Under
1
The
(fi
^^,VM)
x
with a
finite
measure
i/a-
This
feasible
fits
into
(1990).
existence theorem:
the conditions of Proposition 4, there exists a solution to (14J-
Cox and Huang
assertion follows from
are locally equivalent.
The next theorem
(1990, theorem 4.1)
by noting that
Q
and
P
I
is
where the
for the case
utility function is
bounded from above by a multiple
of |u(a,<)|.
Theorem
1.
2 Suppose that
u{c,t)
<
K\u{a,t)\ for
a.e. t;
2.
iaB)-'\u{aJ)\eLP{i^.);
3.
and (13) holds when
u{c,t)
is
unbounded from below
(15)
at the origin
on a set
A
of
t
with strictly
positive Lebesgue measure.
Then there
Proof.
exists a solution to (14)-
By
the
first
hypothesis,
we know that
uicA)
<
'
;
,
;,
K^,
,,
"ic,
a.e.
t.
\u(a,t)\
Given (13) and (15), the assertion follows from a
theorem 4.2) by noting that
P
Before leaving this section,
and
we
Q
(trivial) generalization of
are locally equivalent.
two
give in the following
Cox and Huang
(1990,
I
corollaries sets of explicit conditions
on
the parameters of prices for (12), (13), and (15) to be valid.
Corollary 2 Suppose
exists
that there exists
oo>T>0, €>0
t
is valid.
f
<
oo so that
<
r{t)
<
f,
P-a.s. for a.e.
t
and
there
such that
_lnMM)l
Then (12)
<
When
(16) holds with
.^.,^
lb
6=1,
p
b
(15)
+p
_^^ V^>r.
is valid.
(16)
3
Existence of
Proof.
An Optimal
Policy
15
Note that
M)-'KM,|t"
E
a P martingale
E
=
exp
i-^
X exp
j^
1^
K{s)^dw{s)
+
(^^
-^J^ \K{s)\'ds\
Kh +
l)
r{s)ds
^J^
+
(l
+
f
) In \u{a,t)\
Given that
Fubini theorem then implies that (12)
Identical
always
Proof.
P
is
valid.
arguments proves the second assertion when
Corollary 3 Suppose
is
+p
26
t
valid.
Otherwise,
Suppose
A
that the set
if
that
first
of (13)
<
there exists
A
=
6
of finite Lebesgue measure.
is
r so that r
<
with unity expectation. Since B{t)
>
Then
r{t) P-a.s. for a.e.
We
of finite Lebesgue measure.
is
1.
note that ^
t,
is
if r{t)
>
then (13)
0,
(13)
is valid.
a martingale under
1,
<
1.
[\B{t)J\
Fubini theorem implies that (13)
Next suppose that
A
is
is
valid.
The hypothesis
of infinite measure.
^(0
of this corollary yields
< exp {—rt}
B{t)
Fubini theorem again implies that (13)
When
the utility function
interest rate does not
asymptotically.
we
I
not unbounded from below, for existence,
is
Otherwise, the agent
until
t
=
may
oo.
find
When
maintain a certain level of
may become — oo.
it
suffices that the
utility function significantly discounts the future
it
advantageous to keep accumulating
the utility function
will further require that the interest rate does not
infeeisible to
utility
valid.
go unbounded and the
and postpone consumption
origin,
is
is
unbounded from below
become too low
minimum consumption
his
so that
it
wealth
at the
may become
over time and thus the expected
4
Computation of Optimal
Computation
4
This section
is
Optimal
of
Policies
devoted to the computation of optimal
(1989) and thus
Huang
16
Policies
We
of optimal policies.
we
will
will
partial differential equation
be
show that
and
if
The main
brief.
if
result reported here
computed by taking
the inverse of the marginal utility function f{x~^,t)
easily seen that f{x~^,t)
assume throughout
f{x~^,t)
and (12) holds
if 6
origin, (13) holds.
or
Theorem
The
(10)
G
=
inf{c G J?+
this section that
and (15) holds
and there
/
satisfies
the optimal consumption.
strictly positive constants
if 6
>
Moreover,
it is
1.
if
u
is
Kf,
b,
unbounded from below
easily verified that conditions of either
bond
at the
Theorem
1
From
is
cis)
F
is
the present value of future optimal
can be computed by solving a partial differential equation
strategies are related to the derivatives of F.
Before proceeding, we record in the following proposition the
is
< x~^}, where Uc+
a growth condition:
Thus F{t) = W{t)B{t)
consumption. Under some conditions,
which
define
the value over time of future optimal consumption.
is
this value in units of the
and optimal trading
Then
the strict concavity of u{c,t)
some
Wit) = E'
if c is
c.
exists a solution to (14).
object of computation here
we know
Uc+{c,t)
for
Under these conditions,
2 hold
By
c.
:
concave in
continuous in x.
is
< Kf\u{a,t)\bxb
(0, 1)
a verification theorem
derivatives of the solution.
is strictly
denotes the right-hand partial derivative of u with respect to
We
is
Cox and
the solution satisfies certain conditions, then the optimal trading
For the purpose of this section, we will aissume that u{c,t)
c, it is
idea follows that of
there exists a solution to a second order linear parabolic
strategy in the form of feedback controls can be
in
The
policies.
first
order condition for optimaJity,
the corner stone for the construction of an optimal policy.
Proposition 5 Under conditions of
exists a X
>
this section, there exists a solution to (14) if
and only
if
there
so that
c(0 =
/(^,*)ei>.)
(17)
and
c{t)m
f
Jo
Proof.
follows
The oidy
if
dt
= Wo.
Bit)
part follows from Saddle-Point theorem and Rockafellar (1975).
from the definition of / and the concavity of u
in
c.
I
The
if
part
Computation
4
of
Optimal
Policies
17
For the purpose of computation, we specialize our model of securities prices as follows.
S
that
satisfies
the stochastic integrcJ equation:
S{t)
= S{0)+
f {b{S{3),s)-d{Sis),s))ds+ f (T{S{s),s)dwis),
^0
where d{S{t),
Assume
t) is
>
f
(18)
0,
Jo
the iV-vector dividend rate at time
Thus
further that r{t) can be written as r(5(f),f).
t
when
the risky asset prices are S{t).
K{t) can also
Assume
be written as K{S{t),t).
Next define a process
Z{t)
= Z{0)+ f
^0
some constant Z{0) >
for
{r{S{s),s)
\KiSis),s)\^)Z{s)ds
Using
0.
Ito's
lemma,
it is
f'
K{S{s),sf Z{s)dw{s),
(19)
easily verified that
(1989), (logZ(r)
rate from time
to time
that maximizes the expected continuously
Now
-
Jo
As pointed out by Cox and Huang
compounded growth
+
T
- logZ(0))/T
is
the realized continuously
of the growth-optimal portfolio - the portfolio
compounded growth
rate.
write (18) and (19) compactly together under Q:
(
Z(0 )
=
( Z(0)
]
+ J^^iS{s),Zis),s)ds + J^&iS{s),Z{s),s)dw%s),
(21)
where we note that
Assume throughout
this section that
(^
and a
satisfy a local Lipschitz
and a uniform
linear
growth
condition on any finite time interval [0,r].^° Thus there exists a unique solution to (21) and {S,Z)
has the strong Markov property under
'°The functions C
Af
>
all
<
0,
€
there
[0,
is
md
it
Q
and thus
is
satisfy a local Lipschitz condition
a strictly positive constant
K^
such that for
a diffusion process under Q.
on every
all
j/,
[0,
T\ with
z
€ 9?^ x
-
a{z,
(0,
T <
oo
oo) with
for every
if
\y\
<
M and
T >
|z|
<
and
M and
T]
|C(V, t)
These functions
satisfy a
strictly positive constant
-
C{z,
t)\
< K^\y -
zl
|,t(v, t)
t)\
< K^\y -
uniform growth condition on every [0, T] with T < oo
such that, for edl i G S?'" x (0,oo) and t g [0,T],
if
Kt
\C{x, 01
< Kt{1 +
|i|),
|*(z,
t)\
< Kt(1 +
|x|).
z\.
for every
T >
0,
there exists a
4
Computation of Optimal
following notation wiU be utilized:
The
D!"
for positive integers
its first
N
Here
=
mi, m2,
—=
-^
.
.
.
,
-;:
tun.
If
g
m= mi +
;
7:
JJ^ x [0,T]
:
arguments, the vector {dg/dyi,.
is
Theorem
1.
18
Policies
.
>-> J?
dg/dy^)^
,
is
...
+ m^
has partial derivatives with respect to
denoted by Dyg or ^y.
the main theorem of this section:
3 Suppose that
there exists a function
continuous for
m
<
2,
F
F
:
S^
(0,oo) x
is
X [0,oo)
-
DF^{y,t) and
3?+ such that
Ft{y,t) are
a solution to the partial differential equation
CF{Z, S, t) -
r{S, t)F{Z, S,
t)
+
Ft{Z, S,t)
+ f{Z-\t) =
with the boundary condition
^F(7(T\.S(T^.T)^
F{Z{T),S{T),T)
lim E'
(—»oo
where
C
is
B{T)
the differential generator of
Kj
and 7^ 50
for
that,
\F{y,t)\
2.
Then
and
there exists
the optimal
all
t
|yr^)
Zq such that F(Zo, 5(0),0)
00, that
is,
T >
0,
F
satisfies a
there exists a strictly positive
Vy G (0,oo) x
3ff^;
= Wq.
consumption and portfolio policies are
c{t)
=
e{t)
=[Fs{Z{t),S{t),t)
a{t)
Va-a.e.
f{Z{t)-'^,t)
x[b{Sit),t)
-
+
[a{S{t),t)a{Sit),t)^]-^
riS{t),t)Sit)]Zit)Fz{Z{t),Sit),t)]
=[F{Z{t),S{t),t)-6{tYS{t)\lB{t)
where we have put Z(0)
=
Using the growth condition on / and
equation,
Cox and Huang
(1989,
F{Z{t),S{t),t)
Bit)
itr(Fss<T<T'')
+
u^
-
u,
-
^^^^
a.e.
a.e.,
Zq.
Proof.
'CF =
T <
all
£[0,T], a
Af (1 +
<
0,
{Z,S) under Q;^^ and, for
uniform growth condition on every [0,T] with
constants
=
Theorem
T
- ^
ifzzZ'l/cp
+
I
Fsz<tk
F and the fact that F satisfies the partial differential
2.3) implies that, for every
+ Fs(rS -
/)
— —
FjZinSinT)
f {Z{s)-\s)
"5(ir~
t,
^
+ FzrZ.
W)
'
'
5
Let
A
Special Case
T
-+ oo
19
and use the boundary condition we
F{Z{t),S(t),t)
get
f{zi3r\s)
= E'
ds\J^t
i:
Bit)
Bis)
F gives
where we have used monotone convergence theorem. Thus
By
the hypothesis, Z(0)
=
—m—
F(Z(0),5(0),0)
c(<)
then by Proposition
= /(Z(i)"\i)
5, c is
theorem
5
A
By
a solution to (14).
is
dt
Bit)
^0
initial wealth.
If
we can show that
c
G L^iva)^
the growth condition on /, (12), and (15), one
a solution (14).
as described in (22) follows
is
time.
fizit)-\t)
= ^
exhausts the
quickly verifies that c G L^i^^a) and thus
that finances c
/(Z~\t) over
Zq, thus
^° =
This shows that
the value of
The
fact that the trading strategy
from identical arguments on Cox and Huang (1989,
I
2.2).
Special Case
We now
specialize the
market model developed
in the earlier sections
and consider the model with
constant coefficients examined by Merton (1971) and revisited recently by Karatzas, Lehoczky, and
Shreve (1987). In this case explicit formulas for the optimal consumption and portfolio policies
can be computed just as in Cox and Huang (1990). The method does not use stochastic control
and takes care
results with
way
in a natural
of the
non negative constraint on consumption.
two examples taken from the family of constant absolute
We
illustrate our
risk aversion
and
HARA
utility functions.
We
shall use the following specialization.
The
risky securities follow a geometric
Brownian
motion
Sit)
where b
is
an
=
+ j\ls{,)h-diSis),s))ds +
SiO)
n- vector of constants,
a an n x n
Given some
initial
data
k
Is(,)<Tdwis),
We
furthermore assume that r
for the constant vector
-<T~*(b
-
r).
z, define 4> as
<i>iz)
= E'
f
Z-'iO) =
e-^*fiZit)-\t)dt
z.
Jo
Assuming
uic,t)
=
t;(c)e
^' (cf.
4>iz)
remark
= E'y^°°
2),
t>0
non-singular matrix of constants and
diagonal matrix having 5'(<) in the (t,t)th position.
write r for an n- vector of r's and
J
we can write
alternatively
e-''gie^'Zit)-')dt
;
Z-'iO)
=
z,
is
Is{t)
an n x n
constant and
5
A
20
Special Case
where g
is
Since
the inverse of the time-independent marginal utility function Vc+{c).
it is
easily verified that
e^'Z(t)-^
F
one sees that the function
e^*Z(f)~^
is
= zex^{K^w'{t) +
|k|^. It follows
/•OO
where n stands
mean \ogz +
-r + g'^/2)t
{(i
=
/
'x-logz-{(3-r + g'^/2y
gyt
Q^i
^^^^
/
Q
Jo
-^e
2^'
^=
/3
-
-OO
OO
gy/2ir
ov27r
r
g'^/2
-1-
and a^
^0
=
2r
-1-
^^/p^,
X - log 2
I
^\/27r
dtdx
v2xt
( x-\ot
where we have put
4>{e^^z~^).
1
f+CO
J-oo
=
and variance
standard normal density function. This yields
for the
^(z)
+ |k|V2)0,
that
J-oo
Jo
r
of section 4 can be identified as F{z,t)
lognonnally distributed with
the square root of
-
(/?
z
\
l_
a£i
dt dx,
g'^t,
Under Q,
where g
is
5
A
Special Case
Example
21
Utility
1
functions of constant absolute risk aversion.
Let the time-independent utility function be
= --e-"',
vix)
where ^ >
is
we
the coefficient of absolute risk aversion. In this case,
9{x)
=
find
logx
and so
^(^g-uN
j
-{l/0){\ogz-u)
if
u>
log 2,
otherwise,
similarly for g{ze^). Straightforward computations
and
<i>{z)
and that
if
r
<
=
show that
T{2,n\ogz)
r-M
gaO^"^
gaOfj,
if
z
>
1
'
1
4>{z)
=
gaO^"^
logz
logz
gaOpL
gaOfj,
gamma
where 7 and F are the incomplete
7(a,x)
7
('-')-
7(2,/x'logz-i)
ga6^'i2
functions
=
Te-^^-i
dt
Jo
roo
dt.
It
is
F—
»
easily checked that
00 a^ 2
—
0.
The
F
is
Z{t)
<
el^*
=
isam
=
1
gaOii
Z{t)
>
2
^'>
HARA
utility
1
,/3*i'«
+ 71^(1 -^'"'^(O-'') (aa^)(b-r)
gaOfj,
el^K
Example
= ^^(^)''('''''')(^ -
and by
At
if
F —
vector of optimal dollar amounts invested in the stocks
At
if
twice continuously difFerentiable and that
functions.
as 2
>
is
—
given by
00 and
5
A
22
Special Case
Let V be in turn
p>0,
with
6=1-7
7<1
r?>0,
when 7 >
and 7
In the
0.
7^
first
condition of proposition 4
it
suffices to
take
In this case one finds
0.
-1/1—
»(x)-'-^
p
Computations whose length
is
the sole difficulty give
ganinwhen
>
z
The
2
<
t;c+(0),
itself follows
The
case
this is the
77
-
1/1
7) p
\pj
Vc+{0) and
-'"--^
/z\-i/i-*'
1-7
1
.
I
1
fzV
fzy
i+,-(i_^)
where
inequalities ^
which
=
1-7
,
,,
when
'•'^
^^
pTj
P'
=
>
0,
^ > 1/(1 - 7) and
/x'
> -1/(1 -
from condition 16 of corollary
is
the one that
is
7)
all result
from
/3
> 7r +
(7/I
-
f)g'^/2
2.
usually taken in the approach of
one for which the Bellman equation can be solved quite
dynamic programming,
explicitly.
for
This yields the simpler
expression
from which one derives
Ct
=
consumption and portfolio
falls
.
It is
When
z
is
<
Vc+{0),
i.e.,
we have
z4>'{z)
= ne^*Wti(T<T^)-\h -
when nominal wealth
z approaches zero, however,
and investment
Uc+(0),
=
=
the optimal
-fi(f>{z).
is
Hence when nominal
zero and the optimal
proportional to wealth with
At
z
>
7/
because of the non negativity
below the non-stochastic boundary W_, consumption
dollar investment in the stocks
When
clear that except in the degenerate case
policies are not linear functions of wealth,
constraint on consumption.
wealth e^^Wt
SWtC^^
is
r).
above W_, the optimal policy
is
no longer
linear.
As
which corresponds to large values of wealth, the optimal consumption
policies are "almost" the linear functions of wealth given in
Merton
(1971).
6
Convergence of Finite Horizon to
Horizon Solutions
Convergence of Finite Horizon to
6
We
Infinite
23
Horizon Solutions
Infinite
study two convergence problems in this section. First we show that the optimal consumption
policy in an infinite horizon economy,
sponding
finite
one indeed
if
a pointwise limit of those in corre-
exists, is
horizon economies. Second, under some regularity conditions, the optimal portfolio
policy in an infinite horizon
economy
the limit, with respect to a norm, of the optimal policies of
is
the corresponding finite horizon economies. Thus, there exists a subsequence of the latter that converges pointwise to the former.
limit
It
follows that
if
the latter converges pointwise at
the pointwise
all,
must be equal to the former almost everywhere.
The convergence
where closed-form solutions
optimal policy in
two aspects.
results reported here are useful in
infinite
exist for finite horizon economies,
horizon can be gotten by letting
First, for the class of
models
we know conditions under which the
T —
oo in the finite horizon policies.
Second, in the numerical computation of the optimal policy for the infinite horizon problem, the
horizon will have to be truncated.
It is
therefore imperative to
know
conditions under which the
solutions to finite horizon problems are approximations to that to an infinite horizon problem.
Assume
Theorem
until further notice that conditions of
exists a solution to (14),
denoted by
Also assume that the
c*.
consumption and thus the optimal consumption plan
or
1
Theorem
utility
2 hold
function
is
and thus there
strictly
concave in
unique. Consider a class of finite horizon
is
problems:
where we have used
SUP<:6LP(;,„;T)
E
S.t.
E
L'^{i^a\
JO
B(t)\u(a,t)\ °'^<^(^)\
^
"^0,
T) to denote the positive orthant of the space of processes
c
such that
fT
<
oo.
Jo
It is
clear that
under conditions of Theorem
consumption policy of
as
T —
oo.
will get the
We
first
Lemma
Proof.
Thus,
if
(23),
1
or
Theorem
2,
there exists a unique optimal
denoted by c^. The following theorem shows that c^
we know the functional expression
optimal consumption policy for the
infinite
of
c-^,
—
>
c*, i/a
-
a.e.,
which naturally depends on T, we
horizon case by simply letting
T—
oo.
record a lemma.
5 Let
Xj
be
such that c^(t)
= fi^^^^t)
u^-a.e.
Suppose otherwise that Aj» < Aj. By the
strict
Then
Xr > Xj
ifT'
concavity of u in
c,
> T.
we know that /
is
6
Convergence of Finite Horizon to
strictly decreasing
when
it is
claim that one of the
c^' violates the
a violation.
dt
first
inequality in the above relation
solution to (23).
Theorem
I
that /
c-^
A* be such that c'H)
4 limr-^oo ^T
By Lemma
5,
=
claim that A
We
=
^*
= fi^^^,t),
and c^ -*
its first
=
is
an optimal
T —
as
»
oo, the second assertion follows
strict
concavity of u in
from the
fact
c.
=
lim Aj.
T—oo
A*.
first
that A
>
A*.
identical to those used in the proof of
Then
Lemma
T <
there exists
oo such that
\j >
A*.
5 prove that
^(0/(^,0
f
dt
Bit)
a contradiction.
Next suppose that A <
Lemma
Bit)
c*, Va-a.e.
Jo
is
dt
I
there exists a limit
Wo<E
which
and constitutes
:^a-a.e.
argument by the
take two cases. Suppose
Arguments
[0,T']
This clearly contradicts the fact that c^
i/a-a.e.
A
We
problem with horizon
I
continuous in
is
= E
Bit)
Once we can show that Ay —» A*
Proof.
and hence
strict inequality
^T>m
^m/i^^t) dt
This necessitates that on [0,T],
let
for the finite horizon
must be a
suppose that
E
Now
dt
Bit)
Jo
B{t)
budget constraint
Now
E
r at)fi^,t)
E
Jo
We
<
dt
B{t)
Jo
<
24
nonzero. This implies that
E
Wo =
Horizon Solutions
Infinite
5
A*. Pick
A G (A, A*). Arguments identical to those used in the proof of
show that
Wo = E
-rmiC-^.t)
L
U{t)fi'^,t)
dt
Bit)
> E
I
dt
Bit)
,
VT >
0.
6
Convergence of Finite Horizon to
T—
Letting
Horizon Solutions
Infinite
oo on the right-hand side of the second inequality gives
Mm
Wo>E
If
at)f(
> E
dt
I
B(t)
the second inequality above
an equality, then
is
f
Assumption 3
1.
>
2
must be that
it
strict inequality
If u satisfies conditions of
Theorem
"
u
satisfies conditions of
and
it
=
c*
Wa.
i/a-a.e.,
which
is
clearly
leads to a contradiction.
For this we restrict our
strategies.
Theorem
2,
1,
then
<
\u{a,t)\l dX,{t)
GO.
(24)
then
<
l"^"'*)!''^^"^^
.B(t)
i'^^^Kw)
We
J)
and make the following assumption:
1^)
r«'»(f
If
B{t)
dt
Next we turn our attention to the convergence of trading
attention to the case where p
Villi
B{t)
Jo
suboptimal. Thus the inequality must be a
2.
25
00.
(25)
record below a useful technical lemma.
Lemma
6 Let
c be the
solution to (14) financed by {a, 6) G C{G).
E'
<
u:^/')
Then, under Assumption
3,
00,
and
e{t)^cTit)
f
E'
Jo
Proof.
Using the fact that
P
and
Q
dt
4.1).
we know
that
the proof of Proposition 3,
E'
Lyo
'J
left-hand side
is
assertion follows from similar
J
^/t>0,Q-a.s.
Jo
a square-integrable martingale under Q. Thus Jacod (1979, 2.48) shows that
the second assertion of this
Now we
first
''^^'^^"^'Uw'is)
^
B{s)
B{s)
B{3)
[/:i^-H--[ri^H^/.
'
The
oo.
are locally equivalent, the
arguments of Cox and Huang (1990, theorem
Form
<
Bit)
lemma
is
valid.
I
present our results of convergence of trading strategies.
We
of trading strategies in a metric involving taking expectation under Q.
will
show the convergence
6
Convergence of Finite Horizon to
Theorem
5 Let (a, 9)
^
and (ax.^r)
26
Horizon Solutions
Infinite
the trading strategies that finance c*
and c^
respectively.
,
Then
{0T{t)
-
r->oo
Proof.
Note
=
=
0.
B{t)
Jo
that c^(0
first
9{t)V(Tit)
dt
lim E'
/(^ffff ,Ol[o,r!(0
>
<
and
c'{t)
=
/(^,0
t
>
0.
Triangle
inequality implies
(r
[j[°° \c'{t)
- c^(OI
dt
-«0,„_,M0,,„,
<
+
Recall from
Lemma
limx-.oo ^T-
Thus both
-[ri^*^'"-^<^'"'-<'>i^^'])'-
5 that A^-
is
T
an increasing function of
and from Theorem 4 that A* =
''^'
B{t)
and
decrease to zero
i/g-a-e.
Lebesgue convergence theorem yields that
1
2
cV)-c'{t)
lim
T—"x>
The
I
E'
f
dt
=
0.
B{t)
Jo
assertion then follows from the fact that
c'{t)
E'
f
5 is not
2
c^it)
dt
enough
Theorem
f
for us to conclude that the
T—
we need almost everywhere convergence
be
i0T{t)-0{t)V<T{t)
dt
B{t)
Jo
horizon case can be gotten by letting
for this operation to
= E'
B{t)
Jo
Theorem
-
optimal trading strategy for the
infinite
oo in the optimal strategies for the finite horizon cases as
for this.
The
following theorem gives a sufficient condition
valid.
6 Let {a, 6) and {aT,9r)
Suppose that limx— oo ^t(0 exists
^
the trading strategies that finance c*
i/a-a.e.,
then limx-^oo^Tit)
= ^(0
i^a-o-^-
and c^
,
respectively.
7
Concluding Remarks
Proof.
First recall that
27
^
6t
4.2.3) implies that there exists a
in the sense of
Theorem
subsequence {T„} with T„
Given the hypothesis that limr—oo ^t(0 exists
i/o-a.e., it
j
5.
Chung
(1974, theorems 4.1.4 and
co so that dT„ —* 0,
follows that limr-.oo
t-a-a.e. as
n
^t(0 — ^(0>
—
>
c«.
I'a-ae.
I
In words,
if
the optimal portfolio policies for finite horizon economies do converge pointwise,
they must then converge to the optimal policy for the infinite horizon economy.
Concluding Remarks
7
We have assumed
throughout
this
paper that there are as
many linearly independent
risky securities
as the dimension of the underlying uncertainty, or, markets are dynamically complete.
markets are dynamically incomplete,
and show that when
it is
He and Pearson
straightforward to borrow from
prices of securities together with
some other processes
When
(1989)
follow a diffusion process,
then the optimal policy can be computed by solving a qua^si-linear partial differential equation, and,
in addition, a general existence
theorem
is
available for the case
where the
coefficient of the
Pratt measure of the relative risk aversion^^ of the individual's utility function
is less
Arrow-
than one.
References
8
1.
K. Arrow, Essays in the Theory of Risk-Bearing. Amsterdam: North-Holland, 1970.
2.
J.
Cox and
C. Huang,
A
variational problem arising in financial economics, 1990, to appear
in Journal of Mathematical Economics.
3.
J.
Cox and
C. Huang, Optimal consumption and portfoUo policies
diffusion process, Journal of
4.
C. Dellacherie and P. Meyer, Probabilities and Potential B: Theory of Martingales, North-
P.
New
York, 1982.
Dybvig and C. Huang, Nonnegative Wealth, Absence
tion Plans,
6.
asset prices follow a
Economic Theory 49, 1989, 33-83.
Holland Publishing Company,
5.
when
Review of Financial Studies
1,
of Arbitrage
and Feasible Consump-
377-401, 1989.
L. Foldes, Conditions for optimality in the infinite-horizon portfolio-cum-saving
problem with
semimartingale investments, London School of Economics Discussion Paper No. 53, 1989.
'^See
Arrow (1970) and Pratt
(1964).
8
28
References
7.
I.
Gradshteyn and
I.
Ryzhik.
TaWe of
and Products.
Integrals, Series
Academic
Press,
Orlando, Florida, 1979.
8.
J.M. Harrison and D. Kreps, Martingales and multiperiod securities markets, Journal of
Economic Theory 20, 1979, 381-408.
9.
M. Harrison and
S. Pliska,
Martingales and stochastic integrals in the theory of continuous
trading. Stochastic Processes
10.
and Their Applications, 11, 1981, 215-260.
H. He and N. Pearson, Consumption and portfolio policies with incomplete markets and
short-sale constraints:
The
infinite
dimension case, 1989, to appear
in
Journal of Economic
Theory.
11. J.
Jacod, Calcul Stochastique et Problemes de Martingales, Lecture Notes in Mathematics
714, Springer- Verlag,
New
York, 1979.
12. J.
Jacod and A. Shiryayev, Limit Theorems of Stochastic Processes, Springer- Verlag, 1987.
13.
Karatzas,
I.
J.
Lehoczky, and
S.
Shreve, Optimzd Portfolio and Consumption Decisions for
a "Small Investor" on a Finite Horizon, 1987,
SIAM
J.
Control.
Optimization 25 (1987),
1557-1586.
14. R. Liptser
Verlag,
15.
and A. Shiryayev,
New
R. Merton,
Statistics of
Random
Processes
I:
General Theory, Springer-
York, 1977.
Optimum consumption and
portfolio rules in a continuous time model. Journal
of Economic Theory 3, 1971, 373-413.
16.
R. Merton, optimal portfolio rules in continuous- time
consumption
17.
is
when the nonnegative
constraint on
binding. Harvard Business School Working Paper #90-042, 1989.
H. Pages, Three Essays in Optimal Consumption, unpublished Ph.D. Thesis, Massachusetts
Institute of Technology, 1989.
18. S. Pliska,
A stochastic calculus
model of continuous trading: Optimal
portfolios,
Mati. Oper.
Res. 11, 1986, 371-382.
19. J. Pratt,
Risk aversion in the small and in the large. Econometrica 32, 1964, 122-136.
20. R. Rockafellar, Integral functionals,
linear Operators
New
York, 1975.
normal integrands, and measurable
and the Calculus of Variations, edited by
J.
Gossez
selections, in
Non-
et al., Springer- Verlag,
8
References
29
21. P. Samuelson, Rational theory of warrant pricing, Industrial
Management Review
6, 1965,
13-32.
22. D. Stroock, Lectures on Stochastic Analysis: Diffusion Theory,
Cambridge University
Press,
1987.
23. D. Stroock
and
S.
Varadhan, Multidimensional Diffusion Processes, Springer- Verlag,
New
York, 1979.
24. D.
WiUiams,
Wiley
&
Diffusion,
Sons, 1979.
Markov
Processes,
and Martingales, Volume
1:
Foundations, John
MIT
3
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