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ALFRED
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WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
OPTIMAL CONSUMPTION AND PORTFOLIO POLICIES
WITH AN INFINITE HORIZON: EXISTENCE A^ro CONVERGENCE
Chi-fu Huang and Henri Pages
May 1990
WP# 3172-90-EFA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
»" iv Jt-vP"
OPTIMAL CONSUMPTION AND PORTFOLIO POLICIES
WITH AN INFINITE HORIZON: EXISTENCE AlK) CONVERGENCE
Chi-fu Huang and Henri Pages
May 1990
WP# 3172-90-EFA
%^
E 6
mo
RfcCtWfeD.
Optimal Consumption and Portfolio
with an
Infinite Horizon: Existence
Policies
and Convergence^
Chi-fu Huang'^ and Henri Pages^
March 1988
Revised,
May
1990
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. \N'hen the price processes
for securities are diffusion processes,
differential equation.
problem
is
We
optimal policies can be computed by solving a linear partial
also provide conditions
under which the solution to an
the limit of the solutions to finite horizon problems
infinity.
'The authors would like to thank conversations with Robert Merton.
'Sloan School of Management, Massachusetts Institute of Technology
'Bank of France, Centre de recherche
infinite horizon
when the horizon
increases to
1
Summary
Introduction and
Introduction and
1
Recent advances
in the
1
Summary
understanding of dynamic asset markets have made available a
consumption and portfolio decision of an individual
tools to analyze the optimal
in
set of
new
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
for
an individual
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,
some
situations, optimal policies can even be
The aforementioned
economies with a
The purpose
finite horizon.
some important
in
of this paper
is
economies with an
to
show how
some
same
this set of
infinite horizon.
technical departures, the existence
policy can be analyzed similarly with the
We
directly by evaluating
integrals.
papers, however, address the optimal consumption and portfolio problem in
be brought to bear on the same problem
that, except for
computed
in
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.
show that pointwise convergence always occurs
for
In general, however, portfolio policies converge only in a certain
expectations.
converge at
Note that
The
We
all
On
the other hand,
pointwise, they
if
this type of convergence result
want to consult.
norm
involving taking
the optimal portfolio policies for finite horizon economies do
must converge
analysis in this paper utilizes
optimal consumption
is
to the optimal policy in the infinite horizon economy.
very difficult to get by using dynamic programming.
Cox and Huang
Two other papers complement
(1989, 1990) extensively, which the reader
may
our study. First, Merton (1989) uses the Cox-Huang
technique to solve, in infinite horizon, the optimal consumption and portfolio policy in closed-form
when
for a class utility functions
asset prices follow a geometric
Brownian motion. Second, Foldes
(1989), using a different but related technique, also analyzes the optimal consumption and portfolio
problem with an
infinite horizon in
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
gives results on convergence
The
existence
4, respectively.
Section 6
organized as follows. Section 2 formulates the model.
policies are ancilyzed in Section 3
and Section
and Section
7 contcuns concluding remarks.
Formulation
2
2
Formulation
2
We
will use
an .V-dimensional Brownian motion to model the evolution of exogenous uncertainty. Thus
we take the
"state space"
to be the space of continuous functions from [0,oo) to 3?^ equipped
fi
The
with the topology of uniform convergence on compact subintervaJs of [0,oo).'
distinguishable events
the Borel sigma-field of
is
the agent to be considered
P
under
fi
denoted by
collection of
and the probability
belief of
well-known that
It is
the coordinate process
=
u
continuous function
Q. Since a state of nature
E.
e
Vu;
uj{t),
an A'-dimensional standard Brownian motion^, where
fi,
u){t) is
the value at time
a complete realization of
is
oo) and one learns the true state of nature by observing
[0,
T
the W/ener measure on {^,T) denoted by P.
is
w(u;,t)
is
w
over time,
of the JJ^-valued
t
on the time interval
iv
we model the intertemporal
resolution of uncertainty by an increasing and right-continuous family of sub-sigma-field of
filtration
{u;(5);0
F =
<
{J^t'J
<
5
t}.^
£
^-t-})
where Tt = D^-^t^t ^"d
One can
verify that 7^
=
J^° is the smadlest sigma-field
\/t>o^ty that
all
is.
with probability one,
A
process A'
is
a
J^q
mapping from
the product sigma-field oi
A
respect to J^f
generated by
all
£
3?+.
adapted
A
generated by
the distinguishable events are
see Stroock
is
fi
x 3?+ to
A
B{'^^).
predictable
5?
that
process
if it is
if
the restriction of
X
is
X
to
adapted (to F)
The converse
progressive.
\{
is
and Varadhan (1979, Exercise
f2
x
X
is
is
is
measurable with respect to
adapted (to F)
P-measurable, where
A
[0,t] is
X{t)
process A'
is
=
X{t) P-a.s.
for all s
>
t,
For any integrable random variable
X{t)
will
=
£'[y|.F(] P-a.s.; see
where
Y
X{t)
V
is
7"
measurable with
is
the
field
measurable with respect to
J^t
fi
x
measurable with respect to !Ff Obviously,
A
martingale
jE[-|.?^(]
is
X
under
P
P
is
1.37).
Ji+
for
A'
is
true in our setup;
here
is
a progressive
probability one so that
the expectation under
Jacod and Shiryayev (1987, remark
all
on
x 5([0,<])
P
conditional on
on {il,J^,P), there exists a (P) martingale
be adapted and therefore progressive, and
x B(Ji+),
progressively measurable or
process that has right-continuous paths and has continuous path with
E[X{s)\J^t]
li
not necessarily true in general but
1.5.6).
—
t
that are probability one or zero.
left-continuous adapted processes.
process
if it is
T and
process A'
simply progressive
all t
T
contains only subsets of
or a
.?"
generated by sample paths of w. As a standara Brownian motion starts from zero at time
Tf.
We
consider a securities market under uncertainty in continuous time with an infinite horizon.
X
so that
All the processes to appear
the conditional expectations to appear will be
'See Williams (1979, p.20).
'a Btandard Brownian motion
'Throughout
this
nondecreasing, etc.
paper we
is
a Brownian motion that starts at zero with probability one.
will use
weak
relations.
For example, positive means nonnegative, increasing means
FormuJation
2
3
martingales.
Remark
In the above setup, the probability space
1
does not satisfy the usua.1 conditions^.
This
and Huang (1989, 1990). This departure
follow
a departure
is
P)
not complete* and the filtration
is
from
the earlier literature such as
important for our purpose; otherwise. Proposition
is
F
Cox
1
to
not valid.
is
We
(ft, J",
will use the following notation: if x
is
a matrix, then
\x\
=
trace(ix
(
)
1
where
,
denotes
"transpose."
There are
n
=
0, 1,2,
.
A^
.
.
+
A'.
,
1
securities traded continuously
Security n
is
7^
risky
and
is
on the
infinite
time horizon [0,3o) indexed by
represented by a process of right-continuous and
bounded variation sample paths D^ 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),...,S!^{t)y and D{t)
{Di{t),
.
.
.
,
D^it)y
ex-dividends, assume without loss of generality that Dn{0)
also that
S+
D
is
+
=
D{t)
and a,
=
n
S'(})+ I b(s,u)ds+ I ais,Lj)dw{s),
Jo
6
for all
by S„(0 and
1,2,
.
.
.
,.V.
We
assume
an N-vector process:
S{t)
where
=
t
these securities are traded
.A.s
.
at time
security
respectively, are A^- vector
t
e Pi+.
(1)
Jo
and
N
x A'-matrix processes satisfying
rt
f \b{s)\ds
<
oc,
P-a.s.,Wt>0,
(2)
a(s)\'^ds
<
00,
P-
(3)
I
and the second integral of
We
assume that
a(t)
riskless. Its price at
is
(1)
is a.s.
time
a stochastic integral.
of full rank for
B{t),
t,
is
all
t.
a.s.^Wt
>0,
6
The
zero-th security, called the bond,
is
locally
expl/^ r{s)ds}. One can view this security as a bank account
that pays an instantaneous interest rate r{t) at time
t.
For
B
to be well-defined,
we assume that
the interest rate process r satisfies
rt
i
*A
*A
probability space {Q,T, P)
filtration
F
is
complete
if
any subset of a probability zero
satisfies the usua/ conditions if (i)
complete: Tq contains
all
P-a.s.,Wt>0.
\r{s)\ds< 00,
it
is
right-continuous:
(4)
set is
^t
an element of T.
= ^ix^m
for all
t
6
3?+;
(ii) it is
the P-measure zero sets.
depends on the completion of a probability space and a filtration
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 vaJid, where
we recall that a process A' is a local martingale under P if there exists a sequence of optional times T„ ] cxj, P-a.s.,
*The usual
definition of a stochastic integral
satisfying the usual conditions; see, for example, Liptser
so that the process
{X{Tn A
t);
t
g
3?+
)
is
a (uniformly integrable) martingale for
all
r„.
2
Formulation
Now
consider an agent with a time-additive utility function for consumption, u{c,t) and an
wealth
initial
and
in c,
4
is
Wq >
possibly
0.
Assume throughout
unbounded from below
that u(c,
at c
0.
continuous in
(c,t),
concave and increasing
This agent wants to manage a portfolio of the
and the bond, and withdraw funds out of the portfolio to maximize
risky securities
consumption over time. Our task here
utility of
=
<) is
his
expected
to find conditions on the utility function and on
is
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
The process on
the left-hand side
is
called the gains process
and here
it
is
expressed in units of the
bond. Putting
G{t)
= 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
trading strategy
are the
time
t
number
a A^
+
of shares of the
before trading.
dividends
is
The
1-vector process (q,^^)
bond and of
=
(q, (^'
,
.
.
.
bond
S''(0)
=
5(0).
,0^)), where a{t) and ^"(<)
risky asset n, respectively,
investor's wealth in units of the
1,
owned by the investor
at time
t
at
after the receipt of
is
W{t) = ait) + 0{ty{S'{t) + A£)(0/5(0).
For now, a consumption plan c
is
an adapted process that
negligible with respect to the product
c(t)
measure generated by
denoting the consumption rate at time
plan c
if
t.
A
positive (except on a set that
is
P
and Lebesgue measure on
trading strategy
is
said to finance the
is
Ji+) with
consumption
the following intertemporal budget constraint holds:
(5)
Jo
ij[S)
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 capitaJ gains or losses and minus accumulated withdrawals
stochjistic integral of (5) to
its initial
value plus
consumption. For the
be well-defined, we introduce the class C{G) of trading strategies (a,
the 6 of which satisfies
'
e{s)''a{s)
ds
I
for
for
This equation says
a sequence of optional times T„
B{s)
]
oo, P-a.s.
<
oo,
P-
a.s.,
9),
2
Formulation
5
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
Were
time; see the doubling strategy of Harrison and Kreps (1979).
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 constraint has been analyzed 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
This assumption
all
=
t,
-
r{t)S{t)).
There exists a positive constant
i{t)
Now
<
oo such
models originally considered by Samuelson
P
define a martingale under
that
is
almost surely strictly positive:
= ex^^j\{sYdw{s)-^j^ \K{s)\^dsY
easily verified that £[^(<)]
1, it is
K
P-a.s.
in particular satisfied in the
is
(1965) and Merton (1971).
By Assumption
—<7{t)~^(b{t)
=
«e^V-
Thus
1.
Qt{A)= f a^,t)P{du)
'iAeJ't,
JA
is
a probability measure equivalent to P. The family of probability measures {Qt\t 6 [0,oo)}
"consistent" in that
Proposition
1
Q, equals Qt on
P
on Tt for
Git)
is
where oo >
There exists a probability
to the restriction of
and thus
Tt,
all
=
t
Q
s
>
<
>
0.
We
on {^,T) such that
have the following
its
restriction
on
JT,
is
result.
is
equivalent
G 3?+- Moreover, under Q,
5(0)
+
/'
Jo
1^ dw'is),
t
e
[0,
oo),
Jt>{S)
a local martingale^ under Q, where w'{t)
=
w{t)
-
Jq
K{s)ds
is
an
N -dimensional
standard Brownian motion under Q.
Proof.
Define a consistent family Qt on {0.,Tt) as above. Stroock (1987,
there exists a unique
Q
on (n,.F) so that Q\J^t
^See footnote 6 for a definition
= Qt
for all
<
G 3f+. Since
lemma
Q
is
4.2)
shows that
equivalent to
P
on
2
Formulation
Ti^
we have the
theorem
(see, e.g.,
Since
On
first
P
and
Proof.
1
Q
P
Q may
and
Q
disagree on the
P
and
=
oo)
=
they are said to be locally equivalent.
<,
as
cr-field .F,
are mutually singular
if
shown
and only
Q{^oo < oo)
=
0.
7.1).
P
In the
and
Q
are mutually singular
For
P
and
/
i
The almost
Q
and only
if
Proof.
is
the
same whether
is
it
definition of
G
/
Pm^(s)^T.r,
dG(s)=
/
Jo
Jo
From the
P
Then
2 Suppose {oi,0) 6 C{G).
and
=
random
oo, Q-a.s.
variable
^^
VA6^;
<
J^
if
oo
\
=
dt
Q
and
in the finite
1,
in restriction to J^t^ for all
an
the integml /^ ^(s)^(fG(5) is
computed
I
lemma above P and Q
relative to
(F,P) or
t
£
still
J?^..
ltd process
under both
P
(F,Q).
to
one has
—
B{s)
there exists A'
—-——dw{s).
e{sVa(s)
f^
/
B(s)
J3
<
oo such that
\k\
is
<
well defined
K
.
By
jb{t)-r{t)S{t)
^(0
a{t)K{t)
eit)^ait]
Bit)
Bit)
Kit)
9it)
Bit)
and so
jb{ t)-r(t)S(t)
B{t)
<
K
0(t)^<^it]
Bit)
under P.
On
the definition of k,
have thus
^(0
in fact
horizon case. However, one can
Since (q,^) € C{G), the stochastic integral on the right hand side
other hand by Assumption
necessary and
oc, Q-a.s., as desired.
constant, so by the
f\.
,r b{s)-r{s)Sis) ^
ds+
9{s)
,
is
Q-a.s.,
,
\K{t)\'^
is
it
surely statements on J^ can no longer be applied indifferently
use a/most surely with respect to both
and
\K{t)\^ dt
to be mutually singular,
dt
\K{t)\'^
with respect to either probability, as they can be
Lemma
Q
models of Samuelson or Merton, the process k
are mutually singular.
J^
But theorem 8.19 of Jacod [1979] shows that
{foo<oc} =
and so
if
lemma.
and
Jacod (1979, theorem
sufficient that
in the following
there exists a extended-valued
Q{A) = E[UlA] + Q{An{^=oc}),
see, e.g.,
and Girsanov's
I
4.3)).
Radon-Nikodym theorem,
the
such that P{^oo
assertion follows from the definition of k
are equivalent on (Q,^() for any finite
The measures
By
lemma
Stroock (1987,
the other hand,
Lemma
The second
assertion.
the
we
2
Formulation
7
Thus the Lebesgue
integral
is finite,
assertion follows from Stroock (1987,
We
straint
are ready to
is
in force.
first recall
An
1
consumption plan
is
an Ito process under P. The
Our proof is a
rest of the
I
III. 4. 3).
show that there are no arbitrage opportunities when the
positive wealth con-
Dybvig and Huang (1989, theorem
direct generalization of
arbitrage opportunity
c that is positive
need a technical
Lemma
/q 9{s)''^dG{s)
We
2).
the following usual definition of an arbitrage opportunity.
Definition
We
and
lemma
is
a strategy (a, 6) €
C{G)
with U^(0)
=
that finances a
and nonzero.
to proceed:
3 Let c be a consumption plan. Then
c(0
E'
if
.Jo
dt
dt
[f
.Jo
B{t)
B{t)
where E' denotes expectation under Q.
Proof.
For any finite
t.
cjs)
E'
j: Bis)
the above relation are positive and increase in
theorem we have the assertion.
W{t) >
0,
Proof.
t.
P
and
Q
are locally equivalent, c
is
positive
and Q. Thus the integrands on both
sides of
Thus
letting
<
2 Let c be a consumption plan financed by (a, 6)
P-a.s. for all
From
(5)
t
G 3?+. Then P-almost surely, c
By Proposition
1
—
cx),
by monotone convergence
is
£ C(G) with W{0) =
and with
identically zero.
we have
r 44
B{s)
(^^
+ ^(0 = ^(0) +
and the
known
r e{tfdG{t),
C(G), the right-hand side of the above relation
is
Q
since by the positive wealth constraint the left-hand side
is
that a positive local martingale
Dybvig and Huang (1989).
vi € 3?+.
Jo
fact that {a,0) G
a positive local martingale under
It is
ds
B{s)
i:
I
JO
positive.
P
under either
finite subinterval of [0,oo)
Proposition
c{s)as)
= E
and Meyer (1982, VT.57). Given that
see Dellacherie
on any
ds
is
a supermartingale; see,
e.g.,
lemma
1
of
Formulation
2
By
8
the fact that c
is
a positive process under
Q
a positive process under
c is
on any
P
P
and
Q
and
finite subinterval [0,(].
are locally equivadent,
we know
This implies that
c{t)
E'
f
dt
(-~
B(t)
Jo
is
theorem, the
Q
and
the expectation under Q, the
first
E'
W{0) =
<
where E'[-]
c{s)
Mm
<
first
B(s}
[Jo
ds
f
B{s)
+ W(t)
0,
equality follows from the
inequality follows from the fact that \V(t)
>
0, P-a.s.
monotone convergence
and thus
Q-sl.s. since
P
are locally equivalent, and the second inequality follows from the above mentioned super-
martingale property.
Lemma
3 then implies that
ccm) dt
f
Since ^
is
a
strictly positive process
<
0.
Bit)
Jo
under P,
must be the case that
it
c is identically
zero P-almost
I
surely.
Thus trading
strategies
from C{G) that satisfy the positive wccilth constraint cannot be arbitrage
opportunities.
In
models of a
admissible
finite
horizon
T
such as Cox and Huang (1989, 1990), a consumption plan
c is
if
tWdt <
CO
Jo
for
some given p E
unsatisfactory since
infinity,
[1,cxd).
it
A
direct generalization of this space to our setup by taking
T =
oo
is
does not include consumption plans that do not go to zero as time approaches
which we do not want to rule out a
priori.
We
will let the
space of admissible consumption
plans depends upon the impatience of the agent.
We
assume that the
Assumption
2 For
utility function of the
all a
>
Q,
t
<
s,
agent
satisfies the following
additional condition.
then u{a,t) > u{a,s),
/•oo
\u{a,t)\dt
<oo,
Ua+{a,t)<K^\u{a,t)\,
a.e.
/
(6)
Jo
and
there exists
Ka >
such that
<
G
??+,
where Ua^(a,t) denotes the right-hand partial derivative ofu{a,t) with respect
'Right-hand derivatives of a concave function always
exist.
(7)
to its first
argument.^
Formulation
2
Note that the
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 St
=
2 If u{c,t)
Pick some a
>
for all
and define a
invariant with respect to the choices of a
is
among
t.
>
for all
>
t
0,
then
it
satisfies
Assumption
2.
measure by
finite
K{A)=
V/lGS(R+),
\u{a,t)\dt,
I
Ja
P
and denote the product measure generated by
Lebesgue measure since by impatience
a consumption plan c
will say that
is
E
Then
|u(a,<)|
>
admissible
VM
A^,
Note that A^
Ua-
except possibly for one
<
\c{t)\^\u{a,t)\dt
/
by
t.
is
equivalent to
We
Fix p £ [l,oc).
if
oo.
Jo
the space of admissible consumption plans
ZP(fi X 3?-|.,PA1,^'a), where
and
the positive orthant of the space L^{ua)
is
denotes the progressive sigma-field.^
We
=
denote the positive
will
orthant of L^{ya) by L\{i'a)-
The
following
lemma shows
respect to choices of a
Lemma
4 For any
>
strictly positive scalars, a
=
Proof.
Let c G L'^lua).
Jo
The
first
invariant with
and
a'
such that u{a,t)
>
and
>
u{a',t)
for
all
Ll(i^a').
We
can write
<E
\cit)\''\u{a',t)\dt
/
is
0.
t,Ll{ua)
E
that the space of admissible consumption plans
J
is finite,
then c 6 i^(i'a'). Assume
/oo
r
1
<
\c{t)\''\uia',t)-u{a,t)\dt
/
Jo
E
J
is finite
first
\c(t)\P\u{a' ,t)
/
- u{a,t)\dt
.
L^o
J
term on the right-hand side of the inequality
the second term
+E
\c{t)\P\u{a,t)\dt
/
L-'o
that
by assumption.
a'
>
a.
We
If
we can show that
have
/-oo
/
-
\c{t)\^Ua+{a,t){a'
a) dt
Uo
yoo
<
{a'-a)E
/
\c{t)\n<M{a,t)\dt
<
oc,
Jo
'a
set
X
G
O
X 3?+
is
sigma-field generated by
a progressive set
all
if
1^(0;, <) is a progressive process.
the progressive sets.
A
process
is
The
a progressive process
progressive sigma-field
if
and only
if it is
is
the
measurable
with respect to the progressive sigma-field. Thus any element of i''(i'a) is a progressive process and thus adapted.
Conversely, any adapted process satisfying the L'' integrability condition with respect to fa is an element of L''(i'a)
since in our setup any adapted process is progressive. A good reference for the discussion above is Slroock and
Varadhan (1979, exercise
1.5.6
and
1.5.11).
3
Existence of
where the
first
An Optimal
10
Policy
inequality follows from the concavity of u and the second inequality follows from (7).
Next suppose that
a'
<
a.
Arguments
above prove the assertion by noting
identical to those
that
\u{a\t)
-
u{a,t)\
< u+(a',t)(a -
< Kau{a',t)(a -
a')
Similar arguments prove that c € L^{Ua') implies c 6
Now we
a')
< Kau{a,t){a - a).
I
L''(i/a)-
are ready to specify completely the agent's problem.
He wants
to solve the following
program:
E[f^
sup(a.e)eAG)
W(1)>0
u{c(t),t)dt]
W{0) = a{0)BiO) + e{OfS{0)<Wo,
s.t.
c is financed
and
We
will say that there exists
and
satisfies
the positive wealth constraint.
3
We
€
by (a,0)
L^{i^a)-
a solution to the program (8)
is finite
is
c
rg\
if
the value of the program, val{\Vo),
attained by a consumption plan financed by an admissible trading strategy that
Existence of
An
Optimal Policy
provide in this section sufficient conditions for the existence of a solution to
follows
Cox and Huang
variational problem
(1990).
We
whose solution
first
is
(S).
Our technique
transform the dynamic majdmization 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:
suPc6L^^(..)
s.t.
We
first
E[!^u{c{t),t)dt]
TOO C{t)({t)
Bit)
E
show that the dynamic program
m
Jo
(8)
is
,.
^''
^
-
Txr
^^'^O
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
and with 1^(0) <
H^o-
From
C{G)
the proof of Proposition 2
f ^,ds'rW(t)
i3[S)
Jo
= W{Q)+
if
and only
one
in (9).
satisfying the positive wealth constraint
we know
f 6{tfdG[t),
Jo
if it is
that
Vi>0,
P-
a.s.
Existence of
3
is
An Optimal
Policy
11
a supermartingale under Q. Thus
c{s)
r'
E'
^0
Because W[t) >
and
c{t)
>
-0(5)
J
P
under P, by the local equivalence of
E'
I
and Q, we know
<w{0) v<e3?+,
ds
B{s)
and
cjt)
< W{0),
dt
B{t)
.Jo
where the monotone convergence theorem
Lemma
is
used for the second inequality.
3 that
dt
< W{0) < Wq.
B{t)
.-'0
Thus
then follows from
It
c is feasible in (9).
Conversely, let c G X^(fo) be feasible in (9).
Lemma
(0
''[Jo
B
^(0
3 implies that
< Wo.
dt
Thus
Since
all
theorem
P-local martingales have the representation property relative to
III. 4. 33),
and since
Q
P
and
are locally equivalent,
representation property relative to w' (op.
cit.,
theorem
oo
r„
all
w
(see
Jacod and Shiryaev,
Q-local martingales also have the
III. 5. 24).
Hence there
exists an A'-vector
process p with
\p{s)\
/
ds
<
Q-
t oo,
a.s.,
Jo
such that
c{s)
E'
f
Jo
Let
ds\Tt
B{s)
= E'
r4r-M+
B{s)
.Jo
ti{s)
J
fpi^fdw'is)
ter,+, Q-a.s.
Jo
e{ty = B{t)p{tya{t)-\
W{t) = E'
cjs)
J
>
dslJ't
Wt
P-
a.s..
(10)
Bis)
and
a(t)
Since c
>
under P, W{t) >
restriction to
{T <
0,
= W{t) -
Q
e{t)^{S'{t)
and P-a.s. Let
+
T=
oo} (Jacod and Shiryaev, theorem
AD{t)/B{t)).
lim
T
^n- Since
III. 3. 4),
P{T <
Q
oo)
and
P
are equivalent in
= P(^{T)lf^j^^j) -
0,
Existence of
3
and so Tn
Q we
and
]
is,
We
^.ds
/'
Jo rl(Sj
q(0)
+ ^(0^5(0) +
Finally,
c.
1
/'
we can
this corollary
first
€
(0, 1)
all
and only
a.s.,
te^+.
W{0) < Wq. Thus
then concentrate on
c is feasible in (8).
I
a solution to (9).
if it is
[Wq] to denote the value of
t,
P-
a.s.,
3.
(9).
Note that since
(8)
and
(9) are equivalent,
(9).
provide conditions under which val [Wq)
for almost
Q -
eisfdGis),
easily seen that
it is
c is a solution to (8) if
Proposition 4 Suppose
6
=
record an immediate corollary of Proposition
will also use val
1.
P
Jo
(q,^) finances
Given
We
C{G). Also by construction and the locai equivalence of
{a, 9) £
have
Corollary
we
12
Policy
Hence
oc, P-a.s.
W^(0 +
That
An Optimal
is finite.
that
u{c,t)
is
unbounded from above
such that for some a >
in c
and
there exists
/3i
>
0, /32
>
and
0,
uic,t)<\u(a,t)\{3,
+ ^^ic'-'-l)),
a.e.t-
(11)
and
{aB)-'\u{a,t)\eL^I\u,).
2.
and
that, if u{c,t) is
unbounded from below
at the origin
on a
(12)
set
A
oft with strictly positive
Lebesgue measure,
-dt
L Bit)
Then Da/(VVo)
Proof.
with 6 G
oo.
(13)
is finite.
We first
(0, 1),
<
show that val{WQ) <
oo.
When
there exists a solution c to (9) only
Relation (12) ensures that c G I^(i/o)-
budget constraint and yield a
finite
the utility function
if
there exists 7
>
is
\u{a,t)\{Pi
so that
For c to actually be a solution,
expected
utility.
For the former
that
c(()«i)
^,
Jo
.
B{t)
dt
<
00,
+ ^hi'^^''' ~'^))
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
wealth. Putting p
initial
=
p/(l -b)
> p and l/p+
l/q
=
1,
Holder's inequjility implies
cimt)
1—
'
_i
7
\
r (^]
\B(t)J
b
last inequality follows (12)
utility is finite.
\u{a,t)\''
r
and Assumption
city-^\uia,t)\dt\
I
'
^
'
Thus val{Wo) <
we have
to verify that the expected
/?2
J
^
^U'o <
dt
B{t]
[Jo
00.
00 by (11).
Next we take two
Suppose
cases.
val{Wo) > —00 and thus val (Wq)
on a
set
A
of
t.
is finite.
that u{c,t)
first
is
bounded from below
Otherwise, suppose that u
is finite.
is
for a.e.
'
Then
unbounded from below
Let
k
(13), k
2. Finally,
^L^ir^ADhL
=
^1
<
By
oc,
For this we note that
Jo
at the origin
<
\u[aj)\dt
d\^{t)
Jo
P2
where the
dUt)
\u{a,t)\-'^l
1^)
= E
L B{t}
dt
Thus
Wo
c(0=-yl{,M}>0
is
v<
a feasible consumption plan. Thus
val{Wo) >
I
where the inequality follows from Assumption
2.
1,
then u(c,
our remaining task
is
to give conditions under which
in (11),
if 6
t)
-00,
I
>
Note that,
less
<
u{Wo/k,t)dt>
\u{a,t)\l3i.
Thus the expected
utility is
always strictly
than +00.
Now
Cox and Huang (1990) by
val{Wo)
is
attained.
We
will utilize
rewriting (9) as follows:
(14)
<Wo.
3
Existence of
An Optimal
Policy
14
both the objective functional of the maximization program and the space of feasible
In (14),
objects are defined on a measurable space
the framework of
Here
is
Theorem
Proof.
our
Cox and Huang
The
^+,VM)
x
with a
finite
measure
Va-
This
fits
into
(1990).
existence theorem:
first
Under
1
(fi
the conditions of Proposition 4, there exists a solution to (l^).
Cox and Huang (1990, theorem
assertion follows from
are locally equivalent.
The next theorem
4.1)
by noting that
Q
and
P
I
is
where the
for the case
utility function
bounded from above by a multiple
is
of |u(a,f)|.
Theorem
1.
2 Suppose that
u{c,t)
<
K\u{a,t)\ for
a.e.
t;
2.
{ilB)-Ma,t)\eL^{u,Y
3.
and (13) holds when
u{c,t)
is
unbounded from below
(15)
at the origin
on a
set
A
oft with strictly
positive Lebesgue measure.
Then
there exists a solution to (14)-
Proof.
By
the
first
hypothesis,
we know
that
-—.
\uia,t)\
< A
-
Given (13) and (15), the assertion follows from a
theorem 4.2) by noting that
P
Before leaving this section,
and
we
Q
Vc, a.e.
t.
(trivial) generalization of
are locally equivalent.
Cox and Huang
(1990,
I
give in the following two corollaries sets of explicit conditions on
the parameters of prices for (12), (13), and (15) to be valid.
Corollary 2 Suppose
exists
T>
0, e
>
that there exists
<
f
<
oo so that
When
(16) holds with
<
p
t
is valid.
r{t)
<
f,
P-a.s. for a.e.
t
and
there
such that
-!2^>^A-'
+ ^i' +
26
+
Then (12)
<
6=1,
(15)
is valid.
V,>T.
(16)
3
An Optimal
Existence of
Proof.
Policy
15
Note that
E
B{t)
P martingale
a
= E
exp
I
-^ J^
-
K{s)^dw{s)
^
j[' \K{s)\'ds
xexp^:^ (^-\-l]Kh + ^f\{s)d.5+
+ ^ )ln|u(a,0
1
(
26 \b
MiiM'^-^i'^^^iy^^)'}-
<-
Given that
v<
26
t
Fubini theorem then implies that (12)
Identical
always
Proof.
P
valid.
Otherwise,
Suppose
A
that the set
first
if
that
of (13)
<
is
6=1.
with unity expectation. Since B{t) >
Then
of finite Lebesgue m<^usure.
r so that r
<
r{t) P-a.s. for a.e.
of finite Lebesgue measure.
is
r,
valid.
is
there exists
A
>
+p
arguments proves the second assertion when
Corollary 3 Suppose
is
b
We
note that
,f
t,
is
if r{t)
>
then (13)
0,
(13)
is valid.
a martingale under
1,
at)
<
1.
B{t]
Fubini theorem implies that (13)
Next suppose that
A
is
is
valid.
Fubini theorem again implies that (13)
When
the utility function
interest rate does not
cLsymptotically.
we
is
is
Otherwise, the agent
until
t
I
=
may
oo.
utility function significantly
find
When
it
it
suffices
is
his wealth
unbounded from below
does not become too low so that
minimum consumption
that the
discounts the future
advantageous to keep accumulating
the utility function
will further require that the interest rate
may become — oo.
of this corollary yields
not unbounded from below, for existence,
infeasible to mciintajn a certain level of
utility
valid.
go unbounded and the
and postpone consumption
origin,
The hypothesis
of infinite measure.
it
at the
may become
over time and thus the expected
Computation of Optima]
4
Policies
16
Computation of Optimal
4
This section
is
Policies
devoted to the computation of optimal
Huang (1989) and thus we
We
of optimal policies.
will
will
partial differentiaJ equation
be
show that
and
if
The main
brief.
result reported here
=
easily seen that /(x~',<)
continuous in
is
this section that
f{x~^,t) < Kf\u{a,t)\i>xb
and (12) holds
if
6
origin, (13) holds.
or
Theorem
The
(10)
2 hold
G
(0, 1)
and (15) holds
By
c.
:
Uc+(c,
Then
define
< x~^}, where
Uc-y
a growth condition:
strictly positive constants
if 6
>
Moreover,
it is
c.
x.
satisfies
1.
in
the strict concavity of !/(c,<)
some
if
u
is
Kj,
b,
unbounded from below
easily verified that conditions of either
at the
Theorem
1
this value in units of the
the optimal consumption.
bond
is
cis)
ds\J^t
i:
B(s)
Thus F(t) = \V{t)B{t)
consumption. Under some conditions,
From
the value over time of future optimal consumption.
is
W{t) = E'
if c is
K+
concave
solution.
exists a solution to (14).
object of computation here
we know
strictly
for
Under these conditions,
and there
/
is
inf{c G
denotes the right-hand partial derivative of u with respect to
assume throughout
a verification theorem
computed by taking derivatives of the
the inverse of the marginal utility function f{x~^,t)
We
is
Cox and
the solution satisfies certain conditions, then the optimal trading
For the purpose of this section, we will assume that u{c,t)
it is
idea follows that of
there exists a solution to a second order linear parabolic
if
strategy in the form of feedback controls can be
in c,
The
policies.
F
is
the present value of future optimal
can be computed by solving a partial differentia! equation
and optimal trading strategies are related to the derivatives of F.
Before proceeding, we record in the following proposition the
which
is
first
order condition for optimality,
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;(.^J
(17)
and
c{t)at)
f
JO
Proof.
The only
if
dt
W'0-
B{t)
part follows from Saddle-Point theorem and Rockafellar (1975).
follows from the definition of
/ and the concavity of u
in c.
I
The
if
part
4
Computation of Optima]
Policies
17
For the purpose of computation, we specialize our model of securities prices as follows. Assume
that
S
satisfies
the stochastic integral equation:
S{t)
where d{S{t),t)
= S{0)+ f {biS{s),s)-diSis),s))ds+ f'a{Sis),s)dwis),
^0
Jo
is
the A''-vector dividend rate at time
Thus
further that r{t) can be written as r{S{t),t).
t
when the
t>0,
(18)
risky asset prices are S{t).
Assume
«(<) can also be written as K{S{t),t).
Next define a process
Z{t)
= Z{0)+
f {riSis),s)
+
\K{S{s),s)\^)Z{s)ds
- f K[S{s},sf Z{s}duis),
^0
for
some constant Z{0) >
0.
Using
Ito's
lemma,
^,
As pointed out by Cox and Huang
compounded growth
it is
easily verified that
Z(O)Bit)
,
(1989), (logZ(r)
rate from time
to time
that majcimizes the expected continuously
Now
(19)
Jo
T
- logZ(0))/r
is
the realized continuously
of the growth-optimal portfolio - the portfolio
compounded growth
rate.
write (18) and (19) compactly together under Q:
( z[\] )
=
( Z(0)
]+ J^aS{s),Zis),s)ds + J^d(S{s),Z(s),s}dw'is},
(21)
where we note that
.(5(o,.(o.o
Assume throughout
this section that ^
-
( .if<;>f^;;',;i,
and a
condition on any finite time interval [0,r].'°
has the strong Markov property under
The
M
>
0,
Q
satisfy
a local Lipschitz and a uniform linear growth
Thus there
and thus
is
exists a unique solution to (21)
and
(5,
Z)
a diffusion process under Q.
functions f and & satisfy a local Lipschitz condition on every [0,7^ with T < oo if for every T >
is a strictly positive constant Kj^ such that for all y, z £ 3?'^ x (0, oo) with \y\ <
and \z\ <
M
there
and
M
and
aii<e[o,r]
IC(», t)
-
C{z, 01
<K^\y-
z\,
!*(!/, t)
-
cr{z,
01
< K^'\y -
uniform growth condition on every [0,T] with T < oo
strictly positive constant A'r such that, for all i 6 3?'' x (0, oo) and t g [0,7^,
These functions
satisfy a
|C(i,OI
< Kt{1 +
\z\),
|*(i,OI
<
A'r(l
+
if
|x|).
for
^1-
every
T >
0,
there exists a
4
Computation of Optimal
The
18
Policies
following notation will be utilized:
am
Qmi+m2 + ...,mf/
m
D? =
for positive integers
.
.
,m;v-
If
^'"^
9
'•
.
X [0,7] •—
,dg/dy\)^
.
.
.
+ m;v
has partial derivatives with respect to
i?
is
.
denoted by Dyg or g^.
the main theorem of this section:
is
Theorem
1.
.
arguments, the vector {dg/dyi,.
its first jV
Here
mi,m2,
= mj +
3 Suppose that
there exists a Junction
continuous for
m
<
2,
F
F
:
(0,oo) x
J?'^
—
x [0,oo)
J?^.
such that DF^{y,t) and Fi(y,t) are
a solution to the partial differential equation
is
CF(Z,S,t) - r{S,t)F{Z,S,t) + FtiZ,S,t) + f{Z-\t) =
with the boundary condition
lim £'
rF(7(T).S(T).TU
FiZ{T),S(T).T)]
t— oo
where
C
is
B(T}
the differential generator of
{Z,S) under Q;^^ and, for
uniform growth condition on every [0,T] with
constants
KJ
and
-{^ so that,
for
\F{yJ)\
2.
Then
and
<
all
t
oc, that
is,
0,
F
satisfies a
there exists a strictly positive
Af (1 +
Vj/
\yr^)
£ (0,oo) x 3^^;
= Wq.
consumption and portfolio policies are
c(0
= f(Z{t)-\t)
e{t)
^[Fs{Z{t),S{t),t)
u^-a.e.
+
[a{S{t),t)a{S[t),tf]-'
x[b{S{t),t)-r{S{t),t)S{t)]Z{t)Fz{Z{t),S{t),t)\
q(0
=[F{Z{t),S{t),t)-9{tyS{t)\lB{t)
where we have put Z(0)
=
Using the growth condition on / and
equation,
Cox and Huang
(1989,
Theorem
B{t)
+
\FzzZ^\k\^
F and
^^^^
-a.e.,
the fact that
J.
+
Fszax.
F satisfies the
2.3) implies that, for every
T f{Z{s)-\s)
F{Z{t),Sit),t)
\it(Fss<rv'^)
z/,
u,-a.e.
Zq.
Proof.
^'LF =
T >
£ [0,T], a
there exists Zq such that F{Zo,S{0),0)
the optimal
T <
all
ds +
B{s]
+ Fs{rS -
/)
_,
,
partial differential
t,
F(Z{r),5(r),T)
+ FzrZ.
B[T)
Tt
5
Let
A
T
Special Case
19
—* oo and use the boundary condition
F[Z{t),S(t)J)
we
^E'
B{t)
get
J{Z{s)-\s)
ds\Tt
j:
B{s)
F
where we have used monotone convergence theorem. Thus
By
=
the hypothesis, Z(0)
f(Z(0),5(0),0)
=
then by Proposition
f{Z{t)~^,t) exhausts the
5, c is
theorem
5
A
f
dt
B{t)
Jo
a solution to (14).
quickly verifies that c G L^ii^a) and thus
that finances c
is
wealth.
initial
By
If
we can show that
€ L^{i/a),
c
the growth condition on /, (12), and (15), one
a solution (14).
The
fact that the trading strategy
as described in (22) follows from identical arguments on
is
time.
f{Z{t)-\t)
E'
B(0)
c(t)
/(Z \<) over
Zq, thus
Wc,
This shows that
gives the value of
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
and takes care
results with
in
in
Cox and Huang
(1990).
The method does not
use stochastic control
a natural way of the non negative constaint on consumption.
two examples taken from the family of constant absolute
We
illustrate
risk aversion
and
our
HARA
utility functions.
We
shall use the following specialization.
The
risky securities follow a geometric
Brownian
motion
Sit)
where b
is
an
=
5(0)
+
J (/s(,)b
n- vector of constants,
<t
-
+
d{Sis),s)) ds
We
data
z, define
(f)
= E'
—a~^{h —
r).
as
roo
<f>{z)
"I
e-''f{Z{tr\t)dt
/
;
Z-'(O) -
^.
Jo
Assuming
u(c,t)
=
v{c)e~^'
<i>iz)
(cf.
remark
= E'
/
Jo
t
>
furthermore assume that
write r for an n- vector of r's and k for the constant vector
initial
Is(,)ffdw{s),
an n x n non-singular matrix of constants and
diagonal matrix having S'{t) in the (i,i)th position.
Given some
J
2),
we can write
alternatively
e-^^g(e'^'Zitr) dt
Z-\0) =
z,
r is
/5(()
an n x n
constant and
5
A
Special Case
where g
is
Since
20
the inverse of the time-independent marginal utility function Vc^{c).
it is
easily verified that
= zexp{K^u--(0 +
e^'Z{t)-'^
F
one sees that the function
e^'Z(<)~^
is
=
of section 4 can be identified as F{z,t)
mean logz
lognormally distributed with
the square root of
+ |«|V2)0,
r
(f3-
-|-
(/3
-
+
r
£>^/2)t
Under Q,
4>{e^'z~^).
and variance
g'^t,
where o
is
follows that
\k\^. It
x
-\ogz -{3 -
r
+ 0^/2)^
dt dx.
where n stands
for the
4=e
y/2^t
^-^^-!-
/
/
J-oo
Jo
g
OO gy/2'K
where we have put
eVt
\
standard normal density function. This yields
=
4>{z)
gvt
J-oo
Jo
= P-
r
+
r+oo
'^'
Jo
and q^ = 2r
g'^/2
dtdx
-(-
-'°8')
2ff(e")
3^/g'^,
logz
X
-logz
dx.
A'l
Joo
where
A'1/2
^^ e~^;
cf.
is
^a
£>y27r'
^Q"
2
associated with a Bessel function and takes here the particular form A'i/2(^)
Gradshteyn and Ryzhik (1979, pp. 340 and 967).
We
=
then evaluate the last integral
to obtain the formula:
4>iz)
=
ga
Jo
Jo
where
=
/i
are both positive. Using the formula above,
functions.
Note that when
utility
has a
involve zero consumption. Indeed,
nominal wealth e''^W{t)
is
less
C(
=
r
get Jo
give two examples.
finite
if
g
g^
g
then easy to compute
marginal
and only
if
<p
utility at zero,
fc+(0)
for a
wide range of
[5(vc+(0)e-")e-'^"
^
+
utility
optimal consumption may
< e^'Z(<)~^
that
is if
and only
than the deterministic time-independent boundary given by
W = 4>{vM0)) =—
We now
it is
g^
5(vc+(0)e")e-'^'"] du.
J
if
A
5
Special Case
Example
1
21
Utility functions of constant absolute risk aversion.
Let the time-independent utility function be
where ^ >
is
we
the coefficient of absolute risk aversion. In this case,
find
1
5(x)
= -^logx
and so
-il/e){\ogz-u)
^^''
\
and
similarly for g{z€^). Straightforward computations
=
<^(.)
and that
if
z
<
if
u>
log 2,
otherwise,
show that
if
>
z
1
r(2,^iog2)
log.
"
Qady?
gaOn^
1
4>iz)
1
logz
log 2
Qadfj?
gaOfj,
qo.Ojj,
=
where 7 and F are the incomplete
gamma
7 (a,i)
^
7(2,
,\
/
('-")
/i'
log 2-^
gaOy.
,2
functions
=
r e-H"-^ dt
Jo
/•CO
T{a,x)
F
=
/
e-h"-^
twice continuous and that
F—
It is
easily checked that
The
vector of optimal dollar amounts invested in the stocks
is
At
if
Z{t)
<
e^'
=
ls(i)0{t)
1
1
+
.gadfj,
Z{t)
>
e^'.
Example
2
is
as 2
—
>
00
and
given by
= -^^Z{tr{aa'^){h -
r)
and by
At
if
dt.
HARA
utility
functions.
ga9n
„/3m'(
((7a^)(b-r)
-{l-e^'^'Zity)
F
00 as 2
—
0.
A
5
Special Case
22
Let V be in turn
b
=
7<1
/)>0, T/>0,
with
I
-
1
when 7 >
and 7
7^
In the first condition of proposition 4
it
suffices to take
one finds
In this case
0.
0.
^X
+
1/1-7
S(x)='-^
-
f?
•
p
Computations whose length
is
the sole difficulty give
gQfi{n- 1/1 -7)/) \pj
/hen z
>
=
/>r/-(i-"')
Uc+(0) and
when
2
<
pr
\pJ
^(5
itself follows
The
this
>
inequalities 6
which
is
p \p
Uc+(0), where
1-7
The
+ 1/1-1)
QQfi'ifi'
case
r?
=
0,
p.
> 1/(1-7) and
2(1-7)7-
V
/z'
> -1/(1 -
from condition 16 of corollary
is
the one that
is
7)
all
result
>
from
-jr
+
(7/I
- 7)0^2
2.
usually taken in the approach of
dynamic programming,
for
the one for which the Bellman equation can be solved quite explicitly. This yields the simpler
expression
^
<P(z)
p6
from which one derives
Cj
= SWtC^K
consumption and portfolio
falls
When
z
is
z
<
Uc+(0),
Vc+{0),
we have
z(p'{z)
=
i.e.,
=
the optimal
—p4>{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
It is
VP/
= ^le'^Wt{aa^y\h-T).
when nominal wealth
is
above W_, the optimal policy
is
no longer
linear.
As
approaches zero, however, which corresponds to large values of wealth, the optimal consumption
and investment
policies are "almost" the linear functions of wealth given in
Merton (1971).
Convergence of Finite Horizon to
6
Infinite
Horizon Solutions
Convergence of Finite Horizon to
6
We
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
if
one indeed
exists,
a pointwise limit of those
is
horizon economies. Second, in general, the optimal portfolio policy in an infinite
horizon economy
may
not be the pointwise limit of those in finite horizon economies. Under some
conditions, however,
if
must be the optimal
portfolio policy for the infinite horizon economy.
The convergence
the pointwise limit of the finite horizon optimal portfolio policies exists,
two aspects.
results reported here are useful in
where closed-form solutions
optimal policy
in corre-
in infinite
models
we know conditions under which the
exist for finite horizon economies,
horizon can be gotten by letting
First, for the class of
it
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 untU
further notice that conditions of
exists a solution to (14),
denoted by
Theorem
Also assume that the
c*.
consumption and thus the optimal consumption plan
is
or
1
Theorem
2 hold
and thus there
utility function is strictly
concave
in
unique. Consider a class of finite horizon
problems:
Elir^Si^^M
s"PceLP,K;T)
(23)
where we have used L^{i>a\T) to denote the positive orthant of the space of progressive processes
c
such that
l-T
/ \c{tWdX,{t)
<
oo.
Jo
It
is
clear that under conditions of
consumption policy of
as
T —
oo.
will get the
We
first
Lemma
Proof.
Thus,
if
(23), denoted
Theorem
1
or
Theorem
2,
there exists a unique optimal
by c^. The following theorem shows that c^ —>
c', Ua
-
a.e.,
we know the functional expression of c^ which naturally depends on T, we
,
optimal consumption policy for the
infinite horizon case
by simply letting
T
—
oc.
record a lemma.
5 Let
Xt
be
such that J{t)
= f{^^^,t)
v^-a-e.
Suppose otherwise that Aj- < Aj. By the
strict
Then
X^ > Xj
if
concavity of u in
T'
>
c,
we know that /
T.
is
6
Convergence of Finite Horizon to
strictly decreasing
when
it is
V^o
Bit)
a violation.
dt
first
inequality in the above relation must be a strict inequality and hence
budget constraint
Now
Bit)
Jo
Bit)
Jo
c^' violates the
fi(')
< E
dt
E
claim that one of the
24
nonzero. This implies that
h
<
We
Horizon Solutions
Infinite
for the finite horizon
problem with horizon
[0,r']
and constitutes
suppose that
^mf{^,t)
dt
i
Bit)
This necessitates that on [0,T],
solution to (23).
Now
let
Theorem
j^a-a.e.
A* be such that c'H)
4 limT-^oo
=
-^T
continuous in
is
By Lemma
5,
^'
= /(^^,0>
o^d
—
c
claim that A
We
its first
is
=
T —
-+ A' as
oo, the second assertion follows
strict
concavity of u in
from the
fact
c.
=
lim At.
r-oo
A'.
first
>
that A
A'.
identical to those used in the proof of
<
E
Then there
Lemma
T <
exists
oo such that
Xj >
A'.
5 prove that
-^(0/(^,0
dt
I
Bit)
a contradiction.
Next suppose that A <
Lemma
an optimal
there exists a limit
H'o
which
is
fa-a.e.
argument by the
take two cases. Suppose
Arguments
c
c', i/o-a.e.
A
We
This clearly contradicts the fact that
I
Once we can show that Aj
Proof.
that /
=
c
Bit)
Jo
5
A*. Pick
A G (A, A*). Arguments identical to those used in the proof of
show that
Wo = E
km
him
Tat)fi^j)
I
dt
Bit)
> E
dt
Jo
Bit)
,
vr >
0.
Convergence of Finite Horizon to
6
T—
Letting
25
oo on the right-hand side of the second inequality gives
Wo>E
If
Horizon Solutions
Infinite
hliill
-e(0/(^,0
>
dt
I
B{t)
the second inequality above
E
an equality, then
is
B{t)
must be that
it
suboptimaJ. Thus the inequality must be a strict inequality and
Next we turn our attention to the convergence of trading
attention to the case where p
Assumption 3
l.Ifu
>
it
u
satisfies conditions of
satisfies conditions of
i^a-a-.e.,
which
is
clearly
leads to a contradiction.
For this we restrict our
strategies.
Theorem
1,
then
at)
If
=
c'
and make the following assumption:
2
i:h%
2.
= W.
dt
I
Theorem
2,
\u{a,t)\^ dXait)
oo.
(24)
then
m
fOO
<
)
\uia,t)\'^ dXait]
<
oc.
(25)
B{t)
We
record below a useful technical lemma.
Lemma
6 Let
c be the solution to (I4)
financed by (a, 6) G C{G). Then, under Assumption
<
E'
B{t)
\Jo
3,
00,
J
anc
r
E'
e{t)^ait)
Jo
Proof.
Using the fact that
P
Q
and
dt
4.1).
we know
that
the proof of Proposition 3,
c{s)
E'
f
Jo
The
ds\J^t
B{s)
left-hand side
is
Jo
Bis)
B{s)
'JJ
^
'
Jo
yo
first
assertion follows from similar
'^^'^'"'^'hw'is)
B{s)
lemma
is
valid.
I
present our results of convergence of trading strategies.
In general,
almost everywhere convergence of trading strategies as for consumption
have convergence
in
yt>0,Q-a.s.
a square-integrable martingale under Q. Thus Jacod (1979, 2.48) shows that
the second assertion of this
Now we
= E'
00.
are locally equivalent, the
arguments of Cox and Huang (1990, theorem
Form
<
B{t)
a metric involving taking expectation under Q.
policies.
we do
not have
Rather, we will
6
Convergence of Finite Horizon to
Theorem
5 Let {a, 6)
Infinite
and (07,^7)
Horizon Solutions
26
be the trading strategies that finance c'
and c^
respectively.
,
Then
Proof.
Note
dt
=
0.
Bit)
= /(^^,01{(<T}
that c^(0
first
i9Tit)-9it))^cjit)
r
Jo
E'
lim
<
>
and
c'{t)
= fi^^j)
t
>
0.
Triangle
inequality implies
E'
<
\c'{t)-J[t)\Ut
\
Jo
iE'
B(t)
'
^^
'
''^^'\t}\^dt
5(0
E
+
dt
Jo
Recall from
Lemma
lim7_oo
Thus both
-^r-
5 that
A7
is
T
an increjising function of
and from Theorem
4 that A"
=
A-^(0
l/(
Bit)
'
^^
'
B(t)
'^'
and
At{(0
l/(
decrezise to zero fa-a.e.
5(0"''^ "-^^"^(0"''^^^'-^^'
Lebesgue convergence theorem yields that
c'{t)-c^{t)
lim
I
E'
dt
The
0.
Bit)
Jo
assertion then follows from the fact that
c'{t)-c^{t)
E'
f
dt
B{t)
Jo
Theorem
5
is
{eT{t)-e{t))''a{t)
f
Jo
dt
B{t)
not enough for us to conclude that the optimal trading strategy for the infinite
horizon case can be gotten by letting
T—
we need almost everywhere convergence
for this operation to
Theorem
= E'
be
00 in the optimal strategies for the finite horizon cases as
for this.
The
following theorem gives a sufficient condition
valid.
6 Let {a,B) and {oiti^t)
Suppose that lim7_oo^r(0
^
^^^ trading strategies that finance c'
exists Va-a.e., then limT-^oc^Ti^)
— ^(0
t/a-a.e.
and c^
,
respectively.
7
Concluding Remarks
Proof.
27
First recall that
Ot
^
4.2.3) implies that there exists a
m
the sense of
Theorem
subsequence {Tn} with r„
Given the hypothesis that limr— oo ^t(0 exists
i/a-a.-e., it
t
5.
Chung
(1974, theorems 4.1.4 and
oo so that dT„
—
0, i/o-a.e. as
follows that limj_oo
^t(0 =
—
n
»
oo.
^(i), i^a-a.e.
I
In words,
if
the optimal portfoho policies for finite horizon economies do converge pointwise,
they must then converge to the optimal policy for the infinite horizon economy. For example, the
under Assumption 3 the optimal portfolio policy for the
reader can verify that
utility function
cis
reported in
optimal policy for the
Cox and Huang
infinite horizon case.
HARA
(1989) converges pointwise and thus the limit
This
is
consistent with our result in Section
class
is
the
5.
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
straightforward to borrow from
prices of securities together with
some other processes
When
He and Pearson (1989)
follow a diffusion process,
then the optimal policy can be computed by solving a quasi-linear partial differential equation.
Since
He and Pearson (1989) only provided
has preferences only over
final
in the infinite horizon case
existence theorems for the case where the individual
wealth and not on intermediate consumption, the existence question
when markets
are dynamically incomplete
is still
open.
References
8
1.
2.
J.
Cox and C. Huang, A
in
Journal of Mathematicai Economics.
J.
Cox and C. Huang, Optimal consumption and portfoho
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diffusion process, Journal of
3.
when
asset prices follow a
Economic Theory 49, 1989, 33-83.
C. Dellacherie and P. Meyer, Probabilities and Potential B: Theory of Martingales, North-
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5.
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L. Foldes,
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on
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A
stochcistic calculus
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R. Rockafellar, Integral functionals, normal integrands, and measurable selections,
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in
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20. H.
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Samuelson, Rational theory of warrant pricing. Industrial Management Review
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83
"^
^o&o oot^
&V3T?
Date Due
JUN.
2a n
Lib-26-67
BARC;.