nu/LO
•
M414
no.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Consumption-Portfolio Policies:
An
Inverse Optimal
Problem
by
Hua He and
WP #3488-92
Chi-fu
Huang
November 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Consumption-Portfolio Policies:
An
Inverse Optimal
Problem
by
Hua He and
WP
#3488-92
Chi-fu
Huang
November 1992
M.I
T LIBRARIES
NOV
1
8 1992
Consumption-Portfolio
Policies:
Hua He* and
An
Inverse Optimal Problem"
Chi-fu Huang*
October 1991
Last Revised: October 1992
To Appear
in
Journal of Economic Theory
'The authors thank John Cox, Philip Dybvig, Hayne Leland, Robert Merton, Stephen Ross, Mark Rubinstein,
and Jean-luc Vila for conversations, and an anonymous referee for helpful comments. Research support from the
Batterymarch Fellowship Program (for Hua He) is gratefully acknowledged. Remaining errors are of course their
own.
'Haas School of Business, University of California at Berkeley, Berkeley, CA 94720
'Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139
Abstract
We
study a problem that
policy,
is
the "inverse" of Merton (1971).
we provide necessary and
sufficient conditions for
increasing, strictly concave, time-additive,
it
For a given consumption-portfolio
to be optimal for "some" agent with an
and state independent
asset price follows a general diffusion process.
utility function
These conditions involve a
when the
set of consistency
risky
and state
independency conditions and a partial differential equation satisfied by the consumption-portfolio
policy. We also provide an integral formula which recovers the utility function that supports a
given optimal policy. The inverse optimal problem studied here should be viewed as a dynamic
recoverability problem in financial markets with continuous trading.
1
INTRODUCTION
1
Introduction
1
The study
of an individual's optimal consumption-portfolio policy in continuous time under uncer-
Merton (1971), Cox and
tainty has been a central topic in financial economics; see, for example.
Huang
(1989, 1991), and
Does there
He and Pearson
The main question addressed
(1991).
an optimal consumption-portfolio policy
exist
for
in this literature
is:
an economic agent represented by a
time-additive and state-independent utility function and what are the properties of his/her optimal
policy
if it
indeed exists?
In this paper
we address
the "inverse" of the above consumption-portfolio problem.
general specification of the asset price process,
for a given consumption-portfolio policy to
we
be optimal
time-additive, and state independent utility function.
"supported" by such a
utility function
formula which recovers the
The
is
for
A consumption-portfolio
supports a given
inverse problem studied here can be viewed as a
We
is
to recover an
strictly concave,
policy that can be
also provide
an integral
efficient policy.
dynamic recoverability problem
markets with continuous trading; see Kurz (1969) and Chang (1988)
objective here
sufficient conditions
"some" increasing,
called an efficient policy.
utility function that
and
investigate the necessary
For a
for related
in financial
Our
problems.
economic agent's preferences from the observed consumption-portfolio
policy that has been specified for a given asset price process.
an individual's consumption-portfolio policy
in
Since our emphasis
is
in
analyzing
a continuous time securities market environment,
the inverse problem studied here and the solution method employed in this paper are very different
from those of Kurz (1969) and Chang (1988), who study an inverse problem
in the
theory of optimal
growth.
Cox and Leland (1982)
are the
first
to characterize efficient consumption-portfolio policies
the asset price follows a geometric Brownian motion; also see Black (1988).
this
paper
lies in
when
Our contribution
giving a characterization of the efficient consumption-portfolio policies
when
in
the
asset price follows a general diffusion process. Since our characterization of efficient consumptionportfolio policies
is
derived for a general specification of the price process,
we can
also use the
approach to answer a related question: Can a given consumption-portfolio policy be optimal
same
for
a
given utility function and "some" diffusion price process or for "some" utility function and "some"
diffusion price process?
The motivation
diffusion process
processes,
is
for
studying the inverse optimal problem when the asset price follows a general
twofold.
First,
accumulating empirical evidence suggests that the stock price
and especially the price processes
for portfolios, are not best
described by a geometric
1
INTRODUCTION
Brownian motion;
2
see, for
example, Black (1976) and Lo and MacKinlay (1988). As a result,
for the
study of optimal consumption-portfolio policies one needs to consider price processes more general
than the geometric Brownian motion. Second, when the price process
motion, the calculation of an optimal consumption-portfolio policy
for
is
is
extremely complicated;
example, Cox and Huang (1989) and He and Pearson (1991). As a
rules of
thumb
policies are usually followed.
It is
not a geometric Brownian
see,
result, in practice, certain
thus important to have the necessary and sufficient
conditions for a given policy to be consistent with utility maximization.
Our
strategy to solve the inverse optimal problem consists of two steps. First,
two real-valued functions, the
the dollar
amount invested
first
of which gives the
we
take as given
consumption and the second of which gives
the single risky asset, for any given levels of the individual's wealth,
in
the risky asset price, and the time.
These two functions completely specify the consumption-
We
then characterize the efficiency of this consumption-portfolio
portfolio policy of an individual.
policy through a set of necessary and sufficient conditions that are
imposed
solely
on the given
consumption-portfolio policy. Second, we present an integration procedure that recovers the utility
function supporting an efficient consumption-portfolio policy.
Besides deriving the necessary and sufficient conditions for efficient consumption-portfolio policies,
we
also obtain
For example,
make
it is
some characterizations of an
shown
that,
when
there
is
efficient policy that are of
independent
no intermediate consumption, an
interest.
must
efficient policy
the risk tolerance of the indirect utility function in units of the riskless asset a martingale
(subject to
some
formally later).
regularity conditions) under the so-called risk neutral probability (to be defined
As
will
be made clear
out to be the most significant
later, this result turns
tion for a given consumption-portfolio policy to be efficient.
the price process
is
a geometric Brownian motion and there
present value of the dollar
amount
amount invested
in the stock at
currently in the stock. This holds true for
Our paper
is
It
is
when
implies in particular that,
no intermediate consumption, the
any future date
all efficient
restric-
is
equal to the dollar
consumption-portfolio policies.
related to the characterization of efficient portfolios in a one-period setting due to
Peleg (1975), Peleg and Yaari (1975), Dybvig and Ross (1982), and Green and Srivastava (1985,
1986).
Because of the single period setting, dynamic trading rules are not considered
literature. In contrast, the
emphasis of
this
paper
is
to analyze efficiency of
functions from dynamic trading rules. Finally, our paper
is
and to recover
in
this
utility
related to the classical "integrability"
problem of revealed preference, which asks whether preferences are determined by the entire demand
correspondence; see, for example, Mas-Colell (1977) and Geanakoplos and Polemarchakis (1990).
The
rest of this
paper
is
organized as follows. Section 2 formulates a dynamic consumption-
THE SETUP
2
3
problem and derives necessary conditions
portfolio
shows how to express these necessary conditions
policy
and presents the necessary conditions
utility function. Section 3
terms of the given consumption- portfolio
solely in
This section also gives some examples to
for efficiency.
demonstrate how to use the necessary conditions
by an indirect
satisfied
for efficiency
and provides some characterizations
of an optimal consumption-portfolio policy that are of independent interest. Section 4 shows that
the necessary conditions for efficiency are also sufficient under
then demonstrated
how
more examples
contains
some
to recover the utility function that supports
illustrating our results
regularity conditions.
an
efficient policy.
It is
Section 5
and Section 6 presents some concluding remarks.
The Setup
2
Consider a securities market economy with a
bond available
interest rate.
dynamics
is
for trading.
The
w
is
The bond
T] in which there
[0,
is
one stock and one
price grows exponentially at a constant rate r, the riskless
stock does not pay dividends
2
,
and
price follows a diffusion process
its
whose
described by the stochastic differential equation 3
=
dS(t)
where
1
horizon
finite
n(S(t),t)S(t)dt
+
a(S(t),t)S(t)dw(t),
6 [0,T],
t
a standard Brownian motion defined on a complete probability space (SI,?, P).
notational simplicity,
we assume that
any y >
cess can access
inaccessible. 4 This
starting from any x
over any time interval
[t, t
+ e]
>
for
assumption implies that the stock price
and any time
however small
is
G [0,T), the price pro-
t
e
For
>
strictly positive
0,
and zero
is
always
with probability one. 5
Investors are assumed to have access only to the information contained in historical prices, that
the information the investors have at time
brevity,
We
we
will
sometimes simply use
assume that there
for the price process.
fi(t)
t
is
the sigma-field generated by {5(5);
and a(t)
to denote fi(S(t),t)
and
<
s
<
t}.
this equivalent
For
cr(S(t),t), respectively.
exists an equivalent martingale measure, or a risk neutral probability
Given our current setup,
is,
6
Q
martingale measure must be uniquely
represented by
Q(A)= [ t(u,T)P(du) VAeF,
Ja
'in applications, one can take the risky asset to be an index portfolio.
Nothing
3
will
be affected
Implicit in this
is
if
the stock pays dividends at rates that depend on the stock price only.
the hypothesis that a solution to the stochastic differential equations exists.
many statements to be made so that they are only valid over the range of the stock
This does not change the essence of the results but will complicate the notation, which is already heavy.
5
We will use weak relations throughout. For example, positive means nonnegative, concave means weakly concave,
'Otherwise, we have to qualify
price.
and so
6
forth. For strict relations,
we
use, for
example,
See Harrison and Kreps (1979) for the former and
strictly increasing, strictly concave,
Cox and Ross
(1976) for the latter.
and so
forth.
THE SETUP
2
where
= exp{J\(S(s),s)dw(s)-±£\K(S(3),s)\ 2 ds}
S{t)
(1)
and
ww—*^-'
Under Q, the stock
price
dynamics becomes
=
dS(t)
where w"
A
rS(t)dt
+
<r(t)S(t)dw'(t),
t
e [0,T],
a standard Brownian motion under Q.
is
consumption and portfolio policy (C,A)
a pair of functions, (C(W,S,t),A(W,S,t)), de-
is
noting the consumption rate and the dollar amount invested in the risky asset at time
respectively,
when
the wealth
use C(t) and A(t) to denote
time
From Merton
t.
satisfy
is,
W and the price of the risky asset
is
C(W, 5", t) and A(W, S, t),
policy (C,
Al. the
A)
admissible
is
and the
drift
will often
W(t) denote the wealth
respectively. Let
at
£ [0,7],
t
dW(s)=[rW(s) +
A
we
S. For simplicity,
dynamic budget constraint a consumption-portfolio policy must
(1971), the
starting from any
is
G [0,T],
t
+
A(s)<r(s)dw(s),
term of the wealth process
satisfy a linear
A(s)( li(s)-r)-C{s)]ds
se[t,T).
(2)
if
diffusion
local Lipschitz condition,
7
process can access any y
>
and starting from any x >
over any time interval
[t,t
growth condition and a
and any time
+
e]
for
t
6 [0,T), the wealth
however small
t
>
0.
This condition ensures that there exists a unique solution to the stochastic differential equation
(2).
The
policy (C,
A)
is
said to be efficient,
if it is
admissible and there exist a utility function for
intermediate consumption and a utility function for
and V{x)
and concave
7
A
where
in x,
function /
|i|
Su
-»
9? +
:
:
9i
<
M
— oo},
respectively,
and either u(x,t)
N
x
[0,
T\
—
>
3? is
for
and
i
£
[0,T],
if for
we have
any
\f(y,
all
t
said to satisfy a linear
M
t)
—
>
is
there
f(z,t)\
wealth, u(x,t)
:
3ft
+ x [0,T]
-+SU
{-oo}
which are twice continuously differentiable, increasing
almost
denotes the Euclidean norm of x and A'
a local Lipschitz condition
\z\
{
final
or
V(x)
is
strictly
growth condition
a strictly positive scalar.
is
a constant
< K\i\y —
z\,
Km
where
if
A
concave
\f(x,
all
x such that (C,A)
< K(l +
|z|) for all
x [0,T] —*
n
y,z € SR with |y|
function /
such that for
\y\
t)\
in
:
9?"
denotes the Euclidean norm.
x
and
t,
5R satisfies
<
M
and
THE SETUP
2
is
5
8
the solution to the following dynamic consumption and portfolio problem:
sup c >cM
E? y
[llu(C(s),s)dt
(2) holds,
s.t.
(A.C)
is
admissible,
(3)
W{s)>0
W(t) =
S(t)
any x >
for
>
0, y
0,
W(t) = x and S(t) —
possibility of creating
and
y.
6
t
The
+ V(W{T))
[0,
T],
=
se[t,T],
x,
y,
where E* y [-]
third condition
is
is
the expectation at time
t
conditional on
a positive wealth constraint that rules out the
something out of nothing; see Dybvig and Huang (1988).
Note also that there
is
a vast literature on the existence and the characterization of an optimal
consumption-portfolio policy for a given pair of utility functions (u,V); see Merton (1971), Cox
and Huang (1989, 1991), and He and Pearson (1991),
from that of
What
this literature.
(u,V)1 As we don't know the
suppose that (C, A)
>
0, y
0,
and
t
is efficient.
Then
By
in x,
different
utility functions to
some
pair of
begin with, these necessary and
there exists (u,
V)
so that (C,
A) solves
and the risky asset price
at time
t
strictly
concave
W
in
.
We
will restrict
(3) for
any
(3), or the
W
are
and S,
all t
or V(x)
our attention to
efficient
A) such that the following conditions hold:
A2. A(W, S, t)
S >
to be an optimal policy for
the monotonicity and the strict concavity of either u(x,t) for almost
J must be increasing and
policies (C,
it
6 [0,T]. Let J(W,S,t) be the value of the objective function of
indirect utility function, given that the wealth
respectively.
is
can only involve the given policy (C, A).
sufficient conditions
Now
example. Our purpose here
take a consumption-portfolio policy (C, A) as given and ask:
are the necessary and sufficient conditions for
utility functions
x >
We
for
is
and
0,
continuous and
for all
t
C(W, S,t)
is
continuously differentiable in
G [0,T], and C(W,S,t)
E t
is
W and S
for
W>
0,
such that
2
\C(W(t),S(t),t)\ dt
<
oo
Jo
for the wealth process defined
A3, far
(WW)
and
for
'imposing
later.
by (C,A);
€ [0,oo)x(0,oo)x[0, T],R(W,S,t)
= - f^wA) ^dH(W,S,t) = - %$$%%
(W,S,t) € (0,oo) x (0,oo) x [0,T], N(W,S,t)
strict
=
concavity on u ensures that the consumption function
fyjwffl
is
are well-defined; and for
continuous in
W
,
which we
will
assume
THE SETUP
2
6
(W,S,t) e (0,oo) x (0,oo) x [0,T),
R and
and continuously
and
7/ are
N
twice continuously differentiable in
(W,S)
(W,
5),
where the
a polynomial growth condition 9 and for any feasible policy (C, A) and
its
associated
differentiable in
t,
continuously differentiable
is
in
subscripts denote partial derivatives;
A4. J
satisfies
wealth process defined by (2)
J
\CJ(s)
+ J (s)\ds <
t
r€[0,T)
oo,
where
2
£J = -JwwA 2 o 2 + J ws Aa 2 S +
-Jss<> S
2
+ J w {tW +
A(ji
-
r)
- C] + JS fiS
and
A5. the wealth never reaches zero before time T.
The
Condition
Condition
of J.
A3
A2
requires the policy (C,
implies that
Condition
A4
J
is
A)
to be sufficiently
continuous on
T when
motion. Merton (1990, theorem 16.2) generalizes
two regularity conditions
worked. Thus Condition
Later in this section,
Now
A4, J
let
c>o,a
9
A
\
smooth and
A)
satisfied
A5
we
to be square-integrable.
Condition
u{C,t)
shown that the optimally invested wealth
the stock price follows a geometric Brownian
this result to
any diffusion price process obeying
add another regularity condition A6 that (C, A) must
be understood to
+ J + [rW + A(p t
needs some
by most of the processes with which financial economists have
Bellman's equation
satisfy
(see, for
r)
efficient.
satisfy. Until
Conditions A1-A5.
By Conditions A3 and
example, Fleming and Soner (1992, chapter
- C]J W + nSJs +
I,
—
A5
can be viewed as a necessary condition for (C, A) to be
will
will
C
(C, A) be efficient with corresponding utility functions (u, V).
satisfies the
0= max
Henceforth, subscripts denote
domain and allows us to work with many derivatives
its
(1989, proposition 3.1) have
any (u,V) must not reach zero before
then, an efficient (C,
be given later.
enables us to work with the Bellman's equation.
Cox and Huang
explanation.
will
mentioned otherwise.
partial derivatives unless
for
A3
interpretation of the terms defined in
-<r
2
2
A 2 J ww +
2
<J
SAJWS +
-<r
2
10
3)):
2
S 2 JS s\
growth condition if |/(r,t)| < K(\ + lip) for all
and 7 are two strictly positive scalars.
10
It is certainly not necessary for an indirect utility function to satisfy the Bellman's equation and a polynomial
growth condition. For the former see Cox and Huang (1989) and for the latter see Footnote 20.
i and
function /
(,
where
|i|
:
St^ x [0,T]
5? is
said to satisfy a polynomial
denotes the Euclidean norm of x and
K
J
THE SETUP
2
for all
(W,
S, t)
(E
7
(0,oo) x (0, oo) x (0,T), with the boundary conditions
=V(W),
J(W,S,T)
= tfu(0,s)ds +
J(0,S,t)
[)
V(0),
where we have suppressed the arguments of J, C, and A.
Note that the second boundary condition
it
is
a consequence of the positive wealth constraint and
necessitates immediately that
The
first
are, for all
C(0,5,0
=
0,
A{0,S,t)
=
0.
(5)
(6)
order necessary conditions for the dynamic consumption and portfolio problem (3)
(W,S,t) € (0,oo) x (0,oo) x (0,T),
u c (C(W,S,t),t)
u c (C{W,S,t),t)
A(W,S,t) =
<Jw (W,S,t),
=J w (W,S,t),
r
^' g
(
f)
(
where we have used the notation defined
side of (8)
and
is
in
)
[{C(W,S,t)
=
0,
\iC{W,S,t)>0;
+ SH(W,S,t),
R(W,S,t)
(8)
R(W, S, t) on
Condition A3. Note that
the right-hand
the inverse of the Arrow-Pratt measure of the absolute risk aversion of the indirect
utility function,
henceforth the risk tolerance, and the second term
the "hedging
is
adverse changes in the consumption/investment opportunity set". 11
twice continuously differentiate in
(W,S) €
(0,oo) x (0,oo) and
is
demand
against
By Condition A3, A must be
continuously differentiate
in
= Jww when C >
0.
te(0.T).
Fivtbermore, (7) and the chain rule of differentiation imply that u cc Cw
Since
Jww <
only
ccur
(C,
r
0>
when
we must have C\y >
when
C >
0.
the marginal propensity to consume
The
first
j4'
But these
is
That
is,
strictly positive
consumption can
strictly positive.
order necessary conditions and the Bellman equation place strong restrictions on
know; (as u and
V
restrictions are expressed in terms of the partial derivatives of J,
are unknown). However,
it
will
be shown
in the
which
is
not
next section that these necessary
condi' ons can be transformed so that they can be expressed solely in terms of (C,A). Therefore,
one
c
n directly check whether a given pair (C, /4)
other
iformation. Furthermore, with
cond'
Jits for
Se
some
is
a candidate for an efficient policy without any
regularity conditions,
(C, A) to be efficient are also sufficient.
Breeden (1984) and Merton (1973).
we
will
show that the necessary
2
THE SETUP
Clearly, for
all
8
(W,S,t) € (0,oo) x (0,oo) x [0,T), the assumed
d /1\
0,
S >
and
0,
(9)
d fH\
=
(10)
{a)'
j_or
r 2 dt
ai
dw
W>
(E"
d
dN
Js
on
for all
J implies that
dW\Rj'
dS\Rj
Now,
differentiability of
(11)
'
€ [0,T], define three functions
t
0(W,S,t)
1
=
f
—
dz,
(12)
R[z,S,t)
vl „
X(5
'"
s
K(l)
=
s
where V^ and 5 are two arbitrary
that
fa
=— f£
when
a
>
Jw_
s H(W,Tj,t)
[
4 -mSF*
N(W,S,r)dT,
(14)
['
strictly positive constants,
and where we have used the convention
U,
hiU(W,S,t)= -0(W,S,t) + X(S,t) +
Note that,
for all
W>
\nU{W,S,t)
0,
5 >
=
0,
=
write, for all x
>
first
in the
fl,
Y{t).
(15)
#, and N;
lnJvv(^,5,f)-lnJw(li:,5:,0)
J
\
where we have used the
€ [0,T), by the continuity of
t
t
(13)
In addition, define a function
b.
,
,
In
u c (C(W,S,t),t)-\nJw (K,S,0),
>lnu c (0,0-ln Jw(K,S,0),
order conditions
range of C(W,S,
(7).
For
t
< T,
since
\{
C(\V,S,t) >
0;
if
6(W,S,t) =
0.
Cw
>
for all
C >
(16)
0,
we can
t):
(
ln^C- ^,^),^') =
1
lnu c (x,f)-ln./w(ii:, J£,0),
(17)
<
where
C _1
denotes the inverse of
conclude that
function
for all
is
its
left-hand side
C
with respect to
its first
argument.
0,
S >
0,
and
t
€ (0,T) so that C(W,S,t)
=
t)
in x and-
0,
U(W,S,t)>]imU(C-\x,S,t),S,t)
V5 >
to derive
(
16)— ( 18) involve J and
some conditions
at the
its
boundary of
domain. For
have,
i3
(18)
-
their
we must
0.
derivatives in the interior of their
we
as the utility
(J),
,
r i°
Relations
this relation,
must be independent of S when C(W, S,t) >
state independent. Furthermore, by the concavity of u(x,
W>
From
•
domain
this purpose*,"
\
-,
We now
proceed
€ use a result of
THE SETUP
2
Cox and Huang
9
(1989, section 2.3), which states that, under the square-integrability assumption
of Condition A2, along the optimally invested wealth process, there exists a scalar A
where the inequality holds as an equality when
multiplier and £(t)e~
Tt
of probability P. Since
Arrow-Debreu state price
the
is
W,
W(T) >
S,
and
Note that the A above
0.
at
time
for time
£ are processes with continuous paths,
wealth reaches zero at T, (19) implies that
Jw
t
if
Wm Jw {W,S,t)> J w (0,S,T) =
VS >
V'(0),
so that
12
a Lagrangian
consumption per unit
the optimally invested
continuous except possibly when
is
is
>
W=
T
at
and
(20)
0,
ttr
where the equality follows from
(4).
13
Given the above discussion, we now impose one more regularity condition on (C, A):
A6. R(W,S,t), H(W,S,t), N(W,S,t), and their derivatives are continuous functions of
except possibly for
W=
t
= T
for
W>
A3
and S >
accomplishes two things.
t
=T
First, (16)— ( 18)
can be
and we have
0,
W > 0,5 >
=\nV'{W)-\nJ w (W,S,Q),
\nU{W,S,T)
l\mw io U(W,S,t)
at
0.
This condition together with Condition
extended to
t
0,
>\nV'(0)-lnJw (W,S.,0) = l\mwiolnU{W,S,T),
VS>0,S>0,
(
21 )
IfT
where we note that the
first
relation indicates that
lnU(W, 5, T)
the equality in the second relation follows from the fact that
addition, Q,
in
t
for
W>
X
,
0,
and
N are twice continuously differentiate in
S >
0,
and
t
V
is
is
a function of
W
only,
and
continuously differentiate. In
(W,S) and continuously differentiate
€ [0,T].
Second, we conclude by continuity and
H(W,S,T) =
HmH(W,S,t) =
Q,
that
W>0,
5 >
(22)
0,
(IT
and
1
A(W,S,T)
12
The
(^l^)"
=
Um tT T (/WS,0 + SH(W,S,t)
=
R(W,S,T) = -$$j,
conditions stated below and henceforth implicitly assume that the optimization problem starts at
The
possible discontinuity of
well-defined there.
Jw
at
T
(23)
t
=
0.
and other starting states W(t) = x and S(t) = y.
= is the reason that in Condition A3 N is not assumed to be
when
Similar conditions hold for other starting times
13
(^f)~
W>0, 5>0,
t
W
NECESSARY CONDITIONS FOR EFFICIENCY
3
which
a function of
is
We
will
10
W only.
term relations (9)-(ll) consistency conditions, since they basically require that high
order derivatives of the indirect utility function J exist and that J can be differentiated consistently
with respect to any ordering of
W, S and
Relations (17), (18), (21), (22) and (23) will be termed
t.
and
state independency conditions, as they follow from our assumption that u
Note that
independent.
if
we defined
efficiency
more broadly
V
to include state-dependent utility
functions, then obviously the state independency conditions need not be satisfied.
henceforth the set of efficient policies satisfying Conditions
to
be
efficient if it
is
are both state-
A1-A6 by
We
will
A)
£. For brevity, (C,
denote
said
is
an element of £.
Before leaving this section, we record one well-known fact about (C, A) € £, namely, that the
current wealth plus the cumulative past consumption, both in units of the bond,
a martingale
is
under Q.
Proposition
Let (C, A) G
1
£
.
Then W(t)e~ Tt + f C(W(s),S(s),s)e~ Ta ds
is
a martingale under
Q.
Proof.
See, for example,
Dybvig and Huang (1988).
Necessary Conditions
3
In this section
H
R,
,
As a
result, R,
H
and
a given (C, A) to be
H
R
and
N
are explicit functions of C,
H
efficient.
Our
N
order conditions to express the function
first
conditions derived in Section 2 about R,
C
for
R
C, A, and the derivatives of
derivatives.
for Efficiency
we derive necessary conditions
proceeds as follows. First, we use the
I
,
Second, we write
.
and
N
and
A
H
and
in
terms of C,
derivation
in
A
terms of
and
their
their derivatives. Necessary
are then brought to bear on the given functions
and A.
In this process,
interest. For
we
also derive
some characterizations of (C, A) £ £ that are of independent
example, we show that when there
is
no intermediate consumption, the
risk tolerance
process along the optimally invested wealth in units of the riskless asset must be a local martingale
under the
the
bond
This implies that when the hedging demand normalized by
risk neutral probability.
price
is
a decreasing process and the risk premium per unit of variance
is
positive
and
increasing over time, one expects an efficient portfolio policy to invest less in the stock over time
in present value
terms normalized by the
demand normalized by
the bond price
decreasing over time, the opposite
is
is
risk
premium per
unit of variance.
increasing and the risk
premium per
true under an additional condition.
When
the hedging
unit of variance
is
NECESSARY CONDITIONS FOR EFFICIENCY
3
we
Hereafter,
will
use k(S,
premium on the stock per
possibly on a set of
We
of R,
S and
begin by giving a
H, C, A, and the
or simply k(() to denote (n(S,
t),
unit of the variance on
t
that
11
is
its
- r)/a 2 (S,t), which
t)
rate of return.
is
the risk
Assume k(S, t) /
except
of Lebesgue (product) measure zero.
lemma and
The lemma
a proposition.
derivatives of
R
and
H
.
expresses the function
Then we show
N
in
in the proposition that
terms
R
must
satisfy a linear partial differential equation.
Lemma
Let(C,A)eS. Then,
1
l
N
=
for (W,S,t) 6 (0,oo) x (0,oo)
J>Aifl.\
A
r' \R) w
^
,^Acf}.\
AS
{R) 5
l
x
[0,T],
Jri^
-r' s \-R) s
MrW + A(p-r)~ C)~ - ^5~ - ^j- That
is,
Proof.
N
r.
can be expressed in terms of R, H, C, A, and the derivatives of
We
will
(24)
R
and H.
prove (24) for (W,S,t) G (0,oo) x (0,oo) x [0,T). The assertion for
= T
t
then
follows from continuity.
Differentiating the Bellman's equation with respect to
using the
first
W and simplifying the resulting equation
order conditions (7) and (8), we get
= Jwt + rJw + Jww(rW + A{ii-r)-C) + JwsnS + -a A
where we have used the
implies that the drift of
we conclude
fact that
dJw
is
Cw =
-rJw-
on the
set
Jwww + Aa SJwws + ^o-
{(W, S, t)
:
C(W, 5, t) =
0}.
S Jwss,
This equation
Since (8) implies that the diffusion term of d Jw
is
—aaJw,
that
dJw = —rJwdt — noJwdw.
Hence, the
1
2
2
drift of
We
Jw must
be equal to — r -
2
2
j—- However, the
:L
A2 (JWW\ +(,2 AS (JWW\ + 1JS*({WS\
V Jw J s
\ Jw ) s
\ Jw )w
&
q
,
-
din
Jws
Jw
(25)
+{r
drift of
and
this
Jw
is
WW
W + A(n-r)-C) JJw
Jwt
Jw
-V* (j) w - ° 2AS (3). + r' s2 (i)s- trW+ Ai " ~
get (24)
din
completes our proof.
I
r)
" C)
* + "4
+
*
NECESSARY CONDITIONS FOR EFFICIENCY
3
Proposition 2 For (W,S,t) 6 (0,oo) x (0,oo) x
12
[0,T], the function
R
must
Rt
+ C W R -rR =
satisfy the following
partial differential equation:
-a 2 A 2 R ww +
Ao-
2
SRws +
-^o
Consequently, R(W(t),S(t),t)e~ rt
positive local martingale
Q
14
2
S 2 R Ss + (rW - C)R W + rSR s +
+ fQCw(^)R(s)e~ r3 ds
and thus a supermartingale
,
0.
(26)
(along the optimally invested wealth)
15
is
a
under the equivalent martingale measure
,
on[0,T}.
Proof.
W yields
Differentiating (24) with respect to
wm
R/ww
+(rW +
2
A(fl
+o- 5/l Vv
+ *As(r\
-^s'(f)
2
v/i/ws
\RJws
-r)-C) (1)
- nS
(-) + (t + A w (fi -
r)
+ a 2 AA w (I)
(j£)
w
- Cw )-
s
Using (9) and (11), we can rewrite the above equation as
R2
\R)ww
2
\R/ws
+ M 5 (i)
+ {rW + A (/x -r)-C) (1)
+a 2 5Aw
+(r-Cw
(-J
\RJ ss
2
+ a 2 /!^
(^) w
+ Aw(n-r))-ji
Since
MJxy~
(:
for A",
V = W, 5, we
-<r
2
obtain
A2 Rww + Aa 2 SRws +
= a-iAiXk + 2 a 2
K
2
^S
AS^
K
+A(n-r)R 2 (j)
2
+ (rW - C)R W + rSR s + R + C W R - rR
iiIss
t
+ a 2 S 2 ^- +
R
++a 2 AA w R
2
(r
^
- »)SR S
+a 2 SA w R 2 (j) + A w (^-r)R
^
2
2
=
^{AR W
+ SRs) 2 +
(r
- n)SR s + {n~ r)(A w R -
AR W
)
-a 2 A w {AR w + SRs).
14
The
process
X
is
a local martingale under
{X(t Ar„), t g [0 T]}
Liptser and Shiryayev (1977,
so that
15
The
process
X
is
is
{'
Q
there exists a sequence of stopping times r„ with rn
if
a martingale under
Q
for all n.
p. 25).
a supermartingale under
Q
if
— T
•
Q-a.s.,
For the definition of a stopping time see, for example,
E*[.Y(s)|^"t ]
< X(t)
Q-a.s. for s
>
t.
3
It
NECESSARY CONDITIONS FOR EFFICIENCY
13
remains to show that the right-hand side of the above equation
A
—
R
is
zero.
We
note by (8) and
(9),
_ff
= K + b—.
R
Hence,
\Rj\y
V
R2
RJw
which implies
A w = (SR S + AR W )/R.
Substituting this expression into the right-hand side of (27),
(28)
we confirm
that the right-hand side
is
indeed zero.
For the second assertion,
we
first
show that
R
being a local martingale
is
implied by (26).
Condition A5, we know that the wealth never reaches zero before T. For any
lemma
to R(t)e~
ri
From
6 [0,T), apply
t
Ito's
and use (26) to get
R(W(t),S(t),t)e-
rs
+ f C w (s)R(s)e- ds
Tt
Jo
=
R(W{0),S{0),0)+ I e- TS (R w (s)A(s)a(s) +
R s (s)(T(s)S(s))dw*(s),
te[0,T).
Jo
Since
side
is
R >
for all
W
>
and
Cw
Q
a local martingale under
>
0,
the left-hand side
on [0,T) since
it is
martingale (see Footnote 14), a local martingale on
[0,
T],
In addition,
it is
is
strictly positive.
an Ito integral. By the definition of a local
[0,7") is
automatically a local martingale on
well-known that a positive local martingale
example, Dybvig and Huang (1988,
a supermartingale on [0,T].
Cox and Leland (1982)
lemma
2).
Thus R(W(t),
is
S(t), t)e~
are the
first
is
in fact a general
no intermediate consumption,
it
states that the risk tolerance implied
be a local martingale under the
is
property of an efficient policy. In the case with
risk neutral probability
have constant relative
measure.
and the optimally invested wealth
A) must make the
by an optimal policy must
This
is
certainly true for the
risk aversion, since the risk tolerance in this case
Cw
>
in units of the
bond
is
is
a
a martingale
0,
the above proposition states that an
risk tolerance in units of the
bond a supermartingale under Q on
risk neutral probability. In general, given
efficient (C,
+ JQC w {s)R{s)e- T3 ds
a geometric Brownian motion.
is
under the
Tt
see, for
to point out this local martingale property of the risk
Proposition 2 shows that this
linear function of the wealth
a supermartingale;
I
tolerance process in the context where the stock price process
utility functions that
The right-hand
[0,T] - one expects the risk tolerance in units of the bond, on the average according to Q, to go
down
in
the future.
NECESSARY CONDITIONS FOR EFFICIENCY
3
Remark
1
implicitly
assumes that
In Proposition 2
martingale result
proof
is
the local martingale result
dynamic consumption-portfolio problem of
of course true starting from any
Our
identical.
is
the
we have stated
14
implicit hypothesis
t
on
the whole o/[0, T).
=
0.
This local
x and S(t)
=
y and the
made
henceforth
(3) starts att
€ [0,T), W(t)
=
for notational simplicity and will be
is
This
unless otherwise noted.
We
record an immediate corollary of Proposition 2.
Corollary
Let {C,A) € €.
1
Then
aw - swm e -r, +
K(t)
is
a positive local martingale
Proof. The
AM
l KM
,
c
[ Cwis)
~
•
and thus
deserve attention.
1
H=
0,
k
a constant,
is
Q
a supermartingale under
is
geometric Brownian motion stock price with
zero as
n(t)
assertion follows directly from (8) and Proposition
Several implications of Corollary
ms \-ds
- s
}'
A = kR >
/x
>
when
r.
0,
[0,T].
I
2.
First, consider the special case of a
Note that
W>
on
in this case the
and Condition A5
is
hedging demand
satisfied. Corollary
is
1
implies that
A{t)e~
rt
+ [ C w (s)A(s)e- r3 ds
Jo
is
alocal martingale under
Q
on [0,T]. In addition, by Proposition
-a 2 A 2 A ww + {rW - C)A W -rA + C W A + A =
t
This
is
just proposition 3 of
Cox and Leland
a positive supermartingale. That
future
mean
is
less
is,
(1982).
16
Cw
Since
>
the present value of the dollar
than the current amount invested
in the stock.
(W,t) £ (0,oo)x [0,T],
2, for all
0.
0, these
imply that A(t)e~ rt
amount invested
1
in stock in the
This, however, does not necessarily
To
that one expects to shift value over time from the stock to the bond.
from Proposition
is
see this
we
recall
that
W(t)e- Tt + / C(s)e- r 'ds
t
€ [0,T]
Jo
is
a martingale under Q. This implies that
rt
(W(t) - A(t))e~
+ f\c(s) - C w (s)A(s))e- TS ds
t
€
[0,
T)
Jo
16
The
reader familiar with
and Leland. Here A(t)e~ rt
Cox and Leland
will
+ J Cw{3)A(s)e~ r 'ds
note a difference between the result reported here and that in
is
Indeed, with additional regularity conditions, A(t)e~
below.
shown
r>
to be a supermartingale
+ f Cw(s)A(s)e~ r ' ds
under
Q
Cox
instead of a martingale.
becomes a martingale;
see Corollary 2
/
NECESSARY CONDITIONS FOR EFFICIENCY
3
is
15
a submartingale under Q. Note that the difference between W(t) and A(t)
C - Cw A <
invested in the bond. If
and the optimal policy
C - Cw A >
0,
however, {W(t) - A(t))e~
interpretation of Corollary
premium per
to the
for
in the general case
1
example that
n{t)
H
•
Se~ Tt
is
,
Under
below
this result
demand
Cw ^
When
to consumption.
a bit more complicated as there
Cw =
0.
When
positive),
is
in the
Ae~
rt
is,
Proof. The
that
assertion
demand
H(W,S,T) =
and S
/k
is
a supermartingale
stock in the future, per unit of k,
Interpretations similar to the
k.
Cw —
shift of
indeed a martingale under Q.
as well A.
and
-rJ
a.s.
ds
and
R and
Cw
+
record
5.7.6).
polynomial growth condition. 17
I
we have a sharper interpretation
of Corollary 2,
For example,
W(T) >
left
satisfy a
a martingale on [0,T] under Q.
is
a.s.,
in the
A(t)e~
Tt
to the reader.
We now
of the intertemporal
geometric Brownian motion case discussed above,
becomes a martingale under Q, and thus there
value over time away from or into the stock.
interpretations are
We
Tt
a consequence of the Feyman-Kac representation; see, for example, Karatzas
is
With the conditions
R
W(T) >
CV(s).ft(.s)e
'
and Shreve (1988, theorem
behavior of
the
the second corollary of the proposition:
in
Then R(t)e~ rt +
is
is
the hedging
but with everything normalized by
not only a local martingale but
is
Corollary 2 Suppose
when
on [0,T],
certain regularity conditions, Proposition 2 can be strengthened so that R(t)e~
L Cw(s)R(s)e~ r3 ds
no
is
lower than one's current investment in the stock, per unit of k.
made when
Q
to consumption.
a decreasing process (given that
positive, this implies that the hedging
special case above can be
under
a positive increasing process; that
is
under Q. The present value of one's optimal investment
is
bond and
amount
the dollar
can indeed be a supermartingale under Q. In such a
unit of variance increases over time, and
normalized by the bond price,
is strictly
Tt
is
also a submartingale
is
away from both the stock and the bond
now hedging demand. Assume
risk
A)e~
away from the stock
shifts value
case, the policy shifts value
The
(W -
0,
Tt
This
is
the Cox-Leland result.
is
Other
proceed to complete our derivation of the necessary
conditions for (C, A) to be efficient.
First, consider the special case that
Since the
tions
first
C(W,S,t) >
for all
(W,S,t) € (0,oo) x (0,oo) x (0,T).
order condition (7) holds as an equality, the chain rule of differentiation and Condi-
A2 and A3 imply
that
9
=- ^(?cCW
wg',; VW>0,S><M€[0,r).
t)
H(W,S,t)
,
See Footnote
9.
(29)
NECESSARY CONDITIONS FOR EFFICIENCY
3
Conditions
A2 and A6 immediately
16
necessitate that
CclW
=
1^-5^1
9
E(W,S,Ti.
t\
Note that (30) places nontrivial restrictions on consumption
policy that
for
is
time separable,
some functions
c
and
policies.
C(W,S,t) - c(W,S)f(t)
in that
/, can never
be an
efficient
W>0.
0,
(30)
For example, any consumption
for all
W>
S >
0,
consumption policy unless
and
0,
it
t
6 (0,T)
independent
is
of the stock price.
Now
Cs
R = A+S
We
R
have thus expressed
H
The R,
A6
substituting (29) into (8) and using Conditions A2, A3, and
,
N
and
H
and
so defined
must
VH' > 0,5 > 0,t€
/k,
solely in terms of
give
[0,T].
(31)
C, A and their derivatives. Define
N
and
1
Let (C,A) €
£ with C(W,S,t) >
and
as in (31), (29), (30),
(15), respectively.
We must
= C(0,S,t) =
(1)
A(0,S,t)
(2)
R(W,S,t)
(3)
C w {W,S,t) >
0,
for
0,
and
A(W,S,T)/k(S,T)
S >
0,
C w (W,S,t) >
C(W,S,t) and
is
a function
satisfies the
PDE
0,
then
C =
for
t
W
R
0,
S >
0,
and Q, X, Y, and
and
H
in
>
U(C~
and
U
t
6 (0,T). Dearie R,
as in (12), (13), (14),
x
0;
{x, S,t),S, t) is
< T, U(W,S,T)
ofW
all
and (11)
hold;
is
independent of
S
independent of S for
and S >
S >
only for all
C
0,
lim t|r
for all x
>
W
0,
all
Cwtwf}) =
°>
>
and
W > 0;
and
can be zero at strictly positive wealth levels. For example,
for all wealth levels.
directly using (7) in the region
expressing
>
(26).
Second, consider the general case that
when u =
W
e [0,T];
t
(5) the consistency conditions (9), (10),
A
2.
have
]imwioU(W,S,t) > Kmwioli(W,S,T) for
and
and
>0 forW >0;
in the range of
{C,A)
all
(24), respectively,
(4) the state-independency conditions hold:
(6)
(24).
following theorem summarizes the above discussion.
Theorem
H,
by
satisfy the necessary conditions stipulated in Proposition 2
the consistency conditions and the state-independency conditions derived in Section
The
N
where
In this case
C-
0.
The
H
cannot be expressed
following proposition
terms of C, A, and their derivatives generally.
is
in
terms of
C
instrumental for
NECESSARY CONDITIONS FOR EFFICIENCY
3
Proposition 3 Let (C,A) €
£.
For
W>
R+
TX
S >
0,
0,
17
and
t
€ [0,T],
T 2 H = oK{W,S,t),
(32)
where
Ti
=
--
r2
=
- s vsso S +
2
ksso-
S 2 + 2ksfiS + 2k
2ctso-'S
t
\
,
+ 2a s rS + 2a
t
2
K(W,SJ)
\a 2 A 2 A Ww + o 2 SAAws + \o 2 S 2 A SS + (vsaSA + rW)A w
=
+(a s aS 2 + Sr)A s + -{ass^S 2 + 2ra s S/cr - 2r + 2a t ja)A
+A +
t
We
Proof.
Conditions A2, A3, and
A6 by
For any function / of
tinuously differentiable in
-a 2 A 2
W
i,
—
dW
d 2 Jf
-r
2
£
(!)
€ (0,T).
t
that
t
is
d2
f
+ a 2 SA—-£- +
dWdS
W>
0,
S >
«*-* (s) -' (f H<-
=
and T, the assertion follows from
0,
2
1
-<J
2
2
and
C under Q:
S 2 ^r2 + (rW - C)
dW
ds
t
dS
dt'
£ (0,T),
+•»*,* (i) + (,_cw)(i);
^ (*l <^ (sL-s^
+
(33)
(I ),•
™
fact that
= Rs<?S + RwAa,
(8), (10), (11),
and (24)
for (34).
Following
we have
C{oA) =
Since
'
a consequence of (28), and (26) for (33), and
the definition of A",
t
define the differential generator
aAwR
which
At
.
twice continuously differentiable in (W, S) and con-
is
(i)= -[.•**, (i)
where we have used the
SC s
continuity.
5 and
,
Direct computation shows, for
£
)
prove the assertion for
will
£(/) =
(C w A-CA w +
oK +
ra A -
oACw — oSCs-
z£ = -K + a5^,
['£)-£(*> + *(*!)
(35)
3
NECESSARY CONDITIONS FOR EFFICIENCY
Applying
18
we
substituting (33), (34) and (35) into the above equation,
Proposition 3 plays an important
in
X W YS + a
C(XY) = YC(X) + XC(Y) + a 2 A 2 X w Yw + a 2 AS(X s Yw +
terms of
A and C and
from (8) and (32)
R
for
their derivatives. This
H
and
as functions of
R
get (32).
allows us to express the
role. It
is
A
done
and
=
unknown
functions
+ Y2 k
derivatives, except
a 2 SK-aAT 2
cr r^r
aSVi
+ ki 2
S 2 X s Ys and
I
as follows. If Y\crS
its
2
)
i=-
0,
R
we can
W=
when
and
solve
at T:
,
„ST
aol + k1r 2
r.
36
s
'
oATx + koK
H _
H
I
•
37 )
1
Once
this
is
N
(24) expresses
If
get a
k
=
0.
that
terms
solely in
Y\oS -\-Yik =
PDE
C
0,
and
we cannot
A must
(Here we note that
K
Then we have
when k =
0.
PDE in A and C. In addition,
of C and A and their derivatives.
done, (26) becomes a
=
Then
if
satisfy.
k
/
0,
We
N
in
terms of
take two cases. First, Ti
=
only
if
when k =
0.
Tj
except possibly
(8)
H, and
solve for R,
T2
=
0.
substituting (36) and (37) into
C
and A. Nevertheless, we
= T2 =
except possibly when
In addition, T 2
Second, Ti
/
still
and T 2
=
=
implies Ti
^
0.)
except possibly
and (32) imply that
41-A-?
aS
A
(38,
k
except possibly when k
verify.
We now
Theorem
(1)
(2)
0.
£,
= C(0,S,t) =
A(0,S,t)
we have an equation
In both cases,
collect all the necessary conditions
For (C, A) €
2
5>0,
=
solely in
any (C, A) € £ has to
terms of
A
and
C
to
W
>
0,
satisfy:
we must have
0,
S >
and
e [0,T]; and C\y(W,S,t) >
t
when
C
>
0,
te[0,T\;
+
Suppose T\aS
define Q,
X
,
T2 k
and
Y
^
0.
Define R,
H
,
N
and
as in (36), (37), and (24), respectively; and
as in (12), (13), and (14), respectively.
>0 forW >
(a)
R(W,S,t)
(b)
the state -independency conditions hold:
Then
0;
U(C~
l
(x,S,t),S,t)
is
x>0andt<T,
U{W,S,t) > \im x[0 U(C- 1 (x,S,t),S,t) for
te
[0,T) such that
C(W, 5,0 =
for
all
S >
function of
and S >
W
only;
0,
and
all
H(W,S,T) =
S >
0,
independent of S for
all
W
>
0,
5 >
0,
all
and
limwio U(W,S,t) > ]im wl0 U(W,S,T)
for all (W,S), and
A(W,S,T)/k{S,T)
is
a
NECESSARY CONDITIONS FOR EFFICIENCY
3
(3)
(c)
the consistency conditions (9), (10),
(d)
(C,A)
Suppose
k
(4)
=
=
that Y\
^
=
and (11)
and
hold;
(26);
except possibly
when k =
K
Then
0.
=
except possibly
when
and
0;
Suppose that T\o-S
+
holds except possibly
Note that
PDE
satisfies the
19
Theorems
in
1
^k
=
and
when k —
and
2,
(26)
T\
and r 2 ^
/
except possibly
when k =
0.
Then
(38)
0.
is
the most substantive necessary condition. Other conditions
are either consistency conditions or the state-independency conditions.
We now
present two examples, one to demonstrate the necessary condition for efficiency and
another to demonstrate a price process where TicrS
Section
+ I^k =
1
>
with f(t)
has been asserted in the literature that the pair
It
0,
is
C(W,S,t)
=
f(t)W,
A(W,S,t)
=
^f'^ W,
the optimal consumption-portfolio policy for the log utility function with certain
time preferences captured by f(t); see Merton (1973). 18 Since
must
satisfy the conditions of
First, the
R(W,
S,t)
= W.
Theorem
C >
for
all
W
to
>
0.
1
C
>
for all
W>
and S >
0, this
1.
H(W,S,t) =
that
and thus
Direct computation using (24) gives N(t)
Next we turn our attention
of whether
Theorem
consumption policy implies
that all the conditions of
We
in
5.
Example
policy
More examples can be found
0.
=
—/(<)•
the portfolio policy implies
One can
then easily verify
are satisfied.
Theorem
2.
Note that
this
theorem applies generally independently
Direct computation shows that
K{W,S,t) = Ti(S,t)W/<r(S,t)
.
take cases.
Case
1.
Suppose that YxO~S
+ I^k ^
R
.
H =
18
0.
By
(36) and (37),
«*r,/>
we have
M-,
t«r
aSTi +
2
-* r + *»ri/ a
w=
crSri + kT-2
i
Q
W
solves the Bellman's equation for
This assertion was supported by the fact that J(W, S, t) = g(S, t) + h(t) In
g and h. However, this J function fails to satisfy the polynomial growth condition, which is sufficient
to derive the Bellman's equation. Thus, to our knowledge, it has not been actually shown that an optimal policy
exists for the log utility function using dynamic programming. However, this policy can be verified to be optimal
using other arguments such as those of Cox and Huang (1989, 1991).
some functions
NECESSARY CONDITIONS FOR EFFICIENCY
3
20
These are consistent with our calculation above while using Theorem
2.
Suppose
that T\
Case
3.
Suppose
that Y\o~S
T?
=
K
Then
0.
+ I^k =
-
=
0.
and
and T\ /
T\aS + I^k =
2 Suppose that
Note that T\
is
the drift
Theorem 2
(3) of
and Yz /
— Yi —
and a
straightforward
is satisfied.
Then Kj A — -Ti/k and
0.
(4) of
is
a local martingale.
Conversely, given that a
is
a constant
-a 2 A 2 A W w + a 2 SAA W s +
is
0.
a constant.
term of dk. Thus k must
n
constant, this implies that
K=
is
is satisfied.
following example gives a scenario where T\
Example
Ti
-
Case
The
it
(2a)-(2d) of Theorem 2 are satisfied.
to verify that
Theorem 2
Then
1.
2
2
^SA
and
Ss
p.
Then 1^ =
be a local martingale. Since
^=
=
a local martingale, T\
is
This implies that
0.
0.
a and
r are
In this case,
+ rWA w + rSA s -rA + A + C w A + SCS - CA W =
t
0.
Using the same arguments as in the proof of Proposition 2 we have that
A(t)e-
rt
+ f\c w (s)A(s) + S(s)Cs (s) - C(s)A w (s))ds,
t
€
[0,
T)
Jo
is
a local martingale under
Corollary
2.
In particular,
Q
and
ifC =
we know A(t)e~
in Corollary 2,
is
0,
rt
a martingale under
W(T) >
must
a.s.,
Q
and A
be a martingale
with similar regularity conditions as in
satisfies the
under Q.
growth condition stipulated
Thus
the present value of the
future investment in the stock must be equal to the current investment, a property obeyed by any
optimal policy satisfying the same regularity conditions in the geometric Brownian motion case.
Besides checking whether a given policy (C, A) satisfies the necessary conditions for efficiency
a fixed stock price process, Theorems
for
1
and
2 can also
be used to answer a more general
question: For a given policy to satisfy the necessary conditions for efficiency,
price process?
The
following familiar example demonstrates this.
Before presenting our example,
motion, and when the (direct)
risk aversion, the
a,
what should be the
we note
that
when
the stock price follows a geometric Brownian
utility function exhibits
optimal portfolio policy
and the optimal consumption policy
is
is
a constant Arrow-Pratt measure of relative
a constant mix policy; that
a linear policy C(t)
= f{t)W
is,
for
aW
A{t)
=
some
strictly positive
for
some
function f(t). 20
19
He and Leland (1992) show that when a is a constant and /i is a local martingale, £ is path-independent. Hence,
any optimal policy must be path-independent.
The astute reader will question whether one needs to restrict the coefficient of relative risk aversion to be less
than one since otherwise the indirect utility function fails to satisfy a polynomial growth condition. The linear policy
can however be shown to be optimal using a different technique; see Cox and Huang (1989).
4
SUFFICIENT CONDITIONS FOR EFFICIENCY AND RECOVER ABILITY
On
the other hand,
it
seems quite
likely that for this pair of linear policies to
21
be optimal,
would be necessary that the stock price process be a geometric Brownian motion, and the
function exhibit a constant Arrow- Pratt measure of relative risk aversion.
show
and
in
p.
Using Theorem
it
utility
1,
we
the following example that this pair of linear policies only necessitates that k be a constant,
and a be independent of S, except
Example
3 Let C(W,S,t)
consumption policy
= f{t)W and A(W,S, t) =
independent of S,
is
where a = k (which
in the case
H
=
(n- r)—2
,
where f(t) >
This implies that
0.
—+
N=
aW
Example
and a /
1).
0.
Since the
R(W, S, t) = aW/ n(S,t) and
f{t)--r + a 2 Sn s
a
is
.
a
Thus
\nU(x/f(t),S,t) =
By
^^[-\n(i/f(t)) + \nW) +
Y(S,t).
must
the state-independence of the utility function, the right-hand side
Thus k
is
a function only oft,
Assume without
and hence,
N
=
loss of generality that n(t)
=
This implies K'(t)aW
for
all
Relation (39) thus shows that the
W
(39)
Q
>
utility
0,
^
and
-
(/i
- a)/2
r)(*
+
(r
be
independent of S.
— f(t))n/a —
r.
Then R(W,S,t) = aW/K(t) must
0.
must
n(t)
be
independent oft and
is
satisfy (26).
a constant.
function exhibits a constant relative risk aversion equal
to
fc/a.
The fact
that k
is
a constant does not necessarily
mean
that
fi
and a are constants. Now note
that the consistency condition (10) implies that
Ns =
Suppose k
^
a.
Then
p.s
=
and
p.
is
k— a
Us—
—
0.
independent of S. Consequently, a
In summary, for (C, A) to be efficient,
it
is
necessary that k
function must exhibit a relative risk aversion k/q.
4
=
In addition,
Sufficient Conditions for Efficiency
In this section
we
first
show that with some minor
derived in the previous section are also sufficient.
independent of S.
a constant, and the utility
and a are functions of time
if
and Recoverability
regularity conditions, the necessary conditions
We
then discuss
function that supports a given efficient policy. Specifically,
the utility function.
p.
is
is
how one can
we provide an
recover the utility
integral formula to recover
SUFFICIENT CONDITIONS FOR EFFICIENCY AND RECOVER ABILITY
4
We give
A5
two
A) that
sets of sufficient conditions for a given (C,
C
of Section 2 to be efficient. First, for a
necessary conditions recorded in
Theorem
1
C(W,S,
such that
to be efficient.
Second, in the case where consumption
positive wealth
and where TioS
the
H
same conditions on R,
,
and
N
A)
is
W>
for all
A3 and A6
and S >
the
0,
we construct a
are sufficient for (C, A)
not always strictly positive for strictly
the necessary conditions of
Our proof
are also sufficient.
conditions are through construction;
policy (C,
0,
>
Conditions Al, A2, and
together with the hypothesis that the R, H, and JV
defined in (31), (29), and (24), respectively, satisfy Conditions
+ Tjk ^
t)
satisfies
22
Theorem
2 together with
two
sets of sufficient
for these
pair of utility functions (u,
V)
so that the
solves (3).
Since any efficient policy must be such that C(0,S,
t)
=
and A(0,
=
S, t)
0,
that
the wealth reaches zero there will be neither investment nor consumption afterwards,
our attention to this kind of policies.
Huang (1988)
that
For any one of these policies,
it
is,
we
whenever
will restrict
Dybvig and
follows from
we must have
"
r
E*
/
T
C(W(t),S(t),t)e-
rt
dt
+ W(T)e»e- rT
>
< W(0),
(40)
Jo
that
is,
the present value of future consumption and final wealth
By concavity
wealth.
must be
of the utility functions, sufficient conditions for (C,
less
than the current
A) to be a solution
are that (40) holds with equality and there exists a strictly positive scalar A
>
to (3)
so that, for
all
*€[o,rj,
where we
recall the definition of £(f) in (1)
time
time
for
t
and
its
interpretation as the
consumption per unit of probability P. 21 The following
is
Arrow-Debreu
our
price at
of sufficient
first set
conditions:
Theorem
A5,
3 Let
andC >
So
for all
andQ, X, Y, and U
(1) R,
H,
satisfy a linear
N
W>
growth condition and
and S >
0.
Define R, H,
let
N
(C, A) satisfy Conditions
A I,
.4-',
and
as in (31), (29), and (24), respectively,
as in (12), (13), (14), o.nd (15) respectively. Suppose that
satisfy the continuity
and
differentiability conditions
of Conditions
A3 and
A6; and
=
but our
Here we remind the reader of our implicit hypothesis that the starting date of the optimization
arguments apply to other starting dates t with arbitrary starting states W(i) and S(t).
is
t
4
SUFFICIENT CONDITIONS FOR EFFICIENCY AND RECOVER ABILITY
Theorem
(2) conditions (l)-(6) of
Then (C, A) € S and
are satisfied.
I
the utility functions correspond to (C,
M
When C
(31),
is
U(C- l (x,S,t),S,t) x>0,
\ ]imziQ U{C-\z, S,t),S,t) x = 0;
,J
4 Let So
satisfy a linear
U
X, Y, and
H
,
and
set of sufficient conditions.
whenever VictS + T2K /
and A5, and T^aS + I^k /
(1) R,
°c
not always strictly positive for strictly positive wealth,
applies generally
define Q,
X>
S
"}r sr\
r
I
and we have the second
Theorem
A) are
{
hV(.) =
Proof. See Appendix.
N
(2) conditions (I)
0.
0,
Define R,
cannot be defined through
independently of whether C(W,S,
H
and
,
N
this set of conditions
t)
>
for all
W>
0.
(C, A) satisfy Conditions At, A2,
let
as in (36), (37), and (24), respectively; and
as in (12), (13), (14), and (15), respectively. Suppose that
A3 and A6; and
and (2a)-(2d) of Theorem 2 are
satisfied.
A) are
the utility functions that correspond to (C,
,
R
Note however that
growth condition and
satisfy Conditions
Then (C, A) € £ and
23
M,J
,
,,
_
"
U(C- l (z,S,t),S,t) x>0,
1
\ lim, io W(C- (*,5,0,S,t) * = 0;
j
I
Proof. See Appendix.
hm z ioU(z,$,T)
i
=
0.
I
Using Theorem 3 one easily shows that the linear policies
Example
in
3 are indeed optimal for
the stock price process identified there and the utility function exhibiting a constant relative risk
aversion equal to k/q.
Of
course, this result
well-known from solving the dynamic consumption
is
and portfolio problem using dynamic programming.
The
general procedure to recover the utility function should be obvious from the proofs of
Theorems
3
and
4.
First, define R,
as in (12), (13), (14),
and
H N
,
(15), respectively.
Now,
m zl0 ZV(C=
(..
1
3
and
4.
Second, define Q, X, Y, and
define \J(x,t) and
U(C-
lnU(x,0
InV,x)
Theorems
as in
l
{x,S,t),S,t)
1
(^,5",t),5 ,0
,
«(•*.£
hm z[0 U(z,S,T)
•>«
x
=
0.
x
x
V such
>
=
0,
0;
that
U
FURTHER EXAMPLES
5
Then, the
24
utility functions that
support (C, A) must be
u(x,t)
—
\
V(x)
=
[
U(x,t)dx,
X
V(x)dx.
These are the integral formulae which recover the
utility functions that
support a given
efficient
policy (C,A).
Before leaving this section, we point out that the necessary and sufficient conditions established
in Sections 3
and 4 can be readily extended to
this case, in
Theorems
Theorems
and
3
4,
and
1
conditions relating to time
2, all
problems,
infinite horizon
T
i.e.,
T=
oo and
V=
0.
In
should be removed. In addition, in
one needs to add that the present value of the future wealth goes to zero when
the future extends to infinity; that
is,
E*[M/ (<)e _r( —
as
]
t
—
oo.
22
Further Examples
5
In this section
Example 4
we present two more examples
to
demonstrate our
Consider a pair of consumption and investment functions
c(w,s,t)
=
fttywMs 1 -*®
2
V (j(S,t) p
where f
is
results.
a(t)
a and
a strictly positive deterministic function,
and
values between
1
satisfying \\m t ->T
Q (0 —
1>
an d ^ n ^ P
may
parameters of the wealth process generated by (C,A)
Lipschitz condition for
some functions a and
We
(3.
)
are deterministic functions with
ls
a strictly positive constant.
The
not satisfy a linear growth and a local
will ignore this
problem for now and proceed
with other necessary conditions.
We
t,
and
To
will
show
either p
=
begin, the
that for this pair of policies to be efficient,
1
or
(p.
- r)/a
is
it
consumption policy determines
the hedging
demand function
a(t)
which in turn determines the risk tolerance function
R(W,S,t)
= ±wW>S l -W>.
P
See
Huang and Pages
(1992).
necessary that
independent of S.
"W**)- Cw (W,S,t)
2
is
5'
a =
(3
=
1
for all
=
FURTHER EXAMPLES
5
—
Since a(t)
as
1
—
we have H(W,S,t)
—> T,
t
25
as
t
]
T, which
is
(30)
and
is
part of the
state-independency condition.
Next, for (9) to be satisfied, we need
-
p(/3(t)
This implies that
f3(t)
—
^W-^S^for
1
all
t
2
=
-/>(/?(*)
and thus
\
p
Now, define
N
N
- i)iz^|2w-/»W5/»(«)-».
o{S,typ
a(t)
J
according to (24):
^^
WU].^^
W
1
=
2
2
V
2
J
o(0
V
/
S2
HrW + A(p - r) - C(t)W^S -^)^ + p.S (~j^|)
1
5mce
^w
= Rt/R 2 =
6y
u;e
(//,),
>
the hedging
5 >
0,
and
0,
demand must
f
€ [0,7].
be zero
-
r.
deduce that
- l)W(0-l5»-«W -
-f(t)p(a{t)
/or a// iy
^
-
=
cannot be true unless a(t)
T/iz's
1
for
t.
Consequently,
the
parameters of
all
and
Equation (10) then implies that
Ns
and we must
either have p
—
1
K
(k2cj2)5
1
or (n 2 a 2 )s
the wealth process satisfy a local Lipschitz
conditions,
In
and
summary,
be the case that
of S. Indeed,
S
,
in the latter case the
= {H/R)t =
")
=
0.
Note that
in the
and a growth condition
when p
(3
—
be constant
1, the
former case
if (p.
—
r)
2
/a 2
also satisfies these
parameters of the wealth process are purely deterministic.
in order for the pair of policies to be efficient for
a and
°'
and equal
to one,
and
some
either p
log utility function supports (C, A),
=
1
stock price process,
or (p
- r)/a
and when k 2 o~ 2
is
is
it
must
independent
independent of
the utility function that exhibits a constant coefficient of relative risk aversion equal to
p supports
(C,A).
Note that
Huang
in the case
where n 2 a 2
is
independent of S, k
(1989), there will be no hedging demand.
is
independent of S.
The optimal consumption policy
From Cox and
will be a function
FURTHER EXAMPLES
5
26
only of wealth, and the optimal portfolio policy can be calculated as an explicit integral and can be
represented generally as
——g(W(t),t)
A(W,S,t) =
for
some function
Example
g.
5 Consider a pair of consumption and investment functions for an infinite horizon prob-
lem,
where f(S)
= A 3 - r/(aa 2 ) -
y/A\
C{W,S,t)
=
rsw,
A(W,Sj)
=
f(S)W,
+ A 2 S, where a >
are constants. Note that the marginal propensity to
the proportion of the wealth invested in the stock
also that with (C,
A) defined above,
We
will ignore this
answer
is
a >
may
some
utility
is
proportional to the stock price, and
consume
0,
= aa 2 (A 3 +
1
consumption policy implies
R=
Using relation (24)
~2
23
We
and A3
may
not satisfy a linear
not satisfy the integral condition of Condition
We
ask:
Can
We
(C, A)
will see
- ^Ai + A 2 S(t))S(t)dt + aS(t)dw(t),
the price process satisfy certain restrictions,
ao
-^
0,
a decreasing function of the stock price. Note
this stochastic differential equation,
martingale measure exists for this price process on any finite interval
=
0,
function and for some stock price process?
provided that 1) there exists a solution to
N
A2 >
0,
affirmative for the following price process:
dS{t)
Clearly, the
A\ >
>
problem and proceed with other necessary conditions.
be a pair of optimal policy for
that the
7
parameters of the wealth dynamics
the
growth condition and the consumption policy
A2.
is
0,
f
and 4)
that
Thus
L±lw= l_ w
a
k
k
3) the parameters of
the wealth never reaches zero. 23
H = -W/S.
A-SH_ =
[0,t],
2) the equivalent
we.
write
A3
n.
--1
~-^A
+A
S) -Ua
+ A7—z\
-^-r-VM
2
r
!
1
2
2
ignore the condition that the expected discounted wealth under the equivalent martingale measure goes to
This condition may be checked by simulation. Also, using the boundary conditions discussed in Chapter 5,
Gihman and Skorohod (1972), one can check that zero is inaccessible for this price process.
zero.
A
CONCLUDING REMARKS
6
+q
(r
27
+ aa 2 \A3 -
+a 2 a 2 (a3 +
1
-
-~ -
v^ + A
-a^S - -aa 2 + 2ar We now
a
choose 7 and
a
5mce 7
or
if
5 and
-A
1
is
positive
a constant.
is
and
2
2 <r
A3 +
/ias £0 6e strictly positive,
A3 +
-
s) - ?-£- (a 3
2
-^ - j A
-^ +
x
+ A2 S +
l\
-
- ^Aj + A 2 s) -
1
-ys\
r.
r.
as follows.
=
7
\A3 -
s[A x + A 2 s)
(a-
r/a
2
1),
+ r/a'y - 4(A 3 +
2(^3 + 1)
+
s/(A 3
we choose A3 so
sufficiently small.
The consistency and
that
a >
1.
l)r/o*
if r/a
Tats can 6e achieved
N
For the 7 and a defined above,
2
is large,
independent of
is
the state-independency conditions are easily seen to be
satisfied.
Next, one verifies that (26)
all the regularity
is
satisfied and,
by
Theorem
The
conditions stated above can be verified.
3,
(C,
utility
A)
positive
that if
when
A3 > r/(ao 2 ) +
the stock price is low
din 5(<)
Thus
= aa 2 (a3 +
the log price process follows a
greater than one.
then the proportion of the wealth invested in the stock
is
1
mean
have studied the "inverse" of the
negative
-
when
\]
x
the stock price
+ A 2 S(t)j
reversion process
if
dt
+
A3 + 1 —
is
high.
By
Ito's
is
lemma.
adw{t).
l/2a -
y/A~\
>
0.
classical
optimal consumption- portfolio problem of Merton
This inverse optimal problem should be viewed as a dynamic recoverability problem
(1971).
financial
for
and
is strictly
Concluding Remarks
6
We
\/A~\,
provided that
function that supports (C,A)
exhibits a constant coefficient of relative risk aversion equal to a, which
Note
is efficient
markets with continuous trading.
We
have derived the necessary and
in
sufficient conditions
a given consumption-portfolio policy to be optimal for some agent with an increasing, concave,
time-additive, and state independent utility function in an
asset.
The
constant.
economy with one
risky asset price follows a general diffusion process,
and the
risky
and one
riskless
riskless interest rate
is
a
Using identical arguments, we can generalize the results reported to cases where there
6
CONCLUDING REMARKS
are
more than one
risky asset,
We
the interested reader.
and the
28
interest rate
stochastic.
We
leave this generalization to
can also generalize our results to allow state-dependent utility functions
by simply removing
all
derive our results
dynamic programming.
is
is
the state-independency conditions.
It
The technique we have
exploited to
seems plausible that our method might also be
generalized to allow for non-time-additive utility functions as long as the optimal consumptionportfolio
problem
for these utility functions
be a fruitful direction
functions.
for future research
can be analyzed by dynamic programming. This
may
given the increasing interest in non-time-additive utility
7
REFERENCES
29
References
7
1.
F. Black, Studies of Stock Prices Volatility
Changes,
in
Proceedings of the 1976 meetings of
the American Statistical Association, Business and Economic Statistics Section, 1976.
2.
F. Black, Individual
A Guide
3.
to
Investment and Consumption Under Uncertainty,
Dynamic Hedging, Donald
D. Breeden, Futures Markets and
plete Markets, Journal of
4.
F.
in Portfolio
Insurance:
L. Luskin, Ed., Wiley, 1988.
Commodity Options: Hedging and Optimality
in
Incom-
Economic Theory 32 (1984), 275-300.
Chang, The Inverse Optimal Problem:
A Dynamic Programming Approach, Econometrica
56 (1988), 147-172.
5. J.
Cox and H. Leland, Notes on Dynamic Investment
Strategies, Proceedings of
the Analysis of Security Prices, Center for Research in Security Prices
Seminar on
(CRSP), Graduate
School of Business, University of Chicago, 1982
6.
J.
Cox and
C. Huang, Optimal Consumption and Portfolio Policies
a Diffusion Process, Journal of Economic Theory
7.
J.
49
When
Asset Prices Follow
(1989), 33-83.
Cox and C. Huang, A Variational Problem Arising
in Financial
Economics, Journal of
Mathematical Economics 20 (1991), 465-487.
8.
J.
Cox and
S.
Ross,
The Valuation
of Options for Alternative Stochastic Processes, Journal
of Financial Economics 3 (1976), 145-166.
9.
P.
Dybvig, Inefficient Dynamic Portfolio Strategies or
in the
10.
P.
Stock Market, Review of Financial Studies
1
How
The Review of Financial Studies
Dybvig and
11.
P.
12.
W. Fleming and
1992.
S.
Throw Away a
Million Dollars
(1988), 67-88.
Dybvig and C. Huang, Nonnegative Wealth, Absence
tion Plans,
to
of Arbitrage,
and Feasible Consump-
1 (1988), 377-401.
Ross, Portfolio Efficient Sets, Econometrica 50 (1982), 1525-1546.
H.
M. Soner, Controlled Markov Processes and
Viscosity Solutions, preprint,
7
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13.
A. Friedman, Stochastic Differential Equations and Applications, Vol.
New
14.
30
Academic
1,
Press,
York, 1975.
Geanakoplos and H. Polemarchakis, Observability and Optimality, Journal of Mathematical
J.
Economics 19 (1990), 153-165.
15.
I.
Gihman and A.V. Skorohod,
Stochastic Differential Equations, Springer- Verlag,
New
York,
1972.
16.
R. Green and S. Srivastava, Risk Aversion and Arbitrage, Journal of Finance
XL
(1985),
257-268.
17.
R. Green and S. Srivastava, Expected Utility Maximization and
Demand
Behavior, Journal
of Economic Theory 38 (1986), 313-323.
18.
M. Harrison and D. Kreps, Martingales and Multiperiod
Securities Markets, Journal of Eco-
nomic Theory 20 (1979), 381-408.
19.
H.
He and N. Pearson, Consumption and
Portfolio Policies with Incomplete Markets
and
Short-Sale Constraints, Journal of Economic Theory 54 (1991), 259-304.
20.
H.
21. C.
He and
H. Leland, Equilibrium Asset Price Processes, Working paper,
Huang and
UC
Berkeley, 1992
H. Pages, Optimal Consumption and Portfolio Policies with an Infinite Horizon:
Existence and Convergence, Annals of Applied Probability 2 (1992), 36-64.
22.
I.
Karatzas, and
S.
Shreve, Brownian Motion and Stochastic Calculus, Springer- Verlag,
New
York, 1988.
23.
M. Kurz, On the Inverse Optimal Problem, Mathematical Systems Theory and Economics,
H.
W. Kuhn and
Who
G. P. Szego eds., Springer- Verlag, Berlin, 1969.
Should Buy Portfolio Insurance, Journai of Finance 35 (1980), 581-594.
24.
H. Leland,
25.
R. Liptser and A. Shiryayev, Statistics of
Verlag,
26. A.
New
Random
Processes
I:
General Theory, Springer-
York, 1977.
Lo and C. MacKinlay, Stock Market Prices Do Not Follow Random Walk: Evidence from
a Simple Specification Test, Review of Financial Studies
1
(1988), 41-66.
7
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27. A. Mas-Colell,
31
The
Recoverability of Consumers' Preferences from Market
Demand
Behavior,
Econometrics 45 (1977), 1409-1430.
28.
R. Merton, Optimal Consumption and Portfolio Rules in a Continuous
Time Model, Journal
of Economic Theory 3 (1971), 373-413.
An
Intertemporal Capital Asset Pricing Model, Econometrica 41 (1973), 867-887.
29.
R. Merton,
30.
R. Merton, Continuous
31. B. Peleg, Efficient
32.
B. Peleg
43
Time Finance,
Random
and M. Yaari,
(1975), 283-292.
A
Basil Blackwell,
New
York, 1990.
Variables, Journal of Mathematical
Price Characterization of Efficient
Economics 2 (1975), 243-252.
Random
Variables, Econometrica
APPENDIX
8
32
Appendix
8
Proof of Theorem
t
=
with
Without
3.
we assume that the optimization
loss of generality,
W(0) = W_ and 5(0) =
5. First
we show that
(40) holds as an equality.
starts
from
Under Q, the
discounted wealth process and the discounted stock price process become
d{W{t)e- Tt )
d(S(t)e-
where we
satisfy a linear
+
)
-C(t)e- rt dt
=
S(t)e~
2m )e Lmt
\S\
,
rt
+
A(t)(T{t)e-
rt
for all integers
Al
(see Condition
Lm
exist constants
m
=
1,2,...,.
Given
\A{t)a{t)e-
/
1
6
[0,T],
By
the hypothesis that
Aa
and So
Friedman (1975, theorem 5.2.3)
+ \W\ 2m )e Lmt and E*[|5(<)| 2m <
for the former),
so that E*[|W(f)|
E*
dw'(t),
a(t)dw'(t),
a Brownian motion under Q.
is
growth condition
shows that there
(1
w*
recall that
Tt
=
2m
]
(1
]
we can
this,
Tt 2
<
<
dt
easily
show that
00.
\
Jo
Hence W(t)e~ rt + / C(s)e~ rs ds
(
and Shiryayev (1977,
§4.2).
is
a square integrable martingale under Q; see, for example, Liptser
Consequently, (40) holds as an equality.
Next, by condition (4) of Theorem
by condition
(2) of
there exists a A
Theorem
>
1,
u and
such that the
1,
V
first
u and
V
are well-defined
and are state-independent; and
are strictly increasing concave.
We
remain to verify that
order conditions (41) and (42) hold. For any function /,
define the operator C:
an
L{1)
-
ia
a2 A2
2
^L
.a
2+a, SA-^L
bA
dw +
dwds +
l
2
2
(T
2
S
b
2^12+(
+ (rW
es
dl dl
_ C) df_ + rS
rb
dw + ds + dt
-rf + CwfCondition (5) of Theorem
C{
R
]
1
~
~
can be written as C(R)
aA
-R^
o 2 A\v
and
(29), respectively,
0.
This implies that
2aSA -^- + aS
+
2C W
~R~'
(28),
which
2r
~R~~~R
where the second equality follows from
(31)
+
=
W-l + -R(43)
is
a consequence of the definition of
and the consistency condition
(9).
By Condition A5,
R and
// in
the wealth never
8
APPENDIX
33
reaches zero before T. For any
6 [0,T),
t
dO{W(t),S(t),t)
where C
is
diffusion
term of dO becomes
+
(C (wium)
W and
t
.
1
-a A
R(W(t),S,t)
( f
[L
—
+
)dt
d
s
S under P. 3y
W{t)
aA ~
R(W(t),S,t)
dO
——1—
(C(0) +
the differential generator of
drift of
O
implies that
=
1
The
lemma
Ito's
^
dZ
\R(z,S,t)J ,
H(W(t),S,t)
„,„,.,
—r<Jo
„
,
Si,)dwm
'
the consistency condition (9), the
f H(z,S,t) \
R(W(t),S,t)
aAdw(t)
+
\
ab
)
H(W,S,t)
„, Twr „ 7&S
R(W,S,t)
—
is
rW{t)
/
i
W{t)
L„l
ffiC2 f
+
i
2
1
(
\
,
lj
,
U(*,5,*)^s5
1
,2 f
U(W(*),5,t)
2
™ro) i <™»^
rW(«)
* M5(
+
+
,R(
s
Jw
Using integration by parts, we have
W(t)
f
\
/
w(t)
r
w(t)
rWit)
l
_
i
.
/
i
\
R(z,S,t)J ss
+
1
\* A \wjd)j* +
Ay
f
\R(z,S,t)J
w(t) /
i
zS
L
°*
AA
-(t(im).
^ Jw
\R(z,S,t)J s
\
where
hi(S,t)
=
(rW + A(W,S,t){n-r)-C(W,S,t))-
R(K,S,t)
d°
8
APPENDIX
34
Using the definition of £, we get
_/
rW(t)
f
+
W(t)
/
(/*~
/
+
r
W
^ U;
Av
R(z,S,t)
/
g2
dz+
<J
Jw
Jw
Av
g2
/
Jw_
Jw
a 2 AA z
4 •Mse^),*-'-*^
=
h,(S,t)
a consequence of the definition of
We
condition (9).
and
H
z
\
i
ff r
c,J s
\R(z,S,t)J
d*
(
1
(
dz
)
\R{z,S,t)) z
in (31)
and
(28),
which as we mentioned before
(29), respectively,
and the consistency
can thus write
dO(W(t),S{t),t)
Similarly, Ito's
R
/
5
i
c ,^
R(z,S,t)
-p7
\
1
+
where both the second and the third equalities follow from
is
^
K
R(z,S,t)
*R(z,S,t)
/
(A*- r
/
/-W(0
d2+
A 2__}__ dz+
/ VV C)
,
\
i
cm ) z
\R(z,S,t)J
2
/
x
,w(t)
\
!
M \R(z,S,t)J
q<
cm
/
7vv
=
/
)
7vv
t
,W(t)
N
J
lemma
=
fci(5(t),*)*
-
piw'c^wl 7 ^)^') " kdw W-
implies that
"^ Sll\'%
dX(S(t),t)=
S(t)dw(t)
R(W,S(t),t)
+
h 2 (S(t),t)dt,
where
Using the consistency condition (10) and the definition of
-h
Finally, since
dY(t)
=
x
{S,t)
C(W, S, t) >
h 2 (S,t)
= -N(W,S,t) -
we have
in (24),
^^- -
r.
N{W,£_,i)dt,
\nlt(W(t),S(t),t)
Since
+
N
for all
= -0(W(t),S(t),t) +
W>
0,
X(S{t),t)
+
Y(t) =]n£(t)-rt.
and the wealth never reaches zero before
T
we thus have
\nu c (C(W(t),S{t),t),t) = lnA
+ ln£(0-r«,
a.s.
t
<E
[0,T)
by the hypothesis,
8
APPENDIX
with X
=
W=
at T,
1.
At T, on the
where
the other hand, on the set where
-rT =
Proof of Theorem
But we
still
1.
is
0,
\nU{W(T),S(T),T) = \nV'(W(T)).
we have
is efficient.
V'(0).
I
S and W(0) =
that the consumption
may
W and
start
from
=
0.
The
only
not be strictly positive at nonzero wealth
Naturally,
U(W(t),S(t),t)
if
C >
0, In
=lnA + ln^(0-
u c {C(W(t),S(t),t),t)
a.s.
rt,
t
lnf(t)
-
Theorem
3
=
€ [0,T)
rt.
IfC =
shows that
\nu c (0,t) < ln£(<) Finally,
t
have
In
=
W(T) =
Again take 5(0) =
4.
thing different in this case
with A
by the continuity of \nU(W,S,t) except when
0,
Wm\nU{W(t),S(t),t) > lim \nU(W,S(T),T) =
have thus shown that (C, A)
levels.
W(T) >
-rT = Wmln U(W(t),S(t),t) =
ln£(T)
We
set
we have
ln£(T)
On
35
arguments identical to those used
in
the proof of
rt.
\nV'(W(T)) < ln£(T)-rT
with equality holding for
W{T) >
0.
These prove the assertion.
I
show that
0,
condition (2b)
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