Journal of Mathematical Economics 20 (1991) 465487.

North-Holland

A variational problem arising in financial

economics*
John C. Cox and Chi-fu Huang
Sloan School of Management, MassachusettsInstituteof Technology, Cambridge, MA 02139,
USA

Submitted November 1988, accepted August 1990

We provide suff5zient conditions for a dynamic consumption-portfolio problem in continuous

time to have a solution. When the price processes satisfy a regularity condition, all utility
functions that are continuous, increasing, concave, and are dominated by a strictly concave
power function admit a solution.

1. Introduction

The study of an individual’s optimal consumption and portfolio decisions

over time is a classical problem in financial economics. Traditionally, this
consumption-portfolio problem has been analyzed using stochastic dynamic
programming; for example, see Merton (1969, 1971). More recently, a
number of authors have used a martingale representation technique instead;
see Cox and Huang (1989), Karatzas, Lehoczky and Shreve (1987), and
Pliska (1986).
Cox and Huang (1989) focuses on the explicit construction of optimal
consumption-portfolio policies when they are known to exist. Pliska (1986)
gives conditions to guarantee the existence of an optimal consumption-
portfolio policy when there is no positivity constraint on consumption, while
Karatzas, Lehoczky and Shreve (1987) work on the same problem with a
positivity constraint.’ However, neither Pliska nor Karatzas, Lehoczky and
*We would like to acknowledge helpful conversations with Sergiu Hart, Andreu Mas-Colell,
and Ho-Mou Wu. We are especially grateful to a referee, Kerry Back, who provided many
insightful comments and an almost line by line proof of a key proposition. The December 1985
version of this paper was presented at the Summer Workshop of the Institute of Mathematics
and Its Applications at the University of Minnesota in May 1986. Later versions were presented
at seminars at Brown University, Cornell University, Massachusetts Institute of Technology,
Northwestern University, University of California at Berkeley, University of Chicago, and
University of Pennsylvania. We appreciate the comments made at these seminars. We are also
grateful for the financial support provided by Batterymarch Financial Management. We are of
course solely responsible for any remaining errors.
‘Throughout, we will use weak relations, that is, positive means non-negative, increasing
means non-decreasing, and so forth.

0304-4068/91/%03.50 0 1991-Elsevier Science Publishers B.V. All rights reserved

466 J.C. Cox and C.-f: Huang, A variational problem

Shreve give explicit and easily verifiable conditions on an investor’s utility

function and on parameters of the price processes for existence.
In this paper, we consider a model of a securities market with price
processes that are somewhat more general than those considered by
Karatzas, Lehoczky and Shreve. Explicit conditions on utility functions are
then given to ensure the existence of a solution to the consumption-portfolio
problem. These conditions make admissible any utility function that is
continuous, increasing, and concave, and is dominated by a strictly concave
power function. These conditions thus include most utility functions of
economic interest.
The paper is organized as follows. In section 2 we formulate a model of a
securities market and state the consumption-portfolio problem. Section 3
establishes the connection between the dynamic maximization problem
formulated in section 2 and a static maximization problem of the Arrow-
Debreu type. Section 4 provides sufficient conditions on utility functions as
well as on (Arrow-Debreu) prices to guarantee a solution to the static
problem. The results of sections 3 and 4 are then combined in section 5 to
give suficient conditions for the existence of a solution to the dynamic
problem. Section 6 contains some concluding remarks.

2. The dynamic consumptioo-portfolio problem

We fix a complete probability space (Q,9, P) and a time span [0, a,

where T is a strictly positive real number.2 An element of 62, denoted by w,
is a state of nature, which is a complete description of the exogenous
uncertain environment from time 0 to time T. The sigma-field 9 is the
collection of events distinguishable at time T and P is a probability measure
representing an individual’s beliefs about the likelihood of distinguishable
events.
There is defined on the probability space an N-dimensional standard
Brownian motion denoted by w = {w,,(t); t E [O, 7’j, n = 1,2,. . . , N}. Let Ft be
the smallest sigma-field containing all the P-measure zero sets with respect to
which {w(s); 05s 5 t> is measurable. The increasing family of sub-sigma-fields
of 9, F= {9*; t E [O, T-J}, is usually termed the filtration generated by w. We
assume that PT=9, that is, the true state of nature will be revealed at time
T by observing w from time 0 to time T. Since a standard Brownian motion
starts at zero P-a.s., 9, contains only sets of probability zero or one. All the
processes to appear will be adapted to I; unless otherwise specified.3

*A probability space (QF,P) is said to be complete if AE~ and P(A)=0 imply A’E~ for
any A’ t A.
3A process X = {X(t); t E [0, fl} is said to be adapted to F if X, as a mapping from P x [O, Tl
to 93, is measurable with respect to the product sigma-field generated by 9 and the Bore1
sigma-field of [0, T] and if X(t) is measurable with respect to 6 Vt E [0, T].
 J.C. Cox and C.-f: Huang, A variational problem 467

We will use the following notation: If g is a matrix, (g(* denotes tr (ggT)

and (gl denotes ,/m, where T denotes transpose and ‘tr’ denotes trace.
Consider a frictionless securities market with N+ 1 long-lived traded
securities indexed by n =O, 1,2,. . . , N. A long-lived security is a security
available for trading throughout the period from time 0 to time T. Security
n #O is risky and is characterized by a cumulated dividend process D, =
{DA): CE CO,7% an d a P rice process S,={&(t); t~[0, Tj}, where D,(t) has
right-continuous and bounded variation sample paths and denotes the
cumulated dividends paid out by security n from time 0 to time t, and
S,(t) denotes the ex-dividend price of security n at time t. We will hence-
forth denote (S,(t), . . . ,S,(t))’ and (Dl(t), . . . , DN(t))T by S(t) and D(t), respec-
tively. Since securities will be traded ex-dividend, we assume without loss
of generality that D,(O) =O. We will use AD(t) to denote
(DJ+D&-),...,D&)-D&-))T, where D,(t-) is the left-limit of D, at t,
which exists since D, is of bounded variation. Security 0 is locally riskless,
pays no dividends, and sells for B(t) =B(O) exp {& r(s)ds} at time t, where
B(0) is a strictly positive real number and where r(t) is the instantaneous
riskless interest rate at time t. We assume that r(t) is positive. Henceforth the
0th security will be termed the ‘bond’.
Assume that S+ D is an It6 process:

S(t) + D(t) = S(0) + 5 5(s) ds + j a(s) dw(s) Vt E CO,T], P-a.s., (1)

0 0

where t: and (r are, respectively, an N x 1 vector process and an N x N matrix

process satisfying

(2)

(3)

Note that (2) and (3) are sufficient for the It6 process of (1) to be
well-defined. We will also assume throughout that a(w,t) is non-singular for
almost all w and t.
Now consider an agent with a time-additive utility function for consump-
tion, u( *,t), a utility function for final wealth, V( *), and an initial wealth
W,>O. This agent wants to manage a portfolio of the risky securities and the
bond, and withdraw funds out of the portfolio to maximize his expected
468 J.C. Cox and C.-jI Huang, A variational problem

utility of consumption over time and final wealth. Our task here is to find
explicit conditions on the utility functions, u and V, and on the parameters of
the price processes to guarantee the existence of a solution to the agent’s
problem.
For the agent’s problem to be well-posed, however, we need to first specify
the admissible objects of choices, that is, the admissible trading strategies on
the securities, consumption processes, and final wealth. This is the subject to
which we now turn.
We will use c to denote a consumption rate process with c(t) denoting the
consumption rate at time t and use W to denote a final wealth, We will say
that a consumption-final wealth pair (c, W) is admissible if

where 1 S:p< co, v is the product measure generated by P and Lebesgue

measure, 0 denotes the optional sigma-tield,4 and LP,(Q x [O, T], 0, v) and
LP,(Q, fl, P) denote the positive orthants of E’(s2 x [O, T], 0, v) and U’(sZ,9, P),
respectively. The requirement that c and W be positive is natural since
otherwise would make no economic sense.
A trading strategy is an (IV+ l)-vector of processes, denoted generically by

{4t), e(t)=(e,(t), . . *, emT; t E co,

T7)Y
where a(t) and O,(t) are the number of shares of the 0th and the nth security
held at time t, respectively, satisfying the following conditions:

1. 1 Icr(t)B(t)r(t) +O(t)Tc(t)/ dt < co P-a.s., (4)

2. i (t9(t)Tcr(t)1’dt < 00 P-a.s.,

3. there exists a consumption-final wealth pair (c, W) ELP,(v) x L<(P) such

that, P-a.s.,

ol(t)B(t) + O(t)‘(S(t) + AD(t)) + j c(s) ds

0

?he smallest sigma-field of subsets of ax [0,7’j with respect to which all the processes
adapted to F having right-continuous sample paths are measurable as mappings from D x [0, T]
to R is termed the optional-sigma field and is denoted by 0. It is known that any process
measurable with respect to B is adapted to F; see Chung and Williams (1983, p. 56).
 J.C. Cox and C.-f. Huang, A variational problem 469

=a(O)B(O) + e(0)TS(O)
+ j (a(s)B(s)r(s)+ O(s)=&s))
ds
0

4. tl(T)B( T) + fI(T)T(S( T) + dD( T)) = W P-a.s. (7)

Relations (4) and (5) ensure that the integrals of (6) are well-defined [see
Liptser and Shiryayev (1977, ch. 4)] while relations (6) and (7) are budget
constraints. The consumption-final wealth pair (c, IV) of (6) and (7) will be
said to be financed by the trading strategy (a,@. Note that all the simple
processes5 that finance admissible consumption-final wealth pairs are
trading strategies. Also, a trading strategy (a,@ is associated with a wealth
process

w(t) = am + e(t)T(s(t) + AD(~)),

which is the value at time c of the portfolio plus the dividends received.
So far, we have not put any restriction on the price processes other than
certain regularity conditions on their parameters. For our consumption-
portfolio problem to be well-specified, we certainly do not want the price
processes to allow something to be created from nothing, that is, allow free
lunches,6 when an admissible trading strategy is employed. Harrison and
Kreps (1979), Huang (1985), and Kreps (1981) have shown that for free
lunches not to be available for simple strategies it suBices that S and D are
related to martingales after a change of numeraire and a change of
probability, or equivalently, there exists an equivalent martingale measure.7
However, free lunches can still exist for other strategies that satisfy (4)-(7).8
Dybvig (1980) and Harrison and Pliska (1981) have shown that the natural
economic requirement that the agent’s wealth over time be positive rules out
all the free lunches. This is the approach we will take. With the positive
wealth constraint, we can also weaken the requirement that there exists an
equivalent martingale measure. It suffices that S and D are related to local
sA simple process is one that has bounded values and changes its values at a finite number of
predetermined non-stochastic time points.
6For a formal definition of a free lunch see Kreps (1981).
‘An equivalent martingale measure Q is a probability measure on (C&F) equivalent to P so
that S(t)/B(t)+rb [l/B(s)]dD(s), prices plus cumulated dividends, in units of the bond, is a
martingale under Q. Probability measure Q is said to be equivalent to P if they have the same
measure zero sets. This definition is symmetric and thus we say P and Q are equivalent to each
other. A necessary and sufficient condition for this is that the Radon-Nikodym derivative dQ/dP
is strictly positive.
sAn example is the doubling strategy of Harrison and Kreps (1979).
470 J.C. Cox and C.-f Huang, A variational problem

martingales9 after a change of numeraire and a change of probability. This

is the subject to which we now turn.
The following assumption will be made throughout our analysis.

Assumption 2.1. There exists If< cc such that [o(t)-‘(c(r)-r(r(t)S(t))ls

Rv-a.e.

Put

which is prices plus cumulated dividends, in units of the bond. Ito’s lemma
implies that

G*(t)=d&[&)-r(s)S(s)]ds+d$dw(s).

Now put
it(t)= --o(t)-‘[c(t)-r(t)S(t)]
and

q(t)=exp j~(s)~dw(s)--fjl~(s)1~ds (8)

0 0

By Assumption 2.1, ‘1is well-defined. Putting

Q(A)+ ~(0, T)P(do), VAEF,

A

the following proposition shows that Q is the unique probability measure

equivalent to P and under which G* is a local martingale.

Proposition 2.1. v is a martingale under P with EC?(T)] = 1 and Q is the

unique probability measure equivalent to P that makes G* a local martingale.
Moreover, we have

‘A process is a local martingale under the probability Q if it has right-continuous paths and
there exists a sequence of optional times T.1 coQ-as. so that the process {X(T, A t); t E%+} is a
martingale under Q.
 J.C. Cox and C.;f. Huang, A variational problem 471

G*(t) = jt -4s) dw*(s) t E [0, T-JP-as.,

o B(s)

where

w*(t) s w(t) - j K(S)ds, tECO,Tl

0

is a standard Brownian motion under Q.

Proof. See appendix.

Since P and Q are equivalent and thus have the same probability zero sets,
we will use as. to denote almost surely with respect to both measures from
now on.
A trading strategy (a, ~9)is admissible if it satisfies (4)-(7) and W(t) ZOP-
a.s. for all t, where W(t) is the wealth process associated with (a,8).
Henceforth, we will use H to denote the space of admissible trading
strategies. Given Proposition 2.1, arguments identical to Dybvig and Huang
(1989, Theorem 2) show that there are no free lunches for trading strategies
in H.
The problem facing an agent can now be stated formally in the following
way:

u(c(t), t) dt + V(W)
1
s.t. (c, IV) is financed by (a,0) and lies in L<(v) x Z&(P)

with a(O)B(O)+ e(0)Ts(O) 6 IV,. (9)

3. The correspondence between a dynamic problem and a static problem

It is well-known that the dynamic consumption-portfolio problem formu-

lated in section 2 can be transformed into a static problem of the Arrow-
Debreu type; see Harrison and Kreps (1979) and Huang (1985), for example.
For completeness, we will outline this transformation.
Consider the static variational problem:

sup
(c,W ELP+
E I’u@(t), t) dt + V( IV)
(v)XLq(P) 0 1
472 J.C. Cox and C.-J Huang, A variational problem

s.t.
1
B(0) E ;c(t)q(t),B(t) dt + lQ( T)/B( 7) 5 W,.
[0
(10)
The following proposition shows that (9) and (10) have the same feasible

Proposition 3.1. (c, W) is feasible in (9) if and only if it is feasible in (10).

Proof. Let (c, w) be feasible in (9). Then it lies in LP,(v) XL%(P) and is
financed by some (a, 0) EH. It6’s lemma, (6), (7), and Proposition 2.1 imply

WWW + j cWW ds
0

= a(0)+ 8(0)TS(O)/B(O)
+ j O(~)~a(s)/B(s)
dw*(s).
0

The left-hand side is positive since CELP+(V)and the trading strategy satisfies
the positive wealth constraint. It is known that a local martingale bounded
from below is a supermartingale (this is a simple application of Fatou’s
lemma). Thus the right-hand side is a supermartingale under Q since an It6
integral is a local martingale.” This implies

E*
II 1
W/B(T) + ; c(t),B(t) dt 5 W(O)/B(O).
0

Equivalently,

B(O)E Wrl(T)IP(T) + ;c(r)rl(r)lP(r) dr 5 W(0) 5

[ 0 1 Wo,

where we have used Dellacherie and Meyer (1982, VI.57). We have therefore
shown that (c, W) is feasible in (10).
Conversely, let (c, w) be feasible in (10). Thus it satisfies the integral
constraint. From the previous paragraph, we know that the integral con-
straint can be written as

“For the fact that an It6 integral is a local martingale, see, e.g., Liptser and Shiryayev (1977,
section 4.2.10).
 J.C. Cox and C.-f. Huang, A variational problem 413

E* W/B( 7’) + j c(t),B(t) dt 5 W(O)/B(O).

[ 0 1 -

Thus

Wi~(T)+Tc(t)/B(t)dt~t’o.
0

Define a martingale under Q:

W/B(T)+i~(t)/B(t)dtlF~
0 1 a.s.

where we have fixed a right-continuous version of a conditional expectation,

which exists as F is right-continuous, and the second equality follows from
the conditional Bayes rule. Note that both the numerator and the denomina-
tor on the right-hand side of the second equality of the last relation are
martingales under P. By the martingale representation theorem [see Clark
(1970, 1971)] we know there exists an N-dimensional process 4 with

so that

x(t) = WW W4 T) + v(T) j: 40/B(t)dtl + Sb

MT dw(s)
1+ fi tl(s)~(s)~
W4

It& lemma then implies that a.s. for all t

= 40) + j -&(dWT- Y(s)K(s)~)

dw*(s),

where At)= EWG’YW)+~(T) j:c(r)lB(r) dt(%l. Putting e(t)‘=

(4(t) - Y@)K(t))TB(t)e(r)- ‘/?(r),
474 J.C. Cox and C.-f Huang, A variational problem

4s)t
w(t) = B(t) x(0-d Bo ds 3

and
a(t)=(W(t)-8(t)(S(t)+dD(t))/B(t).

It is easily verified that (a, 13)EH and linances (c, W). Thus (c, IV) is feasible in
(9). 0

The following corollary records an immediate implication of Proposition

3.1.

Corollary 3.1. (c, W) is a solution to (9) g and only if it is a solution to (10).

Thus to investigate whether there exists a solution to (9), it suffices to

study (10).

4. A static variational problem

In this section we will study a class of static variational problems of the
kind described in (10). For expositional purpose, we will first analyze in
detail a problem in which there are only preferences for final wealth and no
preferences for intermediate consumption. Later, we will generalize the results
in this simpler case to the more general case with intermediate consumption.
In the simpler case, the sufficient condition for existence involves whether the
‘inverse’ of the implicit Arrow-Debreu price system has a certain finite
moment. In the more general case, however, one verifies whether the time
integral of certain moment of the inverse of the Arrow-Debreu prices over
time is finite.

4.1. A simple static variational problem

We are interested in finding a solution to the following problem:

sup ECWI
XELP+
(P)

(A,)
s.t. 4(x) 3 E[xt] =<KO,

where 1 Sp< co, r EL’!+(P) with l/p+ l/q= 1, {>O a.s., and K0 is a strictly
positive constant. As usual, we will say that there exists a solution to (A,) if
the supremum is finite and is attained by some x E L%(P). Henceforth, denote
the value of (A,) by val (K,). Assume throughout that V: ‘%+H% u (- co> is
 J.C. Cox and C.-$ Huang, A variational problem 475

a continuous, concave, and increasing function which can be unbounded

from below at zero. It is easily seen that val(K,) > - cc since K0 >O. As will
be shown in the following two examples, however, when I&(P) is infinite
dimensional, (A,) may not have a solution. Consider the following two
examples.
Take L2 to be [O, 11, 4t to be the Bore1 a-field of [O, 11, and P to be
Lebesgue measure on [0, 11.Suppose that u(z) = z2j3 and

&)=bZo*(o)do VXEL~(Q,~F,P).

Consider the sequence

X”(~)=Kon21~o,,,,,(o),
n=1,2 ,...,

where 1t0,l,n, is the indicator function of [O, l/n]. It is easily verified that
X, E Lp(P) for all p >=1 and 4(x,) = K,, Vn. However,

du(x,(o)) do = K;‘3n”3 --f co as n+cO.

Note that in the above example, the utility function is strictly concave,
increasing, and continuous. It has a zero derivative at infinity and an infinite
right-hand derivative at zero - all very nice properties for a utility function
can have. The problem arises because the commodities close to o=O are
worth almost nothing. The agent would like to put all his money in the
commodity indexed by o =O, but his expected utility will be zero since the
event {o =0} is of zero measure. Thus he tries to purchase commodities as
close to o=O as possible. He achieves this by going along the sequence
(&+3, i,.,).“= 1’ However, the utility function grows too fast to infinity as
the consumption increases to infinity and the expected utility explodes.
The following example, which is adapted from Aumann and Perles (1965),
shows that the supremum may not be attained even if it is finite. Take Sz to
be [0, 11, ,F the Bore1 sigma-field of [0,11, and

P(A)=l$(w+l)dw, VAE.K

In addition, let V(z) = z and

476 J.C. Cox and C.-f: Huang, A variational problem

& P(do) = j x(o) do.

0

Note that since prices for commodities, 3/2(0+ l), are bounded above,
4(x) < co for all x E p(P) for all pz 1. In this case, it is easily verified that
s: V(x(o))P(do)<$K,, but its supremum over all budget feasible XELP,(P) is
equal to BK,, which is not attained. In this case, the prices as captured by
3/2(0+ 1) are bounded from below; but the utility function is linear and
again grows to infinity too fast. Hence the agent chooses to concentrate his
wealth buying inexpensive commodities close to CO=1 and the supremum is
not attained.
Aumann and Perles (1965) studied a class of problems very similar to (A,).
Briefly, they considered conditions under which a given finite supremum is
attained in the following program:

sup j fM4 4P*(d4

xcL:(P*) R

6%)

s.t. b(w)P*(do) 5 Ko,

where P* is a finite measure on (52,F) and f( 2,~ ) is increasing, concave,

continuous in z, and bounded from below, for P*-almost every w. The
program (A,) differs from (A,) in that f is state dependent and cannot be
unbounded from below at zero, and the prices of commodities are unity. We
will show below how the results of Aumann and Perles (1965) can be
adapted to provide conditions for the attainment of a finite supremum.
Before that, in the following proposition, we will first give conditions on the
price < and on the utility function so that the supremum in (A,) is always
finite. It will be demonstrated later that for utility functions unbounded from
above, these conditions also ensure the attainment of the supremum.

Proposition 4.1. Suppose that V is unbounded from above and there exist
/I1 20, fi2 >O and b ~(0,l) such that

V(z)g?1 +@-b VZE%+.

Then val (K,) is finite if l-l E Lpib(P).

Proof. It is easily verified that there exists a solution to (A,) when

V(z)=/?1+/?2z1-b if and only if 5-l eLplb(P). Let valb(&,) denote the
supremum in this case. In the general case, let (x,) be a sequence in LP,(P) so
that E[V(x,)]+val (K,). By the hypothesis, V(z) 6pl +&z~-~. Thus,
 J.C. Cox and C.-f Huang, A variational problem 417

E[ V(x,)] 5 /_I1+ /&E[xi -*] $ valb(&,) < CC for all n. The assertion then
follows by letting n-roe. 0

Note that val(K,) for utility functions bounded from above is certainly
finite independently of 4.
Now we turn our attention to the attainment of val(K,). We show that a
generalization of Aumann and Perles (1965) can be brought to bear on our
problem by a change of unit. We first give a definition which generalizes that
of Aumann and Perles to include functions possibly unbounded from below
at zero.

Definition 4.1. Let f: % + x s1~‘% u ( - cc~f be possibly unbounded from

below at zero and be measurable with respect to the product o-field
W(% +) x 9, where @(%+) denotes the Bore1 sigma-field of ‘%+. Then
f(z, o) =0(z) as z+ 00, LP(P)-integrably in o, if for each E>O there exists
y E LP,(P) such that, P-a.s., f(z, o) S EZwhenever z 2 y(o).

The following technical lemma gives an equivalent definition of f(z,~)=

o(z) as z-+co, LP(P)-integrably in o.

Lemma 4.1. Let f: %+ x C&93 v ( - oo} be possibly unbounded from below at

zero and be measurable with respect to 93(%+) x %. For all E >O, there exists
YE LI;(P) such that, P-a.s., f (z, w) 4s~ whenever z >=y(o) if and only if for all
E>O and a2_0, there exists YELP,(P) such that, P-a.s., f (z,o) ss(z-a)
whenever z 2 y(o).

Proof. The proof for the sufficiency part is easy by taking a=O. We now
prove the necessity part. Let E>O and a>0 be given. By the hypothesis,
3y E L%(P) such that, P-a.s., f (z, co)5 (E/~)z if z 2 y(w). Putting

y(w) if y(o) > 2a

y*w=
1 2a

if y(o) 5 2a,

we have, P-a.s., f(z,co) z.s(z-a) whenever zl y*(o). It is easily seen that

y* E L<(P) and the assertion was proved. IJ

The following proposition is a generalization of Aumann and Perles (1965,

Proposition 2.2) to allow f to be possibly unbounded from below at zero.

Proposition 4.2. Let f of (Az) be possibly unbounded from below at zero and
f(z,m)=o(z) as z-+00, LP(P*)-integrably in CO. Suppose that there exists
478 J.C. Cox and C.-J Huang, A variational problem

p ELP+(P*) such that Scf(p(o),w)P*(do) > -co and the supremum of (A,) is
finite. Then there exists a solution to (A,).”

Proof. See appendix.

Before the first main result of this section, we record two technical lemmas.

Lemma 4.2. Suppose that V is unbounded from above and V(z) $ PI +/&z’ -*
for some /?,zO, B2>0 and (-‘EL*‘*(P) f or some b E (0,l). Then V(z)&w) - ’ =
o(z), L*(P)-integrably in CD.

Proof. Since utility functions are defined up to a positive linear transforma-

tion, we assume without loss of generality that /3r =O. We claim that given
E>O, there exists YELP,(P) such that

V(Z)_I&Z<(O) Vz~y(y(w),

for P-almost every WEQ. Note that

V(z)/zQBzz_b vzE%+\{0).
Putting
Y(o)~(&/BZ)-l’b5(W)-l’*,

it is clear that y> 0 a.s. and y E Lp(P) since r-’ E L”‘*(P). For z 2 y(o) we have

V(z)/z6B1Z-*~:82y(0)-*=E5(0).
Hence,
V(Z)5(Cf$‘5&Z, VzZy(w),

which was to be shown. 0

The following is the counterpart of Lemma 4.2 for utility functions

bounded from above.

Lemma 4.3. Suppose that V is bounded from above and t -’ E L*(P). Then
V(z)<(o)- ’ = o(z), L*(P)-integrably in w.

“In an earlier version we had a different argument for the case where V is unbounded from
below. We thank Kerry Back for providing almost a line by line proof of this proposition.
 J.C. Cox and C.-f. Huang, A variational problem 419

Proof. Fix E> 0 and let a > 0 be such that V(z) Sa for all z. Let y(w) =
(u/s)<(o)-‘. Then YE&(P) and V(z)<(w)-‘5s~ if z>=y(o). 0

Here is our main theorem of this section.

Theorem 4.1. Suppose that V is unbounded from above, V(z) 58, +&z’ -* for
all ZE’%+ for some fll 20, pz>O, and 5-l ELM’* for some bE(0, 1). Then
there exists a solution to (A,). Moreover, if 5-l ALL”* for some $21,
where P is a finite measure absolutely continuous with respect to P, then every
solution to (A,) lies in Lp’(P).

Proof. First we note that by Proposition 4.1, val(K,) is finite. Thus our
problem is to show that the supremum is attained.
Define a measure on (52,Y):

P*(A) = j @)P(do) VA ~9.

A

It is easily seen that P* is equivalent to P since c >O P-a.s. We first claim

that V(z)<(o)-’ =0(z), L’(P*)-integrably in o. By the assumption and
Lemma 4.2, for every E>O there exists ye L%(P) such that V(z)<(o)-’ 5.z~
P-a.s. tlz 2 y(o). Letting EP[ ‘1 denote the expectation with respect to P*,
Holder’s inequality implies that

where l/p + l/q = 1 and the last inequality follows since YEI?’ and
r E Lq(P). The assertion then follows since P* and P have the same null sets.
Consider the program:

sup j V(x(w))P(dw)
XEL:(p*) R
(A:)
s.t. k(o)r(o)P(dw) S K,.

We first claim that the value of this program is finite. This can be seen using
arguments of Proposition 4.1 and the facts that 5-l czLp(P), 5 EC(P), and
L?(P) C L’(P*).
The above program can be rewritten as follows:

sup j V(~(~))S(~) - ‘P*(do)

x=L:(p) fi
480 J.C. Cox and C.-j Huang, A variational problem

s.t. k(o)P*(dw) 5 K,.

Thus, Proposition 4.2 insures that (AT) has a solution.

Note that the consumption set in (AT) is L:(P*). Therefore, a solution to
(A:) may not be a solution to (A,), since LP,(P) c L!+(P*) and the inclusion
may be strict. Our task now is to show that, indeed, every solution to (A:)
lies in Lp(P).
Let x* E L\(P*) be a solution to (A:). Since the Slater’s condition [cf.
Holmes (1975)] is obviously satisfied, it follows from the Saddle-Point
Theorem and Rockafellar (1975) that there exists a positive constant I(/ such
that for all XEL:(P*), P*-a.e.,

~~~*(4)5(~) - l- W~))SW - l2 $(x*(4 -x(4).

Note that since V is unbounded from above and concave, it must be strictly
increasing and thus II/>O. Without loss of generality we can assume that
V(a) =0 for some ~~10. Now take x(o) = aVo~f2 in the above relation. We
get Y(x*(o))~(w)-’ 2 $(x*(o) - a) P*-a.e. Since V(z)&o)-’ = o(z), Lp(P)-
integrably in w, and since P and P* are equivalent, Lemma 4.1 implies that
there must exist an YELP,(P) such that

x*(o) 5 y(0) a.s.

This implies that x* E L%(P).

Suppose in addition that 5 -r E Lp”b(p) for some p’ >=1 and p is a finite
measure absolutely continuous with respect to P. Then V(Z)~(W)-’ =0(z),
E”(&-integrably in o by Lemma 4.2. From previous discussions and the fact
that p is absolutely continuous with respect to P we have

V(x*(o))~(o)-’ 2 +(x*(o) - a) P-a.e.

Lemma 4.1 then implies that there exists YE L$(p) such that x* $ y and thus
x*&p;(P). 0

Lemma 4.3 and arguments similar to those used in proving Theorem 4.1
prove the following:

Theorem 4.2. Suppose that V is bounded from above, and 5-l E Lp(P). Then
there exists a solution to (A,). Moreover, if satiation does not occur and
5-l E Lp”b(p) for some $11, where d is a finite measure absolutely continuous
with respect to P, then every solution to (A,) lies in Lp’(p). On the other hand,
when satiation occurs there is a solution to (A,) that is bounded.
 J.C. Cox and C.-j: Huang, A uariational problem 481

Proof. Proofs of the first and the second assertions are identical to those
for Theorem 4.1. Suppose now that satiation occurs. Let zO=inf (~20: V(z)2
V(z’) Vz’ 20). By continuity, the intimum is attained. Since satiation occurs,
z0 < cc and jnzo{(o)P(do) 5 K,. Thus x(w) = z0 is a solution to (A,) and is
obviously bounded. 0

4.2. Generalization
In this subsection, we will give sufficient conditions for there to exist a
solution to the Arrow-Debreu style variational problem of (10). For brevity
we will state only a generalization where both the utility function for
consumption and the utility function for final wealth satisfy conditions of
Theorem 4.1. The utility function for consumption and the utility function
for final wealth, however, can satisfy different combinations of conditions of
Theorems 4.1 and 4.2. We leave this exercise to interested readers.
In the following theorem, and throughout the rest of the paper, we will use
I to denote Lebesgue measure on [O, 7’J and put <(t)=q(t)B(O)/B(t). Here is
the generalization of Theorem 4.1:

Theorem 4.3. Suppose that u(z, t):%+ x [0, TJ is Bore1 measurable, and is
continuous, concave, increasing, and unbounded from above in z for A-almost
every t E [0, TJ Suppose further that there exist b,, bz E (0, l), bounded functions
PI(t), $t(t)>=O, j&(t), &(t)>O, such that for A-a.e. t,

u(z,t) s l%(t)+ Bz(W-b’ and V(z)~~$,(t)+&(t)z’-~~ VZE‘%+.

For there to exist a solution to (IO), it is suflcient that, r-’ EL~‘~‘(v) and
t(T)-’ ELJ”~~(P). Moreover, if c-t ~L~“~l(3) and t(T)-’ EL~“~~(P) for some
p’ 2 1, where p is a finite measure absolutely continuous with respect to P and
0 is the product measure of P and Lebesgue measure, then every solution to
(IO) is an element of L<(O) x L<(b).

Proof. See appendix.

5. Existence of a solution to the dynamic problem

Now we will give conditions under which there is a solution to (10) and
thus is a solution to the dynamic problem of (9).

Theorem 5.1. Suppose that u(z, t) and V(z) satisfy conditions in Theorem 4.3,
and there exists p‘>p such that
482 J.C. Cox and C-5 Huang, A variational problem

~[i/B(t)l~~,/b1dt]<cm and E[/B(T)lp”b2]<co.

Then there exists a solution to (9).

Proof. We show first that t(t)-’ =B(t)q(t)-‘/B(O) EI.~‘~‘(v). Putting #=

p’/p> 1 and l/8+ l/G= 1, we have

IId
E [jB(t)q(t)-llp~bldt
I([
5 E ;((B(t)(PIb’)“dt
0 I>

where the first inequality follows from Holder’s inequality and the second
from the assumption of this proposition and the second assertion of
Proposition 2.1. Thus B(t)q(t)-l/B(0) E JY’~‘(v). Identical arguments show that
B(T)q(T)- ’ eLplb2(P). It then follows from Theorem 4.1 that there exists a
solution to (10). By Corollary 3.1, we conclude that there is a solution to
(9). Cl

Theorem 5.1 utilizes Theorem 4.3, which is a generalization of one of

various combinations of Theorems 4.1 and 4.2. We leave other possibilities to
the reader.

6. Concluding remarks
Throughout our analysis we assumed that (~(t)( is uniformly bounded, but
this is not necessary. In fact, it suffices that v(T) has a unit expectation, q(T)
and q(T)-’ have certain finite moments, and the time integrals of certain
moments of q(t) and q(t)-‘, respectively, are finite. For specific applications,
these integral conditions can be verified through either direct computation or
simulation. We refer the interested reader to Cox and Huang (1989) for
details.
Although we have dealt with state-independent utility functions, our results
generalize easily to certain state-dependent utility functions. For example, in
the context of Theorem 4.1, if a state-dependent utility function V(z,o) is
bounded from above by B1+ /?Zz’-b uniformly across w, then there exists a
solution to (A,).
 J.C. Cox and C.-f. Huang, A variational problem 483

Appendix: Proofs
Proof of Proposition 2.1

By Assumption 2.1, rc is uniformly bounded. Thus EC?(t)] = 1 for all t and

q is a martingale under P; see Liptser and Shiryayev (1977, Theorem 6.1).
The fact that Q is a probability measure equivalent to P follows from
E[q( T)] = 1 and q(T) > 0 P-a.s.
Now, we want to show that G*(t) is a local martingale under Q. By the
Girsanov theorem [see Liptser and Shiryayev (1977, ch. 6)], we know

w*(t) -w(t) - j ic(S(s), s) ds VIE [0,7-l

0

is an N-dimensional standard Brownian motion under Q. Thus we can write

G*(f) =iy dw*(s) Vt E [0,7,-j.

The right-hand side of the above relation is an Ito integral under Q and thus
is a local martingale under Q.”
The uniqueness of Q follows from arguments similar to those of Theorem
3 of Harrison and Kreps (1979).

Proof of Proposition 4.2

We will first prove a technical lemma:

Lemma A.l. Let x,,xEL:(P*), and x,-+x in L’(P*) and x,+x P*-a.e. as
n+oo. Then

lim sup 1 f (x,(o), o)P*(dw) s j f(x(o), o)P*(dw).

R R

Proof. Fix $ E L:(P*) be such that $2 {p,x+ l} and P*-a.s., f(z,~) 4,~
if zzll/(w). Put U,={w:x,(o)~J/(w)}. Then l,,+l P*-a.e. and
limf(x,(o),w) = f(x(o),w) P*-a.e. by the hypothesis that x,+x P*-a.e. and
f (z, o) is continuous in z, where lu, is the indicator function of the set U,.
To simplify notation, we shall henceforth write f(x(o),c~) briefly as f(x) and
the integral j&(x(o), o)P*(d w) as ~J(x). Note that lU,f(~.) 6 lU,f(JI) 5$.
Fatou’s lemma then implies that

“Note again that an It6 integral is a local martingale; see, e.g., Liptser and Shiryayev (1977,
section 4.2.10).
484 J.C. Cox and C.-f: Huang, A variational problem

lim sup J lc,f(x.) 5 j f(x).

R R

We are now left to show that

lim sup J f(x,) SO.

R\U,

For this note that

Thus the assertion is proved. 0

Let val (y) denote the supremum in (A,) when K, is replaced by some
y>O. By the hypothesis we know that - cc < val(K,) < + co. Put c1=
val(K,), D={y~%:y~K,,val(y)=a}, and let y,=inf{y:yED}. Let {b,} be
a sequence of points of D such that b,J y,. For each k, val(b,)=cr. Let {xk}
be a sequence in L:(P*) such that jnxtsbk and Jnf(xk)tcr. We want to
show that {xk} has a weak cluster point.
Since {xk} is L’(P*)-norm bounded, it suffices to show for each decreasing
sequence {S,} of fl such that S, 18 we have js,x,+O uniformly in k
[Dunford and Schwartz (1957, Theorem IV.8.9)]. Fix throughout E~0.
If y, =O then js, x,~~~x,~E for all m, if k is sufficiently large. Thus we
may choose m, such that ss, xks E for all k and all mzrn,. Suppose that
Y,>O.
Assume without loss of generality that s/2 < y,. Let b= y, -.c/4. By the
hypothesis, val (b) c u since b < yo. Since b,J y. and jJ(x,)ta, 3ko such that
b,cy,+&/4 and Jnj&)>[val(b)+a]/2 if k2ko. Put ~=(a-val(b))/2 and
choose C#J lL!+(P*), 42~ such that f(z, o) 5 [y/4Ko]z if zZ 4(o). Note that
f(p) sf(~$) 5 [y/4K,]+. Thus f(d) is integrable. Fix m, such that

j 4~~14 and sS_lf($)l<r/4 VmZmo.

S,
 J.C. Cox and C.-_fHuang, A variational problem 485

Define T,, = {w E S,: xL(o) > &CO)}and set

For k2ko and mzm,,

,Wb)+a -- 7 -~ Y b

2 4 4Ko Ir

zval(b)+y/2,

where the second inequality follows from the fact that b, SK,. Thus
jRxkm > b. Now note that

sb,+E/+ j xk,
TIC,

for m 2 m, and k >=ko. This implies that

s X,<,b,+&/+-bc y,+&f2-b=3&/4
TkRl

and thus

5 &= j xk+ j xk
.%I SWl\Tk??l Tk”

Since we can choose m, 2rno such that js,xks; for k s k. and all mzm,, we
have js,,, xk 5 E for all k whenever m 1 ml, as desired. Thus there exists a weak
486 J.C. Cox and C.-j Huang, A variational problem

cluster point i of the sequence {x~}. That is, there exists a subsequence of
(x~}, (Q} such that ++jz weakly. Then there is a sequence (Ye}, each of
which is a (finite) convex combination of xv’s such that yk+% in L’(P*); see
Dunford and Schwartz (1957, Corollary V.3.14). Thus there exists a subse-
quence of {yk}, denoted by {yk,}, such that y,,+_? a.e. So there exists a
sequence {yk,} so that y,, +2 a.e. and in L’(P*). Lemma A.1 implies that

lim sup i-0 n,) 5 i-02).

By concavity and the fact that Jof(xJ~~J(x3 for k4 1, we know

{J(Y,,) &.0x1) and h ence ~J(R)~~J(xl). But for each k’, the sequence
Xk’,Xk’+l,**.} converges weakly to 2, so we may conclude in the same way
that JJ(Z?))JJ(x,,) for all k’. Hence

J f(i) 2 lim sup J f(xk,) = 01.

R R

Finally, since the set {xEL’(P*): Jox~K,> is weakly closed, Jo_+?SK,. We

have thus shown that Z is budget feasible and achieves the supremum, which
was to be proved.

Proof of Theorem 4.3

Since i%(t), Lb(t), A(t), and 4Jt) are bounded positive functions of time,
there exist constants pi, &, $i, and 6, such that, ;l-a.e. t,

u(z,t)sf11+f12z1-b1 and V(Z)~~,+~,Z’-~* VZE%+.

Without loss of generality, assume p1 = $1 =O. Let l-’ ~E”~l(v) and

r(T)-’ EL.P’~~(P)and fix E> 0. Define

(e/j?2)-1’b1&o,t)-1’b1 VtE[O,T)
Yb t) = (E/&) - ‘lb2@CD,T) - ‘lb2,
t = T.

It is easily verified that YELP+(P) and u(z, t)&o, t)-’ =0(z), E’(v)-integrably in
(t, w) and r/(z)c(o, T)-’ = o(z), LP(P)-integrably in w. Putting u(z, T) = V(z),
the objective function of (10) is equivalent to
 J.C. Cox and C.-f. Huang, A variational problem 487

where 1 is Lebesgue measure plus a point mass at T. Apply Proposition 4.2

by taking the ‘state space’ to be Szx [0, T] equipped with product measure
P x 1 to conclude that there exists a solution to (10). The rest of the assertion
is left for the reader. 0

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