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A variational problem arising in financial economics
Journal of Mathematical
Economics 20 (1991) 465487. North-Holland
A variational economics*
problem arising in financial
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 consumptionportfolio 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 ArrowDebreu 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].
467
J.C. Cox and C.-f: Huang, A variational problem
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, = an a rice process S,={&(t); t~[0, Tj}, where D,(t) has {DA): CE CO,7% d P 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 henceforth denote (S,(t), . . . ,S,(t))’ and (Dl(t), . . . , DN(t))T by S(t) and D(t), respectively. Since securities will be traded ex-dividend, we assume without loss of generality that D,(O) =O. We will use AD(t) to denote where D,(t-) is the left-limit of D, at t, (DJ+D&-),...,D&)-D&-))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) 0
0
Vt E CO,T],
P-a.s.,
(1)
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 consumption, 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
2.
i (t9(t)Tcr(t)1’dt < 00 P-a.s.,
3. there exists a consumption-final that, P-a.s.,
P-a.s.,
(4)
wealth pair (c, W) ELP,(v) x L<(P) such
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
=a(O)B(O)
+ e(0)TS(O) + j (a(s)B(s)r(s)+
problem
469
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 wealth pairs are processes5 that finance admissible consumption-final 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 consumptionportfolio 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).
J.C. Cox and C.-f Huang, A variational problem
470
martingales9 after a change of numeraire and a change of probability. is the subject to which we now turn. The following assumption will be made throughout our analysis. Assumption 2.1. Rv-a.e.
There
exists
If< cc
such
that
This
[o(t)-‘(c(r)-r(r(t)S(t))ls
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 0
(8)
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 equivalent to P and under which G* is a local martingale.
measure
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 there exists a sequence of optional martingale under Q.
under the probability Q if it has right-continuous paths and times T.1 coQ-as. so that the process {X(T, A t); t E%+} is a
J.C. Cox and C.;f. Huang, A variational problem
G*(t) = jt -4s) dw*(s) o B(s)
471
t E [0, T-JP-as.,
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) ZOPa.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) s.t. with
1
(c, IV) is financed by (a,0) and lies in L<(v) x Z&(P) 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 formulated in section 2 can be transformed into a static problem of the ArrowDebreu type; see Harrison and Kreps (1979) and Huang (1985), for example. For completeness, we will outline this transformation. Consider the static variational problem:
1
E I’u@(t), t) dt + V( IV) sup (c,W ELP+ (v)XLq(P) 0
J.C. Cox and C.-J Huang, A variational problem
472
1
s.t.
B(0) E ;c(t)q(t),B(t) dt + lQ( T)/B( 7) 5 W,. [0
The following proposition
(10)
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*
1
W/B(T) + ; c(t),B(t) dt 5 W(O)/B(O). II
0
Equivalently,
1
B(O)E Wrl(T)IP(T) + ;c(r)rl(r)lP(r) dr 5 W(0) 5 [
0
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 constraint 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
1
E* W/B( 7’) + j c(t),B(t) dt 5 W(O)/B(O). 0
[
413
problem
-
Thus Wi~(T)+Tc(t)/B(t)dt~t’o. 0
Define a martingale under Q:
1
a.s.
W/B(T)+i~(t)/B(t)dtlF~ 0
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 denominator 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) (4(t) - Y@)K(t))TB(t)e(r)- ‘/?(r),
j:c(r)lB(r) dt(%l.
Putting
e(t)‘=
474
J.C. Cox and C.-f Huang, A variational problem
t
4s) 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 3.1. Corollary 3.1.
records an immediate implication
of Proposition
(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] =
where 1 Sp< co, positive constant. the supremum is the value of (A,)
r EL’!+(P) with l/p+ l/q= 1, {>O a.s., and K0 is a As usual, we will say that there exists a solution to finite and is attained by some x E L%(P). Henceforth, by val (K,). Assume throughout that V: ‘%+H% u (-
strictly (A,) if denote 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$(w+l)dw,
In addition, let V(z) = z and
VAE.K
476
J.C. Cox and C.-f: Huang, A variational problem
&
P(do) = j x(o) do. 0
Note that since prices for commodities, 3/2(0+ l), are bounded above, 4(x) < co for all x E p(P) for all pz 1. In this case, it is easily verified that s: V(x(o))P(do)<$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
xcL:(P*)
4P*(d4
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 if and only if 5-l eLplb(P). Let valb(&,) denote the V(z)=/?1+/?2z1-b 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,
417
J.C. Cox and C.-f Huang, A variational problem
E[ V(x,)] 5 /_I1+ /&E[xi -*] $ valb(&,) < CC for follows by letting n-roe. 0
all
n. The
assertion
then
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=
y(w)
1 2a
if if
y(o) > 2a 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 finite. Then there exists a solution to (A,).” Proof.
and the supremum of (A,)
is
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 transformation, 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, which was to be shown.
VzZy(w), 0
The following is the counterpart bounded from above.
of Lemma
4.2 for utility functions
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,) problem is to show that the supremum is attained. Define a measure on (52,Y): P*(A) = j @)P(do)
is finite. Thus our
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 $(x*(o) - a) P*-a.e. Since V(z)&o)-’ = o(z), Lp(P)integrably in w, and since P and P* are equivalent, Lemma 4.1 implies that there must exist an YELP,(P) such that x*(o) 5 y(0)
a.s.
This implies that x* E L%(P). Suppose in addition that 5 -r E Lp”b(p) for some measure absolutely continuous with respect to P. E”(&-integrably in o by Lemma 4.2. From previous that p is absolutely continuous with respect to P we V(x*(o))~(o)-’
2 +(x*(o)
- a)
p’ >=1 and p is a finite Then V(Z)~(W)-’ =0(z), discussions and the fact have
P-a.e.
Lemma 4.1 then implies that there exists YE L$(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.
481
J.C. Cox and C.-j: Huang, A uariational problem
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]
and
E[/B(T)lp”b2]
Then there exists a solution to (9). Proof. p’/p>
We show first that t(t)-’ =B(t)q(t)-‘/B(O) 1 and l/8+ l/G= 1, we have
E [jB(t)q(t)-llp~bldt
I([ 5
E ;((B(t)(PIb’)“dt 0
EI.~‘~‘(v). Putting
#=
IId 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,).
483
J.C. Cox and C.-f. Huang, A variational problem
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 G*(f)
standard Brownian motion under Q. Thus we can write
=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. n+oo. Then
Let x,,xEL:(P*),
and x,-+x
in L’(P*)
and x,+x
P*-a.e.
as
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 section
again 4.2.10).
that an It6 integral
is a local martingale;
see, e.g., Liptser
and Shiryayev
(1977,
J.C. Cox and C.-f: Huang, A variational problem
484
lim
sup J lc,f(x.) R
5 j f(x). 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(~$) 5 [y/4K,]+. Thus f(d) is integrable. Fix m, such that j 4~~14 S,
and
sS_lf($)l
VmZmo.
485
J.C. Cox and C.-_fHuang, A variational problem
Define T,, = {w E S,: xL(o) > &CO)}and set
For k2ko and mzm,,
,Wb)+a
--
7
4
2
-~
Y
b
4Ko Ir
zval(b)+y/2, where the second inequality jRxkm > b. Now note that
sb,+E/+
j
follows from
the fact that
b, SK,.
Thus
xk,
TIC,
for m 2 m, and k >=ko. This implies that s X,<,b,+&/+-bc TkRl
y,+&f2-b=3&/4
and thus 5 &= .%I
j SWl\Tk??l
xk+
j Tk”
xk
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 subsequence 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 But for each k’, the sequence {J(Y,,) &.0x1) and h ence ~J(R)~~J(xl). 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’-~*
Without loss of generality, assume r(T)-’ EL.P’~~(P)and fix E> 0. Define (e/j?2)-1’b1&o,t)-1’b1
p1 = $1 =O.
VZE%+. Let
l-’ ~E”~l(v)
and
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 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
(t, w) and
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 References Aumann, R. and M. Perles, 1965, A variational problem arising in economics, Journal of Mathematical Analysis and Applications 11, 488-503. Chung, K. and R. Williams, 1983, An introduction to stochastic integration (Birkhauser Inc., Boston, MA). Clark, J., 1970, The representation of functionals of Brownian motion by stochastic integrals, Annals of Mathematical Statistics 41, 1282-1296. Clark J., 1971, Erratum, Annals of Mathematical Statistics 42. 1778. Cox, J. .and C. Huang, 1985, A variational problem arising in financial economics with an application to a portfolio turnpike theorem, Working paper no. 1751-86 (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA). Cox, J. and C. Huang, 1989, Optimal consumption and portfolio policies when asset prices follow a diffusion process, Journal of Economic Theory 49, 33-83. Dellacherie, C. and P. Meyer, 1982, Probabilities and potential B: Theory of martingales (NorthHolland, New York). Dunford, N. and J. Schwartz, 1957, Linear operators, Part I: General theory, (Wiley, New York). Dybvig, P., 1980, A positive wealth constraint precludes arbitrage in the Black-Scholes model, Unpublished manuscript (Yale University, New Haven, CT). Dybvig, P. and C. Huang, 1989, Nonnegative wealth, absence of arbitrage and feasible consumption plans, Review of Financial Studies 1, 377401. Harrison, M. and D. Kreps, 1979, Martingales and multiperiod securities markets, Journal of Economic Theory 20, 381408. Harrison, M. and S. Pliska, 1981, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and their Applications 11, 215-260. Holmes, R., 1975, Geometric functional analysis and its applications (Springer-Verlag, New York). Huang, C., 1985, Information structure and viable price systems, Journal of Mathematical Economics 14,215-240. Karatzas, I., J. Lehoczky and S. Shreve, 1987, Optimal portfolio and consumption decisions for a ‘small investor’ on a finite horizon, SIAM Journal of Control and Optimization 25, 1557-1586. Kreps, D., 1981, Arbitrage and equilibrium in economies with infinitely many commodities, Journal of Mathematical Economics 8, 1535. Liptser, R. and A. Shiryayev, 1977, Statistics of random processes I: General theory (SpringerVerlag, New York). Merton, R., 1969, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics 1, 247-257. Merton, R., 1971, Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory 3, 373413. Pliska, S., 1986, A stochastic calculus model of continuous trading: Optimal portfolios, Mathematics of Operations Research 11, 371-382. Rockafellar, R., 1975, Integral functionals, normal integrands and measurable selections, in: J. Gossez et al., eds., Nonlinear operators and the calculus of variations (Springer-Verlag, New York).
Economics 20 (1991) 465487. North-Holland
A variational economics*
problem arising in financial
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 consumptionportfolio 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 ArrowDebreu 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].
467
J.C. Cox and C.-f: Huang, A variational problem
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, = an a rice process S,={&(t); t~[0, Tj}, where D,(t) has {DA): CE CO,7% d P 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 henceforth denote (S,(t), . . . ,S,(t))’ and (Dl(t), . . . , DN(t))T by S(t) and D(t), respectively. Since securities will be traded ex-dividend, we assume without loss of generality that D,(O) =O. We will use AD(t) to denote where D,(t-) is the left-limit of D, at t, (DJ+D&-),...,D&)-D&-))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) 0
0
Vt E CO,T],
P-a.s.,
(1)
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 consumption, 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
2.
i (t9(t)Tcr(t)1’dt < 00 P-a.s.,
3. there exists a consumption-final that, P-a.s.,
P-a.s.,
(4)
wealth pair (c, W) ELP,(v) x L<(P) such
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
=a(O)B(O)
+ e(0)TS(O) + j (a(s)B(s)r(s)+
problem
469
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 wealth pairs are processes5 that finance admissible consumption-final 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 consumptionportfolio 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).
J.C. Cox and C.-f Huang, A variational problem
470
martingales9 after a change of numeraire and a change of probability. is the subject to which we now turn. The following assumption will be made throughout our analysis. Assumption 2.1. Rv-a.e.
There
exists
If< cc
such
that
This
[o(t)-‘(c(r)-r(r(t)S(t))ls
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 0
(8)
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 equivalent to P and under which G* is a local martingale.
measure
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 there exists a sequence of optional martingale under Q.
under the probability Q if it has right-continuous paths and times T.1 coQ-as. so that the process {X(T, A t); t E%+} is a
J.C. Cox and C.;f. Huang, A variational problem
G*(t) = jt -4s) dw*(s) o B(s)
471
t E [0, T-JP-as.,
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) ZOPa.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) s.t. with
1
(c, IV) is financed by (a,0) and lies in L<(v) x Z&(P) 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 formulated in section 2 can be transformed into a static problem of the ArrowDebreu type; see Harrison and Kreps (1979) and Huang (1985), for example. For completeness, we will outline this transformation. Consider the static variational problem:
1
E I’u@(t), t) dt + V( IV) sup (c,W ELP+ (v)XLq(P) 0
J.C. Cox and C.-J Huang, A variational problem
472
1
s.t.
B(0) E ;c(t)q(t),B(t) dt + lQ( T)/B( 7) 5 W,. [0
The following proposition
(10)
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*
1
W/B(T) + ; c(t),B(t) dt 5 W(O)/B(O). II
0
Equivalently,
1
B(O)E Wrl(T)IP(T) + ;c(r)rl(r)lP(r) dr 5 W(0) 5 [
0
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 constraint 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
1
E* W/B( 7’) + j c(t),B(t) dt 5 W(O)/B(O). 0
[
413
problem
-
Thus Wi~(T)+Tc(t)/B(t)dt~t’o. 0
Define a martingale under Q:
1
a.s.
W/B(T)+i~(t)/B(t)dtlF~ 0
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 denominator 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) (4(t) - Y@)K(t))TB(t)e(r)- ‘/?(r),
j:c(r)lB(r) dt(%l.
Putting
e(t)‘=
474
J.C. Cox and C.-f Huang, A variational problem
t
4s) 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 3.1. Corollary 3.1.
records an immediate implication
of Proposition
(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] =
where 1 Sp< co, positive constant. the supremum is the value of (A,)
r EL’!+(P) with l/p+ l/q= 1, {>O a.s., and K0 is a As usual, we will say that there exists a solution to finite and is attained by some x E L%(P). Henceforth, by val (K,). Assume throughout that V: ‘%+H% u (-
strictly (A,) if denote 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$(w+l)dw,
In addition, let V(z) = z and
VAE.K
476
J.C. Cox and C.-f: Huang, A variational problem
&
P(do) = j x(o) do. 0
Note that since prices for commodities, 3/2(0+ l), are bounded above, 4(x) < co for all x E p(P) for all pz 1. In this case, it is easily verified that s: V(x(o))P(do)<$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
xcL:(P*)
4P*(d4
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 if and only if 5-l eLplb(P). Let valb(&,) denote the V(z)=/?1+/?2z1-b 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,
417
J.C. Cox and C.-f Huang, A variational problem
E[ V(x,)] 5 /_I1+ /&E[xi -*] $ valb(&,) < CC for follows by letting n-roe. 0
all
n. The
assertion
then
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=
y(w)
1 2a
if if
y(o) > 2a 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 finite. Then there exists a solution to (A,).” Proof.
and the supremum of (A,)
is
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 transformation, 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, which was to be shown.
VzZy(w), 0
The following is the counterpart bounded from above.
of Lemma
4.2 for utility functions
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,) problem is to show that the supremum is attained. Define a measure on (52,Y): P*(A) = j @)P(do)
is finite. Thus our
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 $(x*(o) - a) P*-a.e. Since V(z)&o)-’ = o(z), Lp(P)integrably in w, and since P and P* are equivalent, Lemma 4.1 implies that there must exist an YELP,(P) such that x*(o) 5 y(0)
a.s.
This implies that x* E L%(P). Suppose in addition that 5 -r E Lp”b(p) for some measure absolutely continuous with respect to P. E”(&-integrably in o by Lemma 4.2. From previous that p is absolutely continuous with respect to P we V(x*(o))~(o)-’
2 +(x*(o)
- a)
p’ >=1 and p is a finite Then V(Z)~(W)-’ =0(z), discussions and the fact have
P-a.e.
Lemma 4.1 then implies that there exists YE L$(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.
481
J.C. Cox and C.-j: Huang, A uariational problem
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]
and
E[/B(T)lp”b2]
Then there exists a solution to (9). Proof. p’/p>
We show first that t(t)-’ =B(t)q(t)-‘/B(O) 1 and l/8+ l/G= 1, we have
E [jB(t)q(t)-llp~bldt
I([ 5
E ;((B(t)(PIb’)“dt 0
EI.~‘~‘(v). Putting
#=
IId 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,).
483
J.C. Cox and C.-f. Huang, A variational problem
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 G*(f)
standard Brownian motion under Q. Thus we can write
=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. n+oo. Then
Let x,,xEL:(P*),
and x,-+x
in L’(P*)
and x,+x
P*-a.e.
as
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 section
again 4.2.10).
that an It6 integral
is a local martingale;
see, e.g., Liptser
and Shiryayev
(1977,
J.C. Cox and C.-f: Huang, A variational problem
484
lim
sup J lc,f(x.) R
5 j f(x). 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(~$) 5 [y/4K,]+. Thus f(d) is integrable. Fix m, such that j 4~~14 S,
and
sS_lf($)l
VmZmo.
485
J.C. Cox and C.-_fHuang, A variational problem
Define T,, = {w E S,: xL(o) > &CO)}and set
For k2ko and mzm,,
,Wb)+a
--
7
4
2
-~
Y
b
4Ko Ir
zval(b)+y/2, where the second inequality jRxkm > b. Now note that
sb,+E/+
j
follows from
the fact that
b, SK,.
Thus
xk,
TIC,
for m 2 m, and k >=ko. This implies that s X,<,b,+&/+-bc TkRl
y,+&f2-b=3&/4
and thus 5 &= .%I
j SWl\Tk??l
xk+
j Tk”
xk
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 subsequence 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 But for each k’, the sequence {J(Y,,) &.0x1) and h ence ~J(R)~~J(xl). 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’-~*
Without loss of generality, assume r(T)-’ EL.P’~~(P)and fix E> 0. Define (e/j?2)-1’b1&o,t)-1’b1
p1 = $1 =O.
VZE%+. Let
l-’ ~E”~l(v)
and
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 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
(t, w) and
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 References Aumann, R. and M. Perles, 1965, A variational problem arising in economics, Journal of Mathematical Analysis and Applications 11, 488-503. Chung, K. and R. Williams, 1983, An introduction to stochastic integration (Birkhauser Inc., Boston, MA). Clark, J., 1970, The representation of functionals of Brownian motion by stochastic integrals, Annals of Mathematical Statistics 41, 1282-1296. Clark J., 1971, Erratum, Annals of Mathematical Statistics 42. 1778. Cox, J. .and C. Huang, 1985, A variational problem arising in financial economics with an application to a portfolio turnpike theorem, Working paper no. 1751-86 (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA). Cox, J. and C. Huang, 1989, Optimal consumption and portfolio policies when asset prices follow a diffusion process, Journal of Economic Theory 49, 33-83. Dellacherie, C. and P. Meyer, 1982, Probabilities and potential B: Theory of martingales (NorthHolland, New York). Dunford, N. and J. Schwartz, 1957, Linear operators, Part I: General theory, (Wiley, New York). Dybvig, P., 1980, A positive wealth constraint precludes arbitrage in the Black-Scholes model, Unpublished manuscript (Yale University, New Haven, CT). Dybvig, P. and C. Huang, 1989, Nonnegative wealth, absence of arbitrage and feasible consumption plans, Review of Financial Studies 1, 377401. Harrison, M. and D. Kreps, 1979, Martingales and multiperiod securities markets, Journal of Economic Theory 20, 381408. Harrison, M. and S. Pliska, 1981, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and their Applications 11, 215-260. Holmes, R., 1975, Geometric functional analysis and its applications (Springer-Verlag, New York). Huang, C., 1985, Information structure and viable price systems, Journal of Mathematical Economics 14,215-240. Karatzas, I., J. Lehoczky and S. Shreve, 1987, Optimal portfolio and consumption decisions for a ‘small investor’ on a finite horizon, SIAM Journal of Control and Optimization 25, 1557-1586. Kreps, D., 1981, Arbitrage and equilibrium in economies with infinitely many commodities, Journal of Mathematical Economics 8, 1535. Liptser, R. and A. Shiryayev, 1977, Statistics of random processes I: General theory (SpringerVerlag, New York). Merton, R., 1969, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics 1, 247-257. Merton, R., 1971, Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory 3, 373413. Pliska, S., 1986, A stochastic calculus model of continuous trading: Optimal portfolios, Mathematics of Operations Research 11, 371-382. Rockafellar, R., 1975, Integral functionals, normal integrands and measurable selections, in: J. Gossez et al., eds., Nonlinear operators and the calculus of variations (Springer-Verlag, New York).