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WORKING PAPER
ALFRED P. SLOAN SCHOOL OF MANAGEMENT



A THEORY OF CONTINUOUS TRADING
WHEN LUMPINESS OF CONSUMPTION IS ALLOWED

by
Chi-fu Huang

January 1984
Revised: May 1984



t



WP #1574-84



MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139



A THEORY OF CONTINUOUS TRADING
WHEN LUMPINESS OF CONSUMPTION IS ALLOWED

by
Chi-fu Huang

January 1984
Revised: May 1984



WP #1574-84



+

Conversations with Darrell Duffie, Michael Harrison, and David Kreps

proved to be very helpful. Any errors are of course my own.



Abstract



A theory of continuous trading with a very general commodity space is
developed encompassing all the existing models as special cases. Agents are
allowed to consume at lumps if they choose to. The martingale characteriza-
tion of an equilibrium price system originated by Harrison and Kreps [8] is
extended to our economy. The relationship between the sample paths properties
of a price system and the way information is revealed as studied by Huang [10]
is examined. In particular, we show that if agents' preferences are contin-
uous enough, if the information of the economy is generated by a Brownian
motion, and if the accumulated dividends process of a claim is continuous,
then the equilibrium price process for this claim in units of the consumption
commodity is an Ito process.



1. Introduction and Summary

The purpose of this paper Is threefold. First, a theory of continuous

trading In an economy with a time span [0,T] Is developed. Agents In this

economy are allowed to consume at any time te[0,T]. The objects of choice

are processes of bounded variation representing an agent's accumulated net

trades. The traditional way of modeling consumption over time, namely, the
objects of choice being consumption rates. Is just a special case of our

economy. If an agent chooses, he can In fact consume at lumps. The issues

here are what types of trading strategies are admissible and what Is the

appropriate formulation of the budget constraint.

Second, assuming that there exists an equilibrium in our continuous
trading economy, we characterize the equilibrium price system for traded
consumption claims by demonstrating its connection to martingales. A
relationship similar to the one originated by Harrison and Kreps [8] is
derived.

Third, the relationship between the equilibrium price system for traded
claims and the way information is revealed as studied by Huang [10] is
extended to the context of our economy. Keeping the setup of the economy as
general as possible, we are able to characterize the sample path properties of
equilibrium price processes relative to that of a traded consumption claim,
the numeraire security. Roughly stated, if information is revealed contin-
uously, and if the accumulated dividends process of a claim is contiuous, then
the equilibrium price process of this claim, relative to that of the numeraire
security, is continuous. The behavior of the "relative" price system when
information is not revealed in a continuous manner or when the accumulated
dividends process is not continuous can also be derived. Since this exercise
is a straightforward application of the results in Section 6 of Huang [10] , we
leave It to interested readers.



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In an economy where the consumption good Is not available all the time
as In Harrison and Kreps [8] or Huang [10], It Is not unnatural to choose a
traded asset to be the numeraire. Since the single consumption good Is
available for consumption all the time In our economy. It Is therefore
Important to have characterizations of the equilibrium price system In units
of this natural numeraire. Would the equilibrium price process of a claim
having a continuous accumulated dividends process be continuous In units of
the consumption good when Information Is continuous? The answer is yes If
agents' preferences are continuous enough, the meaning of "enough" to be made
precise. In that case, the price process is an Ito process when the infortn-
ation is a Brownian filtration. Agents here are allowed to consume at lumps,
but as long as their preferences are very "continuous", the Ito processes
representation of equilibrium price processes is an inherent property of the
Brownian motion information.

Section 2 of this paper formulates a continuous time frlctionless pure
exchange economy under uncertainty with a time span [0,T]. Agents in this
economy are endowed with a common Information structure, a filtration,
satisfying the usual conditions (to be defined), denoted by F. It is
assumed that there is one perishable consumption commodity In the economy
which can be consumed at any time in [0,T]. There are a finite number of
agents in the economy. Each agent is characterized by a preference relation ;^.
on V, the space of integrable bounded variation processes adapted to F. We
Interpret V to be the space of net trades. There are a finite number of
consumption claims traded in the economy indexed by n = 0,1,... N. We assume
that agents' preferences are continuous in a topology t (to be defined).
Each traded claim is represented by an element of V that is non-negative and
Increasing, describing accumulated dividends in units of the consumption



- 3 -



conunodlty, denoted by D , n = 0,1,..., N. We assume that the 0th claim does
not pay dividends until T, at which time, it pays a dividend that is bounded
above and below away from zero.

An agent's problem in the economy is to manage a portfolio of traded
claims and a consumption plan such that the budget constraint is satisfied and
that his preferences are maximized. The equilibrium concept used is Radnor's
[19] Equilibrium of Plans, Prices, and Price Expectations . The existence of
an equilibrium in our economy is not an issue to be addressed. We assume that
an equilibrium exists where the consumption good is taken to be the numeraire .
The equilibrium price system for traded claims is denoted by S = {S (t);
te[0,T], n = 0,1,. ..,N}.

Section 3 shows that there is a natural mapping between our dynamic
economy and a static Arrow-Debreu type economy but not necessarily of complete
markets. It is shown that the equilibrium price system for traded claims can
be represented by a conditional expectation of a stochastic integral (Prop-
osition 3.2). This representation turns out to be very useful for the results
to follow. By way of this representation, we are able to show that the
equilibrium price process for the 0th claim is bounded above and below away
:ero.

We show in Section 4 that If we take the 0th claim to be the numeraire

and define S* = S/(S + D ), then S* is an equilibrium price system for

traded claims when the price process for the consumption good is

a* = 1/(S + D ). If we denote the (N+l)-vector of accumulated divi-
o o

dends process for traded claims in units of the "new" numeraire by D*, then
Theorem 4.1 shows that the process S* + D* is an (N+l)-vector martingale under
a probability measure Q that is uniformly absolutely continuous with respect
to P, which is the common probability beliefs held by agents.



- 4 -



Given the martingale characterization of S* + D*, It Is then straight-
forward to apply many results In Huang [10] to characterize the sample path
properties of S* In relation to different Information structures. Unfortu-
nately, the T-contlnulty of agents' preferences Is not strong enough to
allow us to say something Interesting about the sample path properties of S.
Recall that S Is an equilibrium price system using the consumption good as the
numeraire. Since S Is In units of the "natural" numeraire, It Is thus
interesting and important to see what conditions are needed such that we can
answer yes to the following question: Is the equilibrium price process for,
say, the nth claim, a continuous process when information is revealed
cntinuously and when D is a continuous process?

Section 5 fixes a continuous information structure and defines a new
topology on V denoted by t*. An agent's preferences are T*-continuous if
consumptions at adjacent dates in the same state of the nature are almost
perfect substitutes. In that case. Corollary 5.1 renders an affirmative to
the question posed in the last paragraph. In particular, when information is
a Brownian filtration, any continuous price process is an Ito process (Theorem
5.1). Section 6 discusses related works. Arguments in proving the existence
of an equilibrium for an autarchy economy are outlined. Concluding remarks
are in Section 7.

2. The Formulation

In this section we model a contlnous time frictionless pure exchange
economy under uncertainty with a time span (0,T], where T is a strictly
positive real number.

Taken as primitive in this economy is a complete probability space
(J2, J , P), where each weQ, represents a state of nature, which is a
complete specification of the history of the exogenous environment from the



- 5 -

beginning of time (t=0) to the end (t=T), ^ Is the tribe of distinguishable
events, and P Is the common probability measure on the measureable space
(fl,3') held by agents.

Agents In the economy are endowed with a common Information structure
F = {"^r* te[0,T]}, an Increasing family of subtrlbes of J", that is,
3^ c. Ij for all t < S. We assume that the filtration F satisfies the

C S "^ ■ ■ ■

usual conditions ;

(I) U ^ = s>t - ^ ¥ te[0,T); and

(II) OVj contains all the P-negllblble sets;

referred to as right contluous and complete , respectively. We will also

Impose the condition that agents in the end learn the true state of nature and

that now is certain. Mathematically, this means that • R where Z(»,t) is 3" -measurable

for each te[0,T]. A process Z is measureable if Z is JxEC [0,T] )-measurable,

where B([0,T]) denotes the Borel tribe on [0,T]. The process Z is denoted by

{Z(t)}. The random variable Z(» ,t) is also denoted by Z(t). The process is

said to be adapted if Z(t) if o -measurable for each te[0,T]. The process is

said to be right-continuous with left limits if t -► Z(u),t) is right continous

and 11m Z(u),s) exists for each te(0,T] for each weii.
stt

Two processes Z, and Z- are said to be versions of each other if
P{Z (t) = Z (t)}=l for each te[0,T]. We say that Z and Z are indis-
tinguishable if P{Zj^(t) = Z2(t) V te[0,T]}=l. Two processes which are
versions of each other may not be indistinguishable. However, if Z, and
Z- are versions of each other and are right (or left) continuous, then they
are indistinguishable. For all practical purposes, indistinguishable
processes should be regarded as the same.



- 6 -



From now on, the term process means a measurable process adpated to F,
unless otherwise stated.

It is assumed that there Is only one perishable consumption commodity in
the economy which can be consumed at any time t£[0,T]. We thus take the
commodity space to be the space of integrable variation processes that have
right continuous paths, denoted by V. By definition, for each veV we have:

(i) E[| ldv(t)l] < «>=,
[0,T]

where the integral takes into account the jump of v at zero;
(ii) v(t) is •J t:~™^^surable for all te[0,T]; and
(ill) for every ojc£., v(cj,t) is a right coninuous function of t.

The interpretation is that v(ol, t) denotes the accumulated net trades from
time to time t in state oefi. By convention, each veV denotes a family
of indistinguishable processes. Let V denote the space of nonnegative
integrable increasing processes with right continuous paths. It is clear
that V C. V. If we order the vector space V by v^ > v^ If and only if
v^ - V2 t V^, V^ is the positive cone of the ordered vector space V.

REMARK 2.1 ; Since for each veV, v(c ,t) is right continous and of
bounded variation with respect to t for all t.>Er, each vrV is an RCLL (for
right continous with finite left limits) process.

REMARK 2.2 ; It is easily checked that if vcV then E[v(t)] < ^^ by
noting that E( J ^^^^j Idv(t) I ) = E(|jq ^jdv"^(t)) + E(Jjq ^jdv"(t)) =
ECv'^CT)) + E(v~(T)) < oc, where v"*" , v~ e V^ and v"*" - v~ = v, and that
E(v(T)) = E(v'*"(T)) - E(v"(T)).



- 7 -



Before proceeding, some definitions are in order. Let X be the space of
RCLL processes adapted to F with the property that for each xeX

ess sup lx(a),t)l < "^ (2.1)

w,t

Let X be the space of RCLL processes satisfing (2.1) and with the property that

X(u),t) > for all te[0,T] and almost every uefi. Letting xeX, x > means

XeX . Similarly x^ , x- e X, x^^ > X2 means x^ - x^ e X. . Denoting by X.^

the space of processes such that for every xeX. . , we have xeX. and P{x>c} = 1

for some ceR^, c ^ 0.

Now consider the bilinear form 4;: XxV ->■ R:

iKx,v) = E(J x(t)dv(t) ).
lO.T]

Proposition A.l in Appendix I shows that \J; separates points and places X and

V in duality. Let t be the strongest topology on V such that its topological

dual is X, the Mackey topology (cf. Schaefer [20], p. 131).

Coming back to economics, we shall assume that there are a finite number

of agents in the economy indexed by 1= l,2,...l. Each agent is characterized

by his consumption preferences on net trades . Formally, each agent 1 is

represented by a complete and transitive binary relation > on the net

trade space V. We shall assume that >. is convex, t continuous, and strictly

monotone, that is,

(I) the sets

{veV: V >, v} V veV

are convex;

(II) the sets

{veV: V > v} and {veV: v > v} V veV

are x-closed; and



- 8 -

(ill) V + y > V V yeV , y j^ 0, where >. Is the strict preference
relation derived from >..

An example of a convex, x-contlnuous, and strictly Increasing prefer-
ence Is given by the utility function U: V-^R, U(v) = E(J. , x(t) dv(t)),

[0,T J

where xeX and P{x>0} = 1.

We shall assume that there exists one agent, say, agent 1, whose pre-
ferences on net trade space can be represented by a concave functional
U.:V->-R whose gradients exist at all veV. Furthermore, there Is yeX with
the property that

ess Inf Y('JJ»t) >
w,t

such that Y < VU,(v) for all veV, where VU. (v) denotes the gradient of U,,

at v. In words, agent I's "marginal utility" Is bounded below away from zero.

It Is assumed that there are a finite number of contingent claims In

zero net supply traded. Indexed by n=0,l,2, . . . ,N. Each claim n Is represented

by an element D eV , where D (co,t) denotes the accumulated dividends. In units

of the consumption good, that a holder of a unit claim n from time zero to

time t has received during that period in state wefl. We shall assume that

D (t) = V te[0,T) and D-^(T) is bounded above and below away from zero. All

the contingent claims are traded ex-dlvldends. Thus, without loss of

generality, we assume that D (0) = V n=l,2,...,N. Let Vq denote the subspace

of V such that veV„ implies v(0) = 0. Then D eV.PiV. V- n.

"^ n +

An admissible price system for the single consumption good Is a strictly
positive bounded process a = {a(t)} having RCLL paths. A consumption plan
is an element of V.

An admissible price system for traded claims is an (N+l)-vector of Inte-

grable semlmartingalea— adapted to F denoted by S = (S (t); n=0,l, . . . ,N} .

n



- 9 -

Since these claims are traded ex-dividends, S (T) = a.s. V- n .

A trading strategy for claims is an (N+l)-vector of predictable and

2/
locally bounded processes 9 = ^^n^^^' n=0,l, . . . ,N} .— Given admissible price

systems for the consumption good and traded claims (a,S), a trading strategy

9 is said to be admissible if it satisfies the following conditions:



(i) ess sup I9n(u),t)l < oo ; (2.1)

n,a),t

(ii) there exists CeV„ such that if we put

t

D(t) = J a(s)dD(s) V te[0,T] (2.2)







and



t
C(t) = J a(s)dC(s) V te[0,T] , (2.3)



we have

9(t).S(t) =

t t- ^ ^

9(0)»S(0) + / 9(s)»dS(s) + J 9(s)«dD(s) - C(t-) V te[0,T] a.s. (2.4)


and

9(T).AD(T) = AC(T) a.s. (2.5)

where "•" denotes inner product, D = {DQ(t); n=0,l, . . . ,N}, and AD(t)

and AC(t) denote the jumps of D and C at t, respectively.

Since 9 is predictable, we can think of 9(t) as the portfolio held
from t- to t before trading at t takes place. Thus 9(t)»S(t) is the value
at time t of 9(t), excluding dividends received at t and before trading and
consumption take place. The right-hand-side of (2.4) is the initial value of



- 10 -



the strategy 9 plus accumulated capital gains up to and including time t,
and dividends received up to but not Including time t, and minus the
accumulated withdrawal up to but not including time t. The equality between
e(t)«S(t) and the right-hand-side of (2.4) is just the natural budget
constraint. Eq. (2.5) is the budget constraint at T. Here we only hope that
the interpretaton above makes intuitive sense. It will be shown later that
Eqs. (2.4) and (2.5) are just the "right" formulation.

Let 0[a,S] denote the space of admissible trading strategies with
respect to the pair of admissible price systems (a,S). It is easily
verified that 0[a,S] is a linear space by the linearity of the stochastic
Integral. By the definition, for each 9e0[a,S] there is CcVq such
that (2.4) and (2.5) are satisfied, which will be called the net trade process
or the consumption claim generated by 0.

REMARK 2.3 ; Since ee0[a,S] is predictable and locally bounded, the
stochastic Integral In (2.4) is well-defined. See Meyer [18] for details.

An agent's problem in this economy is to manage a portfolio of traded
claims and a consumption plan in order to maximize his preferences on net
trades.

An equilibrium of plans, prices and price expectations (of Radner [19])
is a pair of admissible price systems (a,S) for the consumption good and
traded claims, and I pairs of admissible trading strategies and consumption
plans, (9 ,C ) , one for each agent such that C is > -maximal in the set
{CeV: C - C(0) is generated by some 9e9[a,S], and C(0) = - 9(0)»S(0)}, and
markets clear; that is.



I 9.(t) = V- te[0,T] a.s.
1=1



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REMARK 2.4 ; By the Walras Law, if the contingent claims markets clear
all the time with probability one, then the market for the consumption good
will clear all the time with probability one, too.

The existence of an equilibrium in the economy formulated above is not
an issue to be addressed in this paper. The main purpose opf this paper is to
characterize properties of an equilibrium price system when indeed one exists.
Arguments for proving the existence of an equilibrium for an autarchy case are
outlined, however, in Section 6. Comments are also made there on proving
existence in general.

3. Properties of an Equilibirum Where the Single Consumption Good is the
Numeraire

In this section we assume that there exsits an equilibrium in the
economy formulated in the previous section ((ot,S) ,(9 .,C .) ._p , where
a(t) = IV te[0,T] a.s. That is, there exists an equilibrium using the
single consumption good as the numeraire. (In fact, we will show in the
sequel that there exsits an equilibrium in the economy if and only if there
exists an equilibrium where the single consumption good is the numeraire.) It
will be shown that this dynamic economic equilibrium can be mapped into an
Arrow-Debreu type static economic equilibrium but not necessarily of complete
markets. The prices in these economies are naturally linked. Their rela-
tionship will be formally characterized.

A consumption claim veV^ is said to be marketed at time zero if v is
generated by some ee0[a,S]. In this case we say generates v. The
interpretation is that if one pays 0(0) 'SCO) at time zero and gets into
the strategy 0, one can replicate the claim v over time. When veV_ D V ,
one follows 0, receives dividends, and makes withdrawals of funds out of the



- 12 -



portfolio according to v without making any new Investment Into the portfolio.
When V does not belong to VqOV^, additional funds are, however, needed.
Let R. denote the space of marketed claims at time zero. By the fact that
e[a,S] Is a linear space. It Is clear that M Is a linear subspace of V .

A current price system Is a linear functional tt_: M -*■ R. Let veM
and let 9 be the strategy that generates v. Define ti-.(v) = 9(0) 'SCO). Then
Ti gives the price at time zero of marketed claims. Let M be the linear subspace
of V such that Mf^V = M . It should be clear that tTq has an extension to M
denoted by tt: M ->■ R, For each veM we have ■rT(v) = v(0) + tt_(v - v(0)).
Then tt gives the cost at time zero for all attainable net trade patterns.

A T-contlnuous linear functional i|j: V -► R is said to be strictly
positive if \|;(v) > V veV. and v i^ 0. Each x-continuous strictly posi-
tive linear functional can be represented by an element of X, that is
strictly positive with probability one.

The following proposition shows that ti has an extension to all of V
which is T-continous and strictly positive. Furthermore, this extension can
be represented by an element of X . . .

PROPOSITION 3.1 ; tt: M -► R has an extension to all of V denoted by
4;, wlch is T continuous, strictly positive, and can be represented as:

il;(v) = E(J x(t)dv(t)) , (3.1)

[0,T]



where xeX++, that is.



ess inf x(a),t) >
u,t



PROOF ; The fact that tt has an extension \[) to all of V that is
T-continuous and strictly positive follows from Theorem 1 of Kreps [15].
Then ip can be represented by an xeX, such that P{X>0} = 1. The fact that the



- 13 -

extension can be chosen such that ess Inf z(a},t) > follows from the fact
that agent I's marginal utility is bounded below away from zero. Q.E.D.

Since i) is an extension of tt, the current price of a marketed claim
veMg is given by

T

YCv) = E (J x(t)dv(t)).



In particular, for traded claims, we have

T
S (0) = E (/ x(t)dD (t)) n=0,l,2,...,N.
^ ^

It can be checked that each agent i's optimal net trade C in our
dynamic economy is also > -maximal in the set

{CeM: KO = tt(C) < } .

In this sense, our dynamic economic is equivalent to an Arrow-Debreu type
static economy with a net trade space M and prices given by tt.

The linear functional of (3.1) not only gives prices at time zero for
marketed claims, but also provides a way to represent equilibrium price
processes for marketed claims in our dynamic economy over time.

PROPOSITION 3.2 Let v be a marketed claim and let S (t) be its

y

equilibrum price at time t. Then

T

[
t



S (t) = E(J x(s)dv(s) I O )
V \ t



xTU

■^ < ^

E(J x(s)dv(s) I J" ) - J x(s)dv(s)

= V te[0,T] a.s., (3.2)

x(t)

where an RCLL version of the conditional expectation is taken.



- 14 -



PROOF ; For expository purposes, we shall prove the above assertion for
traded claims .

We first show that the right-hand-side of (3.2) is right continuous.
This follows from the fact that we have taken a right continuous version of

E(J x(8)dv(s) I vTt^» which is possible by the fact that F is right continu-

ous (cf. Meyer [18], VI. T4), and that J x(s)dv(s) is right continuous.


Now consider claim n. At time zero, (3.2) is certainly true, since

x(0) = 1 by the fact that the consumption good is taken to be the numeraire.
Thus if (3.2) is not true, there must exist te[0,T] and Ae J" with P(A) >
such that (3.2) is not true for t on A, since the left-hand-side and the right-
hand-side of (3.2) are both right continuous. Without loss of generality, we
assume that on A:



E (J x(s)dD (s) I J^)
n t

S„(t) >



t
n - ' x(t)



We define

e.(a),s) = V wefi, se[0,T], k=0,l, . . . ,n-l,n+l, . . . ,N,

e (u),s) =0 V ojefl, se[0,t]

= -1 V cogA, se(t,T]

= V- (jOGJi\A, se(t,T]

C(a),s) = V- weJi, se[0,t)

= S (u),t) - (D (co,s) - D (a),t)) -^ weA, se[t,T]
n n n

= V ojeJ^V. se[t,T] .
Thus defined, 6 is predictable and locally bounded since it is left continu-
ous. C is an integrable variation process adpapted to F having RCLL paths.
We claim that 6 generates C. To see this, we first note that for se[0,T]



- 15 -



and on fl\A., and for se[0,t] on A we have:

e(s)»S(s) = e (s)«S (s)
n n

s 8- s-

= e (0).S (0) + f e (u) dS (u) + J r (u) dD (u) - J dC(u)
nn J^n n ^^n n q

s s- s-

= e(0)«S(0) + J e(u)»dS(u) + J e(u)dD(u) - J dC(u)


= .

On A and for se(t,T], we have

s s- s-

e(0)«S(0) + J e (u)»dS(u) + J e(u).dD(u) - J dC(u)


= S (t) - S (s) + D (t) - D (s-) - C(s-)
n n n n

= S (t) - S (s) + D (t) - D (s-) - S (t) + D (s-) - D (t)
n n n n n n n

= - S (s) = e (s).S (s) = e(s).S(s) .
n n n

It Is also clear that 9(T)»AD(T) = AC(T). Thus 9 generates C and C Is
marketed. Since e(0)'S(0) = 0, the price for C at time zero is 0. But


1 3

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