M^i
til
m
mi'
m
,./<b5V-35
/9S3
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
MULTIPERIOD SECURITIES MARKETS
WITH DIFFERENTIAL INFORMATION:
MARTINGALES AND RESOLUTION TIMES*
by
Darrell Duffie
and
Chi-fu Huang
WP //1654-85
January 1985
Revised March 1985
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE. MASSACHUSETTS 02139
MULTIPERIOD SECURITIES MARKETS
WITH DIFFERENTIAL INFORMATION:
MARTINGALES AND RESOLUTION TIMES*
by
Darrell Duffie
and
Chi-fu Huang
WP //1654-85
January 1985
Revised March 1985
*We would like to thank conversations with Michael Harrison and
Andreu Mas-Colell.
Some technical help from Kai-Ching Lin of the
University of Chicago is appreciated.
Table of Contents
1
Introduction and
summary
2 Differential Information
1
2
3
The formulation
3
4
The advantage
7
of better information
5 Free lunches and martingales
6
The
7
An example
resolution times
8 Discussions, generalizations, and concluding remarks
9
12
17
19
Abstract
We
model multiperiod
securities
that admits no free lunches
tations.
We
is
markets with
differential information.
related to martingales
when agents have
A
price system
rational expec-
introduce a concept called resolution time and show that a better informed
agent and a worse informed agent must agree on the resolution times of commonly mar-
they have rational expectations and
keted events
if
follows that
if all
there are no free lunches.
It
then
the elementary events are marketed for a worse informed agent then
any price system that admits no
cally equalize the information
free lunches to a better
informed agent must dynami-
asymmetry between the two.
a dynamically fully revealing price system that
complete markets.
if
is
We
provide an example of
arbitrage free and yields elementarily
1.
Introduction and
summary
This paper addresses differential information
in
a multiperiod model of security markets. Fo-
we
cusing on asset price processes that preclude "free lunches,"
We
Kreps connection between price processes and martingales.
extend the famous Harrison-
first
go on to study the information
that must be revealed by a better informed agent through price processes to a worse informed
agent
free lunches are to be precluded. Finally,
if
we formalize the connection between completely
revealing price processes and dynamically complete markets for elementary contingent claims.
and additional
details
summarized below. The
results are
results of Harrison
and Kreps
The
[1979]
connecting the behavior of price processes with martingales have already opened up a theory of
dynamic equilibrium with symmetric information
(Dufiie
and Huang
[1985]).
The
results here are
a step toward dynamic equilibrium with differential information.
The paper
is
summarized as follows. Section
2 motivates the concepts of differential information
The
details can easily
continuous-time setting.
in a
less
demanding discrete-time
be surmized by
Section 3 presents the formal model and shows that, in a rational expectations model,
readers.
agents agree on the joint variation of stochastic price processes.
A
some
contingent claim
is
marketed
if
financed by
of the advantages of better information:
thus a larger space of marketed claims.
some
security trading strategy. Section 4 shows
a larger space of admisible trading strategies and
Section 5 shows that the absence of simple free lunches
(Kreps [1981]) implies a unique implicit price process for any marketed claim.
A
better informed
agent cannot, in the absence of simple free lunches, attain a consumption claim at a smaller
investment than a worse informed agent.
initial
Kreps
[1981]
simple free luDcb concept of "no arbitrage" to the free lunch
free lunches for a particular agent,
we demonstrate the
shows a need to strengthen the
in infinite-dimensional cases.
Barring
existence of a martingale measure for that
agent: a probability measure, absolutely continuous with respect to the agent's
endowed probability
measure, under which security price processes are martingales.
In Section 6
event
is
to
we introduce
the resolution time of an event, the
random, and of course may be
different for differently
An
time one learns that an
resolution time of an event
informed agents.
has no later resolution time than a worse informed agent.
if
The
happen or not to happen with probability one.
first
event
B
A
is
better informed agent
marketed
for
there exists a marketed elementary contingent claim for this event, or equivalently,
exists a trading strategy paying one unit of
assuming the consumption space
is
consumption
in
event
B and
is
an agent
if
there
nothing otherwise. By
separable and thereby extending a result of Kreps [1981],
we
are able to demonstrate the existence of an equivalent martingale measure, barring free lunches.
This
is
a martingale measure assigning strictly positive probability to any event assigned non-zero
probability by the agent's
endowed probability measure. This allows us to show that the absence
free lunches for the better
informed agent implies that the resolution times of the better and worse
of
informed agents for commonly marketed events must be equal (almost surely).
all
events are marketed for the poorly informed agent and
if
It
follows that,
if
the better informed agent has no free
lunches, then the price system has symmetrized the information, neutralizing the better informed
agent's informational advantage.
Section 7 provides an example of a fully revealing arbitrage-free price system. Uncertainty
modeled by a Brownian Motion. Agent a observes the Brownian Motion
observe the Brownian Motion as
evolves through time, but
it
Two
the final value of the Brownian Motion.
The
riskless.
/3.
2. Differential
which
is
and one
brief concluding remark.
model
of uncertainty
Information
central primitive for a
An
"correct," loosely speaking.
any ordered subset of
[0,1], allowing
Fred receives information
termed cr-algebras) on Q.
knows whether
A
CI is
assumption which
is
^
of
C
is
never forgotten,
Vs >
7,
F
=
{7(;
an element of the tribe
is
T
correct. For this section the time set
an element of A, that
is
Tt
is
of "states of the world," one of
formalized by specifying a Eltration
is
or not the correct state
that the filtration F
Q
a set
one to cover continuous and discrete time,
some subset
If
is
agent, say Fred, receives information through time, refining
formalize the notion that such an occurrence
A further
elementary claims. Section 8 adds
for
and a
Fred's knowledge concerning which of the states in
How
does not
at the beginning of time
long-lived securities are traded, one risky
Markets are dynamically complete
discussion, generalizations,
The
knows even
/?
price process of the risky security symmetrizes the original informational assymetry
between agents a and
some
Agent
directly.
is
is,
(
is
in
a single pass.
€ T}
of tribes (also
7t
then at time
A
has "occurred". To
whether
we say that F
is
t
Fred
increasing, or
(.
often technically convenient
when
T
is
a continuum time set
is
right-continuous, or
s>t
A
different agent, say George,
specified by a filtration
G=
{^,;<
might naturally receive
€ T}
at an informational disadvantage at time
time
t,
and perhaps more.
If
$i
C
Ti
of tribes on C.
t,
Of
course,
if
^(
C
t
E T, we
write
G C
then George
is
George knows
at
It
since Fred "knows" every event that
for all
through time
different information
F and say that Fred
is
better
informed than George.
The
join of
7t
and
^j, denoted
Jt^
Qi,
is
the tribe generated by the union of
represents the total information held by the two agents at time
then at time
The
t,
t.
In other words,
by pooling their information, Fred and George would know whether
joined filtration, denoted F
VG
is,
of course {7t\/ Qt\t
E T}.
and
7i
B
\i
B
€.
Qi,
and
7t\/ Qt
has occurred.
The meet
(7-algebra),
time
t
of
and
whether
denoted
is
B
$i, on the other hand,
and
Tt
Tt
A §f
B€
If
A
7<
the intersection of
is
In the "rational expectations" genre
common
it is
to
is
indeed a
is
and George know independently
^, then Fred
has occurred. The meet of the filtrations
and §i (which
7t
denoted Y
AG =
{7t
A
Si]t
€ T}.
assume that agents have private
mation and also learn information from "market observables"
in particular,
,
at
infor-
the market values of
traded assets. Market observables can be represented as a vector of stochastic processes, which we
S from
take to be nothing more than function
at time
is
t
t
modeled as the
is
tribe
Tf on
process
If
some
to 5R^ for
integer
N. The value
B €
S up
7t^ then
to
an agent
one
will
know
and including time
is
at
generated by the functions {S{s); s
fi
S
corresponding filtration of tribes generated by
if
T
5 up
denoted S{t), a function on n. The information revealed by observing
including time
point,
x
f2
time
is
denoted F^
whether or not
t
€
filtration
is
G
the
the
and learns
of course
Suppose that agents are from a set A, indexed by a, with private information given by
process
The
[0, <]}.
t.
endowed with private information corresponding to a
The market observable
to and
= {7t^\t G T}. To belabor
B has occurred by observing
from a vector of market observables S, then the agent's "total" information
F'*.
S
of
5
is
fully revea.l'mg
GVF
.
filtrations
if
yaeA,
Yf'^cF'^vf^
a
which means that by observing
S
each agent learns
all
privately held information not already
known.
A
S
stochastic process
respect to p/ for
all
t
implicit in G. Indeed,
The
adapted to a
€ T. Roughly, S
S
is
adapted to
is
G
G = {get G T} if S(t) is measurable with
if 5 only conveys information already
to G
filtration
adapted
if
and only
if
F'^
C
G.
question arises: Are market observables such as prices naturally adapted to the join of
private information? This
If
is
is
essentially a question
about the economic structure of price formation.
one believes that market values are a direct consequence of individual strategies, such as bidding,
for example, then
it
would be natural
market observables as adapted.
to treat
One
could also
imagine, perhaps, that certain market structures add "noise" to price formation. This issue
addressed
3.
all
in
Kreps
[1977].
is
also
For our purposes, we are not forced to take a stance.
The formulation
In this section
we consider an intertemporal pure exchange economy under uncertainty with
differential information.
Taken as primitive
u EU denotes a complete description
common probability assessment held
subset of the interval
[0, 1],
of the
is
a complete probability space {Cl,T,P), where each
exogenous uncertain environment, and where
by agents
which includes
in the
and
1.
economy. The time set
T
is
P
is
the
taken to be a
assumed that there
It is
sumption only at time
1.
is
one consumption good
We take V =
An
u>
information structure
is
(17,
V
G U. We equip
The economy
1
l^a
C V;
2
>a
is
K
F" e
We
F"
is
an increasing family of subtribes of
populated with agents from a set
implies that v
K
v{u!) units of
I,
consumption
r.
{7t;t
G T},
+k
>-a v,
where
V
the positive cone of
is
interpret
that
"Va
>-a
is
is
=
>?
E
{>:a-,'Vai^°'](^
£ ^}- We
shall
assumed that there are a
degeneracy,
assume
E V and
the strict preference relation derived from >:q, and
with the origin deleted;
be his preference relation on
>:a to
we
=
\
a.s.,
=
number
finite
1, 2,
.
.
.
,
A''.
further impose that for
Each security n
We
a normalization.
of long-lived securities traded in the
and
"Va,
all n,
represented by d„
is
shall henceforth
P{dn <
1}
=
economy
denote (d„)^_j by
d.
To
rule out
1.
{5„(i); n
a vector of semimartingales with respect to some information structure F such that F"
for all
aE
in
E K. We assume
A price system for long-lived securities is an A''-vector of nonnegative processes 5 =
1,2,. .. ,N;t eT} satisfying the following conditions:
is
a
endowed information structure.
that X^„_i dn
S
is,
A:
strictly increasing in the sense that v
Va to be agent a's net trade space,
zero net supply indexed by n
1
that
F.
to be his
It is
means
with the L'-norm topology denoted by
a preference relation defined on
where
3
EV
J, P). Thus v
that the following conditions hold for each (>:„, 'Va)^")
€
available for con-
is
Let F be a countable collection of information structures.
£ltra,tion.
k
economy which
L^{P) to be the commodity space, where L^{P) denotes the
space of integrable random variables on
at time one in state
the
in
=
C F
A;^
2
S„{t)
<
a.s.
I
^5„(0 =
and
1
a.s.WtET;
(3.1)
n=l
3forn =
l,2,...,iV,
^(''[5„,5„]i)^ <oo,
where
{'"[S'„,5„](} is the joint variation process of
and Meyer [1982,
We
assume
Requiring
loss of generality.
S„ and 5„ with respect to F;
F
is
complete
in
that
Tt
to be a semimartingale with respect to
Any
cf.
Dellacherie
VII.44].
in addition that
5
(3.2)
discrete time process
'For the definition of a semimartingale
see, for
is
contains
all
the P-null sets for
some information structure F
a semimartingale with respect to
example, Jacod [1979, pp.29].
4
its
all
is
t
E
T.
without
natural filtration.
time case, almost
In the continuous
some
The assumption
filtration).
the
all
that F
is
known
finer
processes are semimartingales (with respect to
than any of {F";a £ A} allows the possibility that
an agent can learn from a price system. There exists discussions
Kreps
whether F should be at most as
[1977], on
Our assumption
not an issue that concerns us.
Condition
(3.1) is again
Remark
assumed that agents
denoted by H^iS)
is
We
shall
This
is
general enough to include that as a special case.
is
is
a technical restriction.
For a sufficient condition for (3.2) see Dellacherie and Meyer [1982, VII.98].
3.1:
in
5}tra.tion
A have
it
= {hf{S);te
generates
is
denoted by ¥^
=
G T}.
{7i^;t
rationa/ expectations in that they learn from a price system
The information
to refine their information.
S
fine as the joint of all agents' information.
a normalization and condition (3.2)
Given a price system 5, the
It is
the literature, for example,
in
structure of agent
a
after he observes a price
system
T}, where
assume throughout that {{"{S)
satisfies the usuai conditions:
complete; and
1
Tt
=
For consistency, we need to require also that d be measurable with respect to h°{S) for
all
2 right-continuous:
Aux^u' where
a E A. That
is,
u
let
€
be any half open interval contained
\r,s)
T, then V< e
in
['''is),
[r,s).
at time
when
1
be settled, they will know how
agents' long and short positions of long-lived securities need to
many
units of consumption
good to which they are
entitled or
obligated.
Before proceeding,
Lemma
Proof.
A
3.1.
We
we
price system
first
claim that
some
shall first record
is
S
an A'^-vector of
is
is
an F-semimartingale and
[1979] that
5
is
is
H° {S)-semimartingales
By
ff"(5)-optional.
right continuous, therefore fl^°'(5)-optional;
S
technical results.
cf.
S
Chung and Williams
//''*(5)-optional.
an //'"(5)-semimartingale, since
construction,
It
for all a.
is
[1983,
follows then from
fl''*(5) is
adapted to H°'{S) and
Theorem
Theorem
3.4].
is
Thus
9.19(a) of Jacod
a sub-Sltration of F.
I
If F'^ is
system
S
not coarser than F" and agent
will
a
ignors the information revealed through 5, the price
not be an F"-semimartingale, since
omy, where trading over time
budget constraint
for agents
must involve stochastic
is
it is
not adapted to
F"*.
In an intertemporal econ-
allowed, before anything interesting can be said, an intertemporal
must be formulated. With uncertainty present,
integration. Jacod [1979, p. 278-279] has
integrals to have desired properties,
it is
shown that
this
in
budget constraint
order for stochastic
necessary that the integrators be semimartingales. Thus
-
when a
else,
price system carries information with which an agent
the agent's intertemporal budget constraint
Another lemma
Lemma
is
{^{Sn,S,n]t-te T} and {^"(^^[5„,5„,],;
process of Sn and
Proof.
it.
Or
not well defined.
is
<
<
G T} which
e T}, where
is
a
common
{^"(^'[5„,5„,],;
<
version
€ T}
is
^
of the processes
the joint variation
with respect to H°'{S).
Since 5„ and S„i are semimartingales with respect to F and H°'{S), the two processes
G T} and
{^[5„,5r„],;<
VII.42].
5,,,
not endowed, he must learn
needed:
There exists a process {[5'„,5m](;
3.2.
is
We
first
{"°'^^^[Sn,Sm]t;t
G T}
are well defined;
cf.
Dellacherie and
Meyer
[1982,
note that
and
4
cf.
Jacod [1979], Section 2.25. Theorem 9.19(b) of Jacod [1979] shows that there exists a
version of {^[5„
+ 5m,5„ + 5TO](}
{'^"^^'[5„
Sm]t} and
-
Sm^S,,
-
{^°'('^'[5„
and
+ 5m,5'„ + 5^],},
Sm]t}. Therefore there esists a
and similarly
common
for {*"[5n
common
-5m,5n
version of {^[5„,5m](}
and {^"(^)[5„,5„.],}.
I
We shall henceforth use
for all a. (This
F and H°'{S)
is
{[5'„,5m]/} to denote a
F
possible because
is
a countable
since both filtrations are complete.
right continuous, {[5n,5TO]/}, {'^[5„, 5„,]t},
Indistinguishable processes are indeed the
Lemma
will agree
common
set.)
version of {^[5„,5m]/} and {^"^^^[5'n,5m](}|
Note that {[5„,5,„](}
is
adapted to both
In addition, since joint variation processes are
and {^"^^^[5„,Sm](} are indistinguishable^ processes.
same process
3.2 has an important implication.
As long
on the joint variation processes of security
for
any pratical purpose.
as agents have rational expectations, they
prices.
In the case that price processes are
continuous processes, joint variation processes are covariance processes. Thus rational agents must
agree on the variance-covariance matrix process.
Now
let
processes.
A
Pa{S) be the tribe of subsets of
process
F
:
fi
x
[0, 1]
i-h-
5R is
Cx
[0, 1]
generated by
said to be
left
continuous i/" (5) -adapted
H" {S)-predictable
if it
is
measurable with
respect to Pa{S).
Given a price system 5, an admissible trading strategy
=
{0n{i);
n
=
1,2,.
..
,N;t
GT}
for agent
a
is
an A''-vector of processes
satisfying the following conditions:
^A process {.Y(()} is a version of another process [Y(t)} if X(t) = Y (t) with
'Two processe {A'(<)} and {V'{0} are indistinguishable if ^(i) = Y{t) for all
probability one for all t G T.
J £ T with probability one.
.
9
1
/f"(5)-predictable;
is
2 the stochastic integral
d{s)'^dS{s)
/
Jo
is
3 5
well defined with respect to H°'{S) for
self-financing, that
is
= e{o)^s{o)+ f
with respect to H'^iS), and
n=
all
61(1)"^^
yteT
e{s)^ds{s)
a.s.
By
{er,{t)fd[Sn,S„]y <00.
(3.4)
a natural budget constraint and condition (3.4)
(3.3) is
The advantage
4.
technical.
is
a when the
price system
is
the linearity of the stochastic integral and an application of the Kunita-Watanabe inequality
Meyer
Dellacherie and
(cf.
(3.3)
€ V;
Let 0"[5] denote the space of admissible trading strategies for agent
5.
S;
6
l,2,...,iV
e(J
Condition
G T, henceforth denoted hy
t
is,
e{t)'^s{t)
4for
all
We
we know
0'*[S]
is
a linear space.
of better information
show
shall
[1982, pp.277]),
in this section that a better
informed agent will be better
the sense that
off in
he or she has access to more admissible trading strategies and therefore can enjoy a larger feasible
net trade space.
if
agent
a has
The
direct converse of the above statement
is
not true, however.
access to a bigger space of admissible strategies than agent
information as refined by a price system must be finer than that of agent
/9,
say.
We
show that
Then agent
a's
except possibly at date
13
1.
Following Harrison and Kreps [1979]
agent
6
is
a
given a price system
S
if
there exists 6
a and
said to generate v for agent
we say that
a consumption claim v
E 0"[S] such
^(0)''^5(0)
is
that d'^{l)'^d
=
€V
is
marketed
v a.s. In such event,
an implicit price for v at time zero. Let Ala
denote the space of marketed consumption claims for agent a. By the fact that ©"[5]
space, A/q
We
is,
T,^
is
a linear subspace of
shall say that agent
c
Theorem
7," Vf
G
4.1. If
a
is
is
a linear
V
better informed than agent
/3 if
F^
is
a sub-filtration of F", that
T.
agent
a
is
better informed than agent
/3,
then 0^[5]
C ©"[S] and
Mp c Ma.
Proof. Let 9
for
€ ©^[S]. We want
to
show that
satisfies
7
the defining properties of 0'*[5].
therefore
H°(S)
Firstly, the fact that the stochastic integral 9
assertion of
Theorem
9.26 of Jacod [1979]. Also, ^(l)'^d
Secondly, the fact that
£
5
is
V
since 6
well defined follows from the
G e^[5].
a self-financing strategy with respect to
is
first
follows from the
H^lS)
second assertion of Theorem 9.26 of Jacod [1979].
Lastly, the fact that 6 satisfies (3.4)
As long
M^ C M^.
€ Q^lS] and thus
obvious. Therefore 6
is
as a better informed agent has rational expectations, whatever strategy a worse in-
formed agent can employ can be employed by him.
incorrect, however.
The
converse of the above proposition
Consider the following example. Let a and
/9
be two agents, whose informa-
tion structures cannot be ordered. Suppose that the information generated by the price system
is
H^{S) c i7"(5) and thus e^[5] C e"[5].
identical to F^. Therefore
The
is
following proposition shows that the statement that
9^ [5] C ©"[5]
implies H^[S]
C
ff'*[5] is also incorrect.
C hf{S)
Proposition 4.1. Suppose that h^ {S)
V<
€ T/{0}. Then Pp{S) C Pa(S) and e^[5] C
G"[S].
Proof.
of the
2 of
The
predictable tribe P^(5)
form {0} x Bo and
(s,
Chung and Williams
The
B
x
is
6 h^[S) and B G
with Bq
h^s{S) for
By the hypothesis that 7f C
[1983].
P^(5)-predictable rectangle
of the assertion follows
t]
generated by the collection of H^{S)-predictable rectangles
is
J," V«
s,teT,s <t;
G T/{0}, we know
a ;'„(S) -predictable rectangle. Therefore, Pff{S)
from similar arguments as those of Theorem
intuition behind Proposition 4.1
is
clear.
C
Chapter
cf.
that every
Pa{S).
The
4.1.
In a finite horizon single
commodity economy
where admissible strategies are predictable ones, agents are not able to take advantage
better information revealed to
the final date!
A
Theorem
Suppose that
Proof.
and
4.2.
correct statement
Suppose that there
B^
them
/if
exists
hf{S). The rectangle
Chung and Williams
(5)
<
at the final date of the
/if
(5).
will
be no trading at
Then 6^ [5] C e"[S] implies H^{S) C H"{S).
G T/{l} such
{t, l]
economy. There
of their
as follows.
is
C
rest
x
B
is
that hf{S)
^ hf{S). Then
an element of Pd{S) but
is
there
=
1(,,i]xb('^,s) (1
d„{oj,s)
=
-1(,^,]xb(w,s)5,(cj,0 for
8
-
Si{ojj)) for
all
n
a set
BG
hf{S)
not an element of Pa{S);
[1983, pp. 28]. Define a self-financing trading strategy by
ei{u,s)
is
some
yt i.
i
cf.
It is
To
quickly checked that 6 defined above
see this
c
e^[S]
we note
that, d~^ (5R+/{0})
=
an element of ©''[5]. But 6
is
(t,l]x
e°\S]. Thus we must have H^{S)
B^
is
not PQ(S)-predictable.
Pa{S). This contradicts the hypothesis that
c H"{S).
I
A
corollary
available.
is
=
Corollary 4.1. Suppose that h^{S)
Proof. Using
Theorem
/if
(S)
and that e"[5]
=
e^[5]. Then H''{S)
=
H^{S).
4.2 twice gives the result.
I
It
is
clear
from the above discussion that whether a converse statement of Theorem
true will depend upon whether the information conveyed by a price system
is
4.1
is
a substitute or a
complemeDt to agents' information.
5.
Free lunches and martingales
In this section
we
shall
show that a
price system that admits no free lunches
is
intimately
connected with martingales when agents have rational expectations. The definitions of a simple
free lunch
and a free lunch that we employ are those of Kreps
Definition 5.1.
A
a
simple free lunch for agent
is
[1981].
a strategy 6
€
0°'[S] such that ^(0)''^5(0)
<
and d{0)^d€K.
Definition 5.2.
A free
lunch for agent
a
is
a net {{d^,
f-^);
A
€ A} C e"[5] x
V
and a bundle k e
K
such that
e^{l)'^d-v^eKu{0}ioia\lX,
v^^k,
and
liminf^^ (0)^5(0)
Implicit in the definition of a free lunch
ences.
We
is
<
0.
obviously a sense of continuity of agent q's prefer-
take this definition as a primitive in the analyses to follow and refer interested readers
to Kreps [1981] for a host of related issues.
Here are some direct consequences of the no simple
Proposition 5.1. Suppose that a price system
Then
for all n
=
Proof. Obvious.
1,2,
.
.
.
,
TV,
5„(1)
=
d„
a.s.
free lunch definition.
S admits no
simple free lunches for some agent a.
S admits no
Proposition 5.2. Suppose that
functional Ra
'
Mq
"-^
simple free lunches for agent a. There exists a linear
that gives implicit prices at time zero for every marketed consumption
3?
claim (for agent a).
Proof.
clear that
It is
if
5 admits no
simple free lunches for agent a, then there exists a unique
implicit price for every marketed claim (for agent a). Let 6
m e Ma.
Define
tt^
:
Mo, h^
by
5R
ndm) =
e 0"[5] be a
strategy that generates
e{0)'^S{0).
I
Proposition 5.3. Let
m
£
Ma
he marketed for agent a. If there exist no simple free lunches, then
there exists an implicit price process {S^{t);t
price process
Proof. This
is,
m
for agent a.
In addition, the implicit
a direct consequence of no simple free lunch and the fact that there exists a portfolio
of long-lived securities, available for all agents,
(That
for
uniquely determined up to indistinguishability.
is
is
G T}
whose
implicit price process
hold one share of each long-lived security.) In fact, the assertion
where there
exist
ways
is
is
unit throughout.
valid in any
economy
to transfer strictly positive values across time.
I
A
corollary
is
immediate.
Corollary 5.1. Suppose that agent
a
is
better informed than agent
admits no simple free lunches for agent a. Let
m
€ M^. Then
(3
and that the price system
{5,^,(0} ^"'^ {^m{t)} are indistin-
guishable processes.
Proof. This a consequence of
Theorem
4.1
and Proposition
5.3.
I
When
a price system admits no simple free lunches for a better informed agent, he cannot
dynamically manufacture a consumption claim
corollary
no simple
a complement to
is
free lunches,
in the sense that
When
is
Theorem
4.1.
less costly
than a worse informed agent can. This
The advantage
some consumption claims can be manufactured
$
By the
probability measure
Qq
following proposition
*Here we remark that since
A
on the measurable space {Cl,T) that
such that under which the price system
The
and not
Let
<E>
denote
Riesz representative theorem,"*
we
with the positive cone of L°°{P) with the origin deleted, where L°°{P) denotes the
space of essentially bounded random variables on (n, T,P).
P
there are
less costly.
a price system admits no free lunches we are able to say a bit more.
can identify
when
strictly in the sense that the space of feasible net trades is bigger
the set of nontrivial positive linear functionals on V.
to
of better information,
V
is
is
well
5
is
is
martingaie measure for agent a
a
absolutely continuous with respect
a vector of //°'(5)-martingales.
known.
a Banach lattice, any positive linear functional
10
is
is
continuous.
Proposition 5.4. Suppose that
extends
ttq,
Proof. See
The
Lemma
of
1
4>a
\
admits do free lunches for agent a, then there exists
Ma =
Kreps
4>a
^ ^
that
ttq.
[1981].
following theorem connects the no free lunch condition with martingales.
Theorem
Qa, and
denoted by
S
5.1.
There exists a one-to-one correspondence between martingale measures
linear functionals 4>a
€^
with
(/>a
Qa{B) = ^a[lB)
wiere E^{)
is
Ma =
\
e r and
>^B
The correspondence
tTq.
<l>^{v)
=
lies in
$.
is
for agent a,
given by
VuGV,
E*Jv)
the expectation under Qa-
be an extension of
Proof. Let
(pa
essentially
bounded random variable
to all of
tTq
j/a
V
that
on (n,7) such that
Thus there
<l>a{v)
=
exists a nonzero positive
E{vya) Vv e V. Define a
set
function
Be
Qa{B)^ f ya[oj)P[d^)
Jb
easy to see that
It is
Qa
is
7.
Next we want to show that 5
a probability measure on (0,7).
is
a
vector of /^^(Sj-martingales under Qa.
Fix an integer
i
with
I
<i < N.
<
Let
<i
<
<2
<
and {<i,«2}
1
C
T. For any
BG
/i",(5)
consider the following trading strategy:
ei{u,t)
0„{u!,t)
=
1
=
Si{u,t2)
claim that d
since
cf.
B
G
G 6" [5].
Meyer
=
-S,(w,<i) for
=
5,(w,<2)
1,
-
e
(<i,<2]
Si{u,ti) for
otherwise,
t
e
te
all
n
^
is
and
t
e
tj
(<2,1]
and
(<i,<2]
Si{u,ti) for
otherwise, for
€
B
anduiE
[1982,
wG B
(<2,1]
and
w€ B
i.
5
is
well-defined with respect to H''{S)
Chapter
VIII]. Secondly, the fact that d is self-financing follows
in
Theorem
3.1 of
Huang
[1985a].
we have
^{lo
B
a left-continuous (therefore predictable) simple trading strategy;
from the reasoning similar to that given
bounded by
-
t
Firstly, the stochastic integral ^
hf^{S) and since ^
Dellacherie and
5,(w,<i) for
=
=
We
-
i^"{t)f d{S„,S„]ty
< E {[S„,S„],)^
<
11
oo,
Thirdly, since
|
^
j
is
.
where the second inequality follows from
6{iyd G V. Note
that
element of
=
that 9(1^(1
Is
3.3
(5,(<2)
and Equation
-
(3.2). Finally,
which
5',(<i)),
we want
to
show
bounded and naturally an
is
V
The consumption claim
zero. Since
Lemma
and
<i, ^2
are arbitrary,
i
—
15(5,(^2)
it
5',(ii)) is
marketed and has a implicit price of zero at time
follows that
5
is
an
i7"(S)-martingales under
A'^-vector of
Qa.
Conversely,
positive,
and
be a martingale measure for agent a. Putting y
Qa
let
= dQo/dP,
y
is
nonzero,
bounded. As long as we can show that
essentially
k(J
{On{t)?d[S„,S„],y
<<:x.,
the assertion then follows from arguments similar to those in Theorem 3.1 of
Huang
[1985a].
Now
observe
k(J
{en{t))^ d[S„, Sr^jtV
<(esssup y)E(
<
where we have used that
(^„(0)' c?[5„,5„],y
J
CX),
under a substitution of an
fact that joint variation processes are invariant
absolutely continuius probability measure.
I
A corollary
Theorem
Huang
5.1
immediate:
is
a generalization of the martingale results of Harrison and Kreps [1979] and
[1985a]. If a price
5
then
is
is
system S admits no
free lunches for
a vector /r"(5)-martingale under some probability
respect P.
The converse
We shall say
is
an agent
a with
Qa
is
that
rational expectations,
absolutely continuous with
also true.
that a price system
A. In that case, for any agent
a
S
is
arbitrage-free
if it
admits no free lunches for
there exists a martingale measure Qa-
We
all
agents in
can go on to talk about
completeness of the securities markets. That being straightforward generalization of Harrision and
Kreps [1979] we leave
Remark
portfolio
<
£ T
rate,
is
The
whose value
for interested readers.
driving force behind the martingale result
is
unit throughout.
certainly not necessary.
For example,
available
The
some martingale measure.
upon
We
is
Thus the assumption that
if
the fact that there exists a
X!)n=i '^n(0
—
1
'^^- f°r ^^'
there exists a riskless asset with a zero interest
then a price system that admits no free lunches
respect to
6.
5.1:
it
for, say,
agent
a
is
a //°'(5)-martingale with
prepared detailed derivations of this case.
They
are
request.
resolution times
In the last section
we have demonstrated that an
arbitrage free price system
is
intimately
connected with martingales. In this section we shall show that no free lunches necessitates that a
12
better informed agent and a worse informed agent agree on the resolution times of a particular set of
events.
That
We
equalized.
through an arbitrage free price system some endowed asymmetry of information
is,
is
go on to formalize the linkage between dynamically elementary market completeness,
Under some
to be defined, and a dynamically fully revealing price system.
regularity conditions
a price system that admits no free lunches for a better informed agent and yields elementary
completeness of markets for a worse informed agent must convey
better informed agent
BG
Let
is
endowed with to the worse informed agents over time.
h^{S). Define an optional time
r| =
inf {«
€
[0, 1]
where as usual when the infimum does not
we have E{\b
h")
\
=
Ijg a.s.
T^
:
:
we
Tq
is
the
first
>-^
Tg
exist
[0,oo] by
h^{S))
\
B
mapping from
time that agent
for agent a.
q
its
to
fl
is
range
is [0, l],
B
following proposition formalizes this intuition.
Let
6.1.
Proof. Define
Be
f^
:
fi
h^{S). Then
^
[0,oo]
BG
Some
h"{S),
lies in [0,1].
We
however. Thus without loss of
if
We
agent a
will not
is
to
happen
shall thus call
is
or not to
Tg
the resolution
better informed than agent
be later than those for agent
technical
happen
lemmas
/?.
/?,
The
are first recorded.
h^a{S).
by
=
It is
B
sure that event
€ h^[S)
Tq
B€
[0, 1].
Intuition suggests that
then agent a's resolution times for every
Lemma
or 0},
1
takes the value oo. Note that since
with P-probability one, after observing the price system.
time of event
=
Therefore, except on a set of P-measure zero,
shall treat TJi as a
Literally,
Q
E{Ib
can always redefine Tg on a P-nuU set such that
generality
the information that the
all
Vw ^ B.
oo
clear that
f§ = M{te[0,l]:E{lB\hf{S)) =
Thus Tg
is
and Meyer
an if" (S)-optional random variable.
[1982] that
BG
It
l}.
then follows from Theorem IV.53 of Dellacherie
h^^{S).
I
Remark
For the definitions of optional random variables and hj-a{S), see Chung and
6.1:
Williams [1983, Section
Lemma
6.2. If
Furthermore,
Proof. Let
T
1.7].
H^{S) C H°'{S), then any H^{S)-optional random
let
T
be H^[S)-optional. Then h^{S)
C h^{S).
be /r''(S)-optional. Then we have
{T<t}ehf{S)
13
V<GT
variable
is
H^iSyoptional.
C hf{S)
by the definition of optionality. Since hf{S)
{T<t}eh^{S)
T
This, by the definition of optionality, implies that
Next we note that h^{S) contains
yt
e
B e
Let
T.
C
h^{S). Since H'^S)
VieT.
/f"(5)-optional.
is
B e
the sets
all
e T, we know
Vi
hf(S) such that
we know
H'^iS),
B E
h^{S) and
Bf]{T <
t}
e hf {S)
t}
e hf{S)
BfllT <
WteT. Thus Beh^iS).
I
C
Proposition 6.1. Suppose that F^
Proof.
From the
hypothesis,
it
is
BE
F". Let
claer that
/if
Then T^ < T|
(5).
P-a.s.
H^{S) C /f"(5). From Lemma
6.1
we know that
BEh^.{S). Thus
E{1b
\
^
From Lemma
6.2
E
[\b
/ij,(5))
I
/i°'^(5)) lies in [0,1]
iterative expectation
we
get
B
=e{e [ib
=
|
P-a.s.
1b
B
we know that h^{S) C h"g{S). Therefore by
^B
Since £'(1b
=
h^,{S))
/i;:,(5))
1
/ij,(5))
I
F- a.s.
1b
with P-probability one, we must have
P-a.s.,
E(\B\h''j.,J,S)\^lB
which implies that T^{S) < T^{S) P-a.s.
I
The
following proposition shows that the definition of resolution times
invariant under a
is
substitution of an equivalent probability measure.
Proposition 6.2. Let
Tg be
B E
h"{S),
resolution times for
B
let
Q
be a probability measure equivalent to P, and
with respect to
P
and Q,
respectively.
Then Tg
= Tg
let
T^ and
P-a.s.
and
therefore Q-a.s.
Proof. Putting ^
= dQ/dP
and
fixing a right continuous version ^{t)
absolutely continuous with respect to F, {^(0}
P-null set.
We
is
have
=
^
1b
14
\
h"[S)). Since
Q
is
a strictly positive process except possibly on a
E[\Bi\h''MS))
EQ{lB\h"T^)=
= E[^
P-
a.s.
This implies that
Tq < Tg
and therefore Q-a.s.
P-a.s.
Q
since
absolutely continuous with
is
respect to P.
By
the hypothesis that
Q
and
P
are in fact equivalent,
P
we know
= Tg
P-a.s.
absolutely continuous
Tg > Tg
with respect to Q. Thus we can reverse the above argument and have
Hence we must have Tg
is
F-a.s.
and Q-a.s.
and Q-a.s.
I
Before proceeding to the main theorems of this section, we
give
first
some strengthings
of the
results of Section 5.
We
assmue henceforth that the probability space (n,7,P)
shall
the space of strictly positive linear functionals on
a
is
a martingale measure for him that
Let
follows from an extension of
Theorem
^
denote
equivaient martingale measure for agent
.
equivalent to P.
V'a
I
^^a
=
whose proof
3 of kreps [1981],
S admits do
Proposition 6.3. Suppose that
Proof.
Let
denote the space of strictly positive linear functionals on V. The following proposition
'9
such that
is
V An
separable.
is
is
given in the Appendix.
i/jq
G ^
a separable normed space.
The
Then there
free lunches for agent a.
exists
^a-
Given that {U,7,P)
a separable probability space,
is
V
is
assertion then follows from the theorem in the Appendix.
I
A
direct consequence of the above proposition
and Theorem
5.1
is:
Proposition 6.4. There exists a one-to-one correspondence between equivalent martingale measures
for agent a,
Qa, and
linear functionals V^q
€ ^
with
V'o
|
Ma =
The correspondence
ttq.
is
given
by
Qa{B) =
yBeT
rPa{lB)
and4>a{v)
=
E;{v)
Vt;
€ F,
wiere Ea{) denotes the expectation under Qa-
When
the probability space
a strictly positive extension of
is
tt^
separable, no free lunches for agent
to all of
V
.
This
in
a
implies the existence of
turn implies the existence of an equivalent
martingale measure for agent a.
Since
all
the probability measures to appear will be equivalent probability measures, we shall
henceforth simply use a.s. to denote almost surely under any probability measure involved.
Here
Theorem
is
our
6.1.
first
main
result of this section.
Suppose that agent a
is
admits do free lunches for agent a. Let
better informed than agent
BG
/if
(5) be such tiat 1b
r^ = T^
P-a.s.
15
/3,
€
and tiat the price system
Afg.
Tien
Let {S'^(i)} ^^^ {^'bIOI denote the implicit price processes for Is for agents
Proof.
From Corollary
respectively.
Therefore, from
now on we
From Propositions
5.1
we know that {S%[i)) and {5^(i)}
5.1, 6.3,
and
for agents
Q^
B
El{lB\h^i{S))
V<€T
a.s.
Vi€T
a.s.,
and E*J) denote the expectations under equivalent martingale measures
a and
/?,
respectively.
under {H^{S),Qfi) are equal almost
times of
are indistinguishable processes.
and 6.4 we know that
=
£'„(•)
/9,
use {5b(<)} to denote both.
SB{t)^E*,[lB\h1[S))
where as usual
a and
Thus the
resolution times of
B
Qa
under [H'^{S),Qa) and
then follows from Proposition 6.2 that the resolution
surely. It
under {H°'{S),P) and under {H^{S),P) are equal almost surely.
I
Commonly marketed
If
the resolution time of
events will be revealed to differentially informed agents in the
B
under (F°,F)
strictly earlier
is
theorem implies that the price system that admits no
We shall say
A price system 5
Vf
than that under (F^,F), the above
free lunches for agent
B
information to equalize the assymetry of information about
a must
between agents a and
that markets are elementarily complete ioi agent
is
same time.
a
if
1b G
said to be dynamically fully revealing between agents
Ma
a and
carry enougii
jS.
B G /if (5).
if hf{S) = h^ (5)
for all
/?
G T.
Here
Theorem
our second main result of this section.
is
6.2.
Suppose that agent a
better informed than agent
is
markets are eiementariiy complete for agent
agent
a must
that T{'
=
and that J"
if implies that
pleteness hypothesis implies that any event in
h'^{S)
(5)
/if
a and
is
to
exists
t
Now we
G T
claim that hf{S)
such that hf [S)
know P{B) >
since
all
=
-
h^{S).
marketed.
h^ {S)
^
hf{S). Let
the filtrations are completed. Also,
E{lB\hf{S))==lB
and on a
set of strictly positive
It
I
hf{S))
16
^
1b.
B G
we have
a.s.
F-measure
E{1b
•
If the
The elementary market com-
hf [S) V< G T. Suppose this
C hf{S) and
^i
13.
then follows from Theorem
6.1 that the resolution times for all the events in h^{S) under {H°'{S),P)
are equal.
=
then a price system that admits no free lunches for
/?,
be dynamically fully revealing between
The hypothesis
Proof.
/3
is
and under {H^{S),P)
not the case. Then there
hf{S) and
B ^
hfiS).
We
By the
right-continuity of the filtrations,
we know
the resolution times for
B
under H'^{S) and
under H^{S) are not equal, a contradiction.
I
A
corollary
immediate:
is
Corollary 6.1. Suppose that agent
a
in
is
A, and that niarkets are elementarily complete for agent
system must be dynamically
This section,
in
=
the least informed agents in A, that 7^
T{ for
Then any arbitrage
(3.
all
agent
free price
fully revealing in the sense that
h^{S)
=
our view,
is
e T and
V(
hf{S)
agents in A.
all
the most important contribution of this paper.
recognized by probability theorists that the
way information
revealed over time
is
is
It
has been
closely related
to the behaviors of martingales and optional times. In an economic context, the former has been
extensively studied by
proved
in
Huang
[1985a]; while the latter
is
introduced in this section.
the Appendix and used to demonstrate Proposition 6.3
conditions in
Theorem
of independent interest.
is
7.
we have
The
3 of Kreps [1981] to insure that there exists a strictly positive continuous
extension of a linear functional are sometimes hard to verify in applications.
along as
The theorem
It
turns out that as
separability matters are simple.
An example
In this section
we
present an example along the lines of
securities traded, one risky
and one
standard Brownian motion as
it
riskless
informed and there
is
its
5.1.
There are two long-lived
with a zero interest rate. Agent
comes along. Agent ^ knows
motion at time one but cannot observe
Remark
a
gets to observe a
at time zero the value of the
sample paths over time.
Two
Brownian
agents are differentially
no ordering between the two endowed information structures. Markets are
elementarily complete for both agents and the price system
is
dynamically
fully revealing
between
them.
Formally, there
denoted by
W
.
The
We
W
=
is
defined on the basic probability space (n, /, P) a standard Brownian motion,
{W{t);t
e
set of trading dates of the
assume that F"
=
We
[0,1]}.
denote by F^^
economy
F^^ and that, for
T
all
is
t
if
1
at the very beginning agent
he could observe
W
directly,
There are two long-lived
di{u))
j3 is
{5;"';<
taken to be
e
T, 7f
Note that although consumption only occurs at time
at time
=
1,
=
G
[0,1]} the natural filtration of
[0, 1].
(t{W{1)}, the tribe generated by W{1).
knowing the value of the Brownian motion
not better informed.
The optimal
might well be path-dependent.
securities traded. Their payoff structures are
=
In(o^)
17
net trade for agent
/9,
where 7
The
a real number.
is
price system
Si{uj,t)
=
ln(w)
52(0;,
=
W{uj,
+
1)
is
W{u,t) -
(^7
J^
+
[_s
)
Corollary 1.1 of Jeulin and Yor [1979] implies that
w{t)
Since 7
- t)w{i) =
+ {i-t)(f j^^dS2{s) +
+
(1
is
a constant, this implies that {W{t)
iy(i)
+
-
(1
generate the same filtration, denoted henceforth by F^
J
^
YT's'^^)
^
[0' 1]
t)\V{l)]t
€
=
€ T}. We assume
{7t^:,t
P-
and {S2{t);t G
[0,1]}
^•*-
[0,1]}
that F'^ has
been completed. Corollary 1.1(a) of Jeulin and Yor implies that
£>0
Thus ¥^
H^{S)
right-continuous and
is
= ¥^
That
.
is,
is
finer
the price system
Next we claim that
5 admits
is
than either F" or F^.
dynamically
no free lunches.
It
fully revealing
It suffices
follows then that
between a and
H"{S)
=
/?.
to demonstrate the existence of a
martingale measure. Put
Jo
Theorem
1
of Jeulin
to F'^ under P.
and Yor shows that {W{t);t E T}
Now
=
s
a standard Brownian motion with respect
exp{7'M^(l)-i7'},
set function
Q{B) = [
Jb
Theorem
on
is
-
define
e
and a
1
(r2,
6.1
and lemma 6.5
Tf) equivalent to P.
of Liptser
^{co)P{doj)
and Shiryayev
B G 7f
[1977] ensures that
then follows from the Girsanov theorem
It
Q
is
(cf.
a probability measure
Liptser and Shiryayev
[1977, pp. 225]) that
W*{t)
is
=
W{t)--it
= W{1) - f (1+
imi^I^^ ds teT
a standard Brownian motion adapted to T^ under Q. Thus S2
S2{t)
=
W*{t)
+
a F'^-martingale under Q, since
W{l).
Finally, the fact that
1.1(e) of Jeulin
is
and Yor
markets are elementarily complete for both agents follows from Corollary
[1979].
18
Discussions, generalizations, and concluding remarks
8.
The assumption
that the price system has been normalized cannot be taken lightly.
Walrasian economy, a normalization of prices
economy such a procedure
of a price system
may be
will not
economically neutral. In a rational expectations
is
be economically neutral
Take 5 to be a vector
regularity conditions except that they
sum
to a process
normalize prices to sum to one, denoted by 5*, but
generated by S. Then
The assumption
all
The information
in general.
Our
altered by changing numeraires.
the following direction, however.
analysis can be generalized in
bounded away from
zero.
We
is
made
the general case without further assumption.
condition needed
Other than the criticism
arbitrage
is
a
can now
for the ease of exposition.
economy where the commodity space
[1985b]. Results on resolution times that need an equivalent martingale
No
the
the characterizations in this paper apply to 5*.
that consumption only occurs at date one
The
all
agents have access to the information
the space of bounded variation processes representing agents' accumulated net trades;
longer normable.
content
F-semimartingales satisfying
of
still let
All the results before Section 6 are readily extended to an
in
In a
is
in the first
weak requirement
ment has pervasive implications.
The reason
commodity space
that the
cf.
paragraph, the results presented
common
We
Huang
measure cannot be treated
that a price system be viable;
of a price system.
In a
is
cf.
is
Kreps
in this
is
no
[1981].
paper are robust.
have demonstrated that this require-
information economy,
it is
the connection between
martingales and an arbitrage free price system observed by Harrison and Kreps [1979] that makes
a dynamic equilibrium theory possible;
context, no arbitrage
must
is
the
satisfy. It necessitates
minimum
cf.
DufRe and Huang
[1985].
In a differential information
condition a rational expectations equilibrium price system
that a price system be a martingale under some probability. Readers
should convince themselves that putting this papar and Duffie and
Huang
[1985] together
hard to prove the exietence of a dynamic rational expectations equilibrium with a
price system.
is
Whether there
exists a partially revealing
an open question.
19
it is
not
fully revealing
dynamic rational expectations equilibrium
Appendix
Let A' be a separable normed space and
and X2 be elements
is
We
of A'.
M
a pair (Af,7r), a subspace
markets model
is
shall write xi
of A'
be a cone
let A'
>
X2
xi
if
-
X2
== 1,2,
'f^n '^
.
.
.}
x„ ^f k
Xn
E
that
ip
\
Proof.
M=
A.l. Suppose that
n,
where
From Lemma
^
is
5 of
(M, tt)
n. It follows that
F
weak* topology and
E^=i A„ =
1.
Now
let
on
K
0.
Then there
exists
a.
i)
E
the space of strictly positive continuous linear functionals on
—
A: T
{tpk'ik
=
{A„; n
=
1,2,
.
.
'^
such
X.
exists a collection of equicontinuous
G K} such
that i'kik)
>
a separable metric space in the relative weak* topology;
Let {V')t„;"
5.4.7 of Schaefer [1980].
A securities markets model
M. A free lunch in the securities
A" such that
Kreps [1981] we know that there
is
with the origin deleted. Let xi
U{0}-
a.dmits no free lunches.
positive continuous linear functionals on
M=
A'
tt
c A/ x
and lim infn7r(mn) <
Theorem
G
and a linear functional
a sequence {(m„,a:n);n
in A'
and
i/'Jb
|
cf.
Theorem
1,2,...} be a countable dense subset of F in the relative
.}
be a sequence of strictly positive real numbers such that
define
oo
n=l
It is
quickly verified that
V' is
a strictly positive continuous linear functional on A" with
20
xIj
\
M=
tt.
References
1
K.
Chung and R. Williams, An
Introduction to Stochastic Integration, Birkhauser Boston
Inc.,
1983.
2 C. Dellacherie
and
P.
Meyer, Probabilities and Potential A: General Theory of Process, North-
Holland Publishing Company,
3 C. Dellacherie
and
New
York, 1978.
Meyer, Probabilities and Potential B: Theory of Martingales, North-
P.
Holland Publishing Company,
New
York, 1982.
4 D. Duffie and C. Huang, Implementing Arrow-Debreu equilibria by continuous trading of few
long-linved securities, forthcoming in Econometrica, 1985.
5 D. Duffie, Stochastic equilibria: existence, spanning
number, and the 'no expected gains from
Graduate School
trade' hypothesis. Research Paper No. 762,
of Business, Stanford University,
1984.
6
M. Harrison and D. Kreps, Martingales and
arbitrage in multiperiod securities markets, J.
Econ. Theor. 20 (1979), 381-408.
7 C.
Huang, Information structure and equilibrium asset
prices, J.
Econ. Theor. forthcoming
1985.
8 C.
9
J.
Huang, Information structure and viable price systems,
Jacod, Calcul Stochastique
Springer- Verlag,
10 T. Jeulin and
abilites
New
11 D. Kreps,
A
in
note on
mimeo, 1985.
Problemes de Martingales, Lecture Notes
in
Mathematics 714,
York, 1979.
M. Yor,
XHI, L.N.
et
MIT
Inegalites de Hardy, semi-martingales et faux amis, Seminaire de Prob-
Math. 721, 332-359, 1979.
fulfilled
expectations equilibrium, Journal of Economic Theory, 1977.
12 D. Kreps, Arbitrage and equilibrium in economies with infinitely
many commodities,
J.
Math.
Econ. 8 (1981), 15-35.
13 R. Liptser and A. Shiryayev, Statistics of
Verlag,
New
Random
Processes
I:
General Theory, Springer-
York, 1977.
14 H. Schaefer, Topological Vector Spaces, Springer- Verlag,
21
New
York, 1980.
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