Oewey
HD28
.M414
no. ls'^^-
iH'2h
;035
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
/information structures
and viable price systems
/
by
Chi-fu Huang*
January 1985; Revised - September 1985
Forthcoming in Journal of Mathematical Economics
WP //1574-84
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
/
INFORMATION STRUCTURES
AND VIABLE PRICE SYSTEMS
by
Chi-fu Huang*
January 1985; Revised - September 1985
Forthcoming in Journal of Mathematical Economics
WP #1574-84
would like to thank David Kreps for helpful conversations and Terry Marsh
Also,
for discussion on the evidence of the efficient market hypothesis.
suggestions by two anonymous referees were helpful.
* I
MJ.T. IJ3RARaE:$
SEP 2 4
1985
Abstract
A
dynamic model
time that
is
of capital/financial markets
not foretellable.
We
show that
if
is
developed.
A
surprise
is
a stopping
agents' preferences exhibit a kind of time
complementarity, then between ex-dividend dates a viabie price system can make discrete
changes only at surprises. Under the same preference condition, when the information
the economy can be modeled by a Brownian motion, a viable price system
between ex-dividend dates.
The martingale
originated by Harrison and Kreps (1979)
result
is
is
is
in
an Ito process
characterization of a viable price system
extended to our economy. This martingale
independent of the time complementarity of preferences alluded to above.
Introduction
1.
Much
work
of the empirical
economics and accounting concerns the response of
in financial
The
capital/financial asset prices to information.
null hypothesis in this
work
is
capital/financial markets are efficient in the sense that prices rapidly adjust to
The evidence
surprises.
Fama
(e.g.
typically that the
new information
to date has been largely interpreted as confirmation of this null hypothesis
(1970) and Patell and Wolfson (1984)T^
The
construction
of,
and hence
this interpreta-
tion of, the market efficiency tests has often been justified on intuitive grounds; for example,
stock price of a takeover candidate
of the takeover, stockholders are
them by
or
surprise (see Jensen and
is
found to make big changes
if
the
response to the announcement
in
assumed to react to the information and the announcement takes
Ruback (1983)
for a survey of the empirical
works on the market
for corporate control).
To
market
justify the intuitive interpretation of
capital/financial markets. Is
To answer
or surprises!
it
true that prices can
this question,
considers a continuous time
only at two dates,
and
1.
Huang
is
revealed.
A
first
generalized to
surprise
is
accommodate consumption
is
to
the
is
[0,1],
first
paper
in
that direction.
where consumption
is
available
i.e.
determined by the way
not predictable.
is
A
price system can
surprises.
what extent can the
over time. Intuitively,
if
for long-lived assets
results of
Huang
(1985) be
agents' consumption preferences
can have jumps even at times of
economic grounds, we expect agents' consumption preferences to exhibit some sort
of time complementarity: a unit of consumption
now should be almost
it
(1985)
a stopping time that
question this paper addresses
On
new information
or surprises will have to be mathematically
for long-lived assets are solely
change drastically over time, then the prices
no surprise.
of
Using a long-lived asset as the numeraire, he shows that the sample
have jumps only at non-predictable stopping times,
The
we need a dynamic model
discrete changes only at
economy with a time span
path properties of a viable price system
information
make
new information
defined and their linkage to prices formalized.
He
efficiency test,
perfect substitutes.
With
now and a
this kind of
unit of consumption an instant from
time complementarity of consumption,
can be shown that between lump-sum (random) ex-dividend dates prices for long-lived assets can
make
discrete changes only at non-predictable stopping times.
dates, prices adjust
downward by the amount
At lump-sum (random) ex-dividend
of the dividends only
when
there are no surprises at
that time.
For more than a decade financial theorists have been led to believe that Ito process representation of price processes
is
a good model for capital/financial markets not only because the tools
involved are well-developed, but because the sample paths of an Ito process resemble the price
'Several anomalous and apparently persistent departures from the null must be explained in terms of test design to sustain
June/September, 1978, issue of the Journal of Financial Economics).
this conclusion (see, for discussion, the
common
chart of a typical
time, a typical
common
stock between (lump-sum) ex-dividend dates.
stock price moves up and
down
fast
and
in
That
is,
most
for
of the
small steps.
Given that we have formulated a dynamic asset trading model, the second question addressed
in this
paper
is
under what conditions
will a viable price
system be an
dividend dates. Sufficient conditions are that the information
Brownian motion and that agents' preferences exhibit the
in
the
between ex-
Ito process
economy can be modeled by a
sort of time complementarity alluded to
above.
Note that Huang (1985) has demonstrated the relationship between a Brownian motion
formation and
Ito process representation of a viable price system.
long-lived asset.
The
result reported here
in units of the
is
His result
is
numerated
in-
in
a
natural numeraire, the single consump-
tion commodity.
We also generalize the martingale result originated by Harrison and Kreps
our economy. A price system for long-lived assets is viable if and only if it
of
dividend process, both
measure which
ity
(1979) in the context
and
its
accumulated
a long-lived asset, follow a martingale under some probabil-
in units of
uniformly absolutely continuous with respect to agents' commonly endowed
is
probability. This generalization
is
valid in a very general
economy without the
sort of time
com-
plementarity needed for previously mentioned results.
The
paper
rest of this
are in Sections 3 through
2.
is
organized as follows. Section 2 formulates the economy. Main results
Section 6 contains discussions and concluding remarks.
5.
The formulation
In this section
Taken
[0,1].
we model a dynamic pure exchange economy under uncertainty with a time span
as primitive
is
a filtered probability space (n,7,F,P), where each
u E
fl
denotes a
complete description of the exogenous uncertain environment, where Jis the tribe of distinguishable
events, where the SltratioD,'^ F
=
{ J,;
i
6
[0, l]}, is
agents, and where
P
We
satisfies the usual conditions:
assume that F
1
We will
now
that
any
t
=
[t
There
G
also
is
€
common
the
right-continuous: V<
2 complete: V<
at
is
G
[0, 1), 7t
[0, 1], Ti
=
contains
As>f
all
G
J^, s
[0, 1];
the P-null sets.
Mathematically, these entail that
only one perishable consumption commodity
We
otherwise mentioned.)
is
structure of
impose the conditions that agents learn the true state of the nature at time
0) is certain.
[0, 1].
fihration
commonly endowed information
probability measure on (H, 7) held by agents in the economy.
in
7\
= 7
and that
Jo
is
almost
1
and
trivial.
the economy which can be consumed
take the commodity space to be the space of integrable variation processes
V
that have right continuous paths, denoted by
*A
the
By
definition, for each
an increasing family of siibtribes
of
7.
i'
.
G
(All the processes will
V
we have:
be adapted to F unless
<
dv[t)
(/
I
where E{) denotes the expectation with respect
to P.
denotes the accumulated consumption from time
oo,
(2.1)
The intended
to time
interpretation
of nonnegative integrable increasing processes that have right continuous paths.
is
a generating cone of
It is
V
and we denote the order defined with respect to
assumed that the spot market
Remark
Thus
V
is
2.1. Since
Remark
is
right continuous
(for right
continuous with
each v
RCLL
a space of
€V
consumption good
for the
2.2. It is easily
checked that
eV
v
ii
Before proceeding, some definitions are
is
it
open
that
V+
by >.
all
the time.
it
has
left limits.
processes.
left limits)
<
e
oo Vf
[0, l].
Let A' be the space of
in order.
the space
It is clear
and of bounded variation,
then E{v{t))
that v{u!,t)
V+ denote
Let
in state u.
t
is
RCLL
processes such
that
ess sup^^_,
I
where the
essential
supremum
is
x{u!,t) \< oo,
taken with respect to the probability measure P.
nonnegative elements of A' by A'+, the positive cone of
A'.
The order on
denoted by >. Let A'++ be a subset of A'+ satisfying the following:
such that P{x{t) > c}
c ?£
Now
=
1
V^
consider the bilinear form
G
tp
V'(u,
Proposition A.l
in
Appendix
be the strongest topology on
Vj-
We
denote the
A' generated by
A^
A'++ there exists
€.
c
is
also
6
9f+,
[0, 1].
:
V
a:)
x
X
>-^
^.
=E
I
shows that
ip
V
such that
its
separates points, and places
topological dual
is A',
V
and A'
Let r
in duality.
a Mackey topology;
cf.
Schaefer
(1980, p.l31).
Returning to economics, we shall assume that the economy
set A, a generic element of which
function f/^
:
V
t-^-
9i
that
the sense that UQ{v+y)
Ua
7a e
at V, which
is
-X'++ such that
denoted by a. Each agent a G
zero. Formally, let
y
^
0.
We
also
A
is
characterized by a utility
of
^x E dUa{v).
is in
utilities
set of r-continuous supergradients
Holmes (1975).
V
^ > 7a
assume that Ua has marginal
dUa{v) denote the
nonempty by Theorem 14B
Vv G
populated with agents from a
concave, r-continuous, and strictly increasing, the last of which
> Ua{v) Vy G V+ and
bounded below away from
of
is
is
is
We
assume that there
exists
Remark
An example
2.3.
tion having marginal utilities
of a r-continuous, strictly increasing,
bounded below away from zero
assumed that there are a finite number
=
are indexed by j
Each long-lived security
0,1, 2,..., J.
holder of a share of security j, from time
shall
assume that Do{t)
assume that Dj{0)
=
G
V<
[0, 1)
=
Vj
=
1,2,
.
.
,
.
bounded above and below away from
to time
has received during that period
/,
and that Dq{\)
semimartingales^ denoted by
=
S =
curities 5,
.
.
.
,
zero.
e V+,
in state
ui.
bounded above and below away from
Thus without
loss of generality,
we
also
A
consumption plan
is
=
0,1,..., J}.
is
an
RCLL
process
an element of V.
is
a J
-|-
1-vector of integrable
Since these securities are traded ex-
Vj.
for the single
an admissible trading strategy
J}
=
J.
{Sj{t)\j
Given admissible price systems
0, 1,
is
admissible price system for traded long-lived securities
dividends, Sj{l)
^, U{v)
the consumption commodity, that a
admissible price process for the single consumption commodity {I3{t))
An
i-^
represented by an element Dj
is
in units of
zero. All the long-lived securities are traded ex-dividends.
An
V
:
of long-lived securities traded in the economy, which
where Dj{u.\t) denotes the accumulated dividends,
We
U
given by
utility func-
where x G A'++.
-^(/(o.iJ^COc'i'lO).
It is
is
and concave
is
consumption commodity
/9
and
for long-lived se-
an J-f l-vector of predictable processes'* 9
=
{9j{t)\j
=
satisfying the following conditions:
1
ess supj^^,t
I
where as before the
2 there exists v
e{tfs{t)
gV,
essential
v{0)
supremum
=0 such
is
6j{u!,t) \< oo,
(2.2)
taken with respect to P;
that
=
(2.3)
e(0)'^S{0)+ f e{sfdS{s)+
Jo
[
0{s)e{s)^dD{s)- f
Jo
/3{s)dv{s)
V/G[0,1]
a.s.
Jo
and
e(l)^AZ)(l)
=
Ai;(l)
a.s.,
(2.4)
where ^ denotes inner product, and where AD{() and Av{t) denote the jumps
f,
D
and
v at
respectively.
Equations
tion.
of
(2.3)
The budget
and
(2.4) are the natural
budget constraints, while
constraints can best be understood
when
6
is left
(2.2) is
a technical
restric-
continuous. In that case, we
'For the definition of a semimartingale see Jacod (1979, p. 29). It is also shown there that for stoctiastic integ^rals to be
well-defined and have nice properties it is necessary that the integrators be semimartingales; Jacod (1979, pp. 27S-279).
*For the definition of a predictable process
see
Jacod (1979,
4
p. 4).
interpret 6{t) to be the portfolio held from t— to
is
the value at time
and before time
t
<
t
before time
of a portfolio carried from t— into
trading and consumption. That
it is
t,
trading takes place.
t
Thus
6{t)'^ S{t)
excluding dividends received at time
equal to the right-hand-side of (2.3)
is
t
clear.
Let S[^,S] denote the space of admissible trading strategies with respect to the pair of admissible price systems {/3,5).
easily checked that 0[/?,5]
It is
a linear space by the linearity of
is
stochastic integrals.
In relations (2.3)
that
the strategy 6
if
and
is
(2.4),
the process v
e
A consumption plan
A price of v at time
e[/?,5].
consumption plans. By the
Remark
2.4. Since 6
t;
V
€
is
G
is
+
A/
is
—
v
M
v{0)
aE
if it
A
&V
{veV
:
such that Vq
/?(0)u(0)
Note that the definition
+
is
of viability
is
=
in (2.3) are
VIII) for details.
the numeraire
f/a-maximal
6»(0)'^5(0)
financed by some
a linear space.
can be supported by some agent from the set A. Formally,
and Va
is
be the space of marketed
Following Kreps (1981) we shall say that a pair of price systems {l3,S)
to A)
meaning obviously
and bounded, the stochastic integrals
Meyer (1982, Chapter
is
if
Let
6'(0)"^5(0).
a linear space,
6[/?, 5] is predictable
when the consumption good
marketed
said to be
l3{0)v{0)
is
fact that Q[/3,S]
well defined. See Dellacherie and
3. Viability
said to be financed by 6,
followed, a consumption pattern represented by v can be dynamically
manufactured.
e
is
in
{/S,
S)
is
is
viable (with respect
viable
if
there exists
the set
and u
is
financed by 9 for
taken as primitive
in this
some
paper.
The
d
G 6(^,5]}.
readers are referred to
Kreps' paper for a host of related issues.
In this section
we
shall give characterizations of a viable pair of price systems,
consumption commodity
process
is
is
where the
single
taken to be the numeraire. In this case, the consumption good price
unit throughout, and
we
shall refer to a viabie price
system as a viable price system
for
the long-lived securities.
A
necessary condition for a price system to be viable
for all the
marketed consumption plans.
is
that the single price law must hold
In such event, there exists a linear functional
that gives prices at time zero of marketed consumption plans; that
6(0)^5(0), where
A
and V
(m - m{0),6)
are associated.
r-continuous linear functional
^
0.
t/'
:
T
We
call
tt
i—* 9f is said to
is,
Vm G M,
7r(m)
the current price system for
be strictly positive
if
tt
tp{v)
:
=
M
i—> 3?
m(0)
+
M.
>
Vv E V+
.
Proposition 3.1. Suppose that
S
is
is strictly positive, r- continuous,
viable. Tlien
n has an extension to
ofV denoted by
t/),
which
and can be represented as
ii{v)^E\[
VuGV,
x{t)dv[t)\
V-Mo,il
with X
all
(3.1)
)
€ A'++
Proof. This assertion
is
a direct consequence of the definition of
viability.
I
The
linear functional of (3.1) not only gives prices at time zero for
plans, but also provides a
We
way
to represent a viable price system over time.
assume throughout
shall
Proposition 3.2. For every j
=
marketed consumption
S
this section that
0, 1,
.
.
.
,
viable.
is
J, we have
E[!lx{s)dD,[s)\7t)
Sjit)
=
xit)
(3.2)
E
xis)dDAs)
E(f:i
(/J x{s)dDj{s)
I
I
7^-j'^x[s)dD,{s)
7t] - fn xis)dD,is)
V<g[0,
1],
a.s..
x{t)
where
RCLL
versions of the conditional expectations are taken (this
to be right continuous;
Proof. See
Appendix
cf.
Meyer
is
possible because F
is
assumed
(1966, pp. 95-96)).
II.
I
If
we
just the
interpret the process {x{t)} to be the marginal utility process, then Proposition 3.2
famous
relation: the price of a security at
that time of the
sum
relation underlies
time
t
of products of its future dividends
much
of the
modern
is
is
equal to the conditional expectation at
and the
ratios of marginal utilities. This
asset pricing theories; see, for example, Lucas (1978) and
Rubinstein (1976).
Remark
3.1.
From now on we
shall always take an
RCLL
version of any conditional expectation
that appears.
Several corollaries are immediate:
Corollary 3.1. The process {So{t)
+
.Do(0}
^^
bounded above and below away from
6
zero.
Proof.
From Proposition
From Proposition
3.1
we know that
3.2
5o(0
=
we know that x
is
V<
G
[0, 1)
E{x{l)Do{l)\7t)
-^^
a.s.
(3.3)
bounded above and below away from
then follows from the assumption that Do{\)
is
zero.
The
bounded above and below away from
assertion
zero.
I
Corollary 3.2. The process {x{t)}
Proof.
From
(3.3)
we can
is
a semimartingale.
write
g(x(l)Do(l)|7,)
'^'^=
+
Soil)
The numerator
which
is
above expression
of the
bounded away from
martingale;
zero.
Dellacherie and
cf.
A
_
""
,,
^,^,„
"''^'"''^
Doit)
a martingale and the denominator
is
is
a semimartingale,
generalization of Ito's formula shows that {x{t)}
is
a semi-
Meyer (1982, Mil. 27).
I
Let
m
e
M and
let
{5^(0} denote
its
vector of semimartingales, a natural question
Corollary 3.3. Let
Moreover,
let
m—
m
is
E M. Then {Sm{t)} has
m(0) be financed by
S,„{t)=^e{tyS{t)
whether {5„i(i)}
is
E
The numerator
[t]
of the
a representation as (3.2)
+ e{tyAD{t)-A{m{t)-7n{0))
[Ii^{s)dm{s)
above expression
is
V<
Tij
^
I
-
€
first
and
is
[0, 1]
we can
a semimartingale.
a.s.
easily get
J^ x{s)dm{s)
Vf
a semimartingale, since
G
it is
[0, 1]
the
a bounded variation process. Corollary 3.2 shows that the denominator
The
a
Then
6.
= -^
is
a semimartingale.
Proof. Applying the arguments found in the proof of Proposition 3.2,
Srn
Given that S
implicit ex-dividend price process.
a.s.
sum
is
of a martingale
and
a semimartingale too.
assertion then follows from an application of the generalized Ito's formula.
The second
assertion follows from the fact that
S
is
viable.
I
When
a price system
is
viable,
we have
stochastic integral representations for
it.
Moreover,
the implicit ex-dividend price process for any marketed consumption plan and the shadow price
process {x{t)} are semimartingales.
4.
Changing numeraire and martingales
We
show
shall
in this section
that
if
a numeraire
appropriately chosen, then the viability of
is
a price system for long-lived securities necessitates that 5, the price system, plus
dividend process, both
(n, 7), which
some
is
if
the numeraire,
is
accumulated
a martingale under a probability measure on
The converse
uniformly absolutely continuous with respect to P.
is
also true under
regularity condition.
Henceforth
good
in units of
its
is
let (/?,5)
denote a pair of admissible price systems when the single consumption
is,
0{t)
=
{f3*
,S*)
is
the numeraire, that
the pair of price systems
1
Vf
G
viable,
[0, 1].
We
will first
where V< £
show that S
is
viable
if
and only
[0, 1]
and
S;it)
=
Vj
p*{t)Sj{l)
=
0,l,...,J
a.s.
(4.1)
This could be anticipated. In a Walrasian economy, changing numeraires ought to be economically neutral, since only relative prices are determined.
in
economies with
infinitely
many commodities. We
This statement
will
is
not, however, always true
need to employ some special structures of
our model.
We
first
show that
(/9*,5*)
Proposition 4.1. (/3*,5*)
is
is
a pair of admissible price systems
a pair of admissible price systems
if
if
and only
and only
if (/?, 5")
if {/3,S)
is
is.
a pair of
admissible price systems.
Proof.
Suppose
first
that (/3,5)
is
a pair of admissible price systems.
The
fact that
above and below away from zero and therefore admissible follows from Corollary
of the generalized Ito's formula
that S*
is
fact that
shows that 0*
is
a semimartingale.
a semimartingale. Finally, the fact that S*
/3*
is
3.1.
/?*
An
The same formula
is
bounded
application
also implies
integrable follows from the above-proven
bounded above.
is
The proof
for the converse
is
identical, so
we omit
it.
I
Let 0[/?*,5*] be the space of admissible trading strategies given that the pair of admissible
price systems
The
is
(/9*,5*).
following proposition shows that the two spaces &[/3* ,S*] and ©[/?, S] are equal. Its proof,
being tedious and involving a
lot of
machinery
in stochastic integration, is
delegated to Appendix
III.
Proposition 4.2.
{13,
S)
if
and only
e &[0,S]
if it is
if
and only
if 6
£
6[/3*,5*]. Moreover, v
financed by 9 given (/?*,5*).
-
v{0) is financed by 9 given
,
.
M*
Let
denote the space of marketed consumption plans given (^*,5*).
A
corollary
is
imme-
diate.
mE
Corollary 4.1.
Proof. This
M
if
and only
if
me
i\/*.
=
a direct consequence of the fact that 0[/?, 5]
is
0[/?*,5*].
I
We
The
(/?,
S)
M to denote both M and M*
shall henceforth use
following proposition shows that (/?*,5*)
a pair of viable price systems
is
if
and only
if
one.
is
Proposition 4.3.
{l3*
The
price systems.
,S*)
a pair of viable price systenis
is
if
and only
if (/?,5)
is
a pair of viable
two pairs of viable price systems, n and n*
linear functionals associated with
that give the prices at time zero of marketed consumption plans are linked by
Proof.
is
Suppose
?7Q-maximal
{vEV
By Proposition
that
first
(/?,
in
the set
:
/3{0)v{0)
4.2
+
5)
is
viable.
e{0)'^S{0)
and the fact that
There must
=
(3{t)
exist
and v - v{0)
=
1
e
Vi
is
[0,1],
some a E
financed by
A
and Va E
some
E
we know the above
V
such that Va
e[/3,5]}.
set
is
equal to the
following set:
{vEV
Therefore, Vq
is
:
(3*{0)v{0)
is still
+
e(0)^5*(0)
i/Q-maximal
in
=
and
the above
i-
-
set.
v(0)
is
By the
financed by
some
6
E e[l3\S*]}.
definition of viability, the pair (/?*,5*)
viable.
The proof
for the converse
The proof
for the rest of the assertion
It will
this
main
be demonstrated
result,
in
is
identical.
is
easy.
the sequel that 5*
we need a proposition and
is
closely related to martingales. Before proving
its corollary.
Putting
v*{t)^ I
^'{s)dv{s)
WtE{0,llvEV,
J[o,t]
and denoting the
implicit ex-dividend price process of
m€
Af given
(/?*,5*) by
5^, we have:
—
Proposition 4.4. Suppose that the pair {0,S)
m
Let
is viable.
G M. There
exists x*
€ A'++ such
that
^
E
= —
E(j.^x*{s)dm*{s)
5;(')
I
=
7t]
^^
,
^
(/J x*{s)dm*{s)
Proof. Since
tt*
=
/3*(0)n,
(4-2)
X
/
-
^
I
7,)
/o'
V<e[0,l],
has an extension to
it*
x*{s)dm*{s)
all
of
V
denoted by
t/'*-
It is
a.s.
clear that
xp*
can
be chosen to be /?*(O)0:
^*{v)
where we
Now
recall that
x
€
A'^.+
=
f3*{0)E
is
(4.3)
putting
strictly positive,
We
VvgT,
x{t)dv{t)]
[
.
=
x*(t)
a;*
I
^x{t)
V<G[0,1],
bounded above and below away from
(4.4)
zero.
can rewrite (4.3) as
V'*(i')
=E
i
x*{t)dv*
[
Applying the same arguments to prove Proposition
3.2,
it is
straightforward to verify (4.2).
I
The
interpretation of (4.2)
the consumption commodity.
is
The
identical to that of (3.2), except that the numeraire here
following corollary
is
instrumental to the
this section.
Corollary 4.2. The process x* defined
Proof.
We
in (4.4) is
a martingale under P.
note that almost surely
D*o{t)
=
V«g[0,
1)
=
1
iit=
=
i
/e[o,i)
I
and
s^{t)
=
if f
=
1
by construction. From Proposition 4.4 we have
^(^*(1)I^<)_5.^(^)^1
x*{t)
10
V«€[0,1)
a.s.
first
is
not
main theorem
of
That
is,
E{x*{l)\7,)
Naturally E{x*{l)
\
Ti)
—
=
x*{l) a.s. since x*
x*{i)
is
V<€[0,
a.s.
1)
adapted. These simply say that x*
is
a martingale.
I
When
appropriately numerated, the shadow price process
is
not only a semimartingale,
it is
a
P
if
martingale!
Some
A
definitions are in order.
probability
Q
on (O, 7)
dQ/dP
the Radon-Nikodym derivative
P ?« Q. A martingale measure
is
said to be uniformly absolutely continuous with respect to
is
is
bounded above and below away from
a probability measure
is
Q
Q^P
on (H, T) with
zero, denoted by
such that S*
+ D*
a martingale under Q.
Here
Theorem
is
our
4.1.
first
main theorem of
Suppose that
S)
(/?,
+ m*
measure Q. Moreover, S^
is
is
this section:
Then there
a pair of viable price systems.
a martingale for
all
m
Proof. It follows from Corollary 4.2 and Dellacherie and
exists a martingaie
M under Q.
G
Meyer (1982,
VI. 57) that
_E[l'x*{s)dm*is)\7t)
E[jlx'(l)dm'{s)\7)
_
~
^)
E(x'{l){m*{l)-m'{t))\7t)
_
V<
€
[0, 1]
P
-
a.s.
(4.5)
x*{t)
Next note that E{x*{l))
=
5o(0)
=
1.
Thus
Q{D)= !
if
we put
x*[^,l)P{du)
'iBe7,
JB
Q
is
(4.5)
a probability measure on
(fi,
J) uniformly absolutely continuous with respect to P. Therefore
can be written as
S*^{t)
= E*{m*[\)-m*[t)\7i)
where E*{) denotes the expectation under Q\
and
Q
have the same null
sets,
cf.
Vie
[0,1]
P-
a.s.,
Gihman and Skorohod
the above expression holds with
Q
(1979, p. 149). Since
P
probability one.
I
When
the pair of price systems (y9,5)
is
viable, after an appropriate
change of numeraire, the
implicit price process for a marketed consumption plan plus its accumulated dividends will be a
martingale under some probability uniformly absolutely continuous with respect to P.
11
Remark
4.1. Since
almost surely
P
Q
and
have the same null
for both, unless otherwise
sets,
we
a.s. to
denote
mentioned.
Their proofs being straightforward, the following corollaries
Corollary 4.3. S*
from now on use
shall
will
simply be stated.
a vector of supermartingales under Q.
is
Corollary 4.4. Let mi,
m^
-
he such that mj(l)
mj(^)
=
m^il) -
rn2{t) a.s.
for
some
t
G
[0, 1].
Then
•^niilO
At any time
t
e
—
[0, 1], if
two marketed consumption
Conversely,
The
a.s.
it
^®- ^"^ therefore S^iit)
let (/9*,5*)
must have the same
is first
Sm^ii)
^-s.
+ Do
some
are equal for
price at that time.
be a pair of admissible price systems with SqII)
following theorem shows that, under
proposition
=
the future accumulated dividends in units of So
plans, they
suffices that there exists a
A
•^m^(0
+
D^lt)
=
1
V/
€
[0, 1]
regularity condition, for (/?*,5*) to be viable,
martingale measure.
needed. Let
M denote the space
of
marketed consumption plans given
Proposition 4.5. Suppose that there exists a martingale measure
E{snp |5;(0l)<oo
Q
and that
vy.
<elo,i]
Tien the
S*^{t)
single price law holds for all
G
i\/*
and denoting the ex-dividend price of
m
at
t
by
we have
S*,{t)
Proof. Let
m
m
E
= E*(^l'0*{s)dm{s)\T^.
©[/3*,S*] be a trading strategy that finances m.
(4.6)
Then an
implicit price process for
is
5m(0 =
Equations
(2.3)
e{t)^S*{t)
and
(2.4)
+
(3'{t)e{tyAD{t)- l3*{t)A{m{t)-m{0))
^t
e
[0,1]
then imply that
S;^{t)= f /9*(s)dm*(s) - f e{syd{S*{s)
12
+
D*{t))
V^€[0,l],
a.s.
a.s.
Given the hypotheses that
(/3*,5*)
viable and that E'(sup,gro^i]
is
SUt)
|
|)
<
oo,
it is
easily verified
that
E*l
where E*{)
snp \S;{t)
the expectation under Q.
is
The
+
<oo
D;{t)\]
Vj,
Davis, Burkholder, and
Gundy
inequality
(cf.
Jacod
(1979, 2.34)) then implies that
E*{[S*
where {[S* + D* S* + D*]t}
,
is
+ D\S* +
<oo,
D*]i)-^
(4.7)
the quadratic variation process of {S*{t)
+ D*{t)}.
(For the definition
of the quadratic variation process see Jacod (1979, 2.31).) Equation (4.7) as well as Jacod (1979,
2.48) ensures that
E* (^' e{s)^d{S*{s)
This
+
D*{s))
I
7t)
=
0.
in turn implies
S:,{t)
which
independent of
is
9.
Thus the
= E*^'f3'{s)dm{t)\7ty
single price law holds
and
(4.6) is valid.
I
Here
Theorem
is
our second main result of this section:
4.2.
Suppose that the conditions
Proposition 4.5 are vaiid. Then (/?*,5*)
in
is viable.
Proof. Putting
.^(0-^(^1//)
Vf€[0,l]
U.S.,
we have
E*
r
I
I
/
[
=eI [
1=^1/
m0*it)dm(t)
ai)(3*{t)dm{t)
0*{t)dm{t)]
I3*{t)dm(t)
VmG
I
M*,
where we have again used Dellacherie and Meyer (1982, VI. 57). From Proposition
4.5
we know
that the two sets
{veV
:^(0)^5*(0)
+ /3*(0)u(0) =0,
where v
is
financed by
some eee[/3*,S*]}
and
{m&M*:E'l[
are identical.
Now
^{t)^*{t)dm{t)]
=
0}
define
Ua{v)
= El
f
at)/3'{t)dv(t
(01
13
^veV.
It is
quickly checked that
a G A,
since
^
f3*
is
bounded above and below away from
zero.
It
then
follows from the separating hyperplane theorem that the pair of price systems (/3*,5*) can be
supported by Ua and thus
is
viable.
I
Whenever the
regularity condition posited in Proposition 4.5
martingale measure
is
The combination
equivalent to the viability of (/9*,5*) and therefore the viability of
of
Theorems 4.1 and
4.2
is
is
allowed at any time
(/?,
a
5).
a generalization of the martingale result of Harrison
and Kreps (1979). They consider an economy where consumption
Here, consumption
satisfied, the existence of
is
in [0,
1]
available only at two dates.
is
and agents can consume at rates as well as
in
gulps.
Once we have the martingale characterization
to see that the sample path properties of 5*„ depend both
over time and
upon the sample path properties
generalized in a straightforward
manner
to
S^,m e
of viable price processes
of
m*.
show the
A/,
upon the way information
Huang
All the results of
interrelationship
among
is
it is
easy
revealed
(1985) can be
5*,, ni*
,
and F.
We
leave this exercise to interested readers.
The sample path
In the
properties of
5^
are not the
most interesting objects
economy considered by Huang (1985), consumption
and time
Since the consumption good
1.
is
not available
take a long-lived security as the numeraire. In the economy
consumption good
is
available all the time.
The
our economy, however.
available only at two dates, time
is
all
in
the time,
we
it
is
thus not unnatural to
are currently considering, the single
natural objects of interest ought to be the sample
path properties of S^- In particular, given the popularity of Ito process formulations of financial
markets,
we can
it
is
important to see under what conditions
see that the
depend not only on
Sm
is
an Ito process.
sample path properties of the price process
its
of a
in
accumulated dividend process and the information structure as captured
the sample path properties of
a:
exhibited by preferences from the the set A.
nothing can be said about x except that
therefore about
now
(3.2)
marketed consumption plan
by the conditional expectation, but also on the sample path properties of
embedded
Examining
Sm,rn e A/, we have
it is
is
x.
This
is
natural, since
a sense of time complementarity of consumption
Given that
viability
is
defined with respect to A,
bounded. To say something interesting about x and
to consider a subset of A.
turn.
14
This
is
the subject to which
we
5.
An
alternative formulation
Let
H
^n-^++-
denote a subspace of A' such that each h E
Define a bilinear form
(/>
:
V
x /f
Now
identify equivalence classes of elements of
and only
V
is
a continuous process.^
Let
H+^ =
by
h^^ 9f
= E\ f
4>{v,h)
H
h{t)dv{t)]
such that
.
belong to an equivalence class
ui,t;2
if
if
<f>[vi-V2,h)
Denote the space of equivalence
the strongest topology on
"l^
=
ElJ
classes oi
such that
its
V
h{t)d{vi{t)
M
by
.
Then
(p
topological dual
-
V2{t))\
places
is
"V
=0.
and
H
in duality.
H, a Mackey topology;
Let r* be
cf.
Schaefer
(1980, p. 131).
Let A
A
C
he such that each Ua,<y € A
agents from the set
consumption now
is
so because the
Remark
5.1.
is
A
is
Note that the preferences
r*-continuous.
of
exhibit the following sort of time complementarity of consumption: a unit of
almost a perfect substitute for a unit consumption an instant from now. This
siadow
price process
continuous.
is
Note that time-additive
utility functions are
at adjacent dates are perfect nonsubstitutes;
Suppose from now on that
S
is
cf.
not r*-continuous, since consumption
Huang and Kreps
(1983).
viable with respect to A. Still denote by
M the space
of marketed
consumption plans. The following proposition can be viewed as a corollary of Proposition
Proposition 5.1. Let
m e M.
E
S,nit)
Then
(/o
= -^
{5„,(<)}
h{s)dm{s)
is
J,
I
3.2.
a semimartiDgale and can be represented as
j
-
-^
/o h{s)dm[s)
V<
€
[0, 1]
a.s.,
where h E H.++
Now we
of a
among
the accumulated dividend process
marketed consumption plan, the sample path properties of
information
A
are ready to examine the interrelationship
process h
is
is
revealed.
continuous
Some
if
for
its
price process,
definitions are in order.
almost every
ui
S Q,
h{ij.\
)
15
is
a continuous function of time.
and the way
A stopping
time
a
is
map
used the convention that
5)
T"
n
:
=/
V<
3Ju {00} such that {T <
-*•
>
1.
A
stopping time
sequence of stopping times (r„)^i such that r„
sequence of stopping times
< T
said to foretell T.
is
a.s.
is
t}
E
7i
for all
£
t
9?,
said to be predictable
=T
and r„ < T, limr„
where we have
if
there exists a
on {T > 0}. This
(For detailed discussion of predictable stopping
times see Chapter IV of Dellacherie and Meyer (1978).)
The
idea of a stopping time
is
that of the
T
be more precise, any stopping time
can be interpreted as the debut of the set
"The existence
Dellacherie (1978, IV. 51).
cf.
time some given random phenomenon occurs. To
first
of a foretelling sequence
{{t^co)
means that
this
t
:
>
T(u.')};
phenomenon
cannot take us by surprise: we are forewarned by a succession of precursery signs, of the exact time
the
phenomenon
will occur;"
Thus an event whose
occurrence
first
other hand, an event whose
Dellacherie and
cf.
a predictable stopping time
is
occurrence
first
Meyer (1978, IV. 72).
is
is
not a surprise.
not a predictable stopping time
may
On
the
take us by
surprise with a strictly positive probability.^
We make
the following regularity assumption throughout the rest of this section.
Assumption
The informatioD
5.1.
predictable stopping time
and
{T„
let
structure F
}^i
is quasi-left
continuous.
That
is,
let
T
be a
be any increasing sequence of stopping times foretelling
T. Then
00
n=l
An
information structure satisfying the above assumption
discontinuity" by probability theorists.
generated by a
Markov process
Meyer (1963, Theorem
The
is
It
is
known,
for
is
also
termed "having no time of
example, that an information structure
quasi-left continuous, provided that the
semigroup
is
nice; cf.
10).
following proposition shows that the price process of a marketed consumption plan can
be discontinuous only at either surprises or lump-sum ex-dividend dates.
Proposition 5.2. Suppose that
m
€ M. Let T
:
H
-^
on a set of strictly positive probability, then either
T
[0, l]
is
be a stopping time. IfS„,{T)
not predictable or ni(T)
^ Sm{T-)
^ m{T—)
on a set
of strictly positive probability.
Proof.
Suppose
S„i{T—)
*More
this
— Sm{T)
is
a.s.,
not true.
From Proposition
5.1
and Meyer (1966, VI.T14) we know that
a clear contradiction.
precisely, a totally inaccessible
stopping time
is
a surprise with probability one.
IV. 78) for details.
16
See Dellacherie and Meyer (197
Between (random) lump-sum ex-dividend dates, the price process
of a
marketed consumption
plan can have discrete changes only at non-predictable stopping times, or equivalently, at events
that are not foretellable.
T n
Proposition 5.3. Suppose that
:
a predictable stopping time. Then
i— 9? is
5„.(r)-5„,(r-) = -(m(r)-m(r-))
Proof.
When T
EU
The
we know
predictable,
is
VineM.
a.s.
=
h{s)dm{s) \7t)
E(f
h{s)dm{s)
Jr-)
\
a.s.
assertion then follows immediately from Proposition 5.1.
I
At lump-sum ex-dividend
amount
when
dates, which can be
of the dividends
there are no surprises.
Before proceeding, a definition
Definition 5.1.
An
random, the price process would drop by the
is in
order.
information structure
is
said to be continuous
if
all
the stopping times are
predictable.
Remark
5.2.
structure see
The
For an equivalent definition and extensive discussions of a continuous information
Huang
following
(1985).
is
a direct consequence of the above definition and Proposition
Corollary 5.1. Suppose that F
S,n[T)
When
is
is
continuous. Then for any stopping time
- S„,{T-) = - {m(T) - m[T-))
the information structure
dictated by
behavior of
its
its
dividend process.
price process
is
is
On
solely
5.3.
T
Vm G M.
a.s.
continuous, the ex-dividend date behavior of a price process
the other hand,
when a dividend process
determined by the way information
is
is
continuous, the
revealed.
Thus
it
can be
the case that dividends and information work against each other and there won't be price changes
at
lump-sum ex-dividend
dates.
For stopping times Ti and T2, the set
[TuT2)
is
called a stochastic interval;
The
=
cf.
{(u;,0
e n X
[0,1]
:
Chung and Williams
following proposition shows that
Ti{oj)
<
which the dividend process
<
Tii^o)}
(1983, p.29).
when the information structure
nian motion the price process of a marketed consumption plan
interval within
t
is
continuous.
17
is
is
generated by a Brow-
an Ito process
in
any stochastic
Proposition 5.4. Suppose that F
<
times with
{oj,t)
E
Ti
<
T2
<
1,
is
a Brownian motion filtration, that T\,T2 are two stopping
G
M
is
a continuous process on [Ti,T2). Then {5„,(u),f);
[T'i,T2)} is an Ito process.
Proof. First
we note that a Brownian
tion 5.1.1 of
Huang
{u>,t)
m
and that
G
[Ti,T'2)}
{oj,t)
{5„i(tc',f);
is
G
bounded variation.
(1985).
filtration
is
a continuous information structure;
From the hypotheses and Corollary
we then know
5.1
Proposi-
cf.
that {5„,
(oo'.f);
a continuous semimartingale. By the defining properties of a semimartingale,
[Ti,T'2)} can
It
be written as the sum of a local martingale and a process of
follows from Dellacherie and
Meyer (1982,
VIII. 62) that the local martingale
The bounded
part can be represented as an Ito integral, therefore continuous.
thus also continuous. Hence the price process
is
representable as the
continuous bounded variation process, and by definition
is
sum
variation part
of an Ito integral
is
and a
an Ito process.
I
Huang
(1985) has shown in an economy where consumption
that a price process can be represented as an Ito integral
Brownian
for
filtration.
consumption
is
The above
proposition
is
is
available only at two dates
when the information structure
a generalization.
When
agents'
when the information structure
a continuous process and
Brownian motion, the price process between lump-sum ex-dividend dates
Since there can be at most a countable
of the times, the price process of a
The
results of this section
continuous.
erences
shadow
number
of
is
price process
Ito process.
(random) lump-sum ex-dividend dates,
marketed consumption plan
depend largely upon the
is
for
most
an Ito process.
fact that agents' preferences are r*-
This assumption can be defended on economic grounds. The 7*-continuity
means that consumption
a
generated by a
be an
will
is
at adjacent dates are almost perfect substitutes,
of pref-
which captures
an intuitively appealing notion of time complementarity of consumption preferences over time.
For more discussions on the issues of time complementarity of consumption see
Huang and Kreps
(1983).
6.
Concluding remarks
It
should be clear that results of Sections 3 and 4 do not depend upon the hypothesis that the
time set
is
continuous.
integration signs to be
When
the time set
summation
signs.
is
No
any countable subset of
[0, 1],
we simply change
the
interesting sample path properties of a viable price
system for long-lived securities are available.
All of our analyses can be generalized in a straightforward
neous
beliefs, as
manner
to
accommodate
heteroge-
long as agents' endowed probability measures are uniformly absolutely continuous
with respect to one another.
The
of agents
results in this paper are strongly colored by the assumption that the
is
the whole space. That
is,
negative consumption
18
is
consumption space
allowed. This assumption
is
crucial
in establishing that the current price
system
marketed consumption plans
for
r-continuous linear function when a pair of price systems
results thereafter.
The
fact that relaxing this
evident from Jones (1984).
when
What
agents' consumption set
is
sort oi properness of preferences
(,9,
S)
is
is
viable with respect to
A
and the
assumption may significantly change our results
further restriction on
A
is
needed;
cf.
19
is
needed for our analysis to go through
the positive orthant remains an open question.
may be
representable as a
Mas-Colell (1983).
It
seems that some
Appendix
Proposition A.l. The bilinear form
xl'iv.x)
^ V x
A' i—
:
=E
I
3?
x{t)dv{t)
i
separates points.
Proof. Let a:i,X2
exist
t
e
€
B£
and
[0, 1]
^
A' and xi
>
with P(B)
7t
we assume that
loss of generality
we know that
processes,
there must
such that
xi(oj)
Without
RCLL
and X2 are
xg. Since xi
^
X2{oj)
Voj
e B.
> X2(w) on B.
xi(cj)
Define v
:
11
x
[0, 1] i-^ D?
by
Then
ip{v,xi)
-
^p{v,X2)
=E
- X2(s))dv(s)
(xi(s)
/
V-ao,il
[
/
= E{{xi{t)-X2(t))lB)>0.
Thus
V
separates points
Next
v{u>,t)
71^
eV,v ^0.
let V
Voj
in A".
Then
€ Bi. Without
there
must
exist
loss of generality
€
t
and Bi e
[0, 1]
we assume that
v{t)
Tt
-
with P{Bi)
v{t-)
>
>
such that
on Bi. Take the
following cases.
Case
X
:
n
X
1.
Suppose that
[0, 1] K-. 3?
v{u),s)
=
2.
Suppose that
|
v{s)
-
=
=
('-',«)
almost every
for
u e
Bi. Define
•
3.
There
exists
Te
e
=
>
jBi
- v{t-) Vs G
>0-
[f, 1]
for
almost every
such that the optional random variable T^
inf{s e[t,l]:e <\ v{s)
and ^(^2) >
uj
E
Bi. This
is
v.
where we have used the convention that
^2 e J with B2 C
E{lBA^'{i)-'^'it-)))
v{t) |> v{t)
impossible by the right continuity of
Case
Isj x[M]
and
A' and
ijiv,x)
Case
[t,l]
by
x{u,s)
Then x €
€
v{ui,t) for all s
0.
Tf
=
00
Define x
x{u,s)
=
:
-
x
[0, 1]
h^
1b,1[<_j^)(w,s).
20
-
n
1—» [0,oo] defined
9?
by
by
v{t-)},
the infimum does not exist,
if
fi
v{t) \< v{t)
:
is finite
on some
Since [t,T()
a stochastic interval, x
is
Williams (1983, Chapter
2).
That
is,
rp{v,x)
is
adapted, and by construction
x€
A'.
= E
Chung and
cf.
x{s)dv{s)]
{
v{t-))
we have shown that
cases,
RCLL;
Then
= E (iBAyiT,) Combining the above three
it is
>
0.
A' separates points in
V.
I
Appendix
Proof. Consider long-lived security j.
fact that the consumption
commodity
must
B
exist
t
e
and
(0,1]
e
Tt
II:
Proof
At time
is
for Proposition 3.2.
zero, (3.2)
certainly true, since t(0)
is
taken to be the numeraire. Thus
with P{B) >
such that (3.2)
if
(3.2) is
violated for
is
=
1
by the
not true, there
t
on B.
(This
follows from the fact that both sides of the equality of (3.2) are right continuous.) Without loss of
generality,
we assume
that on B:
E (// x{s)dD,{s)
I
7,)
x{t)
We
As
define
defined, 6
is
Okioj, s)
-
Ojiuj^s)
v{uj,s)
yuen,k =
+
=
-1bx(,,i](w,s)
=
1bx[mi(w,s) {Sj{u,t) - Dj{uj,s)
+
bounded and predictable, the
continuous; and v
is
-
l,i
i,2,... ,j
latter of
1,
.
.
.
,
J,
Dj{ijj)).
which follows from the fact that
6 is left
an integrable variation process.
We claim that 6
n \ B and for s e [0,
finances
t]
on
B
v.
is
The
fact that {d,v) satisfy (2.3)
obvious.
^(0)^5(0)+
On
J5 for s
r0iuydS{u)+
e
[t, 1],
r
we have
e{u)^dD{u)-v{s-)
-
S,{s)
+
Dj{t)
- D,{s-) - vis-)
=5,(0 -
Sj{s)
+
Dj{t)
- Djis-) -
=
-Sj{s)
=
(2.4) for s
Jo
•Jo
=Sj{t)
and
ej{s)Sj{s)
=
Sj{t)
6[s)'^S{s).
21
+
Dj{s~) - D,[t)
e
[0,1]
and on
It is
also clear that 0{l)'^ AD{1)
V at time zero
is
zero.
—
Thus
At;(l).
v
is
financed by 6 and v
is
marketed. The price of
But we note the following
iiv)
=E
i
=E
(x{t)Sj{t)
>
E
[
x{s)dv{s)
)
- f x{s)dDj(s)j
'x{t)E[j^x{s)dD,{s)\Tt)
^ -
^—^^
I
/•!
xis)dDAs)
J^
a contradiction.
Appendix
Proof. Let 9
is
clear that
G
G[/?, 5] finances v
III:
Proof for Proposition
4.2.
We want to show that v is
a.s.. We are left to verify that
EiV.
6{iy AD{1) = Av{l)
financed by 9 given (/9*,5*).
It
9{t)^S*{t)^9{0)"^S*{0)+ f 9{s)^dS*{s)
Jo
+ [
l3*{s)9{sfdD{s)-
Now
I
I3*{s)dv{s)
Vf
G
(A.l)
a.s.
[0, 1]
Jo
Jo
define
= e{tfAD{t) -
G{t)
Av{t)
and
=
G*{t)
The
process
9{tyS{t)
is
^*{t)G{t).
+ G{t)^9{0)^SiO)+ f e{sfdS{s)+ [ e{sfdD{s)-v{t)
Jo
Jo
a semimartingale, since the
IV. 20 of
Meyer
first
integral on the right-hand-side
is
a semimartingale
(1976)), and the second and the third terms are processes of
Here we remark that {^(<)^S(/)} may not be a semimartingale since
then
it is
V<e[0,l]
not right continuous, which
differentiation rule for semimartingales
is
if
D
or v
Corollary IV. 23 of Meyer (1976))
22
(cf.
bounded
is
a defining property of a semimartingale.
(cf.
a.s.
(A.2)
Theorem
variation.
not continuous,
Now
we have
using the
d{e{t)^s*{t)
=
where
/3*{t-)d (e{tfs{t)
+
G*{t))
=
+ G(o) +
d
(r (0
{e{t-)^s{t-)
denotes the joint variation process
[•,]
{e{tfs(t)
(cf.
+ G{t)))
+ G{t-)^ d/3*(t) +
d[/3\e^s
+
g],,
(a.3)
Jacod (1979, 2.31)). The second term on the
right-hand-side of (A.3) can be rewritten as:
(^e{t-fs{t-)
where the
first
+
G{t-)^di3*{t)
=
(^e(t)^s(t)
=
d{t)^S{t-)d/3*{t),
+ G{t) -
e(t)'^AS{t)
- e(t)^AD{t) + Ai>(o)d/?*(0
(AA)
equality follows from (A. 2) and the second equality follows from the definition of
G{t).
We
can rewrite the third term on the right-hand-side of (A.3)
d[l3\d^S
+
G],
=
e{t)^d[/3\S]t
+
as:
e{t)^d[p\D], - d[(3\v]t
by Theorem IV. 18 of Meyer (1976). Applying the differentiation rule
for
(A.5)
semimartingales again we
get:
dS*{t)
Finally,
we
=
d{/3*{t)S(i))
=
/3*{t-)dS(f)
+
S{t-)dl3*{t)
+
(A.6)
d[l3\S],.
substitute (A.4) and (A.5) into (A.3) and use (A. 6) to get:
d (e{tfs*{t)
where we have used the
+ G*(o) =
e{t)^ds*{t)
+
0*{t)e[tYdD(t) -
i3*{t)dv{t),
(aj)
fact that
/3*{t-)dD{t)
+
d[l3\D],= (3*{t)dD{t)
and
I3*{t-)dvit)
cf.
+
d[l3\v]t
Corollary IV. 23 of Meyer (1976). Equation (A.7)
finances
i;
given (0*,S*). This naturally implies that ^
are similar.
The proof
for the rest of the assertion is easy.
23
=
is
e
/3*{t)dv{t);
just the differential
e[/?*, 5*].
form of (A.l). Thus
The arguments
6
for the converse
References
Chung, K. and R. Williams, 1983, An Introduction
to Stochastic Integration (Birkhauser Boston
Inc.).
Dellacherie, C.
and
P.
Meyer, 1978, Probabilities and Potential A: General Theory (North-Holland
Publishing Company,
Dellacherie, C.
and
P.
New
York).
Meyer, 1982, Probabilities and Potential B: Theory of Martingales (North-
Holland Publishing Company,
Fama,
New
York).
A
E., 1970, Efficient capital markets:
review of theory and empirical work, Journal of
Finance 25, 383-417.
Gihman,
and A. Skorohod, 1979, Controlled Stochastic Processes (Springer- Verlag, New York).
I.
M. and D. Kreps, Martingales and
Harrison,
Economic Theory
of
arbitrage in multiperiod securities markets. Journal
20, 381-408.
Holmes, R., 1975, Geometric Functional Analysis and
Its
New
Applications (Springer- Verlag,
Huang, C, 1985, Information structure and equilibrium asset
prices. Journal of
York).
Economic Theory
34, 33-71.
Huang, C. and D. Kreps, 1983, Intertemporal preferences with a continuous time dimension: An
exploratory study,
Jacod,
J.,
Mimeo
(Massachusetts Institute of Technology, Cambridge,
MA).
1979, Calcul Stochastique et Problemes de Martingales, Lecture Notes in Mathematics
714 (Springer- Verlag,
New
Jensen, M., and R. Ruback, 1983,
York).
The market
for corporate control, Journal of Financial
Economics
11, 5-50.
many commodi-
Jones, L., 1984, Special problems arising in the study of economies with infinitely
ties,
Discussion Paper No.
Management
596 (The Center for Mathematical Studies
Economics and
Science, Northwestern University, Evanston, IL).
Kreps, D., 1981, Arbitrage and equilibrium
of
in
Mathematical Economics
in
economies with
infinitely
many commodities.
Journal
8, 15-35.
Lucas, R., 1978, Asset prices in an exchange economy, Econometrica 46, 1426-1446.
Mas-Colell, A., 1983,
University,
Meyer,
P., 1966,
The
price equilibrium existence
problem
in
Decomposition of supermartingales: The uniqueness theorem,
Mimeo (Harvard
Illinois
Journal of
7, 1-17.
Meyer,
P., 1966,
Probability and Potentials (Blaisdell Publishing
Meyer,
P., 1976,
Un
Notes
Mathematics 511 (Springer- Verlag, New York).
Patell J.
lattices,
Cambridge, MA).
Mathematics
in
Banach
cours sur
les integrales
Company, MA).
stochastiques, in Seminaires de Probabilite X, Lecture
and M. Wolfson, 1984, The intraday speed of adjustment of stock prices to earnings and
dividend announcement. Journal of Financial Economics 13, 223-252.
24
Rubinstein, M., 1976,
The
valuation of uncertain income streams and
tlie
Journal of Economics, 407-425.
H. Schaefer, 1980, Topological Vector Spaces (Springer- Verlag,
25
New
York).
pricing of options, Bell
325^ 033
Date Due
itr**'