c 2
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A RATIONAL ANTICIPATIONS GENERAL EQUILIBRIUM
ASSET PRICING MODEL:
THE CASE OF DIFFUSION INFORMATION
by
Chi-fu Huang"
May
1983
Sloan School of Management Working Paper No.
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
1553-8A
A RATIONAL ANTICIPATIONS GENERAL EQUILIBRIUM
ASSET PRICING MODEL:
THE CASE OF DIFFUSION INFORMATION
1
by
Chi-fu Huang"
May 1983
Sloan School of Management Working Paper No.
f
1
553-8A
The author would like to thank Doug Breeden, John Cox, and especially
Darrell Duffie and David Kreps for discussions and comments.
Any
errors are of course my own.
Department of Economics and Sloan School of Management, MIT.
Abstract
We
prove existence of an equilibrium in a continuous trading economy with diffusion infor-
mation. Equilibrium asset price processes are Ito integrals. Under some regularity conditions, equilibrium asset price processes
and the vector of
ing information form a vector diffusion process.
A
"state variable processes" generat-
martingale representation technique
is
used to characterize agents' optimal portfolio rules in an equilibrium context.
Introduction and
1.
summary
In the Intertemporal Capital Asset Pricing
tended by Breeden
[4],
Model, developed by Merton
[34]
and
ex-
the story goes roughly as follows: Let there exist a (perhaps en-
dogenously determined) vector diffusion process (F(t)} that describes "states of the world".
Assume
that prices for traded assets can be represented
by stochastic
of the Ito type with coefficients that are functions of Y(t)
the price processes and
sume
that
asset
prices as given,
tion.
Y
and
t
differential equations
at each time
together form a vector diffusion process.
each agent maximizes his expected
Markovian stochastic dynamic programming
is
utility
of life-time
tions on equilibrium asset prices are derived
first
and
Taking
consump-
exists, restric-
by inverting the "aggregate demand"
for assets,
order optimality conditions for agents' dynamic programs.
The two main assumptions made above
tinuous trading economy and the
finite
are the existence of an equilibrium in a con-
dimensionality of Y.
The
latter
is
an assumption
involving endogenous properties of an economic equilibrium. Cox, Ingersoll, and Ross
in
a production economy, identify the finite dimensionality of
that there
is
as-
then used to characterize agents'
consumption-investment choices over time. Assuming that an equilibrium
calculated by adding the
t,
a single representative agent
which prices are "smooth" functions of
Y
in
.
the
They
Y
.
They assume, however,
economy and that an equilibrium
also rely
[10],
exists in
on Markovian stochastic dynamic
programming methods.
The purpose
of this essay
is
First, the existence of
three-fold.
tablished in a Merton/Breeden-like economy.
1
an equilibrium
is
es-
Equilibrium asset prices are Ito integrals
whose
coefficients are "nonanticipative functionals".
shown to be Pareto
tion will be
efficient.
Furthermore, the equilibrium alloca-
Second, conditions are provided under which the
value of these "nonanticipative functionals" at each time
Z(t) and
t,
where
"state variable processes"
can be written as functions of
a finite-dimensional vector of exogenously specified diffusion
is
{•£(<)}
t
the vector of equilibrium price processes for
In particular,
.
Z
traded assets and the vector of state variable processes
together form a vector diffusion
process. Finally, a martingale-representation technique, similar to one proposed
is
used to characterize agents' optimal portfolio behavior.
representation technique
stochastic
more powerful
a
is
It is
[9],
argued that this martingale-
than Markovian
tool for equilibrium analysis
dynamic programming.
In Section 2 of this paper, a continuous-time frictionless
uncertainty with time span
consumption commodity
in
by Cox
[0,
T]
in the
is
formulated.
It is
consumption preferences.
assumed that there
economy, consumed only
number, are characterized by their endowments
pure exchange economy under
at times
at times zero
common
Agents are endowed with a
is
one perishable
and T. Agents,
T
and
finite
and by their
information structure
generated by an exogenously specified vector diffusion process Z. That
is,
Z
F
describes the
evolution of the exogenous uncertain environment.
It is
assumed that there are
zero net supply.
as to
at
most a
Each agent's problem
is
maximize consumption preferences.
finite
number
manage a
to
The
of traded long-lived securities in
portfolio of long-lived securities so
equilibrium concept used
is
Radner's
[37]
equilibrium of plans, prices, and price expectations.
Given the nice properties of F, a diffusion
select
long-lived securities appropriately,
location
is
sense that
securities.
Pareto
all
if
we
an equilibrium exists and the equilibrium
al-
filtration,
Section 3 shows that
Furthermore, markets are complete
efficient.
in equilibrium
in the
contingent claims not traded can be replicated by trading on long-lived
(Equivalently,
price processes for
all
all
contingent claims are priced by arbitrage.)
The
equilibrium
contingent claims are Ito integrals whose integrands are nonan-
ticipative functionals
(Theorem
machinery developed
in Duffie
3.3.1
and
and Huang
its
corollary).
[13].
2
The
existence proof exploits the
T
time
If
endowment
aggregate
C 3 with
functions which are
utility
(Assumptions
4.2.1, 4.2.2),
independent (Proposition
with a time- additive
is
path-independent, 1
bounded
derivatives,
and
if
if
agents have time-additive
agents are at interior
T consumption
then the societal shadow prices for time
function which
C 3 with
is
bounded derivatives on an open subset
of the real line can be constructed to support the equilibrium at the aggregate
point (Propositions 4.1.1
and
very roughly, the product of
are path-
Here we have exploited the fact that a representative agent
4.2.1).
utility
maxima
4.1.2).
Since the equilibrium price of a contingent claim
payoff and the societal
its
endowment
shadow
prices,
it
is,
then follows that
the equilibrium price process of any contingent claim with a nice payoff structure has a nice
representation (Proposition 4.2.2), which can be described by a partial differential equation
with a boundary condition.
nice representation
Conversely, any contingent claim whose price process has a
must have a nice payoff structure (Proposition
nice in each context will be
made
precise.
In addition,
and when some regularity conditions are
nice,
this claim forms,
satisfied,
dynamic choice either
Ross
[10]).
of
claim's payoff structure
is
the equilibrium price process of
with Z, a diffusion process.
Markovian stochastic dynamic programming
[32,33]) or in
when a
The meaning
4.2.3).
in a
is
a useful tool in characterizing an agent's
purely microeconomic context under uncertainty
an equilibrium setting with a representative agent
Sufficient conditions for the existence of
(cf.
Merton
(cf.
Cox, Ingersoll, and
an optimal control are quite severe;
compact
for
example, the space of admissible controls
[8]
recently proposed an alternative using a martingale-representation argument.
method
Kreps
is
[20,
controls
is
Section
3]
and Kreps
a linear space. (This
Merton
[34]
is
stochastic
and Breeden
[26].)
vector diffusion process
say that time
T
Y
is
[3],
Chapter
IV).
Cox
(This
See, for example, Harrison
and
implicit in Cox's setup.)
dynamic programming
[4],
Bismut
In Cox's work, however, the space of admissible
the purpose
is
choices and to characterize equilibrium relations
'We
(cf.
vaguely foreshadowed in the earlier literature.
When Markovian
as in
is
assumed to
exist,
aggregate endowment
is
3
is
used in an equilibrium setting,
both to depict agents' optimal dynamic
among
asset prices.
A
finite
one whose value at each time
path-independent
if it is
t
dimensional
along with an
a function of Z(T).
agent's wealth are sufficient statistics for this agent's
is
shown
that, in equilibrium, agents hold the
portfolios
relations
portfolio.
dynamic program. In
market
most highly correlated with each component of Y, respectively.
among
The above procedure
is
)' exists.
agents' optimal
flawed, however.
we
fix
after an equilibrium
is
after an equilibrium
established and
agents' optimal portfolio behavior.
of long-lived securities.)
It
is
its
established,
known whether such
a
dynamic programming
is
established and characterized.
is
stochastic nature characterized, the martingale-
Cox
[9]
seems to be a very useful way to describe
(The meaning of version
is
that we choose a particular
shown that agents hold pure hedging
a numeraire security (Proposition 5.1).
This characterization
is
valid
securities
and
when Markovian
not applicable.
Recall that, in our setting,
information.
not
is
a version of the equilibrium established in Section 3 and argue that
representation technique proposed by
set
and the market
Before an equilibrium
it is
Equilibrium
Thus, stochastic control of the Markovian type should be used to depict
dynamic choices
In Section 5,
Y
shown to be determined by
asset prices are then
it
and those
portfolio, the riskless asset,
and equilibrium relations among asset prices are examined,
process
this approach,
Z
is
the vector diffusion process that generates agents'
Denoting the price system for long-lived securities by 5, we show that the
state-price vector process
(Z,S)
is
a diffusion process provided certain regularity conditions
are satisfied (Assumptions 4.2.1, 4.2.2, 4.2.3).
If
an agent's time
T
endowment
is
path-
independent (with respect to Z), the number of shares of a hedging security held by this
agent at each time
t
depends only upon
security held by this agent at each time
optimal portfolio rule
is
t
Z(t).
The number
of shares of the numeraire
depends only upon Z{t) and
path-independent with respect to
Z and
5.
S(t).
The
Thus
this agent's
functional relation
between this agent's optimal portfolio rule for pure hedging securities and
Z
is
described
by a system of partial differential equations with boundary conditions (Proposition
Agents
in the
economy can
all,
however, have path-dependent optimal portfolio rules, with
respect to (Z, 5), without destroying the strong
up agents'
first
5.3).
Markov property
of (Z S). Thus,
,
summing
order conditions from their dynamic programs to characterize equilibrium
asset price relations
may
be ineffective. Concluding remarks are given in Section 6.
The economy
2.
we present a model
In this section
of a continuous-time frictionless pure exchange
economy under uncertainty with time span
[0, 7"].
Let
(ft,
7,P) be a complete
probability
Each u £
ft
denotes a complete description of the exogenous environment. The set
of trading dates
is
[0,
space.
T],
with a
common
2.1.
The information
We
where
T
is
a strictly positive real number. Agents are endowed
probability measure on the measurable space
assume that there
defined on the basic probability space,
is
W = {VK(t);0
the tribe 2
generated by {W(s);0
augmented by
the filtration
fact that
cr
"tr"
Fu
—
'
all
Fw
is
(cr
<
<
s
P-negligible sets
Vt£
=
T)
{7J";0
<
t
<
<
T,P), an TV-
T}. The component processes
T]. It
is
—
assume that 7^
clear that 7$
increasing, that
m ,„)
a matrix,
is
we write
|
2
a
|
is,
almost
is
C
7?
7 and
7",
= tr
(<ro' T ),
where
T
that
trivial
if t
<
e.
o*f
Tf
is
and that
For the
t-Ktahf [Z&\
denotes the inner product,
denotes (race. Let
:
RN X
[0,
T]
- R NxN
t,
3
(a Lipschitz condition),
I
Tribe"
may
and
/*(y,
:
RN X
- RN
T]
[0,
such that
\y-y\
\o-{y,t)-<j{y,t)\<K
\»(yJ)-fi(V,t)\<K \y-y\,
3
We
t).
[0,
is
be given functions, continuous in y and
is
t
a continuous information structure, see the piwviuus £say.-I
tr{y, t)
and
<
(ft,
W}v(<) are independent one-dimensional Standard Brownian Motions. Let 7J" be
Wi(t)
and
denoted by P.
7),
structure
dimensional Standard Brownian Motion
If
(ft,
(2.1.1a)
and
MV.O
2
|
<
2
A' (l+
2
I
y
|
),
|
2
a{y,t)
|
<
2
AT (1+
2
|
y
|
)
(2.1.1b)
be read as "sigma-field" or "sigma- algebra", but the former term seems simpler
more modern.
This implies that a and n are measurable with respect to the product Borel tribe on
RN X [0, T\.
(a "growth condition"
cr(y,t)
),
nonsingular for each y and
is
K
some constants
for
t.
Z
Let
We
and K.
assume that the
N X N matrix
be a (measurable) process adapted to
Fw
satisfying the Ito integral equation
Zm = Z(0)+ Jo/
<
for
Z
that
t
is
<
T, where Z(0)
/ <r(Z(«),«)dH^(«)
(2.1.2)
Jo
4
a constant TV-vector.
is
the unique solution of (2.1.2) and
Theorem
Arnold
9.3.1 in
[1]
ensures
a diffusion process with drift vector ft(y,t)
is
Here we should note that the above statement implies the
diffusion matrix a(y,t).
and
+
tt(Z(a), s)ds
conditions:
T
<
\lt{Z{t),t)\dt
I
oo
a.s.,
and
rT
\a(Z(t),t)
/
Jo
2
|2
<
dt
|
oo
a.s.
io
Let 7\ be the tribe generated by {Z(s);0
fact,
use
Tw
equivalent to
Fw
.
F
F
filtration
7 The
sets of
r
is
to denote both
from the context,
Agents
struction,
in the
F
is
all
F1
<
at least as fine as
is
8
F*
<
<}
augmented by
since the process
For this point see Harrison and Kreps
.
and
Fw
Z
is
[20].
from now on. Furthermore, unless
all
the P-negligible
adapted to F"\ In
We
it is
shall therefore
clearly otherwise
processes will be adapted to F.
economy
are
increasing and
endowed with the common information structure F. By conis
a continuous information structure.
The
interpretation
is
that the exogenous uncertain environment can be described by an ./V-dimensional Brownian
motion
VV,
which agents
in
the
economy may not
actually observe directly.
Agents can,
however, observe a vector of "state variable processes" Z, whose evolution over time depends
upon
W in an unpredictable fashion.
That
is,
the information content of the vector of state
variable processes can only be less than that of
process, however, provides agents with the
W
.
The
same information
structure of the state variable
as
if
they could in fact observe
A vector random process Y = {Y(t);t 6 [0, T\}, is called measurable if, when viewed as a
mapping defined on n X [0,r], for all Borel sets B in its range, {(w,«) Y(w,<) € B) € 7<g)B([0,r]),
where B([0,T]) is the Borel tribe of [0,T]. A vector (measurable) random process Y is said to be
adapted to F w if for every t 6 [0,T] the random variable Y(t) is /("-measurable. For brevity, such a
random process will be denoted Y = {Y(t), 7f,t 6 [0,T]} anc* called F^-adapted or Donant/c/pat/ve.
4
:
W directly,
the evolution of
Z
since
W generate the same information.
and
cautioned to note that the vector of state variable processes
assumed that there
It is
consumed only
7
,
Thus
P).
consumption
and T.
at times
at
x)
G
V
time
T
in state
(r,
The consumption space
The
L2+ (P)
set
R
the set {(r, x)
G
V
if
x
>
L%(P); x
Similarly, for v
=
and
>
i'
z
3>
>
if
r
:
G
r
>
[0,
6
(r,x)
endow
V
There are a
agent
^
V
G L?(P) P{u G
:
ft
:
x{u)
0;
we
»
and x
if
will write v
For
v, z
x
>
L2 (P), we
€ £+(P) and P
>
if
E V, the
v
G
= l}, while
0}
V"+ ;
relation
for the other relations just defined
t;
v
W€
>
>
ft
if
z
:
is
i
finite
number
of agents in the
characterized by a consumption set
V,
x(w)
>
if
0}
and
x G
=
1.
v^O;
taken to mean that
is
on V.
economy indexed by
Vj-,
endowments
{>,•
=
i
=
1,2, ...,/.
(ri,ii)
assume that
(i)v;
£
utility
V{
for every
1
f
u t{r,x{u))P(du).
= 1,2,...,/:
= y+;
(2) u;(r,y)
in r
:
and
denoted
R+ X
y,
/?+
with
^
— R
is
finite right
«i(r,y) and
continuous in
hand
#
r,
concave and strictly increasing
partial derivatives with respect to r
u,(r,y), respectively,
and
^
u,(0, z(w))
and y
G L'(P)
VzG4(P).
(3) there exist
if
Ui(r
(r,-,
z,)
G V+ and
e,-
>
such that, for every
-kr + s,x-kx + y)> U{ {r, x)
then (« 2
(«,
y)
G
V
and
+ /n y 2 (u;)P((/a;))2 >
Jfc
G
k^.
Each
C
V",
function
— R of the form:
Ui(r,x)=
We
>
>
v
V+ denotes
x
will write
and consumption preferences represented by a von Neumann-Morgenstern
:
on
variables defined
Agents
2.3.
Ui
R X
with the product topology r generated by
x G L\{P)}. For any x G
i»0.
and
random
of square-integrable
=
V
is
and the L2 (P)-norm topology on L?(P).
oo),
and x
and likewise
0,
We
oj.
defined as {x
is
for agents
represents r units of consumption at time zero and x(u>) units of
the Euclidean topology on
—
exogenously specified.
is
a single perishable consumption commodity in the economy,
is
L2 (P), where L?(P) denotes the space
v
is
The consumption space
2.2.
(ft,
The reader
i?+,
Note that
+
that Ui(r
above implies that
(2)
+
8,x
y)
>
r-continuous, and strictly increasing in the sense
is
whenever
x)
(/,(r,
{/,•
(«, y)
>
and
0;
(For a general definition of properness see Mas-Colell
condition for (3)
^
+
>
i'i
that for
€ L2 (P),
u,(0,x(u>))
(4)
is
for every
i
where
for
=
#-
all I,
all
Ui(r,y)
i 6 L%(P), and
2
1,2, ...,/,
ui
6
(5) there
is
at least one agent, say agent
all r
We
6
R+
c is
mean agent
There are a
indexed by j
L?(P).
time
T
°»
aQ d
Ei=i
(r,y)
£
J?+,
assume that
^
^•'( u; )
such that lim „_oo §-
1,
l's
marginal
£ f° r almost
Ui(r, y)
>
0, for
utility for
time
T
consumption
is
bounded
finite
=
securities
number
1,2,..., J.
of long-lived securities
of one share of security j
in state w.
The traded
is
Kreps
(cf.
Each long-lived security
The holder
endowments
is
[25])
traded
in
the economy,
represented by an element dj
entitled to dj{ui) units of
securities are in zero net supply.
6
consumption at
Agents have zero
initial
of these long-lived securities.
The admissible
2.5.
^
shall further
all
sufficient
zero.
Traded long-lived
2.4.
We
for
proper.
.
interpret (5) to
away from
vt
-
t
Note also that a
[30].)
bounded away from zero
(2).
is
(/,•
a strictly positive real number; and
every
fi,
is
(3) implies that
price systems and trading strategies
Before any discussion of the admissible price systems and trading strategies, some
technical definitions are in order.
A
continuous random process 7
Motion
W) if there
process
<p,
called
is
an ho process
exists a nonanticipative process c
,
(relative to the vector
and an
Brownian
TV- vector (row) nonanticipative
such that
and, with probability one for
l(t)
p\jo
\c(t)\dt
Pj[
M*)|
6
[0,T],
all
t
= 7(0) +
2
d<
<ool
=
<ool
=
l,
l,
c{s)d8+
c(e)d S + / <p(s)dvV(8).
/
/
/
Jo
Jo
(2.5.1)
(2.5.2)
(2.5.3)
Returning back to economics, an admissible price system 5,
S
is
a ./-vector Ito process
of the form:
Sj(t)
= Sj(0) +
with probability one for
of the
J traded
I
6
all t
Si (a)da
[0,
+
0j{s)dW(a)
/
V/ -
1, 2,
. .
.,
J,
T], with S{t) representing the ./-vector of relative prices
long-lived securities at time
b
t.
Given an admissible price system 5, an admissible trading strategy
$
is
a ./-vector
(column) nonanticipative process such that
El
where the
j'-th
row of
is
0j,
J
for every
t
£ [O.T ], the
the Ito sense, that
1
<
oo,
(2.5.4)
and the following conditions hold:
1
1.
0{t)i0(t)0it]t${t)dt
stochastic integral /„ ^(s)TdS(s)
is
well-defined in
is,
/
W)U(t)\dt
<
oo,
a.s.
Jo
anc
rT
I \0(tW(tWY0(t)\ 2 dt
<
oo
a. 8.;
Jo
2.
for every
t
€
[0,7],
$(t)i S(t)
= 0{O)i S(0) +
/ 6(8JidS(s)
jo
a.s.
(2.5.5)
Let 0\S] denote the set of admissible trading strategies with respect to the price system
5.
By
virtue of (2.5.5), elements of ©[5] are
all
self-financing trading strategies. Equation
(2.5.5) says that the initial value of the trading strategy (portfolio) 6 plus the
6
amount
of
Since F is a Brownian filtration, from thn p»n«innn essay we know any equilibrium price system
can be represented as an Ito integral. Thus taking admissible price systems to be the set of Ito
processes is not restrictive at all, once we know that an equilibrium exists.
9
capital gain or loss by
That
time.
is,
any time
t
is
equal to the value of the strategy (portfolio) at that
after the initial investment, there
investment into the portfolio. (This
is
is
neither withdrawal of funds out of nor
stochastic integrals and a simple application of the
shown to be a
easily be
Cauchy-Schwarz
linearity of
inequality,
0[S] can
linear space.
Equilibrium
2.6.
Each agent's problem
as to
in the
economy
is
to
manage a
portfolio of long-lived securities so
maximize preferences on consumption at times zero and T. An equilibrium of plans,
prices,
and price expectations
trading strategies {(0,) _,
:
at
By the
the obvious budget constraint.)
f,
(of
Radner
an admissible price system S, admissible
[37]) is
one for each agent, and a price a for the consumption good
time zero (relative to the prices of the securities, given by 5(0)), such that, for
all
t
=
1,2,...,/,
I
2>;<<)
and
(r,-
-
{(u
The
3.
+ e'(T^d)
0*(O)TS(O)/a, i,
-
among
in the literature
a vector Ito process
[24]
(Essay
a.8.,
1/,-maximal in the set
+ ocrpd) evr.ee
e\s}}.
existence of an equilibrium
has been popular
is
is
v«e(o,n
o,
*(o)Ts(o)/a, *i
The continuous-trading model
assumed
=
I)
(cf.
of financial markets formulated in the previous section
financial theorists for
more than a decade.
It
has always been
that an equilibrium exists and that the equilibrium price system
Merton
[34],
Breeden
[4],
Cox, Ingersoll, and Ross
studied a similar model and showed that,
if
Huang
indeed an equilibrium exists in the
continuous-trading economy, then equilibrium asset prices are Ito integrals
is
[10]).
if
information
generated by (multidimensional) Brownian motion. The existence of an equilibrium for
an economy of one representative agent was demonstrated.
machinery developed
in Duffie
and Huang
[12]
10
to
show that
In this section
if
the
J
we
use the
traded long-lived
securities are chosen appropriately,
an equilibrium for our economy
The proof
the equilibrium allocation will be Pareto efficient.
sense that a finite
number
is
shown
be constructive in the
will
of long-lived securities are selected, their prices are announced,
a trading strategy for each agent
allocation
Furthermore,
exists.
is
shown to be
assigned, markets are
Before we carry out this proof,
to be Pareto efficient.
clear,
and the
we analyze an
analogous Arrow-Debreu economy.
An Arrow-Debreu economy
3.1.
Suppose that
markets for
all
at time zero not only the spot
"state contingent claims"
6
market
for the
are open in an
space and agents are as described in Sections 2.3 and 2.4.
Arrow-Debreu equilibrium, defined
of an
V'
:
V
r
— R
and v i
is
and an allocation
=
{v,
t/,-maximal in the set {v
functional
gives
£
also
economy whose consumption
We
will
then show the existence
as a strictly positive r-continuous linear functional
(r,-, a;,-)
V,
consumption good but
6
^(v)
:
Arrow-Debreu equilibrium
= 1,2,.. .,/}
Vi,i
<
i'(i>i)},
for each
such that
«
= 1,2,...,/.
Here we should note that
prices.
The
if
linear
L?(P)
is
dimensional, then an Arrow-Debreu style economy consists of an infinite number
infinite
of markets at time zero.
There
zero. (For this point in a finite
is
no incentive, however, for markets to reopen after time
dimensional commodity space, see Arrow
Proposition 3.1.1: Let £
=
(Vj,
the conditions of Sections 2.2 and 2.3.
t/,-,
0,-,i
=
an economy satisfying
1,2, ...,/) be
Then there
exists
[2].)
an Arrow-Debreu equilibrium
(to')U*).
Proof: Given the assumptions of Sections
equilibrium follows from Mas-Colell
linear functional
6
A
$
:
V
—*
state contingent claim
Since
V
is
a reflexive
is
R
[30]
7
.
A
2.2
and
2.3,
quasi-equilibrium
and an allocation
{t;,-
£
Vj-,i
the existence of a quasiis
a r-continuous positive
=
1,2,...,/} such that
an element of L?(P).
Banach space, the Closeness Hypothesis in Mas-Colell
satisfied.
11
[30] is
automatically
==
YL\=i vi
H =i
t
*>i
a most surely and
l
'
>
Ui(v)
To show that V and
strictly positive
=
{f,-,t
V(t')
=
V'(t',')-
form an equilibrium, we must show that
1,2, ...,/}
>
By the assumption that Y2i=i
v(t'i')
V>(t>*).
ip
is
and that
Ui{v)
therefore
>
Ui(v') implies ^>(v)
>
By
for
i'i
some
implies i>{v)
Ui(v])
t'
3>
€
>
((4) in Section 2.3),
{1,2, ...,/}.
we know
V'(H,-
suppose v 6 V+, U{i(v)
<
continuity of preferences, there exists
UP (tjv) >
Vi.
ip(v')
q
<
1
>
>
0,
and
t/i»(t/,i),
and
»'»)
such that
Ui.(v',).
Hence,
^M>
This implies that
ij
>
V'(t')
>
that
t»,
>
0,
that r,
is
^-maximal
V'(*',/)-
Since U?
we know
il>(v)
>0.
Therefore
strictly increasing, i> is strictly positive.
>
Hv
i )
for
all
Arrow-Debreu equilibrium.
In an
=
an obvious contradiction.
1,
is
i'(vl)
i.
for
all
t.
Therefore,
We
?/,•»(
v)
V aQ d
{
v%
€
K">*
=
1)2, ...,/}
comprise an
|
We can
therefore freely
In particular, the
possible.
Proposition 3.1.2: Suppose the Arrow-Debreu equilibrium of Proposition
as
implies
can repeat the above arguments to show
choose any contingent claim having a positive price as the numeraire.
is
f/i»(v,-)
Then, by the assumption
economic equilibrium only relative prices are determined.
following characterization
>
3.1.1 has
T
in
(fi,
7)
numeraire the contingent claim paying one unit of the consumption good at time
every state. That
is,
V'(0,
In)
=
1.
Then
there exists a probability measure
12
Q
on
8
uniformly absolutely continuous with respect to P, such that
WxeL2 (P),
4(0,x)~E\x),
E
where
(•)
denotes expectation under Q.
Proof: Since $
(a,
(3.1.1)
()efiX
is
a r-continuous strictly positive linear functional on V, there exists
L?(P) with a
>
>
and £
ll>(r,x)
such that for
= ar+
all
(r,x)
€
V
z(w)S(u>)P(du),
Ja
= ar +
That
T
is,
<j>{x)
<f>(x).
gives the equilibrium price at time zero for a claim
u, and the price of one unit of consumption good
in state
is a.
which pays x(u) at time
By assumption we know
that
In)
V'(0,
=
/ tW)P(du)
-
1.
Defining
Q(A)= [
JA
Q
is
a probability
measure on
countably additive and Q(Cl)
(CI,
z(u)P(du),
7) equivalent to P.
= E(£) =
Q
1;
is
vie;,
It is
a probability measure since
equivalent to
P
since £
is
it is
strictly positive.
Therefore,
V(o,x)
= «£(*)=
=
Now
consider agent
1.
We know
/ ifuKMPfdu/)
Jn
/ x(w)Q(dw )
= £'(*).
v, = (r ,x
l
l )
must
solve the following concave
programming problem:
max
Q
the
is
Ut(v)
said to be uniformly absolutely continuous with respect to
Radon-Nikodym
derivative
dQ/dP
is
P
if it is
bounded above and below away from
13
equivalent to
zero.
P
and
^(w)-^(wi) <0.
s.t.
Since
V,-
= V+
and
il>(i\)
>
for all
reference for the Slater condition
saddle-point theorem that v l
only
if
is
(v)
t
Section 14E of Holmes
number
-
<
Ui(v])
have
t'j
this
+
t;
is
Since U\
is
-
programming problem
with v
£
V+
i>(v])
We
[23].)
vG^+
not the case. Take any
£ V+.
X, 4<(v)
(3.1.1)
claim that X!
>
0.
Since
F+
we have Ui(v +
strictly increasing,
then follows from the
It
[23].)
Xi such that for any v
(For this point, see Section 14F of Holmes
Suppose
the Slater condition holds. (A good
a solution for the above concave
is
there exists a nonnegative
U
we know that
i,
is
v)
l
strictly positive.
is
a positive cone,
—
U\(v\)
>
0.
we
This
contradicts (3.1.1).
Now we
cone,
proceed to prove that £
we know that
{r l ,x 1
Ux{r\,x\
+
+
kl A )
k\ A )
-
is
6 V+
bounded away from
for
all
<
C ri(rJ,ari)
A6 7
Since
zero.
V+
is
a positive
and k 6 R+- So we have,
X,L*(rJ,xJ
+
k\ A )
-
il>{r\,x\)
= Xi* J/ f(w )P(du).
A
Dividing both side of the above equation by k and letting k approach zero, we get, by (2)
in
Section (2.3) and the Lebesgue convergence theorem:
jf
^
mo-;,
*;m)p(<m <
x, jT e(«)p(du;).
(3.1.2)
Equivalent ly,
+
JA (^^)Since (3.1.3) holds for every
0}. For (3.1.3) to hold
it
A6
7,
fy
we can take
must be that P(A)
*i£M >
>
ui(r\,x\(u))y(d«,)
q-
=
A=
0.
(w €
That
:
Xi£(w)- ^-
+
Ui(r|,xJ(w))
<
is,
oifr,,*^))
14
fi
(3.1.3)
0.
a.«.
(3.1.4)
By
#
we know
(5) in Section 2.3,
Ui(r l ,z l (<*>))
is
bounded away from
above expression by Xi, we have the desired result that £
Next we want to show that £
^
2.3) that J2i *i
c/I}.
(r*,7*
c
Then A, € 7
»
where c
for
all
-^nglE^-
t
So
is
and
is
bounded away from
bounded above. Recall the assumption
is
a strictly positive constant. Let
=
fi
|J,
where
^~
U{(r, y)
{w £
^y
CI
:
Section
x^w)
>
567,
—k
J(u,)P(du).
and letting
it
approach zero gives
«,(r;,x;H)P(rfu;)>X./^ t(u)P(du) V B €
•'B no-
|'
=
((4) in
(3.1.1) implies that
Dividing both sides of the above expression by
/
A,-
zero.
For any real numbers k G (0,c/7) and
A-
Vfrlx'i-kl^J-Uiirlx^K-^k [
.'Bfl-4
zero. Dividing the
denotes the
left
hand
7,
partial derivative of U{(r, y) with respect to y. This
implies that
X,'£(w)<—
«,-(r,*,
x*(u;))
(3.1.5)
5 +
<—
for almost every
u,(''.-,0)
w6
A,-,
at/
the second inequality of which follows from concavity.
By
(2) in Section 2.3,
K
is finite.
Since £
<
Now
if a.«.,the proof
let
is
K = sup,-^complete.
(r t-,0)/X,-.
|
Proposition 3.1.2 states that Arrow-Debreu prices can be normalized so that the price
for any claim x
Q
£ L?(P)
at time zero
given by
E
(x), its
expected value under a probability
uniformly absolutely continuous with respect to P.
The uniform
absolute continuity of
Interpret £ to be societal
£
is
shadow
Q
with respect to
prices for time
T
or otherwise, agents can increase their utilities cheaply.
utilities for
time
T
l's
shadow
prices,
can be illustrated as follows.
consumption. In equilibrium, we know
must be greater than or equal to each agent's shadow
than or equal to agent
P
prices for time
In particular, £
T
must be greater
which are constant multiples of
consumption. Since agent
l's
15
marginal
utilities for
time
consumption,
his
marginal
T consumption
are assumed to be
bounded away from
T
hand, by the assumption that time
(almost) every state there
zero, £
an agent to consume a nontrivial amount
shadow price
his
for
consumption
By the assumption that
£
all
bounded away from
aggregate endowment
at least one agent
is
is
in
who
is
is
be no
the agents' marginal
the other
bounded away from
zero, in
consuming a nontrivial amount. For
a particular state,
in that state
On
zero.
less
utilities
it is
necessary, roughly, that
than the societal shadow
are
price.
bounded above, we then know
bounded above.
is
Some Theorems on
3.2.
known
well
It is
the representation of square-integrable martingales
that any square-integrable martingale adapted to a Brownian motion
can be represented as an
filtration
Ito integral
(Kunita and Watanabe
We
[28]).
will
make
use of the following result, originally developed by Fujisaki, Kallianpur, and Kunita
in
the context of non-linear filtering.
Lemma
3.2.1: Let there be defined on a complete probability space (Q',7',P') an
A -dimensional
r
[0,
[15],
process Y. Let the filtration generated by
y
Assumed that 7 T
T\).
P'-negligible sets,
=
and that 7\
7', that for
is
almost
every
t
6
[0,
Y
be denoted by
T] 7\
Suppose that
trivial.
Y
is
Fy
=
augmented by
{7\,t
all
6
the
can be represented by an
Ito integral as
= r(o)+ Jo/
r(t)
where
{h(t),
W
7y
t
,
y
t
is
6
[0,
T]}
is
an
N-
vector
F v -adapted
•,
\h(t)\
\h{t)\
where Epi denotes the expectation under
defined on (H', 7',P') adapted to
F" can
m (t) = m (0) +
some h
+
w*{t),
an N-dimensional standard Brownian motion adapted to
EP /
\
Jo
for
h( 8 )d 8
l
P*.
process defined on (fi\ 7',
At
<
oo,
Then any
3.1 of Fujisaki, Kallianpur,
16
1
)
and where
satisfying
square-integrable martingale
Vt6
[0,T],
satisfying (3.2.1).
Proof: See Theorem
P
,
(3.2.1)
be represented as
/ h(8)ldW"{a)
Jo
lo
Fv
and Kunita
[14].
I
m
Y
The process
Lemma
of
3.2.1
definition of a generalized diffusion see
3.2.1 states that
respect to the Brownian motion \V y
The information
W defined
adapted to
Watanabe
measure
Q
F
structure
Y
The next
constructed
is
F
Lemma
filtration
.
is
generated by an TV-dimensional Brownian
proposition shows that
if
we substitute
W
for
Kunita and
(cf.
P
the probability
Proposition 3.1.2, then any square-integrable martingale defined
in
F
can
still
be represented as an Ito integral with respect to
7 ,Q) adapted
(f2,
to F.
That
is,
N
the multiplicity of the
invariant under a substitution of a uniform absolute continuous probability
Proposition 3.2.1: There
to
[29].)
T'yP1 ) adapted to the
measure. (For a discussion of multiplicity, see Duffie and Huang
(H, 7,
(For the
can be represented as a stochastic integral with
F for our economy
independent Brownian motions on
F
(0*,
can be represented as an Ito integral with respect to
[28]).
7',/").
on (Q,7,P). Therefore, any square-integrable martingale on (H, 7,P)
on (Q, 7,Q) adapted to
filtration
(ft*,
Chapter 5 of Liptser and Shiryayev
any square-integrable martingale on
generated by the generalized diffusion
motion
a generalized diffusion on
is
Q) adapted
to
F
exists
[13].)
an /V-dimensional Brownian motion
such that any square-integrable martingale
m on
(fi,
W*
defined on
7,
Q) adapted
can be represented as
m(0
= m(0)+
/ h{s^dW\a),
a.s.
Jo
where {h(t),7t,t 6
[0,
T]}
is
an TV-vector nonanticipative functional such that
E*
I
\h(t)\
2
dt
<
oo.
(3.2.2)
Jo
Remark: We
will
henceforth denote the set of F- adapted
on (H, 7,Q) satisfying
measure
to P,
P
H(P)
is
a. 8.,
The analogous
by H(Q).
denoted by H(P). Since
Q
is
set defined
processes defined
under probability
uniformly absolutely continuous with respect
= H(Q).
Remark:
Q—
(3.2.2)
N- vector
Since
P
and
unless a distinction
Q
is
are equivalent,
needed.
17
we use
a. a. to
denote both
P—
a.s.
and
Proof: For £
= dQ/dP,
let
us define
£(<)
where
{£(<)}
is
= £(£
|
3) as.,
taken to be a continuous version of {E((
7 )}. 9 Then
t
\
there exists p
G H(P)
such that
^(t)
(Kunita and Watanabe
^(t)
where
= l(0/f(0-
9(f)
+
1
/
p(8?dW(8)
From Proposition
[28]).
and bounded away from
=
zero. It follows
= exp
|^
from
3.1.2
Ito's
V (s)ldW(s)-
(3.2.3)
we know that £
Lemma
strictly positive
that £(<) can be represented as
l
-jo
is
2
\
V (s)\ ds\,
Let
W\t)
= W(t) -
I V (s)d8,
t
E
[0,
T\.
(3.2.4)
jq
Girsanov's Fundamental
Theorem
1
motion on
clear that
W
(fl,
7
,
Q).
It is
W(t)
a generalized diffusion on (H, 7,
is
It
=
then follows from
on
(Cl,
Lemma
m(t)
is
that
W
is
an ./V-dimensional Brownian
adapted to F. By rearranging the
/ ti(s)d S
Jo
+
W*(t),
t
e
[0,
last expression,
T]
(3.2.5)
Q) and generates F. Furthermore,
3.2.1 that
J,Q) can be represented
([17]) states
any square-integrable martingale {m(t),
T ,t G
t
[0,
T]}
as
= m(0) +
A(«)TdVy*(a),
/
a.a.
•/o
This
is
possible since
F
is
a continuous information structure.
18
For details, see
t he
prwwes
for
some h 6 H(Q). This was
Remark:
to be shown.
Relation (3.2.4) defines
W
|
as an Ito process (under
P)
since
T
s
isrsrW
2
<
^'"i ""
°°
<
3 2 8>
-
-
implies that
[
d<
ti(t)
I
I
< rH jf
2
<
k(«)| d«
oo
P—
a.«.
j
by the Cauchy-Scharwz inequality.
Remark:
Let h
6 H(P). Then
m(t)
where
r?)(0) is
a constant,
is
= m(0) +
/ A(s)T(/Vy(s)
jo
a.«.,
a square-integrable martingale under P.
Similarly, for
h £
H(Q),
m(t)
is
= m(0)+
/ h{a)ldW\8) a.s.
Jo
a square-integrable martingale under Q. See Chapter 4 of Liptser and Shiryayev
3.3.
A
[29].
constructive proof of the existence of an equilibrium
In this sub-section
economy. The proof
recast in an
is
we prove the
essentially that given as Proposition 5.1 in Duffie
economy with
7
diffusion process information.
the tribe
if
the equilibrium price measure
separable,
if
In that
paper
it
and Huang
Q
is
[13],
was shown that
an equilibrium in the Arrow-Debreu economy
if
is
existence of an equilibrium for our continuous-trading
exists,
and
uniformly absolutely continuous with respect to P,
then the Arrow-Debreu equilibrium can be implemented by continuous trading of at most
a countable
number
of long-lived securities.
We
show
here, however, that in our
an equilibrium with a Bnite number of long-lived securities
19
exists.
economy
Theorem
in Section 2
3.3.1:
with TV
An
+
equilibrium exists in the continuous-trading economy formulated
long-lived securities having payoff structures:
1
«j)J!Li-j[ ht)dw\t)
^n+i
where
t
£
TV
[0,
+
1
:
H X
[0,
a.s.
with
T]
T]
— R NxN
£j/
\0(t)\
long-lived securities
is
2
= In,
any function that
<
dt)
oo.
A
is
nonanticipating, nonsingular for
all
set of equilibrium price processes for these
is
Sj(t)=E\dj\T
t )
= jf
^(«)rfW'(«)
(3.3.1)
-y
S'.v+ifO
for
all
£
(
[0,
respectively,
allocation
is
—
T],
0j(s)dW(s)
-
J
a.«.
£,(«)»/(«)</*,
Proof: Let
= 1,2,..., TV,
1«
where
W
and
ij
are defined Proposition 3.2.1 and equation (3.2.4),
and where 0j denotes the j-th row of
Pareto
j
fi.
Furthermore, the equilibrium
efficient.
{(r,-,x,-)
6 V+;i
Debreu economy and V De the
=
1,2, ...,/} be
strictly positive
an equilibrium allocation
and r-continuous
in the
Arrow-
linear functional
on
V
that gives equilibrium prices as in Proposition 3.1.2:
tl)(r,x)
where
a
is
= ar + E'(z),
the price for time zero consumption, and
E
(x) gives the equilibrium price at
time zero for claims x 6 L2 (P). By the definition of an Arrow-Debreu equilibrium we know
that (r i ,x i
)
is
^/.-maximal in the set
{(r,x)6V+ :V(r,x)<V(r,-,i,)}
20
for every
it is
i
=
1, 2,
.
.
.,
By the assumption that
/.
agents' preferences are strictly increasing,
obvious that
Equivalent ly,
The
price for x i
—
r,-,
x\
=a(r' -
f.)
+ E\x] -
time zero
x\ at
-
-
^{r'i
a{r{
is
—
it)
r,).
That
consumption good to buy the claim x i
units of
=
ii)
—
is,
0.
(3.3.3)
at time zero agent
In
£,-.
what
follows
i
pays
(f,-
—
r,
we show that an
equilibrium exists in our continuous trading economy with the Arrow-Debreu equilibrium
allocation {(r,-,Xi)€
V+ ;i
=
1,2,...,/}.
well-defined and
The proof
is
completed by the following program:
1.
Verify that d
2.
Show that 5
3.
Allocate an admissible trading strategy to each agent and show that markets
is
lies in
Show there
is
L?(P).
an admissible price system.
and the Arrow-Debreu allocation
clear
4.
is
is
attained.
no incentive for any agent to deviate from
his or her allocated
trading strategy.
Now we
Step
E*(j
1.
proceed exactly as outlined above.
We want
2
\ft{t)\
dt)
<
to
oo,
show that d
which
uniform absolute continuity of
Q.
It
follows
in
is
turn implies J
P
Note that E(f
well-defined.
|/?(t)|
2
</<
<
with respect to Q. Thus, d
from the second remark after Proposition
\P(t)\
2
<
dt)
oo implies
oo Q-almost surely, by the
is
well-defined under
3.2.1 that
d
is
P
and
square-integrable
under Q, and therefore square-integrable under P, again by the absolute continuity of
P
with respect to Q.
Step
2.
First
we show that S
is
well-defined.
the second remark after Proposition 3.2.1 and
integral in the third line of (3.3.1)
1.
Finally,
it
follows from (3.2.8)
integral in the third line of (3.3.1)
is
5
The second
is
line of (3.3.1) follows
well-defined under Q.
well-defined under P, since
P{J
The
2
|/?(<)|
<fr
from
stochastic
<
oo}
=
and the Cauchy-Scharwz inequality that the Lebesgue
is
well-defined.
21
Thus the second and the
third lines of
(3.3.1) are equal
S
under P, and
Step
3.
We
W
from the definition of
is
Therefore the second
.
line is also well-defined
an admissible price system.
allocate an admissible trading strategy 6
%
£ Q[S]
to each agent
i
and show
that with the announced price system agents obtain their Arrow-Debreu allocations almost
surely and markets clear.
Denoting
x,
—
i,
by
we know from Proposition
e,,
nonanticipative process {h
l
(t),t
H
since,
by the fact that £
6
7]} with h*
[0,
= E*(ti) +
€ H(Q) such
h\tydW\t)
/
an TV-vector
that
a.s.,
Jo
bounded above,
is
3.2.1 that there exists
e,-
£ L2 (Q). For
every
t
£
[0, 7*],
and j
1,2,.. .,7V, let
—i
where 0j
denotes the
j'-th
column of the inverse of 0, and
AT
*kn(0 = £*(«*) +
*
We
claim that the
[0,7]
-
0N+i(l)
N
r(n+i)xn
=QV
<
row vector of
£
N+
[0.7],
where
zeros. First
is
an admissible trading strategy. Let
= 0j(t) V
to
0)(t)Sj(t).
.
t
£
[0,7] o.«. for
0j(t) denotes the j-th
we want
e[]
9)(*)dSj( 8 )
/
'0
1-vector process 6 l
be such that 0j(t)
N
- J2
/•(
£
let
row of
0(t),
all
j
=
1,2,... ,7V,
m)dt\
Note that
rT
/
Jo
*W£( W)
T
*W =
rT
/
Jo
I
Jo
i
h\tf'0 \t)0(t)0(t)T0 \t)lh (t)dt
hHtVkHtot
=
I
Jo
Cl
X
and
and where Q denotes an
show that
{6\tV0(t)0{t
:
2
\h\t)\ dt
by construction. Because h 6
E
Thus, (3.3.4) follows since £
stochastic integral
H(Q) we
T
\L *W/W(') 'W<) <
bounded away from
is
l
6 (t)^dS(t)
/
have
it
follows from
Next, we want to show that the
zero.
Memin
[31]
rT
Once we
well-defined under probability measure Q,
is
that the definition of a stochastic integral
invariant under an equivalent change of measure.
we
is
Under probability measure Q,
rT
=
*{t)US{t)
/
Jo
(3.3.5)
well-defined under probability measure P.
is
show that the above stochastic integral
are done, since
oo.
/
Jo
h'Wfi \t)0(t)dW\t)
(3.3.6)
=
which
is
well-defined, since h
/
Jo
hWdW\t),
6 H(Q). Therefore
$'
£ Q[S]
since
by construction
it is
also
self-financing.
Finally,
by construction and relation
(3.3.6),
*''(0)TS(0)+ /
Jo
hWdw'w
+ /
Jo
=*}v+i(o)
MtydSft)
fT
=£*(e
t )
i
+
h (t)UW'(t)
/
=
ei
a.s.
Jo
Thus, for an
initial
trading strategy
0'
$'
investment of
attains
thus chosen be agent
i's
e,
E
(e,)/o units of
consumption good, the admissible
units of consumption good at time
trading strategy for
i
=
I*'
=
-£*••
23
1, 2,
...,/—
1.
T
almost surely.
For agent
I, set
Let
Since 0{S]
is
1
a linear space, O
admissible.
is
Markets for long-lived
securities then clear
by construction. Furthermore,
/-I
rT
/
^(0
T «w(0
+ 4v+i(o)--5>-ei
v
by the fact that,
If all
in
.
an Arrow-Debreu equilibrium, 5Z,-_i
=
«i
0, a.e.
agents follow these assigned trading strategies, each agent
-
f,
=
E'(ei)/a
f,
-
+
(r*
T
agent
t
at time zero
consumes
f.)
*
where the
first line
follows from (3.3.3).
Xi
+
e,-
At time
Step
4.
Now we want
to
z,-
a.«.
Arrow-Debreu
his or her
consumes
= Xi + (x* - ii)
=
Thus every agent gets
i
allocation.
show that there
no incentive for agents to diverge from
is
their assigned trading strategies. First, let us note that for
any h £ H(Q),
rt
I h(sydw\s) te[o,T]
Jo
is
a square-integrable
Proposition 3.2.1).
It
Q
martingale under
the second remark after the proof of
(cf.
then follows from (3.3.5) that, for any
/
e(s)US(s)
=
Jo
9{s)ip(6)dW'{s)
Jo
is
also a square-integrable martingale
I,
and (r,x) 6 V+, with
under Q. Next suppose there
6 &[S] such that
x
I
6 ©[5],
- Xf
£/,(r,x)
= 9N+ i{0) +
/
>
C/,(rJ,a;*)
${t)US(t
Jo
T
= o(r'-r)+
I
'0
Jo
24
0(t)lJS(t).
and
is
one agent, say agent
Taking expectations with respect to Q,
= a(r] - r) + E*U
E\x - x])
9(t)US(t)\
= o(u ~r),
the second line follows from the fact that J
under Q. But,
in
0(s)T(/S(«)
an Arrow-Debreu equilibrium,
a(r'-r)
a contradiction. Therefore there
is
>
£/,(r,x)
< E\x -
is
a square-integrable martingale
t/,(r,-,
x,) implies
x*),
no incentive for agents to diverge from assigned trading
strategies.
The equilibrium
allocation
equilibrium allocations.
is
Pareto
|
Given an Arrow-Debreu equilibrium, we can find
"support"
it
in
a continuous trading economy.
in selecting securities,
process
and
/?.
As long
satisfies
Pareto
agents get their Arrow-Debreu
since
efficient
we can
as ft(u!,t)
1
long-lived securities
which
should be clear from the above proof that
It
our attention to selecting the nonanticipative matrix
restrict
is
N+
nonsingular for
all
t
£
[0,
T] and for almost every
u £
ft,
the given regularity conditions, a continuous trading equilibrium exists with a
efficient allocation.
Corollary 3.3.1: Under the conditions
securities
complete markets. That
is,
of
Theorem
the price of a claim to x at any time
t
Let x
is
E
(x
6 L2 (P). Under the chosen
\
7 ), which can
t
integral.
in
+
1
long-lived
any contingent claim not traded can be replicated by
continuously trading on these securities.
Proof: See Step 3
3.3.1, the given TV
the proof of the Theorem. |
25
price system,
be represented as an Ito
Properties of equilibrium price system
4.
Morgenstern
utility functions
previous section
In the
we demonstrated an equilibrium
economy and showed that the equilibrium
price system in equation (3.3.1)
processes.
That
is,
past history of the
is
any time
at
t
an
is
€
[0,
allocation
Ito process
[4],
whose
not only an Ito process but
/
is
is
The equilibrium
generally depend upon the
fi
some vector
I
ft(Z(8),a)dW(s)+
s(Z(s), 8 )d3,
Jo
of unspecified diffusion processes.
It is
further assumed that (Z,S)
We now
forms a vector diffusion process, or that (Z,S) has the strong Markov property.
show conditions guaranteeing that
will
Z
can be taken to be the "state variable process"
Z, implying that the values of nonanticipative processes
be written as a (Borel measurable) functions of Z(t) and
More
diffusion process.
for
[34],
of the form:
Jo
Z
efficient.
coefficients are nonanticipative
and
tj
our continuous-trading
assumes, however, that the equilibrium price system analogous to
5(0-5(0)+
where
Pareto
is
T], the values of
in
economy. The Intertemporal Capital Asset Pricing Model of Merton
extended by Breeden
(3.3.1)
when agents have von Neumann-
generally,
we
and
t.
will exhibit conditions
c
at
any time
t
€
Furthermore, (Z,S)
[0,
is
T] can
a vector
under which the price process
any contingent claim having a certain payoff structure, adjoined to the state variable
process, forms a vector diffusion process.
From Essay
I
of
Huang
[24]
the fact that equilibrium prices are Ito integrals
is
an
inherent property of diffusion/Brownian motion information structures. Conditions under
which (Z,S)
is
a vector diffusion process are not as naturally posed.
This section flows as follows. In a market equilibrium with a Pareto
in
which agents have von Neumann-Morgenstern time-additive
efficient allocation
utility functions,
we can
construct a representative agent with a von Neumann-Morgenstern time-additive utility
function
case,
who
"supports" the equilibrium by consuming aggregate endowments.
In that
the properties of the equilibrium price for a particular contingent claim will be
determined entirely by
aggregate endowments.
its
payoff structure, the representative agent's preferences, and
When
these "aggregates" have simple structure, so will equilibrium
26
prices.
Construction of a representative agent
4.1.
In a
market equilibrium of either the Arrow-Debreu or Radner
design a representative agent
endowment
who supports
point (Kreps [26, 27]).
This
themselves give a "representative agent".
agent takes on content (and has value)
When
preferences.
if
[37] type,
one can always
the equilibrium price system at the aggregate
almost vacuous, since the equilibrium prices
is
The statement
that there
a representative
is
one can show that this agent has nicely structured
individual agents have preferences given by von
Neumann-Morgenstern
time- additive utility functions and share given subjective probability assessments over states
and when the equilibrium allocation
of the world,
is
Pareto
the representative
efficient,
agent can be chosen to be an expected utility maximizer having a time-additive utility
function. Prescott
commodity
and Mehra
[36] assert
space, Constantinides
that this
is
We
is
u, is separable.
/?+—>/? such that U{(r,y)
i
and
That
= /,(r) + y,(y).
U (r,x)
(a) /,
is,
infinite
extension of
dimensional commodity spaces
there exist functions
/,•
Then Ui can be represented
= f (r)+
i
I
Jn
(2) in Section 2.3 implies that for
y,
The
conceptually straightforward but needs to be formalized.
assume that
Assumption
a model with a finite dimensional
provides a detailed construction.
[8]
Const ant inides' construction to economies with
(such as ours)
so. In
:
R+ — R and
as
gi ( X ("))P(du).
every
t
= 1,2, ...,/,
are strictly increasing, concave, continuous, and with finite right
hand
derivatives everywhere.
We
also
(b) /,
is
assume that
strictly
for every
*
= 1,2,...,/,
concave and differentiate and
y,-
is
strictly concave, three times
continuously differentiable with bounded derivatives, and the second derivative
of Q{
is
bounded away from
zero.
10
The assumption of bounded derivatives can be relaxed by assuming Lipschitz-like
conditions on the derivatives. See Theorem 3 on page 293 of Gihman and Skorohod [16].
27
y,-
:
For every
=
i
1,2,...,/,
we know
(r,-
i,- )
,
solves the following concave program:
max UAr, x)
(r,*)€V+
(4.1.1)
s.t.
By
(4) in Section 2.3, ari
+
a(r
>
0(i,-)
0.
— f,) +
That
is,
6
+
jn
tA*i(»))P(*a)
gi{x(Lo))P(du)
+
+ ^iUri -
x/a(f,
Using the arguments following (3.1.1)
From the arguments used
-
k
g[
"
r')
+
=
1/n.
From
the saddle-
such that, for every (r,x) £
jf f(wXi.<«)
jf
fl^XM") -
- *.-M)/w)
V+
(4.1.2)
z(u))P(c/a;)\
proof for Proposition 3.1.2, we know
£ fl«X«Kw) " *!(«)VW =
X,-
>
0.
B
(r'.z*
j,-.
Let
<
Xt€(«)
Bn
/~ B
f
]
(4.1.4)
for
some
we
,
H
:
a;*(w)
>
£}, let
A
strictly positive scalar k,
6
/,
with
get
f(a/)P(«M,
X,-
= {u6n: z,*(w) > 0} = (J^ Bn
*'/4
(4.1.3)
o.«.,
be the set {u 6
- fcl^n^J G V+
^M)P(rfw) >
0.
we have
the argument used to obtain (3.1.5)
/n
Since
+
r)
to derive (3.1.4),
denotes the derivative of
finally let (r,x)
<
X,-
+
r,*)
in the
?'(*,V))
and
By
the Slater condition holds.
(4.1.2) implies that
«(r;
where
0.
Jq gAx'^mdu) + b(a(u -r') + jn Zi^i^u) - x\(u))P(du)\
>MU) + jf
Then
<
6 R+,
Mr*) +
>fi{r)
ii)
number
point theorem there exists a nonnegative real
and every
—
<j>(x
n
=
l,2,...
.
we have
tt)W>^/
«M rt
[IB
28
flw)P(<M V>ie7.
(4.1.5)
Let
fl
+
-
Bf\{u> e
n ftzfa)) >
X .£( w )} and
:
Applying the argument used to obtain
0.
That
is,
for those
w£fi
is
= ^DW e
B+
(4.1.4) to the sets
>
such that z'^uj)
and
ft
:
rffoV))
<
X t fl«)}.
=
B~ we get P{B + \J B~}
0,
— *.£(<")
l5(*J(«))
It
B~
o.«.
(4.1.6)
from the calculus that
clear
— X,o
/J(rJ)
>
if
r-
if
r* =
0,
and
(4.1.7)
<
fi(r])
Therefore
and
(4.1.3), (4.1.4), (4.1.6),
X,o
0.
(4.1.7) are necessary conditions for (r*,x*)
6 V+
to
be a solution to (4.1.1).
Now
define
F #+
:
— i? and
—#
G #+
:
F(r)
by
£r
'
=
max
1
/,{j,,)
/
£y,<r,
st.
(4.1.8)
and
'
G(r)—
max
s-t.
1
5^7-0i(y,)
£y,<r.
(
41 9
-
)
t'=l
One can
easily verify that
the following
Lemma
F() and G()
lemma shows
are strictly increasing
that F(-) and
G{)
and
strictly concave.
Furthermore,
are differentiate.
4.1.1: F(-) and G'() are differentiable. In addition,
F'(R)
=o
(4.1.10)
G'(X(u))
where
/?
s J3,-_,
f,
and where
X = X\=i
*»'•
29
£(u>)
a.«.,
Proof: Consider F()
Let
first.
r of (4.1.8)
be any strictly positive real number and
be the solution of the concave program of
let (y,)f=I
{1,2,...,/} such that yj
>
0.
F(r
since giving the "extra
(4.1.8).
+ k)-
F(r)
>
f (ffa + *) -
amount" k to agent j
is
D + F(r) >
D + F(r)
and
>
r
k
get
F
at r,
which
exists since
F
is
concave
we have
lies in (0,j/,),
F(r
Dividing both sides of
^fjiVj),
denotes the right hand derivative of
When
0.
6
ffy,j)),
certainly feasible.
we
there must exist j
Then
Let k be any positive real number.
the above expression by k and letting k approach zero,
where
Then
-k)-
F(r)
>
Uf
} ( yj
—k and
Dividing both sides of the above expression by
IT F[r) <
-k)-
ffa)).
letting
k approach zero, we get
J-fjivj),
A;
where D~F(r) denotes the
left
hand derivative of
F
at r.
A;
Ay
=
Therefore
D + F(r) =
on
Suppose the right hand derivative of F{)
(0, oo).
Thus,
D~F(r)
£fj{yj), and so F'(r) exists. Thus, F()
at zero does not exist.
is
differentiate
Then, by
strict
concavity of F(-), there exists a sequence of strictly positive real numbers {k n }, with kn
such that
F(k n )-F(0)
;
Now
it
follows from monotonicity
>
Vn
n,
and concavity of
,<!M5<f
'
;
30
= l,2,...
/,•
/W
that
Vn = l,2
J
0,
which contradicts the fact that /$(0)
zero. Identical
arguments show that G(-)
Next we show that F'(R)
we know
=
(r, )'
,
is
=a
for
is finite
is
r
- E f#
A
some j £ {1,2,...,/} with
>
r-
0.
-
=
£(u>) a.s.
By
strict concavity of
/,-,
«>-
»
i-yjfrj)
Substituting from (4.1.7) gives
=
F'(R)
Similar arguments show G'(X(u>))
The concavity and
bounded.
differentiable at
0,
F'(R)
for
is
the unique set of positive real numbers such that
t=l
R >
Therefore F(-)
also differentiable.
and that G'(X(ui))
ft*)
Since
all t.
Now we
— £(a>) a.s.
differentiability of
Proposition 4.1.1: (R,X)
is
Proof: This assertion follows
F() and
G(-) also imply that F'() and G'(-) are
results of this subsection:
the unique solution of the following program:
max
s.t.
|
main
are ready to prove the
a.
F(r)+ / G(x(oj))P(du)
ar
+
<f>(x)
directly
<ak + 4>(X).
from
(4.1.10).
|
Proposition 4.1.1 implies that the equilibrium established in the previous section can
be supported by a representative agent whose preferences are represented by the functional
V(r,i)
=
F(r)
+ fQ
G(x(u>))P(duj),
and whose consumption
is
constrained by aggregate
endowments.
If
if
each agent's time
T
equilibrium consumption
is
in the quasi-interior
11
L\{P) and
of
time aggregate endowments satisfy a regularity condition, we can say a bit more.
11
L,\(P) has an empty interior.
The
quasi-interior of
31
L+(P)
is
the set {x 6
L+(P) x
:
> 0}.
Proposition 4.1.2: Suppose
P{X >
s]
=
then G'()
= 1,2,...,/, such that
t
=
i
Let c
1,2, ...,/.
three times continuously differentiable on
is
C2
Furthermore, there exist
derivatives.
R+,
1,
for every
2>
a:,
z,-(u;)
Proof: By the assumptions
x,-
(c,
= ess inf X.
oo) with bounded
functions with bounded derivatives
= c,(X(w)) for almost every € Q.
» V 6 {1,2, ...,/} and P{X >
If
c,-
:
(c,
oo)
—
u>
i
e}
=
1,
we know
the solution of
I
lies in
=
X:
R+
the interior of
on a
if r
>
c,
for otherwise there exists a
measure.
set of strictly positive probability
i'
6
Suppose
{1,2,...,/} such that
r
> c
Necessary and
above program are:
sufficient conditions for (y,)f=1 to solve the
/_1
1
x
for
i
=
1,2,...,/-
1.
Now
;ff'(j/.)=
.
t
rV/(»--X!y.)
X,
let
/_1
for
»==
]/j
is
1,2,
...,/—
Since each
y, is
is
/
first
non-trivial
implicit function
=
x,
x7
—
1
arguments.
from the
theorem
It is
fact that each
(see, for
^i(r).
That
y,-
is
(c.
00)
strictly concave.
example, Hestenes
c,-
:
(c,
00)
[22],
—
It
p. 172)
/?+,
1
=
then follows from the
that there exist /
1, 2,
...,/—
1
—
1
such that
is,
X,
:
Let J be the Jacobian of (]/,)[=' with
.
easy but tedious to check that the determinant
z-g'i (c i (r))=^g'I
Define cj
£j
three times continuously differentiable on /?+, each
twice continuously differentiable functions
J/»
1
-1
twice continuously differentiable on /? +
respect to the
of J
1.
1
—» R+
by
cj(r)
X,
= r — ^Zi=\
(r-^2c
£-
c »'( r )>
32
i (r)).
(4.1.11)
clearly twice continuously differentiable.
For
6
r
we have
(c, oo),
= £ ^,(c,(r)).
G(r)
We
on
want to show that
(c, oo).
G
is
three time9 continuously differentiable with bounded derivatives
From Proposition
4.1.1,
G()
is
differentiable
on
(c,
oo) and
A*
i=i
(4I12)
-f **»!>)
= ^;(c,(r)).
The second and
that J2l=\
c 'i( r )
the third lines of the above expression follow from (4.1.6) and the fact
—
1)
respectively. Equation (4.1.12) reconfirms (4.1.10)
"envelope condition".
Now
from the thrice continuous
of
on
c,
differentiability of
j,-
on
R+
G
on
and twice continuous
has bounded derivatives on
we want
to
show that
we only have
to
show the boundedness
has bounded
first
for
the so called
(c,
oo) follows
differentiability
£'_,
Therefore
c'-
is
G" and G m
bounded
since
{1,2,...,/}. (Recall that g"{
<
it
If
Given Proposition
we can show that each
for
all i.)
bounded above away from
zero.
fact that the price
oo)
(4.1.11),
from
(4.1.13) that
1
>
cj-
>
for
all
i
€
Differentiating (4.1.11) with respect to r again,
g'",
g^, g",
and the
fact that g"
is
|
system
As should be
(c,
c,-
»5(<v(r))/»JMr))
clear
is
From
the boundedness of e" follows from the boundedness of
terization possible.
.
(e, oo).
and second derivatives, then the boundedness of G" and G'" on
follows from the chain rule for differentiation.
The
G
is
(r, oo).
Next,
4.1.1,
the thrice continuous differentiability of
and
(3.1.1)
clear
completes markets makes the above charac-
from Section
33
3,
every complete markets equilibrium
in
our continuous trading economy corresponds to an Arrow-Debreu equilibrium.
In an
Arrow-Debreu economy, equilibrium prices are determined by agents' preferences and (the
distribution of) endowments. Thus, equilibrium prices in the continuous trading
economy
should depend only upon the distribution of endowments and not on the distribution of
wealth over time.
on agents, ^,
In the construction of the representative agent, (4.1.8),
=
i
1,2,...,/, are constants reflecting the distribution of
the weights
endowments.
the distribution of wealth over time were important, then these weights would be
and the representative agent's
variables,
The
prices can be linked directly to aggregate
is
making assumptions about X. This
asset price
4.2.
apparent from (4.1.10).
endowments.
aggregate endowments are exogenously specified.
of £ by
in
random
function would be state dependent.
utility
usefulness of a representative agent
If
We
shadow
Societal
In our pure exchange
economy,
can therefore derive useful properties
turn allows us to deduce conclusions about
behavior and optimal portfolio choice.
Equilibrium price processes and "state variable process" form a vector
diffusion process
In this subsection
we show a
set of conditions ensuring that the price process for a
contingent claim, adjoined to the vector of state variable processes, forms a vector diffusion
process.
We
adopt the notation:
0Oi+O 2 +...+O„
QO,
^"tfp"
for positive integers a,,
is
denoted by
Dyg
a2
,
. .
or by g y
.
.,
dy?...dy°»>
an
The
If
.
g
:
l«l=«i+°2 +
R N X [0, T]
—R
is
C
following assumptions are
1
,
...
+ a„
the vector (dg/dt/i,
made throughout
.
.,
dg/dy n
this subsec-
tion.
Assumption
P{X >
c}
=
1,
4.2.1: In equilibrium, x]
where we
Assumption
recall c
from
4.2.2: There exists a
derivatives, such that
>
for every
»'
=
1,2, ...,/.
Furthermore,
(4) of Section 2.3.
C2
function
X = L(Z(T)) a.8.
34
L
:
R N — R+
with bounded partial
)
Assumption
there exist constants
Ko and K\
D°p(yj)
|
Assumption
D y fi(y,t) and D y (r(y, t) exist and are continuous
4.2.3:
4.2.3
is
aggregate endowment
is
|
|
K (l+
y
\
K >),
Assumption
a regularity condition.
The
path-independent:
|
path-independence
is
The
needed.
value of time
is
the positive orthant of
half of
Assumption
agents'
consumption
Assumption
4.2.1
T
Assumptions
shadow
aggregate endowment
the positive orthant of
lies in
and
A more
V
imply that
4.2.2 together
T
useful result
Proposition 4.2.1: There
bounded
differential equation
Assumption
This latter
Section 4.3), but
4.2.1
fact that each agent's
is
the least
consumption
In the first
not binding.
is
The second
half of
a technical assumption.
4.2.1
are continuous and
(cf.
to establish the existence of an equilibrium.
prices at time zero for time
function of Z(T).
half of
T
however, we assume that in the equilibrium, the constraint that
4.2.1,
is
V
first
we used the
satisfactory part of this essay. In Section 3,
set
and
4.2.2 says that the time
assumption can be relaxed to some extent by enlarging the state space
sort of
2,
\<*\<2.
depends only upon the value of the state variable process at that time.
some
a |<
such that
+ D ay a(y,t) \<
|
for
in
consumption,
is
t;
which we interpret to be
is
societal
path-independent and a smooth
actually possible:
exists a function 6
y and
£,
bt
exists;
:
RN X
and
[0,
T] —*
R such
f>(y,t) satisfies
that 8 y and
Dy6
the following partial
and boundary condition:
6lfi
+
6t
+
-tr(J yy <T<rT)
= o,
(4.2.1)
\jm6(y,t)
= G'(L(y)).
In addition,
£(*)
(where
{£(<).
t
£
[0,
T]}
is
= 6{z(t),t)
the martingale defined in the proof of Proposition 3.2.1.). Finally,
35
Z(t)
can be represented
flf)
where
= expU
t](y,l)
is
as:
1
rt(Z{8),s)UW{8)-
continuous
in
y and
t,
-^
r,(Z(s),e)
\
2
dsla.s.
\
satisfying the Lipschitz
Vte[0,T],
and growth conditions:
\v(y,t)-*l(v,t)\<K\y-y\,
(4.2.2)
2
2
2
h(lM)| <A- (l+|j,| )
for
some
K, and
positive constant
* (z(<) '' )
Proof: From
(4.1.10),
6 y (Z(t),t)MZ(t),t)
=
Assumptions
a twice continuously differentiable function
such that, redefining £ on a null
set,
Z(u})
It
follows from
Skorohod
and
t;
hi
[16]
Assumption
4.2.3
:
RN —
and Proposition
/?
>
,
Dyh
4.1.2, there exists
with bounded partial derivatives
= 9(Z{u,T))
and Theorem
vw
9.4 in
ea
(4.2.4)
Chapter
that there exists a function h(y,t) such that h y
exists; h y
also follows
§
4.2.2,
we have
,
2 of Part
Dyh
II
of
Gihman and
are are continuous in y
are bounded; and
t(t)
It
and
4.2.1
(423)
•
mtu)
from Theorem
8.4 in
= 6{Z(t),t).
Chapter
2 of
Part
II
of
Gihman and Skorohod
[16]
that
£(0 can be represented as
£(t)
= h(Z(t),t)=l+
I p(Z(a),8YdW(8)
a.8.,
Jo
where
P (Z(t),ty
= h y (z(t),ty<7(z(t),t),
36
(4.2.5)
and where S(y,t)
satisfies (4.2.1).
clear that
It is
'T
El
<
i
\p{Z(t),t)\ dt
oo
Jo
'o
since £
is
square-integrable. Ito's
fl<)
= ex P
n
Lemma
then yields
l
-jo
r,(Z(s),syd\V(s)-
\r,(Z{s), 8 )\
2
ds\
a.8.
Vt6
[0,
T],
whcT6
ri(z(t),ty=p(z(t),t)y6(z(t),t)
b {Z(t),t)ta(Z(t),t)
y
6(Z(t),t)
The
fact that tj(y,t)
and
cr(y,t).
is
continuous in y and
Finally, (4.2.2) follows
t
follows from the continuity of 6(y,
from the fact that
away from
zero, the fact that a(y,t) satisfies (2.1.1a)
Sy{Z{t),t).
|
If
we
interpret £(<) as the societal
shadow
£(t)
and
is
then Proposition 4.2.1 shows that the shadow prices for time
path-independent.
It
6 y {y,t),
bounded above and below
(2.1.1b),
prices at time
t),
and the boundedness of
for time
t
T
T
consumption,
consumption over time
is
follows that the payoff structure of a claim determines whether or
not the equilibrium price process forms, with the state variable processes, a vector diffusion
process.
Proposition 4.2.2: For any contingent claim x £ L?(P)
x
= E*(x) +
I
K(Z(t),t)ldW*(t)
satisfying
a.s.,
(4.2.6)
Jo
where
it(y,t)
:
RN X
[0,7]
—
RN
is
such that
E(j
T
\n(Z(t),
t)\
2
dt)
<
oo, the equilibrium
price process for a claim to x, denoted {ar(<)}> can be represented as
x(t)
= E'(x)+
I
k{Z(s),
eydW'( 8
Jo
= £*(*)"
/
)
K(Z(s),8)^(Z(s),s)d8+
Jo
I
k{Z(s),8)UW(s)
0.8.
Jo
(4.2.7)
37
Furthermore, suppose that K(y,t)
is
continuous in y and
Then
K(y,t) r v(y<t) satisfy Lipschitz and growth conditions.
and that both n(y,t) and
{(Z(t),x(t)),t
G
T]}
[0,
is
a
under P.
diffusion process
Proof:
t,
First
we note that by uniform absolute continuity
E'lJo \K(Z(t),t)\~dt)
<
oo.
It
of
P
with respect to Q,
then follows from the corollary after Theorem
3.3.1, the
second remark after the proof of Proposition 3.2.1, and Proposition 4.2.1 that the equilibrium price for x at time
E\x 7
t
I
)
is
t
- E\x) + Elj
«(Z(«),«)W(«)
7
t
|
o
= E'(x)+
J
I K(Z(s),8)ldw\s)
Jo
= E\x)-
I
Jo
Now
suppose that
Arnold
6.2.2 in
[l]
k(j/, t)
and
K(Z(s), S yri{Z{s),8)ds+ I K(Z(s), 8 ydW(s) a.s.
Jo
k(j/, t^tjd/, t) satisfy
ensures that {(Z(t),x(t)),t
€
Lipschitz and growth conditions.
[0,
T]}
of stochastic differential (integral) equations (2.1.2)
/i(i/,/), (r{y,t), ti(y, t),
[0,T]}
is
and K(y,t), Theorem
9.3.1 of
and
is
Theorem
the unique solution for the system
(4.2.7). Finally,
Arnold
[1]
by the continuity of
ensures that {(Z(t),x(t)),t
G
a vector diffusion process under P. |
As an immediate consequence, we have:
Corollary 4.2.2:
as 0(Z(lj,1),
<),
if
Hy<
the
If
{ )
'
s
random matrix
0(u>,t) (cf.
continuous in y and
t,
and
if
Theorem
3.3.1)
can be written
0{y,t) and fl(y,t) J il(y,t) satisfy
Lipschitz and growth conditions, then the equilibrium price system for long-lived securities,
S, can be represented as
5(<)
= 5(0)+
/ 0(Z{s),a)dW(a)- / p{Z{a), e)t){Z{8), e)ds
Jo
Jo
and forms, with the state variable process, a vector
N + 1-th
row of
fl(t) is
diffusion process.
a.s.,
(Recall that the
a vector of zeros.)
Proposition 4.2.2 states that
if
the payoff structure of a claim
librium price has a nice representation. (Here
we use
38
nice to
mean
is
nice,
then
its
equi-
of the forms (4.2.6) and
On
(4.2.7), respectively).
the other hand, the fact that the equilibrium price process of a
claim has a nice representation does not necessarily imply that adjoining to that proces the
state variable processes forms a vector diffusion process, unless
conditions are satisfied.
is
nice
little
whenever
its
One
natural question to follow
some Lipschitz and growth
whether a claim's payoff structure
is
The
price proces has a nice representation.
following proposition
is
a
stronger than the converse of Proposition 4.2.2.
Proposition 4.2.3: Suppose the equilibrium
6 L2 (P) can be
price for a claim to x
represented as
x{t)
where
f
is
= x(0) + J
s(a)d8
x
K (Z(8),sydW(e)
t
6
= E*(x)+
oo.
[0,
J
Jo
Then
a.*.,
RN X
:
(4.2.8)
T]
[0,
—» R N
Borel
is
can be written as $(Z(w,t),
?(u>,<)
T] and almost every
K(Z(t),t)UW*(t)
w£H
,
in
and
a.s.
(
f («)
+
K{Z(8),a)tdW'(a)
K(Z(«),«)TirtZ(«), «))</«+ /
a.s.
(4.2.9)
Jo
Step 2 of the proof of Theorem 3.3.1 show that the above expression
defined under Q.
By Corollary
3.3.1
is
a square-integrable martingale.
E
(J
2
\K(Z(t),l)\ dt)
<
we know
that, under probability
Next note that E(f
oo by the fact that £
is
\n(Z(t),
2
<)|
bounded above.
It
measure Q,
<
^0
is
well-
{x(t)}
oo implies that
then follows from the
second remark after the proof of Proposition 3.2.1 that the stochastic integral of (4.2.9)
a square-integrable martingale under Q.
integrable martingale under
continuous.
variation or
have
c(u;,e)
t)
First let us rewrite (4.2.8) as
= x{0) + Jq/
Arguments
<
2
\it(Z(t),t)\ dt)
(Borel measurable) for almost every
x{t)
J
a nonanticipative process, and where n(y,t)
measurable with E(f
Proof:
+
Now
is
=
Q
as well.
note the following:
a constant throughout
-k(Z(u>,
t),
Thus the Lebesgue
integral in (4.2.9)
Furthermore, as a function of
Any continuous martingale
(cf.
Dellacherie and
tyt](Z(uj,t), t) for almost
39
all t
Meyer
€
[0,
is
[12]).
t,
it
is
is
either of
is
a square-
absolutely
unbounded
Therefore,
we must
T] and almost every
w 6
f2.
Substituting into (4.2.9)
we have
z(t)
= z(0) +
n(Z(e),
/
aydW"{e)
a.a.
Jo
=x
Finally note that x(T)
= E*(x).
a.a., so x(0)
|
Propositions 4.2.2 and 4.2.3 together imply that for the equilibrium price of a claim
to have a nice representation
Note
t
also that, in the
may depend upon
representation.
The
structure of a claim
it is
necessary and sufficient that
payoff structure
its
is
nice.
above characterization, the equilibrium price of a claim at any time
the past realizations of the state variable process, even
if it
following proposition formalizes the intuitive notion that
is
path-independent, then
its
if
has a nice
the payoff
equilibrium price process must also be
path-independent.
Proposition 4.2.4: The equilibrium
claim whose payoff
is
price, at
each time
t
£
[0,T], of a
consumption
a Borel measurable function of Z(T), can be written as a Borel
measurable function of Z(t).
Proof: The arbitrage value
By the
for a claim x
definition of conditional expectation
E\x\7t)
£ L2 (P)
class
argument
(see
Chapter
2 of
= E{xi\7t)IZ{t)
E(xi\7
t
since by
Assumptions
from the
Lemma
c^
4.2.1
and
[19].)
Markov property
Chung
)
[6]
If
(see
= E{xi\Z{t))
[6]
t is
E
(x
\
It)
almost surely.
a.a.
x depends only on Z(T),
Chung
or see Williams
4.2.2, £ also
on page 299 of Chung
time
we have
(For the above fact see, for example, Harrison
follows from the definition of the
at
[7],
[38], p.
p. 2-3.)
it
then
and a monotone
122) that
a.a.,
depends only upon Z{T). Finally,
it
follows
that there exist two Borel measurable functions
and f such that
E{xl\Z(t))
=
40
<t>(Z{t))
a.a.
and
S(t)
Thus the
is
= <p(Z(t))
O.8.
assertion follows from the fact that the ratio of two Borel measurable functions
Borel measurable, and the fact that £(t)
is
strictly positive.
|
Special cases of the above proposition are (multiple contingency) options written on
The equilibrium
the final values of the state variable process.
at each time
t
prices of these call options
are Borel measurable functions of the value of the state variable process at
that time.
Remark:
If
the payoff structure of a claim
with bounded partial derivatives, then this
In this subsection
system
S
is
not only path-independent but
is
C2
is
a special case of Proposition 4.2.2.
we have shown that under some conditions the equilibrium
Z
together with the "state variable process"
continuous trading models of financial markets
it
price
forms a vector diffusion process. In
has always been assumed that the equi-
librium price system together with certain unspecified processes, which
may
be endogenous,
forms a vector diffusion process. Here, however, we have provided a set of conditions under
which those unspecified processes are identified as the vector exogenously specified "state
variable processes" Z.
The
4.2.3.
results of this subsection
As mentioned
earlier,
depend crucially upon Assumptions
Assumption
4.2.2
sumption.
same
are not path-independent neither are societal
In that case, the equilibrium price of a
structure will not have a nice representation.
stated in Propositions 4.2.2 and 4.2.3.
aggregate endowment
is
and
can be relaxed to some extent while main-
taining characterizations of equilibrium prices of the
endowments
4.2.1, 4.2.2,
flavor.
When
shadow
time
T
aggregate
prices for time
T
con-
consumption claim with a nice payoff
Here we
still
use nice to
In the next subsection,
mean
of the forms
we show that
if
time
T
not path-independent but can be represented in a certain form,
then, by enlarging the state space,
we can make everything
41
nice again.
An augmented
4.3.
Let us
system
suppose that the aggregate endowment
first
X=X
where
t,
|
ft(y, t)
RN X
:
2
D°fty,t)
A'o, A'o,
and
a(y,
t)
X(T)
= A'
|
+
|
D ay a(y,t)
:
J
&(Z(t),t)lJW(t)
RN X
[0,
T] -*
D°&(y,
t),
|< A' (l+
A'i are positive constants.
= A'(0) +
X(«)
Thus,
and
+
RN
a.s.,
(4.3.1)
are continuous in y and
exist
t)
of the form:
is
and are continuous
if
with
|
where
—R
T]
[0,
fi(Z(t),t)dt
J
and growth conditions, Dyii{y,
satisfy Lipschitz
q |<
+
X at time T
a.s.
Now
ii(Z{s), s)ds
/
+
Kl
\
),
\
a |<
2,
is
[0,
T}} by
(4.3.2)
the unique solution for the system
and
We
€
&{Z(s), 8 )dW(s).
I
of stochastic differential (integral) equations (2.1.2)
[l]).
y
define the process {X{t),t
Then Ux(t), Z(t)\te [0,T}\
process (Theorem 6.2.2 and 9.3.1 of Arnold
|
(4.3.2),
and
is
a vector diffusion
have the following result, analogous
to Proposition 4.2.2.
Proposition 4.3.1: For any contingent claim x £ L2 (P) of the form:
= E(x)+
;*(x)+
x
/
/
K(Z(t),X(t),tydW
»c
(t)
a.8.,
Jo
where K (y,t)
:
A N+1 X
[0,
T)
RN
-
satisfies
E(J
T
\
K (Z(t), X(t),t)\ 2 dt)
<
oo,
if
in equi-
librium the assumptions of Section 4.2 hold, then the equilibrium price process of a claim
to x, {x(()}, can be represented as
x{t)
= E\x)+
I s{Z(s),X(8),8)ds+
Jo
where f(y,0
:
fl
satisfy Lipschitz
N+1
X
[0,
T]
I
k(Z(s),X(8),8)UW(s)
—
R
is
Borel measurable.
and growth conditions, then {Z(t), X(t),
Proof: The proof
is
a.s.,
Jo
If,
in addition, ?(j/,*)
x(t)}
is
a vector diffusion process.
similar to those for Propositions 4.2.1 and 4.2.2, so
42
and n(y,t)
we omit
it.
I
The above
proposition once again signifies that, with complete markets (or
equilibrium allocation
is
Pareto
efficient),
it
is
when the
the nature of aggregate endowments that
determines the properties of societal shadow prices, which in turn, together with the payoff
structure of consumption claims, determine the properties of equilibrium prices.
The characterization
5.
of optimal portfolio rules
In intertemporal portfolio theory,
where price processes are taken as primitives, agents'
optimal trading strategies are typically computed using stochastic dynamic programming
(cf.
Merton
[32,33]).
Merton
[34]
and Breeden
behavior in an equilibrium context by
their
dynamic programs.
representation argument.
See, for example, Harrison
Cox
[4,5]
characterized agents' optimal portfolio
summing up
agents'
first
order conditions from
recently proposed an alternative using a martingale
[9]
(This method
and Kreps
is
vaguely foreshadowed in the earlier literature.
Section
[20,
3]
and Kreps
[26].)
We
will illustrate, in
our equilibrium setting, a technique similar to that proposed by Cox. Properties of agents'
optimal trading strategies which are
be established.
We
difficult to
come by using dynamic programming
will
contend, and hope herein to demonstrate, that the technique using
martingale representation
is
not just an alternative to the dynamic programming;
it
is
a
better technology.
In establishing the existence of
picked TV
+
1
long-lived securities,
nonsingular matrix process,
/?,
an equilibrium
in
TV of which could be characterized by an TV
satisfying certain regularity conditions (Section 3.3).
mentioned that any such process would do the job.
we analyze a
(The (TV
+
our continuous-trading economy, we
3.3.1 that the equilibrium price
Sj(t)
is still
system
= Wj(t)-
I
=
Vj (s)d8
1
43
a.s.
We
In)
It
to be the identity matrix.
then follows from Theorem
is
'0
Jo
SN+1 {t)
a claim to
TV
For this section, to simplify things,
particular version of the equilibrium by choosing
l)-th long-lived security
X
y
= l,2,...,TV,
for
all
6
t
a claim to x
Once again the above
T\.
[0,
€ L2 (P) has
Some prelimenary
set of long-lived securities completes markets,
a value at time
t
E
of
P
Let
[13].)
almost surely.
in order. Let
They
m
m
and
be two square-
are square-integrable semimartingales
P
by the uniform absolute continuity of
and Duffie and Huang
It)
\
and remarks are
definitions
integrable F-adapted martingales under Q.
under
(x
and
with respect to Q.
(m,m) denote the unique F-adapted
Meyer
(See
[35]
process of bounded
variation vanishing at time zero which satisfies
te
m(t)m(t)-(Tn,rh)t
is
an F-adapted square-integrable martingale under Q. (See Dellacherie and Meyer
The
joint variation process [m,rh]
(Meyer
(cf.
known
is
Since
[35]).
continuous
It
[0,T\,
is
equivalent to (m,fn)
We
will therefore
that the joint variation process
(Memin
is
time
T
Theorem
3.3.1
exchange for the portfolio
zero
in
zero
Arrow-Debreu allocation
T]}, which
in state
u
in
is
m
and m.
e,-
=
xi
#'(0),
pays
i
where a
is
By
at that time.
E
In the proof of
m
are continuous
for the portfolio he
Theorem
3.3.1,
is
is
i,-.
the joint variation process
That
is,
e,-
We know
(ei)/a units of
the difference
is
from the same
consumption
at time
the unit price of time zero consumption,
following the trading strategy {0'(<),<
budget feasible throughout, he gets
exchange
—
is
equilibrium allocation and endowment.
proof that, given the price system S, agent
(0,
and
always use [m,m] to denote [m,rn).
Therefore, under P, [m,rh]
[31]).
Recall the notation from
t's
m
invariant under a substitution of an equivalent
two square-integrable semimartingales
between agent
when
a continuous information structure, any F-adapted martingale
the previous essay).
probability measure
for the
F
is
[12].)
e,-(u;)
units of
consumption
holding at that time and consume
we demonstrated / optimal trading
at time
€
T
z,-(w).
strategies,
one for
each agent. They were chosen from a set of "admissible trading strategies" which forms a
linear space.
The primary
tool used
is
martingale representation (Proposition
3.2.1).
Using
the theory of optimal control, however, sufficient conditions for the existence of optimal
trading strategies usually involve compactness of the admissible set of controls
[3],
Chapter
IV).
44
(cf.
Bismut
Now
define a continuous process
c'(t)
This
is
the value process for agent
defined F-adapted process
The
almost surely.
€
{e,'(t)> ?t,t
t's
= E\
ei
T
time
[®>T]} by
\T )
a.s.
t
optimal net trade.
It is
Q
a square-integrable martingale under
is
clear that the above
=
and tht e,(T)
c,-
following proposition shows that the optimal trading strategies have
hedging properties.
Proposition 5.1: For any agent
in the
economy, say agent
i,
we have
for
all
t
E
[0,
T],
#}(«)« 4&wJ]i/<ft
(5.1)
= d\elWj]
t
j=l,2,...,N,
/dt
and
N
W0==«. (0-£'}(0S/(0!
Proof: The
them
and Shiryayev
in Liptser
from the
fact
Theorems
of (5.1) follows from
first line
[29].
In fact,
it
defines 6j.
is
W
the proof of
Since
from the construction
is
we have taken
Theorem
measure (Memin
[31]),
i
|
l
to be the identity matrix, (0 j)ji=1
=
h* for all
=
1
local joint variability of
is,
d[e it
i
in
3.3.1.
for
all
£
(
[0, 7*].
The number
optimally chooses to hold at time
with Wj (that
is
and the
for the j-th (j
< N)
long-lived
locally perfectly corrected with the j-th underlying source of uncertainty,
since d[VV-, Wj] t /dt
that agent
line in (5.1) follows
in Section 3.3.
Roughly speaking, we could say that the price process
security
The second
.
(5.2) follows
Remark:
and the note after
zero throughout, the fact that the joint variation process
invariant under a substitution of an equivalent probability
Equation
and
5.5
5.4
that the joint variation process of a continuous process and a continuous
bounded variation process
definition of
(5.2)
e,-
and Wj
Wj]/dt
<
0,
t
at that time.
is
If
of shares of the j-th (j
< N) security
equal to, again roughly speaking, the
at time
t,
which means, roughly, that
45
Wj,
e,
if
is
negatively correlated
Wj
goes up in the next
instant, ceteris paribus agent
agent
i
we
perfectly,
N
first
of security /,
call
and vice
versa.
them hedging securities. The (N+l)-th
The above
be termed the numeraire security.
will
the value of his optimal net trade to go down)
like
long-lived securities can hedge against the underlying uncertainty
henceforth
will
would
amount
holds a negative
Since the
i
long-lived security
analysis then implies that agents hold
hedging securities purely for hedging motives and use the numeraire security to transfer
wealth across time.
If
we
define the
market portfolio to be the aggregate value of the
market portfolio
long-lived securities, then the value of the
Our
that long-lived securities are in zero net supply.
zero throughout by the fact
is
results are thus largely consistent
with previous characterizations of agents' optimal trading strategies
stochastic
dynamic programming: Agents,
[34],
Breeden
may
not be applicable.
[4,
5]).
In our present setting,
can be applied
in
Even
if
Markovian stochastic dynamic programming
our present setting, the number of hedging portfolios (securities) held
may
be substantially larger than TV.
point will soon be clarified.
In the
above analysis we have characterized hedging properties of agents' optimal
made
trading strategies without the assumptions
Assumptions
4.2.1, 4.2.2,
as a function of Z{t)
for agents'
not,
Merton
Markovian stochastic dynamic programming
by agents as characterized by dynamic programming
If
(cf.
Neverthless, Proposition 5.1 illustrates the hedging properties of
agents' optimal trading strategies.
t
portfolio, the
Markov dynamic program
agents' wealth, are sufficient statistics for agents'
last
market
in equilibrium, hold the
and portfolios most highly correlated with the variables which together with
riskless asset,
This
made with Markovian
and
4.2.3
We
in Section 4.2.
and ask the following question: Can
and S(t)? (Equivalently,
dynamic choice problems?)
If
is
this
now re-adopt
will
%
$ (t) be written
(Z(t), S(t)) a sufficient statistic at
is
true,
what
is
the functional relation?
what additional assumptions are required to develop such a relationship?
proceeding,
we take note
is
Before
of the following fact.
Proposition 5.2: Given Assumptions
matrix, (Z,S)
time
4.2.1, 4.2.2,
and
4.2.3,
and
equals the identity
a vector diffusion process.
Proof: Since the
identity matrix
is
certainly continuous
46
and
satisfies Lipschitz
and
growth conditions, the assertion follows from the fact that
rj{y, t) is
Lipschilz and growth conditions, and from Proposition 4.2.2.
Note that agent
value at time
t
i's
of e,(f).
vector diffusion process.
The
T
optimal net trade at time
If e
It
i
a contingent claim, with a
meets the conditions of Proposition
may then
follow that $*(t)
satisfies
|
itself
is
continuous and
4.2.4,
then (Z,e*)
is
a
a function of Z(t) and S(t) alone.
is
following proposition formalizes this.
Proposition 5.3: Suppose there
partial derivatives such that
=
£,-(u/)
exists a
7r,(Z(u;,
C2
T))
function
a.s..
7r,-
:
RN — R
with bounded
Then
(W)H -(WM))H
(5.3)
uy (Z{t),
t)l<7(Z(t), t)
v (Z(t),
t)6 y (Z(t),
satisfies (4.2.1); v{y,t)
are continuous in y
and
t;
t/<
RN X
:
exists;
t)
6 2 (Z(t),t)
6(Z(t),t)
where S(y,t)
t)MZ(t),
[0,T]
and v(y,t)
—R
such that
is
satisfies
D\v and D y v
exist
and
the following partial differential
equation and boundary condition:
v\\i
+ Vi +
-tr(i/
yj
,<7<7
T
)
= 0,
(5.4)
lim i'(y,t)
= (c (L{y))-ni(y))g(y),
l
t\T
where
as in Proposition 4.1.2,
c s is
L
as in
Assumption
4.2.2,
and §
As
for
ar,(w) is a
C2
as in (4.2.4).
the numeraire security,
Proof: From Proposition
function of
X(w) with bounded
almost surely a
that ii(uj)
is
that
= x^u))—
e,(cj)
C2
i,-(u;) is
4.1.2
we know that
derivatives.
for almost every
Assumption
u6[),
4.2.2 together with the
assumption
function of Z(u>, T) with bounded partial derivatives implies
almost surely a
C2
47
function of Z(ui,T) with bounded partial
derivatives. Ignoring a null set
we can write
ti
where
jr,
RN — R
:
C2
is
= k{Z{T)),
with bounded partial derivatives. By the definition of conditional
expectation and arguments similar to those in Proposition 4.2.1,
we have
t\{i)=E\ti\7t )
= E(tji\7t)
m
v{Z(t),t)
(5-5)
.
6(Z(t),T)
_ E(Cii) + /q v
l
where 6(y,t)
y and
t,
v
t
satisfies (4.2.2),
and v(y,t)
exists,
are diffusion processes.
By
+
y
(Z(s), 8)l(T(Z(8),s)dW(s)
Ji6 y (Z(s),s)^(Z(s),s)dW(s)
and v{y,t)
is
such that
Lemma we
D^is exist and are continuous in
The numerator and
satisfies (5.4).
Ito's
D y v,
the denominator of (5.5)
have:
V
"«*£**,)
d#- ^dw- ^fdw^
0'
0*
dt
_ «*l°f*,) dL
i
(56)
b
Substituting (4.2.5) into (5.6) yields
where we
recall that
W
is
T
,
It is
r't
»-i
f
(
an F-adapted TV-dimensional Brownian motion under Q. Equivalently,
»y(Z(t),tV<?(Z(t),t)
now obvious that the row vector
,
(6 At)W_ 1
(v y (z(t),t)wz(t),t)
„(Z(t),t)6 y (Z(t),tMZ(t),t) \„ ir ,,^
can be chosen to be
i>(z(t),t)6 y (z(t),tMZ(t),ty
6 2 (Z(t),t)
HZ(t),t)
48
')
or (5.3).
12
The
characterization of
follows from Step 3 in the proof of
6% +i
Theorem
3.3.1.
I
Proposition 5.3 states that
if
agent
then under the assumptions of Section
dynamic choice problem
only Z(t), S(t), and
(This follows since
When
the
t'-th
depend on the
That
process.
t,
E
at time
t.
4.2,
|
T
t
)
is
agent's time
T endowment
i's
Z
is
A
how
this
endowment,
minimum
x\, equals the
over the interval
[0,
T].
positive
may
minimum
of Z\.)
{Z,S) forms a
first
T
time
T
T
endowments. One agent's time
some
state variable, say Z\,
endowment equal
agent's time
above two statements are the conditions:
(ii)
not render past
not destroy the Markovian nature of
of value attained by
The second agent has time
and nonzerod, and
may
might come about. Imagine an economy with
between some smooth function of Z(T) and the
implicit in the
$*(t) will, in general,
Conversely, the fact that agents have
two agents identical except that they have different time
T
be a function of
(Z, S) forms a vector diffusion
a vector diffusion process
portfolio decisions.
simple example illustrates
will
sufficient for his
(cf. (5.5)).)
S even when
path-dependent optimal dynamic decision rules
(Z,S).
t
not path-independent,
is
or
t
is
statistic for the value of his net trade.
a function of Z(t) and
the fact that (Z,S)
history irrelevant for agent
path-independent and smooth,
knowledge of (Z(t),S(t))
with Z(t) being a sufficient
(e,-
is
His "indirect utility function" at
historical realizations of
is,
endowment
t's
(i)
to the difference
T endowment.
The minimum
(Note that
value of Z\
is
aggregate endowments are strictly greater than the
Thus the aggregate endowment
at time
T
is
path-independent, implying
diffusion process (provided certain regularity conditions are satisfied). But,
T
in equilibrium,
each agent must hedge against his or her time
which varies
ways that depend upon the entire path of Z\. Each consumes an amount
in
that depends on
In our
Z(T) only, but
their net trades are
endowment
realization
much more complex.
exchange economy, aggregate endowments at time
T
are the essential deter-
minants of asset price dynamics, rather than properties of agents' optimal portfolio
Agents
in the
economy may
all
have path-dependent decision rules, but so long as Assumptions
The
dt
X dP
over this
rules.
/ optimal trading strategies, (#')'_,, one for each agent, are unique over the measure
(Chapter 5 of Liptser and Shiryayev [29]). But elements of a particular equivalence class
measure may not be indistinguishable processes.
49
4.2.1, 4.2.2,
and
(Z,S) obeys the strong Markov property.
4.2.3 hold,
Given Proposition
4.3.1, the following
Proposition 5.4: Suppose that agent
is
t's
easy to prove.
time
+
iii(Z(t),t)dt+
ii { (Z(t),t)dt
'0
Jo
where M,(lM)
and
t;
KN X
:
satisfy Lipschitz
continuous for
|
a |<
|
where
is
x,(0), A'o,
D Qy
and
-R
[O.rj
2,
£,•
satisfies the Ito integral:
rT
rT
x, = i,(0)+/
T endowment
&i(Z(t),t)TdW(t),
J
Jo
JO
and a { {y,t)
RN X
:
[0,T] -*
and growth conditions; and D^fi^y,
t)
a.8.,
RN
(5.7)
are continuous in y
and Dy&{(y,t)
exist
and are
with
ii t{y,t)
|
+
|
Day &i(y,t)
K
|<
(l+
|
y
Kl
|
).
I
« l<
2,
Furthermore, suppose the process
A'j are positive constants.
(i,(<),
t
G
defined by
= i,(0) +
£,-(*)
Then
!
[0,T]}
(f)
is
/
Jo
fi i
(Z(8),s)da+
I
bi(Z(8),8)UW{s)
a.s.
can be written as a function of Z{t), S(t), and
£,(*),
and
{(Z(t),S(t),Xi(t)),t
When
agent
t's
time
to those of Propositions 4.3.1
T endowment
is
decision rule can be revived by augmenting
that case, agent
t's
j's
and
(Z,S) with the process defined
is
a function of Z(t), S(t), and
dynamic choice
at that time.
is
important
|
in the price
in (5.8).
In
which
&i(t),
Using the traditional
characterization of equilibrium asset prices and optimal portfolio rules,
concluded that the x^t)
5.3.
well-behaved, the path-independence of his
indirect utility for net trade
are sufficient statistics for agent
1
E
a diffusion process.
Proof: The arguments are similar
N+
(5.8)
Jo
we would have
formation process, and that agent
i
faces
sources of uncertainty to hedge against.
In this section,
we have characterized properties
of agents' optimal portfolio rules in
an equilibrium setting, primarily using martingale representation techniques. Some results
have been derived
in
an environment where Markovian dynamic programming cannot be
50
[0,
T}}
applied.
Even when Markovian dynamic programming
order optimality conditions
first
Kreps
[26],
may
"is
management problem". The martingale connection
theorists.
makes
[20]
applicable,
not give the right answer.
that the stochastic control machinery
by Harrison and Kreps
is
We
summing up
agents'
contend, along with
not really necessary in the portfolio
of equilibrium asset prices developed
available a rich theory of martingales to financial
The martingale representation technique demonstrated
an alternative to stochastic control, as suggested by Cox
[9],
in this section
is
not only
but also a more powerful one,
especially in an equilibrium setting.
Concluding remarks
6.
This paper addresses three
in
issues: First,
can we prove the existence of an equilibrium
a Merton/Breeden-like continuous-trading
economy?
Second, under what conditions
do the vector price process and the vector state variable process together form a vector
diffusion process?
Third,
is
dynamic programming an appropriate
agents' optimal portfolio rules in an equilibrium setting?
two questions. With respect to the third one,
it is
tool in characterizing
Answers are provided
for the first
argued that martingale representation
techniques might be more powerful. Dynamic programming has not yet been shown to yield
a consistent characterization of heterogeneous-agent equilibrium asset prices.
our work that dynamic programming
The economy considered
in this
is
seems from
It
indeed an inappropriate tool for this problem.
paper
agents consume only at two time points.
is
a continuous time Radner
economy
in
which
Questions related to the Consumption Capital
Asset Pricing Model, such as whether agents would optimally choose their consumption
rates to be Ito integrals
addressed here.
asset prices
When
is
When
totally
when information
there
is
is
generated by diffusion process, cannot be
no intermediate consumption, the behavior of equilibrium
determined by the arrival of information
agents are allowed to consume continuously,
it
(cf.
ourprev
by how agents choose to consume over time and by the way information
This
ongoing research of the authors.
51
om
c»ay ).
seems that prices would be determined
jointly
is
i
is
revealed.
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