\ssr.
HD28
.M414
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Optimal Consumption and Portfolio Policies
When Markets are Incomplete
by
Henri Pages
WP //1883
April 1987
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Optimal Consumption and Portfolio Policies
When Markets are Incomplete
by
Henri Pages
WP #1883
April 1987
M.I.T.
LIBRARIES
Preliminary, Goniincnts Invited
OPTIMAL CONSUMPTION AND PORTFOLIO POLICIES
WHEN MARKETS ARE INCOMPLETE
by
Henri PaRrs
February 1987
Abstract
We
eonsider a conhnntnis time versinn of the eonsnniptinn jiortfolio
where an agent invests
sumption and possibly
his initial endnwiiK^nt
final
wealth
the general existence cjuestion
ovei- a
on
markets
rajfital
hxal horizon. Usiup;
when markets
tli(^
to
i5rol)leni
under luirertainty,
maximize
his utility of con-
martiiif^ale apjiroarh.
are flynamically incomi)lete.
of the diffusion process followed by the assets prices
motion that describes Ihe uncertain environment. The
less
is
tlian
tli(<
i.(\.
sufficient conditions
and arc exjiressed
primitives of the model. \\c slinw in paitirular that even
when
in the
economy, only that which
is
the dimension
dimension of the Brownian
identical to those j^ut forth in the complete markets case
information
when
wc address
we require
arc virtually
entirely in terms of the
the invcsinr ha^ access to the whole
reflected in the securities price system
sti'ictly
is
relevant to manufacture his liest consumption jtlan.
fnr lii'
nuliiniing fsoarrli a'^^istniirc
I would likr to tliank (."^lii-fu Hiiniis. SInnn Srlinnl nf Maiiacmf nf
and imtirnrc
Inm aNn r«])prially jndo}>trd in D A^' Stroork fr"in i]\r Nfatlirinaf ire Drjinrtiii'^iit NflT. witli whnni iiiaii^ if not nil of tlif* proofs
worked owf Fiii'tu'id 'iiipporf finm flio n:iiik of Fi uir*' is jiUo pr-'^tofnlh
ill tlii*; iiajt'^r w rro «iirrn««;fiill\
know If^dii'^d. All
(
.
.
.
;tt
prrois r^'niain of rrnir<in rm" own.
Introduction
1.
Optimal consumption portfolio
been stvidied
work
in the
Merton
of
Karatzas
undn- dynaiiiir
[1909].
et al. [198G).
and found one
[1971],
A common
as a
two prol)lems with
mathematics
feature of
means
tlie
contributions in this
consumption
The
Brownian motion.
one
first
For instance
can easily
jioiffdliri jiolicies
In'
on
first
to solve a nnn-linpar partinl differential ecjuation, sn that
Another possibility
is
that
to invoke directly
it
is
already emphasized, this assunqitif)n
is
However,
when
financial
Ijut
that
An
it
that stochastic
tlu^
Cox and Huang
and Huang [1987]
in
securities
is
dynamic ])rogramming
Im^cii ])ut
furward
This aiii)roarh takes
behave
Krei)s [1979], then extended by Harri-^on
is
that there must exist
some
So the proof
under
Cox and Huang
[198G]
as
frictionless.
hideec'
no constraints on the amount
clearly iniboiinded
is
in ))lare of
and thus
a very jjowerfid technicnie
the dynamic jirogramming:
like
maitingales was
Chaml)erlain [1985]
advantage of scune tools develojied
full
martingale jirobability theory. That a nnilti])eriod seciuities market
its
and to check
in control theory,
[19SG]. (1987] aiul Tliska [1084] in i.nrtf.,lin theory, or
general efjuilibnuiii
the discoinited prices of
One
general consunqition jiortfolio ])roblem iiosed by economists.
alternative aiiprdarh has recently
notably.
in
i?
not best suiterl td
is
])ut
will
construct a control in
markets are
invest in each security, so that the admissible set
Our conelusinn
nnrt-rnw]i(irt
many economists
unfortunately one has
this
some existence theorem
tnipalatable
given
asstimed to
is
iiulcrfl ojjtimal.
To do
markets with no transactions costs nor restrictions on short-selling
money one can
is
it
vci'v flifficult to
a compactness assunqition of the set of admissible strategies.
of
in the last reference
to ran^tnirt a control
is
characterization.
its
when
an issue under general circumstances.
afterwards— with the vrriBrati<<n tlimmii of control theory
of the existence of a solution hinjies
There seem
Seconflly, the very existence
inteii^st.
dynamic i)rogrammin<j
Ijy
the use of stochastic
field is
therefore uncertain whether
It is
paper
a recent
inherent difficulty, especially
its
is
most situations of practical
in
to solve the existence lunblcni in
general
in
quite elaborate, even though the secin'ities price process
continue to use this technique
way
rontinnon? time have long
to derive the oi')timal ])olicy for an individual.
approach.
this
is
follow a simple geometric
of ojjtimal
in
achievements
of its latest
non-negativity constraints on constnnj)tion are involved.
aljove, the
and
)iiir(Mtainty
economic? literature. Tlie traditional approach originated from the pioneering
dynamic programming
to be
jiolirics
first
arbitiage free whenever
is
foreshadowed by Harrison and
and Tliska [1981] and Duffie and Huang
nu^asme under which the
jirobability
[1985].
The point
risky s<'cuiitie? earn in average
the rate of return on the riskless bond. (This would certainly be true under the original probability
lieliefs if
agents were risk neutral.
What
risk aversion.)
However
a ])ar with
dynamic inograiiimiiig
financial
comjilete
its
markets are
when
the
as
it
staiuls. tins
<lyn:iini<-;illy
number
the change in jn'obability really does
new strand
relative,
roiiiphlr.
because
Roughly
of risky securities
is
of literature cannot
it
rests
sjieakiiu',.
ecjual to the
'
on the
]>v
is
that
it
subsumes
(pute considered on
critical
assumjition that
markets are said to be dynamically
diniensioii
of uncertainty.
The
goal
of
tlii?
i)aper
i?
to
fill
pap hy
thi?
dynamic programming, and more
sporifirally to
show how an
approach
is
oxistrnr*^ jiroof
can bo derived when
Needless to say, we l)elieve that the martingale
markets are dynamically incomplete.
financial
hoped
thn martingale apjiroarli as an alternative to stochastic
extriidiiip;
more appropriate than the dynamic i>rogranuning one
and
in this ajiplicatiou,
it
is
that this paper will contribute to the idea that the former dominates the latter, both in
terms of their respective tractability and of their
int\iitive appeal.
Before we delve into the arcana of probability theory, we jnesent a
lirief
summary. This pa-
per takes the continuous version of the canonical consumption portfolio problem and examines
the behavior of an investor
who endeavors
preferred consumption plan. His prol)lem
to design a best financial strategy to achieve his
maximize
to
is
his exjiected utility of
most
consumption over
time, subject to a budget constraint which equates the agent's revenue (capital gains on assets, div-
somewhat
idends earnings, endowment) and expenses (consumiitidu. net trading nn stocks). To be
more
explicit
single
about
good — taken
oiu-
framework,
rerall that there
to be the numeraire -available for consumjition; there
the ternunal date too,
all
reasons we
Init for ex]iositionaI
confingrnt rjaini or mnf^uinptinn pinu
across
T
a finite horiz.on
is
states of nature
and
at
all
A
times
the space of niarkrtri]
in assets.
income
all
means
wdiole consuiiq)tion sjiace or.
over time that the investor
if
one
niiglif
sjian
that the space of
jirefcrs.
at
Then
a
consumption good
a decision regarding the continuous
There are
M
risky securities, posited
is
riskless.
are either
consumed
or reinvested.
and
Finally,
the contuigent claims that can be manufactured by
withdrawals from the portfolio durin? the time
f/yiiajjiiVai/y inrniiiplcte just
]iresently.
entin^ly generated by the capital gains
is
The proceeds
consists of
/'iin'//es
is
economy and a
and a bond, whose instantaneous rate of return
Exce]>t for his initial enflownient, the agent "s
dividends on his investments
stratof^y
it
a distribution of the
to time T.
trading of the existing securities from time
to follow a vector diffusion juocess,
liy
this
may be some bequest
not bf)ther about
will
characterized
is
in
that
[n.T]. Then to say that capital markets are
marketed bundles
strictly
is
included
are distributions of the consum]ition
tlier(^
want to jnck but which cannot
l)e
in
the
good
delivered by an apjiropriate
trading strategy.
We
first
will
M
call
question we have to address
incomplete,
when
i.e.,
literature obtains
lying
the space of maiketed binidles and A'
when
is
uiidei-
whirh circumstances
.M does not coincide^ with
the
number
Brownian motion which
representation theorem
tli.-\t
M
X.
of risky securities
generat(\= unrertainty
any
The
It
is
iiiarki^ts liajipen to
ty])iral
tli(Mi
is
trnr
when A/
consumption
is
less
than
sjjare so that
A',
.M
N
of the inider-
dimension
be shown
M
be dynamically
so far in the
~
X
martingale
usinp; the
attainable tlirou<rh
trading strategy so that markets are dynamically compK^te and
some
This result
ap]iioi)riate
is
hi this srroiid case, there are too fi-w secui'ities to si)an"
becomes
strictly inrlufled in
A.
The
model considercvl
etjual to the
ran
contiiip;ent rlaiiii in A'
whole consumjition space.
thi'
no longer
the entire
Not surprisiuRly.
cnnsofinpurrs in tniiip of the infoiination ptnirtuir a? well.
tlicrr aro
couveniciirr the infomiation set (n-alpehra) peneiate'l
With
as the "priee information set".
information
tin
liy
system
securities jirire
rlynaniirally eomjilete markets,
it
is
For
referred to
turns out that the price
coincides with the whole information in the eronomy. so that in particular any
set
process observed at the current time will he a function of the past and present realizations of the
securities price process.
become incomplete.
This
The
surely a strong re(iuirement, that
is
relation
more subtle {S stands here
little
T
inclusions
contained
M
C
is
it
can hv shown (though
this
M
becomes then a
Namely, one has the
strict
which are outside the price information
not so easy to j)rove) that
is
Hence
oiu' first
all
i)rice
is
set.
the consumption
ol)servation has to bo that
maximization i)rogram ha])])ens to be
borrow fiom ('ox and Huang [198C)
shall
Their idea
]irog;iam.
to the sjiace
restrictefl
M
of
is
to
marketed
to bear martingale theory to transform
already been studied by mathematirians
prices in units of the riskless
interest rate
if
a
measurable, then
ex])lf)de.
is
j)ositive
The
bond
This
their solution roncejit with regard to the
end)ed the investoi's i)ioblem. wliirh
l)undles.
dynamir problem
a ccunplex
Before we do this we find
is
without
it
intec:;i-aterl
financial gains
tlie
efpial to tlie
/
sum
c'{f)ilt
= W'(0) +
that manufactures
r
The fhscounted
in
sum
/
financial
r.
The
of the (discoiniteil) cajiital stains
for the entire
jirice
this set-uji
manufactures
all
of the initial wealth
and
of the
For any consumjition plan
gam-
7r(c*)
,
Jo
denotes the discounted flow of consumi)tion. n"(0) the
of financial gains the
because
convenient to normalize
write accordingly
.'n
r*
one that has
liudfet constraint then stat(\s the familiar condition that the present
on the risky securities
where
into a static
and bounded from above, so that the juice of the bond can neither vanish
present discounted value of
we can
by assumjition
because we assume that the
loss of generality,
is
Al.
is
above
the whole eonsumjition space, and to bring
in
discountcii value of the stream of consumption
G
markets
automatically marketed.
maximization
c
T
set
letting
Because investors have access to more information than that which
the j^rice information set are market(>d.
in
We
nor
and the juice information
for the vector of securities prices).
jiartirular solution to the investors
it
M
system, there are bundles in
in the price
Conversely however,
goods
C X.
between
we can relax by
initial
wealth, and the integral
and dividends accriuuf: to the strategy
time sjian (n.T].
of the
above consnm])tion sti-eam
the initial investment
the only aiiwnuit
is
at
time
f)
is
ecjual
required by the strategy that
agent's maximization jirotri'am can thcicforc be foiniulated as
max/?
/
^i(r{t).f],lf
s.t,
4
;r(c*)
to H'*(n),
= n"(n)
where u
time
at
some
is
We now
0.
and n the linear
utility funrtion
extend
maximization jirogram
this
mi^xE
where
7r
over
X stands
of
all
as before for the
X. When markets
I
u(r(t).t]dt
in the following
4,(r'
s.t.
prices should one ascrilie to a stream of
(ii)
What
ensures that a solution to the extended program,
question arises from the fact that
commodities have their
tliat
is
extend w over
all
jirice
of A',
system,
i.e..
determined
jiy
it
consumption that
M
when
])riori
such that the shadow pijre of consunqition
this jiarticidar valuation,
always be chosen tn
it
turns out that a solution
lir jii'icc
c
expectation unrler some probability. Let then
of the price
it
preferred as
Q
marketed?
in A',
an abundance of
But
c.
c
is
it
to the r>xtended
when
Thus, as far as the ojitimization juolihun
be the
this,
we have
to recall a result
Q
under
with respect
that
it
is
In
as least as
should be clear that the two solutions
it
concerned,
with our choice
measurable by construction.
jirice
is
r
thei-c^ is
is
strictly concnv(\
nothing more than the securities
prices themselves that the agent needs to oliserve to devisi^ his ojitimal strati^gy.
enrlowed with a very rich information structure, only that which
is
With
maximization program can
])robal)ility associated
the utility function
is
4>
of any contingent claim can he written as
optimal by assumption, so that
are in fact indifferent, or even identical
functiouals
the jirice information set.
itself in
is
same budget constraint and
satisfies the
]irice
iiieasiuable with rep])ect to the price
is
The new consumption plan
can be shown that
only the marketed
any one of them. However, one candidate
measurable valuation, and take the conditional exjiectation of
to the price information set.
addition,
is
included
and thus marki^ted. To see
ineasuralilc,
from option juicing theory which states that the price
its
two questions:
not marketed?
is
any, can he
stictly
is
the (unitiue) one which
is
if
There
arl)itrage.
and one could choose a
of sjiecial interest to us:
(0),
a linear functional that extends
is
<f>
way
are incoini)Iete, the ahove emliedding procedine raises
What
first
W
^
)
whole eonsnmption space and
(i)
The
marketed goods
fniirtional that gives the priee of
relevant to manufacture his ojitimal consiuuption jilan
i.-^
contained
The crux
Even
in a
in thi' capital
of the
argument
is
world
markets
that an
investor can loose nothing in terms of his expected utility by j^rojecting the whole jiroblem onto
the information generaterl by
juici-s.
An
many
given as follows. Since there are infinitely
claims outside
M.
the investor
may
l)e
consumi">tion spare subject to an infinite
valuation which
price
is
the consti-aints are slack.
Hence
c is
But
all
indeed a solution to the
valuations which give the
seen as maximizing his exjx^ctiMl
number
in the pi'ice inf'>i'iiiation set
measurable and thus maiketed
works can tentatively be
intuitive accoinit of jiow this
Now
of budget constraints
iitility
the contingent
over the whole
Simi)Iy jiiek the jiarticular
and observe that the corresponding sohition
exi)(~cterl utility
the valuations agier on
inv(^stor"s
jiriccs of
maximization
can
M
In-
improved ujmn only
so that
])r()blem.
all
if
c
is
some
of
constaints are saturated.
Til''
dynaiiiir
rost
of this jiajirr
ronsumption
markets. Wliat
is
is
orKanizcd as follow?.
jiortfolio prolilcm
new about onr
rrs\unrfl aiui
is
ajijjroarli
In scrtinn
is
that
2.
siiiijiiy
flir
framework
fraditioiial
cxtriKlcd to allow for iuroniplcte
we rhararterizc the
set of
marketed consuinption
biuidles entirely in terms of the orip^inal ])rol)ahility beliefs lather tlian relatively to
reference mcasur(\
The
specifying the null sets.
nice fact about this
The other
is
that beliefs do jirovide something
features of the morlel are standard.
martingale representation theorem to show that
This
is
all jirice
some arbitrary
more than merely
In section 3
measurable juocesses are
pose the investor's maximization in-oblem and directly extend the
The
last section
we use the
in fact
important since otherwise our existence proof would not go through. In section
our economy.
for a
Cox and Huang
4,
marketed.
we formally
[198G] results to
concludes witli a very simjile example carried out for the class of
constant relative risk aversion utility functions.
The Model
2.
111
Before we can define the dynamic maximization
forniuiated.
be useful to characterize the economy
and the trading
securities,
strategies
terms of
in
its
process will be
witli diffiisiou price
faced by an investor,
i)r()l)lem
it
will
information structure, the price system of
its
agents can afford.
its
The Information Structure
2.1.
common
Agents share a
future
information wliich
is
revealed continuously thiough time
They have no uncertainty about the present
surprise.
gradually resolved over time; at
is
<
= T
or before time
Taken
number
=
Ti
The following mathematical model
as a primitive ate a jirobaliility space (Q.
positive real
we take
t.
(t[)J'(,'')
t
<
:
,«
<
to
<]
We
7t-
use
the restriction of
Z
to
[0.
f
:
]
0)
t
PM
X n
there
is
no
about the
their uncertainty
We may
think
can occur at
gives these notions a j)recise meaning.
7 P)
.
and a time span
T]. where
[O.
T
Brownian motion defined on
liy
11'
uji to
time
is
it
some
where
Furthermore
t.
to denote the rr-field generated by the progressively
X
/?
and
—
as the collection of events that
the rr-algebi'a generated
lie
(A funrtion Z
measuiable processes
for each
=
Let {W{t). Ti.P) be an N-dimensional
7 =
assumed that
is
{t
they learn the true state of nature.
of the information structure at an interniediate time
it
time
this section a iiiodrl of scciiritirp inaikots in fontiiiuons
is
—
17
>
/?
is
said to be 7t-prnf^rrsfiivrly nicasrira/»/e
X 7)-measurable. where
Bi
Bi
is
the
rr-field
if
of Borel
subsets of \0.t].)
All the processes to
processes
appear
proiiosition addresses this
to be interpreted as
<
,'
<
f].
be progressively measurable, hi our framewoik, the ^/-adapted
Chung and Williams
(cf.
Proposition
will
2.1.
If a
){'
in
ratliiM'
our
As.'JiijHr-
prnrrss
Proof. See Stroock and
is
are
[1983], p. 8)
/^-]irogressiv(~Iy
pednntir measural)ility
<iui-stioii.
measural)le.
The following
whin'e the generic A' ]irocess
is
setuj).
A'f
.w)
ftilnptrij
/.<;
ti<
ri^lif rontiiiunu:^
7i
.
thrill it
i.s
Varadhan (1979), Exercise
,t/s(i
f<'i
rn<-]i
w E H rwJ
'IrBiir 7i
— oIX{k)
:
J)li<f::irssivrly njea.siira/i/e.
l.G.G.
I
Another
whose
imjiortaiit
descrijition
is
infniiiiation
given next.
stiuctuie
is
that genernteil by the secui-ifies ]irice system,
The
2.2.
Wr
ron?i(lcr
liy rn
=
System
Securities Price
0,
?\
market with A/
seruritir?
The market
A/
1
and no transaction costs (hke
dividends at rate (n(0
pays no divi(h'nd and
riskless,
and
r(t)
and
r(.T.
f )
/?
:
X
[0.
is
T
—
sells at
T?"*"
>
]
precisely,
is
it
= ^^p{Jo
D{f)
foi-
i{S{.'').s)d.'^
/
=
they
:
P^' X
h(S{.o].s)ds+
S({))^
T
\(\
I
of full rank for
r and
all
\n(x.t)
n
is
x.y
E P
a matrix, then
that S{-.ui)
The
it
au'l all
is
\rr\
(.
We
G
[n.
-
V
T and
]
:
\h{x.t)
sub-rr-alpeliia generated by
all
])ri('es
is
<
chosen to
\\c jiriee
terms of the current and possibly
We now
use Ci{i\ to
(locally)
that both in{f)
t)
R
be bounded from
economy
are as.'sumed
jirocess.
More
fl(>iiote
{a.s..P)
(1)
7^'
K
\V<
We
liavc usivi the following notation:
nn
measui-able
jiast
the
l)y
is
a countable
p. 94).
all
are adajited to
i)iic(-s
show
of
y^^'^'''"
Jiieasiirahie
7[.
Let
if
it
continuous] with respect
at
time
tinu'.':
/
at
can be expressed
or before
tiiiu^
i.
that the best consumiition ])lan for the ag(-nt
Thus, his entire stiate<:y
])rice
system.
t
Sit] 4
lie
riplit
number
if
Tliese conditions imply
.
said to
is
5
since
Jtlnrrs!^
at) =
Since
Jf
measurable variable
values of the
^:\)]i'^
''
)
Stroock [198G].
a jnocess
ada])t('(l
jirice
will
(rf
i/|.
peiKMal strictly contained in the orip'inal one.
the following definition:
Stroock (1979). Exercise 1.5.C)
])e
is
P^' X [0,Tl —^
:
Vte[O.T].
< K\x -
h{y.i)\
d<Mioted
in
is
Mathematically, we have that any
can always
-
measiue zero
as a fuii'tion of the ]incr ])roc(-ss evaluated at
(cf,
—
pays
assume that
denotes the Hilbert-Schmirlt norm (Ti'ace
progressively measurable (or. etjnivalently.
7f
Asrents in our
risky,
P^' X [0,7] -> n^'""^' are Borel measurable, and
?f)me constant
follows that this second information set
to
We assume
rT(.S'(..).,9),/H'(.^)
I
shall butlierinore
n[y.i)\
unifiue uj) to a set of P-
us adopt once and for
is
f
bond
i?
Jo
-^ 7?^' and n{x.t)
]
indexed
satisfying;
\n{t.O)\ V\h{t.Q)\
all
— the
on the law of the price
all afcree
,
a ripht continuous, /f-pro^ressiveiy measurable, (a.s.,P)
is
.'n
where h{j.t)
for
A/
I, 2
Security
''(') '''}
away from zero
lielow
in the sense that
Jo
is
t
—
Borel measuralile. Finally, we rerjuire that r{x.
assumed that S E P'
continuous diffusion process
rT{x.f)
time
< N
no constraint on short sclhnp
Security n
for S„(t) ex-dividends.
t
bounded above and
have rational expectations
5(04
time
i?
can be respectively written as /„(.'>(0-0 and r{S(t),t) with i„{x.t)
above. Thus, D{t)
to
will lie frictimile?? in tliat there
"^^
whore A/
lonsj-livcd scnuitirs traclrd.
1
l)rokera;^e fees for instance).
^^"^
'^^'^^
4-
/
li.o]'!.^
will bi- defined entii-ely in
.
and rewrite
(1) in unit? of the O-th sernrity.
=
roirespondiiiRly G'{t)
J^ /*(.«)
(/.i.
G
t
(a..«.,
To be a reasonable
probability measure
Q
Q
assumption on n
aiul
market model
seeuritie?
we need two
before that
liut
2.1.
]rt n
the element o{
of
to /?
honiomorphisms from P'
Lemma
}ir
Lrt n
Next, suppose
a.
G
S'^{P'
N
]
N
X
and Range
,
dnnotr
),
S (P'
tliFit rr
measure when
mFiitinf^a.lr
the set of symmetric non-negative definite
of
P
it
E
such that
We
Harrison and Kreps [1979]).
(cf.
there must exist a
in the arliitraf^e free sense,
[{dQ /dP)^]
an
f^ntififyiuf^
G Hom{P'
.
We
technical lemmas.
Horn
(/?'
some qualifying
P
}
=
use S~^{n
,
/?'
=
na
unci that n
n„.
Then
n.
to denote the set
R
/?'
).
,
onto nangr(a). and
n >—» a"^ /s a //leastiraWe
Then
= n
)
denote
to
)
G Hom(7?'
to denote the range of a
w„ flm nytliognnnl jirojrrtidn of
l<y
We
satisfies those three conditions.
real matrices.
{rr)
< oo and
say that a probabil-
will
the absolutely continuous inartingalr measure? under
all
h.
'
P).
absolutely rontinnous with respect to
momentarily
will elicit
^
Jo
an ahsolntply continuous
is
Z?(.f)
Jn
that (G'(t), Ti.Q) be a martingale
ity
i{t)/D(t) and
f h'{S{s).D{s),s)d!^+ f n'{S(s),D{K)..9)d'W(s)
S'(0)+
[0,r],
=
i'(t)
Ito's fnniiul.i imi)lies that
Jo
for all
S{t)/B{t).
B(s)
Jo
=
+
S'{t)
=
DefiniiiR S'(f)
7?aiifp(a)
= Range
function
{rr
)
and
(T(n'^n) + n'^.
Proof Stroork
We
G
Pio(\f.
II1.2..S.
introduce the notation
Lemma
rri]
lemma
(19801.
2.2.
}
Define h
,
anJ
=
hituitivcly.
.
G Hom{P^' P^'
Let n
Panf^e(rr
rr
m-rtj
(n
=
a)'^
»;
Then
k^.
is
simj)ly tlir inverse of
Then there
).
n^t] fni a//
rr
rr
G
for
/?'
M<'Vei'\'<'V
any
n
exists a uni(]ue
G
r/
P'
rr
i—>
is
rr
rr
ojierating on
G Eom{P^^ .P^'
its
)
range.
such that
a inensural>le function of
there exist?
some
(
E
P'
such that
^oV = ^^- Thxis
rrt]
by lemma
2.1
=
(n
rr)'^
Null
(rr)
= t„t„( =
=
rT{n
rr)
rr
7r„t]
follows from thr fart that fnr two
{n
wliei-e
^
7r„T{
G Range
(rr''")
hi aflditjnn.
rrrrr;
The uniqueness
n
rr
i?
-
the null sjinrr of
rr.rr
G
(Null rr]^
H
The
iiicasui-al
)ilit
n)ri
rr
=
n^n^i]
G Hnm
Null f^)
=
K„ti
p-^')
(7?''^
,
=
and
for
all
t]
G P^
.
0.
y follows alsii diivrtly
rr
from lemma
2 1.
Remark. Whrn
hccomrs
Wo
rrrr
=
a
in
<
rr\S(if)
7^/, thr hf-rJimriisinuHl iijrntity
to find
])o?iti(>ii
s
<
(i.e.,
t]
assumption on
Assumption
X
=
onto, n^
matrix, sn that
above relation
tlir
thr absolutely rontinunii? niaitinj^ale measures.
of
whirh
a(lai)trd
is
We
to the a-field
All our results will be subject to the following
price measurable).
and
rr
all
many such measures, one
that there are infinitely
(}ualifyinf;
7?'^
is
>].
now
arr
shall see
T,^
=
r]
n
h.
2.1.
Lrt K,(x,t)
=
this
assumption
is
—
—rr{j.l){l)(x,t)
Then k
r{x.t)j).
bounded
js
for all {x.t)
G
io,rj.
Note that
Brownian motion
a? in the
model
Proposition 2.2. Suppose
Nikodym
derivative
R =
=
f,
satisfied in particular
—
Q
when 5
follows a multii)licative geometric
considered by Mertnn [1971].
first
Thm
an al^sohitrly mntinunus niaitini^alr niras[irr.
is
its
Radon-
dQ
v{T)
=
is
equal to tiN where
cxp
I
I
k{S{.].
dWi.")
..)
-
1/2
|«(.S'(,s),
I
,)f
d."
I
and
N
= N{T) = exp
/ r{S{s).
dW{s) - 1/2 f
.0
|r(,9(,0.
.')|^ d.o
\
.
I
fnr
snmr
In this
Prnnf, R{f]
Dy
E
,>{t)
Null
rrif) siirh
theorem
= E
/n^i
''(.*'(''), ')|^'/.''
I
.7J
)
is
a inartiii<r^)r nnrler
the martins^ale rejiresentation thmi-t^in (rf
4) there exists
<
as well as in tlir scnjuel. the notation
7?
(
E^
that
a G PML'^iX X T;
/?"•')
P
OO.
y will
.t
whirh can
lie
stand for the inner product
taken to
for instanr(- Kunitrv
lie
continuous
(a..'..
P).
and Watanabr (19C7). Section
such that
R{1)
=
1
+
^f')
/
dWi.-^),
(3)
Jf)
where
PML
Lebespue
(cf.
[X
X P) denotes the
But
iiieasurr.
Dellacherie and
d{R(t)(r{f))
Hence
liy
{f^'{t). Ji
Meyer
JJ-iiin^ressively
.
[1982],
Q]\<
R(t)dCr'{t)
=
R{t)h'(S{i).t)dt
arpumnif
martingale
Lemma
=
a standarrl
^
VII 48)
L
(A
X P] and A
(R(t](V(i].
J,.
P]
measurable jirocesses
if
and only
if
By Itns formula
of
is
is
a martinfrale
then, we have
+ (r{t)dR{f) + dR{t)dn'{t)
it
-¥
must
R(t)<7'{S{t).t]dM'it) 4 (r(()dR{t)
lie
that
R{t)h'{S{t).t) -+n'{S(t).t)n{1)
10
=
+
the
n'(.^(t).t)a{t)dt.
(cf.
Jacod (1979), Proposition
1.43), or cqnivalently
R{t)(h{f)-r(t)S(t))
where
rr(t)n(t)
=
0.
for brevity h{t) stand? for h(S{().t) anri similarly for other varialiles.
To
lemma
solve the last equation, note that by
r{t)S{t))
e Range a''"{«)
n(t)K(t))
=
such that -(h{t)
some
i/(<)
{R{t)K{t)
+
G
Null
rr(<)
nrA/L^(A x
Now
dW{t].
i'(0)
formula to logr7(/) shows that
let
=
r(t)S(t))
rT{t)K.{t).
=
=
solve dt](t)
—a(t){h{t)
then follows that (T(a{t)
It
—
-
1/(0
a
fuid dR{t)
d)V(t).
K{t)
T]{t)
we
in (3),
An
statement of the theorem with
as in the
again K{S(t).()
2 2
+
R(t) K(t)
P). Substituting for
t}(t)
t]{t) is
Note that by lemma
in [0,r].
-
=
unique k(()
2.2 there exists a
so that the general solution for a disjilays the following orthogonal decomposition
n{t)
for
+
a{t)
dlVit)
=
application of Ito's
T
irjilaced
by any
t
meas\u'able and that by the
jirogi-essiveiy
is
=
boTUidedness assumption of k we have
K{S{t).t]\
<
dt
oo;
(4)
/:
hence the integral defining
Now
put iV(()
=
t]{f)
/?(0
We
.
well-defined
is
have
(rf.
Stroock [198C], Section
to sjiow that ti{f)
is
II. 3).
strictly positive {n.s., X
X P). For
this
it
n(t)
suffices to
demonstrate that
rT
fT
/ K{S(t].t)
Jo
But
dM'it)
-
{n.f..
P]
(4) implies
s(t).t)dW(t:
cf.
> -oo
1/2 / \K{S{t).t)f dt
Jo
Lijister
('I.."..
7^)
and Shiryayev (1977). Theorem
By ItnV formula
dN(t.]
=
R{f)
This, toj^ether with (4). ensures that t]{T)
IRlt)
-^rr'^nU) +
>
+ -)
= - dW
dM'
-
N
1
,
i/2^{dn{t)f -
t]^{f)
riii)
K
oo:
again.
dRU)
—
i- -
= {N
7.1.
<
-^j-'ir^if)
n^(t)
tr(t]
1'
t:
dW +
N\K\'dt -
K
+
(
N
JMf)
dt
dt
•
Thus we can write
dN{t)
since r
and
k are orthogonal.
The
=
N{t)i''{t)
dWit).
exi)ression for A'(') can in turn
ajiplied to log A'^(/).
We now
jji'ocerd to
show
tjiat r;(/) is
jince mea<=urab
11
(5)
bi^
derived from
It6"s
formula
Proposition
Proof.
By
The
2.3.
process^
ri{t) is
prirr inrHsurahlc.
proposition 2.2.
/
logr;(t)=
dW -
»c(.S(..),.s)
LemiiiH 2.2 phnw? that the srroiid intrpral of the
approximated
sum
as the
K{S(t)j)
hand
riplit
it
|^/,s.
;.)
measurable.
$\(]c is ])riro
Consider the
of i)rice measvu'aljle functions.)
E Range
definition K{S{t).t)
1/2 I |/c(5(,.).
integral.
first
(It
can be
Since by
(<),
dWU) =
dWit) = K(S{t)j)n„Td'W{t).
7r„T«.(5(0,<)
But
= n^(<TrT^]^ndW(t) = a^ (rrrr^ )^
n„jdM'(t)
and since S(t) +
I
i(S(.'>). s)
df
—
h(S (,<<),
I
i^)
dit is
(^dS{t)
+
,{S(t).t.]dt
-
h(S (t)
.
t)
dt^,
certainly price measurable, the result follows.
Jo
Jo
I
From now on
be denoted by Q.
sets.
It
the jirnbability
Note that since
remains to check that
i]
whose Raflon-Nikodym
r;
is
strictly positive,
T-square integrable.
is
diu'ivative
{(i..<>
.
P
P).
t]
i?
and
jirice
Q
measurable
will
have the same null
In fact, a stronger result holds
imder our
assumptions.
Lemma
2.3.
Prnnf. Let 7
ajci r;~' arc
t]
>
0.
Tj"
We
jjj
L'^{P] fnr any q >
have
=cxpl
I
qr
-/ir
(V
= exp
-
'/H'
1/2 I 7|K|'r;/|
-
I
a martingale
Since k
is
bounded
(say by
denotes the norm of
finitr.
rj
K). E'^[if]
in L'^(P)
^ ly?
1
^li^ jy\^
dt.
I
P
under
< max|
-xp
'/'
expf
"^
^^"
The same argument works
T
A''
j
for r;^
\
<n
\\i]\\^,r
< 00 where
||r?||,,r
.
I
Wc
turn
now our
attenti'in to tradiiif' strategies.
12
2.3.
The Trading
Strategies
The ronsumjition Fpacc
for the ngcnt^
is
PML^X
X P) X L^(r)
= L^\O.T]
where PML'^{\ X P)
A
alt)
t.
^,,,(0 are the
bond B{t) and
an
is
x P) x L^[^.7,P).
M+
1-vector process ('>,")
=
{(it(0.
numl^er of shares of the 0-th and the m-tli
Thus, the vahie of the
of the
U.PM.X
the space of con^umptinn rafe processes and L^{P) the space of final wenlfh.
ip
trading strategy
and
X
stratep;y (a.
{fi,„{t)\
=
rti
at
/?)
time
M)
I
W{t)
=
1.
,
seciu'ity. resj^ectively,
Af}, where
held at time
shares of the risky securities S{t). or
+
a{t)B{t)
For a progressively measiualile process ^
^ =
that nf the portfoho consisting; of q(() shares
is
(
'',ii('))-
E
/?'
(!(t)- S{t).
we
.
make use
constantly
will
of the following
notation:
where
as already
mentionned
||()||,,,r
t^ia. P) =
Note that C^(G P)
.
A
(i).
(ii).
is
|/?
\r{t]\'dt)
/
v-r
r-r
denotes the L''{P)-
e PM{X X
Accordinf:;ly
noi'iii.
let
lll^^^llkr < oo}.
P: R^')
entirely defined in terms of the prol)abi]ity P.
trading strategy
is
said tn hr
.Tifniissi/i/e if
fteC'(rr.P)
There
(7)
phn (r.W) G PML^{X x
exists a rnnfiuinj:fion
a(t}D(t)
+
0{t]
S[t)
+
P) x L'^{P] such
tliat
r {.'<), is
I
Jo
(8)
= a(n)i?(0) +
^(0)
S(0)
+
+
(^n(,'^)dD(s)
0(s}
r/(7(,9
I
for
all t
e
[0.
T]
{a..o..P)
and
ir
The consumption plan
[rt.fl.c.W)
is
is
= a(T)D(T) +
(c.lV) of (S) and (9)
S(T)
0(T)
is
said to he a srH-finnnrinf; sfrafep-j
said to
(rf.
]io
(a.s..
finnurrij liy (n.O) anrl the quadruple
Cox and Huang
just the natural intej^ratrd budsret constraint. Fiisf <i{t)D(t\
W(^altli
valui'd at the current ])rice of the existing securiti(-s.
of consuni])tion
from D to
t.
Of coiuse
rt(f))Z?(())
+
13
P)
^(0)
S{i^]
+
"(/)
The
=
W
[198G].
.S'(/)
=
integral /„
(i))
is
ji.
W
5).
Condition
[t] is
((.•>]
fh
(ii)
the financial
is
the stream
the initial wealth that the
.
a^cnt invrst? at
tiinf'
=
t
The qnanfity
0.
+
J^(n(.*) dD{.'')
capital gain? of the portfolio (n. 0) raniod from
to
the final wealth wliirh ran he viow(-fl a? a l)e(inrst.
is
W
wliile
t.
^ a{T)D{T) +
Condititm
strategy depends on the state only through the information availalilc
constraint
— and
that J^
({(''{/>)
fl(,i)
is
can bo interpreted as the
dG(.'^))
0{.<>)
an informational
time
at that
a well-defiiied stochastic integral, as
is
the trading
that
iinplir?
(i)
= W{T)
0{T)- S(T)
shown
in the
next
lemma.
Lemma
2.4.
Let
Thru
satisfy (7).
L
E
an L'(Q)-j/iarfuifa/r for any q
is
Ptoof
lit 4-
h'
—
rr{h
=
Define X{t)
dW —
n'
= —k
rS)
nIC'
is
—
+
rS)dt
we have
hoiiiidrfi,
-ft
.
We
(.>].
r/lj'j.
rr
I
By
[1.2).
d(r
/o'^i")
da'{s)
(i(s)
h
The
first
pmve
stochastic integi'al
— rS = —nn. and
{h^rS)dt = -
X{T] G
that
is
.1
n"
the Cauchy-Schwar?, inefpiality and the fart that ^ and
sum
thus the
term
tlie first
I
L'^iP).
of
By
(2),
dC =
two terms. Since
is
Kdf.
ft
D
nvv lioinided,
<
h"
L'^{P)-nonn
its
is
such
that
<
K dt
'
(t'
K
for
some constants
A'
rr
So X{T)
7
e
€ L^(r) By
I>rt
^
stochastic
lie
D
•
'/^"(.«).
and lemma
T,.Q]
is
fact
u/ijf.s
B
at tiiiir
f
oo.
< oo.
r
2.3 tins implies that A'(r)
of the nnriii,
we
{>i
.
.
r
\^'
.
see that
]
^
G
[1.2
By the
.
is
G
L''(Q) for any
).
linearity of the
a linear sjiace.
the juiec-s in units nf the O-th seenrity (the Ixnid),
all
about strategies
of
n^d
an L''((V)-martin'Ta!r fnr any 7
(cf.
("ox
and Huang
Lemma 2.5. Suj^pnsr (n.ft.r.W) G ^ and H
r' {t) = r{t)/D(t) drnntc tlir nnrnializrd wraith
{r.,W) in
K
12.
and the suli-additivity
convenient tn normalize
well-known
<
II
denot<' the sjiare consistinp^ of tlie qiiadiMijilrs
intefriv'il
<
2.r
?.r
H.'.ldrrs inociiuvlity
(1.2) so that (/„'^(.')
a^O
7.r
./o
Fur the s(>cnnd term
and K'
./n
Kdt
e
2.r
.(-
..),/.«
+
\V'{t)
=
\V{t]/B(t]
ami rDn^niujitinn
ir
14
will
record a
(198G). rro])osition 3.2).
is
EQ
We
It
T,
-^"'•(0.
at
=
n{t]
tiitir
i
+
0{t]
S'{t)
and
Then the vahir of
fl.cW) E ^-
Proof. Lot (n.
-
dW'it)
But
+
d(n(t)
and
+
+
S'{t)
dfl(t)
S' {t)r{t)
sinre dfl{t)
=
S'(t)]
^(<)
dW'(t) =
=
dt
de{t)
we
+
da{t)
rf^(/)
+
S'{t)
•
+
dS'it)
0(t)
dS'
dO(t)
{t)
D(t) one find?
l)y
+
idS'(t)
+
S'(t)r(t) dt)
=
c'(t) dt
0(t.)
/*(() dt
get
(dS'(t)
()(t)
that
iini)liop
and dividing through
differentiating (8)
dn(t]
fonimlH
Ito's
+
-
i'(t) dt)
=
c'(t) dt
dG'(t)
0(t)
-
c'(t) dt.
Eqnivalently,
W(t)+
for
ail
under
t
€
Q
i^.r). (a
.«.,
/r*(.s)f/,'.
=
+ [ 0{.^)da'(s)
iy*(n)
P). Writing the al)ove equality at
with resjiert to
we
7J.
= T and
/
taking eonditional expectations
onre we have taken
get the desired result
(10)
and leinnra
(9)
2.4 into
areount.
3.
Marketed goods and the price information
marketed consuinjition
In general, a
to notier that in the defiiiitiriu of a strategy
ineasiu'abie with respect to T,"
information
in the
Tlii?
.
need not he
]>lan (r. \V)
economy. However,
nowhere we ass^imed that
natural
is
if tlie
if
we
insist that
wage income flow
measural)Ie consumjition bundle (c.ir) can be manufactun-d
This result
tools
is
aimed
1-ie
at
showing that the
at
f
=
set of
marketed bundles
a by-in-ndnrt, wi' will also
price measurable.
Writing (10)
T, one sees that,
foi-
is
liy
is
t]ic
r'(t)dt
I
suffices
to be progressively
agents have access to the whole
itself i)rice measnral:)le,
any price
an ajiiuojiriate trading strategy.
4.
we present the
hi this section
sufficiently large to contain
that
k'''
it
inartinp;ale mi'asure
all
Q
is
the price
the only
in section 4.
a strategy,
= W; +
.'n
W^
is
had
this,
This certainly restricts the chnice of solutions considered
W+
wdiere
y
essential for the existence proof develojied in section
measurable bundle?. As
one to
measurable. To see
jiriee
()(t)
drr'(t).
I
./n
the initial wealth in units
the
f)f
0-tli
security held by the agent.
This prompts the
following definition;
M
= Ixit.w)
By lemma
th(^
])rice
2.4.
:
X
=
A'n
we know that
it
measurable functions
consumiition goods
will
+
is
<>{
then follow
/
^(0
'
dn'{t)
for
some
a subsjiace n{ L'^(P)
L'^{P).
The
relation
easily.
15
We
X,,
G
/?
and
will first
between
]ii'ice
fl
G
show
£-('.'.
that
M
T)
contains
all
measurable and marketed
Lemma
3.1. Fiupposn
constant
(^
surh
X
€ L^iP) and
H
X, = E'^\X\T,\
€
t
T],
(0.
Tlmn there
exists a
t}iat
<
\\X,\\2.r
From the Bayrs
Pronf.
f«i a//
rule
Jarod (1979). Lenuiia
for iustanrc
(rf.
WXlU.p.
('
7.9)
and the
fart that r]{t)
is
adapted
X, =
-^E^inx
E^\-^x
=
7,]
I
But
—
;;].
I
n(t]
nit)
rT
= exp(
K
/
dW -
1/2 /
Kl,
tl{t)
dt]
\k\'^
./(
••
'
so that
(-^) =
By
exactly
some
K
tlie
same argument
exp{ f 2k
-
f/H'
•
/ |K|^/f} exp /
2
E
a? in leiiiiiia 2.3,
(i]/t]{t)]
Thu? hy the (laufliy-Srhwarz
majoriz-ing k.
<
i?^ (r;/r,(())'
< r
or. takinp:
< ,(r-OA' <
T,
,r/v-
^ ^
f^^
iiuMjuality
1/2
|A',|
|K|'fi«.
ll/2
/?'
7,
X'
r,
l'/2
A'^
e:'^
r,
norm?.
\X,\\2.r <('\\X\\2.i
Lemma
For each
3.2.
X
G L^lf) thnr
is
Xq G
a luiiqur
7?
and
a
imiqitr
px € PML'^{\ X T;
T?''^')
such that
A'
farf
J/i
An = E^\X] awl
tlinr
= A-o4 / px{t)d]r{t)
is
a
C
< oo sudi
III
III
Ilk.v|||2.r
Pn^nf Su]>pose that A„ G
By
is
p
P
aii<i
p
(ii.x..
To
jirr)ve
A
X P).
/''
^
S '
.
a rontinurtus martiiifrale and so A'd
=
tliat
vll
liA||2,rII
G PML^(X X P P^
the Caurhy-Srhwarz inequality |||H||i.r^
Thi.= jiroves the
=
<
/„
(a....P).
)
are
.^ucli
||'7||2.r|||/'|||2.r. i'"
p{t)
dM''{t)
=
uniqueness.
the existenre. consider
S{.>']..9)d.<'
Jn
16
that A'n
(a,,
0.
+ j^
p(i)
+ /p%(0
Henre
dM''(t)
=
'/li'*(0. 7,,
|||/'|||i.g
=
0.
g)
and so
r
the Cameron-Martin-Girsanov throiciii, W*(t)
By
law
Q = r
o
W'^
Q =
Li
=
Q
Q. But
i/^Yli-i
III.4.C).
Thus
But
(1,2), so
if
if
is
to the
-
=
j \,(S{s)...)\' ds]
1/2
flit)
the iniiqne sohition to the martingale problem associated with the operator
is
-ajj
Q
Brownian motion with respect
itself a
defined by
^=exp^.(5(
[j k(S{s),.) dW
so
is
~
extremal for
initiating froju 7,ero at
^""^^
Yli=i '^ig^
=
<
"U^*.
e L^(r) then by Holder's inequality and lemma
Xi = E^[X\T,] we have that (A',. Ti.Q) is an //'(Q)A"
representation theorem
Jacod (1979), Theorem
(cf.
Stroock [1986], Theorem
(rf.
11.2
X
2.3.
By
martingale.
and Corollary
X, = E'^{X]+ [ Pxit)
G L'(Q)
for
any q
G
the martingale
11.4),
dW'(t)
•'0
where px G PML'^iX X Q).
It
follows
rl +
A,+,
rl + /<
ti
-x,=
dW
rx
=
therefore
IrxT'
<nv
rx
J
I
r
1+^
-
rx
K
llfi
I
(if
2,r
7.r
rf+i
" X(
^f+A
+ A
\rx\o
I
i+f
<
Xi^f —
+ AS 1/2
A'/
Irxl'd.f'
12.
('hoosf- f
>
2.r
such that AiS^^' < 1/2 and assume that
2,,„^l/2
<
\rx\'<i
2
|||/'.Y|||2.r
— Xi
Xi^,^
lemma
3.1.
Finally, choose ^
>
1
CO-
Then
< AC
2.r
2.r
1)y
<
so that \/k
<
Tliei
/>
1/2
rx
2.r
<
Irxl'd.^)
/*
IG/rC'
2.r
2.r
or
rx
where C'
<
('•
A
7.r
(12)
2.r
= 4C^/k.
For the general rase, we aiiiirnxiinatr A'
liy
bninided fnurtions
the Leiiesgne dominatrrl cnuvergenrr theorem. A''"'
17
— X
'
in
L'^{P).
A''"'
=
Now
A'lin.nld A'|).
let
From
as Ix'fore A''"'
=
+
Xj"'
dW
/o^/'.y'"'
X
anrl
= Xo +
dW
J^r.x
he
roproscutations for X'"' and
tlir iiiartiu!^al<-
resppctivcly
Tlir Caiirhy-Schwarz inequality iniplirs that |||/'a'"'|||2
BurkhnldrrV
inrqiiality in turn impiiop that
D
stant
Since X^"'
bonuded,
is
|||/'.\-'"'||kQ
follows that
it
< D
q
Ik~'ll2,c?-
xl"^\\i_Q for
some con-
III^.V*"'|||4
-
||X'"'
We may now
Hb.r < CO.
|||/'.\''
—
r
X,
apply (12) to
get
_
j^i")
<(;'
..v'"'-..v""'
x^"'^
2.r
2,r
when n and
any
X
m —
6 L^(P) by
Thus
oo.
converges in the
^.\
•
n'^nn
|||2.r
|||
p\ and we obtain
t^o
(12) for
n -» oo.
letting
I
The
Lemma
t
e
lemma
following last
Let
3.3.
X
e L'(r).
X
Tltni
6
M
M
<=^
and the
PxU) 1
null space of
Null{rj(t))
cr.
P)
(a.,«..
for cacL
\0.T].
Xe
rwof. Say
.Y
=
PML
and
=
(i(r{t)
orthogonal
rr*"'"/i_v(f).
X=
Then
flit]
+ /
=
zero.
rr''^
+ /
1
Null(r7(()).
a''^ pxit)
rr'^(t)0(i)
<
•^'^
<
||k''^''il|2.r
oo.
=
(«..».,
7r^jpx{t)
fl{t)
=
P)
for
each
^^inre
pxi^)
/
G
is
dW'(t).
Tlius
px =
px(t)
rT'{t).lM''(t)
=
[O.T], and
t*"""^
G
0(t)
=
let
€ Range rT"^(0- Hence
Ao+ /
.'n
^(0
dG'(t).
./n
Ijounded
lik^'^llkr
some constant
A'n
of n.
.'n
D
=
./o
dW'it) = A'n+ /
A:o+ / n'^iDflit)
Moreover, since
a' ,nV'(t)
(}{f)
|||rT*"'^fl|||2,r
tjio null sjiace
tf>
Conversely, suppose px{t]
<j''^
A'n
.'n
bounded away from
is
is
M. Then
+ I Hf)
.'0
A'o
B
But since
for
between
gives the relation
K by
<
/^llk'^^llkr = A'lllrvllkr
lemma
3
2.
Thus
A'
<
/^^^' i|A||2.r
<
oo.
G MI
Returning to our main concern, wn can iinw state the fnjlnwing
Proposition 3.1.
rrnn{.
We
n'idlV
-4
If
X
E L^iP]
'^
j'lirr ;neasiira/'/e.
have seen that (H'V 7,.Q)
K dt)
=
n' d'W' so that
C
is
is
a
fiien A'
lesult:
E M.
Brnwniau motion. But by
(2).
-
=
h' dt
+
n'
dW =
a ri^ht continuous. TJ-P'ogressively measural)Ie and (n.n..r)
continuous process satisfying
(^[t]
d(V(i]
,S'-(n)
4
I n'iDdirH)
.In
18
in
..
.P)
.
r
Since
from
L,
Q
and
are eqnivalont. thosr statmirnt? hold also uiidoi- Q.
zero, heiire
=
ifrV*
On
Q
aud
'
i?
from 5*(0)
the other hand,
A',
if
=
at
=
<
Q
Tims
0.
E'^\X\Jf\. (X,. T,^,Q)
extremal for H'
is
an L'(Q)-martingale for any q
is
Hence by Jacods martingale representation theorem and the
X
We
1^'
sums
If
<
we
define
A''
=
'"*("') d."
/^
+
W
we know
.
/ r'it)df =
Xn+
lemma
by
define \V'(t) in accordance with
=
lemma
l)y
3.3,
t
= T shows
that IV
=
Finally
gives H'*(0)
W'[t) +
A'd-
/
we
2.5
W+
E"^
= \V(T]
Al-
is
is
m
I
price measurable
so that (9)
is
is
also satisfied
+
\V
dK
c
/
7,
satisfied.
On
the other
ft.
r.W) G
conclude tins sectinn by
Proposition
3.2.
/?(/)
=
rj(?)
if
we
»"(o)+
7,
hi
t)
-dC'lt).
cIkjosc rt{t) so that H''"(t)
=
a{t) +f)(t] S' (t),
.W.
jn'ovinrr a icsult of
).<;
hand the choice
Jo
then clear that
It is
as
d(r{t).
r*(.')f/.'
/
price
see that
r'[!^)d
then the corresponding {n.
We
G
G L^{P) and
tliat A'
n
so that (10)
X
approximate the integral defining X). Therefore we have (13) which wc rewrite
to
choice
price measurable
(13)
Wit] =
The
is
apply proj)Opitiou 3.1 to the case where the consumption good (c.W)
W+
Now
X
(1,2).
Jn
orthogonal to the null space of n. so that
i?
now
can
measuralile.
(use
^
fart that
G
= At = An +
Jo
But px =
away
l)ouiidcd
is
to the maitingalo i)rol)Iein associated with the operator
tlir niiiqiir polntioii
iiaitiatiug
B
But
thr
unii]'!/^
indejjcndant niterest.
jnjrr mea.-^uj-n/Je
,T/i.-;o/ute/_v
r<intiimnuA niHitingHlc
mea.'Jine.
Proof.
Recall from (5) that dN,
r(0
0.
since
j^
/'
{n.s..r).
and
r:
lie
in
We
have
=
/'(f)
N,
some
dli'it) for
i'(f]
dM"{t) =
r{t)
i'{1)
{dM'lt]
orthogonal subsjiaces. so that N(t)
is
a
-
G
(NulU)-'-. and suppose that
t:{i] dt)
=
rlt)
dVJH) = dMt)
martingale mider both
P
and Q.
In
particular.
A'
= 1+ / N(t]ry{t)
dM''{t)
Jo
with
^.v(/)
resjiect
to
=
tlir
i'{t]
G
Null {n{t)).
price ])roces?
By lemma
and the
3.3 then. A'
result follows.
19
^
M
Therefore. A'
is
not measurable with
4.
Optimization under incomplete markets
wr
In this section
finally
address the maximization program fared
a utility funrtidu on ronsuniiitinn lates
final
\'
wealth
/?+
:
a strategy {n,
0,
—
»
r,W)
U {— oo}.
/?
to
maximize
i;
:
x
7?^
His initial wealth
expected
his
problem
a(0)5(0) +
The problem
H''(0).
is
on
to find
V{W\
-f
S{()]
fl(0)
<
\V(0)
[14
Cnx and Huang
|198G].
we eml)ed
this
dynamic
problem
in the following static variational
E'
SUJ)
rML\{\
anrl a utility function
(n..^..r).
that there exists a snlntjnn to (14), as in
(c.w) e
endowed witb
..'0
W>{)
To prnve
U { — 00}
denoted hy
is
c{t]A)dt
(nJ.c.W) e M
s.t.
/?
is
utility
E
sup
—
[('.r]
an agent. He
liy
u[r(t).i),it
/
+ V{W]
X r) X L\[r
+
t]di
s.t.
H'*
(15)
.'n
where
Q
is
the jirobability defiiieri in section 2.2.
By lemma
constraint of (14)
(15) can
i)e
We
it
=
/
it
is
immediate that
satisfies also tliat of (15).
associated with an [n.
.
c
.\V
)
So
G M
we
onc(-
a trading strategy satisfies the biidget
if
sr>lution
(cH') of
same budget constraint, we
arc done.
liave jjroved tliat
satisfying the
any
record the following useful lemma.
Lemma
])lnrrRfi
with
2 5 ap])lird
4.1.
Y(t)
Lrt X{i) hr
Sljr/] th;it
pi
Thru
juniitivr pinrrsfi
fni ryrry Tf -stcij<j<inf^
an
T/'
•j>rogrrssivrly niea.si!ra/)/e
T
tillir'
E"^ A'r/,-<:
tlirir rxistfi
=
7}
(IG)
y,j,.
Afr)renver,
E'^ I X(i],U
J
Proof Dellacherie and M^'ver
a rlefinition
However, we
will
E^[X{t)\7,^\
=
y{f]<U
/
[1982]. tln-orcnis VI. 43
and
57.
ourselves to the case where
Thr process Y
[i]
is
and the
Chung and Williams
call(><l th--
20
fact tliat
all
oiitional jirocesses
[1983], Section 3.4).
(^huug and Williams [1983], Section
of stojiiiing times, sec for instanrr
ie,'--tiict
Y{i].
£•'-'
.'0
are necessarily progressively measuinblc (rf
For
=
l\
r
=
^
A
T, so that
(IG)
o]>tional luojection of A'(0-
1.7.
becomes simply
Now
of
c
W)
lot (r,
lemma
as in
c(t)/D(t)).
a solution of (15), define
lie
4.1.
Consider
we can rewrite
it
first
W
=
|^) and
£^*^[iy
the budget ronstraint (15).
the oi)tional projection
let r l)e
By lemma
4.1
(choose X{t)
=
as
T
E^
where we have used the
now
(c,H')
initial cost is at
W'(0).
fact that the optional projection of r*
most H'*(0).
It
remains to check that
is
the discoxinted value of
we know that
price measurable so that from section 3,
is
By Jensens
<
+ W'
I c'{t)dt
Jo
at least as
is
it
But
marketed and that
is
it
c.
good
its
as the original (c.W'').
inequality,
u{i{t).t)>E%{r{l).t)\
r,']
and similarly
V(W) > E'^lViW)
I
Tr
Thu
Ht).t)>ii
E'^l
+ V(\V
+ V[W]
u{r{t).t)dt
.•'0
V(W
1
E'
by proposition
since
r;
=
2.4 (choose A'
U.{r{t].t)df
+
—ti{r{t).t]
.'^-measurable (proposition 2.3)
is
V{\V
+
E"^
by
lemma
4.1 again.
=
T
n{c{t).()df
E'
+ V{\V
..'n
ThuF, tuider assumjitinn
two problem? yield
the solution
concave,
is
onr-
mrasurablr
in fart ecjuivalent solutions.
unique so that
may have
good (c.H) where
any so]ntif)n of (15)
2.1.
W
it
will
\\V
(]iro]iosition 2.3), (c,H')
is
jiric(^
than one sfilutmn
'y\r](t)
\
mafchcd by
:
^
<
'
<
a solution of (14), so that the
the utility function u
automatirally be
of course luori-
= E
When
is
T]\
When
measiu'abje
("onsidcr for instance
and similarly
certainly marki-ted,
A
for
verification in
the one abov(^ shows that (r,n') satisH'S the budgif (rmstraint anci that
21
strictly
is
it
tlir
is
all
in c,
not strictly
consiunjition
Sinre q(t)
r.
is
t;
concave
is
jirice
resjiect similar to
at
least as
good
as
the oriRiiial (c.W).
Titility
Wr may
and rhoosr the
tlm? have two
(lirrr, nu^asnrHliIr)
on
inost ronvrni'^iif df^iiendinK
Tlie riear advantage of (15)
that
is
under whirh there exists a solution to (15) can he found
commodity
is
either a
where there agent's discount rate
utility fimctions of jiractical interest
finite inniiiont
luider our
(oj).
cit..
theorems
2
=
is
Suj>jui<ir
Sufipnsr
further thnt thrrr
Hsymptofirnlly fnr aowo
rroof.
Cox and Huang
Thr
ex/sf.s
>
[198C],
last jn'ojuisitinn
a sjiecial case.
2.3.
:
G
/;
>—+
7?"*^
t;
and
When
when
the
the utility
Note that
Theorem
>
0.
price
this last condition
r]{t)
is
has a certain
always satisfied
For the sake of conijileteness, we report two sets of
/?"*"
(0. ]] .surh
Df,
shadow
i.s
bounded
ronfinuou.'>.
that for rvrry
f
or unliounded from below.
ronr^vr hwI
strictly increasing.
>
Tiien thi^rr rxif^ta a sohition to (15).
2 4
includes the utility functions that arc of constant relative risk aversion as
For utility functions that are uubouiuled from below, we have the following.
Proposition 4.2.
Sujtjinfir
Af,
tli^t
[1985],
the final date.
at
that the invrrs(- of the
sufficient conditions for utility fiuictions that are either
4.1.
and Htiang
e~'^''u(T),
9 aiul 2 10)
own assumptions by lemma
Proposition
('ox
Ceneral conditions
hoiuided from below, then a sufficient condition for almost
is
/?,
than (14)
form
fvuictiou displays a time- additive structure of the
all
in
consumption rate process or cons\imption
«(j,<)
same expected
ajiiilirations.
ninrli rasirr to solve
is
it
solutions yielding tbo
.'?n;'/in.'jr
further that tlirrr
that n
rxif^ts
}<
:
/?^\{n}
>
sucli that
\l'{x)
asywptoticpilly fnr snnir Kf.
rronf Cox and Huang
>
[198C].
n
i—» 7?
i>
fi<r
< Ky
(Jiffcimtinl'lc.
rvrry
A
>
concave and
d
-ft+(*
7
Tlirii t}irrr rxists a snlutidll to (15).
Theorem
2.C
22
f^trictly incrcaf^inc.
5.
Wo
we
An example
conclude this paper by giving an apjiliration to a
conpinni)tion portfolio problem. Since
piini)lc
are primarily interested in cxistmcr rather than characterization,
example.
in this
We
where markets are incomplete.
consumption space
=
B(t)
G{t)
1.
The
ft
is
a
i{S{s),s)d.^
1
vector of constants, n
Al
=
|h-
at
[
d.« -^
The
the final date.
constant and equal to zero so that
is
Brownian motion
V(G[0,T)
Is{s)rTdW(.'^)
{a.s.,r),
Jo
is
MX
an
N
constant matrix of
e(jual t" Si{t).
is
i
=
M
1
full
rank
The
(M
< N),
j^rocess ri[t) of
becomes the lognormal variable
0{t)
e
where £.^(G,P)
S(t)
L'^iF)
is
= cxp{-nh
W{t)
- -l^'f }.
A
take a particularly simple form
hi this case strategies
-f
Is{s)h
?-tli eiitiy
,,(0
a(t]
interest rate
Jo
A/ X
jiro])nsition 2.2
=
The
= S(0)+ [
and 75(f) a diagonal matrix wlinsr
W{t)
consumption occurs only
Siipjiose that
then be L^{P).
will
can be extended to the situation
easily its solution
risky securities gain i)rocess follows a nmltiplicative geometric
= S(t)+ f
Jo
where
how
only want to illustrate
we do not seek completeness
;
= W(0) +
W
self-financing strateg)'
is
defined by
dG(t) and the space of marketed commodities
jlj(t)
= W(0) + /
(17)
dG(t)
0(1]
for
some H'(0) G
the set of 7,-prngressiveIy measural)])'
such that
/?
|||rT
and
is
G C^(G,P)
7s(0'^(0lil2,r
<
},
O^i ^^
before.
We now
choose a utility functinu of constant relative
risk aversion:
1
u{x]
=
^
1
l/i
~ ~
X
if
T
G (n,oo)
and
7
>
0;
if
3-
=
and
-y
>
1;
if
3:
=
and
7
G
(0. 1).
I
'^
=
nrit
defined
where we have used the convention that
The
j)rnl)lem faced by
an
investfir
.t
max E^ulW)
su]i
log
.t.
endowcMl with one unit of the Cdnsumjition good
s.t.
new
Consider the extended maximization
=
/n
])rop;raiii
\e^u{X)
:
W
>
and n'(n)
=
at
time
is
1.
coi-respondinj-'; to (15)
X>
23
n
and
E'^
X
=
l]
.
(IS)
whrrr t^
=
{e^X^~^
P.ip
Now
defiup
When
as bofore.
rj.
X
Y =
:
X >0
X
X
<^
'^
>
7
Ca\irhy-Schwarz inequality
1
wr have
and
V
=
X
E'^
=
where 7
E L^(Q),
l]
=
sui'
Y E
L"^
E^\X\'~-^
conjnpatc (i
aiul 7' arc
then
{
:
i;
-f-
E'^IXI
= l]
=
Since by the
1).
(Q) so the above siipreinum
is
.
no greater
than
i}=
{e'\Y\:\\Y\U.,,=
sup
{e'^-Y:\\Y\U.,,=
sup
If?
VeL->'(g)
YeL-^'iQ]
that
supreniuni
tlic
is
i,Q
-»-i
Y =
acliieved for
V
)
->-i
One can check
-
i\=
which yields
7.Q
(19)
-).<?
The
cape D
be solved
<
7
<
1
can
treated in the
l)e
same way with
"
replacing "sup", while
7=1
can
diiectly.
Since
X
G L^(P)
lemma
by
2.3,
A'
indeed the solution of (18).
is
measurable by proposition 2.3 so that by projiosition
strategy that manufactures H'. Substituting (17) for
W
where
iuf
'
as a result
of the
=
exp{7^''
3.1,
=
in (19),
rj
M''{T]
W
A'
marketed.
is
we
hi addition. A'
get after
We now
jjrice
is
elicit
the
some manipulations
- ^I'^H'p
('ameron-Maitin transformation
li'*
is
given by
Tl'*(f)
= Wit) +
aht.
Defining
W{i) = exp{7'^''
ItfVs
formula
- —\^^>?]-
^''(O
imi)lies that
\\\i\
=
e"^
W
7,1
=
+
1
I
7
\V{.'<)rrh,nr(.'>)
Jo
=
+ 7
1
/
\V{-'')n^n^nl,- ,/H'*(.,)
./n
=
since da{t)
=
Is(t)ndV,''(1).
+
1
Hence by lemma
7
/
n'(.') i^^{.-<)<^nh
3.2, Oit)
=
,/r;(..)
il'^ (f)n'^nh\V (t). But hh
G Range rr^
so that
(7"^
because
S'l
7r„
=
/\/
when n
is
nh
=
{rrrr
)" nrrli
of full rank.
thai, a= exix'ctefl. thr piirtfnlui shares
We
ar'-
=
{frrr
tlien
= [nn
)~
h
have
rciustant
24
)~ n^h
and indejiendan*
of the level of wralfli
6.
Concluding remarks
In this pajjer
incomplete.
of
we have oxtrndrd the martinsalr apprnarh
We
obtained essentially two results. First,
it
is
still
One
hedging the
risk associated
with holding positions
in
it
is
consistent with the
has, for instance, a constiiictive technique
any ojjtion written on stocks. Second,
also possible to solve a very general version of the intertemporal jiortfolio problem.
not seek to characterize the optimal individual pohcies,
existence of a solution, which are virtually the
same
dynamically
possible to duplicate the pay-offs
any contingent claim by using an ajipropriate trading strategy, jirovided
information generated by the securities price system.
for
to the rase wliere iiiaikets are
we provided
as in the
it is
Though we did
sufficient conditions for the
complete markets case. Furthermore,
those conditions were entirely defined in terms of the primitives of the model.
The model we have
dealt with
agree on the law of motion
i')rices
are necessarily
a rational expectations one in the sense that
by the seciuitics pnrc
satisfi<vi
have shown that when markets are
fjy
is
incojiijilete, all
''marketivl"
,
in
proc(\>;s.
its
agents
all
Under those conditions, we
actions cfinsistent with the information revealed
the sense that they can be imjilemented by trading the
few long-lived securities continuously through time, and this result has jirovided the clue to the
derivation of the ojitimal strategies.
25
References
1.
G. Chanihrrlaiii, Asset piiring
in iiuiltiixMiod srniritics
markets, Miineo, University of Wis-
consin. 1985.
2.
K.
Chung and
R. Williams,
An
Int induction to StocliHstic bitogrHtinn,
Birkhauser, Boston,
1983.
3.
.1.
Cox and
C. Huang.
A
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Massachusetts histitute of Technology. 198G.
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Cox and
C. Huang, Optimal consumption and portfolio i)olicies
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Hollanrl
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h53k
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[00
26
Prorr.s.sr.s,
Springer- Verlag,
New
York.
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