CANONICAL MARTINGALE MEASURES OF
INCOMPLETE ASSETS MARKETSYOSHIO MIYAHARA
Faculty of Economics
Nagoya City University

Proceedings

PROBABILITY
Editors
S.Watanabe
M. Fukushima,
Yu. V. Prohorov
A. N. Shiryaev

of the Seventh Japn-Russia

Symposium (1995)

THEORY AND MATHEMATICAL STATICSCANONICAL MARTINGALE MEASURES OF
INCOMPLETEASSETS MARKETSYOSHIO MIYAHARA
Facultyof Economics,
Nagoya City University
Mizuhocho Mizuhoku, Nagoya, 461, Japan
E-mail :miyahara@econ. nagoya-cu. ac.jp

ABSTRACT
Weconsider the incomplete assets markets. Then there are many equivalentmartingale measures. Amongthem, the probability
measure, which minimizes
the relative entropy with respect to the original probability
measure P. has a
special meaning. Wecall such a measurethe canonical martingale measure. The
canonical martingale measureis. if exists, unique. We investigate the existence
problem of canonical martingale measures.
1. Introduction
In the theory of the financial
market, equivalentmartingale
measures are often
discussed(Harrisonand Pliska 1981, 1983).
Suppose that the price process of stocks is
the stochastic
process S = (S(t). t > 0) definedon some probability
space (£2. T, P).
Under the assumption of the absence of arbitrage
opportunities,
there is an equivalentprobability
measure Q ~ P such that S is a martingale
under Q.(We call such ameasure Q the equivalent
S-martingale
measure.)If the market is complete,the equivalentmartingale
measure Q is determineduniquely.
But, if the market is incomplete,there are many equivalentmartingale
measures. In such a case, the problem to analyze the mechanism of determining
theprices of contingent
claims remains to be an open problem. An answer to the above
problem is adopting
the minimization
principleof relative
entropy as the criterionofreasonablemartingale
measure.
The equivalent
S-martingale
measure P*. which has the minimum relative entropy
w.r.t. P, is called the canonicalmartingale
measure of the price process S (Definition2). It may be natural to think that the theoreticalprice (= value) of a contingent
claim X is equal to the expectation
Ep*[X]of X with respect to P*. in regard of
Sanov's Lemma of the large deviation
theory. The canonicalmartingale
measure of
the price process 5 is unique,if it exists(Remark1). Therefore,
the existenceofcanonical
martingale
measures is the problem to be investigated.
Our first main result is Theorem 3 in •˜2, which asserts that if the price process
S is bounded,then there exists the unique canonicalmartingale
measure. In •˜3
we investigate
the case where the price process S is given by stochastic
differential
equations(S.D.E.).
The second main theorem is Theorem 4, which asserts that in
the SDE case there exists the unique canonicalmartingale
measure P*, and thatthe measure P* is given by Girsanov's
transformationof measure from the original

343

344

measure P. Our results are obtained withoutof Radon-Nikodym
derivativesof S martingale
measure P (see Remark 4).
2. Preliminaries

and General

the assumption
measures withof square integrability
respect to the original

Results

Let (fi,J7,P) be a probability
space and let {Tt,t> 0} be an increasing
familyof sub (7-fields of T. where we assume that Tt is right continuous
and includes all
P-negligible
sets in T. Suppose that the price process S = (S(t),t> 0), S(t)=(5ri(t),5f2(f):•E•E•E, -?«"(£))
à¬ RK, is given as a K-dimensional^-adapted
stochastic
process denned on (f2,.F, P).
Under the above framework, wedefine V(S) as the set of all equivalent
S-martingale
measures, namely the set of all probability
Q on (Q,T)
such that (S(t),t> 0) is(Ft, <2)-martingale
and Q ~ P (absolutely
continuous with each other),
and we define M(S)as the set of all S-martingale
measures which is absolutely
continuouswith respect to P. From the definitionsof P(S)and M.(S),the following
lemma isobtained easily.
Lemma 1 V(S)and M.(S)are convex subsets of M.\(=theset of all probability
measures on (fi, F)).
We denote the relative
entropy of Q with respect to P by H(Q\P),and thevariation
distance of Qx and Q2 by | Qi - ^lUarDefinition
1 For the convex subset S of Mi, we set H(£\P)= infQ6£H(Q\P).
If
this infimum is attained
at a point in S, then the point is called minimum point in £.
Remark 1 If a minimum point exists, then it is unique. This fact follows from thestrictconvexityof the relative
entropy.
The basic propertiesof the relative entropy are describedin the following
lemmas.
Lemma 2 (Csiszar
1984)
Let£ be a convex subset of M\, and let Qn,n= 1.2,...,
be a sequence from £ such that lim,!-^ H(Qn\P)= H(£\P).
Then(1) there is a probability
measure Q^ such that lim n^^WQn - QoolUar = 0.

wmQ^KHWP).
Lemma 3 (Csiszar
1975, Lemma2.1)
Let Qq,Qi<C P and set Qa = aQi +(1-a)Q0
for a,0<a<1.
Then(1) H(Qa\P)> H(QQ\P),\/a,0< a < 1«ff(Qi|P)> //(QilQo)+ H(Q0\P).
(BJSoiq,0<a0< 1, suchthatH{Qa\P)> H{Qao\P),
Va,0<a<1*=> H(Qa\P)= H(Qa\Qao)+H(Qao\P),
Va.O<a<1.
Lemma 4 Suppose that Qq,Q\< P, Q\ ~ P , and H(Qi\P)< oo. Andset
Qa=olQx+(1-a)Q0
for a,0<a<
I.Then(1) there exist a unique a0,0 < a0 < 1, such that H(Qao\P)= info<a<i H(Qa\P).
(2) Qa0 satisfies
the condition
that Qao ~ P.

345

(Proof)Since the set {Qa, 0 < a < 1} is a convex closed subset of.Mi, the result(1) follows from Lemma 2.
The proof of (2) is as follows.
Case 1. Ifa0 = 1, then (2) is trivial.
Case 2. If a0 = 0, then by Lemma 3 (1) it holds that oo > H{Qi\P)> H{Q1\Q0)+
H(Q0\P).
Therefore
it follows that /f(Qi|Q0)< oo and so Qi < QQ. Using theassumptions that Q\ ~ P and Qo "^ P å we know that Qo ~ f•E
Case 3. If 0 < aQ < 1, then by Lemma3 (2), oo > H(Qi\P)= H{Qx\Qao)+
H(Qao\P).
So we obtain the result in the same way as case 2. (Q.E.D.)
Using the above lemmas, we can prove the following
theorems.
Theorem 1 If H{V(S)\P)< oo, then H(V{S)\P)å =H{M{S)\P).
(Proof)From the definitionsof V{S) and M{S), the inequality
H{P(S)\P)>
H(M(S)\P)
is obvious.
Let {Qn,n = 1,2,...}
be a sequence in M(S)such that H(Qn\P)| H{M(S)\P),
and let Q be a point of ^(S)
such that H(Q\P)< oo. Then we can apply Lemma 4
to Qn and Q, and we know that there is a point Q'n on the segment QnQ such that
Q; ~ P and tf(Q;|P)< /7(Qn|P).
Since SA(S)is convex, it follows that Q'n (E J\A(S).
and from Q'n ~ P it follows
that Q^ G P{S).
Therefore we have proved that
H{V{S)\P)< lirninfH(Q'n\P)< lirninf

H{Qn\P)- //(A4(s)|P)and the proof is completed.(Q.E.D.)
Corollary
1 If H(P(S)\P)< oo an<L V(S) has the minimum point, then that pointis also the minimum point of M(S).
Theorem 2 Suppose that H(P(S)\P)< oo. If there exists the minimum point inM(S),then the point is also the minimum point in V{S).
(Proof)
Let Q be the minimum point in M(S),and let Q be a point in V(S)such that H(Q\P)< oo. Applying
Lemma 4 to the segment QQ, we know that there
exists a point Q' in QQ such that H(Q'\P)< H(Q\P)and Q' ~ P. On the other
hand, from the convexity of M(S) it follows that Q' E Ai(S).
Thus we have obtained
the results that Q' e V(S) and
H{Q'\P)
From the uniquenesscompleted.
(Q.E.D.)< H{Q\P)= H{M(S)\P)

of the minimum point,

it follows< H{P{S)\P).
that

Q = Q', and the proof isDefinition
2 // the ninimum point in V(S)exists,then that point is calledcanonical martingale
measure of S.
Theorem 3 Assume that the price process S is bounded and assume that H(P(S)\P)
oo. Then the canonical martingale
measure P* exists and is unique.

the
<

346

(Proof)If we prove that M(S)has the minimum point, the result of theorem
follows from Theorem 2. Let Qn à¬ JA(S)be a sequence such that H(Qn\P)[
H(M(S)\P),
and put qn(u) = ^-(lj).
From Lemma2 it follows that there exists a
probability
measure Q such that Qn -åºQ in variation (||Qn - Q\\var -> 0). and
H{Q\P)
It is well-known< liminf(see Ihara

1993,
\\Qm

So it follows

that

H{Qn\P)= H{M{S)\P)

Theorem 1.5.3)- Qn\\varthat= Ja \qm - qn\dP(uj)Q < P and\\Qn~QWvar=J|?»~?|^(w)-0,

.
dQ
q=dF-

wk&re

Weshall show next that Q 6 M(S).
Since Qn 6 M.(S),.Ft-measurable bounded function g(u>) and for s > t
EQn[Ss9}= EQn[EQJS3g\Ft}}=( StgqndP->= EQn[EQn[Sa\T3]g\/ StgqdP=
E^[Stg]it holds

thatfor any= EQn[St9](n-oo)

where weuse the assumption of boundednessof St. On the other hand it is obvious
that
EQn\Ssg]= I SsgqndP->/ S.gqdP=
E^[Sag](n-> oo)
ThereforeEe[^^j= En[5't5]
for any bounded Ft -measurable function g[uj).
This
proves that Q is a S-martingale
measure. The fact that Q à¬ V{M) follows from
Theorem 2. and the proof is complete.
(Q.E.D.)
By Theorem 3, the existence of the canonical martingale
measure in the case thatthe price process is bounded is assured.
This may be enough in the practicalpoint
of view, but is not enough in the theoreticalpoint of view. In the next section we
investigate
a case where the price process is not bounded.
3. The Case of Stochastic

Differential

EquationIn this section we investigate
the case where the price process St is given bystochastic
differential
equations(SDE).
Let W(t)= (Wi(t),...
,Wd{t))
be the ddimensional(!Ft, P) Wiener process, and assume that Tt = T^ - cr{W(s), 0 < s < t}and T = J^. Let the price process S(t)= (5i(t),....
Sxit))
satisfy
the following
stochastic
differential
equationsdSk(t)= bk(t,S(t))dt+£akj(t,S(t))dWj(t),k=l,...,K(1)

347

or equivalentlySk{t)the following=Sk{0)+

stochastic

fbk(s,S(s))ds+Y,

we assume

j._1

that

bk andequations

I'a^SMdWjis),

JO

where

integrala,kj,kk==l,...,ff(2)

yo= 1,2,...,K.j- 1,2,...,d,satisfy

theglobalLipschitz
condition,
so that the above equations have a unique solution.
Theorem 4 Suppose that the price process S(t) - (Si(t),...
, 5JK-(f))
is given by (1)or (2) and assume that H(P(S)\P)< oo. Then there exists the canonicalmartingale
measure P*, i.e. P* is the minimum point of V(S),and P* is obtainedby Girsanov
transformationfrom P. The canonical martingale
measure P* is also the minimum
point of M(S).
For the proof of this theorem, we need some preparations.
Let Q be a probability
measure on the space ($7, J7) such that Q <C P. Then the following
facts (a)-(f)arewell-known (see Liptzer and Shiryaev 1974, for example).
(a)Setq = ^ and q(t) - ElqlTt].
Then q(t) is a (not necessarilysquare integrable)(Tt, P) martingale.
(b) q(t)has the following
representation(Liptzerand Shiryaev
p. 171 Theorem
5.8)

q{t)

= l+Jzfg]{s)dWj{SlP(fT\g(s)\2ds=

fTJ2\gj(s)\2ds< oc) = l(c)Set7,(0Then the process

W{t)= (Wi(t)i

wpi(t)
is a d-dimensional(^i, Q)-Wiener(d)Using
the notations of (c),dSk(t)

ifq(t)->0otherwise=•E•E•E, W^(0)definedby= wi(t)- jrl7i(a)daprocess (Liptzer
and Shiryaev p. 225 Theorem 6.2).
we obtain the following formula= bk(t1S(t))dt+
ZUa'>i(t>s(t))dWM={h(t,S(t))+ZUa^S(t)hj(t)}dt+ZUa*J&in the sense of probability
measure Q. Therefore we know that, since we have assumed
the Lipschitz
continuity
of a,kj, the necessary and sufficientconditionfor Q to be a
S-martingale
measure is thatdh{t,S{t))+Y,akj{t,S(t))'yj(t)=O(dtxdQa.s.),k=lt...,K.

348
Fromthe definitionof 7j(f),j= 1,...

,d, theabove condition=0{dt xdQa.s.),

is equivalenttodh(t,S{t))q{t)+J2akj{t,S{t))gj{t)In the case of Q ~ P, the following
results hold true.
(e) If Q ~ P. then q(t) > 0 and Q is obtained
from P, namely, q(t) has the following
representation
Theorem 5.9)q(t) = expiflllMdWjis)
JO

by Girsanov's
transformation(Liptzerand Shiryaev
p. 171- \ f^{s^ds}

j=1

/JO

(f)If Q ~ F, then the necessary
measure is thatk= l,...,K.

and sufficientbk(t,S(t))+J2akj(t,5(i))7i(0=0

J=1

conditionfor Q to be a S-martingale(dtxdPa.s.),k= l,...,K.

or equivalently
M*,5'(0M<)+Ea*i(«.'Sr('))ai(«)=0(dtxdPa.s.),k= l,...,K.

3=1

The following
lemmma is essential
for the proof
Lemma 5 Let {qn{u),n=1,2,...}
be a sequence
0 and !^qn{oj)dP- 1, and assume that
Qn-*q(asn-*oo)

Set qn(t)and q(t)
qn{t)

q(t)

of Theoremin L1(f2,T,P)

4.
such

and using the fact (b),- l +J:ft9{f\s)dWi{s)yP(fT\9W(s)\2ds= fTJ2\9f(s)\2ds= l+ib[t9J(s)dWj(S),P(fT\g(s)\2ds=

fTJ2\9j(s)\2ds

Then it holds
9j

J0

(as

n~^°°)

in>Jo

we represent qn(t)measurew.r.t.

dtxdP,< oo) = l

<oo)=

j=i

that
~~*9j

qn(uj)inLx(fi,J",P).

- E[qn\Tt\
and q(t) = Efgl^i],in the following form

j=iJ0

that

j=1,2,...,d.

l

349

(Proof)

Let r/

be the stoppingbyM{t <T;qn(t)
>1}
T, if
{t<T;qn(t)>l}=T,W =and set qn,i{t)time defined{= qn(t A r/" ). Sinceqnyi{t)is square= l + i:ft9^l)(s)dWj(s)integrable,

it has the following

represent at ion
qn,i(t)and it holds
gj

that
-æfgj

(as

I-æfoo)in measure

Moreover, as Liptser-Shiryaev
can assume thatshowedgf\t.u)=g^+k)(t,u>),(see

k=0,1,...,

w.r.t. dtxdP,

Liptser-Shiryaev
ifj=1....,d.
1974,

pp.

sup|gB(«,w)|</,

168-169),

j=1,...,d.

we
(3)

u<t

In the same way, settingTl=and qi(t)= q(t A fj).

inf{i<T:
T, if{

we know thatq(t) >1}{t<T;q(t)>l}=$17; is square

integrable

and that

and we can assume that
gj{t,uj)=gf+k)(t,uj),

k=0,1,...,

ifsup\q{u,u)\</,

j=l,...,d.

u<t

We next prove that g^ ' converges to g^ in probability
as I -> 00 and that thisconvergence is uniform in n,n = 1,2.... ,. In fact, from the definitionof gj ' and (3)weobtain for any e > 0

Pfsup^l^u)-*}"'(«)I> e)<^(suPu<tQn{u)<1 /sup{qn(u);I<E [qn]II M, M = suVnE[qn]<where we use the martingale
it holds that

inequaliy.

P(sap\gf(v)
u<t

> I)
u<t}>lQn(t)dP(4)

In the same way weknow that for any e > 0-gi{u)\> e) < \E[q]
I

(5)

350

From (4)/0 = /0(e;S)

and (5) it follows
that for any e > 0 and 6 > 0 there
such that for any / > Iq it holds thatP(snp\gf)(u)-gf(u)\>e)<6,

71=1,2,...,

existsa consantj=1,2,...,d,

(6)

and
PP(. (snp\gf\u)-gj(u)\

>e)

<6,

j=l,2,...,d.

(7)

Next we will prove that, when I is fixed, pjn' converges to g^ ' in probabilityasn ->oo. From the definitionsof {qr^;??.
= 1, 2,...}and qt, they are square ntegrable
and qn<i -> qi (as?i-> oo ) in L2(df x dP).
In the square integrable
case, it is wellknown that g^-* gV (as n -» oo) in L2(dtxdP)
follows from qn>i -» ^ (as n -»•E
oo)in L2(dfx dP).Therefore^W)_>^^^

we can result(asn_^qq)inthat

for any fixed I,Imeasurew.r.t. dtxdP,= 1,2,...,
j=1,2,...,d.

(8)

Using the above results,
we next prove that g^ -æf^- (as n -> oo) in measurew.r.t. dt x dP. We set dp= dt x dP on [0,T] x Q. From (6) and (7) it felloes thatfor any e > 0 and f> > O.there exists a consant Zo such that for any / > /0 it holds that
>
n

,'),

=l,2,...,

»<

>e})

<T6,

(9)

j=l,2,...,d,and/z({(t,a;)e[O,r]xn;\gf\t,u)-gj(t.,uj)\

We here fix an /,/ > /0, and then usingthat for any n > no/i({(t)w)e[0)r]xn;>e})<T6,(8)

j=1,2,...,d.

we can take a number n0 = no(e,6,1)|9f'')(f;a;)-^)(t,u;)|>e})<6)

Using the results(9),(10)and
6 > 0 we can choose constants
any n>Hq
^ \9T

-93\

<

(ll),
we obtain such a result that/0 = ^o(ei^)
and n0 - no(e,6,l),I/x(|5Jn)-< T6+6+
Thus we have proved that

g^

such

j=1,2,...,d.

M)|

>3e)

(10)

-> g~j in measure w.r.t. dt x dP.

(ll)for any e > 0 and> l0 such that for
T6=
1)j=
T+
(2
d.
6,
2,...,
l,

f»$f\
{\9)+
)+n
^\
g0>e>e
gy-'\

.

9j\

><0

(Q.E.D.)

Remark 2 Karatzas,
Ocone and Li 1991 discussed
similarproblems.
prove that qn G D^i and that qn converges in D^i (with their notations),
could apply their results.
But it is not clear that qn G Di i.

If we could
then we351(Proof of Theorem 4) Let Qn, 1, 2,...,
be a sequence from M{S) such that H{Qn\P)I
H{M{S)\P).
By Theorem 1, we can assume that Qn G V{S).
Set qn(u) = %-. Just
as we have seen in the proof of Theorem 3, we know that there exists a probability
measure Q such that Q < P and Qn -> Q in variation,
and
H(Q\P)< lirninfH(Qn\P)= H(V(S)\P)- H(M(S)\P)(12)and that\\Qn~QWvar=J\*»~?lrf^(W)

~>°-

- dQ
q=dF-

where

In the followings,
we use the notations{qn,9j,Tj ,Wj ) or (q,Pj,7j,Wj),
for
Qn or Q, corresponding
to the notations(<?, <7j,7j, Wj) used in (a)-(f)for Q.
Since Qn G V{S). it follows from (f) that qn and gf1 satisfy the following
equationsbk(t,S(t))qn(t)+Ylakj(t,S(t))gf(t)=O

By Lemma 5 it holds
lim(Pa.s.),

k= l,...,K.

(13)

that
<tJ-

=9j(inTherefore,taking a subsequence
the following
resultmeasurew.r.t.

if necessary,

dtxdP),

j=1,...,dfrom (13)

and

(14)

(14)

we have obtaineddbk{t,S{t))q{t)+YJakj{^S{t))gj{t)

^Q

(dt

xdPa.8.),k=\,...,K.

j=i

Since Q <C P. from this formula and (d) it follows that Q is S-martingale
measure.
Thus we have proved that Q à¬M(S),and that Q is the minimum point in M{S)by
(12).
The fact that Q is also the minimum point in P(S)follows from Theorem 2.
The fact that Q is obtained by Girsanov's transformationfrom the original probability
measure P, is follows from (e). The proof is complete.
(Q.E.D.)
Remark 3 Follmer and Schweizer 1990 discussed the similar problem in the space
V2{S)- {Q E V(S);Sis square integrable
w.r.t. Q and ^ G L2(P)}which isa subspace of V(S)and A't(S').
They introduced
the concept of minimal martingale measure(mMM) in V2{S),and gave the sufficient conditions
for the existenceof mMM. We can verify that, when the system (1) is of simple form (for exampleb(t, S) and a(t, S) are linear in S), the mMM is identified
with canonical martingale
measure. So we come to the following
conjecture.

352

Conjecture.

If mMM exists,

thenmMMis identified

withthe canonicalmaltingale

measure.
4. Concluding

Remarks

In this paper we have investigatedthe existence problem of canonical martingale
measures. All our results are obtained
without
the assumption of the square integrabilityof Radon-Nikodym
derivative-^. So our methods can be applied to many
more cases. And if it is necessary to extend the class of martingale
measures to theclass of local martingale
measures, then it may be possible to apply our methods tothat of local martingale
measures.
In the case of SDE discussed
in section 3, if d < K, then the equivalent
Smartingale
measure is unique in general. Therefore, in order that the assets markets
are incomplete,
it must hold that d > K. On the other hand, if the price process
S(t)is a jump process, then the equivalent
S-martingale
measures are not unique(without
the assumption of d < K) in general.
So it may be an interesting
problem
to apply our methods to such processes. This is our next subject
to study.
5. Acknowledgements
The author

thanksProfessor

C. Strieker

for his usefullcomments on section

3.

References
1. I. Csiszar,
Annalsof Prob. 3(1)(1975),
pp. 146-158.
2. I. Csiszar,
Annalsof Prob. 12(3)(1984),
pp. 768-793.
3. H. Follmer
and M. Schweizer,
in Applied
Stochastic
Analysis,ed.
and R.J. Elliot( Gordonand Breach,1990).
pp. 389-414.
4. J. M. Harrisonand S. R. Pliska,
Stochastic
Processes
and Appl.

M.H.A. Davisll(1981),pp.

15(1983),pp.

215-260.

5. J. M. Harrisonand S. R. Pliska,

Stochastic

Processes

and Appl.

313-316.

6. S. Ihara,

Information

Theory for Continuous

Systems(WorldScientific,

1993),

p.308.

7. I. Karatzas,(1991),

pp.

D. L. Ocone and J. Li,

Stochastics

and Stochastics

Reports

127-131.

8. R. S. Liptser
and A. N. Shiryaev,
Theory ( Springer-Verlag,1974),

Statistics
P. 394.

of Random

Processes:

1 General37