S ÉMINAIRE DE PROBABILITÉS (S TRASBOURG )
DARRELL D UFFIE
Predictable representation of martingale spaces and
changes of probability measures
Séminaire de probabilités (Strasbourg), tome 19 (1985), p. 278-284
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Representation of Martingale Spaces
Changes of Probability Measure
Predictable
and
Darrell Duffie
Graduate School of Business
Stanford
Stanford,
University
California 94305, USA
ABSTRACT
We
the
predictable representation of martingale spaces under a change of probability
measure. The canonical decomposition of special semimartingales provides a simple route
to the identity and cardinality of a minimal generating subset of martingales under a change
study
of probability.
1.
Introduction
"high"- dimensional spaces of martingales can be generated by a fixed vector
of martingales via predictable representations. This property has found application, for
example, in stochastic control [1], filtering [11], and more recently in the economics of
security trading [5]. The classic study by Kunita and Watanabe [9] of square-integrable
martingales has been widely extended; a book [8] and paper [7] by Jean Jacod cover much
Certain
of the
theory
I
am aware
of.
subspaces under different probability measures, in particular the identity and cardinality of minimal generating subsets
of martingales. This cardinality has been termed multiplicity [2], and more generally,
Here
we
characterize
an
association between martingale
q-dimension [8]. Under regularity conditions on the change of probability measure, the
q-dimension of the space of q-integrable martingales is invariant. A generating vector of
local martingales under one probability measure maps to a generating vector of local martingales under a new probability measure via the transformation specified by "Girsanov’s
Theorem"
.
instigated by a study of multiperiod security markets [4], a setting
predictable representation property under a change of probability plays an
This paper
in which the
was
important role [6].
I would like to thank Ruth Williams and Jean Jacod for comments. An
early draft
appeared as an appendix to my dissertation. For the dissertation in general, I would
to acknowledge the guidance of David Luenberger, David Kreps, and Kenneth Arrow.
like
279
Preliminaries
2.
complete probability space and F {3t; t E R+}
We work exclusively
of sub-tribes of 1 satisfying the usual conditions.
probability space (ft, 3, F, P) for this section.
(n, 3, P)
Let
be
=
a
filtration
be
a
on
the filtered
special semimartingalei X has a unique decomposition of the
.X~ + X°, where .X~ denotes the local martingale part and .X~ denotes the pre-
It is well known that any
form: X
=
Similarly, if X (Xi, ... , Xri) is an R"-valued
whose components are special semimartingales, we write X° for (Xl , ... ,Xn ), and
If I is a set of special semimartingales, we use X ° to denote the set of local
dictable finite variation null-at-zero part.
process
so on.
martingales ~X°;
Xe
=
I}.
.
following general conditions for the existence of real-valued stochastic integrals
with respect to R"-valued semimartingales were developed by Jacod [8]. I use a presentaThe
tion similar to that of Memin
First, let M be
an
[12].
variation real-valued process C
process
c
=
(c;;such that
(optional compensator)
Stieltjes integral
L(M)
martingale. Then there is an increasing finite
and an optional n x n positive semi-definite matrix valued
= c; J ’ C, where as usual [’,’] denotes quadratic variation
R"-valued local
and the raised dot notation A . B is used for the
of A with respect to B
(and
for stochastic
soon
((~;~~
the
as
as
well).
Let
predictable processes H
(Hi, ... , Hn) such that
is locally integrable. If H E L(M) the stochastic integral H’ M is
unique local martingale satisfying
denote the set of R"-valued
defined
integrals
path-by-path
=
[H.M,N] = (03A3HiMi).C,
for every real valued local
(M;, N~
K;
=
In the
increasing
L(A)
j ~;
defined
as
where Ki denotes the
optional
process
satisfying
C.
case
of an R"-valued RCLL finite variation process A
real-valued finite variation process V and
such that A;
Let
.
martingale N,
=
v; V.
If A is
predictable,
denote the set of R"-valued
’V
is
the
Stieltjes integral
we can
predictable
H;u;)
(Al, ... AR), there is an
optional process v (ui, ..., vn)
=
choose
v
processes H
finite variation process. For H E
a
an
L(A),
=
and V to be
=
predictable.
(Hi, ... Hn)
the stochastic
such that
integral
8 ~ A is
’ V.
Finally, let X be an R"-valued semimartingale. Let L(X) denote the set of R"valued predictable processes H such that there exists a decomposition of X as the sum
of an R"-valued local martingale M and an R"-valued finite variation process A with
H E L(M) n L(A). The stochastic integral HX, defined as the sum of HM and HA
.
1
The definitions of
a
special semimartingale and other standard concepts used in this
[8], or Dellacherie and Meyer [3].
paper may be found in Jacod
280
depend
does not
that for the
HE
sum
particular decomposition
of component stochastic integrals
on
the
~~ H; X;, which may not exist for all
L(X).
For any q E
[l,oo)
~X~q
for any
=
positive extended real-valued functional ~~
define the
’
(~q
on
the
semimartingales by
space of
Mo
chosen for X. This definition extends
semimartingale
0
and ~~
~~q
M
denote the
X. Let
oo.
~Lq(03A9,F,P)
~supt0[X,]1/2t
=
We restricition
our
convenience. Extensions of our results to the
4.8 of Jacod
[8].
As is well
known, .M~ is
a
subspace
of local
martingales
M such that
attention to the null-at-zero
general
case are
merely
easily deduced from
Banach space under the
norm
for
Lemma
~~ ’ ~~Q, taking an
equivalence class of indistinguishable processes. A stable subspace
of Mq is a ~ . ~q-closed vector subspace M of MQ such that IAMT E M for every M E M
A E 30, and stopping time T. This is equivalent to stability under stochastic integration,
in the sense that .M is a stable subspace of .M~ if and only if, for any vector local martingale
M whose components are elements of M,
element of .M~ to be
an
,
,
C9(M)
This is
a
is the closure of UMEM
For any set A of
--~
is itself
Of course
a
stable
subspace
of Mq
martingales.
adapted
a.s.
oo
~l.
martingales let ,~q(M)
for all M E M. In fact,
)
subspace of Mq containing
if there is
an
and ATn E ~ for all
For any vector M of local
denote the set of processes which
processes let
in .~. That is, A E
such that Tn
L(M)} ~
For any set .M of local
[8;(4.3)].
C~(M) (8;(4.5)~.
for any vector M of local
HE
(H M E
trivial extension of Jacod
denote the smallest stable
"locally"
-
martingales,
increasing
sequence of
.
times
(Tn )
n.
let
C(M) _ {H ’ M; He L(M)}.
martingales,
LEMMA 2.1. For any vector M of local
stopping
are
=
C(M).
.
and (Tn) be an increasing sequence of stopping times converging
PROOF:, Let X E
1. Define a
to infinity such that there exist Hn E L(M) with XT^ = HnM for all n >
=
=
Let
Let Yl
Hi.
Yn+i Yn on ~O, T"~
sequence (Yn) of processes in L(M) as follows.
=
Yn, the processes (Yn) "paste together" to
and Yn+l = Hn+i on DTn, oo(. Since
Y’ M
form a process Y, which is predictable since Y = limn Yn. Then Y E L(M) and X
=
n.
all
for
I
since XTn
(yM)Tn
.
=
If M is
..
a
stable
whose components
a
subspace
are
of
elements of
q-generator of M and there is
~l is n and M is
a
Mq, a q-generator of M is
q-basis
no
a
vector M of
M. If M
such that
=
martingales
Mn) is
(Ml,
local
...,
q-generator of fewer components, the q-dimension of
for M. Many
examples
are
given by Jacod [7,8].
If M has
no
281
(finite) q-generator,
is defined to be
infinite from
The
its
zero.
This
following result
or
covers
is defined to beinnnite. If ~~!
all cases,
although
it is
possible
{0},
to
its
q-dimension
distinguish countably
=
uncountably infinite q-dimension [8 ;Chapter 4].
the minimum dimension
whether
q-dimension
not the
components of M are in
Suppose, for some
LEMMA 2.2.
.
(A~i,... ,A~,)
q-dimension of a stable subspace M C ,M9 is in fact
of a vector of martingales M which generates M (or M C
shows that the
is
vector of local
a
q E
[1,oo),
martingales
.
that J~t is
a
such that J~!
PROOF: Choose any vector martingale N = (Ni,
with associated dimension process sN, as defined
stable
C
... , N~ )
subspace
,C(M).
.
of
and M
Then
_
whose components
are
=
n.
in M,
denote the
by Jacod [8,p.147]. Let
dimension process associated with M. By assumption, there exists an m x n matrix valued
K~ M, in the obvious sense.
process K whose rows are elements of L(M) such that N
Since the components of both M and N are elements of
Jacod’s Proposition (4.71)
almost surely. Then Jacod’s Theorem (4.74) can be
applies for q 1 and sN
applied to complete the proof. I
=
=
COROLLARY.
Suppose 1 S
oo.
p
q
Then
p-dim(Mp)
PROOF:: This follows from the fact that Mp C Mq.
.
Change
3.
.
of
Probability
Q be any probability measure
Let
I
(i~, 3) absolutely continuous with respect to P. When
defining concepts under Q we work on the filtered probability space (H, 3Q, FQ, Q), where
on
1Q and FQ denote completions for Q. We distinguish definitions for the two filtered
probability spaces (fI, 3, F, P) and (n, 3Q, FQ, Q) by augmenting the notation with "P"
or
"Q",
H 9X,
and
We
in
as
Lp(X)
l~ p and
and
and
LQ(X),
so on.
Let
(~
S(P)
special semimartingales, respectively,
use
the facts that
under P is
a
S(P)
C
and
.
and
Sp(P)
~~
.
and
X~,
and
denote the spaces of semimartingales
under
P, and similarly define S(Q) and Sp(Q).
S(Q) and that the quadratic variation of a semimartingale
Q-version of its quadratic variation under Q. See, for example, Chapter
7 of
[8].
Let the
P-martingale 03BE denote the density process [8, Chapter 7] for the RadonNikodym derivative dQ dP, equating 03BE(t) with the restriction of dQ dP to Ft for all t > 0. For
reference, we identify the mapping
and its domain of definition, the P-local
martingales in Sp(Q). This identification is known as Girsanov’s Theorem, due to Lenglart
[10]
in this
generality. The following form of the theorem is from (8,(7.29)J.
THEOREM 3.1. Let M
°
integrable variation,
E
in which
.
case
Then M e
M~Q
=
Sp(Q) if and only
(03BE-1)-.M, 03BE~P and
if
(M, £J
=
M
is
locally
of
282
The condition
is
trivial
a
bounded
[A~f, $] may be difficult to verify. The following sumcient condition
consequence of [3;yil.39]. As usual, Mp denotes the space of P-essentially
on
martingales.
LEMMA 3.1. Let q E
°
(1, oo)
and q~ E
+ Q~
(1, oo]
=
1.
then
Sp (Q).
C
following result is due
The
to Memin
[12].
PROPOSITION 3.1. Let X be any R"-valued
P-semimartingale. If H ~ Lp(X)
.
then
LQ(Xand H PX is a Q-version of H 9X.
HE
’
The next lemma is
operations
are
technical aid.
a
defined, and so
on
for
whenever the
for other combinations.
Q Sp (P),
LEMMA 3.2. For any M E
.
(a)
We write
both
and
are
in
and:
~p(Q),
M
=
(&#x26;)
(c)
=
~P
(d)
=
0
is predictable, finite
PROOF:: Since
the
M°P
=
claims follow
.
Suppose the components of M = (Mi, ... Mn)
and H E
Sp(Q)
a
Q-version
PM E Sp(Q)
straightforward
LQ(M°Q)
and
elements of
(H
is
a
Q-version
extension of Jacod’s
of H
.
A
and that
proof
of this result for
(H PM)
n
> 1
is
a
proof of [8, (7.26(a))]. Then the result follows from
I
Theorem 3.1.
preliminary result showing
a change of probability.
a
PROPOSITION 3.3.
.
N of local
then HE
= 1, [8,(7.26(a))] shows that H e
case n
(03BE-1)- .(H P.M,03BE~
under
If H
Lp(M).
are
.
PROOF:: For the
We have
(d). Since
remaining
immediately. I
PROPOSITION 3.2.
is
variation, and null-at-zero under P, and Q -~ P,
E Sp ( Q) as well as (c) and
properties hold under Q, proving
forms
the
canonical
M decomposition of M4P under Q, the
same
Suppose
martingales,
the basic
relationship between stable subspaces
~ is essentially bounded.
£qP(M)Q
For any q E
If, in addition,
C
(1, oo)
and any set
dP dQ exists and is essentially
’
bounded, then
~
PROOF::
Only the
proof below.
=
CQ(,M°~).
first assertion is
proved here.
The
proof of the second
is clear
given the
283
and thus
implying a sequence (Xn) converging to X in f I
E MP, where Mn E .M,
such that, for all n, Xn
converging in II
Lemma
E
and
3.2
Hn
3.1,
By Proposition
Hn E
LQ(Mn)and (Hn
By Dellacherie and Meyer [3], (VIL95, Remark (c)),
~
Let X E
=
.
~ X~Q - Hn
for
a
given
Kq depending only
constant
IIqQ-closed, X~
is
x9 II
~qQ
on
q. Thus
X - Hri
Hn
Q.Mn ~qQ
~M~ -+
also
and
=
(A)
in
Since
f.
I
E
’
The
following
may be considered the main result. As
ficient conditions for
measure are
given by
jumps is
a
local
martingale
.
special semimartingale
a
reminder,
special semimartingale
necessary and suf-
under
with convenient sufficient conditions for
given by Theorem 3.1,
Lemma 3.1. It may be worth
a
to be
a
noting
under any
a
a
change
of
q-generator
semimartingale with locally bounded
absolutely continuous change of probability
that
a
measure.
(M1, ...,Mm)) is a q-generator of M p wbose components
are Q-special semimartingales,
and t is locally (P-essentially) bounded, then:
Suppose M
THEOREM 3 . 2.
.
(a)
=
MQ
(&#x26;)
m.
PROOF : {Part (a)1 By definition,
C
MQ. Suppose X E MQ. Let (Tn ) be an
increasing sequence of stopping times such that T,~ oo P-almost surely and ( ~ )T^ is (PE Sp(P) for all n, and by
X E Sp(P).
essentially) bounded for all n. Then
For any n, the quadratic variation (X, X~Q is a P-version of (X, X~p on (0, TnD. Thus, for
--~
.
any n,
K9 II XT^ ~qP ~
B
II XT ~qQ
m
essential upper bound on )T^ and I~q is as given in relation (A). (We have
Q, but for the last claim we can restrict ourselves to (H,
and apply
the results of Jacod jsJ, Chapter ?, in particular Theorem ?.2.) Thus X~ E
and
where B is
(~
an
not assumed P -~
M p~~o~,
by
Lemma 2.2 there exists H E
Lemm a 3.2
we
{Part (b)}
Remark:
have X
such that X~*
3.2 and
1, the assumption that 1/~ is locally bounded
may be
This follows from Lemma 2.2.
For the case q
following corollary
expression (A).
H
By Proposition
=
proving part (a).
=
replaced by the assumption
The
L(M)
=
that
I
MQ C Sp(P).
is verified
.
by applying symmetry and the bound Ifq
used in
284
COROLLARY 1. Suppose Q and P
bounded. Then
.
are
equivalent and ~ as well as
If ~ is
=
M°Q is
a
q-generator
(q-basis)
for
a
t
q-generator
are
locally (essentially)
(q-basis)
for
then
.
References
1.
BISMUT, J., Theorie Probabiliste du Control des Diffusions, Memoirs of The American
Mathematical Society 4 ( 1976), .
2.
DAVIS, M.H. AND P. VARAIYA, The Multiplicity of an Increasing Family of 03C3Fields, The Annals of Probability 2 ( 1974), 958 - 963.
3.
DELLACHERIE, C.
4.
5.
AND P. MEYER, Probabilities and Potential B: Theory of Martingales, North-Holland Publishing Company, New York, 1982.
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AND S.
ematics Journal
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