Introduction
This is the second of three papers on the nonstandard theory
for stochastic integration.
standard theory in its
In the first paper we studied the non-
own right, and we shall now compare that
theory with the standard one.
In the first section we define the
constitute
the class
in this paper.
SL 2 -martingales, which
of hyperfinite martingales we shall work with
This class is a little smaller than the class of A2 2
gales behave more regularly under standard parts.
section we use the fact that an
SL 2 -martingale
In the second
M has
S-right
limits to introduce what we call its right standard part
1 2 -martingale.
we prove that this is an
construct a hyperfinite process
Y
0 M+
; and
X
which
Given a process
is integrable (standard sense) with respect to
0
.
SL -martln-
martingales which we studied in the first paper, and the
M+ ,
we show how to
(called a 2-lifting of
X) such
and we see that standard stochastic inte0 M+
gration with respect to
theory with respect to
M.
may be obtained from the nonstandard
This is a generalization of Anderson 1 s
work in [1], and since the argument is rather similar to his, we
have left some of the more technical lemmas to the reader.
In the
last section we introduce what we have called the "well-behaved''
martingales; that is martingales which on a set of measure one
only one noninfinitesimal jump in each monad.
have
These martingales are
particularly easy to work with 1 and we obtain their basic properties
and show that any
subline
S
SL 2 -martingale
which is well-behaved.
useful in the next paper.
M has a restriction
Ms
to a
Well-behaved martingales will be
- ii -
To prove the equivalence of the standard and nonstandard
theories for stochastic integration, it still remains to show that
all
1 2 -martingales can in a suitable sense be represented as the
right standard parts of
81 2 -martingales.
This is the problem we
shall study in the third paper.
We shall use the same symbols and terminology as in the first
paper, and the reader is refered to the introduction of that paper
for some remarks on the literature.
We only mention that we still
work with polysaturated models of nonstandard analysis. (See[11])
A reference containing the roman numeral
I
is to the first
paper [5]; hence Theorem I-14 is Theorem 14 of that paper; and a
similar convention applies to III and the third paper [6].
- 1 -
SL 2 -martingales
1.
In [ 5]
we studied the class of
A2 -martingales, and we now
define an important subclass:
Definition 1 :
M : T x n -+ *JR
A hyperfini te martingale
internal basis
<O,{Gt},P>
2
Mt E 8L (O,Gt,P)
will be called an
for each finite
adapted to an
SL 2 -martingale
M is called a local
t E T.
SL 2 -martingale if there exists an increasing sequence
0•
such that
n EN •
n
(w)
+
a.e. and
m
The sequence {•n}
Due to the
M•
n
if
{• n }n EJN of s.t.
8L 2 -martingale for each
is an
is then called a localizing sequence for
M~
SL 2 -martingales behave more
S-integrability, the
regularly under standard-parts than the
A 2 -martingales.
However,
S-integrability is difficult to check and consequently some properties of
SL 2 -martingales are harder to prove than the corresponding
properties of
A 2 -martingales.
quence of Lemma I-3
that a hyperfinite martingale
E(M~+[M](t))
tingale if and only if
The corresponding result for
an
t ET .
seem to be stuck.
A 2 -mar-
is finite for all finite
M~ + [M](t)
is
t ET.
M is
S-integrable for
But this is not obvious, and the reason is that
we must now consider
need not be in
M is a
8L 2 -martingales would be that
8L 2 -martingale if and only if
each finite
A
It was, for example, an easy conse-
JA(M~+[M](t))dP
G0 ,
for
AEGt ~
P(A)R:SO.
Since
we cannot use Lemma I-3 as above, and we
Using another characterization of
S-integrability,
we may, however, prove the assertion:
Theorem 2:
Let
internal basis
only if
M be a hyperfinite martingale adapted to the
<O,{Gt},P>.
Then
M~+[M].(t) ESL 1 Cn,Gt,P)
M is an
SL 2 -martingale if and
for each finite
tET.
-
Proof:
Since in either case
2 -
A 2 -martingale,
M is a
it is enough
to prove the theorem for such martingales.
We shall use the following characterization due to Anderson [1 ]:
If
f : Q
+
~
is an internal, *-non-negative, Gt-measurable function)
f ESL 1 (Q,Gt,P)
then
if and only if
0
ffdP =
f 0 fdL(P).
By Proposition I - 17
and taking
*-expectations we get:
E(M~+[M](t))
(1)
since
fMdM
is a martingale.
Define a sequence
Then
= E(M~)
fM,
n
{Tn}nElli
Tn ( w )
= min { sET : IM( s , w ) I ~ n }
dM
is a
'n
A 2 -martingale
follows from Lemma I - 12 that
finite
of internal stopping times by
.
for each
JtM dM,
o 'n
n
is
n E:N ,
and it
S-integrable for each
t ET .
By the result of Anderson quoted above
Since
a. e., we get
For almost all
for
m~ n ,
w there exists an
and thus
The sequence
0
[M, ] (t)
0
[M] (t)
n
0
[M, ](t)
n
which is integrable since
have
+
n EJN
0
(maxJ'-1 2 )
s~t s
and
such that
M, (t)
n
+
'm ( w)
>
M(t)
a.e ..
is increasing and bounded by
E( 0 [M](t)) ~
0
E([M](t)) <
and by Doob's inequality
oo
0
t
[M](t)
We also
~
E ( 0 max M2 ) <
s -
0
s~t
0 (
and thus
3 -
E (max M2 )
s
<
00
s~t
2 .
max Hs)
Applying Lebesgue's Convergence
is integrable.
s~t
Theorem to both sides of (2), we obtain
(3)
Combining (1) and (3) we see that
0 E(M~+[M](t))
OE(M2)
t
= E( 0 (M~+[M](t)))
if and only if
= E(OM2)
t
M~+[M](t)
Anderson's characterization now tells us that
grable if and only if
is
8-inte-
is, and the theorem is proved.
We just mention one other result of the same type which was
proved in [ 4 ] :
If
M
J.S
8L 2 -martingale, then
an
8- integrable for each finite
2
maxMs
s,:::t
is
t ET .
Also notice the combination of Doob's inequality and Lebesgue's
Convergence Theorem in the proof above, it will reappear several
times in the sequel.
Our next result shows that the class of
8L 2 -martingales is
reasonably closed under stochastic integration.
Recall Definition
I - 18.
Proposition 3:
JXdM
is an
X E 8L(M) ,
Proof:
If
M
J.S
an
8L 2 -martingale.
then
JXdM
X E 8L 2 (M) ,
8L 2 -martingale and
If
M is a local
is a local
then
8L 2 -martingale and
8L 2 -martingale.
The second assertion follows from the first by definition of
8L(M)
Assume that
sider the case where
such that
IXI
<
n .
M is an
X E 8L 2 ( M)
Then
8L 2 -martingale, and let us first conis finite, i.e. there is an
n EN
- 4 -
and it follows from Theorem 2 that
fXdM
X E SL 2 (M) .
Let us now consider the general case
exists a sequence
0
that
f
TtxQ
n
+
t
Then there
of finite elements 1n
{Xn}nElli
IX 2 -x 2 jdvM
81 2 -martingale.
is an
o
as
n
+
oo, and
(recall the
comments leading up to Theorem I - 21 , or see Anderson [ 1 ] ) . We have
t
t
o < 0 E([JXdM](t)- [fx dl1](t)) = 0 E( r:x2 ~~-!:X 2 ~M2 ) = 0 f<x 2 -X2 )dvM + o , as
n
o
o n
Ttxn
n
t
n + oo •
Since each [ Jx dM] (t) is S-integrable it follows that
n
[fXdM] (t)
S-integrable~ and hence by Theorem 2
is
that
fXdM
is
SL 2 martingale.
an
2. The right standard part of
According to Theorem I-9, local
and
S-left-limits
A. 2 -martingales
A. 2 -martingales
have
S-right-
a.e., and thus the following definition makes
sense.
Definition 4:
a process
0
Let
M :Txn
+
H : F+ x n + F
S-right-limit of
t
+
+
*JR
be a local
M(t,w)
M
M+( r,w)
at the monad of
is called the right standard part of
the left standard part of
0
by letting
A. 2 -martingale.
M.
to be the
We want to make a martingale of
0
Define
be equal to the
r.
The process
0
M+
In a similar way we define
S-left-limit
tt ,
0
-
M
of
M.
and we first construct
the stochastic basis:
If
Hoo
=
<Q, {G t}, P >
cr(U{L(Gt): t
is the internal basis of
finite});
M,
define
i.e. the completion with respect to
of the
a-algebra generated by all the Loeb-algebras
finite
t ET .
Let
N be the set of all null-sets of
L(Gt)
H00
L(P)
for
We define
- 5 -
a family
{ Ht} tEIR
Ht
A family
=
of a-algebras on Q by:
+
a(NU(U{L(Gs): sET, s R~t})).
{Ht}tEIR
of smaller algebras is defined by
+
=
Ht
Lemma 5:
a ( U{ L (G s ) : s E T , s
For each
Ht
=
H't
=
R~
t}) .
t EJR+
U{cr(L(Gs)UN) :sET, s
R~t}
and
Proof:
Obviously
SR~t}
U{a(L(Gs)UN) :sET,
cHtca(U{a(L(Gs)UN) :sET, SR1t}
and it is enough to prove that
a-algebra.
Let
{A }
Sn
= *[s n ,t+1]
n
R~t}
.
s
E nsn ,
is a
A E a(L(Gs )UN) .
The family
n
n
is a countable family of internal sets with the
nT
Assume
finite intersection property, and by saturation
,...,
R~t}
be a countable family of sets from
n
U{a(L(Gs)Uf'll) :sET, s
U{a(L(G )UN) :sET, s
s
and
then
,..,
s >
s
- n
for all
n •
n Sn
=1=
¢ .
Let
nE.N
Consequently
A E a(L(G ..... )UN)
for each n EJN and hence
n
s
UA E a(L(G ..... )UN) c U a(L(G )UN) .
Since Ua(L(Gs)UN) clearly has
n
s
SR~t
s
got the other properties of a-algebras, the Ht-part of the lemma
is proved.
The
H'-part is similar.
t
Using Lemma 5 and basic properties of the Loeb-measure we have .
Lemma 6:
If
that
Let the family
A E Ft ,
{ Ft} tEIR
there exist an
L(P)(A6B)
=
0 .
be either { Ht} or
+
s E T , s R~ t , and a set
t:J .
{ H.
BEG
s
Then
such
-
= F+ = n F
t
s>t s
F
t
(b)
(c)
6 -
for all
For all null-sets
N
t ER+
in
F"" ,
N EFt
for all
t ER+ .
The properties (b) and (c) above are usually called "Heyer's
usual conditions" and are assumed in most standard theory for stochastic integration (see Metivier ( 8 ]).
The following nonstandard version of Egoroff's Theorem is often
useful:
Proposition 7:
and let
Xn : n
{X }
n
*JR
+
sequence
Let
Let
with
{Yn}
converges
v E
w E n'
we have
Let
0
X = Y
Loeb [ 7].
Yn
P( B ) > 1
m
let
n
k,m
Y
_l_
m
X
and assume that the
n
a.e. to a random variable
a set
&1
n E
Y
on
of Loeb-measure one
1
*J.J 'JN
n
,
<
be an internal random variable on
L(P)-a.e.; such an
X
v ,
and all
m EJN
a set
uniformly on
A
such that
c: A
B
m
m
m
be such that for all
<n,G ,P>
such
exists by Proposition 2
By Egoroff's Theorem,
+
0
G-measurable functions
Xn ( w ) = Y ( w ) •
there exists for each
that
=
n
such that for all
0
X
Y
Then there exist
*JN 'JN
and a
that
be a hyperfinite probability space
be an internal sequence of
n~E;
<&1,L(G),L(P)>.
Proof:
<n,G,P>
~ee
e.g. Royden [ 1 0 ] , page 7 2 ) ,
with L( P) (A ) > 1 _.:!.
m
m
m
We may find a B EG with
A
and
w E
0
Bm
X =Y
on
and all
B
m
n EN ,
For each
n >- nk ,m
Let
llk,m = max{nE*JN: !Xq(w)-X(w)l
such that
such
m
implies
q E *JN
of
<~
for all
nk,m ~ q::, n} .
wE Bm and all
kElN
- 7 -
By definition of
nk
~m
we must have
v E *:N -...N
saturation we may find a
It follows that for all
0
X (w)
n
=
Lemma 8:
Let
H : T x
internal basis
1"2
Q1
*.R
-+
=
,m E *N 'N .
less than
and all
Putting
Y(w).
llk
U B
m Elli m
n
<
v ,
we have
we have proved the proposition.
be a local
:>..
2-martingale adapted to the
Then the right standard part
<Q,{Gt} ,P>.
l<,m EN.
for all
llk,m
n f. *::N'-JJ '
Using
0
M+
is a
right continuous process adapted to the stochastic basis <Q,{Ht},L(P)>.
Moreover, for each
tF'II:1t,
Proof:
t >s
t ER
such that
and each
0
definition.
=
(M(t,w))
0
The right continuity of
s ET ,
0
s
F'll:1
t ,
there exists
t ET ,
H+(t,w) L(P)-a.e ..
M+
follows immediately from the
The rest of the lemma follows by applying Proposition 7
to the sequence
element of
T
{M(t+1/n,w)}nE*::N,
larger than
where
t + 1 /n ,
M :R+ x Z-+ R
<Z, { Ft}
is called an
denotes the least
N c Ht .
and using
Recall from Metivier [ 8 ] that if
basis, a process
t +1/n
>
,l.l
is a stochastic
L 2 -martingale if
is a martingale with respect to the basis and if for all
the random variable
w -+ M(t,w)
M
t ER+ ,
is an element of
We may now conclude:
Proposition 9:
Let
to the internal basis
*R
M :TxQ-+
be a
:>..
2 -martingale with respect
Then
<Q ,{Gt} ,P>.
°M+
is a right continuous
.
1 e w1t
. h respect to t h e stoc h ast1c
. b as1s
.
L 2 -mart1nga
Proof:
We already know that
adapted to
find a
<Q, { Ht}, L( P) > ,
t E T, t F'll:1t ,
0
M+
is a right continuous process
by Lemma 8.
such that
0
E( 0 M+(t) 2 ) = E( 0 H(~) 2 ) ~ 0 E(M(~) 2 )
and hence
0
H+(t)
<
M+(t) € L 2 (n,'Ht ,L(P)).
~
Also, given a
=
0
M(t)
since
a. e..
M
is a
t ER+
we may
But then
:>..
2 -martingale,
-
Let
A E Hs ,
A
find
s, t
ET
= 0 .
s <t .
We shall show that
By Lemma 6 (a) and Lemma 8 we may
M(s)
0
such that
that there exists a
since
s, t EJR + ,
and let
f ( 0 M+(t)- 0 M+(s))dL(P)
8 -
BEG ......
s
=
0
0 ~1Ct)
= 0 M+ ( t)
L(P)(Bl\A) = 0 •
with
M is a martingale and
M+ ( s) ,
BEG .....
s
and such
\tJe have:
To get the standard part
outside the integral we have used Lemma I-12 which implies that
M(t)
and
M(s)
A process
are
S-integrable.
M: JR
X
+
there exists a sequence
Ma
a.e. and each
z
+
{a }
n
is an
n
JR
This proves the proposition.
is called a local
of stopping times such that
L 2 -martingale.
0
an
+
(JO
In view of Proposition 9
it is natural to guess that the right standard part of a local
x2 -martingale
L 2 -martingale.
1s a local
But this is false as the
following example shows:
Example 1 0:
Choose
y E
*JN -....JN
and let
Let
The internal measure
n
P on
consist of
fJ
£
=
1
L 2
n=l n
y
is defined by
then
P{w +}
n
elements:
2y
£ R~II2f 6 •
=P{w n- }
1- -2£n2
.
T = { kf n: k E *1--T, k < n 2 } for some
We shall use the time-line
n E *JN ,1,r •
Gt =
The family
{Gt}
{0,n}
t <1-1/n; G 1 _ 11 n
by the sets
for
for
of internal algebras is defined by:
{wn+'wn_} ;
and
Gt
is the internal algebra generated
is the internal power set of
t >1 .
The martingale
M(t,w) = 0 for all
t > 1 .
w
M
is defined as follows:
if t
< 1
--1n
M(t,w n +)
= -M(t,w n- ) =
n
if
n
- 9 -
M lS obviously a hyperfinite martingale adapted to the
<ri,{Gt},P>;~and
internal basis
using the sequence
-rn(wk+) = Tn (wk - ) = 1 - 1 /n
we see that
M is a local
0
Let us show that
{a
n
}
a. e. .
where
k<n
if
8L 2 -martingale.
L 2 -martingale.
is not a local
Let
be an increasing sequence of stopping times such that a n
For
t <1 ,
Loeb-measure
n E:N
M+
'
k>n
if
Tn(wk+) = Tn (wk - ) = 1 + (n-k)
{-rn}
the
and
0
a-algebra
must have measure zero.
sequence for
+
n
But then
oo
consists only of sets of
L(P){w:a (w) <1} <1,
such that
Consequently
Since
1 .
Ht
+
00
a.e. there must be an
and consequently this set
an ( wk+) :::: 1
for all
k ElJ .
E( 0 M+ (1) 2 ) = oo and {a} is not a localizing
an
n
0 M+ .
This proves that 0 M+ is not a local L 2 -
martingale.
We can get more out of this example:
a subline of
Then the restriction of
T .
A2 -martingale.
Let
= T-.....{1-l}
n
M to
Thus a restriction of a local
necessarily a local
S
S
be
is not a local
SL 2 -martingale is not
A2 -martingale.
3. Stochastic integration with respect to
0
M+
In this section we compare the nonstandard theory for stochastic
integration with respect to
gration with respect to
0
M with the standard theory for inte-
M+ .
Let us briefly review the standard definition of a stochastic
integral.
Let
(Metivier [ B]):
N: JR+
x
Z
+ JR
.
.
b e a rlght-contlnuous
respect to the stochastic basis
<Z,{ft},~>.
1 e Wl. t h
.
L 2 -martlnga
The
a-algebra
P of
- 10 -
predictable sets with respect to
JR
subsets of
for s , t E JR + ,
+
xZ
<Z, { Ft}, Jl >
<s,t) xF
generated by the sets
F
s <t ,
s
EF
We may define a unique measure
X
p
on
ai EJR ,
N
n
:L a. 1
J F (t,w)
<si'ti x s.
i=1 l
l
we define the stochastic integral of
to be the process
JXdN
a 1. 1 F
1=1
If
X
defined by
A2 (N)
( w ) ( Nt , t- Ns . At) =
S•
X with respect
defined by
n
( t , w ) -+ • I:
Let
by
is a predictable process of the form
X(t,w) =
to
{O} x F 0
mN({O}xF 0 ) = 0.
If
for
and
s
P •
predictable if it is measurable with respect to
and
a-algebra of
A process is called
and
s
is the
1A
l
1
lS a predictable process let
(t)
X
( s , w) = X( s , w)
for
X(t)
s <t ,
t
I 0 XdN
.
be the process
(t)
X
( s , w) = 0
be the set of all predictable processes
X
for
such that
A 2 (N)
We extend the stochastic integral to the class
s > t .
by
noticing that the processes of the form
X
are dense in
n
I:
= l=O
a.1
• F
l <s.,t.Jx
2
1
L (JR+xZ,P,mN) ,
1
si
and that the mapping
X -+ ("XdN
-o
is
an isometry from this dense subset into
If we denote
the unique extension of this isometry to the whole of
L 2 (Z,F 00 ,Jl)
also by
XEA 2 (N)
X -+ J 00 XdN ,
0
we define the stochastic integral
to be the process
fXdN
for
- 11 -
Let us return to our nonstandard setting:
be a
:\ 2 -martingale adapted to the internal basis
. .
9,
P ropos1.t1.on
to
0
M+
We write
spect to
{Ht}
P'
and
P
<Q, {G t}, P> .
By
L 2 -martingale with respect
is a right-continuous
<Q,{Ht},L(P)>.
M : T x Q + *.R
Let
for the predictable sets with re-
for the predictable sets with respect to
The difference between the two classes is not large:
Lemma 11 :
For each
mN(A~B) = 0
there exists a
L 2 -martingales
for all
N
B E P'
=0
L(P)(CtlD)
Let
Tf
DE H't
IR xQ
from the power set of
+
T.
into the power
by:
TI(A)
see that
there 1.s a
denote the set of finite elements of the time-line
Tf x Q
Since
<!:2,{Ht},L(P)>.
•
We define a mapp1.ng
set of
such that
adapted to
Follows from the fact that for
Proof:
with
AEP
= {(t,w)
ETxQ
( 0 t,w) EA}.
1.s really an inverse image operation, it is easy to
1T
commutes with all countable Boolean operations.
1T
A c: TxQ
A set
is called adapted (to the basis <1:2 '{G t} 'P >) if
Let A be
Gt ' tET.
the internal algebra of adapted sets, and let L(A) be its Loebeach section
At
is l.n the corresponding
algebra with respect to the internal measure
ition of
vM
vM
(remember the defin-
preceding Definition I-18.)
Our task is to compare stochastic integration with respect to
with stochastic integration with respect to
S-lim M(s,w) ,
s+t
of
0.
A local
if
0
0
M+
0
M+
cannot register what happens to
M
in the monad
To make sure nothing significant happens there, we define:
A2 -martingale
M(O,w)
= 0 M+ (O,w)
M
is said to be
a.e ..
M
S-right-continuous at
0
- 12 -
We may prove:
Lemma 12:
at
0.
Let
Then the image under
H't
respect to
sets
8L 2 -martingale which is
M be an
L(A)
of the adapted
By the definition of the predictable sets, it is enough to
Fs EH~.
n(<s,t]xF )
s
n({O}xf 0 )
or
8L 2 -martingale and that
M
8-right-continuous at
both are needed in the proof.
Lemma 13:
0.
Let
0
M be an
Then the mapping
Proof:
lS ln
We leave the details to the reader, and only remark
that the conditions that
at
with
L(vM) ) .
prove that any set of the form
L(A),
P'
of the predictable sets
is contained ln the Loeb-algebra
(with respect to
Proof:
n
8-right-continuous
is an
8L 2 ~martingale which is
from
n
The measure
to
is
8-right-continuous
L(A)
is
measure-preserving~
is uniquely determined by its values on
the sets of the form
<s,t]xFs,
countable Boolean
operations~
L(vM)(n(A))
A
for
P'
M
{O}xf 0
,
and since
n commutes with
it lS enough to prove
of this form.
Again we leave the details to
the reader, but remark that we make vital use of the two conditions
on
M •
M be as ln Lemma 13.
Let
image
n(P)
of the predictable sets under
From Lemmas 11 and 1 2
set
C E n(P)
that if
f : Tf x
expectation
n(P)
n L(A)
see .that
n L(A)
Q +
JR
is
ECflnCP)nL(A))
and the measure
mo +
M
and
by
TI
we see that for each set
such that
""
m0 +
M
We define a measure
mM+(Bt.C) = 0
0
mo +(nA)
M
B E n ( P)
on the
= mo
+(A).
M
there is a
It follows from this
•
n ( P) -measurable, then the conditional
of
f
m0 +
M
with respect to the sub-a-algebra
equals
agree on
TI(P)
f
a.e ..
n L(A)
.
By Lemma 13 we
- 13 -
X E A2 ( 0 M+) , we define
If
The process
X'
X' :Tf x
will obviously be
the conditional expectation of
X'
the random variables
Definition 14:
2-lifting of
close
X'
Y E SL 2 (M)
Y
if
on
Tf
and
n L(A)
is
,
*JR
x Q -+
and
X"
is called a
are infinitesimally
x Q •
fXd 0 M+
Our purpose is to show that
X,
X"
M
Y : T
L ( vH) -a. e.
2-lifting of
with respect to n(P)
.....
are equal m0 + -a. e •.
An adapted process
X
X' (t,w) = X( 0 t,w) .
by
JR
n(P)-measurable, and if
X"
and
Q -+
= 0 (fYdM)+
when
Y
is a
and hence to derive the standard theory for sto0
chastic integration with respect to
for integration with respect to
M+
from the nonstandard theory
M.
The notion of lifting is central 1n the theory of Loeb-spaces;
liftings of random variables
were first studied by Loeb in [7]
further developed by Anderson in [1]
and [2].
and
Liftings of processes
were introduced by Anderson in [ 1 ]
to treat stochastic integration
with respect to Brownian motions.
The importance of the concept is
perhaps most easily seen from Keisler [ 3 ]
where different classes
of processes are characterized by what kind of liftings they allow,
and this again is used to prove properties
o~
the solutions of
stochastic differential equations.
But let us return to our problem.
enough
We first prove that there are
2-liftings.
Lemma 15:
Let
H
be an
SL 2 -martingale
Then
X
has a
S-right-continuous at
2-lifting with respect to
0 ,
M.
{t } be an increasing sequence of finite elements of T
n
such that ot -+
Let x(n) be the restriction of X" to
n
Then X(n) E L 2 (A ,L(A ),L(vH ) )
A = (*[O,tn]nT) xQ.
where A
n
n
n
n
--n
'
Proof:
Let
CX>
•
- 14 -
and
VM
are the restrictions of
n
By Theorem 11 (ii) of Anderson [1 ]
A
and
respectively to
there exists for each
n
Y(n) E SL 2 (A ,A ,v~1 ) such that 0 Y(n) = X(n) L(v~)- a.e..
n n !n
may choose the y(n)'s such that yCm)~A = y(n) for m > n .
An .
a
~rJe
n
{Y(n)}
saturation we may extend the sequence
sequence
{Y(n)}
nEll
with the same property.
ns.n
By
to an internal
Y
Choosing
= y<n>
we prove the lemma.
The next lemma tells us that which lifting we choose does not
matter:
Lemma 16:
SL 2 -martingale which is
M be an
0 , an d 1 e t
a.e. a t
of
Let
X E
A
lt
2 ( 0 M.+)
•
.
n' en
X, there exists a set
0
( JtY(s ,w)ill1(s ,w))
0
If
Y
and
Y'
S-right continuous
are
2-liftings
of Loeb-measure one
= 0 <ft_Y' (s ,w)dH(s ,w))
such that
for all
0
Proof:
Applying Doob's inequality to the positive *-sub-martingale
t
t
(t,w)-+ I <J Y(s,w)dM(s,w)- f Y' (s,w)dM(s,w) I we get:
0
E( sup
=
0
0
If sY(r,w)dM(r,w) - f s Y
1
s~t
But
0
o
o
t
E( Cf (Y-Y' )dM)2 )
o
=
£0 (Y-Y' ) 2 dL(vM) = 0
Tt n
0
o
o
I2 ) ::
t
E(I:(Y-Y' ) 2 M1 2
o
)
for all finite
s
s
E(sup(J YdM-J Y'dM) 2 ) = 0
ss.t
(r~w)dH(r,w)
t
o
=0r
Tjn
(Y-Y' ) 2 dvM
t E T.
for all finite
2
4E( <J (Y-Y' )dM) )
t ET
It follows that
and the lemma
is immediate.
We may now obtain the main result of this section, the comparison between standard and nonstandard stochastic integration:
- 15 -
Theorem 17:
continuous at
Let
0
with respect to
M.
.
X E A 2 (M)
and let
'
SL 2 -martingale which is
be an
M
y
Let
S-right2-lifting of
be a
X
Then
fXd 0 M+ = °CfYdM)+ .
Proof:
/Xd 0 M+
We first remark that since the process
is only
defined up to equivalence, the equality in the theorem must be
interpreted as equivalence; by Lemma 16 the equivalence class of
0
(/YdM)+
is independent of the choice of
Y .
enough to prove the theorem for some lifting
Let us first assume that
where
s, t E:R+ , s
s E·T , .....s R1 s and a
we may choose
""'
s
<
t ,
and
It is therefore
Y .
X
is of the form
X
F s € Hs .
= 1
.
<s,t]xFs'
By Lemma 6 (a) there are an
such that
such that
•
the same lemma we f1nd
a
0
MCs,w) =
,....
t ET ,
t
R1
t ,
By Lemma 8
0
H+ (s,w)
L ( P) - a . e . , and by
such that
0
,...,
M(t
,w) =
0
M+( t 3 w ) •
Define
We shall prove that
infinitesimally close to
t
For each
L(vM)- a. e..
YR-~1A
Y
obvi-
Y
is
Let
r E:R+
=1
we have
Fs
X", and by the choice of
a.e. and hence that
X.
JrXd 0 M+
o
X; since
it is enough to prove that
is a version of
it follows that
2-lifting of
X"
2-lifting of
U
n [(*<s+.:!.,t+.!.]nT)xG ] ,
nEil'J mEil'J
n
m
s
then the process
and
is a
SL 2 (M)
ously is an element of
A=
Y
( 0 M+ - 0 M+ )
rAt
rAs '
Y
is a
""'
s
- 16 -
r
while for each
ET
and by the choice of
s
and
t
L(P) -n.o.a.
This proves the theorem for
of the form
X
by linearity the theorem holds for all
X
1
<s,t]xF , and
s
of the form
To prove the theorem it is then enough to show that the
mapping
0
X(t) +
(/YdM)+(t)
1 2 -isometry for all
is an
t EF+.
We have
t+l/n
= lim
E(( 0
n+oo
b
t+l/n
Yd.M) 2 )
= lim 0 E((!
n+oo
t+l/n
= lim 0 E( I: Y2 1lli2 ) = lim 0 /
n+oo
0
Y2 dvM =lim
n+ooTt+l/;n
r dmo + = f x< t)
[o,t)xn M E xn
=f
Yd.M) 2 )
J 0 Y2dL(vM)
n+oo Tt+l/nxn
2
=
f
0 Y2 dL(vM)
n([o,t]xn)
elmo + '
+
M
where we have used the usual combination of Doob's inequality and
Lebesgue's Convergence Theorem to introduce the limit; the fact
that
/YdM
is an
SL 2 -martingale (Proposition 3) to get the stan-
dard part outside the expectation; and that
Y E SL 2 (M)
to move it
inside again.
The rest of the equalities follows from the assumption
that
2-lifting of
Y
measures
is a
L( vM)
,
moM+
and
X ,
and the basic relations between the
moM+ .
As we have already noticed, the equality
f x< t )~m 0
JR xn
+
+
M
establishes the theorem.
E( 0 (fYdM)+(t) 2 ) =
- 17 -
So far we have only dealt with integrals of the form
where
N
L 2 -martingale and
is an
L 2 -martingale and
X
X E A2 (N) .
predictable, we define
is
if there exists a localizing sequence
X € A2 ( N
On
)
for each
If
n EN.
{on}
for
N
X
N
fXdN
is a local
to be in A(N)
such that
The stochastic integral
is then
fXdN
defined as the limit
The theory developed above can
lim fXdN
.
0n
n+co
be extended to the case where M is an SL 2 -martingale and
We sketch the construction:
an internal stopping time
<s&,{Gt},P>
X E A2 ( 0 (M
Y : T xn
each
MT
Tn
such that
) +)
*.R
+
.
n
0
for each
such that
Such a
Y
Tn
For each
n
we may find
adapted to the internal basis
Tn =on
n EN ,
Y is a
L(P) -a.e ..
We can then prove that
and we can find an adapted process
2-lifting of
X
with respect to
is called a local 2-lifting of
X.
The
result follows from Theorem 17:
Corollary 18:
Let
continuous at
0 .
Y € SL(M)
M be an
Let
SL 2 -martingale which is
X E A( 0 M+) .
Then
X
0 M+
has a local
2-lifting
and
The statements above were proved in [4 ] •
a local
S-right-
The problem for
M
SL 2 -martingale is more troublesome since by Example 10
need not be a local
~ 2 -martingale, but by using a localizing
sequence of stopping times, we should usually be able to reduce the
problem to the
SL 2 -martingale case.
We end this section by a remark on the right standard part.
It may seem that the use of this process has been a little unnatural,
and that it would have been better to work with the standard part
X n + R
defined by 0 M(t,w) = 0 (M(t,w)).
The
+
advantage of our procedure is that the right standard part is always
process
0
M: R
- 18 -
right continuous, and - since the standard theory for stochastic
integration is developed only for right continuous martingales this makes comparison with standard treatments easier.
Also, our
method leaves the standard martingale invariant under restriction
of
M to a subline, and we shall see in the next section that this
may be of importance.
4. Well-behaved martingales and the quadratic variation
A central notion in this paper and in [5 J
is that of the
quadratic variation of a hyperfinite martingale.
L 2 -martingales
notion for real-valued
N: :R+ x Z -+ :R
~
an increasing sequence of partitions
n = {O=t o <t 1 <···<t k <···}
the positive real numbers such that the diameter
n-+
tends to zero as
quadratic variation
,and
oo
[N]
For a proof that
of
[N]
sup
N
~
n
=
We have a similar
oo ,
o(~n)
= sup(t. -t.)
.t.E~ 1+1 1
1
then the
of
n
1s defined by
exists and is well-defined, see
Metivier [8 J , page 234.
If
M: T
standard part
x
n
0
-+
M+
*R
is an
SL 2 -martingale, we form the right
and its quadratic variation
also form the quadratic variation
increasing process it must have
its right standard part
0
[M]+ .
natural to ask when the processes
[M]
of
[ 0 M+] .
M;
We may
since this is an
8-right-limits, and we can define
In the spirit of Section 3 it is
[ 0 M+]
and
0
[M]+
are equal.
That this question have some real importance may be seen by comparing
the nonstandard version of the Transformation formula (Theorem I-22)
- 19 -
with the standard version on page 265 in Metivier [8]
0 M+
;
applied to
ln the first case we integrate with respect to
second with respect to
[ 0 M+] .
[M]
in the
If we want to deduce the standard
form from the nonstandard, we must know the relationship between
[ M]
[ 0 M+ J
and
•
SL 2 -martingales where
It is easy to make examples of
A closer inspection of such examples makes the
following definition natural.
Definition 1 9:
Let
(t,w) EJR+ x
Q
exists an
s ET ,
0 M(r,w)
M : T x Q -+
M
we say that
= S-lim
utt
s
~
t ,
M(u,w)
*:R
be a hyperfini te martingale.
--
-
such that for all
for
(t,w)
is well-behaved at
r
~
s
r ET ,
0 M(r,w)
and
r
Flz1
For
if there
t ,
= S-lim
we have
M(u,w)
for
U-}t
r >s .
The martingale
subset
of
Q'
and all
w
E Q'
is called
well-behaved if there exists a
of Loeb-measure one, such that for all
Q
M
,
M
is well-behaved at
In particular all
t EJR+
(t ,w) .
S-continuous martingales are well-behaved.
In this section we shall sketch two results about well-behaved
martingales; the first is that if
gale which is
S-right-continuous at
second is that if
line
S
of
M
T
M
is a
SL 2 -martin-
is a well-behaved
0 , then
[ 0 M+] =
0
[M] +
The
A2 -martingale, then there exists a sub-
such that the restriction
MS
of
M to
T
is
well-behaved.
To solve the first problem we first prove the following result
which should be of independent interest:
Lemma 20:
ous at
and
Let
SL 2 -martingale, S-right-continu-
M be a well-behaved
0 . Then the left standard part
M is a local
In particular:
0
2 -lifting of
(fMdM)+
0
0
M-
is an element of
M- with respect to
= f 0 M-d 0 M+
.
0
M+ .
A( 0 M+ ) ,
- 20 -
Proof:
We only outline the idea of the proof and leave the details
to the reader.
Let
f: E
the first component of
N2
+
be a bijection and let
f(n) .
we see by Lemma I-10 that out-
side a set of measure zero, the sequence
and
be
Define internal stopping times
As in the proof of Theorem I-22,
infinitesimal jumps of
(f(n)) 1
{Tn}
enumerates the non-
It follows that the points where ( 0 M- )'
M.
M differ are the union of a null-set and the set
u n *[T ,T +1/m>.
But since M is well-behaved lim 0 M(T +1/m)
n m
n n
m+oo
n
0
0
= M(T n ) = M+(T n ) which implies that m
n*[T ,T +1/m> has Loebn n
0
0
measure zero. Thus ( M-)' = M L(vM)- a. e ..
The last part of the lemma follows from the first by Corollary 18.
Theorem 21:
Let
M be a well-behaved
. ht cont1nuous
.
S -r1g
at
Proof:
N
0 . Th en
Metivier [8]
0 [
SL 2 -martingale which is
= [ 0 M+ ]
M] +
•
proves (Korollar 1 and 2 on page 267) that if
L 2 -martingale, right-continuous and with left limits, then:
is an
[N](t) = N(t) 2 -N(0) 2 -2JtN-dN,
0
where
N-
1s the left limit of
[ 0
M+] (t) =
0
M+(t) 2
-
0
N .
M+(0) 2
Applying this to
-
0 M+
we have
2ft 0 M-d 0 M+
0
= 0 M+(t) 2
- 0 M(0) 2 -
2°<ft MdM)+ = 0 (~-~-f MdM)+(t) = 0 [M]+(t)
a
where we have used Lemma 20 and Proposition I-17.
The proof above is the only place in this paper where we use
a result from the standard theory for stochastic integration beyond
the mere definitions.
[ N ]( t )
= N( t
However, we shall prove the formula
) 2 - N( 0 ) 2 - 2 Jt N- dN
pendently of Theorem 21).
0
by nonstandard methods in [6] (inde-
- 21 -
To prove our second result we need some preliminaries.
[8] , Metivier proves (Satz 17.5) that if
N
1s a right continuous
process with left limits adapted to a stochastic basis
then there exists a sequence
If
M: T
x
Q -+
<Z, { ft}, J.1 > 3
of stopping times adapted to
N has no jumps outside the graphs of the
the same basis such that
a n 's •
{an}nEN
In
*JR
A 2 -martingale, let
is a
oM: R
+
x
n -+ R
be defined by
= sup{ 0 M(s,w):
oM(t,w)
sET
A
s~t} -inf{ 0 t1(s,w): sET
A
SR1t} .
In a way entirely similar to that of Metivier, we may prove that
Q'
there exist a set
of Loeb-measure one
of stopping times adapted to
then
oM(t,w) :t: 0
and a sequence {an 1nEJN
<r2,{Ht} ,L(P)>
such that if
if and only if there exists an
Using the lifting-techniques of Loeb [7 ]
w E Q'
with a n(w)
nEN
and Anderson [2 ]
=t.
it
is not difficult to prove the following:
Lemma 22:
Let
be a stopping time from
a
Q
to
R+
adapted to
Then there exists an internal stopping time
T:
Q-+ T
adapted to
<Q,{Gt},P>
such that
0T
= a
L(P)- a.e ..
We have already tacitly made use of this lemma in the argument
leading up to Corollary 18.
Combining Lemma 22 w:i:t:h what we have already got, we obtain a
sequence
where
{T n }nEJN
oM :t: 0
Theorem 2 3 :
in side a set
Let
exists a subline
to
S xQ
of internal stopping times enumerating the points
M: T xQ
S
of
-+
T
is a well-behaved
Q1 c Q
*R
be a
of Loeb-measure one.
A2 -martingale.
such that the restriction
A. 2 -martingale.
Then there
M8
of
M
- 22 -
Proof:
Let
constructed above.
(t,w)
at
steps.
and
{Tl}lEJN
then
t
be the sequence and the set
nl
Clearly,. if
=
0
T1 (w)
wE
n1
for some
and
1 E:N .
First we prove that there are a
of Loeb-measure one
0
such that if
n2
The proof is in two
y E ~JN
and
and a set
=
t
0
for
s ~ t , s < Tl (w) -1/y
M(s,w) = S- limM(r,w)
for
s ~ t , s > -r 1 (w) + 1 /y .
r+t
n2 c n1
then
'tl (w) '
M(s,w) = S- limM(r,w)
rtt
0
wE
is not well-behaved
M
and
Secondly, we prove that this is enough to construct the subline
S .
To do this it will be enough to construct
S such that the set
{wE rl :3sES3nE:N(-r (w)-1/y<S<T (w)+1/y)}
n
n
has Loeb-measure zero, since
( t, w)
where
n2
of
•
w is not in the union of this set and the complement
We shall prove by a combinatorial argument that we can
always find such an
( i)
Let
then is well-behaved at all points
M
S .
{ Bm} mEJN be a sequence of internal subsets of
5-tuple ( 1 ,m,n ,k ,p) E J.l 5
For each
D(l)
=
m,n,k,p
B
m
and
1
=nu
+
define the set
wE U n D(l)
k p>k m,n,k,p
M(t,w) has S-left
m
n · DCl)
n k p>k
m,n ,k ,p
It is enough to prove that if
t
with
{wEBm:Vr,sET[((T 1 (w)-1/k~r,s~-r 1 -1/p)
We claim that for all
have
n1
wE Bm
then for all
l,m,n
wE n 1
But this is immediate since for
and
8-right-limits at
t
=
0
T1 (w)
we
.
- 23 -
4-tuple (l~m,n~k) EN 4
For each
c<l)
n n<l)
c
m,n,k
B
Since
m
L(P)(
such that
p>k m,n,k,p
=n U n
D(l)
n k p>k m,n,k,p
L ( P )( B
n D(l) ,c(l) ) < 2-(m+n+k) .
p>k m,n,k,p m,n,k
we see that
'n u c <1 )
m n k
m,n,k
and consequently for each
LCP)CU
we pick an internal set
) < 2-m
1 E:N
n uc<l) k) = 1 .
m n k m,n,
4
For each
*:N
in
(l,m,n,k) E:N , let
such that for all
t-11k_::r,s,::t-11p
y
wEC(l)
m,n,k
(1)
k be the largest element
m,n,
and all r,sET if
ort+11p_::r,s_::t+11k
then
p
jM(r,w)-M(s,w)l<~.
C(l)
c n D(l)
this is true for all finite p
m,n,k p>k m,n,k
and hence for some infinite p
since C(l)
is internal. Thus each
m,n,k
y (1 )
is infinite and by saturation we may find an infinite y less
m,n,k
Since by definition
than all of them.
Putting
Q
= n u n u c (1 )
lmnk
2
m,n,k
we have finished the first part of
the pr>oof.
( ii)
We may now construct the new time-line
the only points
are the points
the set
{w E
Q
(t,w) E R
(
+
0 t
1 (w) ,w) ,
xo
where
S .
Since for
wE n 2
M may fail to be well-behaved
s
it is enough to construct
:3sES3nEIN(t (w)-11y<s<t (w)+1/y)}
n
n
such that
has Loeb-
measure zero.
By choosing a smaller
y
if necessary we may assume that
Define a subline
s'0
=
0
and
Obviously
= the
si_+l
21y < s!
can be only one
l+l
least element of
- s! _< 31y ,
s ' ES'
l
in
by putting
T larger than
and for each
*< t n ( w )
S1 c T
- 1I y ,
t
n
wEn ,
(
n EJN
w) + 1 I y > •
there
-
Let
4/5
= Y
n
Then
24 -
. 1n
. f.1n1te,
.
1
1s
n 5/4~:.,
IT=
n
is infinitesimal.
an internal sequence
p!
1
= P{w:
{•n}n<fn.
w
to
In
S :
of the elements of
s 0 -- 0
Let
between
S'
:r
p!
J
and
l
s·+2n;y
l
<
s.
and if
s. + n/y
or more such elements.
s.+n/y
S' ,
we have
In.
<
l
We now define
n
the sets
:rp~
n/3
{•n}nElli
Let
n
1
can only cover less than
be
Extend the sequence
3n<fn(s!E*<T (w)-1/y,T (w)+1/y>)}.
Since for each
the elements of
while
,
is chosen consider
l
s. + 2n;y .
l
There must
Since
In
l
j
there must exist a
s!
choose the corresponding
Let
t ET ;
less than
p.
1
t;(n/ y)
= P{w: 3n
:r
s.<et
3
We
in this interval such that
as
J
s.
.
l+l
the number of elements in
= ty /n
.
S
less than
< ln(s.E*<• (w)-1/y,• (w)+1/y>)}
n
1
n
ty
3t
nJ/4
1
then
This tells us that
l
S
at
t
= n 1/8 ,
then
P { w : 3 n < In 3 s € S ( • ( w ) - 1 I y <s < • ( w ) + 1; y ) } < 3 /n 1/ 8 ~ 0
n
n
and consequently
L(P){wE~ :3s€S3nETI'J(T (w)-1/y<s<T (w)+1/y)}
1
is
If
p. < - · n
if we we cut off
t
n
n
=
0.
We have already observed that this proves the theorem.
- 25 -
If we replace
11
A2 -martingale" by
"local A2 -martingale" in
the hypothesis and conclusion of Theorem 23, the resulting statement
is false.
In fact, by making a slight change in the martingale of
Example 10, we may construct a local
SL 2 -martingale which does not
A2 -martingale.
have any restriction that is a well-behaved local
The point is that to make the martingale well-behaved we must remove
a point on the time-line that is essential for making it a local
A2-martingale.
The well-behaved martingales are "well-behaved" in the sense
that they satisfy Lemma 20 and Theorem 21.
These results are neces-
sary to derive the standard Transformation formula from the nonstandard version.
Theorem 23 shows that the class of well-behaved
martingales is "large enough".
In other respects the class is sadly
irregular; for instance is the sum of two well-behaved martingales
usually not well-behaved.
behaved martingales
lS
M
Furthermore, there are examples of well-
and processes
not well-behaved even when
is a
X
Y E A2 ( 0 M+)
2-lifting of a
X E SL 2 (M)
is bounded.
where
such that
fXdM
X E SL 2 (M)
However, if
M is a well-behaved
martingale, then it was proved in [4] that
fXdM
A2 -
is well-behaved.
Together with Theorem 23 this seems to indicate that the class of
well-behaved martingales is a natural class when making stochastic
models for different kinds of phenomena.
Let us end with an application of the theory:
Example 24:
Let
x
We have proved that
be Anderson's process from Examples I-1 and I-15.
x
is
Moreover,
S-continuous, and we get that
0
[x]+(t) -- t .
it is well-behaved and by Theorem 21
Since
x
is
0
x+Ct)
=
S-continuous,
-
26 -
But by a well-known standard theorem (see
M~tivier
[8], Satz 8.2,
S
is a Brownian
or Nelson [9], Theorem 11 .8), this implies that
motion.
The results and examples only mentioned in this paper,are
given in detail
in [4], where the missing proofs also may be found.
- 27 -
References
1.
R.M. Anderson:
A Nonstandard Representation for Brownian
Motion and Ito Integration.
Israel J. Math. 25(1976)
pp. 1 5-4 6.
2.
R.M. Anderson:
Star-finite Probability Theory.
Ph.D.-thesis
Yale University 1977.
3.
H.J. Keisler:
An Infinitesimal Approach to Stochastic Analysis.
Preliminary Version 1978.
4.
T.L. Lindstr¢m:
Nonstandard Theory for Stochastic Integration.
(Unpublished)
5.
T.L. Lindstr¢m:
Hyperfinite Stochastic Integration I:
Nonstandard Theory.
The
Matematisk institutt,
Universitetet i Oslo, Preprint Series, 1979.
6.
T.L. Lindstr¢m:
Hyperfinite Stochastic Integration III:
Hyper-
finite Representations of Standard Martingales.
Matematisk institutt, Universitetet i Oslo,
Preprint Series, 1979.
7.
P.A. Loeb:
Conversion from Nonstandard to Standard Measure
Spaces and Applications in Probability Theory.
Trans. Amer. Math. Soc. 211(1975), pp. 113-122.
8.
M. Metivier: Reelle und Vektorwertige Quasimartingale und die
Theorie der Stochastischen Integration.
LNM 607,
Springer-Verlag 1977.
9.
E. Nelson:
Dynamical Theories of Brownian Motion,
Princeton University Press, 1967.
10.
H.L. Royden:
Real Analysis.
11.
K.D. Stroyan and H.A.J. Luxemburg:
of Infinitesimals.
Second Edition 5 Macmillan 1968.
Introduction to the Theory
Academic Press 1976.