HYPERFINITE STOCHASTIC INTEGRATION II: COMPARISON WITH THE STANDARD THEORY

for stochastic integration. In the first paper we studied the non- standard theory in its own right, and we shall now compare that theory with the standard one.

In the first section we define the SL²-martingales, which

constitute the class of hyperfinite martingales we shall work with in this paper. This class is a little smaller than the class of A2 -

martingales which we studied in the first paper, and the SL -martln-^{2 .} gales behave more regularly under standard parts. In the second section we use the fact that an SL²-martingale M has S-right limits to introduce what we call its right standard part ⁰M+ ; and we prove that this is an 1²-martingale. Given a process X which is integrable (standard sense) with respect to ⁰ +

M , we show how to construct a hyperfinite process Y (called a 2-lifting of X) such

and we see that standard stochastic inte- gration with respect to ⁰M+ may be obtained from the nonstandard theory with respect to M . 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 have only one noninfinitesimal jump in each monad. These martingales are particularly easy to work with1 and we obtain their basic properties and show that any SL²-martingale M has a restriction Ms to a subline S which is well-behaved. Well-behaved martingales will be useful in the next paper.

To prove the equivalence of the standard and nonstandard

theories for stochastic integration, it still remains to show that all 1²-martingales can in a suitable sense be represented as the right standard parts of 81²-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. SL²-martingales

In [ 5] we studied the class of A²-martingales, and we now define an important subclass:

Definition 1 : A hyperfini te martingale M : T x n -+

*JR

adapted to an internal basis <O,{Gt},P> will be called an SL²-martingale if Mt E 8L (O,Gt,P) 2 for each finite t E T. M is called a local

SL²-martingale if there exists an increasing sequence {• } EJN _{n n} of s.t.

such that 0 • (w) + m a.e. and M•

n n ^{is an 8L2}-martingale for each

n EN • The sequence {•n} is then called a localizing sequence for M~

Due to the S-integrability, the SL²-martingales behave more regularly under standard-parts than the A2-martingales. However, S-integrability is difficult to check and consequently some proper- ties of SL²-martingales are harder to prove than the corresponding properties of A²-martingales. It was, for example, an easy conse- quence of Lemma I-3 that a hyperfinite martingale M is a A2-mar- tingale if and only if E(M~+[M](t)) is finite for all finite t ET.

The corresponding result for 8L²-martingales would be that M is an 8L²-martingale if and only if M~ + [M](t) is S-integrable for each finite t E T . But this is not obvious, and the reason is that we must now consider JA(M~+[M](t))dP for AEGt ~ P(A)R:SO. Since A need not be in G_{0 ,} we cannot use Lemma I-3 as above, and we

seem to be stuck. Using another characterization of S-integrability, we may, however, prove the assertion:

Theorem 2: Let M be a hyperfinite martingale adapted to the

internal basis <O,{Gt},P>. Then M is an SL²-martingale if and only if M~+[M].(t) ESL¹Cn,Gt,P) for each finite tET.

Proof: Since in either case M is a ^A2-martingale, 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) then f ESL¹(Q,Gt,P) if and only if ⁰ ffdP =

f

⁰fdL(P).

By Proposition I - 17

and taking *-expectations we get:

(1) E(M~+[M](t))

=

E(M~) since fMdM is a martingale.

Define a sequence {Tn}nElli of internal stopping times by T n ( w )

=

min { sET :

I

^{M ( s , w )}

I

^{~ n } .}

Then

fM,

dM is a A2-martingale for each n E:N , and i t n 'n

follows from Lemma I - 12 that JtM dM, is S-integrable for each o 'n n

finite t E T .

By the result of Anderson quoted above

Since

a. e., we get

For almost all w there exists an n EJN such that 'm ( w) > t for m ~ n , and thus ⁰ [M, ] (t) ^{+ 0} [M] (t) and M, (t) + M(t) a.e ..

n n

The sequence ⁰ [M, ](t)

n is increasing and bounded by ⁰ [M](t) which is integrable since E(⁰ [M](t)) ~ ⁰E([M](t)) < oo We also have ⁰ (maxJ'-12 )

s~t s and by Doob's inequality

( 0 2 0 2

E max M ) < E (max M )

s - s

s~t s~t

< 00

and thus ^{0 (} max Hs) is integrable. ^{2 .} Applying Lebesgue's Convergence

s~t

Theorem to both sides of (2), we obtain ( 3 )

Combining (1) and (3) we see that

0E(M~+[M](t))

=

E(⁰(M~+[M](t))) if and only if OE(M2)

=

E(OM2)

t t

Anderson's characterization now tells us that M~+[M](t) is 8-inte- grable if and only if 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 an 8L 2-martingale, then

8- integrable for each finite t E T .

maxMs 2

s,:::t

is

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: If M ^J.S an 8L²-martingale and X E 8L 2 (M) , then JXdM is an 8L 2-martingale. If M is a local 8L 2-martingale and X E 8L(M) , then JXdM is a local 8L 2-martingale.

Proof: The second assertion follows from the first by definition of 8L(M) Assume that M is an 8L 2 -martingale, and let us first con- sider the case where X E 8L 2 ( M) is finite, i.e. there is an n EN such that

I

^X

I

^{< n . Then}

and it follows from Theorem 2 that fXdM is an 81²-martingale.

Let us now consider the general case X E SL ² (M) . Then there of finite elements 1n

exists a sequence {Xn}nElli that ⁰

f IX

^{2-x2 jdvM +}

o

TtxQ n t

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))

=

⁰E( r:x²~~-!:X²~M²)

=

⁰f<x²-X²)dvM ⁺

o ,

as

- n o o n Ttxn n t

n ⁺ oo • Since each [

Jx

dM] (t) is S-integrable i t follows that n

[fXdM] (t) is S-integrable~ and hence by Theorem 2 that fXdM is an SL²martingale.

2. The right standard part of A. ² -martingales

According to Theorem I-9, local A.²-martingales have S-right- and S-left-limits a.e., and thus the following definition makes sense.

Definition 4: Let M :Txn +

*JR

be a local A.²-martingale. Define a process ⁰ +

H : F+ x n ⁺ F by letting ⁰ M ⁺⁽ r,w) be equal to the S-right-limit of t ⁺ M(t,w) at the monad of r . The process ⁰M+

is called the right standard part of M . In a similar way we define the left standard part of M to be the S-left-limit ⁰ M ^- of M .

We want to make a martingale of ⁰

tt ,

and we first construct the stochastic basis:

If <Q, {G t}, P > is the internal basis of M , define

Hoo

=

cr(U{L(Gt): t finite}); i.e. the completion with respect to L(P) of the a-algebra generated by all the Loeb-algebras L(Gt) for

finite t E T . Let N be the set of all null-sets of H We define

00

a family { Ht} tEIR of a-algebras on Q by:

+

Ht

=

a(NU(U{L(Gs): sET, s R~t})).

A family {Ht}tEIR of smaller algebras is defined by +

H t

=

a ( U { L (G s ) : s E T , s R~ t}) .

Lemma 5: For each t EJR+

and

Ht

=

U{cr(L(Gs)UN) :sET, s R~t}

H' _t

=

Proof: Obviously

U{a(L(Gs)UN) :sET, SR~t} cHtca(U{a(L(Gs)UN) :sET, SR1t}

and i t is enough to prove that U{a(L(G )UN) :sET, s R~t}

s

a-algebra. Let {A } be a countable family of sets from n

is a

U{a(L(Gs)Uf'll) :sET, s R~t} . Assume A E a(L(Gs )UN) .

n n The family

Sn

=

*[s ,t+1] nT _{n n} is a countable family of internal sets with the finite intersection property, and by saturation n Sn ⁼¹⁼ ¢ . Let

nE.N ,...,

s E nsn , then and ,.., s > s

- n for all n • Consequently A E a(L(G ... n s ^)UN) for each n EJN and hence

UA E a(L(G ... )UN) c U a(L(G )UN) .

n s ^SR~t s Since Ua(L(Gs)UN) clearly has

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: Let the family { Ft} tEIR be either { Ht} or { H ..

t:J .

^Then

+

If A E F t , there exist an s E T , s R~ t , and a set that L(P)(A6B)

=

0 .

BEG s ^such

(b) F = F+ =

n

F

t t s>t s for all t ER+

(c) For all null-sets N 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 sto- chastic integration (see Metivier ( 8 ]).

The following nonstandard version of Egoroff's Theorem is often useful:

Proposition 7: Let <n,G,P> be a hyperfinite probability space and let {X }

n n~E; be an internal sequence of G-measurable functions Xn : n +

*JR

with Let Y = ⁰X

n n and assume that the sequence {Yn} converges a.e. to a random variable Y on

<&1,L(G),L(P)>. Then there exist a set &1¹ of Loeb-measure one and a v E

*JN

'JN such that for all n E

*J.J

'JN , n < v , and all w E n ' we have ⁰ Xn ( w ) = Y ( w ) •

Proof: Let X be an internal random variable on <n,G ,P> such that ^0X = Y L(P)-a.e.; such an X exists by Proposition ² of Loeb [ 7]. By Egoroff's Theorem, ~ee e.g. Royden [ 1 0 ] , page 7 2 ) ,

> 1

_.:!.

such

there exists for each m EJN a set A with L( P) (A )

m m ^m

that Yn + Y P( B ) > 1 _l_

m m

uniformly on A m such that B c: A

m m let n be such that for all

k,m implies

Let

We may find a and ^0X =Y on

w E Bm and all

B EG with m

B For each kElN m

n EN , n > - nk ,m

llk,m = max{nE*JN: !Xq(w)-X(w)l

<~

for all wE Bm and all q E *JN such that nk,m ~ q::, n} .

By definition of nk we must have llk E

*N

'N .

~m ,m ^Using

saturation we may find a v E

*:N

-...N less than llk,m n f. *::N'-JJ ' It follows that for all and all

for all l<,m EN.

n < v , we have

0X (w) = Y(w).

n Putting Q¹

=

U B we have proved the proposition.

m Elli m

Lemma 8: Let H : T x 1"2 -+

*.R

be a local ^:>.. 2-martingale adapted to the internal basis <Q,{Gt} ,P>. Then the right standard part ⁰M+ is a right continuous process adapted to the stochastic basis <Q,{Ht},L(P)>.

Moreover, for each t ER and each s E T , s ^F'll:1 t , there exists t E T , tF'II:1t, t > s such that ⁰ (M(t,w))

=

⁰H+(t,w) L(P)-a.e ..

Proof: The right continuity of ⁰M+ follows immediately from the definition. The rest of the lemma follows by applying Proposition 7 to the sequence {M(t+1/n,w)}nE*::N, where t +1/n denotes the least element of T larger than t + 1 /n , and using N c Ht .

Recall from Metivier [ 8 ] that if <Z, { F t} ,l.l > is a stochastic basis, a process M :R+ x Z-+ R is called an L2 -martingale if M is a martingale with respect to the basis and if for all t ER+ , the random variable w -+ M(t,w) is an element of

We may now conclude:

Proposition 9: Let M :TxQ-+

*R

be a ^:>..2-martingale with respect to the internal basis <Q ,{Gt} ,P>. Then °M+ is a right continuous

2 . 1 . h h h . b .

L -mart1nga e w1t respect to t e stoc ast1c as1s

Proof: We already know that ⁰M+ is a right continuous process

adapted to <Q, { Ht}, L( P) > , by Lemma 8. Also, given a t ER+ we may find a t E T, t F'll:1t , such that ⁰H+(t)

=

⁰M(t) a. e.. But then

E(⁰M+(t) 2 ) = E(⁰H(~)² ) ~ ⁰E(M(~)² ) ^< ~ since M is a ^:>..2-martingale, and hence ⁰M+(t) € L2 (n,'Ht ,L(P)).

s, t EJR + , s < t . We shall show that Let A E H s , and let

f (

^0M+(t)-0M+(s))dL(P) = A

find

s, t

E T such that

0 . By Lemma 6 (a) and Lemma 8 we may

0M(s)

=

⁰M+ ( s) , ⁰~1Ct)

=

⁰M+ ( t) and such that there exists a BEG ... with L(P)(Bl\A) = 0 •

s \tJe have:

since

M

is a martingale and BEG ...

s To get the standard part outside the integral we have used Lemma I-12 which implies that M(t) and M(s) are S-integrable. This proves the proposition.

A process M: JR ^X

z

⁺ JR +

there exists a sequence {a } n

is called a local

of stopping times such that 0 an ^{+ (JO} a.e. and each

Ma

is an L²-martingale. In view of Proposition 9

n

i t is natural to guess that the right standard part of a local

x

²-martingale 1s a local L2-martingale. But this is false as the following example shows:

Example 1 0: Choose y E *JN -....JN and let fJ consist of 2y elements:

Let £ = ^y L 2 ¹ then £ R~II2f 6 • n=l n

The internal measure P on n is defined by P{w +}

=

P{w } - 1- n n- - 2£n2 . We shall use the time-line T = { kf n: k E

*1--T,

k < n 2 } for some

n E *JN ,1,r •

The family {Gt} of internal algebras is defined by:

Gt =

{0,n}

for t <1-1/n; G1 _11 n is the internal algebra generated by the sets {wn+'wn_} ; and Gt is the internal power set of n

for t > 1 .

The martingale M is defined as follows:

M(t,w) = 0 for all w if t < 1

--

¹ _n M(t,w +) _n

=

^-M(t,w _n-⁾ = n if

t > 1 .

M lS obviously a hyperfinite martingale adapted to the internal basis <ri,{Gt},P>;~and using the sequence {-rn}

'

^where

-rn(wk+) = T (wk ) = ^{1 - 1 /n} if k>n n -

Tn(wk+) = T (wk ) = 1 + (n-k) if k<n n -

we see that M is a local 8L²-martingale.

{a } n

Let us show that ⁰M+ is not a local L²-martingale. Let be an increasing sequence of stopping times such that a _n ^{+ oo} a. e. . For t < 1 , the a-algebra Ht consists only of sets of Loeb-measure 0

n E:N such that

and 1 . Since ^{+ 00} a.e. there must be an L(P){w:a (w) <1} <1, and consequently this set

n

must have measure zero. But then an ( wk+) :::: 1 for all k ElJ . Consequently E(⁰M+ (1)2 )

=

^{oo and {a}} is not a localizing

an n

sequence for ⁰M+ . This proves that ⁰M+ is not a local L 2 -

martingale.

We can get more out of this example: Let S

=

T-...{1-l} be n

a subline of T . Then the restriction of M to S is not a local A²-martingale. Thus a restriction of a local SL²-martingale is not necessarily a local A²-martingale.

3. Stochastic integration with respect to ⁰M+

In this section we compare the nonstandard theory for stochastic integration with respect to M with the standard theory for inte- gration with respect to ⁰M+ .

Let us briefly review the standard definition of a stochastic integral. (Metivier [ B]):

b . . ² . 1 . h

Let N: JR+ x Z + JR e a rlght-contlnuous L -martlnga e Wl t respect to the stochastic basis <Z,{ft},~>. The a-algebra P of

predictable sets with respect to <Z, { Ft}, ^Jl > is the a-algebra of subsets of JR xZ

+ generated by the sets <s,t) xF

s and ^{{O} x F}0

for s , t E JR + , s < t , F E F

s s and A process is called predictable if it is measurable with respect to P •

We may define a unique measure on ^p by

and mN({O}xF_{0 )} = 0 .

If X is a predictable process of the form n

X(t,w) = :L a. 1

J

F (t,w) i=1 l <si'ti x s.

l

for ai EJR , we define the stochastic integral of

X

with respect to N to be the process JXdN defined by

If X

n

( t , w ) -+ • I: a_1. 1 F ( w ) ( Nt , t- Ns . At)

1=1 S• 1A 1

l

=

I

t XdN .

0

lS a predictable process let X(t) be the process

defined by (t) (t)

X ( s , w ) = X ( s , w ) for s < t , X ( s , w ) = 0 for s > t . Let A²(N) be the set of all predictable processes X such that

We extend the stochastic integral to the class A2(N) by noticing that the processes of the form

X =

_l=O n ^{I: a.1} _l <s.,t.Jx ^{• F}

1 1 si

are dense in L (JR+xZ,P,mN) , and that the mapping ^{2 X -+} ("XdN

-o

^is

an isometry from this dense subset into

the unique extension of this isometry to the whole of also by

XEA²(N)

X -+ J⁰⁰XdN , we define the stochastic integral

0

to be the process

If we denote L²(Z,F ,Jl)

00

fXdN for

Let us return to our nonstandard setting: Let M : T x Q ⁺ *.R be a :\ ²-martingale adapted to the internal basis <Q, ^{G t}, P> . By

P . . 9 0 +

ropos1.t1.on , ^M is a right-continuous L²-martingale with respect to <Q,{Ht},L(P)>. We write P for the predictable sets with re- spect to {Ht} and

P'

for the predictable sets with respect to

The difference between the two classes is not large:

Lemma 11 : For each A E P there exists a B E P' such that

mN(A~B) = 0 for all L²-martingales N adapted to <!:2,{Ht},L(P)>.

Proof: Follows from the fact that for with L(P)(CtlD)

=

^{0 •}

there 1.s a DE H'

t

Let Tf denote the set of finite elements of the time-line T.

We define a mapp1.ng from the power set of set of T f x Q by:

TI(A)

=

{(t,w) ETxQ (⁰t,w) EA}.

IR xQ

+ into the power

Since ^1T 1.s really an inverse image operation, it is easy to see that 1T commutes with all countable Boolean operations.

A set A ^c: TxQ is called adapted (to the basis <1:2 '{G t} 'P >) if each section At is l.n the corresponding Gt ' ^{tET. Let A be} the internal algebra of adapted sets, and let L(A) be its Loeb-

algebra with respect to the internal measure vM (remember the defin- ition of vM preceding Definition I-18.)

Our task is to compare stochastic integration with respect to M with stochastic integration with respect to ⁰M+

S-lim M(s,w) , ⁰M+ cannot register what happens to M in the monad s+t

of 0. To make sure nothing significant happens there, we define:

A local A²-martingale M is said to be S-right-continuous at 0

if ^{0 0} +

M(O,w)

=

M (O,w) a.e ..

We may prove:

Lemma 12: Let M be an 8L²-martingale which is 8-right-continuous at 0. Then the image under n of the predictable sets

P'

with respect to

H'

_t is contained ln the Loeb-algebra L(A) of the adapted sets (with respect to L(vM) ) .

Proof: By the definition of the predictable sets, i t is enough to prove that any set of the form n(<s,t]xF )

s or n({O}xf_{0 )} lS ln L(A), Fs EH~. We leave the details to the reader, and only remark that the conditions that M is an 8L²-martingale and that M is 8-right-continuous at 0 both are needed in the proof.

Lemma 13: Let M be an 8L²~martingale which is 8-right-continuous at 0. Then the mapping n from P' to L(A) is measure-preserving~

Proof: The measure is uniquely determined by its values on the sets of the form <s,t]xFs, {O}xf_{0 ,} and since n commutes with countable Boolean operations~ i t lS enough to prove

L(vM)(n(A)) for A of this form. Again we leave the details to the reader, but remark that we make vital use of the two conditions on M •

""

Let M be as ln Lemma 13. We define a measure m0 + on the M

image n(P) of the predictable sets under ^TI by mo +(nA)

=

mo +(A).

M M

From Lemmas 11 and 1 2 we see that for each set B E n ( P) there is a set C E n(P)

n

L(A) such that

m

0 +(Bt.C)

=

0 •

M It follows from this

that if f : T f ^{x Q +} JR is n ( P) -measurable, then the conditional

expectation ECflnCP)nL(A)) of f with respect to the sub-a-algebra n(P)

n

L(A)

see .that

and the measure m0 + equals f a.e ..

M mo +

M and agree on ^TI(P)

n

L(A) .

By Lemma 13 we

If X E A2 (⁰M+) , we define X' :Tf x Q -+ JR by X' (t,w) = X(⁰t,w) . The process X' will obviously be n(P)-measurable, and if X" is the conditional expectation of X' with respect to n(P)

n

L(A) , the random variables X' and X" are equal ...

m0 + -a. e •.

M

Definition 14: An adapted process Y : T x Q -+

*JR

is called a 2-lifting of X if Y E SL ² (M)

close L ( vH) -a. e. on T f ^x Q •

and Y and X" are infinitesimally

Our purpose is to show that fXd⁰M+

=

^{0 (fYdM)+ when Y is a}

2-lifting of X , and hence to derive the standard theory for sto- chastic integration with respect to ⁰M+ from the nonstandard theory for integration with respect to M .

The notion of lifting is central 1n the theory of Loeb-spaces;

liftings of random variables were first studied by Loeb in [7] and further developed by Anderson in [1] and [2]. 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. We first prove that there are enough 2-liftings.

Lemma 15: Let H be an SL²-martingale S-right-continuous at 0 , Then X has a 2-lifting with respect to M.

Proof: Let {t } be an increasing sequence of finite elements of T n

such that ot

n ^{-+ CX> •} Let x(n) be the restriction of X" to

A

=

(*[O,tn]nT) xQ. Then X(n) E L²(A ,L(A ),L(vH ) )

'

^{where A}

n n n --n n

and VM

n are the restrictions of A and respectively to An . By Theorem 11 (ii) of Anderson [1 ] there exists for each n a Y(n) E SL²(A ,A ,v~

1

⁾ such that ⁰Y(n) = X(n)

n n !n ^L(v~)- a.e.. ~rJe

yCm)~A = y(n) for m > n . By may choose the y(n)'s such that n

saturation we may extend the sequence {Y(n)} to an internal nEll

sequence {Y(n)} with the same property.

ns.n ^{Choosing Y}

= y<n>

we prove the lemma.

The next lemma tells us that which lifting we choose does not matter:

Lemma 16: Let M be an SL²-martingale which is S-right continuous t 0 d 1 t X E ^{A 2 ( 0} M.+) •

a.e. a , an e ^{l t .} If Y and Y' are 2-liftings of

X,

there exists a set n' en of Loeb-measure one such that

0 Jt 0 t_

( Y(s ,w)ill1(s ,w))

=

<f Y' (s ,w)dH(s ,w))

0 0 for all

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 0

f

^{s s 2 t 2}

E( sup

I

Y(r,w)dM(r,w) -

f

Y¹ (r~w)dH(r,w)

I ) ::

4E(

<J

(Y-Y' )dM) )

s~t o o o

t t

But ⁰E( Cf (Y-Y' )dM)2 )

=

⁰E(I:(Y-Y' )2 M12 )

=

⁰

r

^{(Y-Y' )2dvM}

o o Tjn

= £

^{0 (Y-Y' )2dL(vM)}

=

⁰ for all finite t E T. It follows that Tt n

s s

0E(sup(J YdM-J Y'dM)2 ) = 0 for all finite t E T and the lemma

ss.t o o

is immediate.

We may now obtain the main result of this section, the compar- ison between standard and nonstandard stochastic integration:

Theorem 17: Let M be an SL²-martingale which is S-right- continuous at 0 and let ² Let ^y be

'

^{X E A (M)}

.

^a 2-lifting of X

with respect to M. Then

fXd⁰M+ = °CfYdM)+ .

Proof: We first remark that since the process /Xd⁰M+ 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 . It is therefore enough to prove the theorem for some lifting Y .

Let us first assume that X is of the form ^X = 1 .

<s,t]xFs' where s, t E:R+ , s < t , and F s € H s . By Lemma 6 (a) there are an

s

^{E·T , ... s R1 s and a} such that

0MCs,w) =

By Lemma 8 we may choose ""' s such that ⁰ H + (s,w) L ( P) - a . e . , and by

• ,.... 0 ,..., 0 +( )

the same lemma we f1nd a t E T , t ^R1 t , such that M(t ,w) = M t 3w • Define

We shall prove that Y is a 2-lifting of X; since Y obvi- ously is an element of SL²(M) it is enough to prove that Y is infinitesimally close to X" L(vM)- a. e.. Let

A= U

n

[(*<s+.:!.,t+.!.]nT)xG ] , nEil'J mEil'J n m s

then the process is a version of X", and by the choice of and

t

it follows that YR-~1A a.e. and hence that Y is a 2-lifting of X .

For each r E:R+ we have

JrXd⁰M+

=

1 (⁰M+ -⁰M+ )

o

Fs rAt rAs '

""' s

while for each

r

E T

and by the choice of

s

^and

t

This proves the theorem for X

L(P) -n.o.a.

of the form 1

<s,t]xF s by linearity the theorem holds for all X of the form

, and

To prove the theorem it is then enough to show that the mapping X(t) + ⁰ (/YdM)+(t) is an 1²-isometry for all t EF+.

We have

t+l/n t+l/n

= lim E((⁰

b

^{Yd.M)2 ) = lim0}E((! Yd.M)^{2 )}

n+oo n+oo

t+l/n

= lim ⁰E( I: Y²1lli2 ) = lim ^{0 /} Y²dvM =lim

J

⁰Y²dL(vM) =

f

⁰Y²dL(vM) 0 n+ooTt+l/;n n+oo Tt+l/nxn n([o,t]xn) n+oo

= f r ^dmo

⁺

^{= f x<}

^{t) 2}

^elmo

^{+ '}

[o,t)xn M E xn 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 SL²-martingale (Proposition 3) to get the stan- dard part outside the expectation; and that Y E SL² (M) to move i t inside again. The rest of the equalities follows from the assumption that Y is a 2-lifting of X , and the basic relations between the measures L( vM) , moM+ and moM+ .

As we have already noticed, the equality E(⁰(fYdM)+(t)2 ) =

f x<

^t

)~m

₀ + establishes the theorem.

JR xn M

+

So far we have only dealt with integrals of the form fXdN where N is an L²-martingale and X E A²(N) . If N is a local L²-martingale and X is predictable, we define X to be in A(N) if there exists a localizing sequence {on} for N such that X € A 2 ( N ) for each

On ⁿ EN. The stochastic integral fXdN is then defined as the limit lim fXdN .

n+co ⁰n The theory developed above can be extended to the case where M is an SL²-martingale and

We sketch the construction: For each n we may find an internal stopping time Tn adapted to the internal basis

<s&,{Gt},P> such that ⁰Tn =on L(P) -a.e .. We can then prove that X E A 2 ( ⁰ (M ) +) for each n EN , and we can find an adapted process

Tn

Y : T x n +

*.R

such that Y is a 2-lifting of X with respect to each MT . Such a Y is called a local 2-lifting of X. The

n

result follows from Theorem 17:

Corollary 18: Let M be an SL²-martingale which is S-right-

continuous at 0 . Let X E A ( ⁰M+) . Then X has a local 2-lifting Y € SL(M) and

The statements above were proved in [4 ] • The problem for M a local SL²-martingale is more troublesome since by Example 10

0M+ 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 SL²-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 i t would have been better to work with the standard part process ⁰M: R X n + R defined by

+

0 0 -

M(t,w)

=

(M(t,w)). The

advantage of our procedure is that the right standard part is always

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. We have a similar notion for real-valued L²-martingales N: :R+ x Z -+ :R

an increasing sequence of partitions ~ n = {O=t <t <···<t <···} o 1 k of the positive real numbers such that the diameter o(~n) = sup(t. -t.)

.t.E~ 1+1 1

1 n

tends to zero as n-+ ^oo ,and sup ^~ = ^{oo ,} then the n

quadratic variation [N] of N 1s defined by

For a proof that [N] exists and is well-defined, see Metivier [8

J ,

page 234.

If M : T x

n

-+

*R

is an SL ² -martingale, we form the right standard part ⁰M+ and its quadratic variation [⁰M+] . We may also form the quadratic variation [M] of M ; since this is an increasing process i t must have 8-right-limits, and we can define its right standard part ⁰ [M]+ . In the spirit of Section 3 i t is natural to ask when the processes [⁰M+] 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)

with the standard version on page 265 in Metivier [8] applied to

0M+ ; ln the first case we integrate with respect to [M] in the second with respect to [⁰M+] . If we want to deduce the standard form from the nonstandard, we must know the relationship between [ M] and [ ⁰ M+

J •

It is easy to make examples of SL²-martingales where A closer inspection of such examples makes the following definition natural.

Definition 1 9: Let M : T x Q -+

*:R

be a hyperfini te martingale. For (t,w) EJR+ x Q we say that M is well-behaved at

- - -

(t,w) if there exists an s E T , s ~ t , such that for all r E T , r ^Flz1 t , we have

0M(r,w)

=

S-lim M(u,w) utt

r > s .

for r ~ s and ⁰M(r,w)

=

S-lim M(u,w)

U-}t for

The martingale M is called well-behaved if there exists a subset Q' of Q of Loeb-measure one, such that for all t EJR+

and all w E Q' , M is well-behaved at (t ,w) .

In particular all S-continuous martingales are well-behaved.

In this section we shall sketch two results about well-behaved martingales; the first is that if M is a well-behaved SL²-martin- gale which is S-right-continuous at 0 , then [ ⁰M+] = ⁰ [M] + The second is that if M is a A²-martingale, then there exists a sub- line S of T 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: Let M be a well-behaved SL²-martingale, S-right-continu-

0 - 0 +

ous at 0 . Then the left standard part M is an element of A ( M ) , and M is a local 2 -lifting of ⁰M- with respect to ⁰M+ .

In particular: ⁰(fMdM)+

=

^{f0M-d0M+ .}

Proof: We only outline the idea of the proof and leave the details to the reader. Let f: E + N² be a bijection and let (f(n))₁ be the first component of f(n) . Define internal stopping times

As in the proof of Theorem I-22, we see by Lemma I-10 that out- side a set of measure zero, the sequence {Tn} enumerates the non- infinitesimal jumps of M . It follows that the points where ( M )' ^{0 -} and M differ are the union of a null-set and the set

u n *[T ,T +1/m>.

n m n n

=

^{0M(T )} _n

=

^{0M+(T )} _n

measure zero. Thus

But since M is well-behaved lim⁰M(T +1/m)

m+oo n

which implies that n*[T ,T +1/m> has Loeb- m n n

( 0M-)'

=

⁰M 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 SL²-martingale which is S -r1g . ht cont1nuous at 0 . . Th en 0 [ M ] +

= [

⁰ M+ ] •

Proof: Metivier [8] proves (Korollar 1 and 2 on page 267) that if N is an L²-martingale, right-continuous and with left limits, then:

[N](t) = N(t)² -N(0)² -2JtN-dN,

0

where

N-

1s the left limit

of

N . Applying this to ⁰M+ we have

[ 0M+] (t) = ⁰M+(t)² - 0M+(0)² - 2ft⁰M-d⁰M+

0

=

^{0M+(t)2 - 0M(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. However, we shall prove the formula

2 2 Jt -

[ N ]( t )

=

N ( t ) - N ( 0 ) - 2 N dN

0 by nonstandard methods in [6] (inde- pendently of Theorem 21).

To prove our second result we need some preliminaries. In [8] , Metivier proves (Satz 17.5) that if N 1s a right continuous process with left limits adapted to a stochastic basis <Z, { ft}, J.1 >3

then there exists a sequence {an}nEN of stopping times adapted to the same basis such that N has no jumps outside the graphs of the a 's _{n •} If M : T x Q -+

*JR

is a A ²-martingale, let

be defined by

oM: R x n -+ R +

oM(t,w)

=

sup{⁰M(s,w): sET ^A s~t} -inf{⁰t1(s,w): sET ^A SR1t} .

In a way entirely similar to that of Metivier, we may prove that there exist a set Q' of Loeb-measure one and a sequence {an 1_nEJN of stopping times adapted to <r2,{Ht} ,L(P)> such that if w E Q'

then oM(t,w) :t: 0 if and only if there exists an nEN with a n(w)

=

t.

Using the lifting-techniques of Loeb [7 ] and Anderson [2 ] it is not difficult to prove the following:

Lemma 22: Let a be a stopping time from Q to R+ adapted to Then there exists an internal stopping time

T : Q-+ T adapted to <Q,{Gt},P> such that ^{0 T} = 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 {T } of internal stopping times enumerating the points n nEJN

where oM :t: 0 in side a set Q ₁ c Q of Loeb-measure one.

Theorem 2 3 : Let M : T x Q ^-+ *R be a A ² -martingale. Then there exists a subline S of T such that the restriction M8 of M to S xQ is a well-behaved A. 2-martingale.

Proof: Let {Tl}lEJN and

nl

be the sequence and the set

constructed above. Clearly,. if wE

n1

^and M is not well-behaved at (t,w) then t = ⁰ T₁(w) for some 1 E:N . The proof is

steps. First we prove that there are a y E ~JN and a set of Loeb-measure one such that if wE

n2

^{and t} = ⁰ 'tl (w) '

0M(s,w) = S- limM(r,w) for s ~ t , s < Tl (w) -1/y and r t t

0M(s,w) = S- limM(r,w) for s ~ t , s > -r 1 (w) + 1 /y . r+t

in two

n2

^c

n1

then

Secondly, we prove that this is enough to construct the subline S . To do this i t 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 M then is well-behaved at all points ( t , w) where w is not in the union of this set and the complement

of n_{2 •} We shall prove by a combinatorial argument that we can

always find such an S .

( i) Let { Bm} mEJN be a sequence of internal subsets of n₁ with

have

For each 5-tuple ( 1 ,m,n ,k ,p) E J.l ⁵ define the set D(l) m,n,k,p =

{wEBm:Vr,sET[((T 1 (w)-1/k~r,s~-r 1 ^-1/p)

We claim that for all 1 and m B

=

n u n · DCl)

m n k p>k m,n ,k ,p

It is enough to prove that if wE Bm then for all l,m,n we wE U n ^D(l) But this is immediate since for wE n₁

k p>k m,n,k,p

t + M(t,w) has S-left and 8-right-limits at t

=

⁰ T1(w)

.

For each 4-tuple (l~m,n~k) EN⁴ we pick an internal set c<l) c

n

n<l)

m,n,k p>k m,n,k,p such that L(P)(

n

D(l) ,c(l) ) < 2-(m+n+k) . p>k m,n,k,p m,n,k

Since B

= n

U

n

D(l) we see that m n k p>k m,n,k,p

L ( P )( B

'n u c <

1 ) ) < 2-m m n k m,n,k

and consequently for each 1 E:N

LCP)CU

n

uc<l) k)

=

1 . m n k m,n,

4 (1)

For each (l,m,n,k) E:N , let y k be the largest element p m,n,

in

*:N

such that for all wEC(l) and all r,sET if m,n,k

t-11k_::r,s,::t-11p ort+11p_::r,s_::t+11k then

jM(r,w)-M(s,w)l<~.

Since by definition 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

than all of them.

Putting ^Q =

n u n u

c ( 1 ) we have finished the first part of

2 lmnk m,n,k the pr>oof.

( ii) We may now construct the new time-line S . Since for wE n ₂ the only points (t,w) E R xo where M may fail to be well-behaved

+

are the points ^{( 0}t 1 (w) ,w) , it is enough to construct

s

such that the set {w E Q :3sES3nEIN(t (w)-11y<s<t (w)+1/y)}

n n has Loeb-

measure zero.

By choosing a smaller y if necessary we may assume that

s' ₀

=

0 and Obviously

Define a subline S ¹ c T by putting si_+l

=

the least element of T larger than

21y < s! - s! _< 31y , and for each wEn , n EJN there

l+l l

can be only one s ' E S ' in

*

^{< t} _n ^{( w ) - 1 I y , t} _n ^{( w ) + 1 I y > •}

Let 4/5

n = Y Then n . . f. . 5/4~:., 1

1s 1n 1n1te, n ^{IT= ,} while

is infinitesimal. Extend the sequence {•n}nElli to an internal sequence {•n}n<fn. Let

p!

=

P{w: 3n<fn(s!E*<T (w)-1/y,T (w)+1/y>)}.

1 1 n n

Since for each w the sets

can only cover less than

In

of the elements of S' , we have

:rp~ <

In.

l

We now define S : Let s _{0 -}- 0 and if s .

l is chosen consider the elements of S' between s. + n/y and s. + 2n;y . There must

l l

be n/3 or more such elements. Since s·+2n;y

l

:r

s.+n/y

l

p! J ^<

In

there must exist a j in this interval such that choose the corresponding s !

J as s. . l+l

Let t E T ; the number of elements in S less than t is less than t;(n/ y)

=

ty /n . If

p.

=

P{w: 3n < ln(s.E*<• (w)-1/y,• (w)+1/y>)}

1 1 n n then

We

:r

^p.

s.<et ¹

l

3 ty

< - · - 3t

nJ/4 This tells us that n

if we we cut off S at 1/8

t

=

n , 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)}

=

0.

1 n n

We have already observed that this proves the theorem.

If we replace ¹¹A2-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 have any restriction that is a well-behaved local A2-martingale.

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 non- standard 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. Furthermore, there are examples of well- behaved martingales M and processes X E SL 2 (M) such that fXdM

lS not well-behaved even when X is bounded. However, if X E SL 2 (M) is a 2-lifting of a Y E A2 (⁰M+) where M is a well-behaved A2 - martingale, then i t was proved in [4] that fXdM 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 be Anderson's process from Examples I-1 and I-15.

We have proved that x is S-continuous, and we get that ⁰x+Ct)

=

M oreover, ⁰ [x]+(t) -- t . Since

x

is S-continuous, i t is well-behaved and by Theorem 21

But by a well-known standard theorem (see M~tivier [8], Satz 8.2, or Nelson [9], Theorem 11 .8), this implies that S is a Brownian motion.

The results and examples only mentioned in this paper,are

given in detail in [4], where the missing proofs also may be found.

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: The Nonstandard Theory. 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. Second Edition5 Macmillan 1968.

11. K.D. Stroyan and H.A.J. Luxemburg: Introduction to the Theory of Infinitesimals. Academic Press 1976.