HYPERFINITE STOCHASTIC INTEGRATION III: HYPERFINITE REPRESENTATIONS OF STANDARD MARTINGALES

Introduction

In the second of the papers 1n this series (8], we showed that

0 +

astochastic integral with respect to the right standard part

M

of an SL²-martingale

M

could be obtained from a nonstandard stochastic integral with respect to M • We did so in order to show that the standard and nonstandard theory for stochastic integration are equ1- valent, and we shall now complete our programme by showing that all (standard) L²-martingales can be represented as right standard parts of SL²-martingales.

The argument is in two steps. If <Z,{ ~},~> is the stochastic basis of an L²-martingale N, we first find a hyperfinite probability space <Q ,G , P > , a family { ft '} tEJR of a-algebras on Q , and a

+

measure-preserving a-homomorphism

e:

<Q, F' , ₀₀ L ( P) > + <Z, F , ₀₀ ~ > which maps each f'

t some ⁿ _{':J t C:\,1} ^,., _'

onto Ft . Each F

t

is a sub-a-algebra of L(G t) for and we find an L ²-martingale N°: JR x Q + JR adapted to

e

+

<Q,{L(Gt)},L(P)> such that each Nt is F-t-measurable, and

e

2

e

[Nt:::_a.]

=

[Nt:::,a.] for all t E JR+ and all a. E JR. This L -martingale N⁹ is called a weak Loeb-space representation of N •

The second step is to construct an SL²-martingale M adapted

to such that _N

e

hyperfinite representation of N .

This M is called a weak We shall prove that if X E A ² (N) , then we may find YESL²(M) such that

This will complete our programme, since we may now obtain

fXdN

from fYdM •

The representation map

e

above will not give any correspondence between the paths of N and those of N° , since i t is not induced

by any point-mappin~

e:n

⁺ Z. Working with

e

is thus a rather elusive affair. However, using the representation theorems for Radon- spaces proved by Anderson in [2], one may obtain Loeb-space represen- tations for large classes of L²-martingales where the representation map

e

is defined from a mapping

e:n

⁺ Z by

eCe-

¹(A)) =A. Such representations we call strong Loeb-space representations, and the corresponding hyperfinite representations are called strong hyper- finite representations.

Using the Skorohod Topology on the space D of right-continuous functions with left limits (see Billingsley [4]), the following result was proved in [ 6]: Let N: IR+ x Z + IR be an L ²-martingale which is right-continuous and have left limits a.e .• Let Ft be the a-algebra obtained from the finite dimensional sets up to time t ' and let ,..., be the equivalence relation on

z

defined by: X "'Y if for all t € IR+ , N(t,x)

=

^{N(t,y) Let}

^z

^{I = Zj,..., F' =} ft;,..., ' ^{J.l I} = ^].lj,..., and let

t

N ^{I :}

z

^{I X} IR+ + lR be the canonical process. Then Nl is an L -mart1n-^{2 .}

gale with respect to <Z',{Ft},J.1 ⁹ >, and as such has a strong hyper- finite representation. In other words, if we disregard all information not obtainable from the finite dimensional sets, all L²-martingales have strong hyperfinite representations.

The result on strong hyperfinite representations uses Anderson's representation theorem for Radon-spaces; the methods used to construct weak hyperfinite representations seem to be new. However, similar results are announced to appear in Anderson [3].

The reader is refered to the first paper [7] for further infor- mation on the literature.

We shall use the same conventions and terminology as in the two previous papers; and we only remind the reader that we are working

with polysaturated models for nonstandard analysis (see Stroyan and Luxemburg [13]); the saturation property will be used repeatedly in this paper.

We shall use the notation Theorem I-14 for Theorem 14 of the first paper [7]; and the similar convention applies to the second paper [8] and II.

1. Weak Loeb-space representations of measure-spaces

We want to represent arbitrary

L

²-martingales as the right

standard parts of SL²-martingales, and to do this we first represent the corresponding measure spaces by hyperfinite Loeb-spaces.

Definition 1: Let <Z,F,~> be a probability space. By a weak Loeb-space representation of <Z,F ,~> we shall mean a hyperfinite probability space <Q, G, P > ; a sub-a-algebra F ' of L (3 ) ; and a measure-preserving a-homomorphism 0:<Q,

F'

,L(P)> + <Z,F,~> which is onto F . The representation is called a strong Loeb-space repre- sentation if there is a partial mapping e:Q +

Z,

such that

e-^{1(A) EF'} for all AEF and e(e-^{1CA)) =A} for all such ^A.

Obviously, the domain of e must have measure one.

Using the standard part map of the defining topology, Anderson proved in [2] that all Radon-spaces have strong Loeb-space represen- tations. We shall prove that all probability spaces have weak Loeb- -space representations. To do this we need the following theorem which we take from Sikorski¹s book [12] (pp. 144-145):

Proposition

2:

Let Z and Z¹ be sets and let

A

and

A'

be a-algebras on Z and Z¹ respectively. Let S be a generator set for

A

and let f:S +

A

¹ be a mapping. Then f has an extension to A which is a a-homomorphism if and only if for each countable family {Ai} iEIN of elements from S :

n e(i)A.

iElN ¹

=

0

~ n di)f(A.) =

0 ,

iEIN ¹

where £ : IN + { 1 , -1 } .

We may now use Proposition 2 to prove

Theorem 3: Each probability space has a weak Loeb-space represen- tation.

Proof: Let <Z,F,~> be a probability space and let <*Z *

' '

F *" > ...

be its nonstandard version. Let S = { *F: F E ^{f} ,} and let

F

be the a-algebra generated by S •

Define a mapping f:S ^-+ F by f(*F)

=

F. We shall use

Proposition 2 to extend f to a a-homomorphism h:F -+ F • Since

r

is closed under complements~ it is enough to show that for each countable family {*Fi}iEJN of sets from F such that

n

*F.

= 0

iEJN ^l then

n

fC*F.)

= ⁿ

^F.

= ⁰

^{But this lS} easy since saturation

iEJN l iEJN ^l

and

n

*F.

= 0

^implies

ⁿ

^{*F. ~}

⁰

^{for some n E JN. Since}

iEJN ^l i~n ^l

*c

n

F. )

= ⁿ

^*F.

= 0 '

^{we have}

^n.

^F.

^{= 0}

and consequently

n

F.

=

i~n ^l i~n ^l i~n l iEJN l

...

Hence f can be extended to a a-homomorphism h: F ^-+ F •

Let us show that h is measure-preserving, i.e. L(*~)(A) =

=

~

(

h (A) ) for A E

F .

If A E S this is obvious by definition of the Loeb-measure. Since v(A) = ~(h(A)) defines a measure on

F,

it follows from Caratheodoryis Extension Theorem (Royden [11], page

....

'

0 .

257) that L(*~)(A)

=

~(h(A)) for all A E F , since S is an algebra of sets.

We have thus constructed an internal probability space

<*Z,*F,*~>; a sub-a-algebra ^... F of L(*F) ; and a measure-preserving ...

a-homomorphism h:F ^-+ F. It only remains to turn <*Z,*F,*~> into a hyperfinite probability space:

For each finite set F_{1 , •••} ,Fn of elements of

F,

there is a finite partition P of *z such that if P E P and P

n

*F.

+ 0 ,

l

then P c *F . .

l By saturation there exists a hyperfinite partition P of

*z

such that for each F E F if P

n

*F

* 0

^{for a P c P , then}

P c

*"

^{I: •} Let ...., be the equivalence relation generated by

P

let

*

Q

= Zj-,

and let TI:*z + Q be the quotient map. We may choose

P

such that the equivalence classes of

P

are elements of *F. Let G

=

TI(*F) and let P

=

TI(*v) , then <Q~~ ,P> is a hyperfinite pro-

bability space. Let f' be TI(F) , it follows easily from the defi- ni tions of

F

^{and P that fi c LCG) . Define}

e:F' + F by 0 : ho TI ^-1 •

Then {<Q,G,P>,F',e} is a weak Loeb-space representation of <Z,F,v>9

and we have proved the theorem.

If {<Q,G,P>,f' ,e} is a weak Loeb-space representation of

<Z,F,v>, we have for each set F' in f' a set e ( F' ) E F such that L(P)(F')

=

v(0(F' )) . On the other hand, for each FE F , we have a non-empty subset e-¹({F}) of f' consisting of sets that are equal to L(P)-a.e .. It is easy to prove that we have a similar correspondence for random variables. If f:Q + IR is f'-measurable, then there exists a uniquely determined f :Z

e ^{+ IR} such that for each cxEIR, 0[f~cx]

=

^[f0^~cx]

.

Conversely, if f:Z + IR is F-measurable there is a nonempty set of F'-measurable functions f e such that

Two such f 0 's are different only on a set which e maps on

0 ,

and vJhich consequently has measure zero. Since e is measure-preserving we have ff 0dL(P)

=

ffdv and ffdL

=

_{Jf0dv .}

If {fn} is a sequence of random variables on <Q,F' ,L(P)> which converges a.e. to f , then {(fn)e} converges a.e. to f 0 . If {fn} is a sequence of random variables on <Z,F,v> that converges a.e. to f , then

simple facts about

converges a.e. to _f

e

_.

and _f

e

in the sequel.

We shall use these

2. ~leak Loeb-space representations of martingales

In this section we use the representation of measure spaces found in Theorem 3, to represent arbitrary L²-martingales by L2 -

martingales on Loeb-spaces. These are still real-valued martingales parameterized by m+ , but in the next section we shall see how to replace them by hyperfinite martingales.

We first extend the notion of weak Loeb-space representations to stochastic bases:

Definition 4: By a weak Loeb-space representation for a stochastic basis <Z, { F t}, J.l > , we mean a sequence <Q, {G t} tEJF. ,P > of internal

+

probability spaces with G s c:G t for s ^< t an increasing family {F·Ptem U{oo}

+

of a-algebras on n with F t c: L(G t) for t E m+

and a measure-preserving a-homomorphism e:F' + F _{00 00} such that 8 is surjective and for each t Em+ U{oo} , the tuple {<n,Gt,P>,F t'e rF t}

is a weak Loeb-space representation for <Z,Ft,J.I>.

Let M: JF.+ ^X

z

^{+ m be an L 2}-martingale adapted to the stochastic basis <Z,{Ft},J.I>, and let {<Q,{Gt},P>,{Ft},e} be a weak Loeb-space representation of this basis.

variable Mt:n -+JF. 8 such that

For each t E JF.+

8 [ Mt 8 ::a ]

= [

Mt ::a]

we may find a random for all a E JF., and [Mt::a 8

J

^E Ft Since 8 is measure-preserving, the process

M8

:m

^xQ +

m

+ defined by is an L²-martingale

adapted to the stochastic basis <S'l,{Ft},L(P)>.

Definition 5: Let M:JF.xZ+JF. + be an L²-martingale adapted to the stochastic basis <Z,{ft},J.I>, and let {<S'l,{Gt},P>,{Ft},8} be a weak Loeb-space representation of <Z, { F t}, J.l > . Then

{<n,{Gt} ,P>,{F iJ,e,M8 } is a weak Loeb-space representation of M if M8 is an L²-martingale with respect to <S'l,{L(Gt)},L(P)>.

Notice that the martingale Me in Definition 5 is required to be adapted to the basis <st, { L(G t)}, L( P) > and not only to

<st, { F t' } , L ( P) > . We shall need this to be able to replace Me by a hyperfinite martingale adapted to an extension of <n,{Gt},P>.

But by this requirement it is no longer obvious that M has a weak Loeb-space representation; however, we shall prove in Theorem 7 that it does have one.

Let us make the connection to the setting of [8]: Let {Gt}tET be an extension of {Gt}tEW to an increasing sequence of internal

+

algebras indexed by a hyperfinite time-line T. From Lemma II-5 and Lemma II-6, we see that

Ht

= n

^{cr ( L CG )UN)}

nElN sn

for each sequence {sn} of elements from T such that ^{{OS }} n decreases strictly to t . This tells us that the family {Ht}

does not change if we pass to a sub-line, and that it is uniquely determined by the original family {Gt}tEW and does not depend

+

Lemma 6: Let M:W+xz + W be a right-continuous L²-martingale with a weak Loeb-space representation {<st,{Gt} ,P>~{

ftl

,e,M } .

e

e

2

Then M is an L -martingale adapted to <st,{Ht},L(P)>.

Proof: Let A E H s , then by Lemma II-6 there exists a B such that B € L(Gs+ 1 /n) for all n E lN and L(P)(At.B)

=

0 . For t > s we get:

f

A(Mt-Ms)dL(P) ^{e e}

= f

B(Mt-Ms)dL(P) e e

= I

Blim (Mt-Ms+ 1 /n)dL(P) ^{e e}

n+oo

=

^lim

JB(M~-M:+ 1 ^/n)dL(P) =

0, where we have used

n+oo

since M is right-continuous, and the usual combination of Doob's inequality and Lebesgue's Convergence Theorem to get the limit out- side the integral.

We now prove

Theorem 7: If M is an L²-martingale, then M has a weak Loeb- -space representation.

Proof: Let <Z, { F t} , Jl > be the stochastic bas is of M • is finite, the result is obvious and we hence assume that

If F F _QO

QO

is infinite.

,.... .

For each t E lR+ u { ^QO} let F t be the cr-algebra generated by the sets { *F: F EFt} . In the proof of Theorem 3 we saw that for

each t there exists a measure-preserving a-homomorphism ht:Ft ~Ft, such that ht(*F)

=

F for all FE F t . If t > s , it is clear that ht IF s

=

hs since they agree on the generator set { *F: F € F s}

Hence all ht are obtainable from h00 and we shall write h for h 00

For each t E JR+ we may find an Ft -measurable random variable M' · _{t '}

*z

~

:rn.

such that h [ M ^I > a. ]

=

ht [ M ^I > a. ]

= [

Mt _> a. ]

t - t - ^{for all a. €}

:rn. .

Since

that that

we may find an internal and °K

=

^M' L(*Jl)-a.e.

t t

such that for all t E JR+

F or a 11 s, t E ^TO ..l.l'.+ , s < t , an d a 11 A E F ... s

n

*Fs , we h ave:

Let now

s. < t . .

l l

l.t. '+= PEP

be a finite family of sets from F , and

00

<s. , t. > 6 JR+ 2 such

l l

Then there exists a finite partition P of

*z

such and P

n

*B. _l

* ⁰

^{for some} i€{1, ... ,n}, then P c *B. ;

l

and if A is in the algebra generated by P, and A E

*

f s. ^{for some}

l

i € { 1 , ... ,m} , then

1

< *card(l'5

where *card(P) denotes the internal cardinality of P • To see

that this statement is true, take

P

to be the partition generated

by B_{1 , •••} ,Bn 5 and use the inequality we proved above.

By polysaturation there then exists a hyperfinite partition

P

of

*z

such that for each B E f _co if P E P and P

n

*B

*

^{¢ , then}

P c *B ; and if A is in the intersection of the internal algebra generated by P , and * f s , then for all t > s

IJ<Kt-K )d*lll

A

s

~ *card 1 p •

Since F _co is infinite, *card(P) E *JN,lN.

Let A be the internal algebra generated by P , and for each t € JR+ define

,....

G

=

An*F

t t

By definition of P, we have *BE A for all BE f , and

00

... """

consequently F t c L(G t) .

If ,..,. is the equivalence relation induced by P, let 1r :*Z ⁺ *Zj ...

,...

be the quotient map. ~A/e write Q for *Zj ... ; Gt for TI(3t); and P for TI(*J..d . We also put f'

=

1r(f ) for t E IR+U {co} , and let

t t

0 = ho1r-¹

Then {<n,{G t} ,P>,{~} :;0} is a weak Loeb-space representation hie must prove that M⁰ is a martingale with respect to the basis <Q,L(St),L(P)>:

If A E L(G s) , then there exists For t > s we have

=

I

f

(M'-M')dL(*]J)I

1T -1 (B) t s

= I _

⁰₁

f (Kt-K )d*lll

~

1T (B) s

BEG s such that L(P)Q\t.B) = 0 •

But this proves that M ⁸ 1s an 1²-martingale with respect to the stochastic basis <n,L(Gt),L(P)>, and hence the theorem.

If X is a predictable process with respect to <Z,{ft},~>,

then we can find a predictable version of x 8 , and if X is predict- able with respect to <n,{ Ft} ,L(P)>, then x₈ is predictable. If X E A 2 ( M) , then x⁸ E A 2 ( M0 ) and for r E lR +

I

X²dmM

= I

x⁸ 2 dm ₈

[O,r]xZ [O,r]xQ M

We need these simple facts to prove the following commutation rules for 8 and stochastic integrals:

Proposition 8: Let M be a right-continuous 1²-martingale. If

x

^{E A2}(M) , then x8 E A2(M ) and <IXdM)⁸

=

^{Ix8dM8 •}

If Y is predictable with respect to <Q, { Ft}, L( P) > and

Y E A 2 ( M8 ) , then _{Y 8} E A 2 ( M) and

IY

^dM8

= ( IY

8 dM) 8 •

Proof: Since M 1s right-continuous, for

each sequence {rn} decreasing to zero, and thus stochastic inte- gration with respect to M ⁸ is well-defined.

Assume first that X is of the form X

=

1<s,t]xF where FE Fs Then X⁸

=

1<s,t]xF7 , where F' E e-¹(F)

n F~

and we have

I rxe dMe e e Ir e

0

=

1 F, (M~Ar -Ms/\r)

= (

₀ XdM) . By linearity the assertion holds for all X of the form _X

=

^I: a .1 t ] F , F. E Fs.

1 <si, i x i 1 1

If X is an arbitrary element of A²(M) , there exists a sequence {Xn} of elements of the above form such that

Thus

r

lim(L²><IXndM)

n-+oo 0

lim(L

²

)(fX~dM

r ⁸

) =

n-+oo 0

=

r ^IXdM

0

for all

Moreover, the elements {Xn} are chosen such that lim

f

(X -X )²~ = 0 , but since

n-1-<» [O,r]xZ n

f

_(X-X ^.:2. _{J ~}

= f

^(X

e -x;)

Eh dm

e

[O,r]xZ n [O,r]xn . M

we also have But then

The second part follows immediately from the first since

(Y )

e =

^{Y •}

e

We have similar results for local L -martingales:

Let F" = ( F ^{I )} + =

t t

n

F'

s>t s Then the following result was proved in [ 6]:

Proposition 9: Let M:JR ^X

z

-1- JR

+ be a right continuous local martingale adapted to the basis < Z,{F t} ,JJ >, and let X € A(M) • There exist a weak Loeb-space representation {<n,{Gt},P>,{Ft},e} of

<Z, { ft} , JJ > and a version of _M

e

which is a local with respect to <n,{Ht},L(P)>, and such that

localizing sequence of stopping times for

x

^{0 and M}

^e

^{will be}

adapted to <n, { F~} , L( P) > , and each is F"-measurable.

t Moreover,

Some extension of the algebras F' t

the stopping times will not be adapted to

(like F")

t { F'} t .

is necessary since We need not, how- ever, use the whole of fV' · it is enough to add its null-sets to Ft' .

t ' ^::>

3. L²-martingales as right standard parts

In the last section we saw how we could replace a given martin- gale by a martingale adapted to a basis of Loeb-algebras having the same properties of stochastic integration. In this section we shall

go one step further by showing that each martingale of the latter type may be considered as the right standard part of a hyperfinite martingale.

Before proceeding, the reader should recall the comments on the family <n, {G t}, P > preceding Lemma 6.

Theorem 10: Let <n, {G t} tEJR , P > be an increasing family of hyper- +

finite probability spaces. Let N:JR+xn-+ JR be an L²-martingale with respect to <n,{Ht},L(P)> and assume that N

0 is cr(L(G )UN)- o

-measurable. Assume further that for each t E JR+ and each sequence {tn}n£N decreasing to t we have N(tn)-+ N(t) L(P)-a.e ..

Then there exist a hyperfinite time-line T, an internal basis

<n, {G t} tET, P > extending M: T x n -+ *JR adapted to

<n ~ ^{G t} tEJR , P > , and an SL²-martingale +

<n,{Gt}tET'P> such that for each t E JR+

0M+(t,w)

=

N(t,w) for L(P)-almost all w.

Moreover, we may take M to be well-behaved, S-right-continuous

at 0 , and such that for all t E JR+

0 [M]+(t,w)

=

[N](t,w) for L(P)-almost all w •

Proof: By saturation we may extend the family {Gt}tEJR to an +

increasing internal family {Gt}tES , where S is a hyperfinite time-line.

For each t E JR+ , the random variable Nt: n -+ JR lS H -t -measurable and thus i t is cr(L(G s )U N)-measurable for all s > t , s E JR+ . Consequently there is an

L~

s) E SL2 ( n ,G s, P) such that

0

L~

s)

=

Nt L( P) -a. e. .

~!Je

may extend the sequence {

L~t+

¹ /n)} nEIN by saturation to

where L(t+¹/y) t

an internal sequence

is G -measurable (recall that r is the least element in

(t+1/y)

S larger than r ) . There must be an internal initial

segment such that

111 (t+1/y) _ 1 Ct+1)11 < 1;

t t 2 - y '

for all y ~ 1;. • By construction ^I; must be infinite and it follows that ⁰L(t+ 1/1;) = N

t t

(t+1/1;) ²

L ( P ) -a . e . ; Lt E S L ( Q ,G ( t + 1

7

^{I; ) ,} P ) and

"

tile denote (t+1/t;) by t and choose 0

=

0 ; this is possible since we have assumed that N is cr(L(G )UN)-measurable.

0 0

" " "

Given a finite set S

=

{t 1 , ... ,tn} of such elements we show how to turn the process L(ti) into an SL 2-martingale with respect

t i

~o the basis <Q,{Gt}tES'P>; i.e. we construct an SL²-martingale 8 M:

s

^{X Q +}

:m

adapted to <Q, {G t} tES ,P >

I!

§M(ti) -

L~~i)ll

2 ^Rj 0 for each tiES :

l

A A A

such that

Assume that tl<t2<•••<t

§

^A L(tl) we define M(t ) -_{.. 1" -} t 1

" n

we have constructed for

s " .

j < i , we define M(ti+ 1 ) following way: For each w E Q , let

[ w] .

= n {

AE Gt" . : w EA} ,

l l

and define B by:

If

in the

B

=

{yE*JN:P{w:

I I (L~~i+l)

_§M(ti))dP! > 1/yP([w] .)} < 1/y}.

[w]i 1+l 1

Then B is an internal set and we prove that 1N c B : Assume n E 1N, but n ( B : Then either

P{w:

f CL~ti+1)

_§M(t.))dP >1/nP([w].)} > 1/2n

[ wh

i+ ^{1 l l}

or

P{w:

I

(Lt(ti+l) _§M(t.))dP <-1/nP([w].)} > 1/2n.

[wl· i+l l l

l

Assume the first; the argument in the second case is similar. The

set -

§~1( t .))

^{dP > 1} I nP [ w] . }

l l is in and

P(C)

=

^f<Nt. -Nt.)dL(P)

=

0, which is impossible. Hence INcB,

c

^{l+l l}

and since B is internal we may find an infinite y E B .

"

Let D be the set of internal measure less than 1/y such that

For w ED , we define

for w E D ; then DE G

t . .

l

" "

s " s "

M(ti+l) = M(ti) . For w E ri'D , let

s "

M(ti+l ,w)

= f

(L(ti+l) _SM(t.))dP/P[w.]

[wh ti+l ^{l l}

" "

It follows immediately that for each w En ,

f

(SM(ri+l)- SM(-f:i) )dP

=

0,

" " [w] •

and hence SM is a martingale. Since SM('ti) E SL²

(n

,G t. ,P) by

l

induction.hypothesis, we have JDSM(ti+l)²dP

= JDSM(ti) ² dPR~0

^We

also have

Since obviously

o" "

SM(t" ) -- ⁰L(ti+1 )

=

N

l. +1

ti+l ti+l a.e., we have constructed the martingale by induction.

We now turn to the next step of the proof: So far we have only constructed approximating martingales on finite time-lines, we must extend this to hyperfinite time-lines: Let {qn}nEIN be an enumer- ation of the non-negative rationals, and let sn

=

{ql'' .. ,qn}.

We may extend the sequence {Sn}nEIN to an increasing internal sequence {~n}n$v of hyperfinite subsets of S, and the sequence {§nM}nEIN to a sequence of internal martingales adapted to

<n,{Gt}tES ,P>.

n

If m,n E IN, m < n , then for all qk , k ^" < m , we have:

By saturation there exists an infinite n such that llsnMcqk) _§nMcqk>ll2 < 1fn

for all n E IN , k ~ n . It follows that o§

nM(qk)

=

N(qk) , L(P)-a.e., for

Using this and the assumption that N is right continuous

o § + §nM

we have ( nM) = N a. e.. is right continuous at 0 , since

,.

0

=

0 •

By letting be a suitable sub-line of § , we can make the n

restriction of to T well-behaved by Theorem II-23. Let M be this restriction, then by Theorem II-21 °[M]+

=

[N) • This proves the theorem.

The assumption that N

0 is a(L(G )UN)-measurable was made

0

1n Theorem 10 to ensure us that we could choose M right-continuous at 0. We saw in Theorem II-17 that this was a necessary condition when we compared stochastic integration with respect to M with stochastic integration with respect to ⁰M+ .

Using Theorem 10 one may prove the following analogous statement for local L2-martingales:

Theorem 11 : Let <Q, {G t } tE:ffi. , P > be an increasing sequence of +

hyperfinite probability spaces. Let N::ffi.+xQ be a right continuous local L2-martingale with respect to <Q,{Ht},L(P)>, such that N

0

is a(L(G )UN)-measurable. Then there exists a local SL 2-martingale

0

M: T x Q + *:rn. adapted to an extension <Q, {G t} tET ,P > of

<Q,{Gt}tE:ffi. ,P> such that for all t E :rn.+:

+

0 +

M (t,w)

=

N(t,w) for L(P)-almost all w.

Moreover, we may assume M to be right continuous at 0 and such that for all t E JR.+ :

I don ¹ t knov.r if we can make the martingale M in Theorem 11 well-behaved; application of Theorem II-23 as in the proof of Theorem 10 is not possible since the restriction may fail to be a local SL²-martingale (see Example II-10 and the comments following the proof of Theorem II-23). The equality ⁰ [M]+ =[N] can be

proved directly without using Theorem II-21.

4. Hyperfinite representation of martingales

Time has come to gather our results. In Section 2 we found a representation of arbitrary 1²-martingales as 1²-martingales on Loeb-spaces, and in the last section we saw how we could repre- sent martingales of the latter type as right standard parts of SL2 - martingales. From Theorem II-17 we know the relationship between stochastic integration with respect to an SL²-martingale and sto- chastic integration with respect to its right standard part. The following definition seems natural:

Definition 12: By a weak hyperfinite representation of a martingale

. } ' e}

N we mean a weak Loeb-space representat1on {<n,{Gt ,P>,{Ft},e,N and a hyperfinite martingale M adapted to some extension

is equivalent to

Combining the results mentioned above, we get:

Theorem 13: Let

N

be a right continuous L²-martingale. Then

N

has a weak hyperfinite representation M which is a well-behaved 8L²-martingale 8-right continuous at 0 , such that ⁰ [M]+

=

^{[NJ 0 .}

If X E A ² (N) then there exists a Y E 8L² (M) such that fYdM is a hyperfinite representation of fXdN .

Proof: By Theorem 7, N has a weak Loeb-space representation

{ P} ,

e

,N } •

e

t

By Lemma 6, N

e

satisfies the conditions of Theorzm 10, and by that theorem we obtain a weak hyperfinite repre- sentation M of N, which is well-behaved, 8-right continuous at 0, and such that ⁰ [M]+

=

^[N°]

=

^{[N] 0 .}

If X E A 2 ( N) , then X0 E A 2 ( NA) and

by Proposition 8. By Lemma II-15 and Theorem II-17,

x

⁰ has a 2-lifting Y with respect to M such that

Thus fYdM is a hyperfini te representation of fXdN , and the theorem is proved.

Using Corollary II-18 and Theorem 11, one may prove the corres- ponding result for local 1²-martingales:

Theorem 14: Let N be a right continuous local L²-martingale and let XEA(N). Then there ~xists a weak hyperfinite representation M of N which is a local 8L²-martingale, 8-right continuous at 0

Moreover, there is a Y E 8L(M) such that fYdM is a hyperfinite representation of JXdN

Our purpose in this paper has been to show that everything that can be done by the standard theory of stochastic integration can also be done by the nonstandard theory. Theorems 13 and 14 seem to indicate this, since by using them we can define stochastic integration with respect to any right continuous local L²-martingale from the nonstandard stochastic integral with respect to a certain local SL²-martingale. The construction is complicated and cumbersome and it is not my intention that i t should be used to construct non- standard martingales from standard ones in a concrete situation;

indeed, one of the nicest and most interesting aspects of the non- standard approach to stochastic anclysis is the simple and natural constructions i t gives for certain classes of processes; see e.g.

Loeb's construction of Poisson processes in [9], and Anderson's of a Brownian motion in [1] (see Examples I-1, I-15 and II-24). Also consult Keisler's lifting results in [5]. My intention has rather been once and for all to show that by using nonstandard methods we do not condemn our theory to less generality than what we could obtain by sticking to the standard methods.

5. The transformationformula revisited

As an application of the theory for weak hyperfinite repre- sentations, we shall use the nonstandard version of the transfor- mationformula (Theorem I-22) to prove the standard version. Let

us first prove i t for the right standard part ⁰M+ of a well-behaved SL²-martingale

o[M]+

=

[oM+]'

tp: JR + JR is in

M which is S-right continuous in 0 and such that and then use Theorem 13 to prove it ln general: If

C²(JR) , we get from Theorem I-22:

<.p ( ⁰ M + ( t ) ) - <.p ( ⁰ M + ( 0 ) )

=

¹ im 0 ( * <P ( M ( t + 1 I n ) ) - * <.p ( M ) )

n-+oo 0

t+1/n t+1/n

=

lim^{0 (}

f

*<P' (M)dM + ~

f

*<.p"(M)d[M]

n-+oo 0 0

+ ^{I: 0} (*tp(M ) - *<.p(M _)- *<.p' (M _)(M -M _) -l*<.p"(M _)(M -M _)2 ))

sst+1 /n s s s s s ² s s s

By Lemma II-20, *<.p'(M) is a lifting of <.p'(⁰M-) (where ⁰M- is the left standard part of M), and consequently lim ⁰Jt+1/n *<.p'(M)dM

n-+oo 0

= f\p' (

⁰M- )d⁰M+ .

0 Using that ^M is well-behaved, S-right continuous at 0 and such that ⁰ [M]+ = [⁰M+] , it is not difficult to see that

Jt<P"(⁰M-)d⁰ [M]+ . Since M is well-behaved,

0

the jumps of ⁰M+ are exactly the noninfinitesimal jumps of M , and hence

lim ^I: o(*rn(M ) - *w(M _)- *rn' (M _)(M -M )-l*rn"(M )(M -M _)2)

~ s · s ~ s s s- ² ~ s- s s

n-+oo ss:t+1/n

=

I

Combining these results we have:

which is the transformationformula for ⁰M+ .

Let now N be a right-continuous L²-martingale with left limits. To prove the transformationformula for N, we find a weak hyperfinite representation M of N as in Theorem 13. If N

denotes the left limit of N, N- is predictable, and we know from Theorem 13 that <ft<.p'(oM-)doM+)

=

Jtcp ^{I (} N-) dN .

0 0 ₀

It is easy to see that <J\p"(⁰M-)d[⁰M+])

=

0 0 ft<.p"(N-)d[N]

0 by

approximating <.p"(oM-) by the predictable processes

by the processes

Since is left-continuous,

=

(lim Jtx d[⁰M+])

0 n

e

n-+a>

= ftq>"(N-)d[N]

0

It remains to show that ( I:

s~t

=

^I:

s~t

To do this we need two lemmas:

Yn -+ (p"(N-) • Now

=

lim<Jtx d[⁰M+])

0 n

e

n-+co

Lemma 15: Let N be a right continuous L²-martingale w~th left limits adapted to a basis <Z, { F t}, ll > • Then there exist a sequence

of F co -measurable functions and a set

Z'c Z

of measure one such that if z E

z

^I and N ( z, t)

*

^{N- (} z, t) , then t

=

^f _n ^{(z) for}

some n E

:rn.

Moreover, if z € Z' and n

*

m , then f _n (z)

*

^f _m ^{(z) •}

Proof: Let

F

_co be the completion of F co with respect to. ll , and let N be the null-sets of

F

₀₀ Define Ft =

n

cr(FSUN) •

s>t Then

<Z, { F t}, Jl > satisfies Meyer's ¹¹usual conditions", and since N is right continuous, N is a martingale with respect to this basis.

By Satz 17.5 of Metivier [ 10], there is a disjoint sequence { ^T

J

nEIN of stopping-times enumerating the jumps of N , each Tn adapted to

Let f =E(T I F ) .

n n oo

lemma follows.

Then ^{: T} _n a. e. , and the

Lemma 16: Let N be a right continuous L²-martingale with left

,...

limits, and let ^tD: IR-+ IR Let N be a weak Loeb- -space representation of N which also is right-continuous with left limits. Then for all r E IR+

,....

Proof: Applying Lemma 15 to N , we get a sequence { fn} nElN of ...

f'-measurable functions enumerating the jumps of _co

N,

and hence the second sum above equals:

I: (c.pWf )

-lp(Nf )

-c.p' CN[ )(Nf

-N[ )

nElN n n n n n

fn(w)st

Approximating f

n from above and below with simple functions and ,....

using the right continuity of

N

and N, we see that:

,....

[w + Nf (w)(w)]e = [w ⁺ N(f ) (w)(w)] a. e. , and

n n e

[w +

N'£

< w) < w)

l

e = [w ⁺ N(f ) (w)(w)] a. e ..

n n e

Thus:

< I: <c.p<Nr >

-q~<N£

^{) -c.p'}

c'N[

HNf

-'N£ ) - ~lP"<N£

HN'f -N'£ )²))e

nElN n n n n n n n n

fn (w )st

=

I: (c.p(N(f) )-c.p(N(f) ) -·c.p'CN(f) )(N(f) -N(f) )

-~c.p"(N(f

)»-J(-F)-N;) )²)

nEJN ne ne ne ne ne ne i1e "TTe

To prove the lemma i t only remains to prove that the sequence {(fn)e} enumerates all the jumps of N; by Lemma 15 there exists a

of F -measurable functions which does this. Let

co

Am= rl'U[g =(f )e]. n m n - By approximating the restriction of gm to Am from above and below by simple functions, we get

(Ngm-N~m)e

=

,.... ^...

_

_,.... ,...._

{fn} enumerates Am , and { (fn)e}

enumerates the jumps of N . As we have already observed, this

e Ae

< N e-N e ) on A But N e = N e on since

gm gm m gm gm m

the jumps of N. Consequently N _gm =

N~

^{a.e. on}

m proves the lemma.

Theorem 17: (The transformationformula): Let N be a right con- tinuous local L ²-martingale with left limits, and let q:»:

:m

+

:m

be in C2 ( JR) • Then for all t E JR+ :

t t

= f((J' (N-)dN + ~J((J"(N-)d[N]

0 0

a. e.

where the sum is absolutely convergent.

Proof: It is clearly enough to prove the theorem for L²-martingales.

Let M be a weak hyperfinite representation of N as in Theorem 13.

By the transformationformula for

<Jt<P'(⁰M-)d⁰M+)₀

=

Jt((J'(N)dN and

0M+ and the observations

0 - 0

t t

q:»(Nt) - q:»(No)

=

(cp(o M; )-q>(OM:)) e

=

Cfq:~' (oM-)doM+) e + ~(Jq:~"(oM-)d[oM+] ) e +

0 0

I: (cp(oM+) _ q:»( oM-) ^-q>' (oM-)( oM+_ oM-) _ J.q:~"( oM-)(oM+ _oM-) 2))

=

t

s

s s s s ² s s s e

ss

where we have used the following argument for the equality of the sums:

~

Let

N

be the weak Loeb-space representation corresponding to N . By Satz 20.7 of Metivier, N has an equivalent ^A L²-martingale adapted to <n,{Ft+},L(P)> which is right-continuous and have left limits. There must be a set n' cQ of Loeb-measure one where

~N -- ⁰M+ f or a 11 t . Applying Lemma 16 to

N ,

we get the equality above.

That the sum is absolutely convergent follows from the fact that the sum in Theorem I-22 is absolutely convergent.

Corollary 18: (Ito's formula): Let N be a continuous local L^{2 -} martingale, and let q:»:

:m

⁺

:m

be in C2 ( JR) • Then:

t t

<P ( N t ) ~· <P ( N _{0 )}

= f

^{<P 1 (} N) dN +

! f

^{(J) 11 (} N) d [ N ) .

0 0

We may now prove the result we used 1n the proof of Theorem II-21:

Corollary 19: Let N be a right continuous local L²-martingale with left limits. Then for t E :rn.+ :

t

[NJ(t)

=

N(t)²-N(0)² -2fN-dN a.e ..

0

Proof: Apply Theorem 17 with <P:x + x2 :

Performing the multiplications in each term inside the sum we get zero, and hence we are left with Nt^{2 -} N²

=

2ftN-dN + [N](t) , which proves

0 0

the corollary.

Notice that the proof we have g1ven of Corollary 19 is circular;

we used Corollary 19 to prove Theorem II-21 which we used to prove Theorem 10 which we used to prove Theorem 13 which we used to prove Theorem 17 and hence Corollary 19. This also makes the proof of the transformationformula circular, since Metivier used this formula to prove Corollary 19. Luckily, i t is not difficult to avoid this circu- larity since we can prove Theorem 10 without using Theorem II-21:

The argument would proceed as in the proof we have given up to the use of Theorem II-21; we thus have a well-behaved SL²-martingale M

which is S-continuous at 0 , such that ⁰M+

=

N • Using the difi- nition of

partition

[N] as the L¹-limit of t·i:E1T(N(ti+l) -N(ti))²

l

1T gets finer, i t is easy to find a restriction ,...,

as the

,...,

M of M to a subline such that ⁰ [M]+

=

[N] . This martingale M would then satisfy Theorem 10.

As we have now proved Corollary 19 (and thereby Theorem II-21), the theory developed in these three papers should be reasonably self- -contained. In addition to some simple facts about the paths of real- -valued martingales (i.e. Satz 17.5 and Satz 20.7 of Metivier [10]), we have only used the most basic results from the standard theory for stochastic integration. We should therefore be able to develop by purely nonstandard methods a theory for stochastic integration just as powerful as the standard theory. One may also hope that the dis- creteness of the hyperfinite time-line and the simplicity of the non- standard definition of the stochastic integral will make the hyper- finite theory easier and more intuitive to work with; this seems

indeed to be the case in Keisler's theory for stochastic differential equations.

Statements left unproved in the present paper were proved 1n [6].

References

1. R.M. Anderson: A Nonstandard Representation for Brownian

Motion and Ito Integration. Israel J. Math. 25(1976) pp. 15-46.

2. R.M. Anderson: Star-finite Probability Theory. Ph.D.-thesis Yale University 1977.

3. R.M. Anderson: Star-finite Representations of Measure Spaces.

To appear.

4. P. Billingsley: Convergence of Probability Measures.

John Wiley and Sons, 1968.

5. H.J. Keisler: An Infinitesimal Approach to Stochastic Analysis.

Preliminary Version 1978.

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(Unpublished)

7. T.L. Lindstr¢m: Hyperfinite Stochastic Integration I: The Nonstandard Theory. Matematisk institutt,

Universitetet i Oslo, Preprint Series, 1979.

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Comparison with the Standard Theory. Matematisk

institutt, Universitetet i Oslo, Preprint Series, 1979.

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