- i -

Introduction

For a number of years nonstandard measure theory was in a

somewhat problematic state. Although many interesting papers

appeared (e.g. Bernstein and Wattenberg [5] , Loeb [14] ,[15] and

Bernstein and Loeb [4]), the theory suffered from the fact that

nonstandard measures refuse to be countably additive. This made

application to and comparison with standard measure- and proba­

bility-theory difficult.

Many of these difficulties were resolved when Peter A. Loeb

in [16] showed how to turn *-measures into real-valued,countably

additive measures. This opened the way for the use of nonstandard

constructions in probability theory, and Loeb himself gave the first

example by constructing a Poisson process. Shortly afterwards R.M.

Anderson [1] used Loeb's technique to make a nonstandard construction

of a Brownian motion in a very simple and attractive way. Anderson

also gave a nonstandard treatment of stochastic integration with

respect to this Brownian motion process, and this theory was used

by H.J. Keisler ln [10] to obtain new results on stochastic differ~

ential equations.

Keisler [9] (see also Hoover [7]) based his hyperfinite model

theory on the Loeb-measure, and A.E. Hurd [8] used it in analyzing

equilibrium states in statistical mechanichs. In his thesis [2],

Anderson further developed the theory of Loeb-spaces, found a repre­

sentation of Radon-spaces as inverse images of hyperfinite Loeb-spaces,

and gave applications to economics. This list does by no means exhaust

the applications of the Loeb-measure, but it should be long and

varied enough to indicate the fruitfulness of the idea.

In this paper we shall carry on the work on nonstandard stochas­

tic integration begun by Anderson ln [1 ]. Let us first give a short

- ii -

sketch of standard stochastic integration. The problem is simply

to give a precise meaning to the heuristic formula fXdM where

X,M: :Rt x Q + :R are suitable processes on a probability space Q •

This is nontrivial since "typical" examples of M (e.g. Brownian

motions) have paths of unbounded variation, and the integral fXdM

thus can not be defined by pathwise Stieltjes integration. However,

it was realized by N. Wiener and K. Ito that in the Brownian motion

case the integral fXdM can be defined as an L2 -limit, and this

has later been extended to more general classes of processes M

(so called local L 2 -martingales; see Meyer [20] or Metivier [18]

for expositions.)

We now turn to Anderson' s work: Let n E *N ':N , and let

T = { k/ n: k E *:N, k < n 2} • Anderson constructed a hyperfini te

*-probability space Q and an internal process x : T x Q + *.R -

which in fact was nothing but a hyperfinite random walk - such that

the process 13 : :R+ x Q + :R defined by 0 -S(t,w) = x<t,w) , was a

Brownian motion (here t is the least element of T larger than t .

For a definition of Anderson's pi'Ocess, see Example 1 below). If

Y:TxQ+

integral

*.R

fYdx

is an internal process, he defined the stochastic k-1 . . .

to be the process (k/n ,w) + E Y(l/n ,w)(x(l+1 /n ,w)-x(l/n,w)); i=O

i.e. as a pathwise "Stiel tj es integral'~ Anderson was now able to

show that given a process X such that fXdS was defined by the

standard theory, he could find an internal process Y (called a

2-lifting of X ) such that Thus he could do by

nonstandard methods what had proved impossible by standard ones; to

write the stochastic integral as a pathwise Stieltjes integral. This

intuitive definition of the stochastic integral enabled him to

give a simple proof of Ito's Lemma. The work by Keisler [10] on

stochastic differential equations has later confirmed the importance

- iii-

of this approach to stochastic integration.

The purpose of this paper is to extend Anderson's results

about x to a larger class of internal processes, which we shall

call local A2 -martingales. We shall deal with properties of these

processes such as finiteness and (right- and left- ) continuity of

the paths; we shall define stochastic integrals with respect to

them in a way similar to that of Anderson; and we shall derive the

basic properties of these integrals, and prove nonstandard versions

of the transformation-formula and Ito's formula.

In another paper [12], we shall prove that the nonstandard

theory for stochastic integration with respect to such hyperfinite

martingales, is equivalent to the standard theory for stochastic

integration with respect to '~the standard parts '1 of the martingales.

In a third paper [13], we shall show that all standard local

L2 -martingales can - in a suitable sense - be represented as "the

standard part" of a local A2 -martingale; and we shall argue that

this combined with the results in [ 1 2] shows that anything that can be

done by the standard theory for stochastic integration, can be done

by the nonstandard theory. The hope is -of course - that the non­

standard theory will be simpler and more intuitive to work with.

This and the subsequent papers are based on a common previous

version, the [11] of the references. That version is at times a

little more comprehensive and contains some additional material.

On the other hand it is rather unreadable.

Our main references will be Metivier [18] on stochastic inte­

gration, and Stroyan and Luxemburg [23] on nonstandard analysis.

This paper and the other two mentioned above constitutes my

cand.real.-thesis written at the University of Oslo under the super-

- iv -

vision of Professor Jens Erik Fenstad. I would like to thank

Professor Fenstad for all his encouragement and good advice, and

I would also like -to ext..:.nd my thanks to all the membert?· of the

seminar on nonstandard analysis in Oslo during the last years.

- 1 -

1. Setting

We shall work with polysaturated models for nonstandard ana­

lysis. That is we begin with a set K large enough to include all

the standard entities we shall need (e.g. the real numbers F, the

relevant measure spaces), and we construct the superstructure V(K)

over K. vJe then take *V(K) to be a card(V(K))-saturated non-

standard model of V(K) . This will be our polysaturated model.

For a proof of the existence of such models and a survey of their

properties, see Stroyan and Luxemburg [ 2 3 ] , sections 7. 4 - 7. 6.

Let <Q,G,P> be a *-measure space. We may define a finitely

additive measure 0 P on the algebra G by 0 P(A) = st(P(A)) for

all A EG • Peter A. Loeb proved in [ 16] that 0 P satisfies the

conditions of Caratheodory's Extension Theorem (Royden [22], page257)~

and that 0 P thus may be extended to a measure on the a-algebra

a (G)

L(P)

generated by G . The completion of this measure we denote by

and call the Loeb-measure of P . The a-algebra a (G) is

called the Borel-algebra of G and its completion L(G) with respect

to L(P) is called the Loeb-algebra of G . The measure space

<Q, L(G), L( P) > is called the Loeb-space of <Q ,G , P > . When °P( Q)

is finite, it follows from the uniqueness part of Caratheodory's

Theorem that L(P) is uniquely determined by P; C.Ward Henson has

proved that this still holds when °P(Q) = "" • If Q is *-finite

and P(Q) = 1, <n,G,P> is called a hyperfinite probability space.

The measure- and integration-theory of Loeb-spaces were devel­

oped in Loeb [ 16] and Anderson [ 1 ] . Expositions are given in Loeb

[17] and (without proofs) in Keisler [10].

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2. Hyperfinite Processes

We now define the basic concepts in the theory of hyperfinite

processes:

A hyperfinite time-line is a hyperfinite subset T of *:R +

such that 0 E T and for each a E :R+ there is a t E T such that

tRJa. If S and T are hyperfini te time-lines with S c T , we

call S a subline of T .

By a hyperfinite stochastic process X we mean an internal

mapping X : T x Q + *m, where T is a hyperfini te time-line and

<n,G,P> is a hyperfinite probability space such that the mappings

w + X( t, w) are G -measurable for all t E T .

If T 1s a hyperfinite time-line, an internal basis

<n,{Gt}tET'P> indexed by T consists of a hyperfinite set n ,

an increasing *-sequence {G t} tET of *-algebras of subsets of n ,

and a *~probability measure p on where t co

is the largest

element in T . For convenience we shall often assume that Gt is co

the internal power-set of n such that each one-point set { w} is

G t -measurable. co

Let us illustrate these definitions by an example:

Example 1: In [ 1 ], Anderson constructed the following process:

Choose an n E *N,JJ, and let the time-line be T = {k:kE *N, k<n 2 L n

The set n is the set { -1 ~ 1} T of all internal functions from T to

{-1,1}. If wEn , we shall write w. 1

for i w (-) • The internal n

algebra Gk will be the /n

*-algebra generated by the equivalence

classes of the relation ~ k '

where W"' w' k if w. =w.' 1 1

for all

1 < k • The probability measure P 1s the uniform measure on

-n2 n ; P{ w} = 2 for all w E n . Then <Q, {G t}, P > is an internal

- 3 -

basis. We define a hyperfini te stochastic process x : T x n + ii'JR by

k-1 I: w./.;n.

. 0 l l=

Anderson proved that if we define 13 : JR x n + JR by + 0 -(3(w,t) = x<w,t),

where t is the least element of T larger than t , then 8 is a

Brownian motion.

We see that for each t E T , w + x ( t, w) is G t -measurable, and

this motivates our next definition:

A hyperfinite process X : T x n +*JR is said to be adapted to

the internal basis <n,{Gt},P> if w + X(t,w) is Gt-measurable

for all t E T •

Notation: We shall often write Xt(w) or - suppressing w - X t for X(t,w). Let 0 = t 0 <t 1 < ... <t~ =too be the elements ofT

in increasing order. If X : T x n + *JR is a process, we shall write

6X(ti,w) for X(ti+ 1 ,w)- X(ti,w) . We use the following notation

for summation: If s = t. ' t = t. l J

and

t :E X(r,w) s

j -1

for :E X(tk,w) . k=i

i<j, we write

t Notice that the term X(t,w) lS not included in the sum :E X(r,w) .

s We use similar notation when the limits s and t of the summation

depends on w .

By the quadratic variation of a process X : T x n + *JR , we mean

the process [X] :Txn +*JR defined by

t [X](t,w) = r (6X(s,w)) 2

0

Observe that if X is adapted to a basis <n, {G t}, P > , then so is

- 4 -

[X,]. If x 1s the process of Example 1 , then [ x ]( t ,w) = t .

An internal stopping time adapted to the internal basis

<Q,{Gt},P> 1s an internal mapping T: Q + T such that for all

t E T , the set { w E Q : T ( w) ~ t} is in G t

Definition 2: A hyperfini te process M : T x Q + *JR is called a

hyperfinite martingale (hyperfinite sub-martingale, hyperfinite

super-martingale) adapted to an internal basis <Q,{Gt} ,P>, if

M is adapted to <Q,{Gt},P> as a process and for all s,t ET,

s < t , and all AEG s

In this definition the expectation E(1A(Mt-Ms)) 1s just a

hyperfinite sum I: 1A(w)(Mt(w)-M (w))P{w}. wEQ s

This paper is a study of hyperfinite martingales, and we have

already encountered one; Anderson's process x is a martingale with

respect to the basis <Q,{Gt},P> constructed in Example 1.

We shall need the following technical lemma relating the expec-

tation of the quadratic variation of a martingale to the expectation

of the square of the martingale.

Lemma 3: Let M : T x Q + *JR be a hyperfini te martingale adapted

to the internal basis <Q,{Gt}P>, and let a,T be two internal

stopping times adapted to the same basis. Assume that a(w) > T(w)

for w E Q . Then:

o(w) E(M(a(w),w) 2 -H(T(w),w) 2 ) = E((M(o(w),w) -Mh(w),w)) 2 ) ::; E( I: ~M(s,w) 2 ).

-r(w)

Proof: We have:

E((M(a(w),w)-M(T(w),w))2) = E(M(o(w),w) 2-2M(a(w),w)M(T(w),w)+M(T(w),w) 2 ) = = E(M(o(w) ,w )2-M( T (w) ,w ) 2-2M( T (w) ,w )(M( a(w) ,w )-M( T(w) ,w) )) •

- 5 -

To prove the first equality in the lemma, it is thus enough

to prove that E(M(T(w),w)(M(cr(w),w)-M(T(w),w))) = 0

For each s E T let As = { wEQ : T ( w) = s} . Then As EG s , and

E(M(T(w),w)(M(cr(w),w)-M(T(w),w))) = E( I: 1A (w)M(s,w)(M(cr(w),w)-M(s,w))) sET s

cr(w) = E( I: 1A (w)M(s,w) I: t:.M(t,w)) = I: I: E(1A (w)M(s,w)1[t 1t:.M(t,w)),

sET ·D s SET t~s ·D <cr

where [t<cr] = { w: t<cr ( w)} . Here 1 A is G t -measurable and so s

Moreover, [t<cr] = Q '{w:cr(w)~t} EGt. By the martingale

property all the terms in the double sum above are zero, and the

first equality of the lemma follows.

To prove the second equality, observe that:

cr(w) E((M(cr(w) 7 w) -M(T(w),w))2) = E(( I: t:.M(t))2) =

T(w)

Rewriting the last term, we get:

E(I:{t:.M(s,w)t:.M(t,w): T(w).::; s < t < cr{w)})

cr(w) E( I: t:.M(t) 2) +E(2I:t:.M(s)t:.M(t)).

T(w) T$S<t<cr

= E(I:{1h$s]t:.M(s,w) 1[t<cr]t:.M(t,w) I sET,t>s}).

Here 1 [ T~S] , t:.M( s ,w) and 1 [t<cr] are all G t -measurable, and

consequently the expectation of each term lS zero. This proves

the second equality and hence the lemma.

We end this section by quoting two results from the standard

theory of martingales which we shall use in the sequel. The first

is a famous inequality due to J.L. Doob which is proved in almost

any text on stochastic processes; see e.g. Doob [ 6] or Metivier [18]:

Proposition 4: (Doob's Inequality). Let X: J x Z + JR be a positive

(standard) submartingale with respect to a stochastic basis

<Z, { F t} tEJ, ~ > , where J c JR + Let I be a countable subset of J

- 6 -

and assume that b = sup IE J . Then for all p E <1 ,m > such that

E(Xh) < m , we have:

II sup Xtl! < _E_1 I! xb !l . t E I p - p- p

In this paper we shall use the *-transfer of Proposition 4

to deal with hyperfinite martingales, while in [12] and [13] we

shall need the standard version above.

The next proposition is *-transfer of well-konwn inequalities

1n the theory for discrete martingales.

Proposition 5: Let H : T x Q + *JR be a hyperfini te martingale.

Then for all p E <1 ,oo> and all t E T

11/ M~+ [H] Ct) II _:: 1 Op!lsupCM.JII p s ~t p

and

II sup ( Mt )II _:: p/121!/M"~ + [ M] ( t) liP . s ~t p

For a proof, see Neveu [21] (Corollary vm-3-18, page 199) or

Meyer [19]. Lest the reader should feel discouraged by the length

and difficulty of these proofs, we hasten to assure him that we shall

only need the result for p = 4 , and for some constants c 4 ,d 4 E JR in

stead of 10o4 and 4/12 . A simple proof may then be obtained from

Meyer's comments in [19] , and this was done in [11] .

- 7 -

3. The structure of A2 -martingales

We define the classes of hyperfinite martingales that will

be our principal objects of study in this paper:

Definition 6: Let H : T x n + *:R be a hyperfini te martingale

adapted to the internal basis <D,{Gt},P>. M is called a

A2 -martingale if for all finite t E T, E(M~) is finite. M is

called a local A2-martingale if there exists an increasing sequence

{ • n }nE:N of internal stopping times adapted to <D, {G t}, P > such that

0 T + ~ a.e. in Loeb-measure, and such that for each n the martin­n

gale M defined by M (t~w) = M(T (w)At,w) •n •n n

is a A2 -martingale.

The sequence {Tn} 1s then called a localizing sequence for M.

In this section we shall use Lemma 3 and Proposition 4 to

derive some properties of the paths of A 2 -martingales. An obvious

consequence of Lemma 1 is that a hyperfinite martingale is a A2 -

martingale if and only if E(M~+[M](t)) is finite for all finite

tET.

Proposition 7: Let M: T xQ + *:R be a local A2 -martingale. Then

the set of all w E Q such that M(t ,w) and [M] (t ,w) are finite

for all finite t E T has Loeb-measure one.

Proof: Let {T } be a localizing sequence for M. n For each finite

tET and each nE:N, E([M ](t)) •n

is finite. Consequently

[M ](t) is finite a.e .. Since [MT ] is an increasing process •n n

max[M ](s) is finite L(P)- a.e.. Now 0 Tn + ~ L(P)- a.e. and s::;t •n hence max[M](s) is finite L(P) -a.e .. Since this is true for an

s::;t arbitrary finite t E T, the [M]-part of the proposition follows.

- 8 -

By Proposition 3

ECmax M~n(s)) ~ 4E(M~n(t)) s~t

and consequently 2 max M, (s) is finite L(P)- a.e. for all n E:N s<t n

and all finite t E T . Repeating the argument of the first part of

the proof, we prove the M-part of the proposition.

Proposition 7 tells us that the local A2-martingales live

where we want them to live; in the finite part of *F. Our next

result deals with the behaviour of the paths as we approach a point

t in T . We need some definitions.

Definition 8: If f: T + *F is an internal function we define the

upper S-left-limit of f at t by:

S-llm f(s) = inf sup{ 0 f(r): s < r< t, r( mon(t)} s+t s<t

s(mon(t)

The lower S-left-limit of f at t is defined by:

S-lim f(s) = sup inf{ 0 f(r): s ~ r< t, r (mon(t)} s+t ~<t

sl[mon(t)

The upper S-right limi! s-11m f(s) and the lower s-1-t

S-right limit S-lim f(s) is defined in a similar way. s-1-t

The function f 1S said to have an S-left limit

S-lim f(s) = S-lim f(s) =I= ±co and an S-right limit if s+t s+t

S-lim fW = S-lim f(s) :j: ±co • s-1-t s+t

Notice that these limits only depend on the monad

not on which point in that monad we use.

at

of

t if

t and

- 9 -

A process X : T x Q + *JR lS said to have S-left limits

( S-right limits) a. e. if for L( P) -almost all w E Q the function

t + X(t,w) has S-left limits (S-right limits) for all finite t E T.

We have the following:

Theorem 9: Let M be a local A2 -martingale; then M has S-left-

and S-right limits a.e.

Proof: We prove the S-left limit case; the S-right limit case

is similar.

There are three ways in which M can fail to have S-left

limits on a path s + M( s ,w) , there must exist a finite t E T

such that either S-llm M(s,w) = ~ or stt

that

S-lim M(s,w) = -~ or such stt

S-lim M(s,w) - S-lim M(s,w) > e stt stt

for an e EJR , e > 0 •

In the first two cases max !M<s,w)l must be infinite and s~t

according to Proposition 7 this can only happen on a set of

measure zero.

In the third case we may for each n EJ-.J find a sequence

t 1 < t 2 < ••• < tn < t such that !MCti+l ,w)- M(ti,w) I ~ £ , for

1 ,:: i < n , by definition of S-llm and S-lim • If we can prove stt stt

that this only happens on a set of measure zero, we have proved the

theorem. Since it will be useful in the sequel, we formulate this

statement as a separate lemma:

Lemma 1 0 : Let M : T x Q + *JR be a local A 2 -martingale. Let A

be the set of all w such that there exists an e EJR , £ > 0 , and

a finite t E T such that for all n EJ-.J there are n elements

- 10 -

1 < i < n • Then L(P)(A) = 0 .

Proof: It is enough to prove the lemma for A2-martingales, s1nce

we can then use localizing sequences of stoppingtlines to prove it

for local A2-martingales. Moreover, it is enough to prove that

for all £ > 0 and all finite t E T , the set B of all w E Q

such that for all n EJN there are elements t 1 < t 2 < ••• < tn < t

in T with !MCti+ 1 ,w) -H(ti,w)l > £ has Loeb-measure zero.

So let us assume that M is a A 2-martingale with £ , t and B

as above. Define,an internal sequence of stopping times by: T0 (w) = 0

and

For k E:N let Bk be the set of all w E Q such that there

exist t 1 < t 2 < ••• < tk < t with !M<ti+ 1 ,w)- M(ti,w) I >

<r2,{Gt},P> is the internal basis of M, then BkEGt

£ • If

and is hence

measurable. For each w E Bk and each t. l

w as an element of there is an m E*N

and l.f ~m(w) _< ti+ 1 then ~ (w) < t. L L m+ 1 - 1+2 °

T k- 1 ( W ) ::: tk < t ,

By Lemma 3 we have:

E( I: (M(Tn+1(w),w) ~M(T (w),w))2) nE*N n

in the sequence defining

such that t. < T (w) < t. , 1- m - 1+1

Consequently

= E([M](t))

(These sums are really hyperfinite s1nce •n(w) = t for some nE*N ,

and so there is no problem of convergence.)

For w E Bk , we have by definition of

Tk 1 (w) < t : l: (M(T + (w) ,w)- M(T (w) ,w) ) 2 - nE*N n 1 n

Consequently: P(Bk)o(k-1)(~) 2 ::: E([M](t))

•n+ 1 and the fact that

£ 2 ~ (k-1)(2) •

or P(B ) < 4E([M](t)) k - £2(k-1)

Since M is a A2-martingale E([H](t)) is finite and hence

~ 11 -

L(P)(Bk) = 0 P(Bk) + 0 as k + oo. Since B = n Bk , L(P)(B) = 0 .

This proves the lemma, and completes the proof of Theorem 9.

4. S -continuous :>t 2 -martingales

Definition 11: An internal function f: T + *R is called

S-continuous if for all finite s, t E T such that s ~ t , the values

f(s) and f(t) are finite and infinitesimally close. A hyper-

finite process x : T x n + *JR lS S-continuous if for all w in a

set of Loeb-measure one the function t + X(t ,w) is S-continuous.

Using the internal definition principle it is easy to show

that f : T + *R is S-continuous if and only if f(t) is finite

for all finite t E T and for all such t and all positive e: ER

there is a o ER, o > 0, such that for all sET with !s-tl < o,

we have lfCs)-f(t) l < e: •

In this section we want to prove that a local :>t 2 -martingale

is S-continuous if and only if its quadratic variation is S-con­

tinuous. This is an example of the main strategy of this paper; to

relate properties of a hyperfinite martingale to properties of its

quadratic variation. Since it is an increasing process, the quad­

ratic variation is often much easier to work with than the wildly

oscillating martingale. A first feeble attempt in carrying out

this programme was made ln Le~ma 3 and some of its consequences

gathered in Lemma 10 and Theorem 9. Using Proposition 5 we shall

now launch a more serious attack leading up to the result announced

above (Theorem 14). We shall see more consequences later when we

use Theorem 14 to prove Theorem 21.

- 12 -

Let us return to the problem in question. We shall use a

sequence {Tn} of internal stoppingt~s defined by

T ( w) = min { t E T : I M ( t, w) ! > n} , n

and then apply Proposition 4 to the martingales MT with p = 4 • n

We shall thus consider expressions of the form E([MT ](t) 2 ). n

However, this may be infinite, and in that case we should not be

able to complete our calculations. Hence we look for conditions

that will ensure us that it is finite. One of them is obviously

the following:

A process X : T x n -+ *JR lS said to have infinitesimal

increments if for all finite t E T and all w E n , t.M( t, w) R:t 0 •

We shall show that all S-continuous A2-martingales (or A2-

martingales having S-continuous quadratic variation) can be modi­

fied into A2-martingales with infinitesimal increments, and then

use Proposition 5 and the stopping times above to prove our results.

We shall need some technical preliminaries: Recall from

Anderson [ 1 ] , that a G -measurable internal function f on a hyper-

finite measure space <O,G,m> is called S-integrable if:

(i) 0 fl£1dm < oo

(ii) If A EG and m(A) R:t 0 '

then f If I dm R~ 0 • AI

(iii) If A EG and f(x) R:t0 for all xEA '

then J 1 t ! ctm R:l o . A

Clearly, (iii) is redundant if <O,G,m> is a probability space.

Anderson further defined SLP(n,G,m) to be the set of all internal

G-measurable functions such that !tiP lS S-integrable.

Lemma 12: Let <n,G,P> be a hyperfinite probability space. Let

f : n -+ *JR be a G-measurable internal function such that

0 f I f ' 2 dP < 00 • Then f is S-integrable.

- 13 -

Proof: It is enough to prove that f satisfies (i) and (ii) above.

By Holder 1 s inequality we have for A E G :

Taking A = n we have (i), and letting P(A) ~ 0 we get (ii).

This proves the lemma.

Proposition 13: Let M :Txn+ *JR be a A. 2 -martingale adapted to

the internal basis <n,{Gt},P>, and assume that the set

{wE rl : 3t E T ( t finite and 0 6M(t,w) * 0)} has Loeb-measure zero.

Then there exist an internal probability measure P on Gt and OJ

a >.. 2 -martingale adapted to

(i) M has infinitesimal increments.

(ii) For all B EGt , P(B) ~P(B) . OJ

(iii) There is a subset n' of Q of Loeb-measure one, such

that for all wEn' and all tET, M(t,w) = M(t,w).

Proof: Let a E JR+ and t a finite element of T . We shall

construct a process M such that 16M( s ,w)! ~a for all w E Q

and all s < t , and a probability measure on such that

M 1s a martingale with respect to <n, {G t}, P > . For all

we shall have f[M](s)dP:: f[:M](s)dP. Finally, replacing

s E T , ,.., M and

,.., P by M and P respectively, (ii) and (iii) hold.

Let us first show that if we can construct such M

for all a and t ' then we can prove the proposition:

the set of all n E*:N such that there exist Mn and

the following properties:

-n + *JR a hyperfinite and -n M : T X Q lS process, p

internal probability measure on Gt OJ

such that

pn

is

··n M

and p

Consider

with

an

is a

- 14 -

hyperfinite martingale adapted to

1 exists a set A EGt , P(A) <-, n oo n n

and all w (A , H(t,w) -n = M (t,w). n

have I p (B)- :pn ( B )j < 1. ' and for all n

> f [ Mn ]( t ) d :pn . Finally, for all -I -n I 1 t~M Ct,w) ~n.

There

such that for all t < n

For all B EGt , we 00

t<n '

f [M](t)dP > -wEn and all t < n ,

This is an internal set, and putting 1 a = -n

and t equal to the

least element in T larger than n ) we see that it contains all

finite n. Thus it contains an infinite y E *JN ' and taking

"" f1Y "" -y proposition. M = p = p and n' = n ...... A '

we get the y

Let a and t be given; we shall construct M and p . The

proof is long and tedious, and we divide it into several steps:

1 Definition of M : Let A be the subset of n consisting of

all w such that there exists an s < t with jt~M( s ,w) I >a . The

set A is internal and Gt-measurable. For wE A , let

the first element s 1n T such that lt~M( s, w) I > a .

If wEn '-A, let M(s,w) = M(s,w) for all sET.

If w EA, let M(s,w) = M(s,w) for s < s - w

let

let llM(s,w) = 0

and

for s > s w

s be w

By hypothesis, A has Loeb-measure zero and M thus satisfies (iii).

2 Definition of P : We must change P into P such that M

becomes a martingale with respect to cn,{Gt},P>. This is done by t.

defining a measure P 1 for all t. E T , t. < t , by induction. The l l -

measure Pt obtained by this procedure will be our measure P . t·

Let P0 = P . Assume that P 1 is defined, we shall construct

- 15 -

For each wEn, let [w]t· = n{B EGt· :wEB}. Given such an 1. l

equivalence class we define e: [ ] E *JR by: w t· 1.

.... t . f'V ,..,.

= l:{(I~M(ti,w)j-a)P 1 {w}:wE[w]ti" SO;

t· ti .... If Zi'E[wlt. and s,..,. * t. ' let p 1.+ 1 ({Zi'}) = p {w }/1 +e:

w 1. [w) t. 1. 1.

= t.} l

t· I ,..,. I t. f'V 1 ~M(ti,w) If Zi' E [ w lt. and let p 1.+ 1 ( {Zi'}) )•P 1 {w} sf'V = t. = a(1+e:[ ] w 1. ' 1. w t· 1.

This defines ti+1 t P , and hence P = P by induction.

3 P is a probability measure:

We write:

B = {Zi' E n

c = {wEn

: Zi' E [ w ] t " s,..,. * t . } • w l l

: w E [ t:l ] t . A sw = t i } 1.

16M( ti ,Zi') I a(1+e:[wlt.)

1.

t· Let us caluclate P 1 +1([wlt.)

1.

and

We have:

t· Thus P 1 +1([w]t·) =

1.

t· P 1 ([wlt.), and summing over all equi-

1. pti is a probability measure, so valence classes we see that if

ti+1 is P By induction is a probability measure.

- 16 -

i M is a martingale with respect to <rl, {Gt} ,P > : It is enough

to prove I: flM(t. ,w)Ptr;;:n = o for all wEn and all tiE T . wE [wlt. 1

l

By definition of M and P this is obvious if t. > t l-

or t. > s ' l IJ)

and hence we only have to prove the formula for t 1 <t, t.<s. l - w

In that case

P ti{w} = P{w} I a , where a = . rr • ( 1 + e: [ ] ) , for all w E [ w J t . • J<l w tj l

Let D={[W'J.r. :W'E[wlt.}. :L+l l

Then:

- t...... - t I: tlMCt. ,W')P {w} = I: tlMCt. ,W')P < [wlt > wE [w]t. 1 [W'Jt.E D 1 i+l

l l+l

- ,..., ti+l ,..., = :r l!M(t.,w)P ([w]t ) , [w] E.D 1 i+l

t. 1 l+

since it follows from .the calculation in ~ that Pt<[wl > = ti+l t·

= P l+l<[W'J ) for t > t. ti+I - l+ 1

Let now Dl = { ['Wlt. ED : s ..... * t. } l+l w l

and D2. = {[W'] t. ED : s .... = t.} l+l w l

We get

- ,..., ti+l "' I: l!M(t.,w)P ([w]t ) =

[wlt. ED 1 i+1 l+l

t· t· I: tlMCt. ,W'> P l+l < [W'lt > + I: flMCt. ,Ul') P l+l c [wl > =

["'] ED 1 1·+1 [-:vlt.€ D2 1 tl.+l w t. 1 w l+l l+l

1 I: tlM(t. ;un P c r'WJt >

[w\. ED 1 i+1 l+l

since M is a martingale. This proves that M is a martingale

with respect to <rl,{Gt},P>.

- 17 -

i An estimate: To prove that P(B) ~P(B), we shall need the

following estimate (where

n-1 < :t

i=o

t = t ) : n

n-1 :t

1 t· :r [II C1+e:[ 1 >l<-:t{(!f~M(t.,W'>!-a>·P l{w}:'WE[wlt 1\ g,...=t.}) =

i=o [] ,., .. wt a l . w l w t c.6 J<l . l . J

l

1 n-1

= - :t a i=o :t {(jf!M(t.,W')j-a)oP{w} :'WE [wlt A g,...=t.}

[wlt.cn l i w l l

_lf C!f!M(s ,w>!-a)dP<lfjflM(s ,w)!dP<3.Jrnax!M(s,w)ldP a · w -a w -a · A A A s~t

By Doobis inequality we see that E(max!M<s,w)j 2 ) is finite, s~t

and it follows by Lemma 12 that max!M(s,w>l is 8-integrable. s~t

Since P(A)~o, we get lf max!M<s,w)!dP~O, and consequently

n-1 a A s~t

:t ( .:t e:[ 1 )P{w} ~a. w€n'A l=O w t·

l

Q. P(B)~P(B) for all B€Gt : Let C be the set of all w€n'A CXI

n-1 such that .:t e:[ 1t ~~. It follows from.§_ that P(C) ~1.

l=O w i

Let D c C be an element of Gt . CXI

For each w € D , we have

t t P {w} = P{w}/ II (1+e:[ ] ) , and consequently P (D)=: P(D) • S<t W S

But on the other hand

This may be seen as follows:

< elog( 1 +e:) = 1+e: since the logarithm is concave.

- 18 -

M ( 1 ) n 1 n(n-1) 2+n(n-1)(n-2) 3+ ; ( )k 1 oreover, +~:: = +ne: + 1 •2 e: 1•2•3 e: ;•• ~ "- ne: = 1-ne:

k=o

(since and the sum converges). tAle get

n-1 Pt{D) ~ r P{w}(1-ne:) = P(D)- I: P{w} I: e:[ ] F.::$

wED wED i=o w t. 1

P(D) by 5. Hence

Pt(D) F-::$P(D) .

This proves the assertion when B c C • Let now B be an arbi-

trary element of G t ; then B = ( B n C) ll ( B'C) • By what we have (X)

just proved P(C) F-::$1 , and since P is probability measure

Since BnC c C , we have already seen that

1 f[M](r)dP?:: J[M] (r)dP: It is enough to prove

f t.M(ti,w) 2dP ~ f t.M(ti,w) 2dP for each [wlt. [w]t·

t. < t 1

and all equi-

1 1 valence classes [ w] t. . We define

1

E1 = { [ wl t c r w 1 . : s .... * t.} • -c • w 1 1+1 1

E2 = {[wlt. c[wlt.: s'il:l' = ti} 1+1 1

Then f tiM(t 1. ,w) 2dP = I: t.M(t. ,w) 2P[w]t + I: tiM(t. ,w) 2P[w]t 1 . 1 .

[wlt. [w] EE 1+1 [w] EE 1+1 t. 1 1 t. 2 1 1+ 1+1

If s < t. ' then t.M( t. 'w) = 0 w 1 1 for all wE[wlt. and the inequality

1

is obvious. If not, we have

2 2 t· r t~MCt. ,w) P[W'lt = . rr. <1+e:[ 1 > I: t~M<ti,W'> P 1+ 1[w~.

[w] EE 1 i+1 J <1 w t. [w]t EE. 1+1 ti+l 1 - J i+l 1

since t~A(t.,~) = tiM(t. ,~) by definition of M and 1 1

t· P 1 + 1 [w""] . b h · · 3 W 1 . y t e computat1on 1n . e a so have

ti+l

= 19 -

= . IT • ( 1 +e: [ ] ) ]<1 w tj

-( ,..., 2-["'] = :r t.M t. ,w) P w t [wJt. EE2 1 i+l

1+1

>

Combining our results we obtain the inequality. We have now proved

all assertions about M and P made in the beginning of the proof,

and we are through.

Notice the use of Proposition 13 in the proof of the main

result of this paper:

Theorem 14: Let M : T X n -+ *:R be a local A 2 -martingale. Then

M is S-continuous if and only if [M] 1s.

Proof: (i) Assume that [M] 1s S-continuous. Using a localizing

sequence of stopping times we may assume that M is a A2 -martingale.

Then M satisfies the hypothesis of Proposition 13, and M of that

proposition is S-continuous if and only if M is ; and [~] is

S-continuous since [M] 1s. Let Tn be the internal stopping time

Since

than

,..., M has

If r E *:JR

r. For

A (t) = m,n

Tn(w) = min{tET: CM](t,w) ~ n}

infinitesimal increments, [ f=f ] < n+1 . Tn

-let r be the smallest element of T larger

m,n E *:IN and t E T , let

where [t•n] is the integer part of ton . If A is the set of

all w such that s -+ M ( s ,w) is not S-continuous, then Tl

- 20 -

A = u u n A(k) Since -+ 00 a.e. as 1 -+ 00 ' it 1S kEIN mEJN nEIN m,n

enough for us to prove L(P)( n A(k))=O for all k,m,lE N; and to nEJN m,n

prove this it 1S enough to prove P(A(k))F:::~O for k,m,lEJN, m,y

y E*N '-N . We have:

[ky] . -< md 4 I: E < < [ M l < 1 + 1 1 y ) - [ M ] < i 1 y ) ) 2 )

i=O Tl Tl

[ky] -- --= E ( md 4 I: ( [ M ] ( i + 1 I y ) - [ M J ( i 1 Y ) ) 2 )

i:Q Tl Tl

(By Prop 5, for sui table d 4 E:R)

Since [M ] < 1+1 , we have: Tl

IJ<y] --- - 2 [ky] --rrrl4 I: ([M ]( i +1 1 y) - [11 ]( il y) ) ~ m:14 < 1 +1 ) I: < [!-1 ]( i +1 1 y) - [M ]( il y))

i=O Tl Tl i=O Tl Tl

2 ~ rrrl4 (1+1) 0

There exists a set of Loeb-measure one where

[M ] ( i+1 I y) - 01 ]( il y) < o for an infinitesimal o , when Tl Tl

i+1/y ~ [ky]. For w in this set

[ky] [ky] md 4 I: ([M l<1+11y,w)- ['M ](ily,w)f ~ m~o :E([M ](i+17y,w)-[M ](ilpw))

i:Q Tl Tl i:O Tl Tl

< mod 4 < 1+1 ) F:::l o .

Thus the expression inside the expectation

[ky] E(md 4 I: ([M ](i+1ly)- [M ](ily)) 2 )

i:Q Tl Tl

is finite everywhere and infinitesimal a.e., and hence the expecta­

tion is infinitesimal. It follows that P(A(k)) F:::t0 and as we have m,y '

already noticed, this proves one part of the theorem.

- 21 -

(ii) Assume that M is S-continuous. As above we may assume

that M is a A2 -martingale, and since M satisfies the hypothesis

of Proposition 13, M is S-continuous. It is enough to prove that

[M] is S-continuous.

We define the sets B~~~ for m,n E *JJ , kEN by

B(k) = {wEn: 3i :5. [kon](([M](i+1/n)- [M](i/n)) 2 ) > 1/m)}. m,n

As above it is enough to prove for all m,k E lJ

y E *N ':N .

For g1ven y E *:IN 'N we define a new martingale M( Y) by:

For each w E Q such that there is an i < [ky J with

suP < I 11 < s , w ) - 'f-1( ify , w ) I ) :::. 1 , l?Y:s s:si+17y

let s be the least s w

M(y)(s,w) = M(s,w)

satisfying this formula. We put

for s < s , but let t.M( Y) ( s ,w) = 0 - w

Then M(y) is obviously a martingale, and since and

for s > s w

M(y) have

the same

to prove

paths outside a set of infinitesimal measure, it is enough

PCC(k)) ~o where C(k) is the set obtained from M(y) m,y ' m,y

in the same way that B ( k) was obtained from M . m, Y

Let S = {i/y: iE*JH be a subline of T. Let N : Sxn -+ *R

be the restriction of M( Y) to S . We define a sequence {on} of

internal stopping times by

on ( w) = min { s E S [ N] ( s , w) > n}

By the definitions of

Since -+ 00 as n -+ 00 '

for each n ' where c<k) m,y,n

M(y) and on, we have [N 0 ] :5. n+2 . n (k)

we only have to prove that P( C ) ~ 0 m,y,n

lS the set obtained from M(y) in the on

same way that was obtained from M. We have

- 22 -

0 < P(C(k) ) m,y,n

l)<y] ...... ( ) ""'( ) - 2 < I: P{w: ([My ](i+1/y)- [My ](i/y)) > 1/m}

. a a 1=0 n n

(by Prop. 5)

:: mc4 (4/3)'+ [!J E((M~y)(i+1/y) -M~y)(i/y)) 4 ) (by Doob's inequality) 1=0 n n

[ky] •. [ky] - 4 = mc,,(4/3)'+ I: E((N <i+1/y) -N (i/y)) .. = E(mc,,(4/3)'+I: (N (1+1/y)-N (i/y)) ). .. . a a .. . a a

1=0 n n 1=0 n n

Now (N Ci+1/y) -N (i/y)) 2 < [N ] :: n+2, and thus the an an - an

expression inside the last expectation is always less than:

4 [kyJ - 2 4 2 mc,,(4/3) (n+2) I: (N <i+1/y)-N (i/y)) ::mc,,(4/3) (n+2) which is .. . a a ..

1=o n n ( finite. On the set of measure one where M y) 1s S-continuous

an each (N (1+1/y) -N (i/y)) 2

an an is infinitesimal, and repeating the

argument we see that the expression inside the last expectation is

infinitesimal a.e.

It follows that the expectation is infinitesimal, and so is

P(C(k) ) . Hence [MJ is S-continuous and so is [M] . m,y,n

proves the theorem.

This

We have replaced the original martingale M by M in the

proof above to be able to prove that

[ky] 2 md4 :r C[M J<i+1/y)- [M J<ilr>>

i=O Tl Tl

and

are finite everywhere. This we need to know to deduce from the fact

that these expressions are infinitesimal a.e. that their expectations

- 23 -

are infinitesimal. The difficulty is peculiar to the nonstandard

theory since in standard integration theory integrals over sets of

measure zero are always zero. In other contexts - e.g. in the

theory for (Schwartz-) distributions - this peculiarity is an

advantage.

We end this section with an example:

Example 1 5: Let x be Anderson's process from Example 1 , and let

us see how much information we may obtain about x using the theory

we have developed so far: Obviously x is a martingale and

[x ](t) = t. By Lemma 3 x is a A2-martingale, and from Proposition 7

we see that x is finite a.e.. Since [xJ obviously is S-continuous

it follows from Theorem 14 that so is x .

5. The stochastic integral

Definition 16: Let X, Y : T x n -+ *JR be two hyperfini te processes.

By the stochastic integral of X with respect to Y , we shall mean

the process fXdY defined by

vJe shall write

process fXdY

t (t,w) -+ L X(s,w)6Y(s,w)

0

t t f 0 X(s,w)dY(s,w) or f 0 XdY for the value of the

t 2 2 at time t. Obviously, [fXdY](t)) =EX (s)6Y(s) .

0

We shall mainly be concerned with stochastic integrals of the

form fXdM where M is a hyperfinite martingale with respect to an

internal basis <S'2;{Gt} ,P>, and X is a process adapted to the

same basis. It follows from the martingale property that fXdM

then is a martingale adapted to <Q,{Gt},P>.

- 24 -

The following simple identity will be useful in [12] and [13] .

For X= x (the process of Example 1) it was proved by Anderson in

[1]; the proof in the general case is similar to his:

Proposition 17: Let X be a hyperfinite process. Then

Proof: He have

k-1 k-1 k-1 k-1 i-1 = X~+2X 0 .~ ~xt.+<.r ~~.) 2 -x~-2x 0 .r ~xt.-2.r <.r ~xt.)~Xt.

l= 0 l l= 0 l l= 0 l l= 0 J = 0 J l

k-1 k-1 i-1 k-1 i-1 = < r ~xt.) 2 -2 r r ~xt.~xt·

i=o 1 i=o j=o 1 J Now we have 2 L .'- ~xt.~Xt.

i=o J=O 1 J

=

Thus

which proves the theorem.

k-1 < r ~xt.) 2 =

i=O l

k-1 k-1 r .r ~xt.~xt.

i=O ]=0 l ]

If M is a A2 -martingale, we define A2 (M) to be the set

of all those internal processes X

of M such that 0 E((ftXdM) 2 ) <oo 0

fXdM is a hyperfinite martingale

this is by Lemma 2 equivalent to

adapted to the internal basis

for all finite t E T . Since

with quadratic t

0 E(EX2 ~M2 ) < CX)

0 for all finite t E T .

If M is a local A2 -martingale, we define A(M) to be the

~et of all X adapted to the basis of M such that there exists a

localizing sequence {Tn}nEJN

n El-T •

for M with XEA 2 (M ) for each 'n

The classes A2 (M) and A(M) are perhaps too large to be

really useful, and we now construct two smaller classes SL 2 (M) and

- 25 -

SL(M) which consist - in some sense - of the square S-integrable

elements of A2 (M) and A(M):

For each t E T , let Tt = T n * [ 0, t > and let T t be the

internal power-set of Tt If M is a A2 -martingale with respect

to <S1, {G t}, P > , we define an internal measure

*-measurable space <Tt x S1, T t x G t > by: 00

"M t on the

2 v M { < s , w >} = P { w } ( tJ. M ( s , w ) ) for w E S1 , s E T t . t

If t is finite:

t 2 "H (TtxQ) = L L tJ.H(s,w) P{w} = E([H](t))

t wEQ 0

which is finite.

Definition 18: Let M : T x Q -+ *JR. be a A2 -martingale adapted to

and let X be a process adapted to the same basis. Then <S1,{Gt},P>

XESL2 (M) if and only if for all

finite t E T .

If M is a local A 2-martingale, X E SL(M) if and only if there

is a localizing sequence {Tn}

all n EJN .

for M such that XESL 2 (HT ) n

for

Since the process X 1n Definition 18 is adapted, each XtTtx S1

is measurable with respect to a *-algebra Pt much smaller than

T t X Gtoo •

X~Tt x n E

xt Tt x S1 E

The question naturally arises whether . 2

SL (Ttxa,TtxGt '"M) if and only if 00 t

SL 2 (TtxS1,Pt'"M tP~). The answer is not quite obvious t

since there are more sets of infinitesimal measure 1n Tt xGt 00

than in Pt, but it is not difficult to prove:

Lemma 19: Let <S1,G,~> be a hyperfinite measure-space, and let B

be a *-sub-algebra of G . Let f : S1 -+ *JR. be B-measurable. Then

if and only if 1 f E SL (a, B, ~) •

= 26 -

Proof: Left to the reader.

The next proposition shows us that the class SL(M) is

large enough for most purposes.

Pro:eosition 20: Let H be a local A.2-martingale and X a process

adapted to the internal basis of M. Assume that on a set of Loeb-

measure one max I X( s) I is finite for all finite tET. Then s .st

XESL(M). In particular if X is a local A. 2-martingale,

then X E SL01)

Proof: Let {~n} be a localizing sequence for M, and define

another sequence {crn} of internal stopping times· by

cr ( w ) = min { s E T : I X ( s , w ) ! ~ n} n

By the hypothesis, 0 ~ ( w) n

-+ 00 a. e ..

lS a localizing sequence for M and

Let ~I : n

jX(s,w)J <n for

Consequently X E SL 2 (M~,) ? and X E SL(M) . n

then {~~}

s <~'(w) n

If X is a local A.2-martingale, then X satisfies the hypo-

thesis of Proposition 20 by Proposition 7.

Before we turn to the next theorem let us recall some results

about S-integration. Let <~ ,G , ~ > be a hyperfini te measure space

with ~ ( ~) finite. Anderson ( 1 ] showed that an internal G -mea sur-

able function f : ~ -+ *R is S-integrable if and only if there

exists a sequence {fn} of G-measurable finite functions such that

1 im 0 ( f I f- f n I d ~ ) = 0 •

If f lS *-non-negative it follows from the proof that we may

take f = f 1\ n n Anderson also proved that if f is G-measurable

and *-non-negative, then °ffd~ ~f 0 fdL(~) with equality if and only

if f is S-integrable.

'

- 27 -

We now use Theorem 14 to prove:

Theorem 21: Let M be an S-continuous local A2-martingale, and

let X E SL(M) Then fXdM is S-continuous.

Proof: It is clearly enough to prove the theorem when M is a

A2-martingale and XESI/(M).

For each n EJ.J define an adapted process X : T X Q -+ *:R by: n

X (t,w) = X(tqw) if jXCt,w)! <In; X (t,w) = Tn if X(t,w) > Tn; n ' - n

and X ( t , w ) = - fn if X ( t , w ) < -In . n

Since M is S-continuous, it follows from Theorem 13 that

[M] lS S-continuous. Now [JXndMJ(t)- [JXndM] (s) =

~ X 2 t:~M 2 < n f l:IM 2 = n ( [ M ]( t)- [ M] ( s) s n s

for s,tET, s<t and hence

we see that [JXndM] lS S-continuous for each nEJ.J

By the last of the results of Anderson mentioned above:

0 s s 0 < E(sup ([f XdM]-[J X dM])) <

0 s s E(maxC[J XdM]-[J X dM]))

S5t o o n - s ~t o o n

By definition of X E SL 2 ( M) and the first of the results of

Anderson above, 0 JCx 2-X2 )dvM -+ 0 as n-+ oo. n t

A well-known result

of measure-theory (see e.g. Bauer [3] , Satz 15.5), tells us we may

o Js Js find a subsequence {sup ([ XdM]-[ Xn dM])}kElli which converges s~t o o k

to zero a.e ..

Using the E-o-formulation of S-continuity (recall the comment

following Definition 11) one may prove exactly as in the real-valued

case that the S-uniform limit of a sequence of S~continuous func-

tions is S-continuous. It thus follows that [fXdM] is S-continuous.

- 28 -

Applying Theorem 14 again we see that fXdM lS also S-continuous,

and the theorem is proved.

If v.Je replace X E SL(M) by X E A.(M) ln Theorem 21, the

resulting statement will be false as is easily seen.

Anderson [ 1 ] proved the theorem above for M = x (the process

of Example 1) when X was in a certain subclass of SL 2 (x) (the

class of H2-liftings"). Keisler [1 O] used Bernstein's inequality

to prove the theorem when M = x and X is a finite adapted process.

6. The transformationforrnula and Ito's formula

In this section we prove a nonstandard version of what may

be called the fundamental theorem of stochastic analysis.

If r E T , we write r and + r for respectively the prede-

cessor and the successor of r ln T

Theorem 22: (Nonstandard version of ·the Transformationformula).

Let cp : *JR -.*JR be an internal function and assume that l.D,tp' ,(!)"

all exist and are S-continuous and finite for finite arguments.

Let M: T x Q -+ *JR be a local A. 2 -martingale. Then for all finite tET:

o lP ( Mt ) - o tp ( M o ) - o f; tp v ( M ) dM + ~ o£t<P" ( M ) d [ M]

t +:Z::: 0 (<.;J(M +)-<p(M )-tp'(M )(M +-M )+l<.p"(M )(M -M ) 2 ) a.e.,

0 s s s s s 2 s s+ s

where the last sum is absolutely convergent.

Proof: We begin by enumerating the points s E T such that

0 -6M(s ,w) o~:O. By Lemma 10 this is possible for almost all w E Q ,

since for each finite tET and each positive e: E JR there are only

a finite number of s E T , s < t , such that 6M( s-, w) > e: • To make

- 29 -

the definition precise, it is convenient to define a bijection

f: N+ + N+ x~+ and let (f(n)) 1 denote the first component of

f(n) . Define a sequence {Tn}nElli of internal stopping times by

T n ( w) = min { s € T: I L\M( s-, w) I > ( f (~)) 1 " s * T k ( w) for k<n } .

Let Q 1 be the set of Loeb-measure one where M(t) is finite for

each finite t E T and where the sequence { T n ( w)} enumerates all

finite sET such that 0 L\M(s-,w) * 0.

internal extension of {Tn}nEJN such that Tk * T when k * m . , m

For each finite t E T and each v E *:N '-JN , let

A = {sET: s+ ( {Tn(w):nsv}}. t,v

For all w E Q 1 we get from Taylor's formula:

t lP ( Mt ) - <P ( M 0 ) = I: ( <,o ( M + ) - lP ( M ) )

0 s s

= (q>(M +) -t,o(M )) s s

= :E (r.p(M t) -<.p(M- t)) + r {<P' (M )(M +-M ) T " Tn" SEA s s s nsv n t,v

+ l!1.., 11 (M )(M +-H ) 2 + l(,n''(r: ) •n"(H ))(M M ) 2 } 2 ., s s s 2 't' .., s - "¥ s ~- 8 +- -s '

where I; lies between H + and Ms s s

By the S-continuity of <.p 11 the sum over the very last term

is infinitesimal for w E Q 1 and thus

tp(Nt)- <p(Mo) ~ !: (<.p(M t) -q>(M- t)) n< Tn" Tn" _v

+ I: {<P' (M )(H +-M ) + ~W"(M )(M +-H ) 2 } for wE Q 1 • sEA s s s s 8 s

t,v

Adding and subtracting

- 30 -

we get: t t

<P ( Mt ) - <P ( M 0 ) F:$ f <P I ( M ) dM + ~ f <P II ( M ) d [ M] 0 0

+ L {tp(MT At) -!p(MT-At) -q>Y(MT-At)(MT At-MT-At) n~v n n n n n

on Q' •

Since this J.s true for all v E *JJ '-JJ , the series

00 0 2 :r {tp(M t)- q>(M _ t) - ~,p' (M _ t)(M t-M _ t)- ~"(M _ t)(M t-M - t) } 0 T A T A T A T A T A T A '[ A T A n n n n n n n n

The convergence is absolute since the order of the terms in the

sequence {•n} does not matter. But the sum above is equal on Q 1

to the sum in the theorem, and this completes the proof.

In the S-continuous case Theorem 22 becomes

Corollary 23: (Nonstandard version of Ito's formula). Let

<P : *R-. "'JR be an internal function and assume that lP,!.P' ,<P" all

exist, and are S-continuous and finite for finite arguments. Let

M : T x n -+ *JR. be an S-continuous local A. 2 -martingale. Then for

all finite t E T

t t <P ( Mt ) - lP ( M 0 ) ~ f lP 1 ( M) dM + ! f <P" ( M) d [ M J a . e .

0 0

Applying Corollary 23 to a suitable integral of Anderson's

process x , one may obtain a nonstandard version of Ito's lemma,

as Anderson did in [ 1 ] .

For standard versions of the transformationformula and Ito's

formula, see Metivier [18] , page 264 and page 49 respectively. We

shall deduce these results from the above versions in [13 ].

- 31 -

7. Doob-Meyer decomposition

A process X : T x r2 -+ *JR t

is said to be of finite variation

if I X 0 I + E I ~X( s) I is finite 0 .

a.e. for all finite t ET. Such

a process is easily decomposed into two monotone processes x+

and X , by defining and t E(~X(s,w) AO). 0

Integration with respect to monotone processes

is easy to deal with, and since we in this paper have developed

the theory for integration with respect to martingales, we should

now be able to treat integration with respect to processes of the

form X + M where X is of finite variation and M is a local

:\2-martingale.

Definition 24: Let Y : T x S1 -+ *JR be a hyperfini te process adapted

to the internal bas is <S1, {G t} , P > . Y is said to have a Doob-

Meyer decomposition if there exist a process X of finite variation

and a local :\ 2-martingale M both adapted to <r2,{Gt},P> such

that Y = X+ M .

Proposition 25: Let Y : T x Q -+ "'JR be a hyperfinite process adapted

to the internal basis <r2,{Gt},P> such that Y(O) is finite a.e ..

Assume that there exists a sequence

times such that 0 , -+ oo a.e., and

n

all n E:N and finite t E T . Assume

{•n} of internal stopping tAT

E( E n(~Y(s)) 2 ) is finite for 0 t

further that E!E(~Y(s)!G )! 0' s

is finite for finite t E T, L(P)- a.e.. Then Y has a Doob-Meyer

decomposition.

Before we prove the proposition, we make two remarks on the

conditions which Y must satisfy:

- 32 -

Remark 26: The conditional expectation E(llY(s) !G ) . s

hyperfinite case just an average: As in Section 4 let

n{BEG :wEB} s

Then we have:

E(llY(s)!G Xw) = ( :r llY(s,w)P{w})/P([w] ) s ;e[w] s

s

lS ln the

This definition makes some properties of the conditional expectation

very easy to prove, e.g. will the inequality

which we shall need in the proof of the proposition, reduce to the

algebraic inequality

[(a1p + o•o +a p )/(p + •oo +p )]7.(p + ooo +p ) < a 2p + ooo +a2p 1 n n ·1 n 1 n - 1 1 n n

t Remark 2 7: The condition ~I EC~Y( s) !G s)! <"' a. e. in the proposition

is not much stronger than necessary~ and we show that it is satisfied

when Y has a Doob-Meyer decomposition Y = X + M where X has

reasonable integration properties: Assume E( x;) <"' and E( -X~) < "'

for all finite t ET. Then E(llY(s)!Gs) = E(llX(s)!Gs) +E(llM(s)jGs)

= E ( ll X ( s ) I G s ) and E ( r l E ( ll Y ( s ) I G s ) I ) = E ( r I E ( ll X ( s ) I G s ) I ) ~ E(rEC I llX(s)! ICGs)) = E(r! t~X(s) I) = E(X;) + E(-X~) < "' • Thus t rjE(llY(s)!G )j < "'a.e .. This condition seems to correspond in the 0 s

standard case to the condition that Y is a quasimartingale, i.e.

that the Doleans-measure of Y is of finite variation (see Metivier

[18]). tAT

The condition that E( r nllY(s) 2 ) is finite is just to ensure ·o

that the martingale we construct is a local A2-martingale. Without

this condition we can only prove that M is a martingale.

- 33 -

Proof of Proposition 25: Define M by M(O) = 0 and

t1M(t) = t1Y(t) -E(t1Y(t)jGt) M is obviously a martingale. Define t

X(t) = Y(O) + L:E(t1Y(s) !G ); by hypothesis X is of finite variation. 0 s

Obviously Y = X + M •

It remalns to prove that M is a local A 2 -martingale, and to tAT

do this it is enough to prove that E( [MT ](t)) = E( L: nt1M(s) 2 ) is n 0

finite for all n EJJ and all finite tET.

The first term is finite by assumption and the third is finite since

we saw in Remark 26 that E(E(t1Y(s)!G ) 2 ) < E(t1Y(s) 2 ). s

inequality on the second term we get

Using Holder's

tAT tAT 1 tA~ 1

E( L: 0 lt1Y(s)E(L~Y(s)!G >!) < (E( L: 0 6Y(s) 2 )}2 (E( L:.llE(t1Y(s)!G )2 ))"=2 0 s - 0 0 s

tAT < E( L: n~Y(s) 2 ) , which proves that the second term is finite. - 0

This proves the proposition.

As a consequence we get that all 1'reasonable" hyperfini te

sub-martingales have Doob-Meyer decompositions:

Corollary 2 8: Let Y : T x Q + *JR. be a hyperfini te sub-martingale

such that there exists an increasing sequence {Tn} of internal

stopping times such that:

(i)

(ii)

0 , (w) + oo for almost all w • n tAT

E( L: n(t1Y(s)) 2 ) is finite for all n EN and all finite t€ T. 0

(iii) E(Y(tATn)-Y(O))<oo forall nElJ, andallfinite tET.

(iv) Y(O) is finite a.e ..

Then Y has a Doob-Meyer decomposition.

Proof: Since Y tAT

r nE ( ll Y ( s ) I G ) o

0 s

is a

Now

tAT E niECliY(s)IG )I = O tAT S

E ( Y ( t A T ) - Y ( 0 ) ) = E ( E nE ( ll Y ( s ) I G ) ) n 0 s

tAT

*-sub-martingale

and hence condition (iii) implies that E n!ECliY(s) IG ) I is finite 0 s

aoe. for all nE:N and all finite tET. t

But 0 T

n

it follows that for all finite tET, r!ECt~Y(s)IG )i 0 s

The corollary now follows by Proposition 25o

-+ oo a o e . , and

is finite a.e ..

- 35 -

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