Chapter 7: Stochastic Integration

347

CHAPTER 7:

STOCHASTIC INTEGRATION

In this chapter we study pathwise integration with respect to a process that is a sum of a martingale and a process of bounded variation.

The infinitesimal analysis of the latter is

similar to section (2.3) and analogous to the classical analysis of Lebesgue-Stieltjes path-integrals.

The new feature of this

approach is that infinitesimal Stieltjes sums also work in the general case.

(7.1) Pathrise Stieltjes Sums In section (2.3) we showed how to represent every Borel measure on

[O.l]

by choosing an internal measure on

T.

In

section ( 4 . 1 ) we saw how an internal measure can arise first and how to make a standard Borel measure from it. the classical Stieltjes measure

dF

equals

We also saw that dF

0

st-',

where

u

F = S-lim F(s)

and

dF

is the hyperfinite projection measure.

s lr

We saw a hint of some problems with jumps of (where

F

was increasing and finite).

F

in Chapter 4

In section (5.3) we

resolved similar problems for more general processes by taking At-decent path samples where time increment. chapter.

At

was a coarser infinitesimal

We will use the same basic approach in this

In this section the first step is to show how to

sample a process simultaneously with its pathwise variation.

We

begin by fixing some basic notation that we shall use for the rest of the chapter.

Chapter 7: Stochastic Integration

348

(7.1.1) NOTATION:

$2.

T h e infinitesimal time a x i s ,

U.

P

on

and the u n i f o r m probability

the sample space,

R

are the same as

i n Chapters 5 and 6 except that now w e let infinitesimal element o f

U.

(For example, if

smallest positive element of and

U.

H.

w e might haue

-.

a larger increment.)

H6 =

{t E

1

At = n

6t

For any

denote any

n!

i s the

6t =

2 [i]

6t. A t

in

let

U

: t =

k6t. k E *IN}

U (1)

and U A = {t E U : t = kAt, k E *IN}

If

g : U + *lRd

is internal

(d

denote the formard differences o f and

At.

U (1).

finite) let

g

corresponding to

6t

Also let

16t 16gl . t

6Var g(t) =

for

t E

= )[l6g(s)I

: 6t

<

s

<

t , s E US]

AVar g(t) = )[lAg(u)I

: At

<

u

<

t. u E HA]

and

denote the uariations to time

t.

where

of

1-1

g

in steps o f

denotes the

6t

or

d-dimensional

At

up

7.1

Section

eucLidean norm f : H

349

Pathwise Stielties Intecrrals

* d IR

on

.

FCnaLLy.

* * L ~ ~ ( R ~ . I Rt ~s )a]n

if

f : U + *IR

[or

internat function. Let

and

1 f(u)Ag(u) t

ltfAg = S

,

for

+

s.t E PA

u=s step A t

[ W h e n the uaLues o f

f(u)

are

linear maps.

means the map evaluated at the uector

Our

convention

variations at

after

the

sums

6g(u).]

defining

the

internal

is to make i t compatible with our

At

Bt-decent path samples especially in the case of

processes

whose

liftings are only nonanticipating

6t. Our (artificial

a right-most

this

instant

=

max[U6\{1}] In

start

or

6t

definition of progressive

to

f(u)bg(u)

case

6t = [ l - ~ ]

T

< we

1.

D-space convenience-) convention of having r = 1

causes us an extra headache when

(We could ignore this problem on

take

6g(T) = [g(l)-g(~)]

and

[O.m).) interpret

i f necessary and also let

with a similar convention for

uA*

The last convention will allow us to place a final jump at

ChaDter 7: Stochastic Intearation

350

r = 1

on our internal paths and account for the corresponding

X(l)

We simply l i f t

measure.

Suppose

that

g : T + *IR

we

whose

begin

against

of

Z( )6g

the

st g = 0

so

that

is O.K.

variation

works

too

6t-variation

sampling

along

)dh

when

then

isn't,

so

h = st g.

On

Ag = 0

and

the

then

the standard part f(kAt) = 0

then

is zero.

coarser

The standard

f(k6t) = (-1) k -1.

A t = 26t.

so the standard part

Suppose

the variation

s(

is infinite and

if

We

sampling always works; perhaps

If

well.

Zl6fI

'Borelable'. but

is too simple.

A t = 26t.

Coarser

function

is limited.

Bt16g[ = 2t.

but

does not properly represent let

internal

represents the integral of

is zero, while

function

the other hand, i f we

even

but

an

B116g(s)l.

Zf(s)6g(s)

d(st(g)).

g(k6t) = (-l)k6t. part

with

6t-variation.

would like to say that st(f)

separately.

and

it

the

is not

Af = 0

The following results show how

infinitesimal

time

axes

works

for

Stieltjes integration.

(7.1.2) PROPOSITION:

If and

if

almost

var

X

var

<

R + *Rd

tn

T6

path has

has a

then

such that the

has a of

6t-decent

projection ftnite

a.s..

03,

projectton

Z.

x

surely

(X,AVar X) path

H

its decent

g(m.0)

A t 2 6t

:

At-decent AVar X

i[ : [O.l] x

classtcal

there

path sample

is

an

n

+

IR d

uariatton, tnfinttestmal

(d+l)-dtaenstonal

process

path sample and the decent is

indistinguishable

from

351

7.1 Pathwise Stielties Intearals

Section

Recall that the classical variation of a path

0

r

to

is defined to be the

sup

of all sums

over the set of all finite partitions of

[O.r].

Finiteness of

this sup is equivalent to saying that each component. the vector

2

from

%(a,")

2,.

of

is the difference of two increasing functions.

We c a n supplement (7.1.2) with the hypothesis in the next result.

(7.1.3) PROPOSITION:

X

If

U

:

R

*Rd

W a r X(l)

6t-vartatton,

T.

x

6t

in

Us

such that

and

the projectton

E 0

aLmost

sureLy

of

tndtsttngutshable from

that

(2.

Limtted

Q.s., f o r some LnftnttestmaL

then there ts a n LnftnttestmaL

(X. AVar X)

has

A t 2 6t

in

has a

At-decent path sampLe

sample

of

var

(X. AVar X)

is

2).

PROOF : First we shall prove that the hypothesis of (7.1.3) implies

X

that

has a

At-sample and

x"

has bounded variation.

Then

we shall prove (7.1.2).

A C R

Suppose o E A.

6Var Xo(l)

measures o n

1;

by

is measurable, I s finite.

For each

P[A] = 1. o E R

and whenever define internal

ChaDter 7: Stochastic Integration

352

= 6X(t.o)

u (t) w

+

where

a

+

= (6X,(t,w)

u;(t)

= (6X;(t.w)

= max(a.0)

S

whenever

+

u,(t)

+ ....,6Xd(t,w)) .*...6X,(t,o))

a- = -[min(a.O)].

and

:U

is an internal subset of

We

know

u = +.

and

that

-

or

blank, then

Therefore

w E

whenever

A,

the

P"u =

formulas

d-tuples of Bore1 measures on

define

r

For

of Chapter 2 can be used

the machinery

€

(0.1)

and

finitely decreasing to

pz[O.r]

any

u 0

( u = +,-)

st-1

0

[O.l].

countable

to see that

+ -

pw = j ~ ~ - p ~ .

Let

sequence

r.

= S-lim u~[T6[0.tm]]. m*

(I

= +,-,blank.

and for any sequence strictly finitely increasing to

r,

u = +.-.blank.

= S-lim u~[Ua[0.tm]].

p:[O,r)

strictly

tm

m*

This shows us that

S-lim X(t.o) = pw[0,r],

that the

S-limit

t lr

as of

t

increases to w,[O.r]

increasing S-limits on

is

r the

functions.

A

pw[O.r)

equals

difference

of

Existence of

implies that

X

and that each component

has a

two

right

continuous

increasing and decreasing At-sample whose

7.1

Section

2,

projection, process,

353

Pathwise Stielties Integrals

is

indistinguishable

(5.3.25).

Lemma

This

shows

from

that

a

decent

path

the hypothesis

of

(7.1.3) implies that of (7.1.2). but our sampling convention at At y[O]

means

that

# 0.

example, let at

-

may not equal

Notice that close jumps of

so that we may need

cancel

-1

%(O) = st X(At)

6t.

u+

to choose

w.

-

and

u

At

even

larger.

if

can also For

-21 + 6t and u - be unit mass + - = 0. The proof of y = y -y

be unit mass at

for each

st X(6t)

+ u

Then

(7.1.2) given next completes this part of the argument.

PROOF OF (7.1.2): Suppose

At

>

6t

%(*,w) = stk X(*.w) q.r E [O.l]

= r

and

is infinitesimal

and B

>>

0.

var z(1.0) there exist

and

<

w

OD.

q = ro

i s such that

Then

for

< rl <

* * *

every

'

rm

such that

There are also times X(tj.w)

Z

%(rj.w)

and

s.tj.t E TA X(t,o)

Hence for each infinitesimal

var

-

Z

At

such that

P(r.w),

>

6t,

1 1 ~ x 1a.s. t

var %(q)

so

S

X(s.0)

z %(q,u),

Chapter 7: Stochastic Integration

354

Next we find one infinitesimal time sample satisfying the

V(t)

opposite inequality. Let know

S-lim(X.V) = ( 2 , var 2)

number 0

<

j .( m .

For this

A 1) Z

X(jAt

whenever

At,

-

g(i)

Thus the internal set of

and

T6

in

At Z m

s.t

g.

We

A 1)

V(jAt

such that for Z'

var

%(i) a.s.

.:1

€

At's

2 6t

in

T6

such that

):11

>

t

IAXII

P[max(IV(t)-V(s)-l

var

a.s.. so for every finite natural

there exists

m,

6t-lifting of

be a

:

s.t

E

At]

<

At

S

contains an infinitesimal.

(X(t)sZA:lAXl)

a

Such an infinitesimal

At-lifting of

(2,

var

At

makes

g).

(7.1.4) DEFINITIONS:

If

U

:

T x R

+

(U, 6Var U)

that

*Rd

has a

is an internal process such

6t-decent path sample with

projection indistinguishable from say

U

has

S-bounded

:

[O.l] x R + Rd

(c,

var

6t-variation or

c),

then we 6t-bounded

variation. If

W

variation and

U

has

a.s. has bounded classical.

S-bounded

6t-variation with the

Section

7.1

projection that

U

355

Pathwise Stielties Integrals

fi

is a

When

indistinguishabLe

W,

from

then we scy

6t-bounded variation Lifting o f

U

has

S-bounded

6t-variation. T6

internal. pathwise measures o n

W. we

define

by the weight functions

T6 x R .

as weLL as a measure o n

6u(t.w) = 6pw(t)6P(o).

T

Extend these measures to either all of 6pw(t) = 0

by taking

or all of

T

x R

t Q T6.

if

The measures we have just introduced play a role in showing the connection between internal summation and classical pathwise integration.

The hyperfinite measures

variation measures of the paths of so

that both

f(w) = p,[lT]

pw

<

1

u

and

S-integrable

with

<

fi, 1.

K

0

0

while

% C T x R.

respect

to

denotes the section,

=

{t

E T

:

are the total

p,

is normalized

This makes

weaker conditions would suffice for this).

If

st-l

(t,w) 6 % } .

P

the (of

function course,

Chapter 7: Stochastic Integration

356

(7.1.5) THE ITERATED INTEGRATION LEMMA FOR PATH MEASURES: 6 p : Y x R + *[O.l]

Let For

each

the weight

o

T.

measure o n

be an internal function.

function

Suppose that the function

P.

is S-integrable w i t h respect to the

weight

defines a

6vw(t)

Let

f(w) = vw[U]

be given by

u

6 ~ ( t , o ) = 6po(t)6P(w).

function

The

hyperfinite extension measures satisfy: (a)

(b)

(c)

If

Y

is

Loeb(R)-measurable

If

Loeb(T x Q ) .

E

lr

is

then the function and

a-measurable. then for a.a.

is

p,-measurable.

If

X

for almost all

is

pw(Ww)

[--,-I

: 'U x R

o.

pw(Qo)

Xu

is is

o , lro

P-measurable and

u-integrable. then p,-integrable

.

and

.

E[ Jxo ( t dw, ( t ) 1 = JX ( t o )du ( t o ) . PROOF : (a)

If

91

<

and

PWC91,1

u[91]

= E[~,(91~)].

Monotone

E[lim

V"Cr1. The

Class

p,(91:)]

hypothesis that

Yo

is internal, then

Lemma by

p,[T]

the

is internal for each

w

ECV"(*,)l.

so

=

Moreover.

uC*l

of

follows

rest

(3.3.4). Dominated

(a)

because Convergence

is P-S-integrable.

easily

from

the

l i m ECvw(9:)1

=

Theorem 'and

the

Section

(b)

If

internal

'21

W

is

u-measurable, then by (1.2.13) there is an

such that

v '213 = 0.

u[#

N.

contained in a Loeb null set and since

is

a.s.

Since

= 0

p,[N,]

is

'21

a.s.

has measure zero a.s.

Y,

Ww

we see that

0,

= p,[SCw]

p,[W,]

v

v

W

Therefore

By part (a)

W,

is complete,

p,

Using (1.2.13) for these a.s. and

357

Pathwise Stielties InteFrals

7.1

P

is

p -measurable 0

is complete,

p , [ W , ]

P-measurable i f we take any value for the null set of

0's

where i t may fail to be defined (for example, we may take the outer

measure

Fw[Ww]).

= ~ [ " u ] = E[p,(91,)]

u[W]

Finally.

= EC~,(W,)I.

(c)

If

X 1 0 is

sequence

of

simple

convergence

we

functions

know

JXkdu

Sk

JXkdu = E [ Xwdpw].

S"

for

positive

X = X+-X-

with

by

X.

By

By

part

(b)

monotone

SXidpw] = E[[Xdp,].

integrable

be a monotone

Xk

1 JXdu.

Again

= E[lim

[ X,dpw] lim E

{Xk}

u-integrable. let

Finally, we

and apply the positive part to

X+

and

we

know

convergence

Thus part

functions.

monotone

(c) holds may

X-

write

in order

to finish part (c).

(7.1.6) THE STIELTJES DIFFERENTIAL LIFTING LEWMA:

Let

W

:

bounded vartatton. ltfttng

(a)

U.

Rd

[O.l] x R

If

U

Then

W

a.s. have decent paths o f has a

&it-bounded vartatton

t s such a ltfttng, then for a.a.

the Borel measures

I

w

= 6Uw o st-'

equal the

Lebesgue-Stieltjes measures generated by

(b)

the Ic

W

total 0

( c ) n,(O)

st-1: = 0.

uartatton

measure,

ldW,l

a,

dWw : equals

ChaDter 7: Stochastic Integration

358

PROOF :

U

Let

be a

to obtain a Let

A

At-decent path lifting of

>

6t

so

At

that

U

be the null set where

We know that i f

o Q A.

Apply (7.1.2)

S-bounded 6t-variation. U] # [W, var W].

stk[U.6Var then

= S-lim U(t)-U(6t) t lr

ru[O.r]

has

W.

= Wo(r)-Wo(0)

= dWw[O.r]

and Ir I[o.r] o

so

= S-lim 6Var U(t) t lr

= var W(r)

(a) and (b) hold. = W(0)

lim W(r) r 10

Since

uo(0) = 0.

a.s..

This proves the

1emma. Next we deal with the measures from (7.1.4) that we are most interested in for stochastic integration.

(7.1.7) DEFINITION"

H

Let

G : T x R

[O.l] x R

:

+

*IR

-4

IR

be a function.

such that f o r a.a.

o.

An internal

the hyperfinite

measure :

~"{t

i s

called a

In

in

properties on

to

compute

(7.3). G

we

= 0

# H(st(t).w)}

6U-path lifting o f

order

summation

st G(t.w)

H.

martingale will

need

and hence also on

H.

integrals to

require

by

internal additional

Section

(7.1.8) THE

H

U :

:

H

[o.il

x R

R

x

H

-,

H

measurable, then if

-

bU-PATH LIFTING LEMMA:

Let

rf

359

Pathwise Stielties Integrals

7.1

*Rd IR

S-bounded

is

has a

Gt-variation.

[Borel[O.l]

GU-path Lifting

b.

i s bounded b y

bounded b y

have

we may

x

G. G

choose

Meas(P)]Moreover, so

it

is

b.

PROOF :

K

Let

be

indistinguishable K(st(t).o)

bounded

Let

H

from

(see

(5.4.10)).

G be a

u-lifting of

u-lifting, see (1.3.9)).

function

By

(Loeb x Loeb)-measurable

is

u-measurable.

(Bore1 x Loeb)-measurable

a

(5.4.9).

and

K(st(t).w)

hence

(resp. a

By the Iterated Integration

Lemma (7.1.5).

Except for a null set K

0

A C R.

is a simple multiple of

for a.a.

po

is limited so that

on the Loeb sets of

H.

Hence

w.

~

Finally,

GVar U(1.w)

~

K(st(t),w)

lemma is proved.

:{

stt G(t.o)

= H(st(t),o)

# K(st(t),o)l

for all

= 0.

t.

8.5.

w,

so

our

ChaDter 7: Stochastic Intepration

360

( 7 . 1 . 9 ) THEOREW:

Let measurable

H

x

: [O.l]

R

+

and bounded

lR

by

be b.

x Meas(P)]-

[Borel[O.l]

W

Let

x R +

[O,l]

:

IR

d

be a process wtth a.a. decent paths o f bounded uartatton. U : T x Q

Let

W

of

H

and let

-+

*Rd II

G :

also bounded b y

be a

6t-bounded uariatton lifting

x R -+

*R

be a

6U-path ltfttng o f S(t.o) =

b. T h e n the tnternal process

t

G(s.o)6U(s.w)

ts a

6t-bounded

uartatton

Lifting o f

16, the pathlvtse classical tntegral

I(r.o)

=

s:

€I-dW.

PROOF : First we show that b

so

<<

S G

denote a bound for

S

and

6t-decent path sample.

H.

only has finite jumps where

right-S-continuous where By

has a

(7.1.6).

for

measure generated by

6Var U(t)

8.8.

dWw.

o.

6Uo

If

t.t

6Var U(t)

+ At

€

Let

Us.

does and

S

is

is. 0

st-'

= u

0

is the Bore1

By change of variables,

We apply a similar argument to

ZG-6U. Thfs proves the theorem.

7.1

Section

361

Pathwise Stieltjes Integrals

(7.1.10) DEFINITION: We

caLL

the

G

function

the

of

next

theorem

a

H.

6U-summable path Lifting o f the process

(7.1.11) THE STIELTJES SUHHABLE LIFTING THEOREH:

U

Let

H

is

be a

[Borel[O,l]

H

K

0

has

a

-

-S-integrable for a.a.

w,

a.s.

G

6U-Lifting

If

and i f

x Weas(P)]-measurable

~~IHol=ldW,l < then

W.

6t-bounded variation Lifting o f

such

Go

that

is

In this case

o.

t-6t S(t,w) =

1 G(s.o)6U(s.o)

S-6 t is a

6t-bounded variation Lifting o f

I(r.o) =

J)(q.u)dW(q.o).

PROOF : Let

G'

be any

6U-lifting of

H. We will show that all G'

sufficiently small infinite truncations of S-integrable.

are pathwise

Of course. infinite truncations of a lifting are

also liftings. Define m .

Hm =

{H . -m

for finite

m

,

H > m

IHI < H

<

m m

and

Gm =

{ G' -m

-m

in the case of

H

and all

m

. . .

G > m IGl G

for

<

<

m.

-m

G'.

For a.a.

Chapter 7: Stochastic Intevration

362

w,

{Hm}

for every

L 1 (var W)-norm.

is a Cauchy sequence in the r

in ,'Q

there is a finite

m(r)

>

1

so

that

such that if

k 2 m(a).

Thus, the internal set

contains an infinite

n = n(e).

By saturation the countable

intersection n*m[m(r) .n(r)l contains an infinite n. We claim n that G = G is our summable lifting for H. This follows from the definition of standard

n

because

P[ZlG-Gkl16Ul

> €1 <

r

f o r all

r.

By the bounded lifting theorem (7.1.9) above, for each N

finite

We

m

we know

also know

stkSm

-B

stkZG6U

S,(t.o)

Im

JHdW

= Im(r,w)

where

in probability in

in probability, hence

By the bounded case ( 7 . 1 . 8 ) . that the decent path projection of

D[O,l]

stk I G 6 U = JHdW

for each finite

m

and a.s.

we know

7.1

Section

363

Pathwise Stieltles Inteprals

are indistinguishable.

IGn-Gkl 1

Since

I IGnI-IGkl 1 ,

the same

convergence estimates show that the decent path projection of

are indistinguishable. lifting of

Hence

ItG6U

is a

6t-bounded variation

S'HdW.

(7.1.12) EXAWPLE: Consider

the process

J(t.o)

of

(5.3.8). (4.3.3). and

N

(0.3.6) whose decent path projection process.

one on a

pat)

is a classical Poisson

We wish to calculate

as an example.

6J(t.w)

J

Since

J

is finite and increasing by jumps of

6t-sample, i t has is a function of

and

since

only

6J-liftings. we see that

S-bounded

o t+6t the

alone

times

J(t+6t.o)

6t-variation.

(=1

Since

with probability jumps

count

when

it

ts a

6J-Ltfttng o f

for

N

J(r.o). too.)

(This depends on our right continuous path convention The important fact that we are trying to illustrate is

that the lifting the "coin, "

J(t+Bt,o)

must anttctpate the next

o t+6t. Now we compute

toss

364

ChaDter 7: Stochastic Integration

t-6t

1J(s+6t,w)6J(s,o)

+ 2.1 +

= 1.1

+

0 . 0

J(~~.u)-l

s=6t

where

is the time of the last jump of

T~

time

J(*.o)

at or before

t.

Notice obtained

that

if

we

want

lift

to

from projecting Anderson's

the Brownian

infinitesimal

any infinitesimal time advance or delay can be

B

6J-lifting because the paths of

B(t,w)

are

random walk,

tolerated in a

S-continuous.

6J-lifting of

is a nonanticipating

%

motion

%.

Hence,

Path liftings

need to be done more carefully when the differential process is a martingale

with paths of

infinite variation as we shall see

below. Here

is

structure

to

a

result

that

* finite

our

adds

the

stochastic

representation

of

evolution

Stieltjes

path

integrals.

(7.1.13) THE NONANTICIPATING STIELTJES LIFTING THEOREM: Let

W

: [0,1] x

R

4

IRd

be a progressiue

with a.a. decent paths of bounded variation. a.s.

Let

H

:

[O.l]

x R +R

process

var W(l)

<

03

be a preutsible process

mt th

There is an tnftnttesiaal tit-bounded variation Lifting after

6t

and

H

has a

>

6t

U

0

such that

W

has a

mhich i s nonanttcipating

0-predictable

6U-suamable path

Section

365

7.1 Pathwise Stielties InteErals

lifting

G.

ItG*61J

These Ltftings make

6t-bounded uariation Lifting of

Q

which is nonanticipating after

JrH*dW

6t.

PROOF : Apply

the

Nonanticipating

W

(6.3.8) to

obtaining a n

nonanticipating a f t e r some infinitesimal

6t-variation.

(7.1.4)

H

U

process

which

U

U

to find a n

has

S-bounded

G.

U

with

to

obtain

a

similarly

u

of

bounded

In the integrable case, apply the truncation

the proof

which a r e called

U

W.

is bounded we apply (6.6.8) with the measure

0-predictable

is

At-decent path sample for

that

so

Theorem

T h e coarser sample still has decent paths, so

associated

argument of

Lifting

Next. apply (7.1.2) to

At.

is our lifting of

If

Path

internal

and has a

At

6t E U A

infinitesimal

Decent

(7.1.11) to a sequence of processes

of

0-predictable.

This proves the theorem.

(7.1.14) SUMMARY: There a r e two main

ideas in this section.

T h e first

is

that coarse enough time samples of a process whose standard part has

finite

classical

interchanged

with

variation

have

the standard part

a

variation

(in

that

D[O.l].)

can be

The second

idea is that Iterated Integration allows us to connect a.s. path approximation

to

internal

sums.

The

following

two

exercises

test your understanding of the second idea on internal summands which need not be liftings of any standard process.

Showing

366

Chapter 7: Stochastic Integration

this

internal

stability,

separate

development of more general sections.

from

lifting,

integrals easier

makes

the

in the following

The internal sums also have "standard" applications.

(7.1.15) EXERCISE: U

Let

Gl(t.w)

and

and for

8.8.

have

6t-bounded

G2(t.w)

variation.

that

b E 0,

are internal, bounded by

w ~

T h e n f o r a.a.

~

:{

Stt

Gl(t.w)

f

= 0.

st G ( t . w ) } 2

w

t-6t (Gl(s,o)-G2(s,o))6U(s.o)l

max[I'

Suppose

: 6t

<

t

<

11

Z

0.

s=6t In other words, both

summands give nearly

the same Stieltjes

Bt-bounded variation.

Suppose that

sums.

(7.1.16) EXERCISE: Let

U

internal

is

have and

has

a

limited

bound.

Then

G

S(t.o) =

t

G-6U

a.s. only jumps where

U

does.

'6 t

Hence

S(t)

has a decent path sample.

See the proof of

(7.3.8) i f you have trouble formulating the jump condition. is easy to see that

S(t)

need not have

It

6t-bounded variation.

367

(7.2)

Quadratic Variation of Martingales One of the main ingredients in martingale

integration is

the quadratic variation process associated with a martingale.

It is used in estimates similar to the classical domination of a signed measure by its total variation in the previous section, but there are also surprises.

Consider these curious heuristic

formulas for one-dimensional Brownian motion:

(db) 2 = dt

&

dbdt = 0 ,

so

d(f(b))

= f'(b)db

1 + 5 f"(b)(db)2

= f'(b)db

1 + 5 f"(b)dt.

+

* * -

for example. d(b

No

doubt

our

reader

2

) = 2bdb + dt.

will

see

some

tempting

analogies

for

Andersonn's infinitesimal random walk, for example,

6(B2)

= (6B+B)2 - B2 = (6B)2 + 2B6B = ( ~ +f2B6B i)~ = 2B6B + 6 t . *

Such calculations are made

precise

transformation formula given below. the generalized study of the

(db)2

by

the

(generalized) I t o

In this section we take up term.

Brownian motion is

an important test case. The following is an extension of (7.1.1) where

AM, etc. are defined ( f o r the internal functions

6M

t + M(t,w)).

and

Chauter 7: Stochastic Integration

368

(7.2.1) NOTATION: Let

6t

and

At

*lRd

are internal processes with values in denotes

I f

be t i m e i n c r e m e n t s .

the e u c l i d e a n inner

product

M

and

and if *Eld,

on

N

(x.y)

we define

the joint q u a d r a t i c v a r i a t i o n processes f o r the respective time increments by:

+ )[(6M(s.o).6N(s.w))

:

0

<

s

<

t, s E

f o r 6t

<

H6]

t E

Us,

and

We

also

define

maximal

functions

for

the

respective

increments by:

6

M (t.o) = max[IM(s.o)l

: 0

<

s

<

t, s E

Us]. for

6t

<

t E

At

<

t E UA.

T6,

and

M A (t,o) = rnax[lM(s.o)l

:

0

<

s

<

t, s E

U,], for

7.2 Quadratic Variation of Martingales

Section

All

the

little

details

variation are important.

in

our

We include

369

definition M(6t)

of

quadratic

in the quadratic

variation. while no such term was needed

in first variation.

This term corresponds to the standard term

%(O)

convention of starting

M

If (d = 1)

and

= B(t,w)

because of our

6t-decent path samples at

6t.

is Anderson’s infinitesimal random walk

is as in (5.2.3). then

6t

=: t =

[6B,6B](t,o)

:

0

<

s

<

t , step at]

It is well known that the paths of classical Brownian motions such as

are nowhere differentiable (see one of the

g(r,o)

books by Breiman, Doob or Loeve from the references). various

classical

formulations

of

the

idea

increments of Brownian motion tend toward We would

that

infinitesimal?

when

At

finite

could be given.

like to turn the question around and ask:

[AB,AB](t,w)

In fact,

is much larger than

6t,

What is but still

This is answered by Lemma (7.2.10).

We begin with some simple, but illustrative calculations.

(7.2.2) EXERCISE

(Cauchy’s inequality for quadratic variation):

For interna

mensional processes

M

and

N.

I [6M, 6N] (t HINT:

Apply the

* transform

with

components

6Mi(s).

t

>

s E

r;.

of Cauchy’s inequality to vectors 6Ni(s)

for

l < i < d

and

Chapter 7: Stochastic Intearation

370

The next result frequently allows us to focus our attention on single martingales, yet conclude results about pairs.

(7.2.3) EXERCISE (Polarization identities)

For internal

HINT:

M

d-dimensional processes

* d IR

Sum the corresponding identity of

and

N.

.

(7.2.4) EXERCISE: Let be a

T

be a one-dimensional

M

6t-stopping

time.

6t-martingale and let

Show

that

E[M2(~(~),o)]

= E{[~M.~M](T(w).w)}.

General martingales require some coarser time sampling just as in the last section.

N6

makes

# NA

and

The nasty martingale

then is, M

of (6.5.4)

A main result about

[6N,6N] # [AN,AN].

quadratic variation says that i f

N(t)

(M,N)

is a

6t-martingale.

6t-sampling also works for the quadratic variation, that

[6M,6N] and

N;

has a

Bt-decent path sample for the same

moreover,

infinitesimal

At

in

[6M,sN]

T6.

Z

[AM,AN]

6t

as

a.s. for any coarser

The path property

(7.4.9) using estimates for stochastic integrals.

is proved

in

Stability for

bigger increments is Theorem (7.2.10).

We had a hard time deciding what

level of generality to

Section

7.2

Quadratic Variation of Martingales

present in hyperfinite stochastic integration.

L2

the

case on

371

We shall present

[O,l] in section (7.3).

This reduces the

that would be required for a treatment of the

technicalities

full local theory .

We do offer notes on the extension to local

martingales in sections (7.4) and (7.6) (which our reader may ignore).

L2

Lindstrom [1980] treats local

our outline

toward Hoover 81 Perkins [1983] more

is directed

general theory.

martingales, but

Our reader must consult their paper for more

details of the local case. in the local case,

This section is not very technical

we also give

so

the local results.

reader may ignore the statements such as "t is limited in she is only

interested

[O,l].

in

Some stopping

The

T" i f

times are

needed anyway, so this should cause no trouble. Definition martingale" function.

(6.7.3)

is

automatically

set

has

a

up

so

that

a

"

6t - 1 oca 1

locally-S-integrable

maximal

The lifting theorem (6.7.5) shows that there is no

l o s s in generality with this definition, or, to put i t another

way, the maximal always

locally

functions of integrable

sequence in general).

(but

standard

local martingales are

localizing

with

a

different

Our next result gives integrability of

the quadratic variation in both the local and "global" cases.

(7.2.5) THEOREM:

M

Let

and Let

p 2 1,

T

be a be a

6t-stopping

(M6(~))p

([~M.~M](T))~'~ is

d-dimensional

is is

S-integrable

time.

and

only

after

6t

Then for each finite

S-integrable

S-integrable. if

*m a r t i n g a l e if

and

only

In p a r t i c u l a r . if

[6M,6M](1)

if

M2(1) is

Chapter 7: Stochastic Integration

372

S-integrable. 6t-reducing

If

M

is a

sequence

6t-local martingale with the

S-integrable for each

[6M,6M]1’2(~m)

then

{ T ~ } ,

is

m.

PROOF : p = 1

The last remark follows from the first part with simply applying Both (1.4.17)

(6.7.3)(b).

implications and

given next.

by

of

the

first

part

the Burkholder-Davis-Gundy Assuming that

(M6(~))p

are

proved

inequalities

(resp.

using (7.2.6)

[6#,6M](~)~’~)

is

S-integrable. there is a convex increasing internal function satisfying the conditions of (1.4.17) with (resp. x

[~M.~M](T)~’~).

* C0.a).

E

rk(2x)

<

for all

= (M6(7(w)))’

f(w)

rk(x) = @(x’).

The internal function rk(0) = 0

is convex, increasing, has

krk(x),

x

CJ

and satisfies +1 k = 4’ . The

where

E *[O,m),

inequality (7.2.6) completes the proof of the first assertion (because one implication in (1.4.17) does not require part (b) as noted in its proof). The

S-integrable

M2(1)

is

Doob

inequality

S-integrable. if

and

(6.5.20) shows

only

that

[M6(1)I2

if

is

S-integrable and the first part of the result connects this with quadratic variation using

T

5

1.

(7.2.6) THE BURKHOLDER-DAVIS-GUNDY For every standard real

exist every t E Us :

This completes the proof.

INEQUALITIES: k

standard real constants d-dimensional

-

*martingale

>

and

0

c,C

M

>

0

d

E

such

and every

N,

there

that f o r 6t

and

and for every internal convex increasing function

*[o.-)

*~ 0 . m )

satisfying

Section

7.2

373

Quadratic Variation of Martinaales

q(0) = 0

9(2x)

and

k*(x)

x

f o r all

E *[O,m)

the following inequalities hold:

PROOF : This

result

finite case

of

follows by the

taking

extension o f )

d-dimensional

[1972] Theorem 1 . 1 .

Davis-Gundy's

* transform

the

of

(the

Burkholder-

While this is a cornerstone

of our theory, we shall not give a proof since i t is a "wellknown standard result."

(7.2.7) PATHWISE PROJECTION OF For

process

any

internal

[6M,6M](t)

of squares.

process

each

t

The (local)

is finite when

r E [O,m)

M,

the

quadratic variation

is increasing for all

o

S-integrability of

(7.2.5) means that except for whenever

[6M.6M]:

o

[6M,6M](t,o) o

e A.

since i t is a sum

[6M.6M]

proved in

in a single null set is also finite.

A,

Hence for

the left and right limits along

S-lim[6M.6M](t) t tr

= inf{st[6M,6M](t)},

S-lim[6M.6M](t) t lr

= sup{st[6M,6M](t)}.

t E Us.

tZr

and

both exist in

IR.

t E

Us

tZr

It follows, (5.3.25).

that

[6M,6M]

has a

Chapter 7: Stochastic Integration

374

At-decent path sample for some infinitesimal A t actually has a

[6M,6M]

as the process M.

in

T6,

but

6t-decent path sample for the same

The proof that

has a

[6M,bM]

6t

6t-decent

path sample, Theorem (7.4.9). uses the machinery we develop for stochastic integration.

We believe that there should be simple

direct proofs of this basic fact, but do not know any. We abuse notation and define a pathwise projected process using the extended standard part:

By the preceding remarks a process with paths

[P.%](r,w)

is indistinguishable from

Dt0.m).

The abuse of notation is

in

justified by (7.2.11) The following is a key technical lemma that tells, us some information about paths of quadratic variation processes.

(7.2.8) LEHMA:

Let

M

suppose that part,

;(a) =

be a u

d-dimensional

6t-stopping time whose standard

is a u(u).

6t-local martingale and

satisfies

S-lim M(t) t lo

= st[M(u)]

<

and

m

a.s.

Then

[ % , 1 ] ( ;=)

st{[6M,6M](u)}

a.s.

PROOF : We will show that for any infinitesimal

At

in

H6,

7.2 Quadratic Variation of Martingales

Section

[GM,6M](o+At)

[6M,6M](a)

Z

a.s

The dependence of the exceptional null set on "a.s.")

does not matter.

375

(in the

At

The external almost sure statement

means that the internal probability

-

P{[6M,6M](a+At)

holds

for

all

At

st[6M,GM](a+At) decreases

tends to

finitely

subsequence, but

finite

in a

to zero.

Thus

S-integrable and

a

the

<

SL2,

so

maximal function.

Y6.

Hence

an a.s.

At

convergent

is increasing, the

= st[bM,dM](a)

a.s. M2(1)

is

(6.5.20) has an

SL2

In this case, the martingale

1.

t l a < t

9

{ M(t)-#(a)

=

of

lemma in the case where

M(o+At)-M(a) is also

i t has

[%,%](a)

0

N(t)

B

in probability as

since quadratic variation

shall prove

<

B }

interval

st[6M,6M](a)

whole limit converges a.s. and We

>

[6M,6M](o)

by Doob's

,

a

,

t

2

<

o+At

a+At

inequality

That maximal function

N 6 (1) = max[lM(t)-M(a)l

:

u

<

t

<

o+At]

t

is

infinitesimal

st[M(a)].

yields

by

the

hypothesis

S-integrability means

expected value.

N

a.s.

it

has

S-lim M(t) = t la infinitesimal square that

Finally, applying the BDG inequality (7.2.6) to

376

Chapter 7: Stochastic Integration

CE{[~M,~M](U+A~)-[~M,~M](CJ)}

<

12}

E{maxlM(t)-#(a)

z 0.

t

This proves

the

lemma in the global

SL2[0.1]

case.

(The

reader can easily prove the local case by introducing a reducing sequence.

Moreover, the bounded integrability of the reducing

sequence is all that is needed, not the fact that 6t-decent path sample.

M

has a

This is helpful in Exercise (7.4.4).)

(7.2.9) COROLLARY: If

M

is a

is

[6M.6M]

d-dimensional

6t-local martingale, then

t = 0, a.s.

S-continuous at

PROOF : a(o) = 6t

The stopping time a.s. s o the lemma yields

satisfies

S-continuity of

G(0) = st M(6t)

[6M,6M]

at zero.

(7.2.10) THE QUADRATIC VARIATION LEMMA: Let

M

{tj

: j E

*IN.

of

Ui

with

be a 0 to

<

n}

tl

<

j

<

d-dimensional is any 0 . -

<

tn,

6t-martingale.

I f

S-dense internal subset then

PROOF : The components of a martingale

d

[6M.6M](t)

=

are also martingales and

1 [6Mi.6Mi](t).

Section

7.2

Quadratic Variation of Martingales

377

If we prove the lemma f o r one-dimensional martingales, i t follows for

d-dimensional ones by summing components.

shall assume that

is a one-dimensional

M

Hence we

6t-local martingale

for the rest of the proof.

M

Since S-continuous

l2

IM(tl)

has a

at

zero

a.s. a.s.

IM(6t)I2

6t-decent path along

Corollary

Ti

sample,

so

9

(7.2.9)

Hence, we may a s well assume that

that we only have

to compare

[6M,6M](to) to = 6t

the difference between

large and small increment5 beginning at the same time. a useful

formula

for

comparing

large

and

small

-

M(tO)

= 1[6M(s)

: to

<

s

<

t l , step 6tl.

so

2

=1

+ 2

11 6M(r)6M(s) s>r s-6 t

=I Hence,

t -6t 1

= [6M,6M](tl)

+ 2

1

5=t0

so

summing Here is

increments,

starting with the first large one:

M(tl)

2

=: IM(t0)l

IWt)I2

shows that

is right

it

(M(s)-M(to))6M(s).

Chanter 7: Stochastic Integration

378

In general,

where

(using

o r , letting

our

[s]

convention

= max[t

. t

j .

j

on

<

s,

0

<

j

<

The same sum formula may be used to define summand

t

in

Us

yielding a

n].

N(t)

*martingale

for any upper along

Ui.

By

direct calculation the quadratic variation

t

= 4 )lM(s)-M([s])

[&N,6N](t)

l2

6M(s)

12.

To conclude the proof we need a reduc ng sequence even when

M2

is

S-integrable on

Stopping Lemma ( 6 . 4 . 5 ) 6

M (Tm-6t) $, m.

i(st

T

[O.l].

In this case, apply the Path

to obtain stopping times ~

=) st(M(~,)]

and

T~

such that

~ , f l . In general, if

Section

Quadratic Variation of Martinaales

7.2

{ T ~ } is a

6t-reducing sequence for

1 [6N,6NI2(rm)

<

M

M 6 (rm)

with

-

1 5m[6M.6M] 2 (

379

5

then

m,

T ~ ) .

1 By

(7.2.5) and

[6N.6Nl2(~,)

(1.4.14).

Burkholder-Davis-Gundy's

is

S-integrable.

(7.2.6) inequality and (1.4.13) tell us

that E(max[lN(t)l

: t

I

T~])

1 -

<

2

CE([GN.GN]

<

u

-21

(T,,,)),

CE(st[6NS6N]

C.

for a standard positive constant

(T,,,))

W e will show that

1 E(stC6N.6NI

and

IN(t)(

therefore

Z

0

2

for

( T ~ ) )= 0

finite

t

8.5.

proving

our

lemma.

For each

n

in

IN

1 -

< $ E ( S ~ [ ~ M . ~ M ] ~ ( T ~ )+) BmE(st{I[

: s E

A:(")]}

where

A",")

= { s E Ui : s

<

T ~ ( o& ) IM(s)-M([s])l

> f},

1 -

2) .

380

Chapter 7: Stochastic Integration

estimating ( s

I

T

m

).

,$ 2m

IM(s)-M([s])l

where

is

it

large

It is sufficient to prove that : s E A”,(w)}

E(st{2[lBM(s)12

for every po

A:

on

m,n

= uo = Bt

E

IN.

Define

1 -

2) = 0

6t-stopping

times

as

follows:

and

th

p i = i-

timelM(t)-M(t-Cit)l

>

1

<

n].

and

u

If

M

then

has a

i

= min[t

: tj

j

2 pi,

Bt-decent path for the sample

>

lM(s)-M([s])1

$

and

s

,$ m.

a finite amount infinitely near

Us

path along

number

M(*.w) of

(5.3.4)(c).

w

and

Therefore

U6

[s]

and

s.

could only have jumped by more than s

,$ m

Thus for almost all

A”,(o).

varies

but since i t has a decent

s,

between

times before

s E

M(*.o)

i t must have jumped by an amount

one single time in hand,

0 5 j

by w.

(the A:(o)

2 > 1 at > - n n On the other 1 n

a finite

C0.m)-version

of)

is contained in the

countable (external) union

Hence, by of (7.2.8).

S-integrability and (7.2.6) applied a s in the proof

7.2

Section

Quadratic Variation of Martingales

E(stJP[ 16M(s)

1'

: s E

1

c

I

2

38 1

A:(o)])

st E(max[lM(t)-M(p,Ar,)I

i €IN

Pi A

Tm

<

t

:

<

ai A

T

~

t. E T,])

= 0.

We get zero because

M6(

T ~ )

I

max[ IM(t)-M(piATm)

whenever

M

has

happens a.s. Since

T~

is

: pi A

T,,,

6t-decent paths

This proves that

1

S-integrable while

a.s. as

m +

m,

5 t

<

for

the sample

ui A

max[IN(t)l

: t

T

I

~ Z]

0

which

o,

Tm] Z

0

a.s.

this concludes the proof of the

1 emma.

The primary consequence of this lemma i s the fact that the quadratic

variation

independent of

of

a

standard

the lifting and

local

hypermartingale

the infinitesimal

increment

is

in

particular (that is, once the increment is coarse enough to make the paths of

Fix an

A

r

decent). E

[O.m)

and i f

an increasing sequence. define

0 = ro

<

rl

<

0 . -

<

rk = r

is

Chapter 7: Stochastic Integration

382

(7.2.11) COROLLARY:

%

Let

be a local hypermartingale and let

%.

6t-local martingale lifting o f

S(%,{rj})

converges to

[%.fi](r)

P

standard quadratic variation o f of

lifting

and

indistinguishability)

we

r E C0.m).

tends to zero.

The

does not depend o n the

denote

decent

be a

in probability as the

maxlrj-rj-ll,

mesh o f the sequence,

choice

For each

M

path

the

unique

standard

(up

process

t,o

by

[ii.G](r).

PROOF : Choose any

[%,i](r) 6t = t

such that

= st[6M.6M](t) (7.2.10).

Lemma

H

t E

<

tl

<

a.s.

whenever -**

<

Let

the

tk = t

= st M(t)

G(r)

in

m

mesh

of

a

By

sequence

>>

such that

em

0

max(t -t I < e m , the probability above holds. .i j-1 0 = r < rl < * * - < rk = r be a standard sequence in

whenever Let

choose

be finite.

* finite

is infinitesimal,

This is an internal statement s o there is an

CO.-)

IN

a.s. and

with t

j'

maxlr -r

J

0

<

j

<

k

j-1

I <<

Let

em.

to = 6t.

so

p(rj)

= st M(t

)

. i

a.s.

tk = t

and

Section

7.2

The sequence

Quadratic Variation of Martingales

383

satisfies the internal probability above. s o

{tj}

standard parts yield

proving the corollary. Two

M

6t-martingales

N

and

distinct infinitely close times. M+N

is not a

could have finite jumps at

This would mean that the sum

6t-martingale because the paths no longer have

separated jumps.

M+N

Of course

is a

*martingale,

coarser infinitesimal time sample increment make

M.N

M+N

and

all

If we start with

%

we may apply the

%

lifting theorem to

(%,%). If

(%.N)

lifting of

(i.5).

at all),

so

may define

M+N

is a

is a

[g,%](r,u)

the

and

2d-dimensional

2d-dimensional

M

then

would

At-martingales.

standard local hypermartingales martingale

116

in

At

some

so

N

and

martingale

6t-local martingale

must jump together (or not

6t-local martingale.

In this case we

= S-lim[6#,6N](t.o) tl r

a.s. and use the

polarization identities (7.2.3) and (7.2.11) to see that this is independent

of

the

indistinguishability.

choice

of

the

lifted

pair

This shows how to extend

up

(7.2.11) to

pairs and the following extends (7.2.6) to pairs.

(7.2.12) LEMWA: Let

(%.H)

marttngaLe wtth the

If %(;)

17

is a

= st M(a)

be

LocaL

6t-Local marttngale Lifting

Gt-stopptng a.s..

2d-dimensionaL

a

ttme satisfying

then

[%.HI(;)

G <

to

OJ

= st[BM,6N](o)

hyper-

(M,N). a.s. and a.s.

Chapter 7: Stochastic Integration

384

PROOF : By the preceding remarks and ( 7 . 2 . 8 ) we know that

[1.8](;)

= S-lim[6M.6N](o+At)

a.s.

At10

[i,%](;)

and that

>

0

for

We simply apply the

* finite

[6M,6N](a)

Z

Z

0

infinitesimal

* transform

a.s.

of Cauchy's

dimensional vectors with components

I [ 6M.6N] (u+A t)-[

that

: 1

<

i

[6M,6M](o+At)

S-integrability of finite a.s.

inequality to the

6Mi(t).6Ni(t).

6%.6N] ( a )I =

=I)[6Mi(t)6Ni(t)

know

At

'E6.

in

[6M,GN](u+At)

We

= st[GM,6M](u+At)

Hence i t suffices to show that for every

a.s. At

= st[61,6M](u)

[6N,6N]

<

d.

Z

<

<

o+At, t E

[6M,aM](a)

a.s.

(I

makes

This proves the lemma.

t

[6N,6N](o+At)

T,]l

and

local

- [6N,6N](o)

385

(7.3) Square Martingale Integrals

i

Let

:

[O,l]

R

x

+

IR

P(0) = 0.

integrable and

be a hypermartingale with

We know from the Martingale Lifting

Theorem (6.5.13) that there exists a

M2(1)

S-integrable and

M(6t)

0

E

whose

s.

1.

with

6t-decent path

By Theorem (7.2.5). we

is S-integrable for all

[6M,6M](t)

M

6t-martingale

a.

projection is indistinguishable from know that

g2(1)

t

<

1.

In this

section we use estimates on the quadratic variation to show that the martingale integral

is

well-defined

as

the

6t-decent

path

projection

of

the

Stieltjes sums t-6t 1s=6t G(s.o)6M(sSo).

S(t.o) =

for a pathwise lifting

G

H.

of

"Well-defined" means this

a. s. does not depend on the choice of the martingale lifting,

M. or the path lifting, G. once

M

is chosen.

construction is the analog of (7.1.4).

Our first

The development runs

parallel to section 7.1. except that we use martingale maximal inequalities

(instead

of

the

triangle

inequality)

and

this

requires that our summands be predictable.

(7.3.1)

DEFINITION: Let

M2(1)

M

be a 6t-martingale w i t h

S-tntegrabte.

p a t h uartatton m e a s u r e

For e a c h ho

on

o E R

U

W(6t)

2 0

a. s. and

deftne a quadrattc

by t h e w e t g h t f u n c t i o n

Chapter 7: Stochastic InteFration

386

D e f i n e a total quadratic variation measure as

on

u

the hyperfinite extension o f the measure

Y x R

with weight

function

du(t.o) = 6Ao(t).6P(o).

Since

E{[6M,6M](l)}

hyperfinite measure, u .

extends to a bounded

is limited, u Since

[6M,dM](l)

P[A]

= 0, then

is P-continuous. i. e., i f

Iterated Integration (7.1.5) applies to

u.

is S-integrable. u u[Y

x

A]

Since

continuous at zero, u is continuous at zero, u[st-'(o)

(7.3.2)

G :

= 0. Also,

%

is right x R]

= 0.

DEFINITION: Let

H

:

T x R

+

*IR

[O.l] x R + IR

ho{t

i s called a

be any function.

such that for almost all

:

st[G(t.o)]

# H(st[t].w)}

2 61 -path lifting o f

An internal

o

= 0

H.

We can prove a path lifting theorem like (7.1.8) for the quadratic path variation measure, but unpredistable integrands give the "wrong" answer, as shown in the following exercise.

387

7.3 MartinFale Integrals

Section

(7.3.3) EXERCISE:

B(t,o)

Let

be Anderson's infinitesimal random

w a l k associated w i t h

6t

as above in (5.2.3).

Define

2 2B6B( o) = )[2B(s,o)[B(s+6t,o)-B(s,o)]

:

0

I

s

<

Ir]

t, s E

Show that

Z ~ Z B ~=BB2(t1-t pt2B6B = B2(t)+t and

2 St2B6B = B (t).

(HINT:

Write

B2(t)

Show that

Pt

Show

when

K(t.o)

that

as a double sum and compare.)

a n d St are not

= ZB(t.w).

= B(t.o)+B(t+6t.o)

*martingales.

H ( r ) = 2g(r).

K(t.w) are

then all

= ZB(t+6t.o)

.-.

the

functions

and

6B"-path ltftings o f

The exercise above shows that

Pt is.

but

K(t.o)

H.

6M2-path lifting alone is

not enough to make infinitesimal Stieltjes sums independent of the

infinitesimal

differences

in

liftings.

Moreover,

the

388

Chapter 7: Stochastic Intearation

internal sum

is infinite a. sgn[aB(t)]

for all noninfinitesimal

s.

depends precisely

on

t.

but

w t+6t,

The function

is internal and

bounded.

(7.3.4) DEFINITION: G : H x R + *IR

An internal. process if

G

is 0-predictable

S-tntegrabLe

is

and the function

with respect

6M2-summabLe IG(t,w)I2

is

to the hyperftnite measure 6u =

generated by the weight function

u

16MI2-6P.

This summability condition is equivalent to the condition

by the Iterated Integration Lemma (7.1.5). Our next result is part of a closure law f o r stochastic Stieltjes sums.

(It lacks the decent path property.)

understood that the martingale

M

It is

is a s above.

(7.3.5) PROPOSITION: G

Suppose

ts

6 M2-summabLe

1

(where

M2(1)

is

t-6t

S-integrable).

*marttngaLe

N(t)

Then

=

G(s)BM(s)

s=6t

after

6t

with

N 2 (1)

S-LntegrabLe

is

a

Section

7.3

389

Martingale Intecrals

PROOF : Since

G

is nonanticipating after

E[6Nlwt] = G(t)E[6Mlw

Moving

st

t

6t.

] = 0.

inside always produces the inequalities:

st[I B G26u] = st[E{[6N,6N](l)}] w t

>

E{st[6N,6NI(l)}

- s > E{

=

The

two extremes of

u-S-integrable. [6N.&iN]

st G2d(A,)}

[ st G 2du

by (7.1.5).

these inequalities agree because

Hence

st E{[6N,6N](l)}

G2

= E{st[6N,6N](l)},

is so

is S-integrable and (7.2.5) completes the proof.

Our next result says nearly the same sums.

u-equivalent

Again,

M

summands pathwise give

is a s above.

(7.3.6) PROPOSITION: Suppose

G1

U{(t,w)

*

and

:

G2

I

2 6M -summable and

st Gl(t.w) # st G2(t.w)}

Then the marttngale tnftnttely close to N2(t)

E{

are

= 0.

t

Nl(t) = B G1(s)6M(s) = 2 t G2(s)6M(s), in f a c t ,

max Et[Gl(s)-G2(s)]6M(s) 6t
I 2}

0.

t S

ChaDter 7: Stochastic Integration

390

PROOF : Since GIG2

IG1I2 and 2 2 max[G1.GU], while

<

IG2I2

are

u-S-integrable. so is 2 st GlG2 = st G1 u-a.e. Therefore,

N

Applying the BDG Inequality (7.2.6) to the martingale

=

N 1-N 2

yields

E{[

max IBt[G1-G2]6Mll2} 6t
In particular, the whole path of path of N2

<

CE{BlG1-G2 216M12}

N1

0.

Z

is inf nitely close to the

a.s.

(7.3.7) PROPOSITION: Suppose

B

is

6 -mar ingal

wtth

M2(1)

F.G

a n d the sequence

{Fm}

are all

S-tntegrable.

If

2 6 1 -summabLe.

st F m

IFm!

<

IGl,

tends to

st F

in

u-measure

and

then

tends to zero in probabiltty.

PROOF : Use comprehension from section ( 0 . 4 ) to extend the sequence {F,}

to an internal sequence satisfying

IFm(

<

IGl

for all

Section

m E

7.3 Martingale Integrals

*# .

Use the Internal Definition Principle (in the style of

Chapter 1) to pick an infinite is

391

Fn

infinite, then

Z

n1

F

n 5 n1

such that whenever

u-a.e.

Now

use

the preceding

proposition to show that

max IPtFn6M

E{[

- ZtF6M1I2}

0.

Z

6t
This means that for every standard positive

I(€) = {m E

*IN

<

I m

contains a finite

k

I

the internal set

E

>

n1 3 P{maxlZt[F-Fk]6MI

B }

<

8 )

This proves the claim.

m(e).

Up to this point we have not assumed that of a standard function of any kind.

G

is a lifting

However, we have not been

able to make any conclusions about the paths of The weakest pathwise result is taken up next.

= PtG.6M.

N(t)

In section (7.6)

we describe Hoover 8 Perkins’ [1983] better martingale lifting that makes 2 6M -summable standard

N

have

internal

6t-decent function-not

paths just

when

G

is

the lifting of

any some

H.

(7.3.8) THEOREM: Let

:

6t-martingale a. s.

[O.l] x R

M

Suppose

preuisible

or

with that

even

+

be

the projection

S-integrable a n d

H : [O,l] x R + IR

only

usual agumentatton of

IRd

M 2 (1)

5

preuisible

with

of

M(6t)

is respect

the Z

0

u-almost to

the

a n d that the pathwise S t i e l t j e s

Chapter 7: Stochastic Integration

392

integrals satisfy

<

E[lL H2(r.w)d[T,~](r,o)]

H

Then

G

:

U

x R

has

+*R.

2

6M -summable

a

Such a

G

m.

2 6M - p a t h

lifting

Pt G(s)6M(s)

makes

haue a

Gt-decent p a t h s a m p l e , so t h e s q u a r e s u m m a b l e m a r t i n g a l e

may the

be d e f i n e d independent

of

the p a r t i c u l a r

lifting as

6t-decent path projection o f

t-6t

2

G(s.o)6M(s.o)

s=6 t

PROOF : First, as long as

N(t)

= ZtG-6M

has a

sample, Proposition (7.3.6) shows that any two G's

have

the

same decent path projection.

remaining claims are 6M-sums have a

that such a

G

PROOF OF LIFTING: Change of variables gives us

=

2

6 1 -summable

Hence

exists and

Gt-decent path sample.

H2(r.o)d[~.$](r.w)

at-decent path

H2(st(t).u)dAa(t)

the

two

that

its

Section

for

393

7.3 Martingale Integrals

all

such

o

that

is

[%,%](1)

finite.

Iterated

Integration gives

= E{b2(st(

E[1H2d[%.%]}

E

t) ,w)dXu( t)}

1

<

H2(st(t).o)du

m.

We know by (6.6.8) or (6.6.11) that there is a

0-predictable

F

such that

: st[F(t.o)]

u{(t.o)

For each

m E N,

= 0

# H(st[t].o)}

the truncations of

F

and

H

at

m

satisfy

Thus

by

infinite

Robinson’s

Sequential

Lemma,

8 2 F26u z

st Fndu 2

for

sufficiently

small

n.

=

The function

G =

F n

H2 st du.

is our square-summable predictable path

ChaDter 7: Stochastic Intewration

394

1 i f t ing .

PROOF OF OF 6t-DECENT PATHS:

N(t)

We say that

M(t) w

N(t+At)

N(t)

@

t , t+At E

N(t)

H6

implies

M(t+At)

N(t)

6 M2-path lifting of

M(t)

only jumps where

@

Z

if

0, then

M(t).

M(t)

does and

has a

also has a 6t-decent path

H

We begin by showing that when

a bounded

= 0, such that

0 C At

and

M(t)

a. s. only jumps where

6t-decent path sample, then sample.

n, PCA]

A

if there is a null set

E A , then whenever

If

a. s. only jumps at the same times as

is bounded and

H. then

= ZtG.6M

N(t)

G

is

a.

s.

does.

H

First suppose that

is bounded and

u-equivalent

to a

basic almost previsible process o f the form in (6.6.13) and that

.-.m is the

6M2-path

0-predictable

(6.6.13).

Since

u

is continuous at zero, we may assume that

go = 0. The functions = ZtG*6M

lifting guaranteed by Lemma

gj

are bounded,

that

N(t)

M(t)

has a 6t-decent path sample at

Case 1:

N(t+At)

t

<

t

- N(t)

<

only jumps where

t+At, for

j

tj

w

lgj(w)I

M

<

b.

does we assume that

and consider two cases.

a s above in

G.

= = [N(t+At)

- N(t

+6t)]

j

To show

+ [N(tj+6t)

- N(t)]

7.3

Section

395

Martingale Inteerals

M

A t most one of the terms with a difference in

M

noninfinitesimal because and

has a decent path at

Thus, i f

are bounded.

‘5-1

- M(tj)] and g [M(t+At) j noninfinitesimal. Therefore, M(t+At)

<

i:

t+At

N(t+At)

so i f

*

N(t+At)

These

t

for

j’

- N(t)

N(t)

H

N

Proposition

(7.3.6) shows

N(t)

exactly one

- M(t)]

is

G

above

- M(t)].

M

a .s. only jumps where

is the particular

makes

gj

is a bounded basic almost previsible

G

G

and

M(t).

M(t+At)

two cases show that

process and

does.

as in

j

o

M(t).

= g.(o)[M(t+At) 3

then

does in the case that

lifting

t

N(t).

gj-l[M(tj)

*

t

*

N(t+At)

of the terms

Case 2:

can be

that

lifting obtained in (6.6.13). any

= BtG-6M

other

2

6M -path

bounded

also a. s. only jump where

M

The next step of the proof uses Lemma (6.6.14) to show

that every bounded almost previsible process produces nice path sums.

V

Let

be

processes, H.

N(t)

have

shown

set

of

all

bounded

that have a bounded

= BtG*6M

that

processes

the

that

2 6 M -path

a. s. only jumps where

V

contains

all

in the paragraphs above.

basic The

set

almost

previsible

lifting, G. does.

M(t)

almost

V

such We

previsible is a vector

space, because i f two stochastic sums only jump where

M

does,

CharJter 7: Stochastic Integration

396

so

Proposition (7.3.7) and Lemma

does a linear combination.

(6.6.14)

I

show that

contains all bounded almost previsible

processes.

Hm be the finite truncations of an

Finally, let

H.

integrable unbounded

N,(t)

G

lGml

2

= ZtGm6M

IGl a.

and

6M -summable

Gm + G

M

only jump

s.

Gm

in does.

Us.

A

Let

Hm

and

u-measure.

The

In particular, Nm 6t

Proposition (7.3.7) says that a subsequence of the

N

6M -path

lift

a. s. has a 6t-decent path sample for the same

tends uniformly to

2

2

is a

H. then the finite truncations

lifting of satisfy

If

as Nm

M.

a. s.

in the space of internal functions on

be the countable union of null sets where some 6t-decent path or the subsequence of

does not have a

not tend uniformly to

If

N.

z ~ + ~ ~ *G z- ~~G M- ~ M .

m.

H ~ + ~m ~ G . L ~ ~MG ; ~ M ,

since we may make the uniform error between less than one third of

the difference.

M

a. s. only jumps where

does

and

o Q A

then for sufficiently large finite

Nm

Nm

does, s o does

Gm -sums and Since each LtG*6M.

G-sums ZtG;6M

This proves

half of the main result of the section. Since we have actually shown more than that

ZtG-6M has a

6t-decent path sample, we have the following corollary t o our proof.

(7.3.9) THEOREM: If

M

i s an

S-integrabLe and summabLe

S-conttnuous

M(6t)

6M2 -path

process. then

=: 0

Lifting

ZtG-6M is also

6t-martingaLe luith and i f

of

a

G

standard

S-continuous.

is

a

M2(1) 6 M 2-

preuistble

Section

7.3

397

Martinnale Integrals

PROOF : Since the proof of (7.3.8) shows that

M

where

M

does and

stochastic sum is

HtG-6M

does not jump by

only jumps

S-continuity, the

S-continuous.

The remaining half

of

the well-definedness

question is:

"What i f we take a different martingale lifting?"

(7.3.10) THEOREM: Suppose integrable

%

that

%(O) = 0.

and

6t-martingale lifting of

N

is a

Suppose

% with

At-martingale

S-integrable.

G2(1)

is a hypermartingale with that

M2(1)

F

is

%

with N2(1) 2 6M -suminable 6M2-

is a

path lifting of a standard almost preuisible process

G

while Then

2 6N -suminable

is a

the

At-decent

path

indistinguishable from the

a

S-integrable and

lifting of

Suppose that

M

H.

6N2-path Lifting of

H.

LtG-6N

is

projection

of

6t-decent path projection of

PF-~M. PROOF : First suppose that process and (6.6.12).

F In

and both

H

is a bounded basic almost previsible

G

have

cases

the

the form of decent

path

the

lifting of

projections

are

indistinguishable from

m

1 hj(.o)*[%(r

j + l)

-

%(rj)].

j=1

Proposition (7.3.6) shows that

F

and

G

can be any bounded

398

Chapter 7: Stochastic Integration

H

path-liftings of Next

we

and still have the above projection.

apply

previsible

processes

previsible

H

and

G

V

V

such that if

is a bounded

projections of set

(6.6.13).

Let

consist F

of

subset

all

6N -lifting of

H.

of

almost

bounded

almost

2 6 M -lifting of

is a bounded

2

LSF*6M and

the

H

then the decent path

ZtG*6N are indistinguishable.

The

is a vector space containing the basic almost previsible

processes by the remarks above.

The space

V

is closed under

bounded pointwise convergence by Proposition (7.3.7).

Thus

V

contains all bounded almost previsible processes. Finally, a truncation argument similar to the last part of the proof of Theorem (7.3.8) shows that every square summable integrand produces indistinguishable stochastic sums.

(7.3.11) EXERCISE:

Show that alL the p r o o f s in this sectton actually apply to the case of take

G :

d-dimensional Linear functtonals.

H

x

R

+

IRd

That ts, ure may

and tnterpret our stochastic Stteltjes

sums as sums of tnner products,

The meaning of lifting is clear.

A

6M"-summable vector

valued function is one for which the square of the vector norm, IG(s.0)

12.

is

u-S-integrable.

399

(7.4)

Toward Local Martingale Integrals To extend the treatment of martingale integrals from the

square

summable martingales

local martingales,

we

of

may

use

Theorem (6.7.5) and the special (6.7.3)

the

the Local

stability

to standard

Martingale

Lifting

Gt-reducing sequence

together with Theorem (7.2.5).

infinitesimal

last section

results

{T~}

of

This gives us the basic

without

use

of

Iterated

these

stability

b

Integration

(7.1.5).

This

section

proves

results, but does not give a general standard lifting theorem. (Sections

(7.6) and

(7.7)

sketch the proofs of

the powerful

lifting theorems of Hoover & Perkins [1983].) We do show how these simple extensions of section (7.3) may be applied to prove that paths of the quadratic variation are decent.

This is the main application of the section.

For this section we work in the evolution scheme of (5.5) and (6.7).

(7.4.1) DEFINITION: Let

M

Gt-reducing

be a

d-dimensional 6t-local martingale with

sequence

{T~}

and

M(Gt)

Z 0

a.s.

The

quadratic path uariation measure is defined by the weight function

The hyperfinite extension measure, in fact,

k

E IN.

[GM,6M](k)

Xu

may now be an unbounded

may not be

S-integrable even for

Proof of lifting theorems require that we control the

Chapter

400

growth of version

6hW's

of

so

7:

Stochastic Integration

that the total measure

6ho*6P.

The

simple procedure

6u

is a bounded

that we

used

in

Definition (7.1.4) will not work. s o we postpone this problem to section (7.7). We do know from the definition of

6t-reducing sequence

that

M 6 ( T ~ ) is

S-integrable for each

m

By (7.2.5) we also know that

-/[~M,~M](T,)

is

S-integrable for each

in.

Despite the technical problem with boundedness of the total quadratic variation measure, the notion of path-lifting remains the same.

(7.4.2) DEFINITION:

Let

M

be a

internal fucntion G function H

hu{t

6t-local martingale as above. is a

An

2 6M -path lifting of a standard

prouided

: st[G(t.w)]

#

H(st[t],o)}

= 0

a.s.

W .

Since we do not yet have an analog of the measure

u

from

the last section, we define local summability with the standard part.

Section

401

7.4: Toward Local Martingale Integrals

(7.4.3) DEFINITION:

M

Let sequence

be a

6t-local martingale with and

{T~}

0-predictable process prouided

that

there

Bt-stopping times

<

M(6t} G

0

Z

a.s.

is called Locally is

an

increasing

6t-reducing An internal. 2 6M -summable sequence

of

{urn} satisfying:

(a)

am

(b)

st[M(om)]

(c)

S-lim urn = m

Tm

= %(S~[CJ,])

a

.

~

a.s.

r

Our first result should be a "closure law" for stochastic sums.

Unfortunately, we cannot conclude that internal sums have

Gt-decent

paths

section (7.6).

without

the

stronger

martingale

However, we will now show that

has all the other properties of a

lifting

N(t)

of

= Zt G-6M

Bt-local martingale.

The maximal function

N 6 (urn) is

S-integrable

by (7.2.5) because we have assumed by (d) that

d[BN,6N](um)

Without knowing that can still show that

is

N

S-integrable.

has a

6t-decent path sample, we

Chapter

402

7: Stochastic Integration

= S-lim N(t) t lom

st[N(o,)]

a.s.

First we show that

= [a,a](st

st[6N.6N](am)

a.s.

am)

We have the inequality:

am) - st[6N,6N](am)

[%.%](st

= S-

- [6N,6N](am)

lim[6N,6N](am+At) A t 10

<

S- lim I [ I G ( S ) ~ ~ ~ S M ( S: )am ~~ At10

= lim At10

s

<

s

<

um+At]

st IG(s) I2dXw(s).

{am
The final integral in this inequality tends to zero because we have assumed that

(7.4.4)

%(st

am) = s t H(am)

a.s.

EXERCISE: If

N

is a

*martingale

6t-stopping time such that

[fi,a](st

after

N6(a)

is

a) = S-lim[6N.6N](a+At),

6t

and

a.s..

then = S-lim N(o+At)

At10

is a

S-integrable and

At10

st[N(o)]

a

a.s.

403

7.4: Toward Local Martinpale Integrals

Section

HINT: This is a "converse" to Lemma (7.2.8).

The ideas in that

proof can be used here. This exercise together with the preceding remarks mean that if we show that

N

has a

6t-decent path sample, then

N

is a

6t-local martingale. The

next

result

independent of

implies

choice o f

the

that

stochastic

lifting, but

sums

applies

to

are more

general internal summands.

(7.4.5) PROPOSITION:

M

Let

be

GI

s u p p o s e that

ho{t

that

a

€ 0

and

6t-local

G2

martingale

a r e locally

: st Gl(t.w)

# st G2(t.o)]

T h e n except for a single null s e t o f t E

O ' S

as aboue and 2 6 M -surnmabLe a n d = 0

a.s.

o.

f o r all f t n t t e

T6' It G1(s,o)6M(s,o)

%

Bt G2(s.w)6M

S,W).

PROOF :

If

and

u:

conditions for

2 am

GI

are the stopping times in the summability 1 2 let u = u A am. The am and G2, m m

satisfy the summability conditions for both Let

N

*martingales.j

We

can

see

(t)

= B

t

Gj(s.o)6M(s.o).

for

G1

and

G2.

j = 1.2.

define

The BDG Inequality (7.2.6) says that

that

the right hand

side of

this

inequality is

404

ChaDter

infinitesimal

1)

by

showing

7: Stochastic Intearation

that

the

internal

function

U

mlG1-G21216M12 Summability

S-integrable. Nj(um) 6

are

is P-S-integrable. of

means

Gj

that

[6Nj,6Nj](um)

is

By (7.2.5) this means that the maximal functions P-S-integrable.

Therefore,

max

<

IH(G1-G2)6MI

6 t t
<

N:(u~)+N~(u~)

see that

is

S-integrable.

)umlG1-G21216M12

is

Applying (7.2.5) again. we

P-S-integrable, s o

Finally, condition (d) of the summability hypothesis says that a.s.

a,

= 0.

Hence the right side of the BDG Inequality is infinitesimal. This concludes the proof, since

u

m

+

a.s.

We also have a convergence in measure result similar to the

last section.

Section

7.4:

Toward Local Martingale Integrals

405

(7.4.6) PROPOSITION:

M

Let

that

the

be a

6t-Local martingale a s a b o v e .

F,G

functions

6M2-suminable

locally stopping

all

and each standard

e

>

mith

sequence respect

<

IGI

{Fk}

are

the

same

to

and f o r each

m

0.

u,

then for each finite

- ItF6M]

m a x [XtFk6M

st

the

IF,]

If

{am}.

times

and

Suppose

+0

6t
in probability.

PROOF :

IFk}

Extend

IFkI

<

IGI.

to

For each

and an infinite

k

for every

an

m E IN

between

m

and

E

E Q+

sequence

satisfying

there is a finite

n1

such that

n2

n1

sufficinetly small infinite

all standard

internal

and

E .

and

k

n2.

By

saturation, all

satisfy these inequalities for

Hence, we may apply (7.4.5) to prove

our claim (along the lines of the analogous result from the last section). We

shall not

following

prove

a general

lifting

result may be proved along

theorem, but

the same

lines as

the the

406

Chapter

decent path part

of (7.3.8).

This

7:

Stochastic Integration

in turn has a n interesting

application.

(7.4.7) PROPOSITION: Let that

G

M

be a

6t-local martingale a s above. Suppose is a locally 6 M2-summable process w h i c h lifts a

H, that i s ,

standard preuisible process

ho{t

:

E 0

st[G(t,w)]

# H(st[t],o)}

= 0

a.s.

0 .

T h e n the stochastic Stieltjes sum

a.s. has a

6t-decent path sample

This "half" of the standard stochastic integrat on the0 r em

for local martingales can be applied to the interna

quadratic

variation process as follows.

( 7 . 4 . 8 ) LEHHA

If

the

pair

L = (M.N)

6t-local martingale. then

N

is

2d-dimensional 2 is a locally 6M -suminable a

612 -path lifting o f the left limit process

N

N ( r - ) = S-lim N(t) t tr

= E(r).

7.4:

Section

Toward Local Martingale Integrals

407

PROOF :

Let

be a

T~

6

N (Tm-6t) .$ m

6t-reducing sequence for

on

{Tm

>

6t)

(M,N).

Since

4/C6M.6M1(Tm)

and

is

S-integrable,

or

N

2 6M -summable up to

is

L = (M,N)

Since

define

stopping

times

16L(t-6t)l

> 71 . ' ' J

st M(a i ) =

%("'3)

i.

for

6N(t-6t) i,j.

J

*

0

6t-decent path sample a.s.

= G(st(t)) = " i c time

finite so

Also,

i.j

by if

a.s.

have

then

(7.2.8). t E t =

U6

in

t

E IN,

if

Similarly, i f we

a.s.

on a decent path, then

Therefore, we

We know that

a

aj

8.5.

i = S-lim [GM,6M](u.+At).

Atlo

has

st M(t)

0, then

aL(t-6t)

rm.

T6

if st [

is

that

i u

j

<<

6M. 6M] finite

m,

(03) and

for some finite

Chapter

408

= S-lim [6M,6M](a!(u)+At) 3

Atlo

= 0

7:

Stochastic Integration

- [dM.bM](~;(u))

a.s.

by the remarks above.

Hence

N(t)

is a

6M2-path lifting of

E(r).

(7.4.9) THEOREM: If

L = (M,N)

martingaLe, then

is

[6M.6N]

2d-dimensionaL

a

has a

6t-local

6t-decent path sampLe.

PROOF : This follows easily from (7.4.8) and (7.4.7) together with the

(*transform of the finite) formula for summation by parts:

[6M,6N](t)

= (M(t).N(t))

- Xt(N,6M)

-

Xt(M,6N).

409

(7.5) Notes on Continuous Hartingales In this section we present a few basic facts about local hypermartingales

with

internal objects.

continuous paths

and

the corresponding

The basic references for this material are

Keisler [1984], Panetta [1978], Lindstrom [1980a] and especially Hoover

&

[1983],

Perkins

section 8 .

part

11,

which has

the

strongest results. The first result says that sampling in order to obtain nice path properties of a lifting is not necessary.

(7.5.1) THEOREM: Let

+.

6t = min T

If

M

w i t h a.s. continuous paths and 6t-local martingale that

the

N

M(0)

with a.s.

continuous

indtsttnguishable f r o m

is a Local hypermartingale

path

= 0.

then there is a

S-continuous paths such

M

is

M.

The next result of Hoover & Perkins [1983] of

fi

projection

says continuity

can be measured by the quadratic variation [in contrast

to the decent path case. see (7.6.2)].

It extends results in

Keisler [1984] and Lindstrom [1980a].

(7.5.2) THEOREM: Let is

M

be a

*martingale

S-continuous and locally

d[6M,6M]

is

and

+.

6t = min T

Then

M

S-integrable i f and only i f

S-continuous and locally

S-tntegrable.

This has the following important consequence.

410

7:

Chauter

Stochastic Intevration

(7.5.3)COROLLARY: Suppose

martingale and process.

M

that G

Then

is

an

S-continuous

6t-local 2 6 M -summable internal

is a locally

ZtG*6M

is an

S-continuous

6t-local

martingale.

We add that

G

need not be

process [compare to (7.3.9).] Keisler's

existence

of

internal

This idea plays a key role in

theorem

equations mentioned below. difference

the lifting of a standard

for

stochastic

G

Since

equations

differential

may be internal, solutions have

standard

parts

which

satisfy an associated stochastic differential equation. The criterion in (7.5.6) for continuity plays a role in constructing strong martingale liftings which satisfy the decent path version of Corollary (7.5.3).

(7.5.4) NOTATION: 6t E 1

Suppose

is a positive infinitesimal and

is nonanticipating after

x

(t) = X(6t)

+

16

(7.5.4)EXAMPLE:

Z

Let

Z[Z(W) cr2

:

w

= Z[Z2(w)

E

:

W

W] :

-

*R

= 0.

w E W]/#[W].

6t.

X

We define

E[6X(s)los]. s=6t step 6t

for

t E Ui.

be an internal function with zero mean, and Define a

1 imi ted

*martingale

variance , by

+

6t = min T.

where

41 1

7.5: Notes on Continuous Martingales

Section

I

1'

E[ l6M(t)

In this case

2 = Z (Ut+6t)6t.

l6M(t)I2

but

wt] = a2*6t. so

(t) = u2*t,

[6M,6M]1 6

while

[6M.6M](t) With

some

itself is usually not s o easily computed. extra

[6M,6M]

estimate

&

integrability, Hoover

Perkins

[1983]

a s follows.

(7.5.5) THEOREH:

IF

M

is

S-integrable and

The

following

a

M~

6t-martingale,

sup[stl6M(t)l] t€0

result

uses

= 0.

is

locally

a.s.. then

[6M.aMl]

to

(t)

check

6

continuity.

(7.5.6) THEOREM: Let

M

be a

6t-local martingale.

If

[6M.6M]1

(t)

6

is a.s. then

M

S-continuous and t f is

S-conttnuous.

sup[stlBM(t)l] tEO

= 0

a.s..

412

(7.6) Stable Hartingale Liftings &

Hoover

[1983]

Perkins

show

that

coarse

enough

6t-martingales have the property that all their Stieltjes sums also have a ingredients Example

6t-decent path sample. of

(7.6.2)

necessary.

their

result

helps

explain

Roughly

internal

the

why

into

hyperfinite

coarser

speaking, the

martingale

martingales.

in

This section outlines the

idea

continuous

Since the internal

time

time

is

to

and

framework. sampling

is

decompose

an

discontinuous

line is discrete

this

means we "take a limit." Let

M

be a

6t-local martingale and let

m

be a natural

We define

number.

,t-6t

and

1

' t-6t

Mm(t)

=

6M(s)I s=6t {IbM(s)l>

The conditioned processes

[see

1 ' m)

(7.5.4)]

and

are each

M(t)

*martingales = M(6t)

and

+ [W,(t)-M

,I6

(t)]

+ [Mm(t)-Mm16(t)].

Section

7.6

m

In the l i m i t as jumps of

we expect

m

the martingale,

Mml6(t)

so

[Mm(t)-Mmlti(t)]

is infinite,

to contain all the

should "tend toward a

is an

using Theorem (7.5.6) above.

the limit condition on

MmIti(t)

m

S-continuous process by

[Mm(t)-Mm16(t)]

This means that

tends to a continuous process as

X

Mm(t)

Hoover & Perkins [1983] show that when

continuous process."

If

413

Stable Martingale Liftinas

m + m.

We shall formulate

more precisely.

is any internal process we denote the infinitesimal

X

oscillation of

OX(t)

up to

by

t

= sup[stlX(u)-X(v)l

:

u

=

v

<

tl.

We want to have

'C6Var

-

Mmlti](t)

for each

limited

t.

happen.

However,

there

A t E Hti

in probability, as

0

Unfortunately, is

always

which makes this happen.

TA

coarser time line

m +

this does

a

coarser

A sample of

a,

not

always

infinitesimal

M

along the

produces only decent path sums.

Recall

the special form of a coarser sample of a local martingale from (6.7.6).

(7.6.1) DEFINITION: Let

M

be

a

tit-martingate.

If

LnftnttestmaZ such that

'[AVar

Mm16](t)

-+

0

A t E H6

t s

an

414

Chapter

in probabiLity f o r aLL Limited

M

At-sample o f

Our next

is a stabLe

example

6t-martingale.

Stochastic Integration

then we say that a

t.

At-LocaL martingaLe.

should help

Here is the outline of what

7:

to clarify

the situation.

M

the example contains:

is a

is an internal bounded predictable summand,

G

1G-6M t

yet

=

N(t)

variation

does

= [6N,aN].

has indecent paths.

not

detect

the

A coarser sample of

M

problem

Moreover, quadratic because

[6M,6M]

is stable.

(7.6.2) EXAMPLE: Let

u

:

W + (0.1)

We may think of rate starting at

Recall

that

our

be an internal function such that

as Bernoulli trials with an infinite success

u

t = 1

by summing “successesn as

infinitesimal

approximation

to

the

process in (0.3.7). (4.3.3) and (5.3.8) had a jump rate

Now we have = b*6t , #W

with

b

Z

1

a-

Poisson a

when

415

7.6 Stable Martingale Liftings

Section

6t-stopping time for the first "success,"

Define a

h n.

k

Then the probability of beginning with at least

"failures"

is

PCT-1

>

k*6t] = (1-p)

PCT-1

>

t]

k

or

T

s

means

t at

"success"

limited multiple of

happens

8

7--1

a

Now we define a

<

and

M(t)

=

1 6M(s), p

(-l)k(l-p) 0

Notice

for

6t-martingale using

(-l)k+l 6M(t-6t.o) =

that

[6M,6M](l+t)

6M = 5 f l + o(6t)

=: t

Ii t es ima 1

a nonin

E 01 = 1 .

t

1,

.

within

t = r c ,

To see this, observe that if

t

t/6 t

a. P[O

for

= (1-p)

for

t

r

T .

>

1.

limited, then

Let

<

if

t = l+k6t

,

if

t = l+k6t =

I

if

t

until

= 0.

where

,

>

M(t,w)

T ( W ) T(O)

.(&I).

the first success,

until the first success,

l+t =

T .

so

However,

7: Stochastic Integration

Chapter

416

we have seen that multiple of

1

Z

T

a.s., since

T

is a.s. a finite

a.Hence

*martingale

Next, we define a

1 G-6M. t

N(t)

where

G

=

is deterministic and bounded,

G(l+k6t)

= (-1)

k

.

This makes

aN(t-6t)

= p-1

(-l)k-l(-l)k(l-p)

We know that

p =

6+

noninfinitesimal limited

o(6t)

r

an

N(T) does not have a

first builds up to infinitely near

r

t = 1.

if

t = l+k6t

if

t =

T ( W T

T(O)

<

= l+k6t

= l + r a

=: 1

T(O)

for some

and

= 0

N(T-6t)

N

s

a.s., hence

N(l)

Therefore.

.

k-l(-uk+l P = P

=

2

Z

r

r-1.

6t-decent path sample because i t

and then jumps down by

1

for times

Section

7.6

Stable Martingale Liftinas

N

The quadratic variation o f

l6Ml = 16"

because

for all

417

M.

is the same as that of

t.

Finally, we compute the decomposition

This decomposition is the same for all values o f

2

that

<

we simply let

so

m

6M( s) I

1

i 3)

0

m = 2.

E

*

IN

such

First,

= l+h*6t

,

if

,

otherwise

s

m

<

T-6t

s o that

EC6M(s)I

5)I

WS]

{laM(s)l<

= (-1)

(s-l)/6t p(1-p)

,

if

then (~-1)/6t 6X(s)

0

hence

P

2

.

for

l < S < T

= {(-l)

otherwise,

1

<

S

<

T.

418

X(t)

z 0

Chapter

7: Stochastic Integration

for all

t

The other part of the decomposition of

a.s

M

satisfies

while (s-l)/6t (p-1)p

This means that the variation in steps of

where

A t E U6

satisfies

At

Z

Mm la

At-sample of

M

s

<

T

otherwise.

satisfies

6t

on a time axis

uA*

0, then

AVar Mm16(~) 1 0

and a

if

,

0

On the other hand, i f we sample

,

a.s

makes i t a stable

At-martingale.

general, such coarser time sampling always works.

(7.6.3) LEHHA:

Let

M

inftnitestmal stable

be a A t E Us

6t-local martingale. so that a

At-local martingale.

At-sample

There is an of

M

is a

In

A

419

7.6 Stable Martingale Liftinas

Section

value

of

that makes

At

the

standard

part

of

the

variation equal the variation of the standard part always exists by results in section 7.1.

Such a

At

satisfies the lemma

above. The point of this lemma together with the Local Martingale Lifting Theorem is that every local hypermartingale has a stable At-local

martingale

lifting,

These

are

the

liftings

that

Hoover 8 Perkins [1983] show make internal Stieltjes sums have decent paths.

(This is why we called them "stable.")

(7.6.4) THE HOOVER-PERKINS THEOREM: If

M

Loca Lg

is a stabLe

At-Local martingale and

AM 2 -summabLe process, then

2

G

is a

G-AN

is a

t

N(t)

=

At-local martingale.

This is the main technical result of the Hoover & Perkins [1983]

for

article.

It is applied to give a new existence theorem

semimartingale

equations.

The

decent

path

property

is

needed to show that the internal sum arising from the solution of a

* finite

difference equation has a standard part.

420

(7.7)

Semimartingale Integrals We want to define integrals

for a wide class of integrands and 'differentials'.

About the A

Z

most general kind of process

that we can use for an Ito-

calculus is defined in (7.7.3).

The term "semimartingale" is

not completely standardized in the literature and our use of i t is relative to our own evolution scheme. gale" to warn you of the latter above.) predominant custom and even

(We used "hypermartinOur usage is close to a

sense of humor couldn't bear

out-

'local-semi-hyper' . . . A semimartingale

Z

Z(r)

where

N

is a process which may be written as

is a martingale, with

variation with

W(0)

+ W(r)

= Z(0) + N(r)

N(0)

= 0

W

and

has bounded

= 0. For the time being let us assume that

N2(1)

r

integrable.

The complete carefully developed theory of sections

(7.1) and

E

[O,l]

(7.3) applies

previsible process. and let a

U

where

Let

the

M

be a lifting of

6t-martingale with

W M

lifting

6t-bounded variation for

as in (7.1). and

6t E TA.

[6MS6M](l)

bounded variation with

Z-integration of

to

be a lifting of

At-martingale

and

var W(l)

we work on

6Var U(l)

N

are both

a bounded

as in (7.2-3)

By first choosing

then choosing

we may suppose that

S-integrable and S-integrable.

U

U

with

M has

is a

6t-

Section

42 1

7.7: Semimartinnale Integrals

We may combine the path variation measure for

M

and

U.

Define the weight functions

and

+ 6hw(t).

6pw(t) = 6Kw(t)

In this case

is

p,[U]

(7.1.5) applies

P-S-integrable, so Iterated Integration

to the hyperfinite measure

given by the

u

internal weight function

6u(t,w) = 6pU*6P(o).

The Predictable Lifting Theorems applied to bounded

u

(6.6.8) and

and a bounded previsible process

0-predictable internal

u{st[G(t,o)]

G

(6.6.11) may be

H.

satisfying

# H(st[t].w)}

= 0.

Since this joint total variation measure

u

measures of sections (7.1) and (7.3).

is both

and

G

dominates the path 6Mz-summable

6U-summable and

is well-defined as

of

yielding a

H(O)*Z(O)

plus the decent path projection

422

There is something extra theory

of

Chapter

7:

to prove

in

(7.1) and

sections

Stochastic Integration

this definition.

The

only

the

(7.3)

shows

that

infinitesimal Stieltjes sums give the same answer for different liftings of N. W

€I.

and

Z

The decomposition of a semimartingale plus bounded

variation

terms is not unique.

into martingale

Z

If

is a

Z(r) = J(r)-Xr

martingale of bounded variation [such as

for a

Poisson process as in (5.3.8), (4.3.3), (0.3.7)] then we may view either

Z

N

W

or

is continuous, then

and then are unique.]

[If

as zero in the decomposition above.

N

and

W

may also be chosen continuous

This means that Stieltjes and martingale

integrals must agree when both are defined since classically one defines integrals against trouble

in

the

classical

dZ

by

JdN

approach

+ JdW.

This causes some

which we

at

least

avoid

conceptually since we use infinitesimal Stieltjes sums for both parts

of

the

26X = 26M + 26U.

lifting,

(We do

still

use

separate estimates for the two terms.) The nonuniqueness

in the decomposition

Z = Z(0) + N + W

causes a far more irritating problem when i t comes to trying to identify a space of

(7.7.1) EXAHPLE: Let half of that

p

W

5. 2 3.

:

W

and

-

dZ-integrable processes.

-1

on one

on the other half (see 4.3.2).

We know

{-l,+l}

+1

be internal and equal

2 E U for each 4,***,e m

infinite natural number such that

finite

{G

:

m, m

<

so

let

n} C T.

n

be an

Define a

Section

6t-martingale by

= 0

M(0)

M(t)

and

.

{ P ( w ~ + 1~ ~ )i~f =

6M(t,o)

Since

423

7.7: Semimartinaale Integrals

2'16M1

.

=: H

m

31 <

t =

where

m-l m '

m < n

otherwise.

we

m,

t

= H 6M,

see

M

that

has

bounded

m

variation. as

The semimartingale

X(t) = 0 +

%(t)

+ 0

or

X(t) = G(t) as

may be decomposed

X(t) = 0 + 0 + %(t).

The

deterministic internal function

is not because

m- 1

m ,

if

t = - m .

0 ,

otherwise

m l n

S-integrable with respect to the first variation z'Gl6MI

= Hn

is infinite.

However,

G

ISMI, is

S-

integrable with respect to the quadratic variation,

and

It

turns out

that semimartingale integrability

defined by saying that a process integrable with

respect

decomposition

H

may

H

is integrable if i t is

For another

to .some decomposition. not

be

is well-

integrable.

it

is

integrable, then i t produces an indistinguishable integral.

In

other words, given another decomposition of

Z.

but

if

either we get

424

ChaDter

the same answer or "infinity."

7:

Stochastic Integration

This is the point of (7.7.10).

Strange, but at least consistent . . . .

A simple special case of

this may be proved as follows. Suppose that and that

(Mj,Uj)

Z(r) = 0 + N1(r) lifts

(N..Wj). J associate total variation measures for infinitesimal increments

6tl

+ W,(r)

+ W2(r)

j = 1,2, as above.

for u1

= 0 + N2(r)

and

and

u2

We

to each lifting

6t2.

If

H

is an

almost previsible process satisfying

Then we can find

0-predictable internal processes

G

j

such

that

This just requires a slightly different truncation argument and the bounded that

H

Predictable Lifting Theorem

(6.6.8).

This shows

has liftings for both decompositions.

The decent path property

of

the infinitesimal Stieltjes

sums can be proved in the same manner as the decent path part of the proof of (7.3.8).

Finally. the indistinguishability can be

proved in the style of the proof of (7.3.10).

In fact, these

two parts can be combined in a single proof using 6.6.14) where we let

W

(6.6.12

-

be the set of bounded almost previsible

Section

7.7:

Semimartingale Intenals

425

processes which have liftings for each decomposition producing 6t -decent path j

sums and

indistinguishable

projections.

The

details are left as an exercise. We hope that the separate treatments of sections (7.1) and

(7.3) are clearer even than a combined summable plus

treatment of

"square

For the

integrable variation" semimartingales.

remainder of this section we simply state the full-blown general results needed to define semimartingale integrals by lifting to infinitesimal Stieltjes sums.

Further details must be found in

Hoover & Perkins [1983].

(7.7.2) SEMIHARTINGALE INTEGRALS: Here

a

is

summary

of

the

classical semimartingale integrals. an

S-semimartingale

Z-integrable

process,

then

If

H

and the

of

effect

is

Z an

this with

section Z(0)

= 0

on is

almost-previsible

decomposition

that makes

H

integrable has a

= 0 + M(t)

+ U(t)

The decent

and

H

6t-semimartingale lifting X(t) has a (6M 2 ,6U)-summable lifting G.

path projection of the

Gt-semimartingale

1 G6X t

Y(t)

=

is what we take as our definition of the process

Ji

H(s)dZ(s)

= y(r).

A slight variation on Theorem (7.3.8) shows that the decent path

Chapter

426

Y

projection exists and moreover that

U6.

does on

well-defined path

7: Stochastic Intezration

a.s. only jumps where

X

Theorem (7.7.10) shows that this definition is up

to

projection

indistinguishability, that

is

the

same

no

matter

lifting, which integrand lifting

G,

is.

the decent

which

decomposition

or which

infinitesimal

time sample (subject to all the "lifting" requirements) we take. We

want

to

set o f

variation

up

var W(r.o)

: C0.m)

any

to

W

the paths of

O'S .

x R

r E

*R

[O.r].

a r

<

definition

of

*IRd

with

that

the

process

is defined f o r a.a. paths. to the setting (5.5.4) with

the only formal change that restrictions at need

4

have finite classical or

C0.m).

We extend the notation (7.1.1)

We

x R

: [O.m)

This means that except for a single

locally bounded variation.

null

W

study processes

bounded

r = 1

variation

on

are dropped. each

interval

m.

(7.7.3) DEFINITIONS: We has

say

that

the internal

T x R

* *Rd

(U.6 x Var U)

has a

process

6t-locally bounded variaiton i f

U

6t-decent path sample and its projection is able f r o m

A

(c,var

ndistinguish-

z).

process

z

: C0.m)

x R +R

is

called

a

d-dimensional semimarttngale i f there exist progressively measurable decent path processes

= W(0)

= 0

such that

N

N

and

- Z(0)

with

N(0)

is a local hypermartingale.

has locally bounded variation. and

Z(r)

W

= N(r)

+ W(r)

W

Section

X

An internal process d-dimensional

M(6t)

U(6t)

f

martingale,

:

f x R

*Rd

-P

is called a

6t-semimartingale for an infinitesimal

there exist

if

427

7.7: Semimartinvale Intevrals

such that

0

%

M

internal processes

U

M

and

U

is a stable

is nonanticipating

after

6t-locally bounded uariation, the pair

with

6t-local and has

6t

(M,U)

6t

has a

6t-

decent path sample a.s.. and

X(t)

for

-

X(6t)

= M(t)

+ U(t)

+

t E H6.

X

An internal process lifting

of

measurable

a

is called a semimartingale

process

Z

Gt-semimartingale for some infinitesimal

2

6t-decent path projection

If

X

is a

X

if

6t

is

and if the

is indistinguishable from

6t-semimartingale.

a

the projection

clearly a semimartingale because the projected pair

Z.

2

(8.c)

is is

the required decomposition.

If

Z

is a semimartingale. then there is a semimartingale

lifting

X

for

Z.

(7.7.4) THE SEMIMARTINGALE LIFTING AND PROJECTION THEOREM:

A process

Z

is a semimartingale if and only tf i t

has a semimartingale lifting

Semimartingales differentials'

are

because

a they

X.

"good" contain

class a

of wide

'stochastic class

of

428

Chapter

7:

Stochastic Intepration

traditionally important processes, are closed under integration by a wide class of integrands. and because they are also closed A

under

change

of

variables

in

the

sense

of

Moreover, there are several technical ways

"Ito's

to say that semi-

martingales are the widest possible class of for

example.

see

Metivier

and

formula."

'differentials',

[1980].

Pellaumail

12.12, or

Williams [1981]. p. 68.

(7.7.5) EXAMPLES We would like to indicate how one goes about verifying that the classical

stationary

independent

Z

semimartingales.

Suppose

process, that is,

Z(0) = 0 and

increment

processes

is such an adapted decent path Z(s)

- Z(r)

is independent of

and its distribution is only a function of

3(r)

Z(r) = %(r)

example, we may have

are

(s-r).

For

where

1 6X t

X(t)

for a

* independent

function of

wt+6t

family

=

{6X(t)

: t €

Y}

as in (5.3.21) or (4.3.6).

need not be integrable.

6X(t)

with

The process

a

Z

There is an extensive classical theory

of the characteristic functions of these processes which (4.3.4) hints at. Z

By breaking

or internally with

Z

up as follows (either measurably with

X)

Zb ( r ) = sum of the jumps of

Z bigger than

and

zb(r)

= z(r)

-

zb(r)

b

Section

7.7:

Semimartinaale Integrals

429

Zb

we can show that the characteristic function of so

Zb

that

is integrable.

is a hypermartingale.

Zb

We can also show that

If we let

bounded variation.

is smooth

c = EIZb(l)].

(We may

even

then

[Zb(r)-cr]

Zb

say

has

has bounded

, .

Ixlu(dx) < m for the classical lxl
variaiton i f and only if u

Z decomposes, Z = N+W b = Z (r)+cr; the discussion

would show why

W(r)

and

with is

N(r)

only

= Zb(r)-cr

intended as

background to justify the definition. A

more

modern

justification

to

single

out

the

semi-

martingales is the form of the Doob-Meyer decomposition theorem

Z(r) = Z(0)

that says a decent path submartingale decomposes,

+ W(r),

+ N(r)

N

where

is a local martingale and

W

is

increasing and previsible. We need not use the two decomposition results we just have stated in our development.

If

clearly don’t use.

M = N+W

write

martingale and stating

this

where

W

Now we will state another which we

M

N

is a local hypermartingale, we may is a locally square integrable hyper-

has locally bounded variation.

is

result

to

indicate

why

our

The point of

seemingly

more

general local martingale integrals are no better than a square integrable theory once we combine each with Stieltjes integrals. Let define

X path

be a

Gt-semimartingale lifting of

liftings

of

integrands

analogous to the one in (7.1.4).

relative

Z. to

We shall a

measure

430

Chapter

7: Stochastic InteEration

(7.7.6)NOTATION FOR SEHIHARTINGALE PATH HEASURES: Suppose that

X(t)

6t-semimartingale and as above.

= X(6t) {T~}

+ M(t)

is the

+ U(t)

is a decomposed

M

6t-reducing sequence for

The joint variation process of the decomposition pair

(M.U).

U VM(t,w) = [bM,6M](t,w)

+ 6Var U(t.w)

is finite a.s.. but is not necessarily

,

for

E

U

T b : VM(t.o)

do tend to infinity, [6M.6M](rm)).

lim

n or t

>

t.

bt-stopping times,

n],

for

n

E

*IN,

a.s. (as we see by examining n = U VM(un-6t) n. The unbounded joint

<

and make

variation path measure

>

Y6,

S-integrable for any

However. the internal increasing family of

u n (w) = min[t

t E

qw

of the pair

(M,U)

for

t E Tb

for

t E

is given by the

weight functions

T\T6

for

t € llb

for

t E

T\Hb

and

The internal measures q,[P]

infinite

with

qw

may very well be unbounded, or make

noninfinitesimal

probability.

This

Section

7.7:

431

Semimartinaale Integrals

complicates our use of lifting theorems, but may be technically

*series

remedied by a un.

of truncations with the stopping times

The bounded joint vartation path measure of the

decomposition pair

We know

<

p,(S)

(M.U)

=

p,(U)

is given by the internal series

1 [-2"1 ;{ 1 VM(on(w)} u

:

n E *N].

This measure is analogous to the path measure (7.1).

but the procedure for bounding

of the pair

(M,U)

Yo

measure

u

defined on

as

is the section of an internal is carried on

T

x R.

T6

x R

Y E U

E Ts : (t.,)

in

section

E Y})].

(7.1) ultimately

0

now translate into a.s. properties o f The next proposition shows how to

p,

The

that is,

u-liftings

71,

x R.

although we may consider i t

something about almost surely-lc -almost everywhere,

taking

is more

pa

by

u(Y) = E[p,({t

Just

into

We also define a bounded total variation measure

complicated.

where

q,

of section

p,

works.

7)"

told

us

u-liftings

-

the truncation procedure

ChaDter

432

(7.7.7)

PROPOSITION: Let

then

X

= M+U,

u[Y] = 0

The proof

e t c . b e as a b o u e .

the converse part

Y E Loeb(UxR).

of

w

this proposition

development of

local martingale

integrals in section

u-lifting of a standard process

and then apply the condition

(7.7.8) DEFINITION: Let

= 0 + M(t)

X(t)

+ U(t)

be a

6t-semimarttngale

( d e c o m p o s e d as a b o u e m i t h . 6 t - r e d u c i n g s e q u e n c e

M).

An internal process

G

is

{T~} for

2

(6M , 6 U ) - s u m m a b L e i f

(a)

G

(b)

except f o r a single nulL set o f

a’s,

whenever

is f i n i t e ( t h e u a r i a b l e s

and

t

t

is n o n a n t i c i p a t i n g a f t e r

ouer

is

The importance of the result is seen from the

(7.4). We may take a bounded

H

If

i f a n d o n l y i f f o r a l m o s t all

of

fairly technical. partial

7: Stochastic Integration

U6).

6t.

s

range

Section

433

7.7: Semimartineale Inteizrals

and there is an increasing sequence o f

{a,}

6t-stopping

times

such that

st E [ J Z ( IC(s,o)1216M(s,o)12

<

: 0

s

<

a,(o))

stlG(s,o)l 2dXw(s,o)]

<

I

m.

= E"J[.
Because of Proposition (7.7.7). the proof of (7.4.2) carries over almost intact to show the next result.

(7.7.9) PROPOSITION:

X

Let

X(6t)

2

0

internal

d-dimenstonal

decomposed as aboue.

for a

single

If

F

and

G

are

null

set

of

0's.

for

all

t,

ltF( t)6X( A

6t-sem mart tngale w i t h

2 (61 .6U)-surmable processes a n d

then except infinite

be a

convergence

in

t)

2

ltG( t)6X( t).

u-measure

results

like

(7.4.6) can now be proved for semimartingale sums.

(7.3.7)

and

Chapter

434

M

Since we have chosen

7:

Stochastic Integration

to be a stable

6t-martingale the

infinitesimal Stieltjes sums

have

6t-decent paths. The

peculiar

"integrability"

question

described

after

Example (7.7.1) is taken care of by the next result.

(7.7.10) THEOREM:

X1

Suppose that

= 0

+ M 1 + U1 and

X2

are both decomposed semimartingale liftings

X1

be a

martingale. has

H

If

(AM 2.AU)-summable projections of

of

XI.

is a

Let

At-semi-

is an almost-preuisible process which

(6M 2 ,6U)-summable

a

X2

6t-semimartingale while

= 0 + M2 + U2

u

G2,

u2-Lifting

Yl(s) = 2'

1-lifting

G16X

G1

and

a

then the decent path

and

Y2(t)

=

Zt G2AX

are

indistinguishable.

This

result

is

proved

using

(6.6.12)

-

(6.6.14)

as

described in the special case following Example (7.7.1). The whole standard theory of semimartingale integrals is summarized by

(7.7.11) THE STOCHASTICALLY INTEGRABLE LIFTING THEOREM: Let

X(t)

be a

d-dimensional

which is a lifting of the almost preuisibte process

6t-semimartingale

%-semimartingale

H

: C0.m)

x

R

+

Z(r). IR kxd

An

has a

7.7: Semimartingale Integrals

Section

435

n

0-predictable decomposition pathwise terms

X(t) = 0

Stiletjes

of

+ W(r).

(bML,dM)-summable

the

N =

+ M(t) + U(t)

integrals

projected and

with respect

<

for all

and there is a n increasing sequence o f {p,}

such that

p,

00

An almost-preuisible process

provided

decomposition

these

the

to

the

Z(r) = 0 + N(r)

satisfy:

~iIH(s.w)lIdW(s.o)I

-

below

for

if and only i f the

decomposition

W =

G

Lifting

a.s.

H

r

a.s.

9-stopping

times

while

is

called

conditions

a hold

Z-integrable

for

some

Z(r) - Z(0) = N(r) + W(r).

In this case the stochastic integral

is well-defined (up to indistinguishability) b y the decent path projection o f