1Research sponsored in part by the Office of Naval Research under Contract N00014-67A-0321-0002, Task ~ffi042214 with the University of North
Carolina.
ON THE EXISTENCE AND PATH PROPERTIES OF STOCHASTIC INTEMRALS
by
Olav
Vallen~erg
Depaptment of Statistias
Univepsity of Nopth CapoZina at ChapeZ HiZZ
Institute of Statistics Mimeo Series No. 909
February 1974
l
ON THE EXISTENCE AND PATH PROPERTIES OF STOCHASTIC INTEGRALS l
by
Olav Kallenberg
Universities of GBteborg and North Carolina
SUMMARY. We study integrals of the form Yet) = J: VdX, t > 0, where X is
a process with stationary independent increments while V is an adapted previsible process, thus continuing the work of
Ito and Millar.
In the case
of vanishing Brownian component, we obtain conditions for existence which
2
are considerably weaker than the classical requirement that v be a.s.
integrable.
small
...
t,
We also examine the asymptotic behavior of Yet)
for large and
and we consider the variation with respect to suitable functions
The latter leads us to investigate non-linear integrals of the form
f.
Jf(VdX).
The whole work is based on extensions of two general martingale-type
inequalities, due to Esseen and von Bahr and to Dubins and Savage respectively,
and on a super-martingale which was discovered and explored in a special case
by Dubins and Freedman.
lResearch sponsored in part by the Office of Naval Research under Contract
NOOOl4-67A-032l-0002, Task NR042214 with the University of North Carolina.
AMS 1970 subject classifications.
60G45, 60J30.
Primary 60H05; Secondary 60Gl7,
Key words and phrases. Stochastic integrals, existence and path properties, processes with stationary independent increments, martingales.
J
2
1.
Introduction. The theory of stochastic integrals goes back to the early
work of Paley, Wiener and ltd (see McKean (1969)), who defined the integral
i
VdX in the case when X is a Brownian motion while V is an adapted
fo
[17] random process with paths a.s. in L [0,1], i.e. such that
2
2
V
(V(t))2 dt < m a.s.
J: ~ J:
Later on, Doob, Courrege,
Do1~ans-Dade
(1.1)
and Meyer (1970), and Millar (1968,
1972) extended the definition to quasi-martingales
X and previsible [20]
processes V satisfying
J:
2
V dA
~
J:
(V(t))2dA (t)
< m
(1.2)
a.s.,
where A is the natural increasing process associated with
also studied the path behavior of the process
case when
V is a.s. bounded while
J:
X.
VdX, t , 0,
Millar (1972)
in the special
X has stationary independent increments
(in which case (1.1) and (1.2) become equivalent), and he showed in particular
how several properties known for
The investigation of the case when
X has stationary independent incre-
ments is carried further in the present paper.
tions for the existence of
I
J VdX.
X itself carryover to
VdX,
In Section 3 we give condi-
which in particular cases are shown to
be the best possible and which, in case of vanishing Brownian component, are
substantially weaker that (1.1).
that, if X has index
then
I VdX
In particular, it follows from our results
8 < P in the sense of Blumenthal and Getoor (1961),
can be defined as a random process in
V whose paths lie a. s. in
t
asymptotic behavior of
L [0,1].
p
fo VdX,
D[0,1]
Section 4 is devoted to a s.tudy of the
both as
t
+
0 and as
t +
further restrictions on V), and of the continuity for fixed
ping V +
f VdX.
for any process
00
(without any
X of the map-
In the concluding Section 5, we give some variational
3
results which lead us to introduce non-linear stochastic integrals of the
f f(VdX).
form
The whole work is based on extensions of two general inequali-
ties, due to Esseen and von Bahr (1965) and to Dubins and Savage (1965)
respectively, and on a super-martingale which was discovered and explored in
a special case by Dubins and Freedman (1965).
These extensions are given i.n
Section 2.
The following function classes will be used throughout the paper:
F
= {f:
R+R ;
+
fFO, f
even and continuous with
-
Fl = {fe:F: f concave on R+L
F = {fe:F: f absolutely continuous,
2
F = {fe:F: froo (f(u)) -1 du < ro}
3
£1
f(O)=O, ft
concave on
R+
on R+},
with
fl
(0+ )=O},
.
2.
Some results on martingale-type sequences. In this section, let
AO C Al
c •••
be a-algebras in the sample space, let the sequence
{~n}
of
{A},
and put Xn = ~l + ... + ~ n , ne:Z + .
n
Our first lemma extends a result by Esseen and von Bahr (1965) (cf. [2]).
random variables be adapted [17] to
Lemma 2.1.
ne:N.
a.8.~
Suppose that either fe:F l ,
or that
fe:F 2 and E[;n+ llx]
n =0
Then
n
,
ne:N .
L Ef(~.)
J
j=2
Note that the factor 2 in (2.1) may be replaced by 1 for
Ef(Xn )
2 2- p
for
s; Ef(~l)
(2.1)
+ 2
f(x) ~ Ixl P , pe:(1,2] (cf. [2]) .
fe:F l
and by
Similar improvements are possible
in all subsequent formulae based on (2.1).
Proof.
fe:F
2
The case
and
fe:F
being trivial, we need only consider the case when
l
E[~n+llxn] = 0
a.s., ne:N.
If we can prove the elementary inequa-
lity
f(a+x)
S;
f(a)
+
xff(a) + 2f(x),
a, xe:R,
(2.2)
4
For a, x
then (2.1) will follow as in [2].
= f:[f'(a+U)-f'(a)]du ~
f(a+x) - f(a) - xf'(a)
and (2.2) follows.
F(x)
Then F'(x)
so
F'
f(O)
~
=-
0 on
= 0,
Next define, for fixed
= f(a)
(O,a),
so we get F
f(x)
~
x
f(a)
o.
~
is concave while
= f'(a-)
F'(O+)
[O,a].
~
x € [-a,O].
x.
on
0
[O,a],
Hence for
proving (2.2) for
F
~
But
since otherwise
R+,
x > a
= - f'(a) + 2f ' (x) + fl(a-x)
= [f'(x) - f'Ca)] + [f'ex) -
we get
0,
~
x
f
+
0 on
~
f'(x-a)]
~
o.
[a,oo), proving (2.2) for
This completes the proof of (2.2) for· a ~ O.
a.
S -
+
F'(a-x)
f(x) ,
and hence F is non-decreasing on
F'(X) + f'(x)
Since F(a)
+
~
> 0,
must be non-decreasing on
for large
< 0
a
J:[f'(U)-f'(O)]dU
- xf'(a) + f(x) - f(a-x),
f'(a) + f'(x)
Finally note that f'
0, we get by concavity
~
Its truth for
a s 0
follows by symmetry.
Our next lemma extends a result by Dubinsand Freedman (1965) •. Here and
below,
is to be interpreted as
0/0
Lemma 2.2.
Let g€F
O.
and suppose that eithep f€F
3
{Xn } is a local maptingale [17].
l
op that
f€F 2 and
Define
n
Y =
n
LE[f(~·)IA.
1]'
J
J-
f(x)
Q(x,y) = g(y) + c
whex>e c
= 2.
Then
eaah x€R and y>o,
Proof.
E~
=0
Let
Q(x
oo
fy
du
g(u)'
+ X , Y + Y ),
n
n
and also fox>
x€R and y€(O,oo],
in the case f€F
(2.3)
j=l
let
and put
n€Z,
+
x = y
~
x€R,
1:S
y~O,
(2.4)
a supex>-maptingale fox>
= o.
be a random variable such that
Ef(~)
= a.
If y + a
= 00,
then
0,
5
Q(x +
~,
= 0,
y + a)
and it follows trivially that
But this inequality is also true for
EQ(x+~,
If x
here.
= 0,
y +a <
EQ(x +
~,
y_+ a) S Q(x,y).
since Lemma 2.1
00,
f(x)
2a
2 foo
du
y+a) S g ( y+a ) + g (y+a ) +
y+a ----(
g u)
< f(x) + 2 foo du
{f y+a du
- g(y)
Y g(u) - 2
y g(u)-
it follows by monotone convergence that we may even take
y
=0
Induction completes the proof.
Let f, g, {X} and
Corollary 2.1.
n
{y}
{Xn } is a.s. aonvernent
on {Y00 < oo},
;;:J
n
be suah as in Lemma 2.2.
while on {Y
00
= oo},
Then
Xn /f-lg(Yn ) ~ 0
a. s.
Here and below it is understood that
f(x)
= IxI P ,
p€(O,2],
= sup{y~O:
f- 1 (x)
g(y) ~ y(log y)c, c > 1,
and
fey) S x}.
For
the last result is
close to one in Loeve (1963) page 387 (cf. [2]) which is derived under more
restrictive conditions.
In particular, we get for
p
= 2,
in the notation
of Meyer (1972),
X
1/2 n
c
{X'X)n (log(X,X)n)
~
0 a.s. on
{(X'X)oo
= oo},
c
> 1/2 •
Note that Dubins and Freedman (1965) (cf. [17]) prove the corresponding
result with (X'X)n
Proof.
On
{Yoo <
oo}
in the denominator.
it follows from Lemma 2.2 that
a. s.
ting
= oo} ,
·it follows from the same
n€N we now define
n
gl (u) = (2/3) g(u), U€[t n _1 ,
For large
converges
{Xn } since x is arbitrary. On
lemma that f(X n )/g(Yn ) ~ some p < 00
-n
t > 0 by foot (g(u)) -1 du = 3 . Putn
n
t ), n€N, and ·h(y) = suP{gl (u); usy},
n
a.s., so the same thing must be true for
{Y00
{f(x+X)}
n
6
it is seen that even
abcve for
he:F3'
But since
g.
Finally use the fact that
Lemma 2.3.
so we get
h(y)/g(y)
or
f(2x)/f(x)
f(Xn)/h(Y ) -+- some pI < co a. s OJ as
n
0 as y -+ co, this implies p = 0 a.s.
~
4, x > 0,
to complete the proof.
Let f, g, {X} and {Y} be such as in Lemma 2.2.
+~p g~~~~:: ~ e} f J: g~~)'
Then
e> O. Y ~ O.
S
and in paptiouZa;p
P{sup[lf(X ) ,lIp _ aY ] ~ b} ~
2 ~
n
n
n
(p-1)abP- l '
a, b
>
0, P
>
1.
(2.5)
Thi.s is essentially an extension of a result by Dubins and Savage (1965)
2
(see also [6, 17, 20J), who consider the special case f(x) ~ g(x) ~ x • The
proof is similar to that of (29) in [6J, once Lemma 2.2 is established.
trary to the Dubins-Savage inequality, (2.5) is void for small
ever, it may be shown that, if f(x) ~
Ixl P
in (2.3),
P{sup(X - aY ) ~ b} S [1 + c a 1/(P-l)b]-P+1,
p
n
n
n
(Here and below,
1
ab P - •
ConHow-
pe:(l,2], then
a, b > O.
cp denotes some positive constant depending on p only).
This follows as in [7, 17J from the following result of some independent
interest, (cf. Doob (1973) for the case
p
= 2).
We omit the elementary but
tedious proof.
Lemma 2.4.
Let
~
be an integrabZe random variabZe and define fop pe: (1,2]
.1(l-X)-P+l.
p, .
Then
x < 0,
x
~
o.
7
3.
Existence of stochastic integrals. For the remainder of the paper, let
X be a random process in 0[0,00)
.=.
XeO)
(even
and define the
0,
€
=0
10gEeiUX(1)
.=.
or
00
iu\ -
L~vy
with stationary independent increments and
measure
t u2a2 + . r
oo
I
f(ux)A(dx) +
and the functions
to
[-€,€], €
>
O.
...,
f,
""
f,
€
(e
{a2fll(0)u2/2
iux
-1) A(dx).
= 0,
if f'(O)
if
>
€
(3.2)
defined in the same way w.r.t. the restrictions of
Let us further suppose that At' t
is At - measurable while the process
~
0,
X(t+s) - X(t), s
~
0,
A
is an increasing
t
~
0, X(t)
is independent
By V with or without subscript we denote a random process on R+
which is adapted to
{At}
and previsible [20].
We shall say that
V is
simpZe if it is a step process with all discontinuities belonging to some
non-random finite set.
For brevity, we shall use the short-hand notation
indicated by (1.1) and (1.2), and we shall often write
for the jump size of X at
I
X{t} = X(t) - X(t-)
t.
Following ltd (see [16]) and .Mil1ar (1972), we shall define the integral
VdX by a limiting procedure, starting from simple V (for which
defined in the obvious way as a finite sum).
f VdX
For this purpose we need two
lemmas, the first of which extends a result in McKean (1969), page 23.
Lemma 3.1.
J:foV
<
IlO
Let f€F with
a.s.
0
(3.1)
a=O,
family of a-algebras in the sample space such that, for fixed
of At'
€
+'xi>€
defined by
holf'(o+)lul
_1lO
r
(eiux_l_iux)A(dx)
we associate the function
feu) =
,y,
if possible), by the formula
Ixis€
To any f€F,
a and
A and the parameters
limsup f(2x)/f(x) < 00, and suppose that
x-+oo
Then there exist some simpZe adapted prooesses VI' V2 '
is
8
suah that
fo(V
OO
fo
f:
EI: foV <
If
00
foVn
the V
n
+
f:
- V) + 0 a.s.
n
foV a.B"
(3.3)
may even be ahosen so as to satisfy (3.3) and
in LI •
(3.4)
Proof.
By an obvious truncation argument, we may restrict our attention to
processes V with bounded support, say in [0,1].
f
(3.4)
t<R+,
is strictly increasing on R
+
Let us first assume that
with
f(2x)
~~g f(x)
=c
<
00
(3.5)
•
Then
f(x+y) s cf(x) + cf(y),
since for
x, YER+
f(x+y) s f(2(xvy)) s cf(xvy)
Let us further suppose that
Vx(t)
as
(3.6)
x, yER,
~ Vet-x).
= c[f(x)vf(y)]
S
c[f(x)
+
fey)].
V is non-random and non-negative, and put
By mean continuity ([11] page 199),
f1fOVx - fovi + 0
x + 0, so we get
1
limsup JIh Jh0 foV xdx - fovi S limsup
h+O
h+O
and in particular
Uh
in Lebesgue measure.
{foU }
h
yeidls
~
kJhdX JlfOVx - fovi =
1 Jh
f -1 (h
0 foVxdx) + V
(3.7)
(3.8)
Furthermore, (3.7) implies uniform integrability of
(c.f [15] page 162), and hence by (3.6) of {fo(Uh-V)},
lim sup
h+O t>O
0,
0
IJ0t foUh - It0 fovl
= 0,
lim jfO(Uh-V) = 0
h+O
so (3.8)
(3.9)
9
For general non-random V,
we may approximate V+
= vvo
= -VAO
and V-
the same way by processes
in
By (3.6),
{foUh } is still uniformly integrable so (3.9) remains true.
tE:(j/k, (j+I)/kJ, j€Z, k€N,
We next define
and conclude from uniform continuity that
~~: ~~b IJ: foUhk - J: fOUhI
~~ I fo(Uhk -
= 0,
By (3.6), (3.9) and (3.10),
t
lim limsup sup IJ fOU hk - Jt fovi
h+O k+oo t>O
0
o
= 0,
Uh)
lim limsup
h+O k+oo
= 0,
h > 0 •
Jfo(Uhk - V)
(3.10)
= 0,
and hence, returning to random V and letting n€N be arbitary, we may choose
hand
k such that Vn
P{~~b
~
II: foVn - I: fovi
Uhk
>
satisfies
p{I fo(Vn-V)
n
n
2- } < 2- ,
>
n
n
2- } < 2- ,
Thus (3.3) and (3.4) follow by the Borel-Cantelli lemma, while the last
assertion follows by uniform integrability, since by (3.6), for
foU hk
= t ? fOUh(j/k) ~
= h~
J
f J:
f~
[foU;(j/k) +
J
[foV;Ci/k) +
h-I, k,
foU~(j/k))
foV~Ci/k)ldx = ~k f J: foVxCi/k)dx = cIfo
To get rid of the assumptions on f,
let a > 0 be such that
f(a) > 0
A
and let
f
A
Then f
be the minimal even function
is easily seen to be continuous at
for some
b€(O,a).
Ixl > b,
and note that
Next construct
g belongs to
strictly increasing on R+,
J:
goy
<
~
g(x)
which is concave on
2: f
0,
from
[O,a).
and in particular f' (b) > 0
f(x) by adding l_e b- 1xl for
F and satisfies (3.5), and that
Since clearly
I:
foV
<
~
g is
a.s. implies
a.s. and similarly for the corresponding expectations, the con-
clusions of the lemma hold with
g in place of f.
This proves the asser-
10
tions involving (3.3), while those involving (3.4) carryover to
f
by uniform integrability.
Let f€F be strictly positive outside 0 and satisfY
Lemma 3.2.
limsup f(2x)/f(x) <
x-+co
00.
Then
II
0
l
fo
Proof.
let
f
If f
foV'
n
P
fo(Vm-V n )
-+
~
0
implies
m, n
0,
-+
00
(3.11)
•
satisfies (3.5), then (3.11) follows by (3.6).
be such as in the preceding proof.
l foV
~ f
is
P 0 implies
fo
(3.5), it is enough to prove that
f
Since
For general
f,
and satisfies
JIAfoV
P 0,
and here
nOn
it clearly suffices to consider non-random V. But in that case, the asser-+
-+
n
tion follows easily by continuity and uniform integrability.
We may now state the main result of this section.
If
Theorem 3.1.
t
Jo
,. ,
f l oV <
00
a.s.,
there exists some a.s. unique process
t
>
0,
~
Jt VdX, t
fo VdX/ ~ 0,
2
where
f(x) ~ x Alxl,
then
in D[O,oo)
such that
0,
S
suplfs V dX sst 0 11
t > 0,
(3.12)
for any sequence {Vn } of simple processes satisfying Jt
o
(the existence of which is ensured by Lemma 3.1).
a.s.,
f
t > 0,
g(x) ~
replaced by g,
and equals
f VdX
Ixl
A
1,
If
0=0
f
1
o(V -V)
n t
and
f
o
~
t > 0,
0,
goY <
00
then the above conclusion remains true with
and furthermore,
the Lebesgue-Stieltjes integral exists
a.s.
The last assertion extends with essentially the same proof a result by
Doleans-Dade and Meyer (1970), and by Millar (1972).
As will be seen from the
proof, this is the only part of the theorem that requires previsibility.
any fixed
X with
0
= 0,
For
Theorem 3.1 yields a larger class of integrable
11
processes than the one defined by (1.1) (cf. [20]). In fact, since
,..,
2
fl(u) = o(u) as u ~ 00, it is easy to construct numerical functions
J
I:
S~Ch that
o Ivi
<
fl,Y <
a.s., t
00
>
Similarly, the condition
00.
is too restrictive in the case when
0,
bounded variation and
I:.2.
while
00
Yo
,..,
= 0,
since then
g(u)
integrable whenever
or
(J
p
< 1).
X is stable
If
exists for any V
and the same processes
V are seen to be
X is such that
(if P
= YO = 0
L ,
= o(u).
X has
J VdX
of index p, it follows from the theorem that
whose paths lie a.s. in
I:
v
IxIPA(dx)
<
00
and
(J
=0
(if P
< 2)
We finally point out the interesting fact that,
for subordinators, the function
g
above may be replaced by the exponent
- log Ee -uX(I) = u yo + Joo0 (l-e -ux ) A(dx),
u ;:: 0,
occurring in the canonical representation of X (cf. (3.1)).
Proof.
We may clearly assume that
[0,1].
By a simple truncation argument [19, 20), we may further assume
that
A has bounded support.
V and all the
V
n
have support in
In the proof of the first assertion, we may
,..,
= 0 by [16], and since u/f (u) is bounded as u ~ 0 0 , we
l
may finally assume that EX(t) ~ O. For convenience, let us replace the
also take
function
(J
f
2
above by the equivalent function
clearly belongs to
F
2
and is such that
f(x) = x /(1+lxl),
f(~)
which
is concave on R.
+
If V
is a simple process of the form
k
V
= .L
J=l
a .1 (t .
t .) ,
J
J-l, J
we get by Theorem 4.1 of Millar (1971), for any
i
i=l
j€{l, ... , k},
f
E{f(a.[X(t.) - X(t._l)))IAt
(t.-t. 1) foof(a.x) A(dx)
}S 2
1
1
1
i-I
i=i 1 1_00
1
i
= 2i=l
(t.-t. 1) l(a.)
1
1-
1
=2
(3.13)
t. ,..,
J foV,
Jo
12
so by (2.5) with P = 2, we obtain for any finite subset T c R ,
+
t
1 2
P{SUp1f(J VdX) 1 / ~ 2aJ1 foV + b} s ~,
a, b > 0,
t~T
0
0
a
which extends to
r = R+
by monotone convergence and right continuity.
Returning to the case of general
from Lemma 3.2 that
(3.14)
V and approximating
{V},
conclude
n
lo'"
fo(V -V ) +P 0
f
as m, n + 00. For any € > 0, we
m n
may therefore choose some nO ~ N such that p{Jl fo(V -V ) > E 3/32} S €/2
o
m n
whenever m, n > nO' By (3.14), we get for such m and n
t
P{SUP/f(J (V -V )dX) ,1/2 ~ €}
tOm n
S
P{SUP/f(jt (V -V )dX) 11/ 2
m n
t
0
~
162 J1 fo(V -V ) + 2€} + -2€ S
€
0
m n
E,
proving that
sup / t V dX tOm
f
Jt
0
V dX I +P 0 as m, n
+
00
(3.15)
•
n
By restricting m and n to some suitable sub-sequence
N'
c
N,
it is
even possible to obtain a.s. convergence in (3.15), and so there must exist
some process Y in 0[0,00) satisfying
rt
.
supIJ·· V dX - Yet) / -+ 0 a.s.,
tOm
(3.16)
m € N'.
The proof of (3.12) is now completed by combining (3.15) and (3.16).
see that any two approximating sequences
{Vn } and
{Vi}
n
To
lead to the same
z,
it suffices to consider the mixed sequence VI' Vi, V2' v
The
proof of the second assertion is similar, except that Millar's Lemma 4.2
limit~
(1971) replaces his Theorem 4.1.
The following simple property of
Section 5.
J VdX
will be needed here and in
13
Wi th pr-obabi Zi ty one 3
Lemma 3. 3.
ft VdX = V(t)X{t},
t
o.
~
(3.17)
t-
Proof.
f VdX has a.s. no
By the uniform convergence in Theorem 3.1,
jumps outside those of
independent processes
in (3.1).
X.
For fixed
X
and
e:
Then clearly fVdX
X'
e:
e: > 0,
wher~
= JVdX~
+
write
X as a sume of
X'
corresponds to the. last term
JVdX~
a.s. (cf. Proposition 2.1 in
£
[20]), and here the last ,oterm'may be interpreted as a Lebesgue-Stieltjes
integral, so with. 'probability 1 we get for all
t VdX
J tSince
e:
= Jt
VdX'
t-
£
t ~ 0 with
IX{t}
I' >
e:
= V(t)X'{t} = V(t)X{t}.
£
is arbitrary, this completes the proof.
For fixed non-random V,
a random process in
property of
X.
the existence or non-existence of
f VdX
as
satisfying (3.17) is an interesting sample path
0[0,00)
We shall consider a simple case when it is possible to
obtain necessary and sufficient conditions for existence in this sense.
Theorem 3.2.
v'
< 0
Let v: [0,1]
~ 1,
and vel)
-
or- that (] =
1
o=
Y
<
-+
R+
be absoZuteZy continuous on
and suppose that either-
1"
~:~n
cr
=°
J vdX
.with
and
f tv'(t)
l'
tv'(t)
1
vet) ~ ~:gup vet) < - 2'
(3.18)
° and
' . f tV'(t) < l'
tv'(t)
1
- 00 < l ~:~n
vet) - ~~up vet) < - •
Then
(0,1]
exists as a r-andom process in 0[0,1]
1
f
1
1
v- (lxl- )}"(dx)
(3.19)
satisfying (3. 17) if1
< 00 ,
-1
and in that case it aZso exists in the sense of Theorem 3.1.
(3.20)
14
Thus, in this particular case, the sufficient conditions for existence
given in Theorem 3.1 are also necessary.
For Brownian motion, a proof of the
corresponding statement may be found in [16J. Applying Theorem 3.2 to the
l
product vw where vet) ~ wet) ~ t- / 3 , we obtain a counterexample to the
claim made by Millar (1972) on the top of page 312.
Proof. Suppose that
f(x) ~ x
2
lxi,
/I
and that (3.18) and (3.20) hold.
a = 0
If
we get
1
=
II
2
x A(dx)
J
v
2
+
v-l(/xl-l)
-1
and from (3.18) it is easily seen that
l v2 _- 0(u[V(u)]2),
o(uv(u) ),
Ju
u + 0,
so by (3.20),
l""
.Jl -1 ,-I
Jo f10v = O( -1 v
J vdX
proving the existence of
(Ix
)A(dx))<
00
,
in the sense of Theorem 3.1.
argument proves the existence under (3.19) when
a
= YO = O.
A similar
Conversely,
when (3.20) is false, it follows by the arglment of Fristedt (1971) page
180 (see also Theorem 2.3 in LI2]) that with probability
I
for infinitely many tE(O,l], and so
4.
1,
v(t)IX{t}1 ~ 1
vdX cannot exist.
Asymptotic properties. We first extend Lemma
2.2 to stochastic integrals:
Let gEF 3, and suppose that eithep a = 0 and fEF 1 op that
0 and fEF . Define Q by (2.4) with c = 4. Then
2
Lemma 4.1.
EX(t) ~
Q(x +
f:
VdX, y +
is a supep-maptingaZe for evepy
f:
f'V),
XER and y>O,
t , 0,
and aZso for
(4.1)
x
= y = o.
15
Proof.
R+,
EX(t) ~ 0 and
Let
f€F 2 •
Since f(lK)
is clearly concave on
the assertion follows for simple V by proceeding as in (3.13) and
applying Lemma 2.2.
In the general case, choose simple VI' V , ...
,..,
that (3.4) (with f
EQ(x +
Since
f
r
in place of f) and (3.12) hold a.s., and note that
V dX, y +
o n
and
such
2
r
o foVn)
g are continuous and
Q(x +ft V dX, y +
It
fOV )
o non
~ Q(x
y, t>O,
xER,
Q(x,y),
S
g (y) > 0,
(4.2)
we further have
+ ft VdX, Y +
0
nEN.
,..,
t
Jo foV)
a. s.,
n
so by Fatou's lemma, (4.2) remains true with V in place of V.
= y = O.
a
The proof for
00,
By mono-
n
tone convergence, we may even take x
~
=0
and
fEF 1 is similar.
We may now extend Corollary 2.1 as follows:
Th~orem 4.1.
Let gEF 3 and suppose that eithep a
Then
{f-1g(I: foy)}-l
as
t
~
0,
and this is aZso true with
further assume that EX (t)
converges a.s. on
Proof.
f:
~
t
~
00
~
feF 2 ,
and fe F1
t1zat
01'
0 a.s.
f replaoed
0 in case
{f: fOV < oo} ~hiZe
The assertions for
YdX
= 0
(4.3)
by
f,
E:
then as
t
If we
e:>0.
~
1-
00,
f~
VdX
{I: foy • ~}.
(4.3) hoZds on
follow as in the proof of Corollary 2.1
by applying the a.s. continuity theorem for super-martingales ([15] page
526).
The same theorem applies for
t
~
0 provided
EX(t)
~
0 when
feF 2 ,
so it remains to consider the case when fEF 2 and X is arbitrarily
centered. Let X' be the process obtained from X by deleting all jumps
of sizes outside
[-a,b]
for fixed
a, b > 0,
and put
c
= EX'(l).
16
Suppose that the corresponding truncation of
Applying (4.3) to the process
X'(t) - ct, t
l~~UP {f-1g(f: ;OV))-l
II:
since the limiting behavior of X as °t
If
0,
A ~
~
1f
~
0,
(4.4)
VdX' - c Jt
0
vi
= 0,
0 is independent of the large
it is possible to choose a
ways leading to different values of c.
yields a.s.
t
0
f to f' s f.
I: vi
VdX - c
slim {f-lg(Jt f'oV)}-l
t~O
0
jumps.
A changes
and
b in two different
Applying (4.4) to the correspond-
ing processes, it follows that
{f-1g(f: ioV))-l
J:
and (4.3) follows by combination with (4.4).
~
(4.5) in the case
2
f(x) s x f"(0)/2,
A = O.
(4.5)
a. s.,
V+ 0
We finally have to prove
Since clearly g(x)/x ~ ~ as
it is enough to assume that
g(x)
=x
x ~ 0 and
and
= x2f"(0)/2.
f(x)
But in that case, (4.5) is a consequence of Schwarz' inequality.
Por V ~ 1, Theorem 4.1 contains Millar's Theorem 4.3 (1971), which iu
turn improves some classical results of Blumenthal and Getoor (1961).
It
is interesting to observe that, for stable X and V ~ 1, (4.3) is r.lose
to the results by Hincin (1938).
for
t
~
For arbitrary V and
0 extends Millar's Theorem 3.1 (1972).
our result
Note that the strongest
possible conclusions from (4.3) are obtained by choosing
xllog xl.
X,
g(x)
close to
If we content ourselves with convergence in probability, it is
sometimes possible to get rid of this logarithm (cf. [19, 20]):
Theorem 4.2.
l
J-1
Ixl A(dx)
Suppose that
=
00
eithe~
and f€F 2,
YO = a = 0 and f€F l o~ that
and Zet us f~the~ assume that
0
= 0,
17
(i)
(ii)
Then
eitheJ:' 1imsup IV(t) I
t-+O
either liminf IV(t)1
t-+O
lim sup P{f(sup
t-+O c>O
sSt
lIS
VdXI)
0
~
<
00
a.s.
O!'
xf' (x)/f(x) t
on R ,
>
0 a.8.
OP
xf' (x)/f(x)
t
on R+ •
E:,
t5 > 0,
Ito f e: oV
6c,
= 0,
s c}
+
(4.6)
and if X and V Q:!1e independent" then even
I
I
{f -1 ( t -f oV)} -1 sup.
VdX -+P 0 as
o e:
sst 0
lIs
t -+ 0,
~fillar' s
We need the follwoing extension of
e: > O.
Property
(4.7)
Z.3 (197Z),
which
may either be proved directly from Lemma 4.1 or by a limiting argument £ron
Lemma
Z.3.
Lemma 4.Z.
Let X, f
and g be such as in Lemma 4.1.
f(JtVdX)
f
Then
00
ot
P{sup
~ c} s -4
t
c
. g(y+ ofOV)
f-
du
-()'
c > 0, Y
~
(4.8)
0,
y g u
and in paPticuZaP,
P{SUp[lf(ItoVdX)11/P Proof of Theorem 4.2.
aJtofoV]~b}
Take f€F
Z
II XA(dx) =o
s
4 pI'
(p-l)ab -
obtained from
IOXA(dX)
=
00
(4.lu)
•
-1
AI' hZ' •••
of h such that
and such that the corresponding processes
X by deleting large jumps have all mean
Xl' X2 ' •.•
O. (If A has
point masses, a randomization may be needed to construct these
the corresponding functions
f
by
(4.9;
and let us first assume that
It is then possible to choose truncations
JXZAn(dX) -+ 0
a, b > 0, P > 1.
fl ,
fZ'
Let
t5 > 0
xn .)
Denote
be arbitrary.
18
If
o < liminf IV(t) I S limsup IV(t)1 < ~
a. s. ,
t+O
t~
we rna.y choose m,M and
t
l
0 such that
>
PA=P{msIV(t)ISM, tStl}t::l-o.
~lrthermore, since
f
fn
and the
f n (u)/f(u)
uniformly in
[m,M],
so for
tSt
Finally, choose
4It
1
0
t2~(O,tl]
are non-decreasing on R+,
as
+ 0
we have
(4.12)
n·~ ~
nEN large enough,
fOV s 02
Jt f ovlJt
sup
(4.11)
n ,
on A.
(4.13)
0
such that
P{Xn(t) = X(t), t S t 2 } t:: 1 - O.
(4.14)
Applying (4.8) with y = 0 g = 1/l ,02 ]' (which is clearly a perro c
missible choice although gt F3), and using (4.11), (4.13) and (4.14), we
obtain for
t
P{f(sup
sSt
s t2
IfS
I
-t
VdX/) t:: oc,
0
S
S
uniformly in c > 0,
J
o
foV S
c}
If
VdX
P{f(sup
sst
P{sup
sst
f(fS
0
5
0
I)
that
o<
fCux)/f01Y)
x < y.
ec,
ft f
oV S 02c}
VdX )/gCJS f oV) t:: ec} + 20 S
nOn
proving (4.6) for
€ =
ment yields the same result for arbitrary
If xf'(x)/f(x)
t::
nOn
E
00.
>
>
20
4~2c
c
+
20
= 60,
An obvious truncation argu-
O.
is non-decreasing in x > 0,
is non-increasing in u
+
it is easily verified
0 for fixed
Assurningfor simplicity of writing that
An(dx)
x and
~
y
with
ACx)lr_a,b]'
19
we have
f:f(UX) A(dx)
fn(U)
f~.f(UX)A(dX)
.......,...- ~ - . ; . . - - - - - + - - - - - -
fiu)
J:f(U Y) A(dy)
= Jb {J~
o
b
J~:f(UY)A(dY)
f(uy)A (dy) }-1 A(dx) +
f(ux)
JO
-a
{J-a f(uY)A(dy) }-1 A(dx)
f(ux)'
_(lO
and here the right hand side is non-increasing in u > 0 and tends to
as a, b
~
0 for any fixed u > O.
extends to any interval
above argument.
e
[m,oo), m > 0,
= 2,
and we may take M =
(lO
in the
If exactly one of the integrals in (4.10) is finite,
X, X'
and
X"
apply (4.6) to the processes
c
Therefore, the uniformly in (4.12)
The symmetric argument applies to the case of non-increas-
ing xf'(x)/f(x).
suppose that
0
are independent and distributed as
X.:!:.. (X' - XH).
it follows that (4.6) is also true for
X,
and
Since (3.6) holds with
X.
This completes th6
proof of (4.6) for
feF . The proof for feF
is similar but much sim2
l
pler, since no centering is needed in that case. Finally, when X and V
are independent, (4.7) follows from (4.6) by Fubini's theorem and dominated
convergence.
When discussing the asymptotic behavior of X,
JVdX,
it is often useful
variable
V.
n
O.
to consider integrals of the form
J VndX
for
This leads us to investigate the continuity of the mapping
V ~ J VdX for fixed
tinuity at
or more generally of
X.
By linearity, we may confine ourselves to con-
20
Theorem 4.3.
Suppose that eithep cr
and f€F 2 • Then
(i) f~ fOV
(ii)
op that EX
=0
t
n
0
E;(:~P if: Vndxl' ~ 0;
(iii)
and f€F ,
l
implies sup 1f V dxl ~ 0;
tOn
if f€F l op if f€F wi.th II x- 2f(x)dx < ~ and
2
liminf xf'(x)/f(x) > I, then °EJ~ foV + 0 implies
o
~
=0
0
n
if gn and hn , n€N, are stpictly inopeasing functions on
R onto itself such that I h-l(y)/g-l(x) < ~, 0 < y < x, then
+
n n
n
t
V dXI) ~ limsup h (J~ foV )
limsup g of (sup
a.s.
(4.15)
n+~
n
tOn
n+~
nOn
1f
gn (x) -= hn (x) -= xlin . A
related result for Brownian mction may be found in [16] page 25. It should
Note in particular that (4.15) holds with
be observed that (i) does not hold in general with the convergence in probability replaced by a.s. convergence.
For Brownian motion, a counter-
example is easily constructed from the law of the iterated logarithm, and
by using arguments from [14], it is even possible to obtain counterexamples
with
Proof.
cr
= o.
To prove (i), proceed as in the proof of Theorem 3.1, using (4.9)
in place of (2.5).
As for (ii), let
feF 2 and conclude from Lemma 2.1
and the argument in (3.13) that for simple V
Ef(f: VdX)
For general
V with
EJ foV
<
~,
S
4E J: foV.
choose simple
(4.16)
VI' V2 , •••
such that
~
f
satisfies (3.4) in
Ll and (3.12) holds a.s., and conclude from Fatou's
lemma that (4.16) is generally true. Our next step is to show that J VdX
21
is a martingale.
Again this is clear for simple V.
In the general case,
choose approximating simple VI' V2' .•. as above. Then Ef(Joo V dX) is
o n
bounded by (4.16), and assuming f to satisfy the assumption under (ii)
we. have f(x)
~
xl+€
for some
> 0 and large x,
€
so
foo V dX is
o
n
uniformly integrable and we may proceed to the limit in the martingale
defining relations.
Let us now define the function
hex)
Ixl
fo fl(~)dU ,
= lxl
I:
which is finite and continuous whenever
xe:R,
-2
x f(x)dx <
00
•
By the argu-
ments in Meyer (1972) pages 28-29, we get
EfCSi'
~
If: VdX/l ~ 4EfC~ s~p II: VdX/l ~ mcI: VdXl,
so by (4.16) it remains to prove that
~
Eh(f VndX)
O.
~
implies
0
But this follows by continuity and uniform integrability,
since hex)
= O(f(x))
xf'(x)/f(x)
~
hex)
Ef(f VndX)
as x
c > 1, x
= f(x)
~
+
S O(f(x))
y,
In fact, choosing y > 0 such that
00.
and integrating by parts, we get for large x
xfx
o
+
~
f(U~dU = O(f(x))
JX f(U~dU
Y
u
c- l x
+
JX
fl (u)du
u
y
This completes the proof of (ii) for
fe:F 2 .
u
S O(f(x))
The case
+
c- 1 hex).
fe:F 1
is similar
but simpler.
To prove (iii), write
~n = fCS~P
If: VndX/l
ne:N,
and note that
P{lni~UP gn(;n) > 1imsup hn(nn)} S
~
n~oo
~
x,y,m
P{limsup gn(~n) > x>y>hk(n k), k~m},
22
where the sum extends over all rational
mEN.
0,
x and
°< Y < x
y with
and all
Since the sum is coutable, it suffices to prove that each term equals
so let us consider fixed
and applying (4.9) with
P(An{g
n
(~
n
x, y and
p = 2,
m.
we get for
Writing A = {hk(n ) < y, k
k
n ~ m
~
m}
»x}) = P(An{~1/2 ~ (g-1(x))1/2})
n
$
n
p{~~/2 ~ i(g~1(X))1/2[1+nn/h~1(y)]
$
-1
-1
16 hn (y)/gn (x) ,
so by the Borel-Cantelli lemma,
P(An{limsup
n-+- oo
gn(~n»x}) $ P(An{gn(~n»x
i.o.}) = 0,
and the proof is complete.
By
the method for proving (ii), \I1e may also obtain the following result
of some independent interest, (cf. Property 2.4 of rn.i11ar (1972)):
Let pE(0,2],
Corollary 4.1.
and cr
= YO = 0
and suppose that
when p S 1.
ES~P
cr = Yoo =
° when
p
>
1
Then
If: vdxl P s
Cp
Jro
!xl P A(dx)
Ef:
IvlP,
_00
The modifications required in the cases
Yo
~
5.
p
= 2,
cr
~
°
and
p
= 1,
° are obvious.
Variation and non-linear inteqrals. For nEN,
<t
=l} be a finite partition of
nk n
the measure
k
let
TIn = {O=tnO<tnl <...
[0,1], and define for any f: [0,1] -+- R
n
TI f = I o(f(t .) - f(t . 1))'
n
j=l
nJ
n,Jwhere
o A = o(x)A = IA(x).
x -
-
We assume that
We further define, for any X and
max.(t . - t
J
V,
nJ
. 1) -+- 0 as
n, J-
the point processes
on
R/{O}
23
= I
(V'X)
t - O<s~t
o(V(s) X{s}),
t
0,
~
w
We shall write -+
for weak con-
and we put in particular
vergence of finite measures on R,
i.e.
for any bounded continuous function
~
f:
n
w
-+ II
means that
R-+R.
Here
+
llf
II f -+ llf
n
= If(X)
ll(dx),
fll as the measure with (fll) (dx) ~ f(x) ~(dx). For
xkll(dx) ~ II k (dx), k = 1, 2. If the II n and II are
and we further define
brevity, we also put
wP P (P -+
w
random, then lln -+
n l.l in probability) means that
a.s. for any bounded continuous f.
P
l.l f ....
n
pf <
CXl
The following theorem extends results by Wong and Zakai (1965), and by
Millar (1972), (see also [9, 18]).
The idea to state variational results
as limit theorems for random measures comes from [12].
whiZe if cr
If either f€F l
00
2
J
= 0,
I (TIn
f:f1oV <
l
If
Theorem 5.1.
0 V
<
(TIn
J
J1
CXl
then
a.8.~
VdX)2
~ 0 2°0 J: v2
Ixl A(dx) <
CXl
f10
and
-1
J~x)ll !
and cr
I~I
= Yo = 0
00
J: lvi
+
Ivi
+
(5.1)
(V'X)2,
<
a.s.~
CXl
1~'X)ll
a.s.
= 0,
or f€F 2 and cr
(5.2)
and if moreover
a.s., then
f(IT n JVdX)
~
(5.3)
fey-X)
hoZdS a.s. or in probabiZity respeativeZy.
Proof.
By Lemma 3.3, we have in the space of measures on R\{O}
IT
JVdX
n
~ I
O<t~l
o(JtVdX)
t-
= v,x
a.s.
(cf. [12]), where X stands for vague convergence.
f€F
2
and cr
= 0,
it therefore suffices to show that
(S.4)
To prove (5.3) for
~
(ITn JVdX)f
We need the fact that
If
:24
(5.5)
(V·X)f.
is concave on R+.
To see this, we have to verify
is non-increasing, Le. that h = 2ff" - (f,)2 ~ O.
2
f(x) s xf' (x) - x {f"(x)/2 since f' is concave, and so
that
(f,)2/f
But
hex) s 2xf' (x)f"(x) - x2 (fll(X))2 - (f' (x))2 = - (xft'(x) - f' (x))2
S 0
as desired.
As in [12], it now follows by Minkowski's inequality that, for
any x j ' Yj'
j~N,
I(l
f(x.))1/2 J
(l
f(y.))1/2/ S
J
(L
f(x._y.))l/2,
J
and applying this to the decomposition X = XE: + X'e;
(5.6)
J
in the proof of Lemma 3.3,
we get
I{(IT
fVdXl)f}1/21 S {(IT JVdX )f}1/2.
n
E:
JVdX)f}l/2 - {(IT
(5.7)
nnE:
Now clearly
(IT n JVdX')f
E:
~
(5.8)
a. s.,
(V'X')f
E:
and it will be shown below that
(IT n JVdX)f
E:
~
0 as
n
+
~
Combining (5.6)-(5.8), it is seen that
lity as
e:
+
0,
which proves that
and then
(V
'X')f
e:
(V 'X)f <
00
E:
O.
+
(5.9)
is fundamental in probabia.s.
Using this fact, (5.5)
follows by another application of (5.6)-(5.8).
To prove (5.9), note first that
-floV
is, and hence that the process
of bounded variation.
Since
fICO)
V is a.s. integrable on [O,lJ since
Jt V,
o
= 0,
t~[O,l],
is a.s. continuous and
it follows that
a.s., so by (5.6) if suffices to prove (5.9) with
XE:
(IT n JV)f
+
0
replaced by the mar-
tingale Y (t) = X (t) - tEX (t). By an obvious stopping time argument,
E:
- e:
E:
1
(cf. Lemma 1.1 of Fisk (1966)), we may further assume that EJofloV < 00 •
Using (4.16) we then get
25
f:
sup E(TI n fVdY<)f S 4E
n
and (5.9) follows.
that
(V'X)f <
~
For
fEF l
a.s.,
and
~
f<oV
O.
= Yo = 0,
(J
<
~ O.
it may be shown as above
and then the a.s. convergence in (5.3) follows by
concavity (cf. [20, 12]).
To prove (5.1), it suffices by the above argument to consider the case
when
A[-E, E]
=0
for some fixed
E > 0,
finitely many jumps, we may even assume that
further take
EX = 0,
I
and since
A = O.
VdX has then only
As before, we may
so after a normalization it only remains to prove
that, when X is a standard Brownian motion,
(5.10)
I 4
(This was proved by Wong and Zakai (1965) under the assumption E V <
but we prefer to start afresh.)
Again we may assume that
l 2
E oV <
J
by Lemma 3.1, we may then choose simple processes VI' V2' ..•
k +
I
0
~,
~ ,
and
such that as
~
(5.11)
But by the classical result for Brownian motion,
(5.12)
as
n
+ ~
and then
k
Using Minkowski's inequality, we obtain
+ ~.
(5.10) from (5.11) and (5.12).
To prove (5.2),
we may use a similar argu-
ment with (5.10) replaced by the well-known relation
I (TIn
JV)ll = ?IJt
J
nj
Vi
+
t n,J. 1
(which may e.g. be proved from Lemma 3.1).
fllVI
0
a.s.,
Our proof is now complete.
Further assumptions must be added in order to obtain a.s convergence
fEF 2 .
in (5.1) and in (5.3) for
proceeds by successive refinements, and. that
every fixed
t
~
0, (X,V)
{IT } is nested if it
n
(V, X) j.5 symmetria, if for
Let us say that
has the same distribution as
where
s < t,
X (s),
=
Yes)
{
2X(t) -
xes),
s
~
Note that the last notion reduces to symmetry of
independent.
(Y,V)
t.
X(l)
when
X and
V are
We shall prove the following extension of results by Levy,
Cogburn and Tucker (1961), Millar (1971) and myself (1974).
Theorem 5.2.
is
If in Theorem 5.1 we add the assumptions that {IT}
n
nested and that either
(X,V)
is symmetl'ia
X and X aPe independent,
01'
then (5.1) and (5.3) hoZd even in the sense of a.s. aonvergenae.
Proof.
For independent
X and
V,
we may use Fubini's theorem to reduce
the proof to the case of non-random V.
But then
f VdX
has independent
and infinitely divisible increments, and our assertions are contained in
Theorems 4.1 and 4.2 of [12].
(Note that the assumptions on
f
in [12],
fEF , If and f(~) are both con2
is sYmmetric, and assume without loss
Theorem 4.2, are fulfilled since for
cave.)
that
Next suppose that
A has bounded support and that
respectively).
(or
(X,V)
EJ: y2
<
~
(or EJ: foy
Proceeding as in [3] and [19], it is seen that
{CITnJVdX)f})
fore converges a.s.
is and
Ll-bounded
<
~
{(ITnfVdX)2R}
reversed super-martingale and there-
This proved in particular that (5.5) and (5.10) remain
true in the sense of a.s. convergence,.and so the preceding proof yields
the present stronger conclusions.
27
It is suggestive to write ffCVdX)
for the f-variation of jVdX,
whenever it exists in the sense of Theorem 5.1.
for any X and V and for
More generally, we define
fe:F
02 f "(0)J>2
(V'X) t f
+
if fl (0) = 0,
(5.13)
IYO If' (0') J: Ivi
(Note the formal similarity with (3.2).)
if
(J::
o·
As a simple example, it may be seen
from (5.13) that, for any p > 0,
a. s. ,
where the integral on the right is the Lebesgue-Stie1tjes integral of
Ivl P
J:ldXI P, t
w.r.t.
> O.
Since the latter process has stai'ionary inde-
pendent increments, the above theory applies to this case.
Many previous results of this paper may be extended to integrals of the
form
ff(VdX).
For brevity, we restrict our attention to'the following
partial extension of Theorems 3.1 and 4.1.
If
Theorem 5.3.
2
J>WJ1ov
he:F
a.s';t t
0
r
fe:F and either fl (0) = 0
and ge:F3'
1
>
0,
then
f(VdX) <
J:f(VdX) -> 0
as t
+ 0,
and this is also true with
As t
+
jtf(VdX)
00
holdS orl
2
{fo(~f)OV = oo}.
~~.
= 0,
a. S'J
and
t >
if
o. If
moreover
(hof)
is typed
~~
(5.14)
a.s.
replaced by
converges a.s. on {f:Ch:f)OV <
0
For convenience,
00
(J
0
then
(h-lg(J:(~)OV))-1
00',
0'1'
oo},
C~),
E
E >
while (5.14)
O.
28
Proof.
We first extend Lemma 4.1 by showing that the process
t
with
Q defined by (2.4) with h in place of f
super-martingale for every x€R and
simple V,
y>O,
~
0,
(5.15)
= 1, is a
= y = O. For
and with c
and also for
x
this follows from Lemma 2.2 and the easily verified fact that
EhCJ>(VdX)) S EJ:Chof)Ov.
(cf. (3.13)).
For general
t
~
0,
It(hof)OV <
V satisfying
co
o
a. s.,
1
write D = {s: IX{s}1 > n- },
n
u > 0 so large that the process Vi = uAV satisfies
t > 0 and
sider fixed
n€N,
t > 0,
con-
and choose
n
P{
P{V'(s)
n
t
~N~
1fo(hof)oVin
= V(s),
-
sED}
n
ft (hof)oV
--~ I > n-1 }
~
2
-n ,
(5.16)
0
n
~ u, SED } ~ 1 _ 2- ,
= p{IV(s)!
being possible by monotone convergence.
(5.17)
n
According to Lemma 3.1 and Theorem
3.1, we may next choose some simple process
V
n
such that
(5.13)
(5.19)
(Note that the assumption 1imsup f(2x)/f(x) <
co
x+m
in Lemma 3.1
is not
needed when V is bounded, since in that case the construction in the proof
Combining (5.16)-(5.19), we get
leads to uniformly bounded processes Vn .)
p{lft(hof)OV - ft(h';;'f)Ovl
o
n
0
>
2n- 1 }
~
2-n +1 ,
p{lvn(s) - V(s)1 ~ 2n- 1 , sED } ~ 1 _ 2- n +1 ,
n
29
and so, by the Borel-Cantelli lemma,
t
(~)oV ~
Jo
(V.X)t f
n
~
00.
= L
O<s~t
~
(5.20)
°
n
Vn(s)X{s}
a.s. as
Jt (h;f)oV,
V(s)X{s},
SE[O,t],
(5.?!.)
By Fatou's lemma, it follows from (5.21) that a.s.
f(V(s)X{s}) ~ 1iminf
n~
I
= liminf
f(Vn(s)X{s})
O<s~t
(Vn.X)tf ,
n~
and by combination with (5.20),
fotf(VdX)
S
1iminf Jtf(V dX)
n~
0
(5.22)
a. s.
n
Using Fatou's lemma once more, we get from (5.20) and (5.22) for
y >
0
EQ(x+ftf(VdX), Y+ft(h;f)OV) S E 1iminf Q(x+ftf(V dX), y+Jt(~f)OV )
o
0
n~oo
0 nOn
~
1iminf E Q(x+ftf(V dX), y+ft(~f}OV )
n~oo
0 nOn
~
Q(x,y),
where the last inequality follows from the fact that (5.15) is a supermartinga1e for
simple V.
This extends the super-martingale property to
arbitnry V•. The last two assertions may now be proved as were Corollary
2.1 and Theorem 4.1.
In particular, it follows that
a.s., or even provided
Jt(h~)
o
e
oV
J:f(VdX)
<
at most finitely many jumps in {O, t]
IxlAI
00
<
~
a,s. when
a.s. for some
of modulus
>
e
e.
>
0,
J:(ho'f).V
since
<
~
X has
Choosing hex)
~
yields the first assertion and hence completes the proof.
We may even go a step further and consider more complex integrals such
as
fUf(VdX), Jf(V; dX)
etc.
Though the formulae now become. more compli-
cated, the methods remain essentially the same.
30
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Lecture
Trans. Amer. Math. Soa.
Department of Mathematics
Fack
8··402 20 G6teborg 5,
SWEDEN
Department of Statistic5
University of North Carolina
Chapel Hill, NC 27514