Journal
of Applied
Mathematics and Stochastic Analysis, 12:2
(1999),
133-150.
EXISTENCE OF MOMENTS OF INCREASING
PREDICTABLE PROCESSES ASSOCIATED WITH
ONE- AND TWO-PARAMETER POTENTIALS
YU. MISHURA and YA. OLTSIK
Kiev University
Department of Mathematics
Kiev, Ukraine 252501
(Received March, 1997; Revised May, 1998)
The criterion and sufficient condition for the existence of moments of oneparameter increasing predictable processes is presented in terms of an associated potential. The estimates of moments of special functional connected with two-parameter increasing predictable processes are given in the
case when the associated potential is bounded. The application of these
estimates to the local time for purely discontinuous strong martingales in
the plane is also presented.
Key words: Increasing Predictable Process, Associated Potential,
Moments of all Orders, Local Time, Discontinuous Strong Martingale.
AMS subject classifications: 60J45, 60G60, 60G48, 60J55.
1. Introduction, Definitions and Notations
We estimate the moments of one- and two-parameter increasing processes in terms of
associated potentials. It is well known that a one-parameter increasing predictable
process has moments of all orders if the associated potential is bounded. In Section 2
give the criterion and sufficient conditions for the existence of moments of all
orders for an increasing process in the case when the associated potential is
unbounded, with examples that demonstrate the optimality of the criterion. In
Section 3 we give estimates of moments for special functionals connected with
increasing predictable two-parameter processes in the case where the associated
potential is bounded. The problem of estimating moments of increasing predictable
processes associated with two-parameter potentials arises from a question whether the
local time for two-parameter purely discontinuous strong martingales has all finite
moments. Note that in a one-parameter case Bass [1] gave a positive answer for this
question. In Section 4 we apply the results of Section 3 to the local time for purely
discontinuous strong martingales on the plane. We estimate moments of special
functionals connected with the local time, and formulate the conditions sufficient for
the local time to have all finite moments.
we
Printed in the U.S.A.
(C)1999 by North
Atlantic Science Publishing Company
133
YU. MISHURA and YA. OLTSIK
134
The following are necessary definitions and notations: For two points s- (81,82)
and t- (tl,t2) in
s < t means that s I < t and s 2 < t2, and s < t means that
s l<t land s 2<t2. Ifs<t, then (s, t] is the rectangle (sl,tl]X(s2, t2]. Let (f,
be a complete probability space with the filtration {5t, t E
} which satisfies the
following properties [2]"
R2+,
Rz+
(F1) ift<t’thentCSt,;
contains all null sets of
(F2)
for
each t,t U t’;
(F3)
o
t<t
(F4)
where
for each t,
V t,
2
s
V
l<t I
_> 0
1
5
independent given t,
t2 are conditionally
2
Let t zJ:t1 V J:t,
zJ:t-a : 2 V t
0
_>
I
and
V
t
8
V as, 1
slt2, t--s<t
V
t-
s l<tl,t
2>_0
aJ:Sl ’t2’
ht2_
V
s 2<t2,t
1>_Otls2"
The
definitions of strong, weak 1- and 2-martingale, and increasing process in the plane
will follow those of [2]. The definitions and notation of two-parameter predictable, 1and 2-predictable processes, predictable and dual predictable projection will follow
those of [7]. The definition of one-parameter potential will follow that of [3]. If
n
we denote
[0, t] C
{0 t 0 < t 1 <... <
ti} i- 1,2, )tn /1n X )2’
n
,k D ,k if m > n, the partition of [0, t]. We also denote tik- (t
Xik
R2+,
Bik
t
(tik, ti + lk + 1],
1
Ak +1 X -AikX.
1
lim
s-,t, s
QI
AIXt
X
X
s--t, s
+,
A2Xt
Qt1
X
s-,t,s
is
X +
Q3
X s with
Q4 -{s:ti < s,s 2 < t2}.
constant on F
A functional
Xtik’
AikX Xi + lk- Xi k AikX Xik + 1 Xi k [ik X
The increment of process X on the rectangle (s,t]
X(8, t] X Xslt2 Xtls2 + X s. Let
I’]Xt- Xt-Xt_ -t- -Xt+
X +
il, tk2)’
2
({0} x [0
All processes are assumed to have these limits, to be
o))U ([0 o) x {0}), and to be continuous in Q1
XtdY
[0,t]
,.
is called a stochastic integral
A functional
partitions
,.
’
of the first
Xtd 1Ytd2Zt
[o,t]
P- lim
I--,o
E XikUikY
i,k
0
kind if this limit exists for any sequence of
P lim
a’ I-o
n-l XikAikY AikZ
1
2
i,k=o
is called a stochastic integral
of the second kind if this limit exists for any sequence of
partitions
A process X {Xt, E
} is called a weak submartingale if X is integrable for
each t, X is adapted and E(X(s, t]/Js) >_ 0 for each s <_ t.
If {At, t G
-lim At,
} is an increasing process, then define
t2cx
lira At, A (xz =tli_,mAt.
R+
R2+
oo
Atl
Acxt2
Existence
of Moments of Increasing
/i2+
Predictable Processes
135
A process {Xt, t E
} is called a two-parameter potential if X is a nonnegative
weak submartingale such that
and X (tl) are one-parameter potentials. If a
potential X is bounded, then it as associated2with the increasing predictable process
A in the following way [5]
Xt(t2)
+ E(Ao- AtloO- At2/t).
A
X
(1)
2. Some Properties of One-Paxazneter Potentials
Let {Xt, t, t E R + } be a one-parameter potential associated with predictable increasing process A t. It means that X -E(A-Atilt) [3]. All increasing processes are
assumed to be integrable.
Theorem 1" The following statements are equivalent"
(a) for every k >_ 1, EsupXkt <_ c k < o,
EA
<_ d k < cx.
(b) for every k >_ 1,
Proof: Suppose condition (a) holds. Let
n). Then for any k _> 2
ciated with
A ) -A A n, X n) be
A
E[A)]
EA)[A)] -1
/ (X
E
)
a potential asso-
+ AI’))d[A)] -1
0
g
J (xn_)
-t-
An_.))d[An)] k- 1,
0
On the other hand,
we have from
2E[A()] k
integration by parts that
E
/ (A
n) -t-
An_))d[An)] k-1
0
-t- E
/
([Aln)] k-1 + [Aln_)] k- 1)dA n),
(2)
0
hence,
E
/ (XI
’) +
xln__))d[Al)]/ -1
0
We
can estimate the
E
/ ([An)]
k-1
0
integral in the left-hand side of
E
/ (XI
0
n)
(3)
+ Xln)__ )d[Aln)] k-1
-t-[Aln2]k-1
(3)
YU. MISHURA and YA. OLTSIK
136
E
(Xn)+ Xln--))d[An)]k-I{s pXIn)>
4k }
(Xn)+ Xn)-)d[Aln)]k-lI{supXn) <
}
0
+E
(4)
4]
0
<s
(supXl )) +
Note that
A
o
n)d[A n)]k-1_I,xlimn
I--*o
[AI .n)]k-l)
oAti
cx
n-1
_< (k-l)
liT
I-*o
E [A 7 )]k -lhA 7
(k 1)
k=o
i [Ai")]k-ldA(n)
o
From (2)-(4)we have
0
or
.[A(2)]
Since
X n)
E(Aoo A n- A
A n/Jt)
<
<_ Xt,
we have that
E[A)] k _< 16k2EsupXkt.
While
n---,cx3, we
obtain
EA k <
Suppose condition (b) holds.
16k2EsupXkt.
Then obviously X
< E(A/t)
and by Doob’s
inequality
EupXf, <_
-< ( k k 1 )k EA.
The theorem is proved.
mark 1" Theorem 1 generalizes the well known result that if an associated
potential is bounded, then the increasing process has all bounded moments. We give
examples to demonstrate that the existence of moments of X does not supply the
existence of moments of A t. This means that the conditions of Theorem 1 are quite
optimal.
Example 1: Suppose {n,n 1} is a sequence of independent random variables,
and each of them has gamma-distribution with parameters
and
an-n
nThen E+ k- 1)..... (an + 1)an < " Consider the filtration n a{k,
(a
,n-1
k
n}
and an mcreasing predictable process A n
k"
k=l
Hence,
a
"
potential associat-
of Moments of Increasing
Existence
ed with A n is
Xn E(Ao An/Jn)
E
Predictable Processes
E
k-n+l
137
k + n.
Note that X n is unbounded, but it has all bounded moments. It is easy to
z-1 < cx, but EA >--n
EA-n
Let a
n
ln n
n-_ and fin-l"
One
1
gamma-distribution with parameters
2
EA-(+k-1).....-W<,
2
moments.
Example 2:
2
a- and
Consider a standard Wiener process
(k
<_ 2--b2, then (tk
a
-
(ksdw s +
+ ]ca
1
a
2b> 1
Suppose
(-+b) fsds.
0
(ksds.
,
ds
t
n-i<<2’.
Aoo-
Consider
<_ c I + c2E
-0,
i<_k, and from
(6)
sdws
0
Moreover, by Burkholder’s inequalities
(ds
E
o
(f
(sds
(/2dws
<_ c4E
(sdws
0
< c 5 + ere
0
(2sds
0
(;)
2n
s
0
(ds
we obtain
<_ c 5 + c6E
<c8+c9E
(;)r.
o
o
From (6) and the second part of (7)
then
a
is a potential with
is a potential,
0
c3E
t
a
Let k-2 n. Then k<2-b2,
i<_k
(i)
a
(5) (the constants depend on and r)
E
,
0
Suppose there exists such hEN that 2
the following cases:
E
1
{wt,t, R+}. Let
kb
Then
oo
1.
Therefore, for any k
-’2b then (kt is a martingale; and if k >
is a potential; if k
is a submartingale.
_>
has
E- e(-kb)t <, >0, 1, i.e., t
0
for every r
-(n
n--1
k2a 2
a,bR, b>0. Then
has all moments. By It6’s formula
2
1.
see that
i.e., the increasing process has M1 bounded
e awt-bt where
If k
-
-
can check that a random variable
n
b
+
3
1
ds
c
8-4-c9E
;
0
ekaws- kbsds
<_...
YU. MISHURA and YA. OLTSIK
138
cx
C8 + C9
EA < ec.
(ii) Let k 2
Thus,
n
,
+ 1. Then
Hence, from the first part of (7)
property and from
c
E(f /4dws) 4
,
J
k2a2
e(------- kb)Sds < c.
o
_> k is a submartingale,
E(fks/2dws)2-cx.
0
Further, from the potential
(5) E(f sk/2 ds) 2 -c. Again, from the first part of (7)
x
0
and from (5) E(f k/4ds)4
Continuing in the same way,
8
0
k
E( f sds) ; i.e.,
0
From
i)and (ii)we have that
we obtain that
0
k
EA
.
EA <
.
for i<
2n-l<
and
EA- for
EA,
Now we will prove a sufficient condition for the existence of moments
k 1
in the case when a potential X -E(A-Atilt) and associated increasing process
A are continuous. Denote a martingale M E(A/t) (M)t is its quadratic characteristic.
Threm 2: Let X be a continuous one-parameter potential, associated with
continuous increasing process
_
A t. Assume that for every k
Then
EA < ,
Proof: By It6’s formula for every k
k
1, E f Xsd
o
>_ l.
2
0
0
Therefore,
supXtk<ksup]
ldM s +
xks
2
0
Xs 2d(M) s"
0
If the condition of the theorem holds, then by Gundy’s inequality
X2s k- 2d(M)s
EsupXt _< kE
0
hence, by Theorem 1 we have the proof.
Pmark 2: Suppose (M)t- f a2sds, where
cx
(x
f0 a2sds < oe.
Then E
0 cx
f0 Xksd(M}s f0 EXska2sds.
a s is a nonrandom function with
Since
EXsk0 when soc, then a
Existence
continuous function
of Moments of Increasing
Predictable Processes
139
EXks is bounded and, therefore, E f Xksd(M}s <
0
3. Two-Parameter Potentials Associated with Increasing Processes
R2+
We
assume in this section that {Xt,Jt, E
} is a two-parameter, nonnegative,
bounded potential associated with increasing predictable integrable process At; i.e.,
(1) holds.
Theorem 3: If X <_c, then for every p >_ l
E(E(A- at/t))P <_ 2 p + lcPp!.
Proof: Fix t 2
>_ O.
Consider a process
(8)
Ztl(t2) E(A tlo At/t), t I >_ 0.
It is obvious that Ztl(t2) is 1-predictable with respect to the filtration {ff,t 1 >_ 0}.
Since
AtlCx -A is F-measurable and from the condition (F4) [2] we have
Zt l (t2) E(E(At I At/1)/) E(AI At/2t
Therefore, Ztl(t2) is predictable in t Moreover, if s I < tl, then
Ztl(t2)- Zsl(t2) E(Atl A Aslcx Aslt2/j2t) O.
-
1.
Thus, Z
_
oo
(t2) is a predictable increasing integrable process as a one-parameter process
1
Then from
with a parameter t 1. Denote
a potential associated with
(s)
Xtl(t2)
Ztl(t2).
Xt l (t2) E(Zc(t2) Ztl (t2)/t)
E(Acx Acxt 2 AtloO -+- At/Jt) X t.
_
(9)
Consequently, X (t2) is bounded; and we can apply the Garsia inequality
eds to the estimlt E(Z(t))" < cv! or
E(E(Ao- At:/))" <_ c"!,
>
1.
It follows easily from the last inequality and from (F4) [2] that for every t E
E(E(Atlcx
[3]
Atilt)) p cPp!,
and by a symmetric argument
((A,:- A,/)) < c!.
Finally, we obtain
E(E(A- A/,)) <_ E(E(A- A/,))
R2+
that
YU. MISHURA and YA. OLTSIK
140
+ 2PE(E(At 2 At/t)) p <_ 2 p + lcPp!"
The theorem is proved.
The next statement follows directly from Theorem 1, (8) and
Corollary 1: The following conditions are equivalent:
< c k for every k >_ 1,
(a)
< d k for every k > 1.
(b) E sup(E(At
EA
Let A be
2
an
2
(9).
-At/Jt))k
integrable predictable increasing process. Denote
vn
inf{t E R
+’Atltl >_ n}
and
/ HI [0, rn[2(s)dAs"
(10)
[o,t]
Note that
ing to [7],
,
process and has all bounded moments.
A is a predictable
I
where we denote
A-(f
[o, -[ 2(s)dAs)
()
In fact, accord
the dual predictable pro-
[o, t]
jection [7] of the corresponding increasing predictable process. Further, again according to [7], for every p _> 1 there exist constants cp such that
Suppose
A and B are two different increasing predictable processes;
Denote
(10).
n} and
n inf{tl R
now that
is defined by
+’Btltl
B2-
f HI[o,n[(s)dB
(11)
s.
The next auxiliary result will be used for the proof of the main theorem.
Lemma 1" Let At, B be two predictable increasing processes, A2 is
(10), By is defined by (11). Then
lim /
[o,t]
Using
(10), (11),
B s dA
E
/ BsdA
[o,t]
and the definition of predictable projection
rn[2(s)dAs
[0,
thatsi_ftI[o, rn]2(s)---,1 a.s.
defined by
s.
write
Note
A
rn[2(s)dAs"
Thus, it is sufficient to prove that
[7],
we can
of Moments of Increasing
Existence
sup
s
<
The sequence
Bs-
/
III
[o,]
{I
2
[0, rn[2 (u)dBu[
+ \[o, ,[
2(s), n > 1}
Predictable Processes
J IIIt2+ \[o,n[2(s)dBs
--O
141
a’s"
[o,t]
is decreasing; hence, it is sufficient to prove
that the last integral converges to zero in probability. But
V1
and we obtain the proof.
and
are continuous on the right.
Further, we assume that r-fields
Lemma 2" Suppose X is a bounded potential associated with increasing predictable process At, that has all finite moments, and B is an increasing predictable process that has all moments. Then the following formula holds
Y
ff
E/ XtdBt+E/ (Bt+_+Bt__ -Bt-)dAt
[o,)
[o,)
+El dlXtd2Bt + /
E
[o,)
O.
d 1Btd2Xt
[o,)
Proof:
Denote M 1
E(A 1 /Jt)
E(A/ZYt)
E(At 2 /Vt) M 2
2.
1"
Then M
1-martingale, M s 2-martingale, M s martingale, and from (1) X
+ M + M + A t. The process B is 1-predictable, hence,
Mr-
t21s
M
1
o
EMBt-E/Mst2dlBslt2-Elim
I)t?lo
0
where
Ms- t2 M(sl,t2
[0, t 1]. We can
+, and
MI" AI" B a.s.,
tlt 2 tit2
0-t<tl<...<t-t I
is the partition of
write
n-1
n-1
n--1
n--1
n--1
i- 0
j-0
=0
j-0
E Mj[-]ij-1B + E
E Aj_ 1MIAmi_ 1B
(Here and further we replace index tit by index ij.) Under the assumption of continand fit on the right, the process
satisfies the conditions of
uity the -fields
Theorems 3.4 and 4.2 [8]. Therefore,
has a modification with limits in Q, i2, 3, 4 which is continuous in Q. For such modification
M
M
n--1
lim
n
IX I--,o
n--1
E-o E=o Meial"lia
j
B-
/ Me
[o,t]
+ dBs.
YU. MISHURA and YA. OLTSIK
142
Consequently, there exists
n-1
n-1
.2. M /ki
E /ojE d-o
I:l--+Oi-o
1
lim
Furthermore,
i=oMt2l’tt2 B
n-1
i=0
1 B--
dlBsd2M
:.
[o,
and the right-hand side of this
A 1 B is uniformly integrable. Similartlt 2 tlt2
Mij[]ij_ 1 B and obtain their uniform integra-
M
n--1
0
n-1
i=0
j=O
ly, we can consider the sums
n-1 n--1
a
BtspE(A/t);
inequality has all moments Hence
bility. Thus,
1
1
y Aj_ 1MIAj_ 1B is uniformly integrable and
j=O
EMB
E
/ M_ +
[o,t]
f
! dlB*d2Ml"
[0, t]
(12)
2
s.
dlMsd2B
(13)
dB s -’l- E
In the same way we obtain that
El M +
EM2Bt
dB s + E
[o,t]
Also,
[o,t]
f
! MsdBs.
[o,t]
EMtB
E
(14)
It is easy to check that the formula of integration by parts for two increasing processes A and B holds
EAtB
E
/" AsdB + E / (B +
s
[O,t]
E
/ (M
Mls
E
s.
(5)
[o,t]
)dBs
E
/
(M s M s_ +
[o,t]
[o,t]
=E
(Mls t-Ms_+ )dB
[o,t]
E/ MlsdBs -EJ Mls_ +dB
Similarly,
B s_ )dA s
/ dlAsd2Bs + / dlBsd2A
[o,t]
Note that
+ Bs
[O,t]
+E
Therefore,
s
[O,t]
[O,t]
s.
(16)
Existence
of Moments of Increasing
Predictable Processes
El M2sdBs-El M2s+
[0, t]
(16)-(17),
j"
+
E
(17)
_dB s.
[0, t]
Adding (12)-(15)and using
EXtBt
143
we have
[o,t]
/
+
+
Bs_ )dA s
+
[o,t]
+El dlXsd2Bs + / dlBsd2X
E
[o,t]
s
[o,t]
Letting t---+cx in the last equality and using the definition of the potential, we obtain
the proof.
Consider the following sequence of predictable increasing processes
BO) -1, B 1) --At, B 2)
/ B!l)dAs,...,Bl S Bk-l)dAs’
k)
[o,t]
[o,t]
The next theorem is the main result of this section
Theorem 4: Let X < c. Then for any
Proof: We use the induction in k.
(1) It is obvious that
< cx.
EB
EA
EBb)< o for some
Denote O’n, k --inf{t I E R+ B (k) >n} and
tit1
Bk’n)= III[o,rn, k[2(s)dBk)
(2)
Suppose that
/
k
>
1
n
>
1
n) be defined by (10),
n) be_
associated with A n). If
prove
A
XI
n)
L
+
that under inductive hypothesis EB
and using
< C < oo, then, letting
Lemma 1,
obtain the proof.
the representation
Thus, prove that EB + 1,n)< . For any process B
Let
a potential
we
k
noc
we
use
B t-B t_
+AIBt+A2Bt-DB t_C.
+
EI
AIBk’n)dAn)+
[o,)
Ef
Since
A2Bk’n)dA n)
FIB
k,
n)dAt
[o,)
it is sufficient to prove finiteness of each term in the right-hand side of the last equali-
ty.
(a) By Lemma 2
we can write
YU. MISHURA and YA. OLTSIK
144
[o,)
-FE/ dlXn)d2Blk’n)-FE/ dlB k’n)d2x(n)’t
[o,)
(18)
[o,)
n) is
n) is defined by (10), and
A
X
(1) into account, obtain that
where
a
potential associated with
we
Taking
2
tl(x)]
2
[0,)
An).
[0,)
By the definition of the stochastic integral of the second kind
d1
(A(n)lcx A))d2Bk, n)_ lim
IAI--*0
[0, o)
n-1
i,j:O
E AI(Ai_() A!y))AjB(k,)
i,j--0
E Vlil, (’)A2 B (, ’)
n-1
ij
I’1
n--1
j
n-1
lim
A
---,0
E lil,(,+’1,kill A(n)
1
i,
0
Bln)+dA ")
j
[o,)
Hence,
(19)
[o,)
Similarly,
[o,)
(20)
[o,)
Substituting
(19)
and
(20)into (18),
[o,)
we obtain
(21)
[o,)
[o,)
[o,)
Since the left-hand side and the first term of the right-hand side of
have by virtue of boundedness of X and inductive hypothesis, that
(21)
are
finite, we
(22)
Existence
(b)
of Moments of Increasing
Predictable Processes
145
First, prove that
El [:]X(n)dB(k’n)t
E
[o,)
/ YlA(n)dB
Consider the following martingales
M
A
X
+ E(At- Aoct2/Jt) E(A/Jt)
and
Mn) xn)- An)-[(n)
where Mo- A and M
A
( )
tlc
2
/v,)
Therefore, from [4]
f
[o,)
E
] (X
n)-
n) + ((n)tl + (n)t21/t))d k’n)"
[0,
Note that
M ) is
a
nonnegative measurable process, hence, by
[7]
nlM n) E(Mn)/_ E(E(A)/t)/_
M M
n)
Similarly,
able rejection we have
and
M)
[o,)
rom the definition of the predict-
[o,)
[o,)
Thus,
M).
[o,)
EA)B ’n) in the four following ways:
,1
E A ( B (,)
A () /y
(Xl n)- AI n)-[- E(A()
tlOC + oct2/ ))dB
we can write
o
EA)B( )
E
( )
EA)Bo’
E
+
-[-
E(Aln)lcXZ A(n)-oot
(X () + AI n--) + + E(A n--)
f
[o,)
J
t_
-[-
oc
2
/t -t-
))dBl k’n)
+ A(n)
oct2//t + ))dBl
(23)
(24)
(25)
YU. MISHURA and YA. OLTSIK
146
EA()B(ko’n)
E
i (Xn-)
Aln)- mr- E(Aln-)l
oo
A- /’-fJ
-Jr-
[0, oo)
))dB k’n)
(26)
Subtracting (24) and (25) from (23) and adding (26), we obtain, using predictability
of k’n) and properties of conditional expectation, that
BI
E
i (V1xn)-V1An))dB’n)-O
[o,)
or
[o,)
n)
n) admits the representation
[o,)
A
n)c
The process
where
An)
dc
A
A A n)c + A n)cd + A n)dc + At n) d,
is continuous
AI n)cd is continuous in t
is purely discontinuous in I and continuous in
discontinuous. Therefore, only the integral E
and purely discontinuous in
t2, and, finally,
t2,
At n)d is purely
f []Bk’n)dA n)d is nonzero, and
E
[0,)
n-1
E
Bk’)dA n) -lira
Ip+0
ijB (k’n )ij A ()
(28)
[o,)
From (27)and (28) we have
E
i HBk’n)dAln)- f Hxln)dB
E
[0,)
< 4cEB) < oc
(29)
[0,)
by the inductive hypothesis.
(c) Subtracting (25)from
E
k’ n)
(23),
we obtain
(A1XI n)- AIAn)+/klAllo)dBk’n)- 0
or
(3o)
Consider the integral in the right-hand side of
(30)
f (AIAIn)_ AIA
lim
(n)
dB(k, n)
tl o]
E AI(A! )- A!2)VlijB(k’n)
of Moments of Increasing
Existence
-
-,t
n,lim
n-1
lim
I-o
From (30)and (31)
n-1
n-1
i,-0
=j
Predictable Processes
E E []iIA(n)vliJ B(k’ n)
E rliIA(n)Ai + 1B(k,n)
i,t-0
147
AIBk,n)dAt
(31)
[o,)
we have
(,)
A1xn)dBt!
E
An absolute value of the right-hand side of the last equality does not exceed
2cEBt)’’ < oc, therefore,
E
(d)
Similarly, using
J A1Blk’n)dAl
(24)instead of (25),
n)
(32)
< o.
one can prove that
(33)
Now, from (a)-(d)we get
(ck + 1,n)
/ Bllc, n)dAn) < 8cEB) <
[0,)
Letting noe and using Lemma 1, we obtain the proof of the theorem.
In Theorem 4 we have proved that the boundedness of the potential implies the
<_
integrability of the process
B A.
4. Apphcation to the Local Time
R2+
The process {Yt, t C
} is said to have the local time Lt(x if Lt(x is an increasing
and for all Borel locally integrable functions f on R
process such that for all t E
R2+
/ Lt(x)f(x)dx /f(Ys)ds
R
(34)
a.s.
[0, t]
A weak predictable set [4] D(w)C
R+t"
is called a stopping set if [0, t]C D(w) when
t D(w), and the event {t D(w)}
We say that the process Y belongs to a
class g locally if there exists a sequence {Dn(w),n >_ 1} of stopping sets and a sequence of processes {yn, n >_ 1} C g such that D n C D n + 1 for every n >_ 1, [.J Dn=
R+,
n>l
and
(Yt-Y)IDn(t)- O.
Let X be
a
local purely discontinuous
stro-ng
mar-
tingale that has local characteristics (as, us). This means that
a s and u s are -adapted,
for each s and w v s is a r-finite measure on R2\F,
for all Borel B such that
R2\F is compact, s<t IB(V1Xs)- f vs(B)ds is
[0, t]
YU. MISHURA and YA. OLTSIK
148
a local strong martingale,
X
VIXsI{I[:]X s > 1}- f asds is a local strong martingale,
[0d]
proce-ss f u s(B)ds is a weak
predictable projection [4] of the
<t
where the
E
X
[0,
purely
process
discontinuous means that X is a uniform limit of
st
f
X ] asds + s<t
E V1XsI{IFIXs
>
d
[o,t]
/
-]"
{Zt, t E R2+}
Eexp(isZt) exp(- tlt2 s
tric process
(A)
v--O"
be the Levy measure for a stable symmeaofhlindex(x + a,()dhwhere
is the constant such that
< a < 2, let O a(dh)
We
as
_< Ihl _1
[0, t]
If 1
hus(dh)ds
a>0
assume that the following conditions hold"
X is a local purely discontinuous strong martingale such that
g 1 a.8.,
h 2 A lus(dh)
(a) for some /’1 sup[
s
R
> 1,
(b) for some 1 < < 2 u(dh) O(dh), if
(c) for some 1< < 2, K 2>1, any O < < 2
f
hl
sup/s
h
la-’lu s-Oa[(dh)_<K 2
a.s.,
-1
(d)
X does not have
more than one jump along any line parallel to
coordinate axes.
be a Poisson field with parameters EN
Example 3: Let Nt, t
=
,tlt 2 (see, for example, [6]). Then N t- tit 2 is a purely discontinuous local strong
martingale with local characteristics us(B
IB(1) and a s 1. Consider local purely
discontinuous strong martingale N with local characteristics "s(B)- IB(1)+Oa(B)
for all norel B such
e Re\r is compact, where Oa(dh)is the Levy measure described above. Then N satisfies conditions (A).
Let Qt(w,.) be a regular conditional probability distribution for 3 t. That is for
every A 3 Qt(" ,A)is 3t-measurable for every w Qt(w,. )is a probability measure
on 3, and Qt(" ,A)
P(A/3t) a.s. for every A 3. Qt exists since 3 is the completion of a countably generated (r-fields and X is real-valued. Denote QtY(w)-
-EN2t
R2+
that
f Y(w’)Qt(w dw’)) E(Y(w)/3t).
Fix t o R2+, > 0. It was proved in [9] that there exists a nonnegative Borel in
x bounded function
Uto(,,x)(w
e
such that for every Borel A E R
Vto(, ,x)(w)dx.
A(Xt + to)dr (w)
A
[0, x))
At
)t
Further, the process {Vt(A,x -e
2Vt(A,x ), 3t, e Q2+} for a.e. x is a weak
submartingale, supermartingale, and a.s. for any s Q2+ and for any sequence
{tn, n >_ 1} C Q2+ such that tns noc, for a.e. x EU n (A,x)---EUs(),x), n---<x.
Here we denote Q2+ Q + x Q +, Q + Q gl [0,oc).
of Moments of Increasing
Existence
Predictable Processes
149
Under the assumption
is a weak submartingale,
(B) for a.e. x the process (Ut(. ,x), t E
supermartingale and for every s E
EUt(,,x)--EUs(,x) if t--s, t > s,
tE
The process Ut(,,x for a.e. x allows the representation [9]
t
Rz+
R2+.
R2+
Mt(, x) + Lt(, x),
Ut(, x)
where Mt(,,x is a weak martingale, and Lt(,,x is an increasing predictable process.
Note that Ut(,x is a bounded potential associated with Lt(,,x ).
Fix > 0 and let
,
/ e)Sl
Lt(x
A"
,XS2dLs(,,x)"
(35)
Then from [9] Lt(x is a local time for Xt, and it does not depend on
sequences of increasing predictable processes
,.
Consider the
/
C)(,x)- 1, cl)(,,x)- Lt(,x), C2)(,,x)- c!l)(,,x)dLs(,,x),
[0,t]
c)(a, )
f c!
1)(a, )a.(a,.),...
and
J
C)(x)- 1, cl)(x)- Lt(x), C2)(x)- c!l)(x)dLs(x),...,
[0,t]
cl)()
j’[0, t] c!-l)(.)a.(x),
Since potential Ut(,,x is bounded, from Theorem 4 for any k >_ 1
Therefore, using (35), we have that EClk)(x)<
Suppose that one more condition holds:
(C) E sup(E(Lo (A,x)- Lt(A,x)/t)) k < d k for every k > 1.
t2
2
Then, from Corollary 1
k,k(t 1
e
EC)(,,x)<
E[L(,,x)] < oc
+ t2)E[Lt (,, x)] k < oc, k >_
for every k
>_
1.
Hence, E[Lt(x)]
1.
Now
we can formulate the result:
Theorem 5: Suppose X is a purely discontinuous local strong martingale and
X
satisfies conditions (A)-(B). Then
increasing process Lt(x
(a) there exists jointly measurable continuous in
such that (34) holds,
<
()
a >1
and for every x
in
addition, condition (C) holds, then for every
if,
(c)
has
all
orders.
moments
Lt(x
of
Remark 3: The existence and estimations of moments of local time for oneparameter purely discontinuous local martingales were obtained in [1].
Q
o
ECI)()
,
R2+
YU. MISHURA and YA. OLTSIK
150
Acknowledgement
The authors thank the associate editor, Professor A. Rosalsky, and the anonymous
referee for their valuable suggestions.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
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Cairoli, R. and Walsh, J.B., Stochastic integrals in the plane, A cta Mathematica
134:1-2 (1975), 111-183.
Dellacherie, C. and Meyer, P.A., Probabilits et Potentiel, Hermann, Paris 1980.
Gushchin, A.A., On the general theory of random fields on the plane, Russian
Math. Surveys 37:6 (1982), 55-80.
Mazziotto, G. and Merzbach, E., Regularity and decomposition of two-parameter supermartingales, J. Multiv. Anal. 17 (1985), 38-55.
Mazziotto, G. and Szpirglas, J., Equations du filtrage pour un processus de
Poisson mfilangfi deux indices, Cotochastics 4 (1980), 89-119.
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Millet, A. and Sucheston, L, On regularity of multiparameter amarts and martingales, Z. Wahrsch. verw. Gebiete 56 (1981), 21-45.
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