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A REMARK ON A BMO MARTINGALE
2000,20B(4):511-514
.«,~cta,sRcientia
1'4mJJ1,m A REMARK ON A BMO MARTINGALE
1
Xiang Kainan ( ~*lfJ ) Institute of Applied Mathematics, Academia Sinica, Beijing 100080, China Department of probability and statistics, Peking University, Beijing 100871, China Abstract
In this paper, a negative answer to a question raised by Durrett(1984)[1] about
a BMO martingale is given.
Key words
Continuous martingale, uniformly integrable, BMO martingale
1991 MR Subject Classification
Let M 1 ::; p
= (Mt , Ft )
< 00, we set
60KIO
be a uniformly integrable continuous martingale with M o
IIMllBMO
p
= sup II[E[IMoo T
-
O. For
MTIPIFT]]l/Plloo,
where the supremum is taken over all stopping times T. Set BMO p = {M : Furthermore, all
11.IIBMO
IIMllBMO < oo}. It is well known that p
p
BMO p = BMO q ('V1 ::; p::; q).
norms are equivalent and
IIMII BMO p
sup T
IIMoo
MTllp 1 , P(T < oo)p -
where the supremum is taken over all stopping times T satisfying P(T
< 00) > O. In the later
we shall simply write BMO for BMOp • Let L oo be the class of all bounded continuous martingales. Let a(M) be the supremum of the set of all a such that sup E[eaIMoo-MTIIT
< 00] < 00,
T
where the supremum is taken over all stopping times T. Then there exist two constants c, C E
(0,00) such that C
a (M In fact
C
= ~,C =
2ee~11
)::;
inf 11M -
NEL oo
= 2.42. This
NIIBM0
1
::;
(C).
a M
is well known Garnett-Jones theorem. The following
example was discussed in [1] page 214.
Example 1
Let B = (B(t)) be a standard one-dimensional (Ft)-Brownian motion. Set
Ro = 0, and for n ~ 1, set N
n; =
inf{t
= inf{n; B(Rn)
> Rn- 1; IB(t) - B(Rn-1)1 > 1}. Let - B(Rn- 1) = -1}, X,
= B(t 1\ RN ) .
1 Received Oct.9,1998; revised Oct.10,1999. Research supported in part by CNSF, Tianyuan foundation, and the Mathematical Center of Ministry of Education
512
ACTA MATHEMATICA SCIENTIA
Vo1.20 Ser.B
Then X E BMO and IIXIIBM0 1 = ~. It is easy to see that a(X) = log 2 = 0.693, so the G~rnett-J ones theorem gives 0.537
1 = 0.693e
. f II II :::; Z~nLoo X - Z BM0 1
:::;
2.42 0.693
= 3.49.
After the above discussion, Richard Durrett raised the following unsolved question. Question Do we have inf z EL oo IIX - ZIIBM0 1 = IIXIIBMo 1 ? In this paper,we give a negative answer to the question as stated below. Theorem 1 Let X be as in Example 1, the following holds:
inf
ZEL oo
IIX - ZIIBM0 < IIXIIBMOI. 1
Proof Let 1 z, 2n [B(Rn /\ t) - B(Rn-l/\ t)]I[N?n)
= 'E
+ B(R1 /\ t) -
£[B(RN /\ t) - B(RN-l/\ t)],
n~2
where -21(, < e < O. Noticing [N 2: n] = nj~t[B(Rj) - B(Rj-1) = 1] E FRn _ 1 , and B(R n + .) - B(Rn) is independent of FRn ' we can see that Z = (Zt) is a martingale. Clearly, Z E L oo • We claim that 9
IIX - ZIIBM0
(1.1)
1
If T is a stopping time, then on {Rk :::; T order to prove (1.1), we calculate
< Rk+1, N > k}, XT
= k + a for some a E (-1,1). In
The calculation is devided into several steps. (i) Suppose k = 0, then XT = a. We have the following calculation.
IXoo -
(XT - ZT)I
Zoo -
-1
-(l-a)£
o
1+£-212 1 23
1
+e
2 (ii) Suppose k = 1, then XT
0
1+a1 2 2
1+a(!)2 2
2
j(O a) = 1 + a (1 + ~
X oo
2
2
l+a(!)n+l
n
,
probability 1-a
_ ~ + ~) _ (1 - a)2e < 1 < ~.
2 22 24 = 1 + a. We have
IXoo
-
2
Zoo -
2
(XT - ZT)I
1(1 ~ 212 )(1 + a) + e - a£1
1
1(1 - 2\)a + 213 + £1
2
1- (1 - 2\)a + 1 - ~ + 2~ - £1
n
-(1- 212 ) a + (n-1) - ~ + 2n3+2 - £
8
probability 1-a
-2
1+a1 2 2 1 +a(!)2 2 2
l+a(!t 2
2
513
Xiang: A REMARK ON A BMO MARTINGALE
No.4
/(1, a)
= -£ +
a(1 - a) 2 e
< -2£ < 1 <
s·9
3 e ~ - e - 214 2 (1 - 2£ - 213 ) 2 1 e /(1 a) = -( - - -)(a ) + + 1 - -22 + -2 , 8 2 ~ - e 6 8£ 4 (1 - 2£ - 213)2 6 - 8£
<
+ 1-
1 2
-2
e
1
+ -2 + -24
1
+ -24
9 = b1 (c) < -. 8
(4) If a > a3, then 1 e e 3 1 1 e 1 a2 /(1, a) = -(1 - 22) 22 + (2 - 2 + 22 - 26 )a + 1 - 22 - 22 + 2 e 3 3 a(1 - a) + 22 + 24 - 26 2 e 9 e e 9
< - + - - - < -.
8 2 8 8 (iii) Suppose k = 2, then ~T = 2 + a. We have
Xoo
IXoo
1(1 - 213 ) (1 + a)
1
3 n
+e -
1(1 - 213)a + 214
2
I-
probability 1-a -2 1+a1 --2 2
Zoo - (XT - ZT)I
-
ae 1
+ eI
(1 - 2~)a + 1 - 212
+ 23"
-(1 - "!")a 23 + n - 2 _ ..!.. 22
-
1 + a (!)2 2 2
£1
+ _3_ 2 +2 n
£
1+a(!t- 1 2
2
Now similarly to step (ii), we can prove that /(2, a) :S b2 (£) < ~ for a function b2 (£) depending on e. (iv) Let k
~
3. Then XT X oo
= k + a". We have IXoo
k-1
-
Zoo - (XT - ZT)I
1(1 - 2k~1 )(1
+ a) + e -
1(1 - 2k~1)a + 2k~2
k k+1
I-
k+n
-(1 - 2k~1)a
(1 - 2k~1)a
+1-
+n -
2\ 21k
a£1
+ £1
+ 2k~3
- £1
+ 2k+3n +2 -
£
probability 1-a -2 1+a1 --2 2 1 + a (!)2 2 2 1 + a (!t+ 1 2 2
514
ACTA MATHEMATICA SCIENTIA
Vo1.20 Ser.B
Similarly to step (ii), we have the following estimations. 1 - -1- +e (1) If a < - 1 - ~-e = a1, then 2k
+1
f(k, a)
= -[ +
a(l - a) 9 2 e < -2£ < 1 <
s·
£2
f( k a) < 1 + , 3
(4) If a3 < a < 1, then
9 8
f(k a) < ,
e
9
e
9
- -8 < -. 8
e
+ -2 - -8 < -. 8
(vi) By (i)-(iv), we have
Furthermore, for any stopping time T,
Thus we can prove for any stopping time T,
E[lXoo - Zoo - (XT - ZT)IIT < 00] ~ b < this shows that IIX - ZIIBM0 1 < ~, namely infz EL oo /IX
-
ZIIBMo 1
9
S' <
/lXIIBMo 1 •
References 1 Durrett R. Brownian motion and martingales in analysis. Wadsworth Mathematics Series,1984 2 He S W, Wang J G, Yan J A. Semartingale theory and stochastic caculus. Science Press and CRC Press INC, 1992
.«,~cta,sRcientia
1'4mJJ1,m A REMARK ON A BMO MARTINGALE
1
Xiang Kainan ( ~*lfJ ) Institute of Applied Mathematics, Academia Sinica, Beijing 100080, China Department of probability and statistics, Peking University, Beijing 100871, China Abstract
In this paper, a negative answer to a question raised by Durrett(1984)[1] about
a BMO martingale is given.
Key words
Continuous martingale, uniformly integrable, BMO martingale
1991 MR Subject Classification
Let M 1 ::; p
= (Mt , Ft )
< 00, we set
60KIO
be a uniformly integrable continuous martingale with M o
IIMllBMO
p
= sup II[E[IMoo T
-
O. For
MTIPIFT]]l/Plloo,
where the supremum is taken over all stopping times T. Set BMO p = {M : Furthermore, all
11.IIBMO
IIMllBMO < oo}. It is well known that p
p
BMO p = BMO q ('V1 ::; p::; q).
norms are equivalent and
IIMII BMO p
sup T
IIMoo
MTllp 1 , P(T < oo)p -
where the supremum is taken over all stopping times T satisfying P(T
< 00) > O. In the later
we shall simply write BMO for BMOp • Let L oo be the class of all bounded continuous martingales. Let a(M) be the supremum of the set of all a such that sup E[eaIMoo-MTIIT
< 00] < 00,
T
where the supremum is taken over all stopping times T. Then there exist two constants c, C E
(0,00) such that C
a (M In fact
C
= ~,C =
2ee~11
)::;
inf 11M -
NEL oo
= 2.42. This
NIIBM0
1
::;
(C).
a M
is well known Garnett-Jones theorem. The following
example was discussed in [1] page 214.
Example 1
Let B = (B(t)) be a standard one-dimensional (Ft)-Brownian motion. Set
Ro = 0, and for n ~ 1, set N
n; =
inf{t
= inf{n; B(Rn)
> Rn- 1; IB(t) - B(Rn-1)1 > 1}. Let - B(Rn- 1) = -1}, X,
= B(t 1\ RN ) .
1 Received Oct.9,1998; revised Oct.10,1999. Research supported in part by CNSF, Tianyuan foundation, and the Mathematical Center of Ministry of Education
512
ACTA MATHEMATICA SCIENTIA
Vo1.20 Ser.B
Then X E BMO and IIXIIBM0 1 = ~. It is easy to see that a(X) = log 2 = 0.693, so the G~rnett-J ones theorem gives 0.537
1 = 0.693e
. f II II :::; Z~nLoo X - Z BM0 1
:::;
2.42 0.693
= 3.49.
After the above discussion, Richard Durrett raised the following unsolved question. Question Do we have inf z EL oo IIX - ZIIBM0 1 = IIXIIBMo 1 ? In this paper,we give a negative answer to the question as stated below. Theorem 1 Let X be as in Example 1, the following holds:
inf
ZEL oo
IIX - ZIIBM0 < IIXIIBMOI. 1
Proof Let 1 z, 2n [B(Rn /\ t) - B(Rn-l/\ t)]I[N?n)
= 'E
+ B(R1 /\ t) -
£[B(RN /\ t) - B(RN-l/\ t)],
n~2
where -21(, < e < O. Noticing [N 2: n] = nj~t[B(Rj) - B(Rj-1) = 1] E FRn _ 1 , and B(R n + .) - B(Rn) is independent of FRn ' we can see that Z = (Zt) is a martingale. Clearly, Z E L oo • We claim that 9
IIX - ZIIBM0
(1.1)
1
If T is a stopping time, then on {Rk :::; T order to prove (1.1), we calculate
< Rk+1, N > k}, XT
= k + a for some a E (-1,1). In
The calculation is devided into several steps. (i) Suppose k = 0, then XT = a. We have the following calculation.
IXoo -
(XT - ZT)I
Zoo -
-1
-(l-a)£
o
1+£-212 1 23
1
+e
2 (ii) Suppose k = 1, then XT
0
1+a1 2 2
1+a(!)2 2
2
j(O a) = 1 + a (1 + ~
X oo
2
2
l+a(!)n+l
n
,
probability 1-a
_ ~ + ~) _ (1 - a)2e < 1 < ~.
2 22 24 = 1 + a. We have
IXoo
-
2
Zoo -
2
(XT - ZT)I
1(1 ~ 212 )(1 + a) + e - a£1
1
1(1 - 2\)a + 213 + £1
2
1- (1 - 2\)a + 1 - ~ + 2~ - £1
n
-(1- 212 ) a + (n-1) - ~ + 2n3+2 - £
8
probability 1-a
-2
1+a1 2 2 1 +a(!)2 2 2
l+a(!t 2
2
513
Xiang: A REMARK ON A BMO MARTINGALE
No.4
/(1, a)
= -£ +
a(1 - a) 2 e
< -2£ < 1 <
s·9
3 e ~ - e - 214 2 (1 - 2£ - 213 ) 2 1 e /(1 a) = -( - - -)(a ) + + 1 - -22 + -2 , 8 2 ~ - e 6 8£ 4 (1 - 2£ - 213)2 6 - 8£
<
+ 1-
1 2
-2
e
1
+ -2 + -24
1
+ -24
9 = b1 (c) < -. 8
(4) If a > a3, then 1 e e 3 1 1 e 1 a2 /(1, a) = -(1 - 22) 22 + (2 - 2 + 22 - 26 )a + 1 - 22 - 22 + 2 e 3 3 a(1 - a) + 22 + 24 - 26 2 e 9 e e 9
< - + - - - < -.
8 2 8 8 (iii) Suppose k = 2, then ~T = 2 + a. We have
Xoo
IXoo
1(1 - 213 ) (1 + a)
1
3 n
+e -
1(1 - 213)a + 214
2
I-
probability 1-a -2 1+a1 --2 2
Zoo - (XT - ZT)I
-
ae 1
+ eI
(1 - 2~)a + 1 - 212
+ 23"
-(1 - "!")a 23 + n - 2 _ ..!.. 22
-
1 + a (!)2 2 2
£1
+ _3_ 2 +2 n
£
1+a(!t- 1 2
2
Now similarly to step (ii), we can prove that /(2, a) :S b2 (£) < ~ for a function b2 (£) depending on e. (iv) Let k
~
3. Then XT X oo
= k + a". We have IXoo
k-1
-
Zoo - (XT - ZT)I
1(1 - 2k~1 )(1
+ a) + e -
1(1 - 2k~1)a + 2k~2
k k+1
I-
k+n
-(1 - 2k~1)a
(1 - 2k~1)a
+1-
+n -
2\ 21k
a£1
+ £1
+ 2k~3
- £1
+ 2k+3n +2 -
£
probability 1-a -2 1+a1 --2 2 1 + a (!)2 2 2 1 + a (!t+ 1 2 2
514
ACTA MATHEMATICA SCIENTIA
Vo1.20 Ser.B
Similarly to step (ii), we have the following estimations. 1 - -1- +e (1) If a < - 1 - ~-e = a1, then 2k
+1
f(k, a)
= -[ +
a(l - a) 9 2 e < -2£ < 1 <
s·
£2
f( k a) < 1 + , 3
(4) If a3 < a < 1, then
9 8
f(k a) < ,
e
9
e
9
- -8 < -. 8
e
+ -2 - -8 < -. 8
(vi) By (i)-(iv), we have
Furthermore, for any stopping time T,
Thus we can prove for any stopping time T,
E[lXoo - Zoo - (XT - ZT)IIT < 00] ~ b < this shows that IIX - ZIIBM0 1 < ~, namely infz EL oo /IX
-
ZIIBMo 1
9
S' <
/lXIIBMo 1 •
References 1 Durrett R. Brownian motion and martingales in analysis. Wadsworth Mathematics Series,1984 2 He S W, Wang J G, Yan J A. Semartingale theory and stochastic caculus. Science Press and CRC Press INC, 1992