Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 4, Article 62, 2002
ON SOME INEQUALITIES OF LOCAL TIMES OF ITERATED STOCHASTIC
INTEGRALS
LITAN YAN
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCE
T OYAMA U NIVERSITY
3190 G OFUKU , T OYAMA 930-8555
JAPAN .
yan@math.toyama-u.ac.jp
litanyan@dhu.edu.cn
litanyan@hotmail.com
Received 06 July, 2001; accepted 25 June, 2002
Communicated by N.S. Barnett
A BSTRACT. Let X = (Xt , Ft )t≥0 be a continuous local martingale with quadratic variation
process hXi and X0 = 0. Define iterated stochastic integrals In (X) = (In (t, X), Ft ) (n ≥ 0),
inductively by
Z
t
In (t, X) =
In−1 (s, X)dXs
0
with I0 (t, X) = 1 and I1 (t, X) = Xt . In this paper, we obtain some martingale inequalities for
In (X), n = 1, 2, . . . and their local times at any random time.
Key words and phrases: Continuous local martingale, Continuous semimartingale, Iterated stochastic integrals, Local time,
Random time, Burkholder-Davis-Gundy inequalities, Barlow-Yor inequalities.
2000 Mathematics Subject Classification. 60H05, 60G44, 60J55.
1. I NTRODUCTION
Let X = (Xt )t≥0 be a continuous local martingale with quadratic variation process hXi and
X0 = 0, defined on some filtered probability space (Ω, F, P, (Ft )). Consider the corresponding
sequence of iterated stochastic integrals,
In (X) = (In (t, X), Ft )
(n ≥ 0),
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
The author would like to thank Professor N. Kazamaki for his guidance and encouragement in the study of martingales and related fields.
The author wishes also to thank Professor N. Barnett and an anonymous earnest referee for a careful reading of the manuscript and many
helpful comments.
The author’s present address : Department of Mathematics, College of Science, Donghua University, 1882 West Yan’an Rd., Shanghai
200051, China.
055-01
2
L ITAN YAN
defined inductively by
Z
In (t, X) =
(1.1)
t
In−1 (s, X)dXs ,
0
where I0 (t, X) = 1 and I1 (t, X) = Xt .
It is known that there exist positive constants Bn,p and An,p depending only on n and p, such
that the inequalities (see [2, 8])
n
n
2
2
(1.2)
An,p hXiT ≤ sup
|I
(t,
X)|
≤
B
hXi
(0 < p < ∞),
n
n,p T
p
0≤t≤T
p
p
hold for all continuous local martingales X with X0 = 0 and all (Ft )–stopping time T .
On the other hand, M.T. Barlow and M. Yor have established in [1] (see also Theorem 2.4
in [7, p.457]) the following martingale inequalities for local times:
1
1
2 2 cp hXi∞
(0 < p < ∞),
≤ kL∗∞ (X)kp ≤ Cp hXi∞
p
p
where (Lxt (X); t ≥ 0) is the local time of X at x and L∗t (X) = supx∈R Lxt (X). It follows that
for all 0 < p < ∞
n
n
(1.3)
cn,p hXiT2 ≤ kL∗T (n, X)kp ≤ Cn,p hXiT2 p
p
for all (Ft )–stopping times T , where (Lxt (n, X); t ≥ 0) stands for the local time of In (X) at x.
However, it is clear that the inequalities (1.2) and (1.3) are not true when T is replaced by
an arbitrary R+ –valued random time (see, for example, [12] when n = 1). In this paper we
extend (1.2) and (1.3) to any random time.
2. P RELIMINARIES
Throughout this paper, we fix a filtered complete probability space (Ω, F, (Ft ), P ) with
the usual conditions. For any process X = (Xt )t≥0 , denote Xτ∗ = sup0≤t≤τ |Xt | and X ∗ =
sup0≤t<∞ |Xt |. Let c stand for some positive constant depending only on the subscripts whose
value may be different in different appearances, and this assumption is also made for ĉ.
From now on an F–measurable non–negative random variable L : Ω → R+ is called a
random time and we denote by L the collection of all random times, i.e.,
L = {L : L is an F–measurable, non–negative, random variable} .
For any L ∈ L, let (GLt ) be the smallest filtration satisfying the usual conditions which both
contains (Ft ) and makes L a (GLt )–stopping time. Define
ZtL = E 1{L>t} |Ft
and
JL = inf ZsL .
s<L
Then Z L = (ZtL ) is a potential of class (D). Assume that the Doob–Meyer decomposition for
Z L is
Z L = M − A.
(2.1)
For simplicity, in the present paper we assume throughout that L ∈ L avoids (Ft )–stopping
times, i.e.,
for every (Ft )–stopping time T , P (L = T ) = 0.
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S OME INEQUALITIES OF LOCAL TIMES
3
Thus, under the condition, Z L is continuous and so M is also continuous. Furthermore, for any
e with
continuous (Ft )–local martingale X there exists a continuous (GLt )–local martingale X
e t such that
hXiL∧t = hXi
Z L∧t
dhX, M is
e
XL∧t = Xt +
,
ZsL
0
where L∧t = min{L, t}. For more information on X L = (XL∧t )t≥0 and (GLt ), see [10, 11, 12].
Lemma 2.1 ([10]). Let 0 < p < ∞ and L ∈ L. Then the inequalities
p
∗ p
p 1
(2.2)
E (XL ) ≤ cp E 1 + log 2
hXiL2 ,
JL
h
pi
p 1
∗ p
2
2
(2.3)
(X )L
E hXiL ≤ cp E 1 + log
JL
hold for all continuous (Ft )–local martingales X vanishing at zero.
It is known that the inequalities in Lemma 2.1 are the extensions to the Burkholder-DavisGundy inequalities. For the proof, see Proposition 4 in [10, p.122] (or Theorem 13.4 in [12,
p.57]).
Let X now be a continuous semimartingale. Then for every x ∈ R the following Meyer–
Tanaka formula may be considered as a definition of the local time {Lxt (X); t ≥ 0} of X at
x
Z t
|Xt − x| − |X0 − x| =
sgn(Xs − x)dXs + Lxt (X).
0
One may take a version L : (x, t, ω) → Lxt (ω) which is right continuous and has a left limit at
x, and is continuous in t. In particular, if X is a continuous local martingale, then Lxt (X) has a
continuous version in both variables. In this paper, we use such a version of local time.
The fundamental formula of occupation density for a continuous semimartingale is:
Z t
Z ∞
(2.4)
Φ(Xs )dhXis =
Φ(x)Lxt (X)dx
0
−∞
for all bounded, Borel functions Φ : R → R, which gives
∗
hXi∞ ≤ 2X∞
L∗∞ (X)
(2.5)
since Lx∞ = 0 for all x 6∈ [−X ∗ , X ∗ ]. It follows that (see [3]) for all continuous (Ft )–local
martingales X, and all t ≥ 0, x ∈ R and L ∈ L
LxL∧t (X) = Lxt (X L )
(2.6)
if M is continuous, where X L = (XL∧t ). So, we have
(2.7)
hXiL = hX L i∞ ≤ 2L∗∞ (X L )XL∗ = 2L∗L (X)XL∗
by (2.5). Furthermore, the following lemma which can be found in [3] extends the Barlow–Yor
inequalities.
Lemma 2.2. Let 0 < p < ∞ and L ∈ L. Then the inequalities
h
p
p i
p 1
p 1
∗
∗ p
2
2
(2.8)
E LL (X)
hXiL , E 1 + log
(XL )
≤ cp min E 1 + log
JL
JL
and
n h
p io
p
p 1
∗ p
∗
2
LL (X)
(2.9)
max E (XL ) , E hXiL ≤ cp E 1 + log
JL
hold.
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4
L ITAN YAN
Remark 2.3. In [3], C. S. Chou proved that (2.8) and (2.9) hold for 1 ≤ p < ∞. In fact, when
0 < p < 1 (2.8) and (2.9) are also true from the proof in [3].
3. I NEQUALITIES AND
PROOFS
In this section, we shall extend (1.2) and (1.3) to any random time L ∈ L.
Theorem 3.1. Let 0 < p < ∞ and L ∈ L. Then the inequalities
h
np
p i
np 1
∗
(3.1)
≤ cn,p E 1 + log 2
hXiL2 ,
E In (L, X)
JL
h
p i
np 1
∗ np
∗
2
(3.2)
(XL )
,
E In (L, X)
≤ cn,p E 1 + log
JL
h
pi
np
np 1
2
2
E hIn (X)iL ≤ cn,p E 1 + log 2
hXiL ,
(3.3)
JL
h
pi
np 1
∗
np
2
(3.4)
E hIn (X)iL ≤ cn,p E 1 + log 2
(XL )
JL
hold for all continuous local martingales X with X0 = 0 and n = 1, 2, . . ..
Proof. Let n ≥ 1, L ∈ L and let X be a continuous local martingale.
(3.1) can be verified by induction. In fact, when n = 1 (3.1) is true from (2.2). Now suppose
that (3.1) is true for 2, . . . , n − 1. Then we have
h
np
np i
n−1
np 1
∗
2
E In−1 (L, X)
≤ cn,p E 1 + log 2
hXiL .
JL
On the other hand, from (1.1) we see that
Z t
2
2
hIn (X)it =
In−1 (s, X) dhXis ≤ sup In−1 (s, X) hXit
0≤s≤t
0
for all t ≥ 0, which gives
2
∗
hIn (X)iL ≤ In−1
(L, X) hXiL .
(3.5)
Thus, by applying (2.2), (3.5) and then applying the Hölder inequality with exponents s = n
n
, and noting
and r = n−1
(a + b)n ≤ cn (an + bn )
(a, b ≥ 0),
we find
E
h
p
In∗ (L, X)
i
p
1
2
hIn (X)iL
≤ cp E 1 + log
JL
p
p
p 1
∗
2
≤ cp E 1 + log 2
In−1 (L, X) hXiL
JL
n1 h
n
np
np i n−1
p 1
n
∗
2
E In−1
(L, X) n−1
≤ cp E 1 + log 2
hXiL
JL
np
np 1
≤ cn,p E 1 + log 2
hXiL2 .
JL
p
2
This establishes (3.1).
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S OME INEQUALITIES OF LOCAL TIMES
5
Now, we verify (3.2). From the well-known correspondence of iterated stochastic integral
In (X) and the Hermite polynomial of degree n (see [4, 7])
In (t, X) =
1
Hn (Xt , hXit ),
n!
n
2 dn
−x2
where Hn (x, y) = y 2 hn √xy (y > 0) and hn (x) = (−1)n ex dx
is the Hermite polynone
mial of degree n, more generally, Hn (x, y) can be defined as
x2
Hn (x, y) = (−y)n e 2y
∂ n − x2y2
e ,
∂xn
we see that iterated stochastic integrals In (X), n = 1, 2, . . . have the representation
n
In (t, X) =
(3.6)
[2]
X
Cn(j) Xtn−2j hXijt ,
j=0
j
(j)
1
where Cn = − 21 (n−2j)!j!
and [x] stands for the integer part of x.
On the other hand, for 0 < j < n2 , by using the Hölder inequality with exponents s =
n
and r = 2j
, we get
E
h
(XL∗ )(n−2j)p hXijp
L
i
n
n−2j
2j
h
np i
∗ np n−2j
n
2
n
≤ E (XL )
E hXiL
2jn
∗ np n−2j
np 1
∗
np
n
≤ cn,p E (XL )
E 1 + log 2
(XL )
JL
np 1
∗
np
≤ cn,p E 1 + log 2
(XL )
.
JL
Clearly, the inequality above is also true for j = n2 and j = 0.
Combining this with (3.6), we get for 0 < p ≤ 1
[n]
2
h
i
X
p i
|Cn(j) |p E (XL∗ )(n−2j)p (hXiL )jp
E In∗ (L, X)
≤ cp
h
j=0
≤ cn,p E
1 + log
np
2
1 ∗ np
(XL )
JL
and for 1 < p < ∞
[n]
2
h
i1
p i p1 X
jp p
∗
(j)
∗ (n−2j)p
≤
|Cn |E (XL )
hXiL
E In (L, X)
h
j=0
≤ cn,p E
1 + log
np
2
1 ∗ np
(XL )
JL
p1
.
This gives (3.2).
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L ITAN YAN
Next, from (3.5) and (3.1) we see that
h
h
pi
pi
p
∗
E hIn (X)iL2 ≤ E In−1
(L, X) hXiL2
n−1
h
h
np i 1
np i
n−1
n
n
∗
E hXiL2
≤ E In−1 (L, X)
n−1
h
n
np i 1
np
np 1
n
2
2
E hXiL2
hXiL
≤ cn,p E 1 + log
JL
np
np 1
2
hXiL .
≤ cn,p E 1 + log 2
JL
Finally, from (3.5), (3.2) and (2.3), we have
h
h
pi
pi
p
∗
E hIn (X)iL2 ≤ E In−1
(L, X) hXiL2
n−1
h
n
np i 1
np 1
n
∗
np
≤ cn,p E 1 + log 2
(XL )
E hXiL2
JL
np 1
∗ np
2
.
≤ cn,p E 1 + log
(XL )
JL
This completes the proof of Theorem 3.1.
Theorem 3.2. Let 0 < p < ∞ and L ∈ L. Then the inequalities
h
np
p i
np 1
∗
2
2
E LL (n, X)
≤ cn.p E 1 + log
hXiL ,
(3.7)
JL
h
p i
np 1
∗ np
∗
(3.8)
E LL (n, X)
≤ cn,p E 1 + log
(XL )
,
JL
h
np
p i
np 1
∗
∗
(3.9)
E LL (n, X)
≤ cn,p E 1 + log
LL (X)
,
JL
h
p i
np
np 1
∗
∗
(3.10)
LL (X)
,
E In (L, X)
≤ cn,p E 1 + log
JL
h
pi
np
np 1
∗
2
(3.11)
E hIn (X)iL ≤ cn,p E 1 + log
LL (X)
JL
hold for all continuous local martingales X with X0 = 0 and n = 1, 2, . . ..
Proof. Let n ≥ 2, 0 < p < ∞ and let X be a continuous local martingale.
First we prove (3.7). From (2.8), (3.5), (3.1) and the Hölder inequality with exponents s = n
n
and r = n−1
, we have
h
p
p i
p 1
∗
2
2
E LL (n, X)
≤ cp E 1 + log
hIn (X)iL
JL
p
p
p 1
∗
2
≤ cp E 1 + log 2
In−1 (L, X) hXiL
JL
n1 h
n−1
np
np i
n−1
np 1
n
∗
2
2
≤ cp E 1 + log
hXiL
E In−1 (L, X)
JL
np
np 1
2
2
≤ cn.p E 1 + log
hXiL .
JL
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S OME INEQUALITIES OF LOCAL TIMES
7
Now, by using (3.7), (2.7) and Lemma 2.2, we have
h
np
p i
np 1
∗
2
E LL (n, X)
≤ cp E 1 + log 2
hXiL
JL
np
np 1
np
∗
∗
(XL ) 2 LL (X) 2
≤ cp E 1 + log 2
JL
12 h
2
np i 12
np 1
∗
∗ np
2
E LL (X)
(XL )
≤ cp E 1 + log
JL
np 1
∗ np
≤ cn,p E 1 + log
(XL )
JL
and
E
h
p
L∗L (n, X)
i
np
1
2
≤ cn,p E 1 + log
hXiL
JL
np
np 1
∗
∗ np
2
2
2
≤ cn,p E 1 + log
LL (X) (XL )
JL
np
np 1
∗
≤ cn,p E 1 + log
LL (X)
,
JL
np
2
which give (3.8) and (3.9).
Next, from (3.1), (2.7) and (2.9), we have
h
np
p i
np 1
∗
E In (L, X)
≤ cn,p E 1 + log 2
hXiL2
JL
np
np 1
∗
≤ cn,p E 1 + log
LL (X)
.
JL
Finally, from (3.3), (2.7) and (2.9) we have
h
pi
np
np 1
2
2
E hIn (X)iL ≤ cn,p E 1 + log 2
hXiL
JL
np
np 1
∗
≤ cn,p E 1 + log
LL (X)
.
JL
This completes the proof of Theorem 3.2.
Now, we consider the reverse of the inequalities in Theorem 3.1 and Theorem 3.2. Let L ∈ L
and 0 < p < ∞. Then the inequalities
p
p
np 1
1 ∗
np
2
(3.12) E 1 + log 2
hIn (X)iL ≤ cn,p E 1 + log
In (L, X)
(n ≥ 1)
JL
JL
follow from (2.5) and Lemma 2.2 for all continuous local martingales X with X0 = 0. Furthermore, in [11, p.161] M. Yor showed that for any non-increasing continuous function g :
(0, 1] → R+ the inequality
h
i
h
1i
∗
(3.13)
E g(JL )XL ≤ cg E (gg 1 )(JL )hXiL2
2
holds for all continuous local martingales X with X0 = 0, where gγ (x) = 1 + logγ
0, x ∈ (0, 1]). As a consequence of the inequality, we have
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1
x
(γ ≥
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L ITAN YAN
Lemma 3.3. Let 0 < p < ∞ and L ∈ L. Then the inequality
p
γp 1
(γ+ 12 )p 1
∗ p
2
(3.14)
E 1 + log
(XL ) ≤ cγ,p E 1 + log
hXiL
JL
JL
(γ ≥ 0)
holds for all continuous local martingales X with X0 = 0.
Proof. Let γ ≥ 0 and let X be a continuous local martingale. Then we have from (3.13)
1
γ+ 12 1
γ 1
∗
2
hXiL ,
XL ≤ cγ E 1 + log
(3.15)
E 1 + log
JL
JL
1
since gγ is non-increasing and (gγ g 1 )(x) ≤ cγ 1 + logγ+ 2 x1 .
2
Now, denote for t ≥ 0
1
γ+ 12 1
γ 1
∗
2
At = 1 + log
XL∧t and Bt = 1 + log
hXiL∧t
.
JL
JL
Then for any couple (S, T ) of stopping times S, T with T ≥ S ≥ 0
γ 1
∗
∗
E AT − AS = E 1 + log
(XL∧T − XL∧S )
JL
γ 1
sup |XL∧t − XL∧S |1{S<T }
≤ E 1 + log
JL S≤t≤T
γ 1
L
L
sup |X
− XS |1{S<T }
= E 1 + log
JL t≥0 T ∧(S+t)
γ 1
L
≡ E 1 + log
sup |(XT ∧(S+t) − XS ) |1{S<T } ,
JL t≥0
where XtL ≡ Xt∧L .
Observe that (X(S+t)∧T − XS )1{S<T } , t ≥ 0 is a continuous (FS+t )–local martingale, we find
by (3.15)
1
γ+ 21 1
2
E AT − AS ≤ cγ E 1 + log
hXiL∧T 1{S<T }
JL
h
i = E cγ BT 1{S<T } ≤ cγ BT ∞ P (S < T ).
It follows from Lemma 7 and Lemma 8 in [5] with C = cγ B, α = β = 1 that for all 0 < p < ∞
p
p
p
γ 1
γ+ 21 1
∗ p
2
E 1 + log
(XL ) ≤ cγ,p E 1 + log
hXiL .
JL
JL
Thus, (3.14) follows from the inequalities
ĉp (ap + bp ) ≤ (a + b)p ≤ cp (ap + bp )
(p, a, b ≥ 0).
This completes the proof.
On the other hand, in [2], E. Carlen and P. Krée obtained the identity
n
X
(n − j)! 2
2
In (t, X)In−2 (t, X) = In−1
(t, X) −
In−j (t, X)hXij−1
t
n!
j=1
(n ≥ 2)
for all t ≥ 0 and all continuous local martingales X with X0 = 0. It follows that
1
n−1 2
hXin−1
≤
I (t, X) − In (t, X)In−2 (t, X) (n ≥ 2).
t
n!
n n−1
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S OME INEQUALITIES OF LOCAL TIMES
9
Integrating both sides of the inequality above on [0, t] with respect to the measure dhXit , we
get
Z t
1
n
2
hXit ≤ (n − 1)hIn (X)it − n
In (s, X)In−2 (s, X)dhXis (n ≥ 2)
n!
0
Rt 2
since hIn (X)it = 0 In−1
(s, X)dhXis , which gives
n
(3.16)
√
1 √
1
hXit2
∗
√
≤ n − 1 In (X) t2 + n In∗ (t, X)In−2
(t, X)hXit 2
n!
(n ≥ 2).
1
Theorem 3.4. Let 0 < p < ∞ and L ∈ L. If V is one of the three random variables XL∗ , hXiL2
and L∗L (X), then the inequalities
np p
np 1
∗
(3.17)
E V
≤ cn,p E 1 + log
In (L, X)
,
JL
p
np (n+ 12 )p 1
2
(3.18)
hIn (X)iL ,
E V
≤ cn,p E 1 + log
JL
np p
(2n+1)p 1
∗
(3.19)
E V
≤ cn,p E 1 + log
LL (n, X)
JL
hold for all continuous local martingales X with X0 = 0 and n = 1, 2, . . ..
Proof. Let n ≥ 2, 0 < p < ∞ and let X be a continuous local martingale.
For n ≥ 3, by applying the Hölder inequality with exponents s = n and r =
Theorem 3.1 we have
h
h
h
np i 2
p i
np i n−2
n
n
∗
∗
E In−2
(L, X)hXiL
≤ E In−2
(L, X) n−2
E hXiL2
h
np i
np 1
hXiL2 .
≤ cn,p E 1 + log 2
JL
n
n−2
and
Clearly, the inequality above is also true for n = 2.
It follows from (3.16) that for n ≥ 2
1 p np
np 1
2
√
E 1 + log 2
hXiL
JL
n!
√
1
1 p
√
np 1
∗
∗
≤ E 1 + log 2
n − 1hIn (X)iL2 + n In (L, X)In−2 (L, X)hXiL 2
JL
p
np 1
2
≤ ĉn,p E 1 + log 2
hIn (X)iL
JL
1
p i 12
p 2 h ∗
np 1
∗
E In−2 (L, X)hXiL
+ cn,p E 1 + log
In (L, X)
JL
p
np 1
≤ ĉn,p E 1 + log 2
hIn (X)iL2
JL
1 12
np
p 2
np 1
np 1
∗
2
2
+ cn,p E 1 + log
In (L, X)
E 1 + log
hXiL
.
JL
JL
J. Inequal. Pure and Appl. Math., 3(4) Art. 62, 2002
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10
L ITAN YAN
Combining this with (3.12), we get the quadratic inequality as follows
np
np 1
2
E 1 + log 2
hXiL
JL
1
p
p 2
np 1
np 1
∗
∗
≤ ĉn,p E 1 + log
In (L, X)
+ cn,p E 1 + log
In (L, X)
JL
JL
12
np
np 1
.
hXiL2
× E 1 + log 2
JL
Solving the above quadratic inequality leads to the inequality
np
p
np 1
1 ∗
np
2
(3.20)
E 1 + log 2
hXiL ≤ cn,p E 1 + log
In (L, X)
.
JL
JL
Consequently, by Lemma 3.3
np
p2
np 1
1
1 (n+
)p
2
hXiL ≤ cn,p E 1 + log 2
In (X) L ,
(3.21)
E 1 + log 2
JL
JL
and so by (2.5) and (2.9)
np
p
np 1
1 ∗
(2n+1)p
2
(3.22)
hXiL ≤ cn,p E 1 + log
LL (n, X)
.
E 1 + log 2
JL
JL
Now, the inequalities (3.17) – (3.19) are consequences of (3.20) – (3.22) by Lemma 2.1 and
Lemma 2.2. This completes the proof.
Remark 3.5. Let 0 < p < ∞ and L ∈ L. As some special cases of the inequalities in
Theorem 3.4, we can show that the inequalities
h
i
p
p 1
p
∗
(3.23)
E hXiL ≤ cp E 1 + log 2
I2 (L, X)
,
JL
h
i
p
p 1
p
2
2
E hXiL ≤ cp E 1 + log
(3.24)
hI2 (X)iL ,
JL
h
i
p
p 1
∗ 2p
∗
(3.25)
E (XL )
≤ cp E 1 + log 2
I2 (L, X)
,
JL
h
i
p
p 1
∗ 2p
2
E (XL )
≤ cp E 1 + log 2
hI2 (X)iL ,
(3.26)
JL
h
i
p
p 1
∗ 2p
∗
(3.27)
E (XL )
≤ cp E 1 + log
LL (2, X)
,
JL
h
i
p
p
p 1
∗
(3.28)
E hXiL ≤ cp E 1 + log
LL (2, X)
,
JL
h
2p i
p
2p 1
∗
∗
(3.29)
E LL (X)
≤ cp E 1 + log
I2 (L, X)
,
JL
h
2p i
p2
3p 1
∗
(3.30)
E LL (X)
≤ cp E 1 + log 2
I2 (X) L ,
JL
h
2p i
p
3p 1
∗
∗
(3.31)
≤ cp E 1 + log
LL (2, X)
E LL (X)
JL
J. Inequal. Pure and Appl. Math., 3(4) Art. 62, 2002
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S OME INEQUALITIES OF LOCAL TIMES
11
hold for all continuous local martingales X with X0 = 0. In fact, from (3.16) we have
h
i
h
h
p i
p i
E hXipL ≤ ĉp E I2 (X) L2 + cp E I2∗ (L, X)hXiL 2
for 0 < p < ∞ and so
i
h
i 21
h
h
p i
p i 12 h
E hXipL .
E hXipL ≤ ĉp E I2 (X) L2 + cp E I2∗ (L, X)
Combining this with Lemma 2.1, we find
h
i
p
p 1
p
∗
E hXiL ≤ ĉp E 1 + log 2
I2 (L, X)
IL
1
i1
p 2 h
p 1
p 2
∗
2
E hXiL
+ cp E 1 + log
I2 (L, X)
IL
and
E
h
hXipL
i
≤ ĉp E
p
1 1 + log
I2 (X) L2
IL
p
2
+ cp E
p
1
1 + log
hIn (X)iL2
IL
p
2
12
h
i 12
E hXipL .
The above quadratic inequalities lead to (3.23) and (3.24).
Next, observe that from (3.6)
(XL∗ )2 ≤ 2I2∗ (L, X) + hXiL ,
we obtain the inequalities (3.25) – (3.28).
Finally, combining (3.16) with Lemma 3.3, we get
p
p 1
E 1 + log
hXiL
JL
√
1
12 p
p 1
∗
2
≤ E 1 + log
2hI2 (X)iL + 2 I2 (L, X)hXiL
JL
p
p 1
2
≤ ĉp E 1 + log
hI2 (X)iL
JL
1 12
p 2
p
p 1
p 1
∗
+ cp E 1 + log
I2 (L, X)
E 1 + log
hXiL .
JL
JL
p
3p 1
≤ ĉp E 1 + log 2
hI2 (X)iL2
JL
12 12
p
3p 1
p
p 1
2
2
hXiL ,
+ cp E 1 + log
hI2 (X)iL E 1 + log
JL
JL
which gives a quadratic inequality
x2 − ĉp y 2 − cp xy ≤ 0
(ĉp , cp ≥ 0)
with
x=E
1
1 + log
hXipL
JL
p
J. Inequal. Pure and Appl. Math., 3(4) Art. 62, 2002
12
and y = E
1 + log
3p
2
p
1
hI2 (X)iL2
JL
12
.
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12
L ITAN YAN
Solving the quadratic inequality leads to
p
3p 1
p
p 1
E 1 + log
hXiL ≤ cp E 1 + log 2
hI2 (X)iL2 ,
JL
JL
which gives (3.30) and (3.31).
Thus, we obtain the inequalities (3.23) – (3.31).
R EFERENCES
[1] M.T. BARLOW
237–254.
AND
M. YOR, (Semi–)Martingale inequalities and local times, Z. W., 55 (1981),
[2] E. CARLEN AND P. KRÉE, Lp –estimates on iterated stochastic integrals, Ann. Probab., 19 (1991),
354–368.
[3] C.S. CHOU, On some inequalities of local time, J. Theoret. Probab., 8(1) (1995), 17–22.
[4] K. L. CHUNG AND R. J. WILLIAMS, Introduction to stochastic integration, Second Edition,
Boston, Basel and Stuttgart, Birkhäuser 1990.
[5] S. D. JACKA AND M. YOR, Inequalities for non-moderate functions of a pair of stochastic processes, Proc. London Math. Soc., 67 (1993), 649–672.
[6] E. LENGLART, D. LÉPINGLE AND M. PRATELLI, Présentation unifiée de certaines inégalités
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New York, Springer 1980.
[7] D. REVUZ AND M. YOR, Continuous Martingales and Brownian Motion, Third edition, Berlin,
Heidelberg and New York, Springer–Varlag 1999.
[8] L. YAN, Some inequalities for continuous martingales associated with the Hermite polynomials,
Kobe J. Math., 17 (2000), 191–200.
[9] L. YAN, Two inequalities for iterated stochastic integrals, Archiv der Mathematik, to appear.
[10] M. YOR, Inégalités de martingales continues arrétées á un temps quelconque (I): théorémes géraux.
Lect. Notes in Math., 1118, 110–146, Springer, Berlin 1985.
[11] M. YOR, Inégalités de martingales continues arrétées á un temps quelconque (II): le rôle de certains
espaces BM O, Lect. Notes in Math., 1118, 147–171, Springer, Berlin 1985.
[12] M. YOR, Some Aspects of Brownian motion, Part II: Some recent martingale problems, Lect. in
Math. ETH Zürich, Birkhäuser 1997.
J. Inequal. Pure and Appl. Math., 3(4) Art. 62, 2002
http://jipam.vu.edu.au/