Journal of Inequalities in Pure and
Applied Mathematics
ON SOME INEQUALITIES OF LOCAL TIMES OF ITERATED
STOCHASTIC INTEGRALS
volume 3, issue 4, article 62,
2002.
LITAN YAN
Department of Mathematics
Faculty of Science
Toyama University
3190 Gofuku, Toyama 930-8555
Japan.
EMail: 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
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
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
In (t, X) =
t
In−1 (s, X)dXs
0
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
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.
Litan Yan
2000 Mathematics Subject Classification: 60H05, 60G44, 60J55
Key words: Continuous local martingale, Continuous semimartingale, Iterated
stochastic integrals, Local time, Random time, Burkholder-Davis-Gundy
inequalities, Barlow-Yor inequalities
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Inequalities and proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
9
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1.
Introduction
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),
defined inductively by
Z
In (t, X) =
(1.1)
t
In−1 (s, X)dXs ,
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
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
2
(1.2)
An,p hXiT ≤ sup |In (t, X)|
p
0≤t≤T
p
n
2
≤ Bn,p hXiT (0 < p < ∞),
p
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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
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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
∗
2
2
(1.3)
cn,p hXiT ≤ kLT (n, X)kp ≤ Cn,p hXiT 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.
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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2.
Preliminaries
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.,
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
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
and
JL = inf ZsL .
ZtL = E 1{L>t} |Ft
s<L
Then Z L = (ZtL ) is a potential of class (D). Assume that the Doob–Meyer
decomposition for Z L is
(2.1)
Z L = M − A.
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.
Litan Yan
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Thus, under the condition, Z L is continuous and so M is also continuous. Furthermore, for any continuous (Ft )–local martingale X there exists a continuous
e with hXiL∧t = hXi
e t such that
(GLt )–local martingale X
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
(2.2)
hXiL ,
E (XL ) ≤ cp E 1 + log 2
JL
h
pi
p 1
p
∗
2
(2.3)
(X )L
E hXiL ≤ cp E 1 + log 2
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-Davis-Gundy 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
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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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 ∞
Φ(Xs )dhXis =
Φ(x)Lxt (X)dx
(2.4)
−∞
0
for all bounded, Borel functions Φ : R → R, which gives
(2.5)
∗
hXi∞ ≤ 2X∞
L∗∞ (X)
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
(2.6)
LxL∧t (X)
=
Lxt (X L )
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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L
if M is continuous, where X = (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 i
(2.8) E L∗L (X)
p
p 1
1 ∗ p
p
2
≤ cp min E 1 + log 2
hXiL , E 1 + log
(XL )
JL
JL
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and
(2.9)
n h
p io
1
p
p
∗ p
∗
max E (XL ) , E hXiL2
≤ cp E 1 + log
LL (X)
JL
hold.
Remark 2.1. 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].
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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3.
Inequalities 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
∗
2
E In (L, X)
≤ cn,p E 1 + log 2
hXiL ,
(3.1)
JL
h
p i
np 1
∗
np
∗
(3.2)
E In (L, X)
≤ cn,p E 1 + log 2
(XL )
,
JL
h
np
pi
np 1
2
2
2
(3.3)
hXiL ,
E hIn (X)iL ≤ cn,p E 1 + log
JL
h
pi
np 1
∗ np
2
2
(3.4)
E hIn (X)iL ≤ cn,p E 1 + log
(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 Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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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
0≤s≤t
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for all t ≥ 0, which gives
(3.5)
2
∗
hIn (X)iL ≤ In−1
(L, X) hXiL .
Thus, by applying (2.2), (3.5) and then applying the Hölder inequality with
n
exponents s = n and r = n−1
, and noting
(a + b)n ≤ cn (an + bn )
(a, b ≥ 0),
we find
h
p
p i
p 1
∗
2
hIn (X)iL
E In (L, X)
≤ cp E 1 + log 2
JL
p
p
p 1
∗
2
≤ cp E 1 + log 2
In−1 (L, X) hXiL
JL
n1 h
n−1
n
np
np i
n−1
p 1
n
∗
2
2
hXiL
E In−1 (L, X)
≤ cp E 1 + log
JL
np
np 1
2
2
≤ cn,p E 1 + log
hXiL .
JL
This establishes (3.1).
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])
1
In (t, X) = Hn (Xt , hXit ),
n!
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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n
2 dn
−x2
where Hn (x, y) = y 2 hn √xy (y > 0) and hn (x) = (−1)n ex dx
is the
ne
Hermite polynomial of degree n, more generally, Hn (x, y) can be defined as
n
Hn (x, y) = (−y) e
x2
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 ,
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
j=0
(j)
Litan Yan
1 j
1
where Cn = − 2 (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
n
n
exponents s = n−2j
and r = 2j
, we get
2j
h
i
h
np i
∗ np n−2j
n
2
n
E (XL∗ )(n−2j)p hXijp
≤
E
(X
)
E
hXi
L
L
L
2jn
∗ np n−2j
np 1
∗
np
n
≤ cn,p E (XL )
E 1 + log 2
(XL )
JL
np 1
2
(XL∗ )np .
≤ cn,p E 1 + log
JL
Clearly, the inequality above is also true for j =
n
2
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and j = 0.
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Combining this with (3.6), we get for 0 < p ≤ 1
[n]
2
h
i
X
p i
E In∗ (L, X)
≤ cp
|Cn(j) |p E (XL∗ )(n−2j)p (hXiL )jp
h
j=0
≤ cn,p E
1 + log
np
2
1 ∗ np
(XL )
JL
and for 1 < p < ∞
1
p
[n
]
2
h
h
i
X
p i
E In∗ (L, X)
≤
|Cn(j) |E (XL∗ )(n−2j)p hXijp
L
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
1
p
Litan Yan
j=0
≤ cn,p E
1 + log
np
2
1 ∗ np
(XL )
JL
p1
.
This gives (3.2).
Next, from (3.5) and (3.1) we see that
h
h
pi
pi
p
∗
(L, X) hXiL2
E hIn (X)iL2 ≤ E In−1
n−1
h
h
np i 1
np i
n−1
n
n
∗
≤ E In−1 (L, X)
E hXiL2
n−1
h
n
np
np i 1
np 1
n
2
hXiL
≤ cn,p E 1 + log 2
E hXiL2
JL
np
np 1
2
≤ cn,p E 1 + log 2
hXiL .
JL
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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
2
≤ cn,p E 1 + log
(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
(3.7)
E LL (n, X)
≤ cn.p E 1 + log
hXiL ,
JL
h
p i
np 1
∗
∗ np
(3.8)
E LL (n, X)
≤ cn,p E 1 + log
(XL )
,
JL
h
p i
np
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
1
np
np
∗
E hIn (X)iL2 ≤ cn,p E 1 + log
(3.11)
LL (X)
JL
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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hold for all continuous local martingales X with X0 = 0 and n = 1, 2, . . ..
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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
n
, we have
exponents s = n and r = n−1
E
h
p
L∗L (n, X)
i
p
1
2
≤ cp E 1 + log
hIn (X)iL
JL
p
p
p 1
∗
2
In−1 (L, X) hXiL
≤ cp E 1 + log 2
JL
n1 h
n−1
np
np i
n−1
np 1
n
∗
2
2
E In−1 (L, X)
hXiL
≤ cp E 1 + log
JL
np
np 1
2
2
≤ cn.p E 1 + log
hXiL .
JL
p
2
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
Title Page
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
≤ cp E 1 + log
(XL )
E LL (X)
JL
np 1
∗ np
≤ cn,p E 1 + log
(XL )
JL
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and
E
h
p
L∗L (n, X)
i
np
1
2
≤ cn,p E 1 + log
hXiL
JL
np
np 1
np
∗
∗
≤ cn,p E 1 + log 2
LL (X) 2 (XL ) 2
JL
np
np 1
∗
LL (X)
,
≤ cn,p E 1 + log
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
∗
2
2
hXiL
E In (L, X)
≤ cn,p E 1 + log
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
E hIn (X)iL2 ≤ cn,p E 1 + log 2
hXiL2
JL
np
np 1
∗
≤ cn,p E 1 + log
LL (X)
.
JL
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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This completes the proof of Theorem 3.2.
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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
(3.12) E
1 + log
np
2
p
1
hIn (X)iL2
JL
p
np 1
∗
In (L, X)
≤ cn,p E 1 + log
JL
(n ≥ 1)
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
Lemma 3.3. Let 0 < p < ∞ and L ∈ L. Then the inequality
(3.14) E
1 + log
γp
1 ∗ p
(XL )
JL
≤ cγ,p E
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holds for all continuous local martingales X with X0 = 0, where gγ (x) =
1 + logγ x1 (γ ≥ 0, x ∈ (0, 1]). As a consequence of the inequality, we have
On Some Inequalities of Local
Times of Iterated Stochastic
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1 + log
(γ+ 12 )p
p
1
hXiL2
JL
holds for all continuous local martingales X with X0 = 0.
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Proof. Let γ ≥ 0 and let X be a continuous local martingale. Then we have
from (3.13)
1
γ 1
γ+ 12 1
∗
2
(3.15)
E 1 + log
XL ≤ cγ E 1 + log
hXiL ,
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
1 1
1 ∗
2
.
hXiL∧t
At = 1 + logγ
XL∧t and Bt = 1 + logγ+ 2
JL
JL
Then for any couple (S, T ) of stopping times S, T with T ≥ S ≥ 0
γ 1
∗
∗
(XL∧T − XL∧S )
E AT − AS = E 1 + log
JL
γ 1
≤ E 1 + log
sup |XL∧t − XL∧S |1{S<T }
JL S≤t≤T
γ 1
L
L
= E 1 + log
sup |X
− XS |1{S<T }
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
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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martingale, we find by (3.15)
1
γ+ 12 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+ 12 1
∗ p
2
hXiL .
E 1 + log
(XL ) ≤ cγ,p E 1 + log
JL
JL
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
Thus, (3.14) follows from the inequalities
ĉp (ap + bp ) ≤ (a + b)p ≤ cp (ap + bp )
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(p, a, b ≥ 0).
Contents
This completes the proof.
On the other hand, in [2], E. Carlen and P. Krée obtained the identity
In (t, X)In−2 (t, X) =
2
In−1
(t, X)
−
n
X
(n − j)!
j=1
n!
2
In−j
(t, X)hXij−1
t
(n ≥ 2)
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for all t ≥ 0 and all continuous local martingales X with X0 = 0. It follows
that
n−1 2
1
hXin−1
≤
I (t, X) − In (t, X)In−2 (t, X)
t
n!
n n−1
JJ
J
(n ≥ 2).
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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
√
1
hXi 2
(3.16) √ t ≤ n − 1 In (X) t2
n!
1
√
∗
(t, X)hXit 2
+ n In∗ (t, X)In−2
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
(n ≥ 2).
Litan Yan
Theorem 3.4. Let 0 < p < ∞ and L ∈ L. If V is one of the three random
1
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
hIn (X)iL ,
(3.18)
E V
≤ cn,p E 1 + log
JL
np p
(2n+1)p 1
∗
(3.19)
LL (n, X)
E V
≤ cn,p E 1 + log
JL
JJ
J
hold for all continuous local martingales X with X0 = 0 and n = 1, 2, . . ..
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Proof. Let n ≥ 2, 0 < p < ∞ and let X be a continuous local martingale.
J. Ineq. Pure and Appl. Math. 3(4) Art. 62, 2002
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For n ≥ 3, by applying the Hölder inequality with exponents s = n and
n
r = n−2
and Theorem 3.1 we have
E
h
p
∗
In−2
(L, X)hXiL
i
n−2
h
np i 2
np i
n−2
n
n
≤E
E hXiL2
h
np i
np 1
≤ cn,p E 1 + log 2
hXiL2 .
JL
h
∗
In−2
(L, X)
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
np 1
n − 1hIn (X)iL2
≤ E 1 + log 2
JL
21 p i
√
∗
∗
+ n In (L, X)In−2 (L, X)hXiL
p
np 1
2
≤ ĉn,p E 1 + log 2
hIn (X)iL
JL
1
p 2 h ∗
p i 12
np 1
∗
+ cn,p E 1 + log
In (L, X)
E In−2 (L, X)hXiL
JL
p
np 1
2
≤ ĉn,p E 1 + log 2
hIn (X)iL
JL
1 12
np
p 2
np 1
np 1
∗
2
In (L, X)
E 1 + log 2
hXiL
+ cn,p E 1 + log
.
JL
JL
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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Combining this with (3.12), we get the quadratic inequality as follows
np
np 1
2
E 1 + log 2
hXiL
JL
p
np 1
∗
≤ ĉn,p E 1 + log
In (L, X)
JL
1
p 2
np 1
∗
+ cn,p E 1 + log
In (L, X)
JL
12
np
np 1
.
× E 1 + log 2
hXiL2
JL
Solving the above quadratic inequality leads to the inequality
np
np 1
1
p
np
∗
.
In (L, X)
hXiL2 ≤ cn,p E 1 + log
(3.20) E 1 + log 2
JL
JL
Consequently, by Lemma 3.3
np
p
np 1
1
1
(n+
)p
(3.21) E 1 + log 2
hXiL2 ≤cn,p E 1 + log 2
In (X) L2 ,
JL
JL
and so by (2.5) and (2.9)
np
np 1
2
(3.22) E 1 + log 2
hXiL
JL
≤ cn,p E
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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1 + log(2n+1)p
p
1 ∗
LL (n, X)
.
JL
J. Ineq. Pure and Appl. Math. 3(4) Art. 62, 2002
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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 3.1. Let 0 < p < ∞ and L ∈ L. As some special cases of the
inequalities in Theorem 3.4, we can show that the inequalities
i
h
p
p 1
p
∗
(3.23)
I2 (L, X)
,
E hXiL ≤ cp E 1 + log 2
JL
h
i
p
p 1
p
2
(3.24)
hI2 (X)iL ,
E hXiL ≤ cp E 1 + log 2
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
(3.26)
hI2 (X)iL2 ,
E (XL )
≤ cp E 1 + log 2
JL
h
i
p
p 1
∗
∗ 2p
(3.27)
LL (2, X)
,
E (XL )
≤ cp E 1 + log
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
∗
E LL (X)
≤ cp E 1 + log 2
I2 (X) L ,
(3.30)
JL
h
2p i
p
3p 1
∗
∗
(3.31)
LL (2, X)
E LL (X)
≤ cp E 1 + log
JL
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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hold for all continuous local martingales X with X0 = 0. In fact, from (3.16)
we have
h
i
h
h
p2 i
p2 i
p
∗
E hXiL ≤ ĉp E I2 (X) L + cp E I2 (L, X)hXiL
for 0 < p < ∞ and so
h
i
h
h
i 12
p i
p i 12 h
E hXipL ≤ ĉp E I2 (X) L2 + cp E I2∗ (L, X)
E hXipL .
Combining this with Lemma 2.1, we find
h
i
p
p 1
p
∗
I2 (L, X)
E hXiL ≤ ĉp E 1 + log 2
IL
12 h
i1
p 1
p
p 2
∗
2
+ cp E 1 + log
I2 (L, X)
E hXiL
IL
and
h
i
E hXipL ≤ ĉp E
p2
p 1
1 + log 2
I2 (X) L
IL
12 h
i 12
p
p 1
2
+ cp E 1 + log 2
hIn (X)iL E hXipL .
IL
The above quadratic inequalities lead to (3.23) and (3.24).
Next, observe that from (3.6)
(XL∗ )2 ≤ 2I2∗ (L, X) + hXiL ,
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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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
≤ ĉp E 1 + log
hI2 (X)iL2
JL
12 12
1
p
p
p 1
p
+ cp E 1 + log
I2∗ (L, X)
E 1 + log
hXiL .
JL
JL
p
3p 1
2
2
≤ ĉp E 1 + log
hI2 (X)iL
JL
12 12
p
3p 1
1
p
p
2
hXiL ,
hI2 (X)iL E 1 + log
+ cp E 1 + log 2
JL
JL
which gives a quadratic inequality
x2 − ĉp y 2 − cp xy ≤ 0
(ĉp , cp ≥ 0)
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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with
12
12
p
3p 1
1
p
2
x = E 1 + log
hXiL
and y = E 1 + log 2
hI2 (X)iL .
JL
JL
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
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which gives (3.30) and (3.31).
Thus, we obtain the inequalities (3.23) – (3.31).
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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References
[1] M.T. BARLOW AND M. YOR, (Semi–)Martingale inequalities and local
times, Z. W., 55 (1981), 237–254.
[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.
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
[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 de la théorie des martingales, Sém. Proba. XIV,
Lect. Notes in Math., 784, Berlin, Heidelberg and 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.
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[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.
On Some Inequalities of Local
Times of Iterated Stochastic
Integrals
Litan Yan
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