LetX = (Xt,Ft)t≥0 be a continuous local martingale with quadratic variation processhXiandX0 = 0

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http://jipam.vu.edu.au/

Volume 3, Issue 4, Article 62, 2002

ON SOME INEQUALITIES OF LOCAL TIMES OF ITERATED STOCHASTIC INTEGRALS

LITAN YAN

DEPARTMENT OFMATHEMATICS

FACULTY OFSCIENCE

TOYAMAUNIVERSITY

3190 GOFUKU, TOYAMA930-8555 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

ABSTRACT. LetX = (Xt,Ft)t≥0 be a continuous local martingale with quadratic variation processhXiandX0 = 0. Define iterated stochastic integralsIn(X) = (In(t, X),Ft) (n≥0), inductively by

I_n(t, X) = Z t

0

I_n−1(s, X)dX_s

withI0(t, X) = 1andI1(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. INTRODUCTION

LetX = (X_t)t≥0 be a continuous local martingale with quadratic variation processhXiand X₀ = 0, defined on some filtered probability space(Ω,F, P,(F_t)). Consider the corresponding sequence of iterated stochastic integrals,

I_n(X) = (I_n(t, X),F_t) (n ≥0),

ISSN (electronic): 1443-5756 c

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

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defined inductively by

(1.1) In(t, X) =

Z t

0

In−1(s, X)dXs, whereI₀(t, X) = 1andI₁(t, X) = X_t.

It is known that there exist positive constantsB_n,pandA_n,pdepending only onnandp, such that the inequalities (see [2, 8])

(1.2) A_n,p hXi

n 2

T

_p ≤

sup

0≤t≤T

|I_n(t, X)|

p

≤B_n,p hXi

n 2

T

_p (0< p <∞), hold for all continuous local martingalesXwithX₀ = 0and all(F_t)–stopping timeT.

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:

c_p hXi∞¹²

p

≤ kL^∗_∞(X)k_p ≤C_p hXi∞¹²

p

(0< p <∞),

where(L^x_t(X);t ≥0)is the local time ofX atxandL^∗_t(X) = sup_x∈_RL^x_t(X). It follows that for all0< p <∞

(1.3) c_n,p

hXi_Tⁿ² p

≤ kL^∗_T(n, X)k_p ≤C_n,p

hXi_Tⁿ² p

for all(F_t)–stopping timesT, where(L^x_t(n, X);t ≥0)stands for the local time ofI_n(X)atx.

However, it is clear that the inequalities (1.2) and (1.3) are not true when T is replaced by an arbitraryR+–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. PRELIMINARIES

Throughout this paper, we fix a filtered complete probability space (Ω,F,(F_t), P) with the usual conditions. For any process X = (X_t)t≥0, denote X_τ^∗ = sup_0≤t≤τ|X_t| and X^∗ = sup_0≤t<∞|X_t|. Letcstand for some positive constant depending only on the subscripts whose value may be different in different appearances, and this assumption is also made forˆc.

From now on an F–measurable non–negative random variable L : Ω → R+ is called a random time and we denote byLthe collection of all random times, i.e.,

L={L : Lis anF–measurable, non–negative, random variable}.

For anyL∈ L, let(G^L_t)be the smallest filtration satisfying the usual conditions which both contains(F_t)and makesLa(G^L_t)–stopping time. Define

Z_t^L=E

1{L>t}|F_t

and J_L= inf

s<LZ_s^L.

ThenZ^L = (Z_t^L)is a potential of class (D). Assume that the Doob–Meyer decomposition for Z^Lis

(2.1) Z^L=M −A.

For simplicity, in the present paper we assume throughout thatL ∈ Lavoids(F_t)–stopping times, i.e.,

for every(F_t)–stopping timeT,P(L=T) = 0.

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Thus, under the condition,Z^Lis continuous and soM is also continuous. Furthermore, for any continuous (F_t)–local martingale X there exists a continuous(G^L_t)–local martingale Xe with hXiL∧t=hXie tsuch that

XL∧t=Xe_t+ Z L∧t

0

dhX, Mi_s Z_s^L ,

whereL∧t = min{L, t}. For more information onX^L= (XL∧t)t≥0and(G^L_t), see [10, 11, 12].

Lemma 2.1 ([10]). Let0< p <∞andL∈L. Then the inequalities E

(X_L^∗)^p

≤c_pE

1 + log^p² 1 J_L

hXi

p 2

L

, (2.2)

Eh hXi

p 2

L

i ≤c_pE

1 + log^p² 1 J_L

(X^∗)^p_L

(2.3)

hold for all continuous(F_t)–local martingalesXvanishing 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 everyx ∈ R the following Meyer–

Tanaka formula may be considered as a definition of the local time {L^x_t(X);t ≥ 0} of X at x

|X_t−x| − |X₀−x|= Z t

0

sgn(X_s−x)dX_s+L^x_t(X).

One may take a versionL : (x, t, ω)→ L^x_t(ω)which is right continuous and has a left limit at x, and is continuous int. In particular, ifX is a continuous local martingale, thenL^x_t(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:

(2.4)

Z t

0

Φ(X_s)dhXi_s= Z ∞

−∞

Φ(x)L^x_t(X)dx for all bounded, Borel functionsΦ :R→R, which gives

(2.5) hXi∞ ≤2X_∞^∗ L^∗_∞(X)

since L^x_∞ = 0 for all x 6∈ [−X^∗, X^∗]. It follows that (see [3]) for all continuous (Ft)–local martingalesX, and allt≥0, x∈RandL∈L

(2.6) L^x_L∧t(X) = L^x_t(X^L)

ifM is continuous, whereX^L = (XL∧t). So, we have

(2.7) hXiL=hX^Li∞≤2L^∗_∞(X^L)X_L^∗ = 2L^∗_L(X)X_L^∗

by (2.5). Furthermore, the following lemma which can be found in [3] extends the Barlow–Yor inequalities.

Lemma 2.2. Let0< p <∞andL∈L. Then the inequalities (2.8) Eh

L^∗_L(X)pi

≤c_pmin

E

1 + log^p² 1 J_L

hXi

p 2

L

, E

1 + log^p 1 J_L

(X_L^∗)^p

and

(2.9) maxn

E

(X_L^∗)^p , Eh

hXi

p 2

L

io ≤c_pE

1 + log^p 1 J_L

L^∗_L(X)p

hold.

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Remark 2.3. In [3], C. S. Chou proved that (2.8) and (2.9) hold for1≤ p <∞. In fact, when 0< p <1(2.8) and (2.9) are also true from the proof in [3].

3. INEQUALITIES AND PROOFS

In this section, we shall extend (1.2) and (1.3) to any random timeL∈L. Theorem 3.1. Let0< p <∞andL∈L. Then the inequalities

Eh

I_n^∗(L, X)pi

≤c_n,pE

1 + log^np² 1 J_L

hXi

np 2

L

, (3.1)

E h

I_n^∗(L, X)pi

≤cn,pE

1 + log^np² 1 J_L

(X_L^∗)^np

, (3.2)

Eh

hI_n(X)i

p 2

L

i ≤c_n,pE

1 + log^np² 1 JL

hXi

np 2

L

(3.3) ,

Eh

hI_n(X)i

p 2

L

i ≤c_n,pE

1 + log^np² 1 J_L

(X_L^∗)^np (3.4)

hold for all continuous local martingalesXwithX₀ = 0andn= 1,2, . . ..

Proof. Letn ≥1,L∈Land letX be a continuous local martingale.

(3.1) can be verified by induction. In fact, whenn= 1(3.1) is true from (2.2). Now suppose that (3.1) is true for2, . . . , n−1. Then we have

Eh

I_n−1^∗ (L, X)_n−1^np i

≤c_n,pE

1 + log^np² 1 J_L

hXi

np 2

L

.

On the other hand, from (1.1) we see that hI_n(X)i_t =

Z t

0

In−1(s, X)2

dhXi_s≤ sup

0≤s≤t

In−1(s, X)2

hXi_t for allt≥0, which gives

(3.5) hI_n(X)i_L≤ I_n−1^∗ (L, X)2

hXi_L.

Thus, by applying (2.2), (3.5) and then applying the Hölder inequality with exponentss=n andr = _n−1ⁿ , and noting

(a+b)ⁿ≤c_n(aⁿ+bⁿ) (a, b≥0), we find

Eh

I_n^∗(L, X)pi

≤c_pE

1 + log^p² 1 J_L

hI_n(X)i

p 2

L

≤c_pE

1 + log^p² 1 J_L

I_n−1^∗ (L, X)p

hXi

p 2

L

≤cpE

1 + log^p² 1 J_L

n

hXi

np 2

L

¹_n E

h

I_n−1^∗ (L, X)_n−1^np iⁿ⁻¹_n

≤c_n,pE

1 + log^np² 1 JL

hXi

np 2

L

. This establishes (3.1).

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Now, we verify (3.2). From the well-known correspondence of iterated stochastic integral I_n(X)and the Hermite polynomial of degreen(see [4, 7])

In(t, X) = 1

n!Hn(Xt,hXit), whereH_n(x, y) = yⁿ²h_n

√x y

(y > 0)andh_n(x) = (−1)ⁿe^x²_dx^dⁿne^−x² is the Hermite polynomial of degreen, more generally,H_n(x, y)can be defined as

Hn(x, y) = (−y)ⁿe^x

2 2y ∂ⁿ

∂xⁿe⁻^x

2 2y,

we see that iterated stochastic integralsIn(X),n = 1,2, . . .have the representation

(3.6) In(t, X) =

[ⁿ₂]

X

j=0

C_n^(j)X_t^n−2jhXi^j_t,

whereCn^(j) = −¹₂j 1

(n−2j)!j! and[x]stands for the integer part ofx.

On the other hand, for0 < j < ⁿ₂, by using the Hölder inequality with exponentss = _n−2jⁿ andr = _2jⁿ, we get

Eh

(X_L^∗)^(n−2j)phXi^jp_Li

≤E

(X_L^∗)^np^n−2j_n Eh

hXi

np 2

L

i^2j_n

≤c_n,pE

(X_L^∗)^np^n−2j_n E

1 + log^np² 1 JL

(X_L^∗)^np ^2j_n

≤c_n,pE

1 + log^np² 1 J_L

(X_L^∗)^np

. Clearly, the inequality above is also true forj = ⁿ₂ andj = 0.

Combining this with (3.6), we get for0< p ≤1

Eh

I_n^∗(L, X)pi

≤c_p

[ⁿ₂]

X

j=0

|C_n^(j)|^pEh

(X_L^∗)^(n−2j)p(hXi_L)^jpi

≤c_n,pE

1 + log^np² 1 J_L

(X_L^∗)^np

and for1< p <∞

Eh

I_n^∗(L, X)pi¹_p

≤

[ⁿ₂]

X

j=0

|C_n^(j)|Eh

(X_L^∗)^(n−2j)phXi^jp_Li¹_p

≤c_n,pE

1 + log^np² 1 J_L

(X_L^∗)^np ¹_p

. This gives (3.2).

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Next, from (3.5) and (3.1) we see that E

h

hI_n(X)i

p 2

L

i

≤E h

I_n−1^∗ (L, X)p

hXi

p 2

L

i

≤Eh

I_n−1^∗ (L, X)_n−1^np iⁿ⁻¹_n Eh

hXi

np 2

L

i_n¹

≤c_n,pE

1 + log^np² 1 JL

hXi

np 2

L

ⁿ⁻¹_n Eh

hXi

np 2

L

i_n¹

≤c_n,pE

1 + log^np² 1 J_L

hXi

np 2

L

. Finally, from (3.5), (3.2) and (2.3), we have

Eh

hI_n(X)i

p 2

L

i≤Eh

I_n−1^∗ (L, X)p

hXi

p 2

L

i

≤c_n,pE

1 + log^np² 1 JL

(X_L^∗)^np ⁿ⁻¹_n

Eh hXi

np 2

L

i¹_n

≤c_n,pE

1 + log^np² 1 J_L

(X_L^∗)^np

.

This completes the proof of Theorem 3.1.

Theorem 3.2. Let0< p <∞andL∈L. Then the inequalities Eh

L^∗_L(n, X)pi

≤c_n.pE

1 + log^np² 1 J_L

hXi

np 2

L

(3.7) ,

Eh

L^∗_L(n, X)pi

≤c_n,pE

1 + log^np 1 JL

(X_L^∗)^np

, (3.8)

Eh

L^∗_L(n, X)pi

≤c_n,pE

1 + log^np 1 J_L

L^∗_L(X)np , (3.9)

Eh

I_n^∗(L, X)pi

≤c_n,pE

1 + log^np 1 J_L

L^∗_L(X)np , (3.10)

Eh

hI_n(X)i

p 2

L

i ≤c_n,pE

1 + log^np 1 J_L

L^∗_L(X)np (3.11)

Proof. Letn ≥2,0< p <∞and letX be a continuous local martingale.

First we prove (3.7). From (2.8), (3.5), (3.1) and the Hölder inequality with exponentss=n andr = _n−1ⁿ , we have

E h

L^∗_L(n, X)pi

≤cpE

1 + log^p² 1 J_L

hIn(X)i

p 2

L

≤c_pE

1 + log^p² 1 JL

I_n−1^∗ (L, X)p

hXi

p 2

L

≤c_pE

1 + log^np² 1 J_L

hXi

np 2

L

_n¹ Eh

I_n−1^∗ (L, X)_n−1^np iⁿ⁻¹_n

≤c_n.pE

1 + log^np² 1 J_L

hXi

np 2

L

.

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Now, by using (3.7), (2.7) and Lemma 2.2, we have Eh

L^∗_L(n, X)pi

≤c_pE

1 + log^np² 1 JL

hXi

np 2

L

≤c_pE

1 + log^np² 1 J_L

(X_L^∗)^np² L^∗_L(X)^np₂

≤c_pE

1 + log^np² 1 J_L

2

(X_L^∗)^np ¹₂

Eh

L^∗_L(X)npi¹₂

≤c_n,pE

1 + log^np 1 J_L

(X_L^∗)^np

and

E h

L^∗_L(n, X)pi

≤cn,pE

1 + log^np² 1 J_L

hXi

np 2

L

≤c_n,pE

1 + log^np² 1 JL

L^∗_L(X)^np₂

(X_L^∗)^np²

≤c_n,pE

1 + log^np 1 J_L

L^∗_L(X)np , which give (3.8) and (3.9).

Next, from (3.1), (2.7) and (2.9), we have E

h

I_n^∗(L, X)pi

≤cn,pE

1 + log^np² 1 J_L

hXi

np 2

L

≤c_n,pE

1 + log^np 1 JL

L^∗_L(X)np .

Finally, from (3.3), (2.7) and (2.9) we have E

h

hIn(X)i

p 2

L

i

≤cn,pE

1 + log^np² 1 J_L

hXi

np 2

L

≤c_n,pE

1 + log^np 1 J_L

L^∗_L(X)np .

This completes the proof of Theorem 3.2.

Now, we consider the reverse of the inequalities in Theorem 3.1 and Theorem 3.2. LetL∈L and0< p <∞. Then the inequalities

(3.12) E

1 + log^np² 1 J_L

hI_n(X)i

p 2

L

≤c_n,pE

1 + log^np 1 J_L

I_n^∗(L, X)p

(n ≥1) follow from (2.5) and Lemma 2.2 for all continuous local martingales X withX₀ = 0. Fur- thermore, in [11, p.161] M. Yor showed that for any non-increasing continuous function g : (0,1]→R+the inequality

(3.13) Eh

g(J_L)X_L^∗i

≤c_gEh (gg¹

2)(J_L)hXi_L¹²i

holds for all continuous local martingales X withX0 = 0, where gγ(x) = 1 + log^γ ¹_x (γ ≥ 0, x∈(0,1]). As a consequence of the inequality, we have

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Lemma 3.3. Let0< p <∞andL∈L. Then the inequality

(3.14) E

1 + log^γp 1 JL

(X_L^∗)^p

≤c_γ,pE

1 + log^(γ+¹²^)p 1 JL

hXi

p 2

L

(γ ≥0)

holds for all continuous local martingalesXwithX₀ = 0.

Proof. Letγ ≥0and letX be a continuous local martingale. Then we have from (3.13)

(3.15) E

1 + log^γ 1 J_L

X_L^∗

≤c_γE

1 + log^γ+¹² 1 J_L

hXi_L¹²

, sinceg_γis non-increasing and(g_γg¹

2)(x)≤c_γ

1 + log^γ+¹² ¹_x . Now, denote fort≥0

A_t =

1 + log^γ 1 J_L

X_L∧t^∗ and B_t=

1 + log^γ+¹² 1 J_L

hXi_L∧t¹² .

Then for any couple(S, T)of stopping timesS, T withT ≥S ≥0 E

A_T −A_S

=E

1 + log^γ 1 J_L

(X_L∧T^∗ −X_L∧S^∗ )

≤E

1 + log^γ 1 J_L

sup

S≤t≤T

|X_L∧t−X_L∧S|1_{S<T_}

=E

1 + log^γ 1 J_L

sup

t≥0

|X_T^L_∧(S+t)−X_S^L|1{S<T}

≡E

1 + log^γ 1 J_L

sup

t≥0

|(X_T∧(S+t)−X_S)^L|1{S<T}

, whereX_t^L ≡Xt∧L.

Observe that(X(S+t)∧T−X_S)1{S<T}, t≥0is a continuous(F_S+t)–local martingale, we find by (3.15)

E

A_T −A_S

≤c_γE

1 + log^γ+¹² 1 J_L

hXi_L∧T¹² 1{S<T}

=E h

cγBT1{S<T}

i

≤ cγBT

_∞P(S < T).

It follows from Lemma 7 and Lemma 8 in [5] withC =c_γB,α=β = 1that for all0< p <∞ E

1 + log^γ 1 J_L

p

(X_L^∗)^p

≤c_γ,pE

1 + log^γ+¹² 1 J_L

p

hXi

p 2

L

. Thus, (3.14) follows from the inequalities

ˆ

c_p(a^p+b^p)≤(a+b)^p ≤c_p(a^p+b^p) (p, a, b≥0).

This completes the proof.

On the other hand, in [2], E. Carlen and P. Krée obtained the identity I_n(t, X)In−2(t, X) = I_n−1² (t, X)−

n

X

j=1

(n−j)!

n! I_n−j² (t, X)hXi^j−1_t (n≥2) for allt≥0and all continuous local martingalesXwithX0 = 0. It follows that

1

n!hXiⁿ⁻¹_t ≤ n−1

n I_n−1² (t, X)−I_n(t, X)I_n−2(t, X) (n ≥2).

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Integrating both sides of the inequality above on [0, t]with respect to the measure dhXi_t, we get

1

n!hXiⁿ_t ≤(n−1)hI_n(X)i²_t −n Z t

0

I_n(s, X)I_n−2(s, X)dhXi_s (n≥2) sincehIn(X)it=Rt

0I_n−1² (s, X)dhXis,which gives

(3.16) hXi

n 2

√ t

n! ≤√ n−1

I_n(X)¹₂

t +√

n I_n^∗(t, X)I_n−2^∗ (t, X)hXi_t¹₂

(n≥2).

Theorem 3.4. Let0< p <∞andL∈L. IfV is one of the three random variablesX_L^∗,hXi

1 2

L

andL^∗_L(X), then the inequalities E

V^np

≤c_n,pE

1 + log^np 1 J_L

I_n^∗(L, X)p , (3.17)

E V^np

≤cn,pE

1 + log⁽ⁿ⁺¹²^)p 1 J_L

hIn(X)i

p 2

L

, (3.18)

E V^np

≤c_n,pE

1 + log^(2n+1)p 1 J_L

L^∗_L(n, X)p (3.19)

Proof. Letn ≥2,0< p <∞and letX be a continuous local martingale.

For n ≥ 3, by applying the Hölder inequality with exponents s = n and r = _n−2ⁿ and Theorem 3.1 we have

E h

I_n−2^∗ (L, X)hXiL

pi

≤E h

I_n−2^∗ (L, X)_n−2^np iⁿ⁻²_n E

h hXi

np 2

L

i_n²

≤cn,pE h

1 + log^np² 1 J_L

hXi

np 2

L

i . Clearly, the inequality above is also true forn= 2.

It follows from (3.16) that forn ≥2 1

√n!

p

E

1 + log^np² 1 J_L

hXi

np 2

L

≤E

1 + log^np² 1 J_L

√n−1hI_n(X)i

1 2

L+√

n I_n^∗(L, X)I_n−2^∗ (L, X)hXi_L¹₂p

≤cˆn,pE

1 + log^np² 1 J_L

hIn(X)i

p 2

L

+c_n,pE

1 + log^np 1 JL

I_n^∗(L, X)p¹₂ Eh

I_n−2^∗ (L, X)hXi_Lpi¹₂

≤cˆ_n,pE

1 + log^np² 1 J_L

hI_n(X)i

p 2

L

+c_n,pE

1 + log^np 1 J_L

I_n^∗(L, X)p¹₂ E

1 + log^np² 1 J_L

hXi

np 2

L

¹₂ .

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Combining this with (3.12), we get the quadratic inequality as follows E

1 + log^np² 1 J_L

hXi

np 2

L

≤cˆ_n,pE

1 + log^np 1 J_L

I_n^∗(L, X)p

+c_n,pE

1 + log^np 1 J_L

I_n^∗(L, X)p¹₂

×E

1 + log^np² 1 J_L

hXi

np 2

L

¹₂ .

Solving the above quadratic inequality leads to the inequality

(3.20) E

1 + log^np² 1 J_L

hXi

np 2

L

≤c_n,pE

1 + log^np 1 J_L

I_n^∗(L, X)p .

Consequently, by Lemma 3.3 E

1 + log^np² 1 J_L

hXi

np 2

L

≤cn,pE

1 + log⁽ⁿ⁺¹²^)p 1 J_L

In(X)^p₂

L

, (3.21)

and so by (2.5) and (2.9) E

1 + log^np² 1 J_L

hXi

np 2

L

≤c_n,pE

1 + log^(2n+1)p 1 J_L

L^∗_L(n, X)p . (3.22)

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

E h

hXi^p_Li

≤cpE

1 + log^p² 1 J_L

I₂^∗(L, X)p , (3.23)

Eh hXi^p_Li

≤c_pE

1 + log^p² 1 J_L

hI₂(X)i

p 2

L

(3.24) ,

Eh

(X_L^∗)^2pi

≤c_pE

1 + log^p² 1 J_L

I₂^∗(L, X)p , (3.25)

Eh

(X_L^∗)^2pi

≤c_pE

1 + log^p² 1 J_L

hI₂(X)i

p 2

L

, (3.26)

Eh

(X_L^∗)^2pi

≤c_pE

1 + log^p 1 J_L

L^∗_L(2, X)p , (3.27)

Eh hXi^p_Li

≤c_pE

1 + log^p 1 J_L

L^∗_L(2, X)p (3.28) ,

Eh

L^∗_L(X)2pi

≤c_pE

1 + log^2p 1 JL

I₂^∗(L, X)p , (3.29)

Eh

L^∗_L(X)2pi

≤c_pE

1 + log^3p² 1 J_L

I₂(X)^p₂

L

, (3.30)

Eh

L^∗_L(X)2pi

≤c_pE

1 + log^3p 1 J_L

L^∗_L(2, X)p (3.31)

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hold for all continuous local martingalesXwithX₀ = 0. In fact, from (3.16) we have Eh

hXi^p_Li

≤cˆ_pEh

I₂(X)^p₂

L

i

+c_pEh

I₂^∗(L, X)hXi_L^p₂i

for0< p <∞and so E

h hXi^p_Li

≤ˆcpE h

I2(X)^p₂

L

i +cpE

h

I₂^∗(L, X)pi¹₂ E

h

hXi^p_Li¹₂ .

Combining this with Lemma 2.1, we find Eh

hXi^p_Li

≤cˆ_pE

1 + log^p² 1 I_L

I₂^∗(L, X)p

+c_pE

1 + log^p² 1 I_L

I₂^∗(L, X)p¹₂ Eh

hXi^p_Li¹₂

and Eh

hXi^p_Li

≤cˆ_pE

1 + log^p² 1 I_L

I₂(X)^p₂

L

+c_pE

1 + log^p² 1 I_L

hI_n(X)i

p 2

L

¹₂ E

h

hXi^p_Li¹₂ . The above quadratic inequalities lead to (3.23) and (3.24).

Next, observe that from (3.6)

(X_L^∗)² ≤2I₂^∗(L, X) +hXi_L, we obtain the inequalities (3.25) – (3.28).

Finally, combining (3.16) with Lemma 3.3, we get E

1 + log^p 1 J_L

hXi^p_L

≤E

1 + log^p 1 J_L

√2hI₂(X)i_L¹² + 2 I₂^∗(L, X)hXi_L¹₂p

≤ˆc_pE

1 + log^p 1 J_L

hI₂(X)i

p 2

L

+c_pE

1 + log^p 1 J_L

I₂^∗(L, X)p¹₂ E

1 + log^p 1 J_L

hXi^p_L ¹₂

.

≤ˆcpE

1 + log^3p² 1 J_L

hI2(X)i

p 2

L

+c_pE

1 + log^3p² 1 JL

hI₂(X)i

p 2

L

¹₂ E

1 + log^p 1 JL

hXi^p_L ¹₂

, which gives a quadratic inequality

x²−cˆpy²−cpxy≤0 (ˆcp, cp ≥0) with

x=E

1 + log^p 1 J_L

hXi^p_L ¹₂

and y=E

1 + log^3p² 1 J_L

hI₂(X)i

p 2

L

¹₂ .

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Solving the quadratic inequality leads to E

1 + log^p 1 J_L

hXi^p_L

≤c_pE

1 + log^3p² 1 J_L

hI₂(X)i

p 2

L

, which gives (3.30) and (3.31).

Thus, we obtain the inequalities (3.23) – (3.31).

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