Iterated integrals with respect to Bessel processes

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Statistics & Probability Letters 74 (2005) 93–102 www.elsevier.com/locate/stapro

Iterated integrals with respect to Bessel processes Litan Yan, Jingyun Ling Department of Mathematics, College of Science, Donghua University, 1882 West Yan’an Rd., Shanghai 200051, PR China Received 2 November 2003; received in revised form 14 January 2005 Available online 23 May 2005

Abstract Let X ¼ ðX t ÞtX0 be the square of a d (X0)-dimensional Bessel process starting at zero. Define iterated stochastic integrals I n ðt; dÞ, tX0 inductively by Z t I n ðt; dÞ ¼ I n1 ðs; dÞ dX s 0

with I 0 ðt; dÞ ¼ 1 and I 1 ðt; dÞ ¼ X t . Then the inequalities      pC n;p;d ktn kp cn;p;d ktn kp p sup jI ðt; dÞj n   0ptpt

and

p

    n  cn;p;d kG d ðtÞ kp p sup jI n ðt; dÞj=ð1 þ tÞ  pC n;p;d kG d ðtÞn kp n

0ptpt

p

are proved to hold for all 0opo1 and all stopping times t, where c; C are some positive constants depending only on the subscripts, and G d ðtÞ ¼ logð1 þ d logð1 þ tÞÞ. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60H05; 60J25; secondary 60G44 Keywords: Bessel processes; Iterated stochastic integrals; Brownian motion; Martingales; Itoˆ’s formula

Corresponding author.

E-mail addresses: [email protected] (L. Yan), [email protected] (J. Ling). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.04.026

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1. Introduction and results Throughout this paper, we shall work with a filtered complete probability space ðO; F; ðFt Þ; PÞ satisfying the usual conditions, and let B ¼ ðBt ÞtX0 be a standard Brownian motion starting at zero. For any continuous process X we denote X t ¼ sup0pspt jX s j and X ¼ X 1 . Let C stand for a positive constant depending only on the subscripts and its value may be different in different appearance, and furthermore, this assumption is also adaptable to c. Carlen and Kre´e (1991) considered iterated stochastic integrals I n ðBÞ ¼ ðI n ðt; BÞ; Ft Þ

ðnX0Þ,

defined inductively by Z t I n1 ðs; BÞ dBs I n ðt; BÞ ¼ 0

with I 0 ðt; BÞ ¼ 1 and I 1 ðt; BÞ ¼ Bt . They established Lp -estimates on I n ðBÞ as follows: cn;p ktn=2 kp pkI n ðt; BÞkp pC n;p ktn=2 kp for all stopping times t, where the right side holds for pX1 and the left side for p41. In this paper, we consider the iterated stochastic integrals with respect to the square of a d (X0)-dimensional Bessel process starting at zero. Our aims are to prove the following theorems. Theorem 1.1. Let X ¼ ðX t ÞtX0 be the square of a d (X0)-dimensional Bessel process starting at zero. Define iterated stochastic integrals I n ðt; dÞ, tX0 inductively by Z t I n ðt; dÞ ¼ I n1 ðs; dÞ dX s (1.1) 0

with I 0 ðt; dÞ ¼ 1 and I 1 ðt; dÞ ¼ X t . Then the inequalities     n n  cn;p;d kt kp p sup jI n ðt; dÞj  pC n;p;d kt kp 0ptpt

(1.2)

p

hold for all 0opo1 and all stopping times t. Theorem 1.2. Let G d ðtÞ ¼ logð1 þ d logð1 þ tÞÞ, d40. Under the condition of Theorem 1.1 we have    jI n ðt; dÞj n  pC n;p;d kðG d ðtÞÞn kp  cn;p;d kðG d ðtÞÞ kp p sup (1.3) ð1 þ tÞn  0ptpt

p

for all 0opo1 and all stopping times t. 2. Bessel processes with nonnegative dimension In this section, we first recall that the square representation of the dimension dð40Þ-Bessel process. Consider the stochastic differential equation pffiffiffiffiffiffiffiffiffi (2.1) dX t ¼ d dt þ 2 jX t j dBt ; X 0 ¼ xX0,

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where dX0. Clearly, this equation has a unique strong solution X, i.e., such that, for each tX0, the random variable X t is FBt ¼ sfBs ; sptg-measurable. Furthermore, if d ¼ x ¼ 0, the solution to (2.1) is X 0, and the comparison theorem ensures that for all d40, X X0 a.s. Thus the absolute value in (2.1) may be discarded. According to Revuz and Yor (1999), the process X is called the the square of a d (X0)-dimensional Bessel process (in symbol, X 2 BESQd ðxÞ). The expression n ¼ d=2  1 is called the index of the process. For d ¼ 0, the process X is a martingale and for d40 it is a submartingale. For every X 2 BESQd ðx2 Þ with xX0; dX0 we define the process pffiffiffiffi Z ¼ X. Then Z is a non-negative Markov process with the infinitesimal generator   1 d2 d1 d L¼ þ , 2 dx2 x dx and the process Z is called the Bessel process of dimension d40 starting at x (in symbol, Z 2 BESd ðxÞ). This is the simplest way of constructing a definition of Bessel process Z. If d41, the d-dimensional Bessel process satisfies the stochastic differential equation Z t d1 Zt ¼ Z0 þ ds þ Bt . 0 2Z s The Bessel processes of dimension dX1 are submartingales, however, the Bessel processes of dimension 0odo1 are not semimartingales. The main sources of general information concerning these processes are Dubins et al. (1993); Go¨ing-Jaeschke and Yor (2003); Itoˆ and McKean (1996); Pitman and Yor (1981); Rogers and Williams (1987); Shiga and Watanabe (1973), and, in particular, Revuz and Yor (1999). Proposition 2.1. Let X 2 BESQd ð0Þ with d40 and let 0opo1. Then the inequalities 1=2 cp;d khX i1=2 t kp pkX t kp pC p;d khX it kp

(2.2)

cp;d ktkp pkX t kp pC p;d ktkp

(2.3)

and

hold for all stopping times t. The inequalities (2.3) are first considered by Rosenkrantz and Sawyer (1977) (see also DeBlassie, 1987; Dubins et al., 1993; Graversen and Peskir, 1998). On the other hand, as some extensions to the inequalities obtained by Graversen and Peskir (2000), in Yan and Zhu (2003) we showed that the inequalities    Xt   pC p;d kG d ðtÞkp ,  (2.4) cp;d kG d ðtÞkp p sup  0ptpt 1 þ t p hold for all stopping times t and all 0opo1, where G d ðtÞ ¼ logð1 þ d logð1 þ tÞÞ is as in Theorem 1.2.

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The proof of inequalities (2.2). Let X 2 BESQd ð0Þ such that d40. It follows from (2.1) that Z t pffiffiffiffiffiffiffiffiffi dþ2 2 X s jX s j dBs þ (2.5) Xt ¼ 4 hX it . 2 0 This implies that equality E½X 2t  ¼

dþ2 E½hX it  2

holds for all bounded stopping times t. Combining this with Proposition 4.7 (Chapter IV, p. 163) in Revuz and Yor (1999) by taking k ¼ 1=2, we obtain the following inequalities: dþ2 3ðd þ 2Þ E½hX i1=2 E½hX i1=2 t pE½X t p t  6 2

(2.6)

for all stopping times t. It follows that for any pair S; T of stopping times with SpT, " #   E½X T  X S pE sup jX tþS  X S j pE sup jX tþS  X S j 0ptpTS

" pE

sup

# jX t j p

0ptpT1fSpTg

p

0ptpðTSÞ1fSpTg

3ðd þ 2Þ 1=2 E½hX iT1fSpTg  2

3ðd þ 2Þ 1=2 khX iT k1 PðSpTÞ, 2

which shows that the inequality kX t kp pC p;d khX i1=2 t kp

(2.7)

holds for all stopping times t and all 0opo1 by applying Lemma 4.1 in Barlow and Yor (1982) (see also Lemmas 7 and 8 in Jacka and Yor (1993) with a ¼ b ¼ 1). By the same way we can obtain the left inequality in (2.2). &

3. Proof of Theorem 1.1 In order to derive Theorem 1.1 we first show a more general result. For a continuous semimartingale X with the decomposition X ¼M þA

(3.1)

and 0opo1, we define the functionals    Z 1 Z    1=2   hMi þ jdA j ; j ðM; AÞ ¼ M þ j p ðM; AÞ ¼  s  1 p   0

p

0

1

  jdAs j . p

Then the following inequalities hold (see Dellacherie and Meyer, 1982): cp j p ðM; AÞpj p ðM; AÞpC p j p ðM; AÞ.

(3.2)

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For a continuous semimartingale X with the decomposition (3.1) we define iterated stochastic integrals I n ðX Þ inductively by Z t I n1 ðX ; sÞ dX s (3.3) I n ðX ; tÞ ¼ 0

with I 0 ðX ; tÞ ¼ 1 and I 1 ðX ; tÞ ¼ X t . Denote Z t Z t I n ðX ; M; tÞ ¼ I n1 ðX ; sÞ dM s ; jI n ðX ; A; tÞj ¼ jI n1 ðX ; sÞ dAs j 0

0

and jI n ðX ; AÞj ¼ jI n ðX ; A; 1Þj: Proposition 3.1. Let 0opo1. Then the inequality  Z 1 n    n=2 kI n ðX ; MÞ þ jI n ðX ; AÞjkp pC n;p  jdAs j   hMi1 þ 0

(3.4) p

holds for all continuous semimartingale X. Proof. Clearly, for all nX2 we have khI n ðX ; MÞi1=2 1 þ jI n ðX ; AÞjkp    Z 1   1=2 p jdAs j   I n1 ðX Þ hMi1 þ 0 p  n 1=n Z 1   1=2 pkI n1 ðX Þknp=ðn1Þ  jdAs j    hMi1 þ 0

p

 Z  1=2  pkI n1 ðX ; MÞ þ jI n1 ðX ; AÞjknp=ðn1Þ  hMi1 þ

1

0

by

I n ðX ÞpI n ðX ; MÞ þ jI n ðX ; AÞj: kI n ðX ; MÞ þ jI n ðX ; AÞjkp pC p kI n1 ðX ; MÞ

n 1=n  jdAs j   p

Combining this with (3.2), we get

 Z  1=2  þ jI n1 ðX ; AÞjknp=ðn1Þ  hMi1 þ

1 0

n 1=n  jdAs j   . p

Thus, by the induction method one can obtain inequality (3.4). & By applying the idea of Carlen and Kre´e (1991) we can show that the following result holds: Proposition 3.2. Let 0opo1. If for any stopping time t,  Z t   1=2 1=2  cp khMit kp p jdAs j  pC p khMit kp , 0

(3.5)

p

then the inequality khMin=2 1 kp pC n;p kI n ðX Þkp

holds for all continuous semimartingale X.

(3.6)

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Proof. Let nX2. Then we can easily find that the equality n X ðn  mÞ! 2 I nm ðt; X ÞhMim1 I n ðt; X ÞI n2 ðt; X Þ ¼ I 2n1 ðt; X Þ  t n! m¼1 holds for every continuous semimartingale X (see (18) in Carlen and Kre´e, 1991). It follows that for nX2 Z t Z t 1 n 2 I n1 ðs; X Þ dhMis  n I n ðs; X ÞI n2 ðs; X Þ dhMis . hMit pn n! 0 0 Hence  Z t 1=2 n=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hMit 2 pffiffiffiffi p n I n1 ðs; X Þ dhMis þ nI n ðt; X ÞI n2 ðt; X ÞhMit . n! 0

(3.7)

n and r ¼ n2 On the other hand, for n42, by applying the Ho¨lder inequality with exponents s ¼ n2 and then applying the inequality in (3.4), we get 2=n kI n2 ðX ÞhMi1 kp pkI n2 ðX Þknp=ðn2Þ khMin=2 1 kp  n2  Z 1     pC n;p  hMi1=2 jdAs j  1 þ   0

2=n khMin=2 1 kp np=ðn2Þ

pC n;p khMin=2 1 kp , which gives for n42 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 k I n ðX ÞI n2 ðX ÞhMi1 kp pkI n ðX Þk1=2 p kI n2 ðX ÞhMi1 kp n=2 1=2 pC n;p kI n ðX Þk1=2 p khMi1 kp .

Clearly, the above inequality is also true for n ¼ 2 with C n;p ¼ 1. Combining this with (3.7), (3.2) and (3.5), we get for nX2, 0opo1 1=2 n=2 1=2 khX in=2 1 kp pC n;p kI n ðX Þkp þ C n;p kI n ðX Þkp khX i1 kp ,

which shows that inequality (3.6) holds. & Now, Theorem 1.1 is a simple consequence of the above propositions.

4. Proof of Theorem 1.2 In order to drive Theorem 1.2, we need some lemmas. Lemma 4.1. Let X 2 BESQd ð0Þ with d40 and let 0opo1. Then the inequality   1=2  1=2    hX it    1=2  t   pC p;d k logð1 þ d logð1 þ tÞÞkp   sup 0ptpt 1 þ t  1 þ tp p

holds for every stopping time t.

(4.1)

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Proof. Observe that Z t hX it ¼ X s dsptX t , 0

we establish the lemma by the right inequality in (2.4) and Cauchy-Schwarz’s inequality. & Lemma 4.2. Let 0opo1. Then the inequality     n       sup jI n ðt; dÞj XC n;p;d  t n n   ð1 þ tÞ  0ptpt ð1 þ tÞ p p

(4.2)

holds for every stopping time t and every n ¼ 1; 2; . . . . Proof. Let 0opo1 and nX1. It is easy to check that for every stopping time tX0      ð1 ^ tÞn   tn      ð1 þ 1 ^ tÞn  XC n;p ð1 þ tÞn  , p p where 1 ^ t ¼ minf1; tg. Combining this with (1.2) and Proposition 2.1, we get         jI ðt; dÞj jI ðt; dÞj n n     sup n  X sup n  0ptpt ð1 þ tÞ 0ptp1^t ð1 þ tÞ p p     1 n X n  sup jI n ðt; dÞj  XC n;p;d kð1 ^ tÞ kp 2 0ptp1^t p      ð1 ^ tÞn   tn      XC n;p;d  XC n;p;d  ð1 þ 1 ^ tÞn p ð1 þ tÞn p for every stopping time tX0. This completes the proof. & Lemma 4.3. Let xX0 and let nX1 be integer. If xn 

n1 X

aj xj p0

(4.3)

j¼0

for some constants aj X0, j ¼ 1; 2; . . . ; n  1 and a0 40, then there exists a constant C40 depending only on n; a0 ; a1 ; . . . ; an1 such that xpC: Proof. The lemma can easily be verified by induction. Clearly, the lemma is true for n ¼ 1; 2. Suppose that the lemma is true for 1; 2; . . . ; n  1, and set b  1 ¼ maxfa0 ; a1 ; . . . ; an1 g. Then we have by (4.3) xn þ xn1  b

n2 X j¼0

xj þ

n2 X j¼0

xj  bxn1 p0.

(4.4)

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2 2

b On the other hand, it is easy to see that there exists a positive constant C n;b Xðn1Þ 4nðn2Þ depending only on n; b such that

xn þ C n;b

n2 X

xj  bxn1 X0

j¼0

for all xX0. Combining this with (4.4), we get xn1  ðb  1 þ C n;b Þ

n2 X

xj p0.

j¼0

It follows from induction hypothesis that there exists a constant C40 depending only on n; b such that xpC: This completes the proof. & Now, we can drive Theorem 1.2 by using the method in Yan (2003). Proof of Theorem 1.2. Clearly, the theorem is true for n ¼ 1 from (2.4). In the following discussion we let nX2. From the Ho¨lder inequality with exponents s ¼ n=ðn  2jÞ and r ¼ n=ð2jÞ for nX3, 1pjon=2 it follows that       n=2 2j=n  n ðn2jÞ=n  jX t jn2j hX ijt  jX j hX i      t t   p sup   sup  sup 0ptpt ð1 þ tÞn  0ptpt ð1 þ tÞn p 0ptpt ð1 þ tÞn  p

p

n

pC n;p;d klog ð1 þ d logð1 þ tÞÞkp with 0opo1 for all stopping times t and nX3 by Lemma 4.1 and (2.4). Clearly, the above inequality is also true for n ¼ 2. Combining this with the equality  ½n=2  X 1 j 1 hX ijt X n2j ,  I n ðt; dÞ ¼ 2 ðn  2jÞ!j! j¼0 we establish the right inequality in (1.3). To drive the left inequality in (1.3), for nX3 and 1pjon=2 by applying the Ho¨lder inequality with exponents s ¼ n=ðn  2jÞ and r ¼ n=ð2jÞ and then applying the second inequality in (1.3) and Lemmas 4.1 and 4.2, we get       n=2 2j=n j      jI ðt; dÞhX i j jI ðt; dÞj hX i    n2j n2j t  t  sup  p   sup  sup  0ptpt ð1 þ tÞn2j np=ðn2jÞ 0ptpt ð1 þ tÞn  0ptpt ð1 þ tÞn p p j=n   jI n ðt; dÞj n ðnjÞ=n   pC n;p;d klog ð1 þ d logð1 þ tÞÞkp  sup ð1 þ tÞn  . 0ptpt p On the other hand, it is easy to check that the identity ðX t Þn ¼

½n=2 X n! j j I n2j ðt; dÞhX it 2 j! j¼0

(4.5)

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holds for every n ¼ 1; 2; . . .. It follows that for nX3; 1ppo1       jX t jn  jI n ðt; dÞj n     cn;p;d kðG d ðtÞÞ kp p sup n  pn! sup n 0ptpt ð1 þ tÞ p 0ptpt ð1 þ tÞ p j=n  ½n=2 X n!  jI n ðt; dÞj n ðnjÞ=n   þ  sup ð1 þ tÞn  . j C n;p;j;d kðG d ðtÞÞ kp 0ptpt p j¼1 2 j! Thus, we have established the following inequalities: cn;p;d x pC n;p;d y þ n

n

½n=2 X

C n;p;j;d xnj yj

(4.6)

j¼1

for nX3, 1ppo1, where x ¼ kðG d ðtÞÞn k1=n p ;

1=n   jI n ðt; dÞj  . y¼ sup n  0ptpt ð1 þ tÞ p

Clearly, inequality (4.6) is also true for n ¼ 2. It follows from Lemma 4.3 that there exists a constant cn;p;d 40 such that xpcn;p;d y for all 1ppo1. This shows that the left inequality in (1.3) holds for 1ppo1. Similarly, it is also true for 0opo1. This completes the proof. & Finally, for a Bessel process Z 2 BES d ð0Þ with dX1 we consider the iterated integrals J n ðt; dÞ, tX0 inductively by Z t J n ðt; dÞ ¼ J n1 ðs; dÞ dZs 0

with J 0 ðt; dÞ ¼ 1 and J 1 ðt; dÞ ¼ Z t . From the proofs of Theorems 1.1 and 1.2 we have: Theorem 4.1. Let Z 2 BESd ð0Þ with dX1 and let 0opo1. Assume that J n ðt; dÞ, tX0 is defined as above. Then the inequalities     n=2 n=2  cn;p;d kt kp p sup jJ n ðt; dÞj  pC n;p;d kt kp 0ptpt

and cn;p;d kðG d ðtÞÞ

n=2

p

   jJ n ðt; dÞj    kp p sup  pC n;p;d kðGd ðtÞÞn=2 kp 0ptpt ð1 þ tÞn=2  p

hold for every stopping time t.

Acknowledgements The authors wish to thank an anonymous earnest referee for a careful reading of the manuscript and many helpful comments.

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