- Email: [email protected]
On the Burkholder–Davis–Gundy inequalities for continuous martingales
Statistics and Probability Letters 78 (2008) 3034–3039
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
On the Burkholder–Davis–Gundy inequalities for continuous martingales Yao-Feng Ren Department of Applied Mathematics, The Naval University of Engineering, 430033 Wuhan, PR China
article
info
a b s t r a c t
Article history: Received 6 December 2006 Received in revised form 29 February 2008 Accepted 1 May 2008 Available online 25 May 2008
Some improvements of the Burkholder–Davis–Gundy inequalities for continuous martingales are obtained. New constants for 0 < p < 2 are derived and p-independent constants for p ≥ 1 are sharpened. © 2008 Elsevier B.V. All rights reserved.
MSC: 60E15 60G44
1. Introduction Let W = (Wt )t ≥0 be a standard Brownian motion, τ be a stopping time of W . It is known that there exist positive constants Ap and ap depending only on p such that
kWτ kp ≤ Ap kτ 1/2 kp ,
if 0 < p < ∞,
ap kτ 1/2 kp ≤ kWτ kp ,
if 1 < p < ∞
(1)
and and kτ 1/2 kp < ∞.
(2)
In his celebrated paper, Davis (1976) obtained the best possible values for the constants Ap and ap Ap = zp∗ ,
2 ≤ p < ∞;
ap = z p ,
2 ≤ p < ∞;
Ap = zp , Ap = zp∗ ,
0 < p ≤ 2, 1
where zp∗ be the largest positive zero of the parabolic cylinder function Dp (x), and zp be the smallest positive zero of the confluent hypergeometric function Mp (x). Martingale theory has wide applications in many areas of modern probability and analysis, martingale inequalities play a basic role in martingale theory. Continuous martingales are very important stochastic processes closely related to Brownian motion, some authors have established many important inequalities for continuous martingales. Let M = (Mt )t ≥0 be a continuous martingale with M0 = 0, hM i be the quadratic variation of M. Novikov (1973) established the following inequality
√ kMτ kp ≤ B pkhM i1τ /2 kp ,
1
E-mail address: [email protected]. 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.05.024
(3)
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
3035
where B is a p-independent constant. Notice that the inequalities for Brownian motion can be translated to martingale inequalities via time change arguments, we have
kMτ kp ≤ Ap khM i1τ /2 kp , 1/2
ap khM iτ kp ≤ kMτ kp ,
if 0 < p < ∞, if 1 < p < ∞ and
(4) 1/2
khM iτ kp < ∞
(5)
and Davis’ constants are best possible. Carlen and Kree (1991) gave explicit information on the constant, they proved that 2 is the best possible p-independent constant in the following inequality
√ kMτ kp ≤ 2 pkhM i1τ /2 kp ,
1 ≤ p < ∞.
(6)
The inequalities mentioned above are all on the Lp norms between Mτ and hM iτ , but one often need to consider the Lp norms between Mτ∗ and hM iτ (cf. e.g. Banuelos, 1988; Peskir, 1996), where Mt∗ = sups≤t |Ms |. In fact, for every 0 < p < ∞, there exist universal constants cp and Cp such that, for all continuous local martingales M vanishing at zero cp−1 khM i1τ /2 kp ≤ kMτ∗ kp ≤ Cp khM i1τ /2 kp . (7) These are the famous Burkholder–Davis–Gundy inequalities of great importance in martingale theory. Many authors have studied the values of the constants cp and Cp , however, the best values are unknown. Liptser and Shriyaev (1989) proved the following inequalities 2−p 4−p
E [hM ipτ /2 ] ≤ E [Mτ∗p ] ≤
4−p 2−p
E [hM ipτ /2 ],
0 < p < 2.
(8)
Barlow and Yor (1982) (and Jacka (1988)) proved that there exists a positive constant B such that
√ kMτ∗ kp ≤ B pkhM i1τ /2 kp ,
p ≥ 2.
(9)
Peskir (1996) gave estimates
√ √ π √ khM i1τ /2 kp ≤ kMτ∗ kp ≤ 2 10 pkhM i1τ /2 kp ,
11 p
p ≥ 1.
(10)
In this paper, we study the constants in the Burkholder–Davis–Gundy inequalities. We give some improvements on the values of the constants appearing in the inequalities (8) and (10), our new constants are sharper than the earlier constants. We deal only with continuous (Ft ) local martingales. 2. Preliminaries We now introduce some notations and conventions concerning martingales needed for the proofs of our results. For more details we refer to Revuz and Yor (1998) or He et al. (1992). Let (Ω , F , Ft , P ) be a filtered probability space with filtration (Ft )t ≥0 satisfying usual conditions, M = (Mt )t ≥0 be a continuous local martingale with M0 = 0 based on (Ω , F , Ft , P ), hM i is the quadratic variation of M. For a stopping time τ , we write Mτ∗ = sup |Ms |, s≤τ
M τ = (Mt ∧τ )t ≥0 .
For a random variable Y , k Y kp = (E | Y |p )1/p is the Lp -norm of Y . If a, b are real numbers, we write a ∧ b = min{a, b}. The theorems and lemmas in this section we need for the proofs are well known (cf. e.g. Burkholder, 1988; Garsia, 1973; Lenglart et al., 1980), but we present them here for the convenience of the reader. An adapted cadlag process X is dominated by an adapted increasing process A with A0 ≥ 0, if E | Xτ |≤ EAτ for any bounded stopping time τ . Lemma 1 (Lenglart). If X is dominated by A, A is continuous, then for any stopping time τ , any constants c > 0, d > 0 P (Xτ∗ ≥ c ) ≤
1 c
E (Aτ ∧ d) + P (Aτ ≥ d).
(11)
Lemma 2 (Garsia). Let A = {At , t ≥ 0} be a positive continuous increasing process with A0 = 0. If there exists a positive random variable Y such that E (A∞ − Aτ /Fτ ) ≤ E (Y /Fτ )
(12)
for any stopping time τ , then E (Ap∞ ) ≤ E (Y p ),
1 < p < ∞.
(13)
3036
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
Theorem A (Doob). Let M = (Mt )t ≥0 be a cadlag martingale, k M kp = supt ≥0 k Mt kp , then for p > 1 p
∗ k M∞ kp ≤
p−1
k M kp .
(14)
Theorem B (Burkholder). Let M = (Mt )t ≥0 be a continuous local martingale, hM i = (hM it )t ≥0 be its quadratic variation process, then for 1 < p < ∞ 1
1
2 2 (p∗ − 1)−1 k hM i∞ kp ≤k M kp ≤ (p∗ − 1) k hM i∞ kp
(15)
where p = max[p − 1, 1/(p − 1)]. ∗
These inequalities have fundamental applications in obtaining limit theorems and other inequalities for stochastic processes. 3. Martingale inequalities In this section we will prove our main results. Theorem 1. Let M = {Mt , t ≥ 0} be a continuous local martingale with M0 = 0, then for 0 < p < 2, the following inequalities hold:
p 2p 2 − p 2
2
p/2
E hM iτ
E (Mτ∗ )p
≤
2p ≤
2
2
p
2−p
E hM ipτ /2
(16)
for any stopping time τ . Proof. As in the proof of Theorem 5 in Liptser and Shriyaev Chapter I Section 9. For any stopping time T (T < ∞), EMT2 ≤ E hM iT , M 2 is dominated by hM i. By Lemma 1, for any positive constants a and λ, we have P (Mτ∗ ≥ a1/p ) ≤ a−2/p E (hM iτ ∧ a2/p λ) + P (hM iτ ≥ a2/p λ).
(17)
Integrating over the interval [0, ∞) with respect to a, we obtain E (Mτ∗ )p =
∞
Z 0
≤ λE
P (Mτ∗ ≥ a1/p )da ( hMλiτ )p/2
"Z
da + hM iτ
0
p = λ− 2 +1 + p
= λ− 2
p 2−p
2λ 2−p
#
∞
Z
( hMλiτ )p/2
a
−2/p
∞
Z
da +
P (hM iτ ≥ a2/p λ)da
0
p p λ− 2 +1 + λ− 2 E hM iτp/2
+ 1 E hM iτp/2 .
(18)
Let p Cp (λ) = λ− 2
2λ 2−p
+1 ,
we have
p p Cp0 (λ) = λ− 2 −1 λ − . 2 Cp (λ) is strictly decreasing in (0, p/2), and strictly increasing in (p/2, ∞). The choice of λ = p/2 renders the right-hand side of (18) as small as possible and yields the desired inequality E (Mτ∗ )p
2p ≤
2
2
p
2−p
E hM ipτ /2 .
(19)
Notice that hM i is also dominated by M ∗2 , by the same approach we have E hM ipτ /2 ≤
2p 2
2
p
2−p
E (Mτ∗ )p .
This completes the proof of Theorem 1.
(20)
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
3037
Remarks. (1) From the proof of Theorem 1, we have
2p
2
2
p
2−p
= Cp
p 2
4−p
< Cp (1) =
2−p
.
The new constant (2/p)p/2 [2/(2 − p)] is sharper than the earlier (4 − p)/(2 − p). (2) For the important case p = 1, the inequalities widely used is from (8) 1 3
E [hM i1τ /2 ] ≤ E [Mτ∗ ] ≤ 3E [hM i1τ /2 ].
(21)
By Theorem 1 we can get better inequalities
√ 1 √ E [hM i1τ /2 ] ≤ E [Mτ∗ ] ≤ 2 2E [hM i1τ /2 ].
(22)
2 2
(3) Notice that
2p lim
p→0
2
2
p
2−p
= 1,
and lim
p→0
4−p
= 2.
2−p
The bound (2/p)p/2 [2/(2 − p)] is much sharper than (4 − p)/(2 − p) for small p. With the help of Theorem 1, we can get better constants for inequalities (10), the new constants obtained in Theorem 1 play an important role in the proof of Theorem 2. Theorem 2. Let M = {Mt , t ≥ 0} be a continuous local martingale with M0 = 0, then
√ √ 1 √ √ khM i1τ /2 kp ≤ kMτ∗ kp ≤ 2 2 pkhM i1τ /2 kp ,
p ≥ 1.
2 2 p
(23)
for any stopping time τ . Proof. We give the proof in two parts. (i) For 1 < p < 2, by Doob’s inequality and Burkholder’s inequality we have ∗ k M∞ kp ≤
p p−1
√
p
k M kp ≤
(p − 1)2
k hM i1∞/2 kp =
p
√
(p − 1)2
p k hM i1∞/2 kp .
(24)
By Theorem 1
k
∗ M∞
s kp ≤
2
2
p
2−p
1/p
√
1/2
k hM i∞ kp =
2
2
1/p
2−p
p
√
p k hM i1∞/2 kp .
(25)
Hence we obtain
k
∗ M∞
2
p
2−p
2
kp ≤ min
Let f (p) = we have f (p) = 0
"√
√ 2
1/p
2 2−p
p
√ 2
,
1 p
+
1 p(2 − p)
√
1/2 p k hM i∞ kp .
(p − 1)2
1
p
p(2 − p)
Let g (p) = −
#
p
1
− +
2−p
p
√
,
1/p
2
1/p
−
1 p2
ln
2 2−p
,
−
1 p2
ln
2 2−p
.
(26)
3038
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
then p(p − 1)
g 0 (p) =
+
p2 (2 − p)2
2 p3
ln
2
> 0.
2−p
g (p) is strictly increasing on [1, 2), g (1) = − ln 2 < 0, g (3/2) = 2/3[1 − (2/3) ln 4] > 0. Hence there exists an unique p0 ∈ (1, 3/2), such that g (p) < 0, when p ∈ (1, p0 ); and g (p) > 0, when p ∈ (p0 , 2). i.e. f 0 (p) < 0, when p ∈ (1, p0 ); and f 0 (p) > 0, when p ∈ (p0 , 2). Therefore f (p) is strictly decreasing in (1, p0 ) and strictly increasing in (p0 , 2). Let
√
p
h(p) =
(p − 1)2
,
then
−3p − 1 < 0. √ 2 p(p − 1)3
h0 (p) =
h(p) is strictly decreasing in (1, 2]. Taking p = 7/4, we have
√
7
f
4
=
r
7
h
4
4
2·
=
7 4
7
√ 7 · 8 4 ≤ 2.66 ≤ 2 2 = f (1),
2 ·
4 3
≤ 2.36 ≤ f
7 4
.
By (26) we obtain
√ √ ∗ kM∞ kp ≤ 2 2 pkhM i1∞/2 kp ,
1 < p < 2.
(27)
1 < p < 2.
(28)
By the same argument, we have
√ √ ∗ khM i1∞/2 kp ≤ 2 2 pkM∞ kp , Thus, for 1 < p < 2, we proved
√ √ 1 √ √ khM i1τ /2 kp ≤ kMτ∗ kp ≤ 2 2 pkhM i1τ /2 kp .
(29)
2 2 p
(ii) For p ≥ 2, by Doob’s inequality and Burkholder’s inequality we have ∗ k M∞ kp ≤
p
k M kp ≤ p k hM i1∞/2 kp =
p−1
√ √
p p k hM i1∞/2 kp .
(30)
By Doob’s inequality and inequality (6) ∗ k M∞ kp ≤
p
k M kp ≤
p−1
√
p p−1
2 p k hM i1∞/2 kp .
(31)
Hence we obtain
k
∗ M∞
kp ≤ min
Since f1 (p) = f2 (3 +
√
p,
2p
p−1
√
p k hM i1∞/2 kp .
(32)
√
√
p is strictly increasing on [2, ∞), f2 (p) = 2p/(p − 1) is strictly decreasing on [2, ∞), and f1 (3 +
√
8) =
8). Thus we have
k
∗ M∞
kp ≤
2(3 +
√
1 8) √ 2 p k hM i∞ kp , 8
√
2+
p ≥ 2.
(33)
The left side of inequality (23) for p ≥ 2 has been proved by Garsia (1973) and Burkholder (1973). For the completion of the paper, we prove it here by using Garsia’s lemma, this approach yields the best constant. ∗ 2 ∗ 2 Suppose E (M∞ ) < ∞, if E (M∞ ) = ∞, we interpret inequality (23) for τ = ∞ as equality. For any stopping time T , define
ft = (MT +t − MT )I (T < ∞), M
G t = F T +t ,
t ≥ 0.
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
3039
e = {M ft , Gt , t ≥ 0} is a continuous local martingale with M f0 = 0, and Then M e it = (hM iT +t − hM iT )I (T < ∞), hM
t ≥ 0.
By the martingale inequality
e i∞ = E (M e∞ )2 = E [(M∞ − MT )2 I (T < +∞) E hM 2 = E (E [(M∞ − 2M∞ MT + MT2 )I (T < +∞)/FT ]) 2 = E (M∞ − MT2 )I (T < +∞)) ≤ sup E (Mt2 ) t ≥0
≤
∗ 2 E (M∞ ) .
(34)
We get ∗ 2 E (hM i∞ − hM iT ) ≤ E (M∞ ) .
(35)
Notice that the inequality (34) can be conditioned with respect to G0 , inequality (35) can be conditioned with respect to FT . Set
ξ = hM i∞ ,
A = hM i,
∗ 2 η = (M∞ ) .
Taking Φ (t ) = t p/2 (2 < p < ∞), from Lemma 2 we obtain ∗ p E hM ip∞/2 ≤ (p/2)p/2 E (M∞ ).
Thus we proved
r
1/2
k hM i∞ kp ≤
p 2
∗ k (M∞ ) kp .
(36)
Hence for p ≥ 2, we get
s
2 p
1/2
k hM i∞ kp ≤k √
∗ M∞
kp ≤
2(3 +
√
8) √ 1/2 p k hM i∞ kp . 8
√
2+
√ √
√
(37)
√
Notice 2(3 + 8)/(2 + 8) < 2.42 < 2 2, 2 > 1/2 2, combine (29) with (37), we proved (23) for τ = ∞. For any stopping time τ , replace M by M τ , we obtain (23). This concludes the proof of Theorem 2.
√
√
√
Remarks. (1) π /11 < 1/2 2; 2 √2 < 2 10, inequality (23) is sharper than inequality (10). (2) From Wang’s (1991) proof, p/2 is also the best possible constant in inequality (36) for continuous local martingales. Acknowledgements The author would like to thank the referee for careful reading of the manuscript and for suggestions. This work is supported by the National Nature Science Foundation of China (10571176). References Banuelos, B., 1988. A sharp good-λ inequality with an application to Riesz transforms. Michigan Math. J. 35, 117–125. Barlow, M.T., Yor, M., 1982. Semi-martingale inequalities via the Garsia–Rodemich–Rumsey lemma and applications to local times. J. Funct. Anal. 49, 198–229. Burkholder, D.L., 1973. Distribution function inequalities for martingales. Ann. Probab. 1, 19–42. Burkholder, D.L., 1988. Sharp inequalities for martingales and stochastic integrals. Asterisque 157–158, 75–94. Carlen, E., Kree, P., 1991. Lp estimates on iterated stochastic integrals. Ann. Probab. 19, 354–368. Davis, B., 1976. On the Lp norms of stochastic integrals and other martingales. Duke Math. J. 43, 697–704. Garsia, A.M., 1973. Martingale inequalities. Seminar Notes on Recent Progress. Benjamin. He, S.W., Wang, J.G., Yan, J.A., 1992. Semimartingale Theorems and Stochastic Calculus. Science Press & CRC Press, Beijing & Boca Raton. Jacka, S.B., 1988. A note on the good lambda inequalities. Lecture Notes Math. vol. 1372, 57–65. Lenglart, E., Lepingle, D., Protelli, M., 1980. Presentation unifiee de certaines inegalities de la theorie des martingales. Lecture Notes Math. 784, 26–47. Liptser, R.Sh., Shriyaev, A.N., 1989. Theory of Martingales. Kluwer Academic Publishers, Dordrecht. Novikov, A., 1973. On moment inequalities and identities for stochastic integrals. In: Proc. Second Japan-USSR Symp. on Probab. Theory. In: Lecture Notes Math., vol. 330. pp. 333–339. Peskir, G., 1996. On the exponential Orlicz norm of stopped Brownian motion. Studia Mathematica 117, 253–273. Revuz, D, Yor, M., 1998. Continuous martingales and Brownian motion, 3rd ed. Springer-Varlag. Wang, G., 1991. Sharp inequalities for the conditional square function of martingale. Ann. Probab. 19, 1679–1688.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
On the Burkholder–Davis–Gundy inequalities for continuous martingales Yao-Feng Ren Department of Applied Mathematics, The Naval University of Engineering, 430033 Wuhan, PR China
article
info
a b s t r a c t
Article history: Received 6 December 2006 Received in revised form 29 February 2008 Accepted 1 May 2008 Available online 25 May 2008
Some improvements of the Burkholder–Davis–Gundy inequalities for continuous martingales are obtained. New constants for 0 < p < 2 are derived and p-independent constants for p ≥ 1 are sharpened. © 2008 Elsevier B.V. All rights reserved.
MSC: 60E15 60G44
1. Introduction Let W = (Wt )t ≥0 be a standard Brownian motion, τ be a stopping time of W . It is known that there exist positive constants Ap and ap depending only on p such that
kWτ kp ≤ Ap kτ 1/2 kp ,
if 0 < p < ∞,
ap kτ 1/2 kp ≤ kWτ kp ,
if 1 < p < ∞
(1)
and and kτ 1/2 kp < ∞.
(2)
In his celebrated paper, Davis (1976) obtained the best possible values for the constants Ap and ap Ap = zp∗ ,
2 ≤ p < ∞;
ap = z p ,
2 ≤ p < ∞;
Ap = zp , Ap = zp∗ ,
0 < p ≤ 2, 1
where zp∗ be the largest positive zero of the parabolic cylinder function Dp (x), and zp be the smallest positive zero of the confluent hypergeometric function Mp (x). Martingale theory has wide applications in many areas of modern probability and analysis, martingale inequalities play a basic role in martingale theory. Continuous martingales are very important stochastic processes closely related to Brownian motion, some authors have established many important inequalities for continuous martingales. Let M = (Mt )t ≥0 be a continuous martingale with M0 = 0, hM i be the quadratic variation of M. Novikov (1973) established the following inequality
√ kMτ kp ≤ B pkhM i1τ /2 kp ,
1
E-mail address: [email protected]. 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.05.024
(3)
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
3035
where B is a p-independent constant. Notice that the inequalities for Brownian motion can be translated to martingale inequalities via time change arguments, we have
kMτ kp ≤ Ap khM i1τ /2 kp , 1/2
ap khM iτ kp ≤ kMτ kp ,
if 0 < p < ∞, if 1 < p < ∞ and
(4) 1/2
khM iτ kp < ∞
(5)
and Davis’ constants are best possible. Carlen and Kree (1991) gave explicit information on the constant, they proved that 2 is the best possible p-independent constant in the following inequality
√ kMτ kp ≤ 2 pkhM i1τ /2 kp ,
1 ≤ p < ∞.
(6)
The inequalities mentioned above are all on the Lp norms between Mτ and hM iτ , but one often need to consider the Lp norms between Mτ∗ and hM iτ (cf. e.g. Banuelos, 1988; Peskir, 1996), where Mt∗ = sups≤t |Ms |. In fact, for every 0 < p < ∞, there exist universal constants cp and Cp such that, for all continuous local martingales M vanishing at zero cp−1 khM i1τ /2 kp ≤ kMτ∗ kp ≤ Cp khM i1τ /2 kp . (7) These are the famous Burkholder–Davis–Gundy inequalities of great importance in martingale theory. Many authors have studied the values of the constants cp and Cp , however, the best values are unknown. Liptser and Shriyaev (1989) proved the following inequalities 2−p 4−p
E [hM ipτ /2 ] ≤ E [Mτ∗p ] ≤
4−p 2−p
E [hM ipτ /2 ],
0 < p < 2.
(8)
Barlow and Yor (1982) (and Jacka (1988)) proved that there exists a positive constant B such that
√ kMτ∗ kp ≤ B pkhM i1τ /2 kp ,
p ≥ 2.
(9)
Peskir (1996) gave estimates
√ √ π √ khM i1τ /2 kp ≤ kMτ∗ kp ≤ 2 10 pkhM i1τ /2 kp ,
11 p
p ≥ 1.
(10)
In this paper, we study the constants in the Burkholder–Davis–Gundy inequalities. We give some improvements on the values of the constants appearing in the inequalities (8) and (10), our new constants are sharper than the earlier constants. We deal only with continuous (Ft ) local martingales. 2. Preliminaries We now introduce some notations and conventions concerning martingales needed for the proofs of our results. For more details we refer to Revuz and Yor (1998) or He et al. (1992). Let (Ω , F , Ft , P ) be a filtered probability space with filtration (Ft )t ≥0 satisfying usual conditions, M = (Mt )t ≥0 be a continuous local martingale with M0 = 0 based on (Ω , F , Ft , P ), hM i is the quadratic variation of M. For a stopping time τ , we write Mτ∗ = sup |Ms |, s≤τ
M τ = (Mt ∧τ )t ≥0 .
For a random variable Y , k Y kp = (E | Y |p )1/p is the Lp -norm of Y . If a, b are real numbers, we write a ∧ b = min{a, b}. The theorems and lemmas in this section we need for the proofs are well known (cf. e.g. Burkholder, 1988; Garsia, 1973; Lenglart et al., 1980), but we present them here for the convenience of the reader. An adapted cadlag process X is dominated by an adapted increasing process A with A0 ≥ 0, if E | Xτ |≤ EAτ for any bounded stopping time τ . Lemma 1 (Lenglart). If X is dominated by A, A is continuous, then for any stopping time τ , any constants c > 0, d > 0 P (Xτ∗ ≥ c ) ≤
1 c
E (Aτ ∧ d) + P (Aτ ≥ d).
(11)
Lemma 2 (Garsia). Let A = {At , t ≥ 0} be a positive continuous increasing process with A0 = 0. If there exists a positive random variable Y such that E (A∞ − Aτ /Fτ ) ≤ E (Y /Fτ )
(12)
for any stopping time τ , then E (Ap∞ ) ≤ E (Y p ),
1 < p < ∞.
(13)
3036
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
Theorem A (Doob). Let M = (Mt )t ≥0 be a cadlag martingale, k M kp = supt ≥0 k Mt kp , then for p > 1 p
∗ k M∞ kp ≤
p−1
k M kp .
(14)
Theorem B (Burkholder). Let M = (Mt )t ≥0 be a continuous local martingale, hM i = (hM it )t ≥0 be its quadratic variation process, then for 1 < p < ∞ 1
1
2 2 (p∗ − 1)−1 k hM i∞ kp ≤k M kp ≤ (p∗ − 1) k hM i∞ kp
(15)
where p = max[p − 1, 1/(p − 1)]. ∗
These inequalities have fundamental applications in obtaining limit theorems and other inequalities for stochastic processes. 3. Martingale inequalities In this section we will prove our main results. Theorem 1. Let M = {Mt , t ≥ 0} be a continuous local martingale with M0 = 0, then for 0 < p < 2, the following inequalities hold:
p 2p 2 − p 2
2
p/2
E hM iτ
E (Mτ∗ )p
≤
2p ≤
2
2
p
2−p
E hM ipτ /2
(16)
for any stopping time τ . Proof. As in the proof of Theorem 5 in Liptser and Shriyaev Chapter I Section 9. For any stopping time T (T < ∞), EMT2 ≤ E hM iT , M 2 is dominated by hM i. By Lemma 1, for any positive constants a and λ, we have P (Mτ∗ ≥ a1/p ) ≤ a−2/p E (hM iτ ∧ a2/p λ) + P (hM iτ ≥ a2/p λ).
(17)
Integrating over the interval [0, ∞) with respect to a, we obtain E (Mτ∗ )p =
∞
Z 0
≤ λE
P (Mτ∗ ≥ a1/p )da ( hMλiτ )p/2
"Z
da + hM iτ
0
p = λ− 2 +1 + p
= λ− 2
p 2−p
2λ 2−p
#
∞
Z
( hMλiτ )p/2
a
−2/p
∞
Z
da +
P (hM iτ ≥ a2/p λ)da
0
p p λ− 2 +1 + λ− 2 E hM iτp/2
+ 1 E hM iτp/2 .
(18)
Let p Cp (λ) = λ− 2
2λ 2−p
+1 ,
we have
p p Cp0 (λ) = λ− 2 −1 λ − . 2 Cp (λ) is strictly decreasing in (0, p/2), and strictly increasing in (p/2, ∞). The choice of λ = p/2 renders the right-hand side of (18) as small as possible and yields the desired inequality E (Mτ∗ )p
2p ≤
2
2
p
2−p
E hM ipτ /2 .
(19)
Notice that hM i is also dominated by M ∗2 , by the same approach we have E hM ipτ /2 ≤
2p 2
2
p
2−p
E (Mτ∗ )p .
This completes the proof of Theorem 1.
(20)
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
3037
Remarks. (1) From the proof of Theorem 1, we have
2p
2
2
p
2−p
= Cp
p 2
4−p
< Cp (1) =
2−p
.
The new constant (2/p)p/2 [2/(2 − p)] is sharper than the earlier (4 − p)/(2 − p). (2) For the important case p = 1, the inequalities widely used is from (8) 1 3
E [hM i1τ /2 ] ≤ E [Mτ∗ ] ≤ 3E [hM i1τ /2 ].
(21)
By Theorem 1 we can get better inequalities
√ 1 √ E [hM i1τ /2 ] ≤ E [Mτ∗ ] ≤ 2 2E [hM i1τ /2 ].
(22)
2 2
(3) Notice that
2p lim
p→0
2
2
p
2−p
= 1,
and lim
p→0
4−p
= 2.
2−p
The bound (2/p)p/2 [2/(2 − p)] is much sharper than (4 − p)/(2 − p) for small p. With the help of Theorem 1, we can get better constants for inequalities (10), the new constants obtained in Theorem 1 play an important role in the proof of Theorem 2. Theorem 2. Let M = {Mt , t ≥ 0} be a continuous local martingale with M0 = 0, then
√ √ 1 √ √ khM i1τ /2 kp ≤ kMτ∗ kp ≤ 2 2 pkhM i1τ /2 kp ,
p ≥ 1.
2 2 p
(23)
for any stopping time τ . Proof. We give the proof in two parts. (i) For 1 < p < 2, by Doob’s inequality and Burkholder’s inequality we have ∗ k M∞ kp ≤
p p−1
√
p
k M kp ≤
(p − 1)2
k hM i1∞/2 kp =
p
√
(p − 1)2
p k hM i1∞/2 kp .
(24)
By Theorem 1
k
∗ M∞
s kp ≤
2
2
p
2−p
1/p
√
1/2
k hM i∞ kp =
2
2
1/p
2−p
p
√
p k hM i1∞/2 kp .
(25)
Hence we obtain
k
∗ M∞
2
p
2−p
2
kp ≤ min
Let f (p) = we have f (p) = 0
"√
√ 2
1/p
2 2−p
p
√ 2
,
1 p
+
1 p(2 − p)
√
1/2 p k hM i∞ kp .
(p − 1)2
1
p
p(2 − p)
Let g (p) = −
#
p
1
− +
2−p
p
√
,
1/p
2
1/p
−
1 p2
ln
2 2−p
,
−
1 p2
ln
2 2−p
.
(26)
3038
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
then p(p − 1)
g 0 (p) =
+
p2 (2 − p)2
2 p3
ln
2
> 0.
2−p
g (p) is strictly increasing on [1, 2), g (1) = − ln 2 < 0, g (3/2) = 2/3[1 − (2/3) ln 4] > 0. Hence there exists an unique p0 ∈ (1, 3/2), such that g (p) < 0, when p ∈ (1, p0 ); and g (p) > 0, when p ∈ (p0 , 2). i.e. f 0 (p) < 0, when p ∈ (1, p0 ); and f 0 (p) > 0, when p ∈ (p0 , 2). Therefore f (p) is strictly decreasing in (1, p0 ) and strictly increasing in (p0 , 2). Let
√
p
h(p) =
(p − 1)2
,
then
−3p − 1 < 0. √ 2 p(p − 1)3
h0 (p) =
h(p) is strictly decreasing in (1, 2]. Taking p = 7/4, we have
√
7
f
4
=
r
7
h
4
4
2·
=
7 4
7
√ 7 · 8 4 ≤ 2.66 ≤ 2 2 = f (1),
2 ·
4 3
≤ 2.36 ≤ f
7 4
.
By (26) we obtain
√ √ ∗ kM∞ kp ≤ 2 2 pkhM i1∞/2 kp ,
1 < p < 2.
(27)
1 < p < 2.
(28)
By the same argument, we have
√ √ ∗ khM i1∞/2 kp ≤ 2 2 pkM∞ kp , Thus, for 1 < p < 2, we proved
√ √ 1 √ √ khM i1τ /2 kp ≤ kMτ∗ kp ≤ 2 2 pkhM i1τ /2 kp .
(29)
2 2 p
(ii) For p ≥ 2, by Doob’s inequality and Burkholder’s inequality we have ∗ k M∞ kp ≤
p
k M kp ≤ p k hM i1∞/2 kp =
p−1
√ √
p p k hM i1∞/2 kp .
(30)
By Doob’s inequality and inequality (6) ∗ k M∞ kp ≤
p
k M kp ≤
p−1
√
p p−1
2 p k hM i1∞/2 kp .
(31)
Hence we obtain
k
∗ M∞
kp ≤ min
Since f1 (p) = f2 (3 +
√
p,
2p
p−1
√
p k hM i1∞/2 kp .
(32)
√
√
p is strictly increasing on [2, ∞), f2 (p) = 2p/(p − 1) is strictly decreasing on [2, ∞), and f1 (3 +
√
8) =
8). Thus we have
k
∗ M∞
kp ≤
2(3 +
√
1 8) √ 2 p k hM i∞ kp , 8
√
2+
p ≥ 2.
(33)
The left side of inequality (23) for p ≥ 2 has been proved by Garsia (1973) and Burkholder (1973). For the completion of the paper, we prove it here by using Garsia’s lemma, this approach yields the best constant. ∗ 2 ∗ 2 Suppose E (M∞ ) < ∞, if E (M∞ ) = ∞, we interpret inequality (23) for τ = ∞ as equality. For any stopping time T , define
ft = (MT +t − MT )I (T < ∞), M
G t = F T +t ,
t ≥ 0.
Y.-F. Ren / Statistics and Probability Letters 78 (2008) 3034–3039
3039
e = {M ft , Gt , t ≥ 0} is a continuous local martingale with M f0 = 0, and Then M e it = (hM iT +t − hM iT )I (T < ∞), hM
t ≥ 0.
By the martingale inequality
e i∞ = E (M e∞ )2 = E [(M∞ − MT )2 I (T < +∞) E hM 2 = E (E [(M∞ − 2M∞ MT + MT2 )I (T < +∞)/FT ]) 2 = E (M∞ − MT2 )I (T < +∞)) ≤ sup E (Mt2 ) t ≥0
≤
∗ 2 E (M∞ ) .
(34)
We get ∗ 2 E (hM i∞ − hM iT ) ≤ E (M∞ ) .
(35)
Notice that the inequality (34) can be conditioned with respect to G0 , inequality (35) can be conditioned with respect to FT . Set
ξ = hM i∞ ,
A = hM i,
∗ 2 η = (M∞ ) .
Taking Φ (t ) = t p/2 (2 < p < ∞), from Lemma 2 we obtain ∗ p E hM ip∞/2 ≤ (p/2)p/2 E (M∞ ).
Thus we proved
r
1/2
k hM i∞ kp ≤
p 2
∗ k (M∞ ) kp .
(36)
Hence for p ≥ 2, we get
s
2 p
1/2
k hM i∞ kp ≤k √
∗ M∞
kp ≤
2(3 +
√
8) √ 1/2 p k hM i∞ kp . 8
√
2+
√ √
√
(37)
√
Notice 2(3 + 8)/(2 + 8) < 2.42 < 2 2, 2 > 1/2 2, combine (29) with (37), we proved (23) for τ = ∞. For any stopping time τ , replace M by M τ , we obtain (23). This concludes the proof of Theorem 2.
√
√
√
Remarks. (1) π /11 < 1/2 2; 2 √2 < 2 10, inequality (23) is sharper than inequality (10). (2) From Wang’s (1991) proof, p/2 is also the best possible constant in inequality (36) for continuous local martingales. Acknowledgements The author would like to thank the referee for careful reading of the manuscript and for suggestions. This work is supported by the National Nature Science Foundation of China (10571176). References Banuelos, B., 1988. A sharp good-λ inequality with an application to Riesz transforms. Michigan Math. J. 35, 117–125. Barlow, M.T., Yor, M., 1982. Semi-martingale inequalities via the Garsia–Rodemich–Rumsey lemma and applications to local times. J. Funct. Anal. 49, 198–229. Burkholder, D.L., 1973. Distribution function inequalities for martingales. Ann. Probab. 1, 19–42. Burkholder, D.L., 1988. Sharp inequalities for martingales and stochastic integrals. Asterisque 157–158, 75–94. Carlen, E., Kree, P., 1991. Lp estimates on iterated stochastic integrals. Ann. Probab. 19, 354–368. Davis, B., 1976. On the Lp norms of stochastic integrals and other martingales. Duke Math. J. 43, 697–704. Garsia, A.M., 1973. Martingale inequalities. Seminar Notes on Recent Progress. Benjamin. He, S.W., Wang, J.G., Yan, J.A., 1992. Semimartingale Theorems and Stochastic Calculus. Science Press & CRC Press, Beijing & Boca Raton. Jacka, S.B., 1988. A note on the good lambda inequalities. Lecture Notes Math. vol. 1372, 57–65. Lenglart, E., Lepingle, D., Protelli, M., 1980. Presentation unifiee de certaines inegalities de la theorie des martingales. Lecture Notes Math. 784, 26–47. Liptser, R.Sh., Shriyaev, A.N., 1989. Theory of Martingales. Kluwer Academic Publishers, Dordrecht. Novikov, A., 1973. On moment inequalities and identities for stochastic integrals. In: Proc. Second Japan-USSR Symp. on Probab. Theory. In: Lecture Notes Math., vol. 330. pp. 333–339. Peskir, G., 1996. On the exponential Orlicz norm of stopped Brownian motion. Studia Mathematica 117, 253–273. Revuz, D, Yor, M., 1998. Continuous martingales and Brownian motion, 3rd ed. Springer-Varlag. Wang, G., 1991. Sharp inequalities for the conditional square function of martingale. Ann. Probab. 19, 1679–1688.