Applied Mathematics and Computation 254 (2015) 252–265
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
Existence, uniqueness and almost surely asymptotic
estimations of the solutions to neutral stochastic functional
differential equations driven by pure jumps
Wei Mao a,⇑, Quanxin Zhu b, Xuerong Mao c
a
School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing 210013, Jiangsu, PR China
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing 210023, Jiangsu, PR China
c
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
b
a r t i c l e
i n f o
a b s t r a c t
In this paper, we are concerned with neutral stochastic functional differential equations
driven by pure jumps (NSFDEwPJs). We prove the existence and uniqueness of the solution
to NSFDEwPJs whose coefficients satisfying the local Lipschitz condition. In addition, we
establish the p-th exponential estimations and almost surely asymptotic estimations of
the solution for NSFDEwJs.
Ó 2015 Elsevier Inc. All rights reserved.
Keywords:
Neutral stochastic functional differential
equations
Pure jumps
Existence and uniqueness
Exponential estimations
Almost surely asymptotic estimations
1. Introduction
Stochastic delay differential equations (SDDEs) have come to play an important role in many branches of science and
industry. Such models have been used with great success in a variety of application areas, including biology, epidemiology,
mechanics, economics and finance. In the past few decades, qualitative theory of SDDEs have been studied intensively by
many scholars. Here, we refer to Mohammed [1], Mao [2–5,9], Buckwar [6], Kuchler [7], Hu [8], Xu [10], Wu [11], Appleby
[12], Gyongy [13] and references therein. Recently, motivated by the theory of aeroelasticity, a class of neutral stochastic
equations has also received a great deal of attention and much work has been done on neutral stochastic equations. For
example, conditions of the existence and stability of the analytical solution are given in [14–20]. Various efficient computational methods are obtained and their convergence and stability have been studied in [21–25].
However, all equations of the above mentioned works are driven by white noise perturbations with continuous initial
data and white noise perturbations are not always appropriate to interpret real data in a reasonable way. In real phenomena,
the state of neutral stochastic delay equations may be perturbed by abrupt pulses or extreme events. A more natural mathematical framework for these phenomena has been taken into account other than purely Brownian perturbations. In particular, we incorporate the Levy perturbations with jumps into neutral stochastic delay equations to model abrupt changes.
In this paper, we study the following neutral stochastic functional differential equations with pure jumps (NSFDEwPJs)
d½xðtÞ Dðxt Þ ¼ f ðxt ; tÞdt þ
Z
hðxt ; uÞNp ðdt; duÞ;
U
⇑ Corresponding author.
E-mail address: [email protected] (W. Mao).
http://dx.doi.org/10.1016/j.amc.2014.12.126
0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
t 0 6 t 6 T:
ð1Þ
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
253
To the best of our knowledge, there are no literatures concerned with the existence and asymptotic estimations of the
solution to NSFDEwPJs (1). On the one hand, we prove that Eq. (1) has a unique solution in the sense of LP norm. We do
not use the fixed point Theorem. Instead, we get the solution of Eq. (1) via successive approximations. On the other hand,
we study the p-th exponential estimations and almost surely asymptotic estimations of the solution to Eq. (1). By using
the It b
o formula, Taylor formula and the Burkholder Davis inequality, we have that the p-th moment of the solution will grow
at most exponentially with exponent M and show that the exponential estimations implies almost surely asymptotic estimations. Although the way of analysis follows the ideas in [2], however, those results on the existence and uniqueness of
the solution in [2] cannot be extended to the jumps case naturally. Unlike the Brown process whose almost all sample paths
are continuous, the Poisson random measure N p ðdt; duÞ is a jump process and has the sample paths which are right-continuous and have left limits. Therefore, there is a great difference between the stochastic integral with respect to the Brown
process and the one with respect to the Poisson random measure. It should be pointed out that the proof for NSFDEwPJs
is certainly not a straightforward generalization of that for NSFDEs without jumps and some new techniques are developed
to cope with the difficulties due to the Poisson random measures.
The rest of the paper is organized as follows. In Section 2, we introduce some notations and hypotheses concerning Eq. (1).
In Section 3, the existence and uniqueness of the solution to Eq. (1) are investigated. In Section 4, we prove the p-th moment
of the solution will grow at most exponentially with exponent M and show that the exponential estimations implies the
almost surely asymptotic estimations.
2. Preliminaries
Let ðX; F ; PÞ be a complete probability space equipped with some filtration ðF t ÞtPt0 satisfying the usual conditions (i.e. it is
right continuous and ðF t0 Þ contains all P-null sets). Let s > 0, and Dð½s; 0; Rn Þ denote the family of all right-continuous functions with left-hand limits u from ½s; 0 ! Rn . The space Dð½s; 0; Rn Þ is assumed to be equipped with the norm
pffiffiffiffiffiffiffiffi
jjujj ¼ sups6t60 juðtÞj and jxj ¼ x> x for any x 2 Rn . If A is a vector or matrix, its trace norm is denoted by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jAj ¼ traceðA> AÞ, while its operator norm is denoted by jjAjj ¼ supfjAxj : jxj ¼ 1g. DbF 0 ð½s; 0; Rn Þ denotes the family of all
almost surely bounded, F 0 -measurable, Dð½s; 0; Rn Þ valued random variable n ¼ fnðhÞ : s 6 h 6 0g. Let
t 0 0; p P 2; LpF t ð½s; 0; Rn Þ denote the family of all F t0 measurable, Dð½s; 0; Rn Þ-valued random variables
0
u ¼ fuðhÞ : s 6 h 6 0g such that E sups6h60 juðhÞjp < 1.
¼p
ðtÞ; t P t0 g be a stationary F t -Poisson
Let ðU; BðUÞÞ be a measurable space and pðduÞ a r- finite measure on it. Let fp
point process on U with a characteristic measure p. Then, for A 2 BðU f0gÞ, here 0 2 the closure of A, the Poisson counting
measure N p is defined by
ðsÞ 2 Ag ¼
Np ððt0 ; t AÞ :¼ ]ft0 < s 6 t; p
X
ðsÞÞ;
I A ðp
t 0 <s6t
where ] denotes the cardinality of a set. For simplicity, we denote: N p ðt; AÞ :¼ N p ððt0 ; t AÞ. It follows from [26] that there
exists a r- finite measure p satisfying
E½Np ðt; AÞ ¼ pðAÞt;
This measure
PðNp ðt; AÞ ¼ nÞ ¼
eðt pðAÞÞðpðAÞtÞn
:
n!
e p is defined by
p is called the Levy measure. Then, the measure N
e p ð½t0 ; t; AÞ :¼ Np ð½t 0 ; t; AÞ tpðAÞ;
N
t > t0 :
We refer to Ikeda [26] for the details on Poisson point process.
The integral version of Eq. (1) is given by the equation
xðtÞ Dðxt Þ ¼ xt0 Dðxt0 Þ þ
Z
t
f ðxs ; sÞds þ
Z
t0
t
Z
t0
hðxs ; uÞNp ðds; duÞ;
ð2Þ
U
where
xt ¼ fxðt þ hÞ : s 6 h 6 0g
is regarded as a Dð½s; 0; Rn Þ-valued stochastic process. f : Dð½s; 0; Rn Þ ½t 0 ; T ! Rn and h : Dð½s; 0; Rn Þ U ! Rn are both
Borel-measurable functions. The initial condition xt0 is defined by
xt0 ¼ n ¼ fnðtÞ : s 6 t 6 0g 2 LpF t ð½s; 0; Rn Þ;
0
e p ðdt; duÞ is the compensated Poisson
that is, n is an F t0 -measurable Dð½s; 0; Rn Þ-valued random variable and Ejjnjjp < 1. N
random measure given by
e p ðdt; duÞ ¼ Np ðdt; duÞ pðduÞdt;
N
here
pðduÞ is the Levy measure associated to Np .
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W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
To study the existence and asymptotic estimations of the solution to Eq. (1), we consider the following hypotheses.
(H1) Let Dð0Þ ¼ 0 and for all u; w 2 Dð½s; 0; Rn Þ, there exists a constant k0 2 ð0; 1Þ such that
jDðuÞ DðwÞj 6 k0 jju wjj:
ð3Þ
n
(H2) For all u; w 2 Dð½s; 0; R Þ; t 2 ½t 0 ; T and u 2 U, there exist two positive constants k and L0 such that
jf ðu; tÞ f ðw; tÞj2 _
Z
jhðu; uÞ hðw; uÞj2 pðduÞ 6 kjju wjj2 :
ð4Þ
U
jf ð0; tÞj2 _ jhð0; uÞj2 6 L0 :
ð5Þ
n
(H3) For all u; w 2 Dð½s; 0; R Þ; p P 2 and u 2 U, there exists a positive constant L such that
jhðu; uÞ hðw; uÞjp 6 Ljju wjjp jujp ;
where
pðUÞ < 1 and
R
ð6Þ
p
U
juj du < 1.
Clearly, (H2) and (H3) implies the linear growth condition
jf ðu; tÞj2 _
Z
jhðu; uÞj2 pðduÞ 6 L1 ð1 þ jjujj2 Þ;
ð7Þ
U
and
Z
jhðu; uÞjp pðduÞ 6 L2 ð1 þ jjujjp Þ;
ð8Þ
U
where L1 and L2 are two positive constants.
In fact, for any u 2 Dð½s; 0; Rn Þ and t 2 ½t0 ; T, it follows from (4) and (5) that
jf ðu; tÞj2 6 2½jf ðu; tÞ f ð0; tÞj2 þ jf ð0; tÞj2 6 2ðkjjujj2 þ L0 Þ 6 L1 ð1 þ jjujj2 Þ:
and
Z
U
jhðu; uÞj2 pðduÞ 6 2½
Z
jhðu; uÞ hð0; uÞj2 pðduÞ þ
U
Z
jhð0; uÞj2 pðduÞ 6 2ðkjjujj2 þ 2L0 pðUÞÞ 6 L1 ð1 þ jjujj2 Þ:
U
where L1 ¼ maxf2k; 2L0 ; 2L0 pðUÞg. Similarly, for any u 2 Dð½s; 0; Rn Þ and t 2 ½t 0 ; T, it follows from (5) and (6) that
Z
U
jhðu; uÞjp pðduÞ 6 2p1
Z
ðjhðu; uÞ hð0; uÞjp þ jhð0; uÞjp ÞpðduÞ 6 2p1
U
Z
U
p
jujp duk1 jjujjp þ 2p1 L20 pðUÞ
6 L2 ð1 þ jjujjp Þ;
n
o
p
R
where L2 ¼ max 2p1 U jujp duk1 ; 2p1 L20 pðUÞ . Hence, the linear growth conditions (7) and (8) are satisfied.
Now we present the definition of the solution to Eq. (1).
Definition 2.1. A right continuous with left limits process x ¼ fxðtÞ; t 2 ½t 0 ; Tg ðt 0 < T < 1Þ is called a solution of Eq. (1) if
(1) xðtÞ is F t -adapted and x ¼ fxðtÞ; t 2 ½t0 ; Tg is Rn -valued;
RT
(2) t0 jxðtÞj2 ds < 1; a:s.;
(3) xðtÞ ¼ n and, for every t0 6 t 6 T,
xðtÞ Dðxt Þ ¼ xt0 Dðxt0 Þ þ
Z
t
f ðxs ; sÞds þ
t0
Z
t
t0
Z
hðxs ; uÞN p ðds; duÞ a:s:
U
A solution xðtÞ is said to be unique if any other solution yðtÞ is indistinguishable from it, that is,
PfxðtÞ ¼ yðtÞ; t 2 ½t 0 ; Tg ¼ 1:
3. The existence and uniqueness theorem
In this section, we establish the existence and uniqueness of the solution to Eq. (1) under the Lipschitz condition and the
local Lipschitz condition.
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W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
Define x0t0 ¼ n and x0 ðtÞ ¼ nð0Þ for t 2 ½t 0 ; T. Let xnt0 ¼ n; n ¼ 1; 2; . . . and define the sequence of successive approximations to
Eq. (1)
Z
xn ðtÞ Dðxnt Þ ¼ nð0Þ DðnÞ þ
Z
t
t0
f ðxn1
; sÞds þ
s
Z
t
U
t0
hðxn1
; uÞNp ðds; duÞ;
s
n P 1:
ð9Þ
Theorem 3.1. Let p P 2 and suppose that the coefficients of Eq. (1) satisfy conditions (H1)–(H3), then Eq. (1) has a unique
solution xðtÞ on ½t0 ; T in the sense of Lp -norm.
In order to prove this theorem, let us present three useful lemmas.
Lemma 3.1 [2]. Let p P 2; e > 0 and a; b 2 R, then
h
ip1 1
jbjp
:
ja þ bjp 6 1 þ ep1
jajp þ
ð10Þ
e
Lemma 3.2. Under conditions (H1)–(H3), there exists a positive constant c such that
E sup jxn ðtÞjp 6 c;
ð11Þ
t 0 6t6T
where c ¼ ðc3 þ c4 ðT t0 ÞEjjnjjp Þec4 ðTt0 Þ ; c3 and c4 are two positive constants of (25).
Proof. For any
e > 0, it follows from Lemma 3.1 that
h
ip1 1
jDðxnt Þjp
:
jxn ðtÞjp ¼ jDðxnt Þ þ xn ðtÞ Dðxnt Þj 6 1 þ ep1
jxn ðtÞ Dðxnt Þjp þ
ð12Þ
e
By (H1), one gets
p
h
ip1 1
k jjxn jjp
:
jxn ðtÞjp 6 1 þ ep1
jxn ðtÞ Dðxnt Þjp þ 0 t
Letting
e¼
h
k0
1k0
ð13Þ
e
ip1
and taking the expectation on both sides of (13), we have
E sup jxn ðsÞjp 6 k0 E sup jjxns jjp þ
t 0 6s6t
t 0 6s6t
1
ð1 k0 Þ
p1
E sup jxn ðsÞ Dðxns Þjp :
ð14Þ
t 0 6s6t
On the other hand, we have
E sup jjxns jjp 6 E sup jxn ðsÞjp 6 Ejjnjjp þ E sup jxn ðsÞjp :
t 0 s6s6t
t 0 6s6t
ð15Þ
t 0 6s6t
Combing (14) and (15), we obtain
E sup jxn ðsÞjp 6
t 0 6s6t
k0
1
n
n p
Ejjnjjp þ
p E sup jx ðsÞ Dðxs Þj :
1 k0
ð1 k0 Þ t0 6s6t
ð16Þ
By (9) and using the inequality ja þ b þ cjp 6 3p1 ½jajp þ jbjp þ jcjp , we have
!
E sup jxn ðsÞ Dðxns Þjp 6 3p1 E
t 0 6s6t
Z
s
Z
t0
U
sup jnð0Þ DðnÞjp
t 0 6s6t
t0 6s6t
t 0 6s6t
t0
p 3
X
:¼ 3p1 Ii :
hðxn1
;
uÞN
ðd
r
;
duÞ
p
r
ð17Þ
i¼1
Let us estimate the terms introduced above. Letting
p
p
Z s
þ E sup
þ E sup f ðxn1
;
r
Þd
r
r
e ¼ kp1
0 , then it follows from Lemma 3.1 that
p
I1 6 ð1 þ k0 Þ Ejjjnj :
ð18Þ
By using the Höder inequality and the linear growth condition (7), one gets
I2 6 ðt t 0 Þp1 E
Z
t
t0
jf ðxn1
; sÞjp ds 6 ðt t 0 Þp1 E
s
Z
t
t0
2
p
2
p
½L1 ð1 þ jjxn1
jj2 Þ ds 6 ðt t0 Þp1 ð2L1 Þ2 E
s
Z
t
t0
ð1 þ jjxn1
jjp Þds:
s
ð19Þ
For the third term I3 in (17), we have
Z
I3 ¼ E sup t 0 6s6t
62
p1
s
t0
Z
U
Z
E sup t 6s6t
0
e
hðxn1
r ; uÞ N p ðdr; duÞ þ
s
t0
Z
Z
s
t0
Z
U
p
hðxn1
r ; uÞpðduÞdr
p
Z
e p ðdr; duÞ þ 2p1 E sup hðxr ; uÞ N
t 6s6t
s
n1
U
0
t0
Z
U
p
;
hðxn1
;
uÞ
p
ðduÞd
r
r
ð20Þ
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W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
e p ðdt; duÞ :¼ N p ðdt; duÞ pðduÞdt. For the last term of (20), using the Höder inequality and condition (7), we obtain
where N
Z
E sup t 0 6s6t
Z
s
t0
U
p
Z t p1 Z
hðxn1
ds
r ; uÞpðduÞdr 6 E
t0
Z
p
hðxn1 ; uÞpðduÞ ds
s
t
t0
6 ðt t0 Þp1 E
U
Z t Z
t0
6 ðt t0 Þ
U
p
2
p1
pðduÞ
½pðUÞ E
2p Z
U
Z
t
t0
p
2
2p
jhðxn1
; uÞj2 pðduÞ ds
s
p
½L1 ð1 þ jjxn1
jj2 Þ2 ds
s
p
6 ðt t0 Þp1 ð2L1 Þ ½pðUÞ2 E
Z
t
t0
ð1 þ jjxn1
jjp Þds:
s
ð21Þ
Now let us estimate the martingale part in (20). By the Kunita’s estimates (see Kunita [27] and Applebaum [28]), conditions
(7) and (8) and properties of stochastic integral with respect to a Poisson random measure, we have a positive real number cp
such that the following inequality holds:
Z
E sup t 6s6t
Z
s
U
t0
0
( Z Z
p
2p
Z
t
e
hðxr ; uÞ N p ðdr; duÞ 6 cp E
jhðxn1
; uÞj2 pðduÞ ds þ E
s
t
n1
U
t0
t0
Z
U
jhðxn1
; uÞjp
s
)
pðduÞds
Z t
Z t
p
½L1 ð1 þ jjxn1
jjÞ2 ds þ L2 E ð1 þ jjxn1
jjp Þds
6 cp E
s
s
t0
p
2
6 cp ½ð2L1 Þ þ L2 E
Z
t0
t
ð1 þ
t0
jjxn1
jjp Þds:
s
ð22Þ
Inserting (21) and (22) into (20), we obtain that
I3 6 c1 E
Z
t
t0
ð1 þ jjxn1
jjp Þds:
s
ð23Þ
p
p
p
where c1 ¼ 2p1 ½ð2L1 Þ2 ðt t0 Þp1 ðpðUÞÞ2 þ cp ðð2L1 Þ2 þ L2 Þ. Therefore,
p
E sup jxn ðsÞ Dðxns Þjp 6 3p1 ð1 þ k0 Þ Ejjnjjp þ c2 E
Z
t 0 6s6t
t
t0
ð1 þ jjxn1
jjp Þds;
s
ð24Þ
h
i
p
where c2 ¼ 3p1 ðt t 0 Þp1 ð2L1 Þ2 þ c1 . Combing (16) and (24) together, we have
E sup jxn ðsÞjp 6 c3 þ c4 E
t 0 6s6t
where c3 ¼
h
k0
1k0
Z
t
t0
i
p
jjxn1
jjp ds:
s
c2
0Þ
Ejjnjjp þ ð1k
þ 3p1 ð1þk
ð1k Þp
p
0Þ
0
max E sup jxn ðsÞjp 6 c3 þ c4
16n6r
t 0 6s6t
Z
t
ð25Þ
c2
ðT t 0 Þ and c4 ¼ ð1k
. For any r P 1,
Þp
0
max E sup jxn1 ðrÞjp ds 6 c3 þ c4
t 0 16n6r
t 0 6r6s
Z
t
!
Ejjnjjp þ max E sup jxn ðrÞjp ds:
16n6r
t0
t 0 6r6s
From the Gronwall inequality, we derive that
max E sup jxn ðtÞjp 6 ðc3 þ c4 ðT t 0 ÞEjjnjjp Þec4 ðTt0 Þ :
16n6r
t 0 6t6T
Since r is arbitrary, we must have
E sup jxn ðtÞjp 6 ðc3 þ c4 ðT t 0 ÞEjjnjjp Þec4 ðTt0 Þ ;
ð26Þ
t 0 6t6T
which shows that the desired result holds with c ¼ ðc3 þ c4 ðT t 0 ÞEjjnjjp Þec4 ðTt0 Þ . h
Lemma 3.3. Let the conditions of Theorem 3.1 hold. Then fxn ðtÞg ðn P 0Þ defined by (9) is a Cauchy sequence in Dð½t 0 ; T; Rn Þ.
Proof. For n P 1 and t 2 ½t 0 ; T, it follows from (9) that,
xnþ1 ðtÞ xn ðtÞ ¼ Dðxnþ1
Þ Dðxnt Þ þ
t
Z
t
t0
½f ðxns ; sÞ f ðxn1
; sÞds þ
s
Z
Z
t
U
t0
½hðxns ; uÞ hðxn1
; uÞNp ðds; duÞ:
s
ð27Þ
Similar to the analysis of (14), by Lemma 3.1 and taking the expectation on jxnþ1 ðtÞ xn ðtÞjp , we have
!
E
sup jx
t 0 6s6t
nþ1
n
p
ðsÞ x ðsÞj
!
6 k0 E
sup jx
t0 6s6t
nþ1
n
ðsÞ x ðsÞj
p
þ
1
p1
ð1 k0 Þ
E sup j½xnþ1 ðsÞ xn ðsÞ ½Dðxnþ1
Þ Dðxns Þjp :
s
t0 6s6t
ð28Þ
257
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
Consequently,
!
nþ1
E
sup jx
n
p
ðsÞ x ðsÞj
6
t 0 6s6t
1
nþ1
ðsÞ xn ðsÞ ½Dðxnþ1
Þ Dðxns Þjp :
p E sup j½x
s
ð1 k0 Þ
t 0 6s6t
ð29Þ
The basic inequality ja þ bjp 6 2p1 ðjajp þ jbjp Þ implies that
"
Z s
p
p
p1
n
½f ðxn ; rÞ f ðxn1 ; rÞdr
E sup j½xnþ1 ðsÞ xn ðsÞ ½Dðxnþ1
Þ
Dðx
Þj
6
2
E
sup
r
r
s
s
t 6s6t
t 6s6t
0
t0
0
Z
þ E sup t 6s6t
E sup j
Z
t 0 6s6t
s
t0
p
p1
½f ðxnr ; rÞ f ðxn1
E
r ; rÞdrj 6 ðt t 0 Þ
U
t0
0
Applying the Höder inequality and (H2), we obtain
Z
s
Z
t
t0
p #
:
½hðxnr ; uÞ hðxn1
;
uÞN
ðd
r
;
duÞ
p
r
p
jf ðxns ; sÞ f ðxn1
; sÞjp ds 6 ðt t 0 Þp1 k2
s
Z
t
t0
ð30Þ
Ejjxns xn1
jjp ds:
s
ð31Þ
By the Kunita’s estimates, Höder inequality and (H2)–(H3), there exists a positive constant c5 such that
E sup j
Z
t 0 6s6t
s
Z
U
t0
Z
p
p1
½hðxnr ; uÞ hðxn1
;
uÞN
ðd
r
;
duÞj
6
2
E
sup
p
r
t 6s6t
þ2
U
t0
0
p1
Z
s
Z
E sup t 6s6t
p
e p ðdr; duÞ
N
½hðxnr ; uÞ hðxn1
;
uÞ
r
Z
s
t0
0
U
p
½hðxnr ; uÞ hðxn1
;
uÞ
p
ðduÞd
r
r
p
6 2p1 ½ðt t0 Þp1 ðpðUÞÞ2 E
Z t Z
t
U
2p
2
jhðxns ; uÞ hðxn1
;
uÞj
p
ðduÞ
ds
s
0
( Z Z
2p
t
2
þ cp 2p1 E
jhðxns ; uÞ hðxn1
;
uÞj
p
ðduÞ
ds
s
Z
þE
t
t0
Z
U
t0
U
Z t
6 c5 E
jhðxns ; uÞ hðxn1
; uÞjp pðduÞds
jjxns xn1
jjp ds;
s
s
t0
ð32Þ
p1
where c5 ¼ 2
E
p1
f½ðt t0 Þ
nþ1
sup jx
p
2
p
2
ðpðUÞÞ þ cp k þ cp L
!
n
p
ðsÞ x ðsÞj
6 c6
Z
t 0 6s6t
R
U
juj
p
pðduÞg. Hence, inserting (30)–(32) into (29) yields
!
t
E
n
sup jx ðrÞ x
t 0 6r6s
t0
n1
ðrÞj
p
ds;
ð33Þ
p
where c6 ¼ k2 2p1 ð1k1 Þp ½ðT t 0 Þp1 þ c5 .
0
Setting un ðtÞ ¼ E supt0 6s6t jxnþ1 ðsÞ xn ðsÞjp , we have
un ðtÞ 6 c6
Z
t
t0
un1 ðs1 Þds1 6 c26
Z
t
ds1
0
Z
s1
t0
un2 ðs2 Þds2 6 6 cn6
Z
t
ds1
t0
Z
s1
ds2 t0
Z
sn1
t0
u0 ðsn Þdsn :
ð34Þ
From the Kunita’s estimates, Höder inequality and conditions (7) and (8), we have
u0 ðtÞ ¼ E sup jx1 ðsÞ x0 ðsÞjp 6 c0 :
ð35Þ
t 0 6s6t
Substituting (35) into (34) and integrating the right hand side, we obtain
!
E
nþ1
sup jx
n
p
ðsÞ x ðsÞj
6 c0
ðc6 ðt t 0 ÞÞn
:
n!
ð36Þ
6 c0
ðc6 ðT t 0 ÞÞn
:
n!
ð37Þ
t 0 6s6t
Taking t ¼ T in (36), we have
E
sup jxnþ1 ðtÞ xn ðtÞjp
!
t 0 6t6T
Then using the Chebyshev inequality, one gets
P
sup jx
nþ1
t 0 6t6T
Since R
P
1
ðtÞ x ðtÞj > n
2
1 c0 Mðc6 ðTt 0 ÞÞn
n¼0
n!
sup jx
t 0 6t6T
nþ1
!
n
6 c0 M
ðc6 ðT t 0 ÞÞn
:
n!
< 1, and by the Borel-Cantelli lemma, we have
1
ðtÞ x ðtÞj 6 n
2
n
!
¼ 1:
ð38Þ
258
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
(38) implies that for each t; fxn ðtÞgn¼1;2 is a Cauchy sequence on ½t 0 ; T under sup j j. However, the space Dð½t 0 ; T; Rn Þ is not a
complete space under sup j:j and we cannot get the limit of the sequence fxn ðtÞgnP1 . So we need to introduce a metric to
make the space Dð½t0 ; T; Rn Þ complete. For any x; y 2 Dð½t 0 ; T; Rn Þ, Billingsley [29] gives the following metric
(
dðx; yÞ ¼ inf
k2K
)
kðtÞ kðsÞ
sup jxt ykðtÞ j þ sup log
;
ts t 0 6t6T
t0 6s6t6T
where K ¼ fk ¼ kðtÞ : k is strictly increasing, continuous on t 2 ½t0 ; T, such that kðt 0 Þ ¼ t 0 ; kðTÞ ¼ Tg. So we have that
Dð½t0 ; T; Rn Þ is a complete metric space. Taking kðtÞ ¼ t, we can see that fxn ðtÞgnP1 is a cauchy sequence under dð; Þ. The proof
is completed. h
Proof of Theorem 3.1. Uniqueness. Let xðtÞ and yðtÞ be two solutions of Eq. (1). Then, for t 2 ½t0 ; T, by the Kunita’s estimates,
Höder inequality, we have
E sup jxðsÞ yðsÞjp 6 c
Z
t 0 6s6t
t
t0
E sup jxðuÞ yðuÞjp ds:
ð39Þ
t 0 6u6s
Therefore, using the Gronwall inequality, we get
E sup jxðsÞ yðsÞjp ¼ 0;
t 2 ½t 0 ; T;
t 0 6s6t
which implies that xðtÞ ¼ yðtÞ for all t 2 ½t 0 ; T. Therefore, for all t 2 ½t0 ; T; xðtÞ ¼ yðtÞ a.s. h
Existence. We derive from Lemma 3.3 that fxn ðtÞgn¼1;2 is a Cauchy sequence in Dð½t 0 ; T; Rn Þ. Hence, there exists a unique
solution xðtÞ 2 Dð½t 0 ; T; Rn Þ such that dðxn ðÞ; xðÞÞ ! 0 as n ! 1. For all t 2 ½t 0 ; T, taking limits on both sides of (9) and letting
n ! 1, we then can show that xðtÞ is the solution of Eq. (1). So the proof of Theorem 3.1 is completed.
Next, we relax the Lipschitz conditions (H2)–(H3) and replace them by the following the local Lipschitz conditions.
(H4) For all u; w 2 Dð½s; 0; Rn Þ; t 2 ½t 0 ; T; u 2 U and jjujj _ jjwjj 6 n, there exist two positive constants kn and L0 such that
jf ðu; tÞ f ðw; tÞj2 _
Z
jhðu; uÞ hðw; uÞj2 pðduÞ 6 kn jju wjj2 :
ð40Þ
U
(H5) For all u; w 2 Dð½s; 0; Rn Þ; p P 2; u 2 U and jjujj _ jjwjj 6 n, there exists a positive constant Ln such that
jhðu; uÞ hðw; uÞjp 6 Ln jju wjjp jujp :
where
pðUÞ < 1 and
R
ð41Þ
p
U
juj du < 1.
Then, Theorem 3.1 can be generalized as Theorem 3.2.
Theorem 3.2. Let conditions (H1), (H4) and (H5) hold. Then Eq. (1) has a unique solution xðtÞ on ½t 0 ; T in the sense of Lp -norm.
Moreover, there exists a constants c such that
E sup jxðtÞjp 6 c:
t 0 6t6T
for any t 2 ½t 0 ; T.
Proof. For each n 1, define the truncation function
(
f n ðt; xÞ ¼
f ðt; xÞ;
if jjxjj 6 n;
nx
f ðt; jjxjj
Þ;
if jjxjj P n;
ð42Þ
and
(
hn ðx; uÞ ¼
hðx; uÞ;
if jjxjj 6 n;
ð43Þ
nx
hðjjxjj
; uÞ; if jjxjj P n:
Then f n and hn satisfy the conditions (H1)–(H3). From Theorem 3.1, we have that the following equation
xn ðtÞ ¼ nð0Þ þ Dððxn Þt Þ DðnÞ þ
Z
t
t0
f n ððxn Þs ; sÞds þ
Z
t
t0
Z
U
hn ððxn Þs ; uÞNp ðds; duÞ
has a unique solution xn ðtÞ. Moreover, xnþ1 ðtÞ is the unique solution of the equation
ð44Þ
259
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
Z
xnþ1 ðtÞ ¼ nð0Þ þ Dððxnþ1 Þt Þ DðnÞ þ
Z
t
f nþ1 ððxnþ1 Þs ; sÞds þ
t0
By (44) and (45), we have
xnþ1 ðtÞ xn ðtÞ ¼ Dððxnþ1 Þt Þ Dððxn Þt Þ þ
Z
t
t0
Z
U
hnþ1 ððxnþ1 Þs ; uÞNp ðds; duÞ:
t
t0
½f nþ1 ððxnþ1 Þs ; sÞ f n ððxn Þs ; sÞds þ
Z
t
t0
Z
U
ð45Þ
½hnþ1 ððxnþ1 Þs ; uÞ
hn ððxn Þs ; uÞNp ðds; duÞ:
ð46Þ
For any fixed n P 1, define the stopping time
sn ¼ T ^ inf ft 2 ½t0 ; T : jðxn Þt j P ng:
Taking the expectation on jxnþ1 ðtÞ xn ðtÞjp and by Lemma 3.1, it deduces that
E
sup jxnþ1 ðsÞ xn ðsÞjp 6
t 0 6s6t^sn
1
ð1 k0 Þ
þ k0 E
E
p1
sup j½ðxnþ1 Þs ðxn Þs ½Dððxnþ1 Þs Þ Dððxn Þs Þjp
t 0 6s6t^sn
!
sup jðxnþ1 Þs ðxn Þs j
p
ð47Þ
:
t 0 6s6t^sn
Therefore,
(
Z s
Z s
1
p1
E sup jxnþ1 ðsÞ xn ðsÞj 6
Eð sup j
½f nþ1 ððxnþ1 Þr ; rÞ f n ððxn Þr ; rÞdrjp Þ þ Eð sup j
p2
ð1 k0 Þ
t 0 6s6t^sn
t 0 6s6t^sn
t 0 6s6t^sn
t0
t0
Z
1
p
p1
½hnþ1 ððxnþ1 Þr ; uÞ hn ððxn Þr ; uÞNp ðdr; duÞj ¼
ðJ 1 þ J 2 Þ:
p2
ð1 k0 Þ
U
p
ð48Þ
By the Hölder inequality and rearranging the terms on the right-hand side by plus and minus technique, we have
J 1 6 ðt ^ sn t0 Þp1 E
6 ðt ^ sn t0 Þp1 E
Z
t^sn
t0
Z
t^sn
t0
jf nþ1 ððxnþ1 Þs ; sÞ f n ððxn Þs ; sÞjp ds
f2p1 jf nþ1 ððxnþ1 Þs ; sÞ f nþ1 ððxn Þs ; sÞjp þ 2p1 jf nþ1 ððxn Þs ; sÞ f n ððxn Þs ; sÞjp gds:
ð49Þ
The Kunita’s estimates implies that
J 2 6 c7 E
6 c7 E
Z
t^sn
½
t0
Z t^sn
Z
U
p
2
jhnþ1 ððxnþ1 Þs ; uÞ hn ððxn Þs ; uÞj2 pðduÞ ds þ cp 2p1 E
Z
t0
U
Z
t^sn
t0
Z
U
t^sn
Z
t0
U
jhnþ1 ððxnþ1 Þs ; uÞ hn ððxn Þs ; uÞjp pðduÞds
o2p
½2jhnþ1 ððxnþ1 Þs ; uÞ hnþ1 ððxn Þs ; uÞj2 þ2jhnþ1 ððxn Þs ; uÞ hn ððxn Þs ; uÞj2 pðduÞ ds
Z
þ cp 22p2 E
þE
Z
t^sn
t0
Z
U
jhnþ1 ððxnþ1 Þs ; uÞ hnþ1 ððxnþ1 Þs ; uÞjp pðduÞds
jhnþ1 ððxnþ1 Þs ; uÞ hn ððxn Þs ; uÞjp pðduÞds ;
ð50Þ
p
where c7 ¼ 2p1 ½ðt ^ sn t 0 Þp1 ðpðUÞÞ2 þ cp . Combing (48)–(50) together, it follows that
E
sup jxnþ1 ðsÞ xn ðsÞjp 6
t 0 6s6t^sn
1
p1
ðT t0 Þp1 E
p2
ð1 k0 Þ
Z t^sn n
o
2p1 jf nþ1 ððxnþ1 Þs ; sÞf nþ1 ððxn Þs ; sÞjp þ 2p1 jf nþ1 ððxn Þs ; sÞ f n ððxn Þs ; sÞjp ds
t0
Z t^sn Z
1
p1
2
c
E
½2jhnþ1 ððxnþ1 Þs ; uÞ hnþ1 ððxn Þs ; uÞj2 þ 2jhnþ1 ððxn Þs ; uÞ
7
p
ð1 k0 Þ
t0
U
Z t^sn Z
o2p
jhnþ1 ððxnþ1 Þs ; uÞ hnþ1 ððxnþ1 Þs ; uÞjp pðduÞds
hn ððxn Þs ; uÞj2 pðduÞ ds þ cp 22p2 E
þ
þE
Z
t^sn
t0
Z
U
t0
U
jhnþ1 ððxnþ1 Þs ; uÞ hn ððxn Þs ; uÞjp pðduÞds :
ð51Þ
For t 0 6 t 6 sn , we have
f nþ1 ððxn Þt ; tÞ ¼ f n ððxn Þt ; tÞ ¼ f ððxn Þt ; tÞ;
hnþ1 ððxn Þt ; uÞ ¼ hn ððxn Þt ; uÞ ¼ hððxn Þt ; uÞ:
ð52Þ
260
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
By (52), we get from (51),
E
Z t^sn
1
2p2
ðT t 0 Þp1 E
jf nþ1 ððxnþ1 Þs ; sÞ f nþ1 ððxn Þs ; sÞjp ds
p2
ð1 k0 Þ
t0
Z t^sn Z
o2p
1
p1
2
c
E
½2jhnþ1 ððxnþ1 Þs ; uÞhnþ1 ððxn Þs ; uÞj2 pðduÞ dsþcp 22p2 E
þ
7
p
ð1 k0 Þ
t0
U
Z t^sn Z
ð53Þ
jhnþ1 ððxnþ1 Þs ; uÞ hnþ1 ððxnþ1 Þs ; uÞjp pðduÞds :
sup jxnþ1 ðsÞ xn ðsÞjp 6
t 0 6s6t^sn
t0
U
By the local Lipschitz conditions (H4) and (H5), we have
E
sup jxnþ1 ðsÞ xn ðsÞjp 6 c8 E
t 0 6s6t^sn
where c8 ¼ ð1k1
E
Z
t^sn
t0
p
p
0Þ
jjðxnþ1 Þs ðxn Þs jjp ds 6 c8
3
2
fknþ1
½22p2 ðT t0 Þp1 þ 22p2 c7 þ cp 23p3 Lnþ1
R
U
Z
t
E
t0
sup jxnþ1 ðrÞ xn ðrÞjp ds;
t 0 6r6s^sn
ð54Þ
jujp pðduÞg. From (54) and the Gronwall inequality, we get
sup jxnþ1 ðsÞ xn ðsÞjp ¼ 0;
ð55Þ
t 0 6s6t^sn
which yields
xnþ1 ðtÞ ¼ xn ðtÞ;
for t 2 ½t0 ; sn :
ð56Þ
It then deduced that sn is increasing, that is as n ! 1; sn " T a.s. By the linear growth condition (7) and (8), for almost all
x 2 X, there exists an integer n0 ¼ n0 ðxÞ such that sn ¼ T as n P n0 . Now define xðtÞ by xðtÞ ¼ xn0 ðtÞ for t 2 ½t0 ; T. Next to
verify that xðtÞ is the solution of Eq. (1). By (56), xðt ^ sn Þ ¼ xn ðt ^ sn Þ, and it follows from (44) that
Z
xðt ^ sn Þ Dðxt^sn Þ ¼ nð0Þ DðnÞ þ
f n ðxs ; sÞds þ
t0
Z
¼ nð0Þ DðnÞ þ
Z
t^sn
t^sn
f ðxs ; sÞds þ
Z
t0
t^sn
t0
t^sn
t0
Z
Z
hn ðxs ; uÞNp ðds; duÞ
U
hðxs ; uÞNp ðds; duÞ:
ð57Þ
U
Letting n ! 1 on both sides of (57), we obtain
xðt ^ TÞ Dðxt^T Þ ¼ nð0Þ DðnÞ þ
Z
t^T
f ðxs ; sÞds þ
Z
t0
t^T
t0
Z
hðxs ; uÞNp ðds; duÞ:
U
that is
xðtÞ Dðxt Þ ¼ nð0Þ DðnÞ þ
Z
t
f ðxs ; sÞds þ
t0
Z
t
t0
Z
hðxs ; uÞNp ðds; duÞ;
U
which implies that xðtÞ is the solution of Eq. (1). By stopping our process xðtÞ, uniqueness of the solution to Eq. (1) is obtained.
Moreover, by the proof of Theorem 3.1, we can easily obtain that E supt0 6t6T jxðtÞjp 6 c. The proof is completed. h
4. Asymptotic estimations for solutions
In this section, we will give the exponential estimate of the solution to Eq. (1).
e p ðdt; duÞ :¼ N p ðdt; duÞ pðduÞdt, we can rewrite Eq. (1) as the following equation
According to the definition of N
d½xðtÞ Dðxt Þ ¼ Fðt; xt Þdt þ
Z
e p ðdt; duÞ;
hðxt ; uÞ N
ð58Þ
U
R
where Fðt; xt Þ ¼ f ðxt ; tÞ þ U hðxt ; uÞpðduÞ.
2;1
n
Let C ðR ½t 0 s; TÞ; Rþ Þ denote the family of all nonnegative functions Vðx; tÞ on Rn ½t0 s; TÞ which are continuously
twice differentiable with respect to x and continuously once differentiable with respect to t. For a V 2 C 2;1 ðRn ½t 0 s; TÞ; Rþ Þ,
one can define the Kolmogorov operator LV as follows:
LVðx; y; tÞ V t ðx DðyÞ; tÞ þ V x ðx DðyÞ; tÞFðt; yÞ þ
Z
½Vðx DðyÞ þ hðy; uÞ; tÞ Vðx DðyÞ; tÞ
U
V x ðx DðyÞ; tÞhðy; uÞpðduÞ;
where
V t ðx; tÞ ¼
@Vðx; tÞ
;
@t
V x ðx; tÞ ¼
@Vðx; tÞ
@Vðx; tÞ
:
;...;
@x1
@xn
First, we establish the p-th exponential estimations of the solution to Eq. (1).
ð59Þ
261
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
Theorem 4.1. Let fxðtÞ; t 0 6 t 6 Tg be a solution of Eq. (1) whose coefficients satisfy conditions (H1) and (H2). For a given integer
p P 2 and any 2 6 q 6 p, there exists a positive constant K such that
Z
jhðu; uÞjq pðduÞ 6 Kjjujjq :
ð60Þ
U
Then, for any t0 6 t 6 T,
E sup jxðsÞjp 6 ½1 þ ð1 þ c12 ÞEjjnjjp eMðtt0 Þ ;
ð61Þ
t 0 s6s6t
þc10 þc11 Þ
where M ¼ 2ðc9ð1k
. c9 ; c10 ; c11 ; c12 are four positive constants of (68), (74), (80), (82).
Þp
0
Proof. Let VðxðtÞ Dðxt Þ; tÞ ¼ 1 þ jxðtÞ Dðxt Þjp , then V t ðxðtÞ Dðxt Þ; tÞ ¼ 0. Applying the It b
o formula to VðxðtÞ Dðxt Þ; tÞ, we
obtain that
VðxðtÞ Dðxt Þ; tÞ ¼ Vðxðt 0 Dðxt0 ÞÞ; t 0 Þ þ
Z
t
LVðxðsÞ; xs ; sÞds þ
t0
Z
t
Z
½VðxðsÞ Dðxs Þ þ hðxs ; uÞ; sÞ VðxðsÞ
U
t0
e p ðds; duÞ:
Dðxs Þ; sÞ N
ð62Þ
By (59), we have
1 þ jxðtÞ Dðxt Þjp ¼ 1 þ jxðt 0 Þ Dðxt0 Þjp þ p
Z
t
jxðsÞ Dðxs Þjp2 ½xðsÞ Dðxs Þ> Fðs; xs Þds þ
Z
t0
t
Z
t0
fð1 þ jxðsÞ
U
Dðxs Þ þ hðxs ; uÞjp Þ ð1 þ jxðsÞ Dðxs Þjp Þ pjxðsÞ Dðxs Þjp2 ½xðsÞ Dðxs Þ> hðxs ; uÞgpðduÞds
Z t Z
e p ðds; duÞ:
þ
fð1 þ jxðsÞ Dðxs Þ þ hðxs ; uÞjp Þ ð1 þ jxðsÞ Dðxs Þjp Þg N
t0
ð63Þ
U
Taking the expectation on both sides of (63), one gets
E sup ð1 þ jxðsÞ Dðxs Þjp Þ 6 þE sup jn þ DðnÞjp þ pE
t 0 6s6t
Z
Z
t 0 6s6t
s
Z
t0
s
Z
Z
t0
t
jxðsÞ Dðxs Þjp1 jFðs; xs Þjds þ pE sup
t 0 6s6t
t0
p2
jxðrÞ Dðxr Þj
e p ðdr; duÞ þ E sup
½xðrÞ Dðxr Þ hðxr ; uÞ N
>
t 0 6s6t
U
p
p
fjxðrÞ Dðxr Þ þ hðxr ; uÞj jxðrÞ Dðxr Þj pjxðrÞ
U
p
Dðxr Þjp2 ½xðrÞ Dðxr Þ> hðxr ; uÞgNp ðdr; duÞ 6 1 þ ð1 þ k0 Þ Ejjnjjp þ
3
X
Q i;
ð64Þ
i¼1
where
Z
Q 1 ¼ pE
t
jxðsÞ Dðxs Þjp1 jFðs; xs Þjds;
t0
Z
Q 2 ¼ pE sup
t0 6s6t
Q 3 ¼ E sup
s
e p ðdr; duÞ;
jxðrÞ Dðxr Þjp2 ½xðrÞ Dðxr Þ> hðxr ; uÞ N
U
t0
Z
t 0 6s6t
Z
s
Z
fjxðrÞ Dðxr Þ þ hðxr ; uÞjp jxðrÞ Dðxr Þjp pjxðrÞ Dðxr Þjp2 ½xðrÞ Dðxr Þ> hðxr ; uÞgN p ðdr; duÞ:
U
t0
Let us estimate Q 1 . By the basic inequality
ar b
1r
6 ra þ ð1 rÞb;
r 2 ½0; 1;
we derive that
ap1 b 6
e1 ðp 1Þ
p
ap þ
1
pep1
1
p
b ;
where a; b; e1 > 0. Hence,
Q 1 6 pE
Z t"
t0
e1 ðp 1Þ
p
p
jxðsÞ Dðxs Þj þ
1
pep1
1
#
p
jFðs; xs Þj ds 6 pE
Z t"
e1 ðp 1Þ
t0
p
p
p
ð1 þ k0 Þ jjxs j þ
1
pep1
1
#
p
jFðs; xs Þj ds:
ð65Þ
262
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
By using Lemma 3.1, we have
E
Z
t
jFðs; xs Þjp ds 6 E
t0
p
Z th
p
ip1 Z
1
hðxs ; uÞpðduÞ þ jf ðxs ; sÞj ds
1 þ ep1
t0
p
6 ð2LÞ2 E
Z
e
U
t
1
½1 þ ep1 p1
p
1
½ððpðUÞÞ2 þ Þð1 þ jjxs jjp Þds:
Letting
E
ð66Þ
e
t0
e ¼ ð2LÞp1 , then we get
Z
t
3p
p
jFðs; xs Þjp ds 6 ð1 þ 2LÞ 2 1 ðpðUÞÞ2 E
Z
t0
t
ð1 þ jjxs jjp Þds:
ð67Þ
t0
Inserting (67) into (65) and letting
Q 1 6 pE
Z
2
e1 ¼ 1þ2L
, we obtain that
1þk0
3
3p
p
Z t
p
1
2
2
4e1 ðp 1Þð1 þ k0 Þ jjxs jjp þ ð1 þ 2LÞ p2ðpðUÞÞ ð1 þ jjxs jjp Þ5ds 6 c9 E ð1 þ jjxs jjp Þds;
p
t0
pe12
t
t0
ð68Þ
p
p1
where c9 ¼ ð1 þ 2LÞp ð1 þ k0 Þ ½p þ ðpðUÞÞ2 . For the estimation of Q 2 . By using the Burkholder-Davis inequality, there exists a
positive constant ~cp such that
Q 2 6 p~cp E
Z
t
Z
t0
12
jxðsÞ Dðxs Þj2p2 jhðxs ; uÞj2 pðduÞds
U
"
p
6 p~cp E sup jxðsÞ Dðxs Þj
Z
t 0 6s6t
Letting
p2
jxðsÞ Dðxs Þj
eE sup jxðsÞ Dðxs Þj
p
t0 6s6t
#12 1
e
E
Z
1
1
E sup jxðsÞ Dðxs Þjp þ p2 ~c2p E
2 t0 6s6t
2
t
Z
ð69Þ
Z
t
t
jxðsÞ Dðxs Þj
Z
t0
Z
p2
12
jhðxs ; uÞj pðduÞds
2
U
t0
p~cp e
p~cp
E sup jxðsÞ Dðxs Þjp þ
E
2
2e
t 0 6s6t
e ¼ p1~cp , we obtain
Q2 6
#12
jhðxs ; uÞj pðduÞds
:
2
U
t0
"
6
Z
e > 0, the Young inequality implies that
Further, for any
Q 2 6 p~cp
t
jxðsÞ Dðxs Þjp2 jhðxs ; uÞj2 pðduÞds:
ð70Þ
jxðsÞ Dðxs Þjp2 jhðxs ; uÞj2 pðduÞds:
ð71Þ
U
Z
t0
U
By the following inequality (see Mao[2]),
2
ap2 b 6
e2 ðp 2Þ
p
ap þ
1
p2
2
p
b ;
a; b; e2 > 0;
pe2
and condition (60), we have
E
Z
t
t0
Letting
E
Z
jxðsÞ Dðxs Þjp2 jhðxs ; uÞj2 pðduÞds 6
U
e2 ¼ ð1þk1 Þ2 ,
Z
t
t0
Z
0
jxðsÞ Dðxs Þjp2 jhðxs ; uÞj2 pðduÞds 6
U
Z t Z
ðp 2Þe2
2
jxðsÞ Dðxs Þjp pðduÞds þ p2 E
E
p
U
t0
pe22
Z t Z
p
ðp 2Þe2 ð1 þ k0 Þ
E
jhðxs ; uÞjp pðduÞds 6
p
U
0
Z t
Z t Z
2
jjxs jjp pðduÞds þ p2 KE
jjxs jjp ds
2
U
t0
0
pe2
Z t
ðp 2Þ
2K
p2
ð1 þ k0 Þ E
pðUÞ þ
jjxs jjp ds:
p
p
t0
ð72Þ
ð73Þ
Inserting (73) into (71), we obtain
Q 2 6 c10 E
Z
t
t0
jjxs jjp ds þ
1
E sup jxðsÞ Dðxs Þjp ;
2 t0 6s6t
ð74Þ
p2
e p ðdt; duÞ þ pðduÞdt and
where c10 ¼ 12 p~c2p ½ðp 2ÞpðUÞ þ 2Kð1 þ k0 Þ . Finally, let us estimate Q 3 . Since N p ðdt; duÞ ¼ N
e
N p ðdt; duÞ is a martingale measure, we get
263
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
Q3 ¼ E
Z
Z
t
t0
fjxðsÞ Dðxs Þ þ hðxs ; uÞjp jxðsÞ Dðxs Þjp pjxðsÞ Dðxs Þjp2 ½xðsÞ Dðxs > hðxs ; uÞgpðduÞds:
ð75Þ
U
We note that it has the form
E
Z
t
Z
0
ff ðxðsÞ Dðxs Þ þ hðxs ; uÞÞ f ðxðsÞ Dðxs ÞÞ f ðxðsÞ Dðxs ÞÞhðxs ; uÞgpðduÞds;
ð76Þ
U
t0
where f ðxÞ ¼ jxjp . Using the Taylor formula, there exists a positive constant M p , such that for p P 2
0
f ðxðsÞ Dðxs Þ þ hðxs ; uÞÞ f ðxðsÞ Dðxs ÞÞ f ðxðsÞ Dðxs ÞÞhðxs ; uÞ
p
p
¼ jxðsÞ Dðxs Þ þ hðxs ; uÞj jxðsÞ Dðxs Þj pjxðsÞ Dðxs Þjp2 ½xðsÞ Dðxs Þ> hðxs ; uÞ
6 Mp ½jxðsÞ Dðxs Þ þ hðxs ; uÞjp2 jhðxs ; uÞj2 :
ð77Þ
Again the basic inequality ja þ bjp2 6 2p3 ðjajp2 þ jbjp2 Þ and the Young inequality implies that
Z
Q 3 6 Mp E
Z
t
t0
½jxðsÞ Dðxs Þ þ hðxs ; uÞjp2 jhðxs ; uÞj2 pðduÞds
U
6 Mp 2p3 E
Z
Z
t
6 Mp 2p3
½ðjxðsÞ Dðxs Þjp2 þ jhðxs ; uÞjp2 Þjhðxs ; uÞj2 pðduÞds
U
t0
p2
p2
ð1 þ k0 Þ pðUÞE
p
Z
t
jjxs jjp ds þ M p 2p3 1 þ
t0
2ð1 þ k0 Þ
p
p2
! Z
E
t
t0
Z
jhðxs ; uÞjp pðduÞds:
ð78Þ
U
By (60), we have
E
Z
t
Z
t0
jhðxs ; uÞjp pðduÞds 6 KE
Z
t
jjxs jjp ds:
ð79Þ
t0
U
Substituting (79) into (78),
Q 3 6 c11
Z
t
Ejjxs jjp ds;
t0
where c11 ¼ Mp 2p3
ð80Þ
h
i
p2
p2
p2
K . Combing (64), (68), (74) and (80) together, we obtain that
ð1 þ k0 Þ pðUÞ þ 1 þ 2ð1þkp0 Þ
p
p
E sup ð1 þ jxðsÞ Dðxs Þjp Þ 6 2 þ 2ð1 þ k0 Þ Ejjnjjp þ 2ðc9 þ c10 þ c11 Þ
t 0 6s6t
Z
t
Eð1 þ jjxs jjp Þds:
ð81Þ
t0
On the other hand, by Lemma 3.1, we have
k0
1
p
Ejjnjjp þ
p E sup ð1 þ jxðsÞ Dðxs Þj Þ
1 k0
ð1 k0 Þ
t 0 6s6t
Z
2ðc9 þ c10 þ c11 Þ t
Eð1 þ jjxs jjp Þds;
6 c12 Ejjnjjp þ
p
ð1 k0 Þ
t0
E sup jxðsÞjp 6
t 0 6s6t
ð82Þ
p
k0
0Þ
where c12 ¼ 1k
þ 2þ2ð1þk
. Consequently,
ð1k Þp
0
0
2ðc9 þ c10 þ c11 Þ
Eð1 þ sup jxðsÞj Þ 6 1 þ ð1 þ c12 ÞEjjnjj þ
p
ð1 k0 Þ
t 0 s6s6t
p
p
Z
!
t
t0
p
E 1 þ sup jxðrÞj
ds:
ð83Þ
t 0 s6r6s
Therefore, we apply the Gronwall inequality to get
!
p
E 1 þ sup jxðsÞj
t0 s6s6t
6 ½1 þ ð1 þ c12 ÞEjjnjjp eMðtt0 Þ ;
þc10 þc11 Þ
where M ¼ 2ðc9ð1k
. This completes the proof.
p
0Þ
The next result shows that exponential estimations implies almost surely asymptotic estimations, and we give an upper
bound for the sample Lyapunov exponent. h
Theorem 4.2. Under the conditions (H1)–(H2), we have
lim sup
t!1
2
1
L½ð1 þ k0 Þ þ 3 þ 2pðUÞ þ 2~c22 log jxðtÞj 6
;
2
t
ð1 k0 Þ
a:s:
That is, the sample Lyapunov exponent of the solution should not be greater than
ð84Þ
L½ð1þk20 Þþ3þ2
pðUÞþ2~c22 ð1k0 Þ2
.
264
W. Mao et al. / Applied Mathematics and Computation 254 (2015) 252–265
Proof. For each n ¼ 1; 2; . . ., it follows from Theorem 4.1 (taking p ¼ 2) that
!
E
2
sup
jxðtÞj
6 becn ;
t 0 þn16t6t 0 þn
k0
where b ¼ 1k
Ejjnjj2 þ
0
lows that
c22 LðTt 0 Þ
2½3þ2pðUÞþ2~
ð1k0 Þ2
(
and c ¼
2L½ð1þk20 Þþ3þ2pðUÞþ2~c22 ð1k0 Þ2
. Hence, for any
e > 0, by the Chebysher inequality, it fol-
)
P x:
sup
jxðtÞj2 > eðcþÞn
6 ben :
t 0 þn16t6t 0 þn
en
Since R1
< 1, by the Borel–Cantelli lemma, we deduce that, there exists a integer n0 such that
n¼0 be
jxðtÞj2 6 eðcþeÞn a:s: n P n0 :
sup
t0 þn16t6t 0 þn
Thus, for almost all x 2 X, if t0 þ n 1 6 t 6 t0 þ n and n P n0 , then
1
1
ðc þ eÞn
log jxðtÞj ¼
logðjxðtÞj2 Þ 6
:
t
2t
2ðt 0 þ n 1Þ
ð85Þ
Taking lim sup in (85) leads to almost surely exponential estimate, that is,
2
lim sup
t!1
1
c þ e L½ð1 þ k0 Þ þ 3 þ 2pðUÞ þ 2~c22 log jxðtÞj 6
¼
;
2
t
2
ð1 k0 Þ
Required assertion (84) follows because
a:s:
e > 0 is arbitrary. h
5. Conclusion
In this paper, we prove the existence and uniqueness of the solution to NSFDEs with pure jumps under the local Lipschitz
condition. Meanwhile, by using the It b
o formula, Taylor formula and the Burkholder–Davis inequality, we establish the p-th
exponential estimations and almost surely asymptotic estimations of the solution to NSFDEs with pure jumps.
Acknowledgements
We thank the referees for careful reading of our paper and for helpful and valuable comments and suggestions. The
research of the first author is supported by the National Natural Science Foundation of China (11401261), Qing Lan Project
of Jiangsu Province (2012), NSF of Higher Education Institutions of Jiangsu Province (13KJB110005), the grant of Jiangsu Second Normal University (JSNU-ZY-02) and the Jiangsu Government Overseas Study Scholarship. The second author is supported by the National Natural Science Foundation of China (61374080) and the Priority Academic Program Development
of Jiangsu Higher Education Institutions. The third author is supported by the Royal Society of Edinburgh (RKES115071)
and the State Administration of Foreign Experts Affairs of China (MS2014DHDX020).
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