Iran. J. Sci. Technol. Trans. Sci. DOI 10.1007/s40995-016-0067-y

RESEARCH PAPER

A Note on pth Moment Estimates for Stochastic Functional Differential Equations in the Framework of G-Brownian Motion Faiz Faizullah1

Received: 4 February 2016 / Accepted: 25 July 2016 Ó Shiraz University 2016

Abstract This paper presents the study of path-wise and moment estimates for the solutions to stochastic functional differential equations in the framework of G-Brownian motion. Under the linear growth condition, the pth moment estimates for the solutions to GSFDEs are investigated. The properties of G-expectations, Ho¨lder’s inequality, Bihari’s inequality, Gronwall’s inequality and Burkholder–Davis– Gundy (BDG) inequalities are used to develop the theory of pth moment estimates. In addition, the continuity of pth moment and path-wise asymptotic estimates for the solutions to GSFDEs are shown. Keywords G-Brownian motion  pth moment exponential estimates  Stochastic functional differential equations  Continuity of pth moment  Path-wise asymptotic estimates

1 Introduction Described by Peng (2006), nonlinear G-expectation is induced by the factors related to problems of stochastic volatility and risk measures in the field of finance (Peng 2006, 2010). Eventually Peng derived G-Brownian motion which is a comparatively new stochastic process; it is different from the conventional Brownian motion as it is not based on a specified peculiar probability space. In other words, G-Brownian motion sets itself for a fresh and rich structure which subsequently generalizes the conventional

& Faiz Faizullah [email protected] 1

Department of BS&H, College of E&ME, National University of Sciences and Technology (NUST), Islamabad, Pakistan

one. A few notable related stochastic calculus recognized by him were G-Itoˆ’s formula, G-Itoˆ’s integral, and G-quadratic variation process hB; Bi (Peng 2008). A comparatively new and appealing occurrence pertaining to the G-Brownian motion is the fact that its continuous quadratic variation process includes stationary and independent increments. Hence, it persistently meets the criteria to be called as a Brownian motion. Considering the enormous applicability of the theory, numerous authors, in a very short period of time, have undertaken publications on this rising phenomenon (Li and Peng 2011; Qian 2011; Ren and Hu 2011; Xua and Zhang 2009). Notwithstanding the fact that stochastic functional differential equations (SFDEs) have a pertinent role to play in mathematics, it is also highly cherished in analysis and formulation in economics, social sciences, physical sciences, control engineering, electrical engineering and mechanical engineering (Cho et al. 2012; Mao 1997; Xiaodi and Xilin 2010; Diop et al. 2012). The idea of G-framework-related stochastic differential equations were commenced by Peng (2006, 2008) and then by Gao (2009). Later on extended by Bai and Lin (2014) to integral Lipschitz coefficients and then by Faizullah with discontinuous coefficients (Faizullah 2012). Ren et al. 2013 introduced stochastic functional differential equations (SFDEs) in the G-framework. Afterward, SFDEs in the G-framework were studied by Faizullah with Cauchy–Maruyama approximation scheme (Faizullah 2014). The stated theory was generalized by Faizullah, Mukhtar and Rana to SFDEs in the G-frame with discontinuous drift coefficients (Faizullah et al. 2016). Recently, Faizullah established the existence-and-uniqueness of solutions to SFDEs in the G-framework with non-Lipschitz conditions (Faizullah 2016). The pth moment estimate is the most basic and useful technique of analyzing dynamic behavior of stochastic systems, for instant see (Kim 2014; Mao

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Iran. J. Sci. Technol. Trans. Sci.

1997; Zhang et al. 2016). It is also worth noting that the pth moment of solution for SFDEs driven by G-Brownian motion has not been fully investigated, which remains an interesting research topic. This article will fill the mentioned gap. Let 0  t  T\1 and s [ 0: Assume that B(t) is an m-dimensional G-Brownian motion defined on complete probability space ðX; F ; PÞ, that is, BðtÞ ¼ ðBt1 ðtÞ; Bt2 ðtÞ; . . .; Btm ðtÞÞ, with the filtration fF gt  0 satisfying the usual conditions. Let BCð½s; 0; Rd Þ denote a collection of continuous and bounded Rd -valued functions u defined on ½s; 0 with norm kuk ¼ sups  h  0 j uðhÞ j. Let j : ½0; T d d k : ½0; T  BCð½s; 0; Rd Þ ! BCð½s; 0; R Þ ! R , dm R and l : ½0; T  BCð½s; 0; Rd Þ ! Rdm be Borel measurable. Consider the following SFDE in the G-framework dZðtÞ ¼ jðt; Zt Þdt þ kðt; Zt ÞdhB; BiðtÞ þ lðt; Zt ÞdBðtÞ;

t 2 ½0; T;

ð1:1Þ having initial data Zt0 ¼ f, which satisfies that Zt0 ¼ f ¼ ffðhÞ : s\h  0g is F 0 -measurable, BCð½s; 0; Rd Þ-val  ued random variable such that f 2 MG2 ½s; 0; Rd : The initial condition fð0Þ 2 Rd is given, the coefficients j; k; l 2 MG2 ð½s; T; Rd Þ are given functions and fhB; BiðtÞgt  0 is the quadratic variation process of G-Brownian motion fBðtÞgt  0 . Under the linear growth and Lipschitz conditions, the G-SFDE (1.1) with the above given initial data has a unique solution Z(t) such that E½sup0  t  T jZðtÞj2 \1 (Faizullah 2014; Ren et al. 2013). Moreover, if Z0 2 Lp ðX; Rd Þ then under the monotone condition, E½sup0  t  T jZðtÞjp \1 for p  2. Also, note that the assumptions of existence and uniqueness theorem hold on every finite subinterval [0, T] of ½0; 1Þ (Faizullah 2014; Ren et al. 2013). Hence, Eq. (1.1) has a unique solution on the whole interval ½0; 1Þ. This type of solution is known as a global solution (Mao 1997). Throughout the paper, we assume that the linear growth and Lipschitz conditions hold and Eq. (1.1) has a unique solution Z(t) satisfying E½sup0  t  T jZðtÞjp \1 for p  2. The rest of the paper is organized as follows. Section 2 is devoted to some basic results and notions. In Sect. 3, pth moment estimates are studied. In Sect. 4, continuity of pth moment is shown. In Sect. 5, path-wise asymptotic estimates for SFDEs driven by G-Brownian motion are given.

2 Preliminaries This section presents some basic definitions and results, which are used in the forthcoming sections of this paper. For more detailed literature on G-expectation, see the book

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by Peng (2010) and papers (Denis et al. 2010; Faizullah 2012; Song 2011). Definition 2.1 Let H be a linear space of real valued functions defined on a nonempty basic space X: Then a sublinear expectation E is a real valued functional on H with the following features: (a) (b) (c) (d)

For For For For

all Y; Z 2 H, if Y  Z then E½Y  E½Z. any real constant c, E½c ¼ c. any h [ 0; E½hZ ¼ hE½Z. every Y; Z 2 H, E½Y þ Z  E½Y þ E½Z.

The triplet ðX; H; EÞ is known as a sublinear expectation space. If only properties (a) and (b) hold then E is said to be a nonlinear expectation. Let X ¼ C0 ð½0; 1ÞÞ, that is, the space of all R-valued continuous paths ðwt Þt2½0;1Þ with w0 ¼ 0 equipped with the norm   1 X 1 1 2 1 2 qðw ; w Þ ¼ max jw  wt j ^ 1 ; 2k t2½0;k t k¼1 and assume Cb:Lip ðRn Þ denotes the set of bounded Lipschitz functions on Rn . Consider the canonical process Bt ðwÞ ¼ wt for t 2 ½0; 1Þ,w 2 X. Then for each fixed T 2 ½0; 1Þ we have Lip ðXT Þ ¼ fuðBt1 ; Bt2 ; . . .; Btn Þ : t1 ; . . .; tn 2 ½0; T; u 2 Cb:Lip ðRn Þ; n 2 Ng; where Lip ðXt Þ  Lip ðXT Þ Lip ðXÞ ¼ [1 m¼1 Lip ðXm Þ:

for

tT

and

Definition 2.2 Let Z and Z be two n-dimensional random variables defined on the spaces ðX; H; EÞ and ðX ; H ; E Þ respectively. They are said to be distributed identically if E½uðZÞ ¼ E ½uðZ Þ; for all u 2 Cb:Lip ðRn Þ: If Z is identically distributed to Z then we write Z Z . Definition 2.3 Let ðX; H; EÞ be a sublinear expectation space, Z 2 H, r2 ¼ E½Z 2  and r2 ¼ E½Z 2 : The random variable Z is said to be G-distributed or Nð0; ½r2 ; r2 Þ-distributed if for every Z 2 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aZ þ bZ a2 þ b2 Z; where Z is independent from Z and Z Z . G-expectation and G-Brownian motion: consider a sequence fni g1 i¼1 of random variables on a sublinear ^ H ^ p ; EÞ ^ such that niþ1 is independent expectation space ðX; of ðn1 ; n2 ; . . .; ni Þ for each i ¼ 1; 2; . . . and ni is G-normally distributed for each i 2 f1; 2; . . .g. Then a sublinear expectation E½: defined on Lip ðXÞ is introduced as follows.

Iran. J. Sci. Technol. Trans. Sci.

For 0 ¼ t0 \t1 \. . .\tn \1 ðt0 ; t1 ; . . .; tn 2 ½t; TÞ, u 2 Cl:Lip ðRn Þ and each

denoted by MGp ð0; TÞ and for 1  p  q; MGp ð0; TÞ  MGq ð0; TÞ:

Z ¼ uðBt1  Bt0 ; Bt2  Bt1 ; . . .; Btn  Btn1 Þ 2 Lip ðXÞ;

Definition 2.5 For each gt 2 MG2;0 ð0; TÞ; the Itoˆ’s integral of G-Brownian motion is defined as

E½uðBt1  Bt0 ; Bt2  Bt1 ; . . .; Btn  Btn1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ E½uð t1  t0 n1 ; . . .; tn  tn1 nn Þ:

IðgÞ ¼

For Z 2 Lip ðXt Þ, the conditional sublinear expectation is defined by E½ZjXt  ¼ E½uðBt1 ; Bt2  Bt1 ; . . .; Btm  Btm1 ÞjXt 

ZT

gu dBu ¼

0

Definition 2.6 An increasing continuous process fhBit : t  0g with hBi0 ¼ 0; defined by hBit ¼

B2t

2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ wðz1 ; . . .; zj Þ ¼ E½uðz tjþ1  tj njþ1 ; . . .; tn  tn1 nn Þ: 1 ; . . .; zj ;

The sublinear expectation E : Lip ðXÞ ! R, defined above is called a G-expectation. The corresponding canonical process fBt ; t  0g, defined as follows is called a G-Brownian motion. Definition 2.4 A d-dimensional process ðBt Þt  0 defined on a sublinear expectation space ðX; H; EÞ is a G-Brownian motion if the following properties hold. (i) (ii)

B0 ðxÞ ¼ 0: For every t; s  0, the incrementBtþs  Bt is Nð0; ½r2 ; r2 Þ-distributed and is independent from ðBt1 ; Bt2 ; . . .; Btn Þ, for every n 2 N and 0  t1      tn  t:

The completion of Lip ðXÞ under the norm kXkp ¼ ðE½jXjp Þ1=p for p  1 is denoted by LpG ðXÞ and LpG ðXt Þ  LpG ðXT Þ  LpG ðXÞ for 0  t  T\1: The filtration generated by the canonical process ðBt Þt  0 is denoted by F t ¼ rfBs ; 0  s  tg, F ¼ fF t gt  0 . Itoˆ’s integral of G-Brownian motion: for any T 2 ½0; 1Þ; a finite ordered subset pT ¼ ft0 ; t1 ; . . .; tN g such that 0 ¼ t0 \t1 \. . .\tN ¼ T is a partition of [0, T] and

ni ðBtiþ1  Bti Þ:

i¼0

¼ wðBt1 ; Bt2  Bt1 ; . . .; Btj  Btj1 Þ; where

N1 X

Zt Bu dBu ; 0

is called the quadratic variation process of G-Brownian motion. We now state three important inequalities known as Ho¨lder’s inequality, Bihari’s inequality and Gronwall’s inequality, respectively (Mao 1997). If 1q þ 1r ¼ 1 for any q; r [ 1; g 2 L2 and h 2

Lemma 2.7

L2 then gh 2 L1 and Zd

gh 

 Zd

c

jgj

q

1q  Zd

c

jhjr

1r :

c

Lemma 2.8 Let C  0; hðtÞ  0 and w(t) be a real valued continuous function on [c, d]. If for all c  t  d, wðtÞ  Rd C þ c hðsÞwðsÞds, then Rt hðsÞds wðtÞ  Ce c ; for all c  t  d. The following two lemmas are borrowed from the book by Mao (1997). Let c; d  0 and  2 ð0; 1Þ: Then

Lemma 2.9 2

c d2 þ :  1

lðpT Þ ¼ maxfjtiþ1  ti j : i ¼ 0; 1; . . .; N  1g:

ðc þ dÞ2 

A sequence of partitions of [0, T] is denoted by pNT ¼ ft0N ; t1N ; . . .; tNN g such that limN!1 lðpNT Þ ¼ 0: Consider the following simple process: let p  1 be fixed. For a given partition pT ¼ ft0 ; t1 ; . . .; tN g of [0, T],

Lemma 2.10 Let p  2 and ^; c; d [ 0: Then we have the following two inequalities

gt ðwÞ ¼

N 1 X

(i) (ii)

ni ðwÞI½ti ;tiþ1  ðtÞ;

ð2:1Þ

i¼0

LpG ðXti Þ,

i ¼ 0; 1; . . .; N  1. The collection where ni 2 containing the above type of processes, that is, containing gt ðwÞ is denoted by MGp;0 ð0; TÞ: The completion of RT MGp;0 ð0; TÞ under the norm kgk ¼ f 0 E½jgu jp dug1=p is

p ðp1Þ^  cp þ p^dp1 : p p cp cp2 d2  ðp2Þ^ þ 2dp2 : p 2 p^ 

cp1 d 

The following lemma can be found in Denis et al. (2010). Theorem 2.11

Suppose that Z 2 L2 . Then for each  [ 0;

E½jZj2  ^ CðjZj [ Þ  : 2

123

Iran. J. Sci. Technol. Trans. Sci.

In the above theorem 2.11, C^ is known as capacity defined ^ by CðHÞ ¼ supP2P PðHÞ, where P is the collection of all probability measures on ðX; BðXÞ and H 2 BðXÞ, which is ^ Borel r-algebra of X. Also, we remind CðHÞ ¼ 0 means that the set H is polar and a property holds quasi-surely (q.s . in short) means that it holds outside a polar set.

where 

Zt

I1 ¼ E

sup 

Zt

I2 ¼ E

sup

I3 ¼ E

ð3:1Þ

K [ 0:

Theorem 3.1 Let Eq. (1.1) satisfy the linear growth condition (3.1). Let Ekfkp \1 and p  2. Then

sup

jZðtÞjp1 jjðt; Zt Þj 

p

ðp  1Þ^ kZðtÞkp ½Kð1 þ kZt k2 Þ2  þ p p^ p1 p

ðp  1Þ^ kZðtÞkp ½Kð1 þ kfk2 þ kZðtÞk2 Þ2 þ p p^ p1 p ðp  1Þ^ kZðtÞk  p 

p

½Vv ðv; ZðvÞÞ þ VZ ðv; ZðvÞÞ

VZ ðv; ZðvÞÞlðZv ; vÞdBðvÞ

 1 þ VZ ðv; ZðvÞÞkðZv ; vÞ þ tracelT ðZv ; vÞ 2 0  VZZ ðv; ZðvÞÞlðZv ; vÞ dhB; BiðvÞ; which yields, 

   Zt sup jZðtÞjp  Ejfð0Þjp þ pE sup jZðvÞjp1 jjðv; Zv Þjdv 0vt

0vt

 þE

Zt sup

pjZðvÞjp1 jlðv; Zv ÞjdBðvÞ



p

þE

sup 0vt

þ

Setting ^ ¼ jZðtÞj

p1

pffiffiffiffiffiffi 3K we get that

pffiffiffiffiffiffi pffiffiffiffiffiffi 3K ð1 þ kfkp Þ þ 3K kZðtÞkp : jjðt; Zt Þj  p

pI1 ¼ pE

Zt sup 0vt

½pjZðvÞjp1 jkðv; Zv Þj

0

 pðp  1Þ jZðvÞjp2 jlðv; Zv Þj2 dhB; BiðvÞ 2

ð3:2Þ

ð3:3Þ

On similar arguments as above we obtain pffiffiffiffiffiffi pffiffiffiffiffiffi 3K ð1 þ kfkp Þ þ 3K kZðtÞkp ; jZðtÞjp1 jkðt; Zt Þj  p pffiffiffiffiffiffi pffiffiffiffiffiffi 3K ð1 þ kfkp Þ þ 3K kZðtÞkp ; jZðtÞjp1 jlðt; Zt Þj  p 6K ð1 þ kfkp Þ þ 3KkZðtÞkp : jZðtÞjp2 jlðt; Zt Þj2  ð3:4Þ p 

¼ Ejfð0Þjp þ pI1 þ pI2 þ pI3 ;

123

p

By the inequality (3.3) we obtain

0

Zt

p

0

0vt



p

ð3KÞ2 þ ð3KÞ2 kfkp þ ð3KÞ2 kZðtÞkp p^ p1

ð3KÞ2 ð1 þ kfkp Þ ðp  1Þ^  ð3KÞ2 ¼ þ½ þ p1 kZðtÞkp p^  p^ p1 p

0

Zt

E

ðp  1Þ^ jZðtÞjp jjðt; Zt Þjp þ p p^ p1

þ

Zt

ðp  1Þ jZðvÞjp2 2

Next we use Lemma 2.10 and linear growth condition (3.1), for any ^ [ 0 to obtain

0

jðZv ; vÞdv þ

½jZðvÞjp1 jkðv; Zv Þj þ

 jlðv; Zv Þj2 dhB; BiðvÞ :

Proof Let p  2. Apply G-Ito’s formula to Vðt; ZðtÞÞ ¼ jZðtÞjp : Then we proceed as follows: Vðt; ZðtÞÞ ¼ Vð0; Zð0ÞÞ þ

jlðv; Zv ÞjdBðvÞ ;

0

E½sup0  s  T jXðsÞjp   L1 eL2 T ; pffiffiffiffiffiffi where L1 ¼ T½ 3K pc2 þ 3Kfpð1 þ c23 Þ  1gð1 þ Ekfkp Þ pffiffiffiffiffiffi and L2 ¼ ½2 3K pð1 þ c2 Þ þ 3Kpfðp  1Þc2 þ pc23 g; c2 and c3 are positive constants.

Zt



0

Zt 0vt

jjðt; uÞj2 þ jkðt; uÞj2 þ jlðt; uÞj2  Kð1 þ juj2 Þ;

jjðv; Zv Þjdv ;

0



Assume the Eq. (1.1) has a unique global solution Z(t). Let the following linear growth condition holds. For all u 2 BCð½s; 0; Rd Þ and t 2 ½0; T,

jZðvÞj

p1



0vt

0vt

3 Moment Estimates for G-SFDEs

jZðvÞj

p1

jZðvÞj

p1

 jjðv; Zv Þjdv

0

 Z t  pffiffiffiffiffiffi pffiffiffiffiffiffi 3K p p ð1 þ Ekfk Þ þ 3K EkZðtÞk dv p p 0

pffiffiffiffiffiffi Z pffiffiffiffiffiffi  3K ð1 þ Ekfkp ÞT þ 3K p EðkZðvÞkp Þdv: t

0

Iran. J. Sci. Technol. Trans. Sci.

Using the inequalities (3.4) and the Burkholder–Davis– Gundy (BDG) inequalities (Gao 2009), we have  pI2 ¼ E

sup j

Zt

½pjZðvÞj

p1

We now use the values of pI1 , pI2 and pI3 in (3.2) and proceed as follows:  E

jkðv; Zv Þj

t  pffiffiffiffiffiffi Z pffiffiffiffiffiffi sup jZðvÞjp  3K ð1 þ Ekfkp ÞT þ 3K p EðkZv kp Þdv 0vt

0vt 0

pffiffiffiffiffiffi þ c2 p 3K ð1 þ Ekfkp Þ þ 3Kðp  1Þ  ð1 þ Ekfkp Þ T   pffiffiffiffiffiffi 3Kpðp  1Þ þ c2 p 3K þ 2   Zt 1  EkZðtÞkp dv þ E sup jZðvÞjp 2 0vt

pðp  1Þ jZðvÞjp2 jlðv; Zv Þj2 dhB; BiðvÞ j 2 Z t  pffiffiffiffiffiffi pffiffiffiffiffiffi  c2 p 3K ð1 þ Ekfkp Þ þ p 3K EkZðtÞkp þ

0

 pðp  1Þ 3KEkZðtÞkp dv þ ðp  1Þ3Kð1 þ Ekfkp Þ þ 2 pffiffiffiffiffiffi p  c2 ½p 3K ð1 þ Ekfk ÞT þ 3Kðp  1Þð1 þ Ekfkp ÞT  t  pffiffiffiffiffiffi 3Kpðp  1Þ Z EkZðtÞkp dv þ c2 p 3K þ 2

0

þ pc23 3Kð1 þ Ekfkp ÞT Zt 3 2 2 þ Kp c3 EkZðtÞkp dv 2

0

0

Next we use the BDG inequalities (Gao 2009) and then mean value theorem (Mao 1997) as follows:  pI3 ¼ pE

sup j

Zt

jZðvÞjp1 jlðv; Zv ÞjdBðvÞj



simplification yields,     pffiffiffiffiffiffi p 2 E sup jZðvÞj  T 3K pc2 þ 3Kfpð1 þ c3 Þ  1g 0vt

 ð1 þ Ekfkp Þ  pffiffiffiffiffiffi þ 2 3K pð1 þ c2 Þ þ 3Kpfðp  1Þc2

0vt 0

 pc3 E

 Zt

jZðvÞj2p2 jlðv; Zv Þj2 dv

12

0



1

sup ½jZðvÞjp 2

 pc3 E

Zt

0





þ

12 jZðvÞjp2 jlðv; Zv Þj2 dv :

pc23 g

 Zt   p E sup jZðvÞj dv 0

0vt

0vt 0 2

2

By using the inequality jajjbj  jaj2 þ jbj2 and then the inequality (3.4), we obtain  1 p2 c23 pI3 ¼  E sup jZðvÞjp  þ 2 0vt 2   Zt p2 2 jZðvÞj jlðv; Zv Þj dv E 1  E 2

0



sup jZðvÞjp  þ 0vt

E 1  E 2



 Zt 

p2 c23 2

  6K ð1 þ kfkp Þ þ 3KkZðtÞkp dv p

Finally, by Gronwall’s inequality   E sup jZðvÞjp  L1 eL2 t ; 0vt

pffiffiffiffiffiffi where L1 ¼ T½ 3K pc2 þ 3Kfpð1 þ c23 Þ  1gð1 þ Ekfkp Þ pffiffiffiffiffiffi and L2 ¼ ½2 3K pð1 þ c2 Þ þ 3Kpfðp  1Þc2 þ pc23 g: By taking t ¼ T; we have   E sup jZðvÞjp  L1 eL2 T : 0vT

The proof is complete.

h

The statement of Theorem 3.1 now follows from the following result.

0

sup jZðvÞjp  þ pc23 3Kð1 þ Ekfkp ÞT

0vt

3 þ Kp2 c23 2

Z 0

t

EkZðtÞkp dv

Remark 3.2 Let Eq. (1.1) satisfy the linear growth condition (3.1). Let Ekfkp \1 and p  2. Then   E sup jXðsÞjp  C eL2 T ; s  s  T

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Iran. J. Sci. Technol. Trans. Sci.

pffiffiffiffiffiffi where C ¼ Ekfkp þ L1 , L1 ¼ T½ 3K pc2 þ 3Kfpð1 þ pffiffiffiffiffiffi and L2 ¼ ½2 3K pð1 þ c2 Þ þ c23 Þ  1gð1 þ Ekfkp Þ 3Kpfðp  1Þc2 þ pc23 g; c2 and c3 are positive constants. Proof Since 0  Ekfkp \1, eL2 T  1 and E½sups  s  T jXðsÞjp   Ekfkp þ E½sup0  s  T jXðsÞjp . Therefore by Theorem 3.1     p p p E sup jXðsÞj  Ekfk þ E sup jXðsÞj s  s  T p L2 T

 Ekfk e

¼ ½Ekfkp þ

0sT þ L1 eL2 T L1 eL2 T

Zt

p

ð1 þ kZq k2 Þ2 dq þ 3p1 K2 Kðt  sÞp1

s

Zt

p

ð1 þ kZq k2 Þ2 dq

s

þ 3p1 K3 Kðt  sÞp1

h

4 Continuity of pth Moment

p

ð1 þ kZq k2 Þ2 dq

s

 3 Kð1 þ K2 þ K3 Þðt  sÞp1 Zt p 221 ð1 þ EkZq kp Þdq 3

where C ¼ Ekfkp þ L1 . The proof is complete.

t

p1

s 2p2

¼ C e L2 T ;

Z

Kð1 þ K2 þ K3 Þðt  sÞp

þ 32p2 Kð1 þ K2 þ K3 Þðt  sÞp1 Zt  EkZq kp dq s

In the next theorem, we show that the pth moment of the solution to G-SFDE (1.1) is continuous. Theorem 4.1 Let Eq. (1.1) satisfy the linear growth condition (3.1). Let Ekfkp \1 and p  2. Then

 32p2 Kð1 þ K2 þ K3 Þðt  sÞp þ 32p2 Kð1 þ K2 þ K3 Þðt  sÞp1 Zt  ½Ekfk þ E½ sup jZðrÞjp dq

E½jXðtÞ  XðsÞjp   aðtÞðt  sÞp ; where aðtÞ ¼ 32p2 Kð1 þ K2 þ K3 Þ½1 þ Ekfk þ L1 eL2 T : Proof Using the inequality ða þ b þ cÞp  3p1 ðap þ bp þ cp Þ, Eq. (1.1) yields jZðtÞ  ZðsÞjp ¼ 3p1 j

Zt

jðq; Zq Þdqjp þ 3p1 j

3

s 2p2

0srq

Kð1 þ K2 þ K3 Þðt  sÞp

þ 32p2 Kð1 þ K2 þ K3 Þðt  sÞp Ekfk þ 32p2 Kð1 þ K2 þ K3 Þðt  sÞp1 p Zt   E sup jZðrÞj dq s

0srq

s

Zt

kðq; Zq ÞdhB; BiðqÞjp þ 3p1 j

s

Zt

By using the above Theorem 3.1 we get lðq; Zq ÞdBðqÞjp :

s

þ 32p2 Kð1 þ K2 þ K3 Þðt  sÞp Ekfk

Taking G-expectation on both sides, using Holder’s inequality, BDG inequalities and linear growth condition, it follows EjZðtÞ  ZðsÞjp  3p1 ðt  sÞp1 E Z t jjðq; Zq Þjp dq þ 3p1 K2 ðt  sÞp1 s

Zt

K3 ðt  sÞ

s 2p2

3

Kð1 þ K2 þ K3 Þðt  sÞp

þ 32p2 Kð1 þ K2 þ K3 Þðt  sÞp Ekfk ¼32p2 Kð1 þ K2 þ K3 Þ½1 þ Ekfk

s

þ3

þ 32p2 Kð1 þ K2 þ K3 Þðt  sÞp1 Zt L1 eL2 T dq

þ 32p2 Kð1 þ K2 þ K3 Þðt  sÞp L1 eL2 T

jkðq; Zq Þjp dq

p1

EjZðtÞ  ZðsÞjp  32p2 Kð1 þ K2 þ K3 Þðt  sÞp

p1

Zt

p

jlðq; Zq Þj dq

þ L1 eL2 T ðt  sÞp ¼ aðtÞðt  sÞp ;

s

 3p1 ðt  sÞp1 KE

123

where aðtÞ ¼ 32p2 Kð1 þ K2 þ K3 Þ½1 þ Ekfk þ L1 eL2 T : The proof is complete. h

Iran. J. Sci. Technol. Trans. Sci.

5 Path-Wise Asymptotic Estimate

6 Conclusion

We now use Theorem 3.1 to establish path-wise asymptotic estimate for the solution of GSFDE (1.1). Also, revise that limt!1 sup 1t logjZðtÞj is called the Lyapunov exponent (Kim 2014). Next, we show that the pth moment of Lyapunov exponent should have an upper bound of pffiffiffiffiffiffi 2 3K pð1 þ c2 Þ þ 3Kpfðp  1Þc2 þ pc23 g:

Usually, it is very difficult to find explicit solutions to the SDEs that are nonlinear. Ultimately there is a requirement of the detailed analysis to come up with the solutions to these equations where as moment estimates and existence are the most significant characteristics for resolutions of SDEs. For the purpose of investigating the pth moment estimates for SFDEs driven by G-Brownian motion, some important inequalities are utilized for the current study. These inequalities include Ho¨lder’s inequality, Bihari’s inequality, Gronwall’s inequality and Burkholder–Davis– Gundy (BDG) inequality. Ultimately the asymptotic estimates are developed and the continuity of pth moment for these equations is shown. The G-Brownian motion theory in other words may be regarded as the generalization of the classical Brownian motion theory. For the estimation of the pth moment for SDE, the methodology used is appealing and may be used in numerous applications that are of practical nature. For instance, pth moment estimates are applicable in distributed system control (Shang 2012, 2013b) and biological population models (Shang 2013a). The systems and methods of the pth moment estimation, established in our paper, can be utilized to extend the pertinent theory in above-mentioned papers.

Theorem 5.1 Then

Let the linear growth condition (3.1) holds.

pffiffiffiffiffiffi 1 lim sup logjZðtÞj  2 3K pð1 þ c2 Þ þ 3Kpfðp  1Þc2 þ pc23 g q:s: t

t!1

Proof For each k ¼ 1; 2; . . .; Theorem 3.1 follows   E sup jZðtÞjp  aebk ; k1  t  k

pffiffiffiffiffiffi where a ¼ T½ 3K pc2 þ 3Kfpð1 þ c23 Þ  1gð1 þ Ekfkp Þ pffiffiffiffiffiffi and b ¼ ½2 3K pð1 þ c2 Þ þ 3Kpfðp  1Þc2 þ pc23 g: Thus by Theorem 2.11 for any arbitrary  [ 0;    E supk1  t  k jZðtÞjp  C^ w : sup jZðtÞjp [ eðbþÞk  eðbþÞk Þ k1  t  k 

aebk eðbþÞk Þ

¼ ae

k

Acknowledgments The author acknowledges and deeply appreciates the careful reading and useful suggestions of the antonymous reviewer, which has improved the quality of this paper.

:

The Borel–Cantelli lemma follows for almost all w 2 X, there exists a random integer k0 ¼ k0 ðwÞ such that sup

jZðtÞjp  eðbþÞk

whenever

k  k0 ;

k1  t  k

consequently, we have 1 bþ lim sup logjZðtÞj  t!1 t p pffiffiffiffiffiffi ¼ ½2 3K ð1 þ c2 Þ þ 3Kfðp  1Þc2  þ pc23 g þ ; q:s: p But  is arbitrary, so we have pffiffiffiffiffiffi 1 lim sup logjZðtÞj  2 3K ð1 þ c2 Þ þ 3Kfðp  1Þc2 þ pc23 g; t

t!1

The proof is complete.

q:s:

h

In the above theorem if p ¼ 2; then pffiffiffiffiffiffi 1 lim sup logjZðtÞj  2 3K ð1 þ c2 Þ þ 3Kðc2 þ 2c23 Þ; t!1 t

Remark 5.2

which show that the Lyapunov coefficient should not be pffiffiffiffiffiffi greater than 2 3K ð1 þ c2 Þ þ 3Kfc2 þ 2c23 g:

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