On the pth moment estimates of solutions to stochastic functional differential equations in the G framework

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1 Faizullah SringerPlu (16)5:87 DOI / x RESEARCH Oen Acce On the th moment etimate of olution to tochatic functional differential equation in the G framework Faiz Faizullah * *Correondence: faiz_math@ceme.nut.edu.k Deartment of Baic Science and Humanitie, College of Electrical and Mechanical Engineering, National Univerity of Science and Technology (NUST), Ilamabad, Pakitan Abtract The aim of the current aer i to reent the ath-wie and moment etimate for olution to tochatic functional differential equation with non-linear growth condition in the framework of G-exectation and G-Brownian motion. Under the nonlinear growth condition, the th moment etimate for olution to SFDE driven by G-Brownian motion are roved. The roertie of G-exectation, Hölder inequality, Bihari inequality, Gronwall inequality and Burkholder Davi Gundy inequalitie are ued to develo the above mentioned theory. In addition, the ath-wie aymtotic etimate and continuity of th moment for the olution to SFDE in the G-framework, with non-linear growth condition are hown. Keyword: th Moment etimate, G-Brownian motion, Stochatic functional differential equation, Path-wie aymtotic etimate, Non-linear growth condition Background Stochatic dynamical ytem have a wide range of alication inide a well a outide the field of mathematic. The quantitative tudie of different field uch a hyic, engineering, ecological cience, ytem cience and medicine have been driven by tochatic dynamical ytem. Stochatic differential equation (SDE) are often ued to model financial quantitie uch a aet rice, interet rate and their derivative. Thee equation have become tandard model for oulation dynamic and biological ytem. Stochatic functional differential equation (SFDE) in the G-framework were initiated by Ren et al. (13). Then tudied by Faizullah (14), he develoed the exitence-and-uniquene theorem with Cauchy Maruyama aroximation cheme (Faizullah 14). Later, he roved the comarion reult, with the hel of which he etablihed the exitence theory for SFDE in the G-framework with dicontinuou drift coefficient (Faizullah et al. 16). G-exectation, which i a nonlinear exectation, defined by Peng (6), ha been motivated by tochatic volatility roblem and rik meaure in finance (Gao 9; Peng 8, 1). Thi led him to derive G-Brownian motion that i a novel tochatic roce. Being different from the claical Brownian motion a it i not baed on a given articular robability ace, G-Brownian motion qualifie itelf for a new and extremely rich tructure which nontrivially generalize the claical one. Some of the ertinent tochatic calculu 16 The Author(). Thi article i ditributed under the term of the Creative Common Attribution 4. International Licene (htt://creativecommon.org/licene/by/4./), which ermit unretricted ue, ditribution, and reroduction in any medium, rovided you give aroriate credit to the original author() and the ource, rovide a link to the Creative Common licene, and indicate if change were made.

2 Faizullah SringerPlu (16)5:87 Page of 11 which were etablihed by him included G-Itô integral, G-Itô formula and G-quadratic variation roce B. A new and intereting henomenon that i related to the G-Brownian motion i the fact that it quadratic variation roce, which i alo a continuou roce, ha got tationary and indeendent increment. Therefore, it continue to qualify for being termed a a Brownian motion. Thu, the idea of G-framework-related tochatic differential equation wa initiated (Peng 6, 8). Due to the alicability of the theory, many author ublihed their work on thi emerging henomenon in a hort an of time (Bai and Lin 14; Deni et al. 1; Xua and Zhang 9). A imortant a the exitence theory, moment etimate i one of the mot ueful and baic cheme of analyzing dynamic behavior of SFDE. It i alo worth noting that the th moment of the olution for uch SDE driven by G-Brownian motion with non-linear growth condition ha not been fully exlored, which remain an intereting reearch toic. Thi article will fill the mentioned ga. We reent the analyi for the olution to the following SFDE in the G-framework dy (t) = κ(t, Y t )dt λ(t, Y t )d B, B (t) µ(t, Y t )db(t), with initial data Y t = ζ atifying t, ), (1) Y t = ζ = {ζ(θ): τ<θ} i F -meaurable, BC( τ,; R n )-valued random variable uch that ζ MG( τ,; R n ). () It i undertood that Y(t) i the value of tochatic roce at time t and Y t ={Y (t θ): ρ θ, ρ>}, indicate BC( ρ,; R)-valued tochatic roce, which i a collection of continuou and bounded real valued function ϕ defined on ρ, having norm ϕ =u ρθ ϕ(θ). The coefficient κ, λ and µ are Borel meaurable real valued function on, T BC( ρ, (Faizullah et al. 16). The ret of the aer i organized a follow: Preliminarie ection i devoted to ome baic definition and reult. th Moment etimate for SFDE in the G-framework ection reent the th moment etimate for SFDE in the G-framework, under non-linear growth condition. Continuity of th moment for SFDE in the G-framework ection how that the th moment of olution to SFDE i continuou. The ath-wie aymtotic etimate are given in Path-wie aymtotic etimate ection. Preliminarie In thi ection ome fundamental notion and reult are given, which are ued in the forthcoming ection of thi aer. For more detailed literature of G-exectation, ee the aer Deni et al. (1), Faizullah (1), Li and Peng (11), Song (13) and book Peng (1). Definition 1 Let H be a linear ace of real valued function defined on a nonemty baic ace. Then a ub-linear exectation E i a real valued functional on H with the following feature: (a) (b) (c) (d) For all Y, Z H, if Y Z then EY EZ. For any real contant γ, Eγ =γ. For any θ >, EθZ =θez. For every Y, Z H, EY Z EY EZ.

3 Faizullah SringerPlu (16)5:87 Page 3 of 11 Let C b.li (R l d ) denote the et of bounded Lichitz function on R l d and { } L G ( T ) = φ(b t1, B t,..., B tl /l 1, t 1, t,..., t l, T, φ C b.li (R l d )). Let δ i L G ( t i ), i =, 1,..., N 1 then MG (, T) denote the collection of rocee of the following tye: η t (w) = where the above roce i defined on a artition π T ={t, t 1,..., t N } of, T. Aociated with norm η ={ T E η u du} 1/, M G (, T), 1, i the comletion of MG (, T). Definition Let (B t ) t be a d-dimenional tochatic roce defined on (, C l,li (H), E), uch that B =. The increment B tm B t i G-normally ditributed for any t, m, n N and t 1 t,, t n t, it i indeendent from B t1, B t,...b tn. Then (B t ) t i known a G-Brownian motion. For every η t M, G (, T), the G-Itô integral I(η) and G-quadratic variation rocee { B t } t are reectively given by I(η) = N 1 i= T δ i (w)i ti,t i1 (t), η u db u = N 1 i= B t = Bt B u db u. δ i (B ti1 B ti ), We now tate three imortant inequalitie known a Hölder inequality, Bihari inequality and Gronwall inequality reectively (Mao 1997). Lemma 3 If 1 q 1 r = 1 for any q, r > 1, g L and h L then gh L 1 and d c ( ) 1 ( d q ) 1 d r gh g q h r. c Lemma 4 Let C, h(t) and w(t) be a real valued continuou function on c, d. If for all c t d, w(t) C d c h()w()d, then w(t) Ce c h()d, c for all c t d. The following two lemma are borrowed from the book Mao (1997). Lemma 5 Let a, b and ǫ (, 1). Then (a b) a ǫ b 1 ǫ.

4 Faizullah SringerPlu (16)5:87 Page 4 of 11 Lemma 6 Aume and ˆǫ, a, b >. Then the following two inequalitie hold. (i) (ii) a 1 b ( 1)ˆǫa b ˆǫ 1. a b ( )ˆǫa b. ˆǫ Theorem 7 Let Y L. Then for each ǫ >, Ĉ( Y > ǫ) E Y. ǫ In the above Theorem 7, Ĉ i known a caacity defined by Ĉ(H) = u P P P(H), where P i a collection of all robability meaure on (, B( ) and H B( ), which i Borel σ-algebra of. Alo, we remind Ĉ(H) = mean that et H i olar and a roerty hold quai-urely (q.. in hort) mean that it hold outide a olar et. The ret of the aer i organized a follow. In Preliminarie ection, the th moment etimate are tudied. In th Moment etimate for SFDE in the G-framework ection, continuity of th moment i hown. In Continuity of th moment for SFDE in the G-framework ection, ath-wie aymtotic etimate for SFDE driven by G-Brownian motion are given. th Moment etimate for SFDE in the G framework Let Eq. (1) admit a unique olution Y(t). Aume that a non-linear growth condition hold, which i given a follow. For every ψ BC( τ,; R d ) and t, T, ( κ(t, ψ) λ(t, ψ) µ(t, ψ) ϒ 1 ψ ), (3) where ϒ( ): R R i a non-decreaing and concave function uch that ϒ() =, ϒ(x) > for x > and dx (4) ϒ(x) =. A ϒ i concave and ϒ() =, there exit two oitive contant α and β uch that ϒ(x) α βx, (5) for all x. Theorem 8 Aume that the non-linear growth condition (3) hold. Let E ζ < and. Then E u Y (v) E ζ α 3 e β3t, τvt where α 3 = Tα 1 (1 c 1 ) α (c 1 ( 1) c 3 ), β 3 =β 1 (1 c 1 ) β ( 1 c 3 ), α 1 = 1 () ˆǫ 1 1 (α β)) (β) E ζ and β 1 = ( 1)ˆǫ (β), α ˆǫ 1 = 1 () ˆǫ (α β) (β) E ζ, β =( 1)ˆǫ (β), c ˆǫ and c 3 are oitive contant.

5 Faizullah SringerPlu (16)5:87 Page 5 of 11 Proof Alying G-Itô formula to Y (t), for, we roceed a follow E u Y (t) E ζ() E vt E E u vt u vt u vt Y (v) 1 κ(v, Y v ) dv Y (v) 1 µ(v, Y v ) db(v) Y (v) 1 λ(v, Y v ) ( 1) Y (v) µ(v, Y v ) d B, B (v) = E ζ() I 1 I I 3, (6) where I 1 = E I = E I 3 = E u vt u vt u vt Y (v) 1 κ(v, Y v ) dv Y (v) 1 µ(v, Y v ) db(v) Y (v) 1 λ(v, Y v ),, ( 1) Y (v) µ(v, Y v ) d B, B (v). By non-linear growth condition (3) and Lemma 6, for any ˆǫ >, we get Y (t) 1 κ(t, Y t ) ( 1)ˆǫ Y (t) ( 1)ˆǫ Y (t) ( 1)ˆǫ Y (t) κ(t, Y t) ˆǫ 1 ϒ(1 Yt ) ˆǫ 1 α β(1 Yt ) ˆǫ 1. Uing the inequality (a b) 1 (a b ) and the fact u τvt Y (v) ζ u vt Y (v), we roceed a follow Y (t) 1 κ(t, Y t ) ( 1)ˆǫ Y (t) ( 1)ˆǫ Y (t) ( 1)ˆǫ Y (t) ( 1)ˆǫ Y (t) α β β Y t ˆǫ 1 = () (α β) (β) ζ ˆǫ 1 () 1 (α β) (β) Y t ˆǫ 1 () 1 (α β) (β) ζ (β) Y (t) ˆǫ 1 () (α β) (β) ζ (β) Y (t) ˆǫ 1 ( 1)ˆǫ (β) ˆǫ 1 Y (t),

6 Faizullah SringerPlu (16)5:87 Page 6 of 11 which yield E Y (t) 1 κ(t, Y t ) α 1 β 1 E Y (t), (7) where α 1 = () (αβ)) (β) E ζ and β ˆǫ 1 1 = ( 1)ˆǫ (β). In a imilar fahion a ˆǫ 1 above we get Y (t) 1 λ(t, Y t ) α 1 β 1 Y (t), Y (t) 1 µ(t, Y t ) α 1 β 1 Y (t). (8) Next by uing Lemma 6, non-linear growth condition (3), inequality (a b) 1 (a b ) and the fact u τvt Y (v) ζ u vt Y (v), we have Y (t) µ(t, Y t ) ( )ˆǫ Y (t) µ(t, Y t) ˆǫ which give ( )ˆǫ Y (t) ( )ˆǫ Y (t) ( )ˆǫ Y (t) ( 1)ˆǫ Y (t) ( )ˆǫ Y (t) ( )ˆǫ Y (t) ϒ(1 Y t ) ˆǫ = () (α β) (β) ζ ˆǫ α β(1 Y t ) ˆǫ α β β Y t ˆǫ () (α β) (β) Y t ˆǫ () (α β) (β) ζ (β) Y (t) ˆǫ () (α β) (β) ζ (β) Y (t) ( 1)ˆǫ ˆǫ (β) ˆǫ Y (t), E Y (t) µ(t, Y t ) α β E Y (t), where α = () (αβ)) (β) E ζ and β ˆǫ = a follow I 1 = E u vt Y (v) 1 κ(v, Y v ) dv α 1 β 1 E Y (t) dv α 1 T β 1 E( Y (v) )dv. ( 1)ˆǫ (β) ˆǫ (9). Then I 1 can be written

7 Faizullah SringerPlu (16)5:87 Page 7 of 11 By inequalitie (8) and the Burkholder Davi Gundy (BDG) inequalitie (Gao 9), I can be written a follow I = E c 1 u vt Y (v) 1 λ(v, Y v ) E Y (v) 1 λ(v, Y v ) α 1 β 1 E Y (t) Next we ue the BDG inequalitie (Gao 9), inequality (8), mean value theorem and the inequality a b a b a follow I 3 = E u Y (v) 1 µ(v, Y v ) db(v) Uing the value of I 1, I and I 3 in () we get ( 1) Y (v) µ(v, Y v ) d B, B (v) ( 1) E Y (v) µ(v, Y v ) dv ( 1) c 1 (α β E Y (t) ) dv c 1 (α 1 1 ( ( 1)α )T c 1 β 1 1 ) ( 1)β E Y (t) dv vt c 3 E Y (v) µ(v, Y v ) dv c 3 E u Y (v) 1 vt 1 E u Y (v) c 3 vt E 1 E u Y (v) c t 3 vt 1 c 3 α T 1 E 1 Y (v) µ(v, Y v ) dv 1 Y (v) µ(v, Y v ) dv α β E Y (t) dv u Y (v) 1 vt c 3 β E Y (t) dv E u Y (v) α 1 T β 1 E ( Y (v) ) dv vt c 1 (α 1 1 ( ( 1)α )T c 1 β 1 1 ) ( 1)β E Y (t) dv 1 c 3 α T 1 E u Y (v) 1 vt c 3 β E Y (t) dv = 1 E u Y (v) T α 1 (1 c 1 ) 1 α (c 1 ( 1) c 3 ) vt β 1 (1 c 1 ) 1 β ( 1 c 3 ) E( Y (v) )dv,

8 Faizullah SringerPlu (16)5:87 Page 8 of 11 imlification yield, E u Y (v) T α 1 (1 c 1 ) α (c 1 ( 1) c3 ) vt β 1 (1 c 1 ) β ( 1 c3 t ) E( Z(v) )dv. By the Gronwall inequality E u Y (v) α 3 e β3t, vt (1) where α 3 = Tα 1 (1 c 1 ) α (c 1 ( 1) c3 ) and β 3 =β 1 (1 c 1 ) β ( 1 c3 ). By taking t = T, we have E u Y (v) α 3 e β3t. (11) vt Noting the fact that u τvt Y (v) ζ u vt Y (v), we roceed a follow E u Y (v) E ζ E τvt E ζ α 3 e β 3T. u Y (v) vt The roof i comlete. Continuity of th moment for SFDE in the G framework In the next theorem, under non-linear growth condition, it i hown that the th moment of the olution to SFDE in the G-framework (1) i continuou. Theorem 9. Then Aume the non-linear growth condition (3) hold. Let E ζ < and E Y (t) Y () γ(t)(t ), where γ(t) = 3 3 (1 c c 3 )α β β E ζ β α 3 e β 3T, c, c 3, α, β, α 3 and β 3 are oitive contant. Proof By uing the inequality (a b c) 3 1 (a b c ), Eq. (1) follow t Y (t) Y () = 3 1 t κ(q, Y q )dq 3 1 λ(q, Y q )d B, B (q) t 3 1 µ(q, Y q )db(q). Alying G-exectation on both ide, uing the BDG inequalitie (Gao 9), Holder inequality and non-linear growth condition, we roceed a follow

9 Faizullah SringerPlu (16)5:87 Page 9 of 11 E Y (t) Y () 3 1 (t ) 1 E κ(q, Y q ) dq 3 1 c (t ) 1 λ(q, Y q ) dq 3 1 c 3 (t ) 1 µ(q, Y q ) dq 3 1 (t ) 1 E ϒ(1 Y q ) dq 3 1 c (t ) c 3 (t ) 1 ϒ(1 Y q ) dq = 3 1 (t ) 1 1 c c 3 E 3 1 (t ) 1 1 c c 3 E 3 1 (t ) 1 1 c c 3 E ϒ(1 Y q ) dq α β(1 Y q ) dq α β β Y q dq ϒ(1 Y q ) dq 3 1 (t ) 1 1 c c (α) (β) (β) E Yq dq t 3 1 (t ) 1 1 c c α β β E ζ β u E Y (r) dr rq 3 3 (t ) 1 c c 3 α β β E ζ 3 3 (t ) 1 1 c c 3 β u E Y (r) dr rq By uing the inequality (11), it follow E Y (t) Y () 3 3 (t ) 1 c c 3 α β β E ζ 3 3 (t ) 1 1 c c 3 β α 3 e β3t dr 3 3 (t ) 1 c c 3 α β β E ζ where γ(t) = 3 3 (1 c c 3 )α β β E ζ β α 3 e β3t. The roof i comlete. In the above theorem c, c 3, α, β, α 3 and β 3 are oitive contant. The value of α 3 and β 3 are given in Theorem 8. Path wie aymtotic etimate Next, by uing Theorem 8 we tudy the ath-wie aymtotic etimate for the olution of SFDE in the G-framework (1). It i undertood that lim t u 1 t log Y (t) i the Lyaunov exonent (Kim 14). It i hown that the th moment of Lyaunov exonent hould not be greater than 1 β 1(1 c 1 ) β ( 1 c3 ), where c 1, c 3, β 1, β are oitive contant and. 3 3 (t ) 1 c c 3 β α3 e β 3T = γ(t)(t ),

10 Faizullah SringerPlu (16)5:87 Page 1 of 11 Theorem 1 Aume that the non-linear growth condition (3) hold. Then lim u 1 t t log Y (t) 1 β 1 (1 c 1 ) β ( 1 c3 ) q.. Proof For each k = 1,,..., uing the non-linear growth condition in a imilar fahion a in Theorem 8, Eq. (1) we obtain, E ( ) u Y (t) α 3 e β3k, k 1tk where α 3 = Tα 1 (1 c 1 ) α (c 1 ( 1) c3 ) and β 3 =β 1 (1 c 1 ) β ( 1 c3 ). Recall that E i a ub-linear exectation. Unlike a claical exectation, it i not baed on a articular robability ace. So, intead of robability, we ue a different concet known a caacity. Thank to Theorem 7 for any arbitrary ǫ >, we have Ĉ ( w: u Y (t) > e (β 3ǫ)k k 1tk ) E u k 1tk Y (t) αeβ 3k e (β 3ǫ)k = αe ǫk. e (β 3ǫ)k The Borel Cantelli lemma follow for almot all w, there exit a random integer k = k (w) uch that u Y (t) e (β 3ǫ)k k 1tk whenever k k, conequently, we get lim u 1 t t log Y (t) β 3 ǫ = 1 β 1 (1 c 1 ) β ( 1 c3 ) ǫ, q.. But ǫ i arbitrary, o lim u 1 t t log Y (t) 1 β 1 (1 c 1 ) β ( 1 c3 ), q.. The roof i comlete. Remark 11 In the above theorem if =, then lim u 1 t t log Y (t) β 1(1 c 1 ) 1 ( ) β 1 c3, Hence β 1 (1 c 1 ) 1 β (1 c3 ) i the uer bound for econd moment of Lyaunov exonent.

11 Faizullah SringerPlu (16)5:87 Page 11 of 11 Concluion Generally, we cannot find exlicit olution to nonlinear SDE. Thu one need to reent the analyi for olution to thee equation. Exitence and moment etimate are the mot imortant characteritic for olution to SDE. Here, we have ued ome imortant inequalitie uch a Bihari inequality, Hölder inequality, Gronwall inequality and Burkholder Davi Gundy (BDG) inequalitie to invetigate the th moment etimate for SFDE driven by G-Brownian motion. Then the aymtotic etimate for thee equation have been develoed. Furthermore, continuity of th moment for the olution to SFDE in the G-framework ha been roved. The G-Brownian motion theory i the generalization of the claical Brownian motion theory. The methodology ued to etimate th moment for SDE i intereting and alicable in variou ractical alication. For examle, th moment etimate are ueful in biological oulation model (Shang 13a) and ditributed ytem control (Shang 1, 13b, 15). The method of the th moment etimation, develoed in our aer, can be ued to extend the related theory in above mentioned aer. Acknowledgement The author acknowledge and areciate the financial uort of NUST reearch directorate for thi reearch work. We are very grateful to the anonymou reviewer for hi/her ueful uggetion, which have imroved the quality of thi aer. Cometing interet The author declare that he ha no cometing interet. Received: 6 May 16 Acceted: 3 May 16 Reference Bai X, Lin Y (14) On the exitence and uniquene of olution to tochatic differential equation driven by G-Brownian motion with integral-lichitz coefficient. Acta Math Al Sin 3(3): Deni L, Hu M, Peng S (1) Function ace and caacity related to a ublinear exectation: alication to G-Brownian motion ath. Potential Anal 34: Faizullah F (1) A note on the caratheodory aroximation cheme for tochatic differential equation under G-Brownian motion. Z Naturforchung A 67a: Faizullah F (14) Exitence of olution for G-SFDE with Cauchy Maruyama aroximation cheme. Al Anal Abtr. doi:1.1155/14/89431 Faizullah F, Mukhtar A, Rana MA (16) A note on tochatic functional differential equation driven by G-Brownian motion with dicontinuou drift coefficient. J Comut Anal Al 5(1): Gao F (9) Pathwie roertie and homeomorhic flow for tochatic differential equation driven by G-Brownian motion. Stoch Proc Al : Kim YH (14) On the th moment etimate for the olution of tochatic differential equation. J Inequal Al 395:1 9 Li X, Peng S (11) Stoing time and related Ito calculu with G-Brownian motion. Stoch Proc Al 11: Mao X (1997) Stochatic differential equation and their alication. Horwood Publihing Chicheter, Coll Houe England Peng S (6) G-exectation, G-Brownian motion and related tochatic calculu of Ito tye. In: The abel ymoium, Sringer, Berlin, Peng S (8) Multi-dimentional G-Brownian motion and related tochatic calculu under G-exectation. Stoch Proc Al 1:3 53 Peng S (1) Nonlinear exectation and tochatic calculu under uncertainty. arxiv:1.4546v1 Ren Y, Bi Q, Sakthivel R (13) Stochatic functional differential equation with infinite delay driven by G-Brownian motion. Math Method Al Sci 36(13): Shang Y (1) Synchronization in network of couled harmonic ocillator with tochatic erturbation and time delay. Ann Acad Rom Sci Ser Math Al 1(4):44 58 Shang Y (13a) The limit behavior of a tochatic logitic model with individual time-deendent rate. J Math. doi:1.1155/13/5635 Shang Y (13b) Grou conenu of multi-agent ytem in directed network with noie and time delay. Int J Syt Sci 14(46): Shang Y (15) Conenu of noiy multiagent ytem with Markovian witching toologie and time-varying delay. Math Probl Eng. doi:1.1155/15/4537 Song Y (13) Proertie of hitting time for G-martingale and their alication. Stoch Proce Al 8(11): Xua J, Zhang B (9) Martingale characterization of G-Brownian motion. Stoch Proce Al 119:3 48