Well-posedness and regularity for solutions of caputo stochastic fractional differential equations in L

Full Terms & Conditions of access and use can be found at

https://www.tandfonline.com/action/journalInformation?journalCode=lsaa20

Stochastic Analysis and Applications

ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/lsaa20

Well-posedness and regularity for solutions of

caputo stochastic fractional differential equations in L ^p spaces

Phan Thi Huong, P.E. Kloeden & Doan Thai Son

To cite this article: Phan Thi Huong, P.E. Kloeden & Doan Thai Son (2021): Well-posedness and regularity for solutions of caputo stochastic fractional differential equations in L^p spaces, Stochastic Analysis and Applications, DOI: 10.1080/07362994.2021.1988856

To link to this article: https://doi.org/10.1080/07362994.2021.1988856

Published online: 27 Oct 2021.

Submit your article to this journal

View related articles

View Crossmark data

Well-posedness and regularity for solutions of caputo stochastic fractional differential equations in L

^p

spaces

Phan Thi Huong^a, P.E. Kloeden^b, and Doan Thai Son^c

aLe Quy Don Technical University, Ha Noi, Viet Nam;^bDepartment of Mathematics, Universit€at

T€ubingen, T€ubingen, Germany;^cInstitute of Mathematics, Vietnam Academy of Science and Technology, Ha Noi, Viet Nam

ABSTRACT

In the first part of this paper, we establish the well-posedness for solutions of Caputo stochastic fractional differential equations (for short Caputo SFDE) of order a2¹₂, 1

in L^p spaces with p2 whose coefficients satisfy a standard Lipschitz condition. More pre- cisely, we first show a result on the existence and uniqueness of sol- utions, next we show the continuous dependence of solutions on the initial values and on the fractional exponenta. The second part of this paper is devoted to studying the regularity in time for solu- tions of Caputo SFDE. As a consequence, we obtain that a solution of Caputo SFDE has a d-H€older continuous version for any d2

0,a¹₂

: The main ingredient in the proof is to use a temporally weighted norm and the Burkholder-Davis-Gundy inequality.

ARTICLE HISTORY Received 1 December 2020 Accepted 22 September 2021 KEYWORD

Stochastic fractional differential equations;

existence and uniqueness of solutions; well- posedness; regularity

1. Introduction

In this paper, we consider a Caputo stochastic fractional differential equation of order a2¹₂, 1

on the interval ½0,T of the following form

CD^a_0þXðtÞ ¼bðt,XðtÞÞ þrðt,XðtÞÞ dWt

dt , (1)

where b:½0,T R^d!R^d, r:½0,T R^d!R^d are measurable and ðW_tÞ_t2½0,1Þ is a standard scalar Brownian motion on an underlying complete filtered probability space ðX,F,F:¼ ðFtÞ_t2½0,1Þ,PÞ:

The stochastic differential equations involving Caputo fractional time derivative oper- ator as in (1) give good models to investigate the memory and hereditary properties of various branches of physical and biological sciences more precisely (see [1–3]). The main reason for this fact is that fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and proc- esses. For more details we refer the reader to the monographs [2,4,5] and the referen- ces therein.

According to the authors’ knowledge, the main achieved results for Equation (1) are limited to problem of the existence and uniqueness of strong solutions [6–8] and mild

CONTACT Doan Thai Son dtson@math.ac.vn Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Ha Noi, Viet Nam.

ß2021 Taylor & Francis Group, LLC

https://doi.org/10.1080/07362994.2021.1988856

solutions [9] in L² spaces. A proof of coincidence of strong and mild solution of (1) in L²spaces under some natural assumptions on the coefficients has recently been proved in [10]. An Euler-Maruyama type scheme for Caputo stochastic fractional differential equations and the convergence rate of this scheme have been established in [11]. Very recently, several results on well-posedness and continuity on fractional exponent a for solutions of (1) in L² spaces have been developed in [12]. However, the well-posedness as well as the regularity of solutions of Equation (1) in L^p spaces with p2 were not investigated systematically in any article. Thus, in this paper, we are trying to fill this gap. Precisely, our first aim in this paper is to establish the well-posedness for the solu- tions of Caputo stochastic fractional differential equations with generalized Lipschitz- type coefficients driven by general multiplicative noise in L^p spaces. Secondly, we are interested in the H€older continuity of the solutions for Equation (1)inL^pspaces.

The paper is organized as follows: In Section 2, we introduce briefly about Caputo SFDE and state the main results of the paper. The first main result is about global exist- ence and uniqueness of solutions (Theorem 1(i)), continuous dependence of solutions on the initial values (Theorem 1(ii)) and on the fractional exponent (Theorem 1(iii)).

The second main result is about regularity in time of solutions (Theorem 2). The proof of the first main result and the second main result are given in Sections 3 and 4, respectively.

2. Preliminaries and main results

2.1. Caputo fractional stochastic differential equations and standing assumptions For p2,t2 ½0,1Þ, let X^p_t :¼L^pðX,Ft,PÞ denote the space of all Ft-measurable, p^th integrable functions f ¼ ðf1,f_2,:::^,f_dÞ^T:X!R^dwith

kfk_p:¼ X^d

i¼1

Eðjfij^pÞ

!¹_p :

A measurable process X:½0,T !L^pðX,F,PÞ is said to be F-adapted if XðtÞ 2 X^pt

for all t0: For each g2 X^p₀, a F-adapted process X is called a solution of (1) with the initial condition Xð0Þ ¼g if Xð0Þ ¼g and the following equality holds on X^p_t for t2 ð0,T

XðtÞ ¼gþ 1 CðaÞ

ð_t

0

ðtsÞ^a1bðs,XðsÞÞdsþ ð_t

0

ðtsÞ^a1rðs,XðsÞÞdWs

!

, (2)

where CðaÞ:¼Ð₁

0 s^a1expðsÞdsis the Gamma function, see [7, p. 209].

2.2. Well-posedness and regularity of solutions

In this article, we assume that the coefficients b and r of (1) satisfy the follow- ing conditions:

(H1) Global Lipschitz continuity in R^d of the drift and diffusion term: For all x,y2 R^d,t2 ½0,T, there exists L>0 such that

jbðt,xÞ bðt,yÞj_pþ jrðt,xÞ rðt,yÞj_pLjxyj_p:

(H2) Essential boundedness in time for drift and diffusion term: rð, 0Þ is essentially bounded, i.e.

esssup

s2½0,Tjbðs, 0Þj_p<M, esssup

s2½0,Tjrðs, 0Þj_p <M:

Note that the contents of assumptions (H1) and (H2) are independent on the choice of the norm on R^d: However, for a convenience in several estimates below, we equip R^d with the p norm, i.e. for any vector x¼ ðx1,:::^,x_dÞ^T2R^d, the p norm jxj_p of x is defined by jxj_p:¼ ðP_d

i¼1jxij^pÞ^1p:

The first main result in this article is the well-posedness of solutions of Caputo SFDE.

Theorem 1. (Well-posedness of solutions of Caputo SFDE). Suppose that (H1) and (H2) hold. Then the following statements hold:

i. Existence and uniqueness for solutions of Caputo SFDE: for any g2 X^p₀, the ini- tial value problem (1) with the initial condition Xð0Þ ¼g has a unique solution on½0,Tdenoted byu_að,gÞ:

ii. Continuous dependence on the initial values for solutions of Caputo SFDE: for anyf,g2 X^p₀, the solution u_að,gÞdepends Lipschitz continuously on g, i.e. there exists L₁ >0such that

kuaðt,fÞ u_aðt,gÞk_pL₁kfgk_p for all t2½0,T:

iii. Continuous dependence on the fractional exponent a for solutions of Caputo SFDE: The solutionu_að,gÞdepends continuously on a, i.e.

^lim

a!aesssup

t2½0,T kuaðt,gÞ u_^aðt,gÞk_p¼0:

The second main result in this article is the regularity of solutions of Caputo SFDE.

Theorem 2. (The regularity of solutions). Suppose that (H1) and (H2) hold. Then, there exists D>0 depending ona,p,L,M,T such that

kuaðt,gÞ u_aðs,gÞk_p Djtsj^a12, for all s,t2½0,T: (3) Corollary 3. For alld20,a¹₂

, there exists a modification Y of X withd-H€older con- tinuous paths, i.e.

PðXt ¼Y_tÞ ¼1 for all t2½0,T:

Proof. Thanks to (3) and using Kolmogorov test, see e.g. [13, Theorem, p. 51], X(t) has and-H€older continuous modification for anyd20,a¹₂

: ^w

3. Well-posedness of solutions

3.1. Existence and uniqueness of solutions

Our aim in this subsection is to prove the result on existence and uniqueness of solu- tions to the Equation (1). For this purpose, let H^pð½0,TÞ be the space of all the proc- essesX which are measurable,FT-adapted, whereFT:¼ ðFtÞ_t2½0,T and satisfies that

kXk_H^p:¼esssup

t2½0,TkXðtÞk_p <1:

Obviously, ðH^pð½0,TÞ,k k_H^pÞ is a Banach space. For anyg2X^p₀, we define an oper- ator Tg:H^pð½0,TÞ !H^pð½0,TÞ by Tgnð0Þ:¼g and the following equality holds for t2 ð0,T

TgnðtÞ:¼gþ 1 CðaÞ

ð_t

0

ðtsÞ^a1bðs,nðsÞÞdsþ ð_t

0

ðtsÞ^a1rðs,nðsÞÞdWs

!

: (4)

The following lemma is devoted to show the well-defined property of this operator.

In the proof of this result and also several results below, we use the following elemen- tary inequality

jxþyj^p_p 2^p1ðjxj^p_pþ jyj^p_pÞ for all x,y2R^d: (5) Lemma 4. For anyg2 X^p₀, the operatorTg is well-defined.

Proof. Let n2H^pð½0,TÞ be arbitrary. From the definition of Tgn as in (11) and the inequality (13), we have that for all t2 ½0,T

kTgnðtÞk^p_p 2^p1kgk^p_pþ2^2p2 C^pðaÞ

ð_t

0

ts

ð Þ^a1bðs,nðsÞÞds ^p

p

þ2^2p2 C^pðaÞ

ð_t

0

ts

ð Þ^a1r sð ,nðsÞÞdWs

^p

p

:

(6)

By the H€older inequality, we obtain that ð_t

0

ts

ð Þ^a1bðs,nðsÞÞds ^p

p

X^d

i¼1

E ð_t

0

ts

ð Þ^a1jbiðs,nðsÞÞjds

!_p

X^d

i¼1

E ð_t

0

ts ð Þ^ða1Þpp1 ds

!_p1 ð_t

0

b_iðs,nðsÞÞ j j^pds 0

@

1 A T^ðap1Þðp1Þ^p1

ðap1Þ^p1 ð_t

0

kbðs,nðsÞÞk^p_pds:

(7)

FromðH1Þ, we derive

jbðs,nðsÞÞj^p_p2^p1jbðs,nðsÞÞ bðs, 0Þj^p_pþ jbðs, 0Þj^p_p 2^p1L^pjnðsÞj^p_pþ2^p1jbðs, 0Þj^p_p:

Therefore, ð_t

0

kbðs,nðsÞÞk^p_pds2^p1L^p esssup

s2½0,T jjnðsÞjj_{p p}ð_t

0

1dsþ2^p1 ð_t

0

jbðs, 0Þj^p_pds

2^p1L^pTjjnjj^p_H^pþ2^p1 ð_T

0

jbðs, 0Þj^p_pds which together with (7) implies that

ð_t

0

ts

ð Þ^a1bðs,nðsÞÞds ^p

p

T^ðap1Þð2p2Þ^p1

ðap1Þ^p1 L^pTknk^p_H^pþ ð_T

0

jbðs, 0Þj^p_pds

!

: (8)

Now, using the Burkholder-Davis-Gundy and the H€older inequalities, we obtain ð_t

0

ts

ð Þ^a1r sð ,nðsÞÞdWs

^p

p

X^d

i¼1

E ð_t

0

ðtsÞ^a1r_iðs,nðsÞÞdWs

^p

X^d

i¼1

C_pE ð_t

0

ðtsÞ^2a2

r_iðs,nðsÞÞj²dsj^p2 X^d

i¼1

C_pE ð_t

0

ðtsÞ^2a2jriðs,nðsÞÞj^pds ð_t

0

ðtsÞ^2a2ds

!^p2

2

C_p T^2a1 2a1 ^p2

2 ð_t

0

ts

ð Þ^2a2kr sð ,nðsÞÞk^p_pds, where C_p¼ _2ðp1Þ^ppþ1p1

^p

2: From (H1) and (H2), we also have

jrðs,nðsÞÞj^p_p2^p1L^pjnðsÞj^p_pþ2^p1jrðs, 0Þj^p_p2^p1L^pjnðsÞj^p_pþ2^p1M^p: Therefore, for allt2 ½0,T we have

ð_t

0

ts

ð Þ^2a2kr sð ,nðsÞÞk^p_pds 2^p1L^p

ð_t

0

ts

ð Þ^2a2 esssup

s2½0,T knðsÞk_p

_p

dsþ2^p1M^p ð_t

0

ts ð Þ^2a2ds

2^p1T^2a1

2a1 L^pknk^p_HpþM^p :

This together with (6), (8) and (H2) implies that kTgnk_H^p <1: Hence, the map Tg

is well-defined.

To prove existence and uniqueness of solutions, we will show that the operator Tg

defined as above is contractive under a suitable weighted norm (cf. [14, Remark 2.1] for the same method to prove the existence and uniqueness of solutions of stochastic differ- ential equations). Here, the weight function is the Mittag-Leffler function E2a1ðÞ defined as follows

E_2a1ðtÞ:¼X¹

k¼0

t^k

Cðð2a1Þkþ1Þ for allt2R:

For more details on the Mittag-Leffler functions we refer the reader to the book [2, p. 16].

We are now in a position to prove Theorem 1(i).

Proof of Theorem 1 (i). Choose and fix a positive constant csuch that

c>j2^p1Cð2a1Þ, (9)

where

j¼2^p1L^p C^pðaÞ

p^pþ1 2ðp1Þ^p1

!^p

2 T^2a1

2a1 ^p2₂

þ T^ðp2Þaþ1

ðp2Þaþ1 p1

_p1 0

B@

1

CA: (10)

On the space H^pð½0,TÞ, we define a weighted normk k_c as below kXk_c:¼esssup

t2½0,T

kXðtÞk^p_p E_2a1ðct^2a1Þ

!¹_p

for all X2H^pð½0,TÞ, (11) Obviously, two norms k k_H^p and k kc are equivalent. Thus, ðH^pð½0,TÞ,k kcÞ is also a Banach space. Choose and fix g2X^p₀: By virtue of Lemma 4, the operator Tg is well-defined. Now, we will prove that the map Tg is contractive with respect to the norm k kc: For this purpose, let n,^n2H^pð½0,TÞ be arbitrary. From (4) and the inequality (5), we derive the following inequality for allt2 ½0,T:

kTgnðtÞ Tg^nðtÞk^p_p 2^p1 C^pðaÞ

ð_t

0

ðtsÞ^a1ðbðs^,nðsÞÞ bðs^,^nðsÞÞÞds ^p

p

þ 2^p1 C^pðaÞ

ð_t

0

ðtsÞ^a1ðrðs,nðsÞÞ rðs,^nðsÞÞÞdWs

^p

p

: Using the H€older inequality and ðH1Þ, we obtain

ð_t

0

ðtsÞ^a1ðbðs^,nðsÞÞ bðs^,^nðsÞÞÞds ^p

p

X^d

i¼1

E ð_t

0

ðtsÞ^a1jbiðs,nðsÞÞ b_iðs,^nðsÞÞjds

!_p

X^d

i¼1

E ð_t

0

ðtsÞ^{ða1Þðp2Þp1} ds

!_p1 ð_t

0

ðtsÞ^2a2jbiðs,nðsÞÞ b_iðs,^nðsÞÞj^pds 0 !

@

1 A L^pT^ðap2aþ1Þðp1Þ^p1

ðap2aþ1Þ^p1 ð_t

0

ðtsÞ^2a2knðsÞ ^nðsÞk^p_pds:

On the other hand, by the Burkholder-Davis-Gundy inequality and ðH1Þ, we have ð_t

0

ðtsÞ^a1ðrðs^,nðsÞÞ rðs^,^nðsÞÞÞdWs

^p

p

¼X^d

i¼1

E ð_t

0

ðtsÞ^a1ðriðs^,nðsÞÞ riðs^,^nðsÞÞÞdWs

^p X^d

i¼1

C_pEj ð_t

0

ðtsÞ^2a2jriðs^,nðsÞÞ r_is,^nðsÞ j²dsj^p2

X^d

i¼1

C_pE ð_t

0

ðtsÞ^2a2jriðs,nðsÞÞ r_is,^nðsÞ j^pds

ð_t

0

ðtsÞ^2a2ds

!^p2

2

L^pC_p T^2a1 2a1 ^p2₂ ð_t

0

ðtsÞ^2a2knðsÞ ^nðsÞk^p_pds:

Thus, for allt2 ½0,Twe have

kTgnðtÞ Tg^nðtÞk^p_p j

ð_t

0

ðtsÞ^2a2knðsÞ ^nðsÞk^p_pds,

where j is given as in (10). This estimate with the definition ofk kc as in (11) implies that

kTgnðtÞ Tg^nðtÞk^p_p E_2a1ðct^2a1Þ

jÐ_t

0ðtsÞ^2a2knðsÞ ^nðsÞk^p_p

E2a1ðcs^2a1Þ E2a1ðcs^2a1Þds E_2a1ðct^2a1Þ

j ess sup

s2½0,T

knðsÞ ^nðsÞk^p_p E2a1ðcs^2a1Þ

!¹

p

0

@

1 A

pÐ_t

0ðtsÞ^2a2E_2a1ðcs^2a1Þds E2a1ðct^2a1Þ jCð2a1Þ

c kn^nk^pc:

where in the final step, we used the inequality c

Cð2a1Þ ð_t

0

ðtsÞ^2a2E2a1ðcs^2a1ÞdsE2a1ðct^2a1Þ see [6, Lemma 5]. Consequently,

kTgn Tg^nkc jCð2a1Þ c

¹

pkn^nkc:

By (9), we have ^jCð2a_c^1Þ<1 and therefore the operator Tg is a contractive map on ðH^pð½0,TÞ,k k_cÞ: Using the Banach fixed point theorem, there exists a unique fixed point of this map in H^pð½0,TÞ: This fixed point is also the unique solution of (1) with the initial conditionXð0Þ ¼g: The proof of this theorem is complete. w

Remark 5. (i) Results about the existence and uniqueness of solutions of Caputo SFDE in case p¼2 have been shown in [6].

(ii) In [8], the author considered the following form inL^p XðtÞ ¼Xð0Þ þ

ð_t

0

ðtsÞ^a1bðt,s,XðsÞÞdsþ ð_t

0

ðtsÞ^a2rðt,s,XðsÞÞdWs, (12) where Xð0Þ 2R^d,r:RþRþR^d!R^dR^m,b:RþRþR^d!R^d are Borel measurable functions anda1 2 ð0, 1Þ,a220,¹₂

:He obtained the existence and unique- ness of solutions with non-Lipschitz coefficients in L^p with p>max _1a¹

1,_12a²

2

: In

this paper, we prove that the result still holds for all p>2 and random initial condi- tionXð0Þ ¼g2X^p₀:

3.2. The continuous dependence of the solutions on the initial values

Our next result is to evaluate the distance between two different solutions. As a conse- quence, we obtain the Lipschitz continuity dependence of solutions on the initial values.

Proof of Theorem 1 (ii). Choose and fixf2X^p₀:Let g2X^p₀ arbitrarily. Since u_að,gÞand u_að,fÞ are solutions of (1) it follows that

u_aðt,gÞ u_aðt,fÞ ¼gfþ 1 CðaÞ

ð_t

0

bðs,u_aðs,gÞÞ bðs,u_aðs,fÞÞ ðtsÞ^1a ds þ 1

CðaÞ ð_t

0

rðs,u_aðs,gÞÞ rðs,u_aðs,fÞÞ ðtsÞ^1a dWs:

Hence, using (5), the H€older and the Burkholder-Davis-Gundy inequalities and (H1) (see proof of Theorem 1(i)), we obtain

jjuaðt,gÞ u_aðt,fÞjj^p_p 2^p1j ð_t

0

ðtsÞ^2a2jjuaðs,gÞ u_aðs,fÞjj^p_pds þ2^p1jjgfjj^p_p:

Applying the Gronwall inequality for fractional differential equations, see [15, Lemma 7.1.1] or [16, Corollary 2], we arrive at

ku_aðt,gÞ u_aðt,fÞk^p_p2^p1E_2a12^p1jCð2a1Þt^2a1

kgfk^p_p: Hence,

limg!fkuaðt,gÞ u_aðt,fÞk_p¼0:

The proof is complete. w

Remark 6. A result about the continuity dependence of solutions of SFDE on the ini- tial values in casep¼2 has been shown in [6].

3.3. The continuous dependence of the solutions on the fractional exponent a In this part, we shall prove the continuous dependence on aof the solution.

Proof of Theorem 1 (iii). Let a,^a2¹₂, 1

be arbitrarily but fixed. Choose and fix g2 X^p₀:Sinceu_að,gÞandu_^að,gÞ are solutions of (1) it follows that

u_aðt,gÞ u_^aðt,gÞ

¼ 1 CðaÞ

ð_t

0

ðtsÞ^a1ðbðs,u_aðs,gÞÞ bðs,u_^aðs,gÞÞÞds þ

ð_t

0

ðtsÞ^a1

CðaÞ ðtsÞ^{^a1} Cð^aÞ

bðs,u_^aðs,gÞÞds þ 1

CðaÞ ð_t

0

ðtsÞ^a1ðrðs,u_aðs,gÞÞ rðs,u_^aðs,gÞÞÞdWs

þ ð_t

0

ðtsÞ^a1

CðaÞ ðtsÞ^{^a1} Cð^aÞ

rðs,u_^aðs,gÞÞdWs:

Using the inequality (5), the H€older and the Burkholder-Davis-Gundy inequalities and (H1) (see proof of Theorem 1(i)), we obtain

kuaðt,gÞ u_^aðt,gÞk^p_p 2^p1j

ð_t

0

ðtsÞ^2a2kuaðs,gÞ u_^aðs,gÞk^p_pds þ2^2p2

ð_t

0

ðtsÞ^a1

CðaÞ ðtsÞ^{^a1} Cð^aÞ

bðs,u_^aðs,gÞÞds ^p

p

þ2^2p2 ð_t

0

ðtsÞ^a1

CðaÞ ðtsÞ^{^a1} Cð^aÞ

rðs,u_^aðs,gÞÞdWs

^p

p

:

Now, let

gðt,s,a,^aÞ:¼

ðtsÞ^a1

CðaÞ ðtsÞ^{^a1} Cð^aÞ

: Applying the H€older inequality, (5), (H1) and (H2), we obtain that

ð_t

0

ðtsÞ^a1

CðaÞ ðtsÞ^{^a1} Cð^aÞ

bðs,u_^aðs,gÞÞds ^p

p

X^d

i¼1

E ð_t

0

gðt^,s,a,^aÞjbiðs^,u_^aðs^,gÞÞjds

!_p

X^d

i¼1

E ð_t

0

ðgðt,s,a,^aÞÞ^p1pds

!_p1 ð_t

0

jb_iðs,u_^aðs,gÞÞj^pds 0

@

1 A

ð_t

0

ðgðt^,s,a,^aÞÞ²ds

!^p

2 ð_t

0

1ds

!^p2

2 ð_t

0

kbðs,u_^aðs,gÞÞk^p_pds

ð_t

0

ðgðt^,s,a,^aÞÞ²ds

!^p

2

T^p22 ð_t

0

2^p1L^pku_^aðs,gÞk^p_pþ kbðs, 0Þk^p_p ds

ð_t

0

ðgðt^,s,a,^aÞÞ²ds

!^p

2

T^p22^p1ðL^pesssup

s2½0,Tku_^aðs,gÞk^p_pþM^pÞ:

On the other hand, using the Burkholder-Davis-Gundy inequality, (H1) and (H2), we have

ð_t

0

ðtsÞ^a1

CðaÞ ðtsÞ^{^a1} Cð^aÞ

rðs,u_^aðs,gÞÞdWs

^p

p

X^d

i¼1

E ð_t

0

gðt^,s,a,^aÞjriðs^,u_^aðs^,gÞÞjdWs

^p X^d

i¼1

CpE ð_t

0

ðgðt^,s,a,^aÞÞ²jriðs^,u_^aðs^,gÞÞj²ds

p 2

X^d

i¼1

CpE ð_t

0

ðgðt^,s,a,^aÞÞ²jriðs^,u_^aðs^,gÞÞj^pds ²_p ð_t

0

ðgðt^,s,a,^aÞÞ²ds ^p2_p

" #^p

2

¼Cp

ð_t

0

ðgðt,s,a,^aÞÞ²krðs,u_^aðs,gÞÞk^p_pds ð_t

0

ðgðt,s,a,^aÞÞ²ds ^p2₂

C_p ð_t

0

ðgðt,s,a,^aÞÞ²ds

!^p

2

2^p1ðL^pesssup

s2½0,Tku_^aðs,gÞk^p_pþM^pÞ:

Combining the above calculations and by the definition of k k_cyields the estimate kuaðt,gÞ u_^aðt,gÞk^p_p

E_2a1ðct^2a1Þ

j2^p1Ð_t

0ðtsÞ^2a2kuaðs^,gÞ u_^aðs^,gÞk^p_p

E2a1ðcs^2a1Þ E_2a1ðcs^2a1Þds E2a1ðct^2a1Þ

þ2^3p3ðL^pesssup

s2½0,Tku_^aðs,gÞk^p_pþM^pÞ ð_t

0

ðgðt^,s,a,^aÞÞ²ds

!^p

2

T^p2

þ2^3p3ðL^pesssup

s2½0,Tku_^aðs,gÞk^p_pþM^pÞCp

ð_t

0

ðgðt^,s,a,^aÞÞ²ds

!^p

2

j2^p1Cð2a1Þ

c ku_að,gÞ u_^að,gÞk^pc

þ2^3p3ðL^pesssup

s2½0,Tku_^aðs,gÞk^p_pþM^pÞ ð_t

0

ðgðt,s,a,^aÞÞ²ds

!^p

2

T^p2

þ2^3p3ðL^pesssup

s2½0,Tku_^aðs,gÞk^p_pþM^pÞC_p ð_t

0

ðgðt,s,a,^aÞÞ²ds

!^p

2

,

where in the finall step, we have used Lemma 5 in [6]. Thus, we have

1j2^p1Cð2a1Þ c

kuað,gÞ u_^að,gÞk^p_c

2^3p3ðL^pesssup

s2½0,Tku_^aðs,gÞk^p_pþM^pÞ ð_t

0

ðgðt^,s,a,^aÞÞ²ds

!^p₂ T^p2

þ2^3p3ðL^pesssup

s2½0,Tku_^aðs,gÞk^p_pþM^pÞCp

ð_t

0

ðgðt^,s,a,^aÞÞ²ds

!^p₂ :

By (9) and p2, therefore to complete the proof it is sufficient to show that

^lim

a!a sup

t2½0,T

ð_t

0

ðgðt,s,a,^aÞÞ²ds¼0:

Indeed, we have ð_t

0

ðgðt,s,a,^aÞÞ²ds¼ ð_t

0

ðtsÞ^2a2 C²ðaÞ dsþ

ð_t

0

ðtsÞ^{2^a2} C²ð^aÞ ds2

ð_t

0

ðtsÞ^{aþ^a2} CðaÞCð^aÞ ds

¼ t^2a1

ð2a1ÞC²ðaÞþ t^{2^a1}

ð2^a1ÞC²ð^aÞ 2t^{aþ^a1}

ðaþ^a1ÞCðaÞCð^aÞ: Thus, we conclude that

^lim

a!a sup

t2½0,T

ð_t

0

ðgðt,s,a,^aÞÞ²ds¼0:

The proof is complete. w

Remark 7. To prove Theorem 1(i) and Theorem 1(ii), we only require the assumption (H1) and the following assumption (weaker than (H2)):

(H3) L^p-integrable in time for drift term and essential boundedness in time for diffu- sion term:

ð_T

0

jbðs, 0Þj^p_pds<1, esssup

s2½0,Tjrðs, 0Þj_p<M:

4. Regularity of solutions

This section is devoted to proving the regularity of solutions to Caputo SFDE.

Proof of Theorem 2. Choose and fixt,s2 ½0,Twith t>s.

Using (5), we derive that C^pðaÞ

2^2p2ku_aðt,gÞ u_aðs,gÞk^p_p

ð_t

s

bðs,u_aðs,gÞÞ ðtsÞ^1a ds

^p

p

þ ð_t

s

rðs,u_aðs,gÞÞ ðtsÞ^1a dWs

^p

p

þ ð_s

0

1

ðtsÞ^1a 1 ðssÞ^1a

bðs^,u_aðs^,gÞÞds ^p

p

þ ð_s

0

1

ðtsÞ^1a 1 ðssÞ^1a

rðs^,u_aðs^,gÞÞdWs

^p

p

:

Now, applying the H€older and the Burkholder-Davis-Gundy inequalities, we arrive at C^pðaÞ

2^2p2 kuaðt,gÞ u_aðs,gÞk^p_p ðtsÞ^ap1ðp1Þ^p1

ðap1Þ^p1 ð_t

s

kbðs,u_aðs,gÞÞk^p_pds

þC_p ð_t

s

krðs,u_aðs,gÞÞk^p_p ðtsÞ^22a ds

ð_t

s

1 ðtsÞ^22ads

!^p2

2

þT^p22 ð_s

0

kbðs,u_aðs,gÞÞk^p_pds ð_s

0

1

ðtsÞ^1a 1 ðssÞ^1a

²ds

!^p

2

þC_p ð_s

0

1

ðtsÞ^1a 1 ðssÞ^1a

₂

krðs,u_aðs,gÞÞk^p_pds

ð_s

0

1

ðtsÞ^1a 1 ðssÞ^1a

₂

ds

" #^p2

2

:

On the other hand, since u_að,gÞ 2H^pð½0,TÞ, there exists M1>0 such that ess sup_t2½0;Tku_aðt,gÞk^p_pM₁:This together with (H1) and (H2) implies that

kbðs,u_aðs,gÞÞk^p_p 2^p1ðL^pku_aðs,gÞk^p_pþ kbðs, 0Þk^p_pÞ 2^p1ðL^pM₁þM^pÞ, krðs,u_aðs,gÞÞk^p_p 2^p1ðL^pku_aðs,gÞk^p_pþ krðs, 0Þk^p_pÞ 2^p1ðL^pM₁þM^pÞ:

Moreover, we have ð_s

0

1

ðtsÞ^1a 1 ðssÞ^1a

2

ds ð_s

0

1

ðssÞ^22a 1 ðtsÞ^22a

ds

¼s^2a1t^2a1

2a1 þðtsÞ^2a1 2a1 ðtsÞ^2a1

2a1 :

Combining the above calculations yields the following estimate C^pðaÞ

2^2p2kuaðt,gÞ u_aðs,gÞk^p_p ð2p2Þ^p1ðL^pM1þM^pÞT^p2

ðap1Þ^p1 ðtsÞ^ð2a1Þp2 þC_p2^p1ðL^pM₁þM^pÞ

ð2a1Þ^p2 ðtsÞ^ð2a1Þp2 þ2^p1ðTÞ^p2ðL^pM₁þM^pÞ

ð2a1Þ^p2 ðtsÞ^ð2a1Þp2 þCp2^p1ðL^pM1þM^pÞ

ð2a1Þ^p2 ðtsÞ^ð2a1Þp2 : Thus, we have

kuaðt,gÞ u_aðs,gÞk_p DðtsÞ^a12, where

D^p:¼ 2^2p2 C^pðaÞ

ð2p2Þ^p1ðL^pM₁þM^pÞT^p2

ðap1Þ^p1 þCp2^p1ðL^pM1þM^pÞ ð2a1Þ^p2

!

þ2^2p2 C^pðaÞ

2^p1ðTÞ^p2ðL^pM₁þM^pÞ

ð2a1Þ^p2 þC_p2^p1ðL^pM₁þM^pÞ ð2a1Þ^p2 0

@

1 A,

which together the fact that a2¹₂, 1

and p2 implies that lims!tkuaðt,gÞ u_aðs,gÞk_p ¼0:

The proof is complete. w

Remark 8. When p¼2, then result of Theorem 2 coincides with the result in [17] in the special case that bðt,XðtÞÞ:¼AXðtÞ þgðtÞandrðt,XðtÞÞ:¼fðtÞ:

Acknowledgement

The authors would like to thank anonymous reviewers for carefully reading the first draft and for many constructive comments that lead to an improvement of the paper.

Funding

The work of Doan Thai Son is supported by the Project QTRU01.08/20-21 of Vietnam Academy of Science and Technology.

References

[1] Hilfer, R. (2000). Applications of Fractional Calculus in Physics. Singapore: World Scientific.

[2] Podlubny, I. (1999). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. San Diego, CA: Academic Press, Inc.

[3] Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y. (2018). A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64:213–231. DOI:10.1016/j.cnsns.2018.04.019.

[4] Diethelm, K. (2010). The Analysis of Fractional Differential Equations. An Application- Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics. Berlin: Springer-Verlag.

[5] Oldham, K.B., Spanier, J. (2016). The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Mineola, NY: Dover Publications.

[6] Doan, T.S., Huong, P.T., Kloeden, P.E., Tuan, H.T. (2018). Asymptotic separation between solutions of caputo fractional stochastic differential equations. Stoch. Anal. Appl. 36(4):

654–664. DOI:10.1080/07362994.2018.1440243.

[7] Wang, Y., Yejuan, X., Kloeden, P.E. (2016). Asymptotic behavior of stochastic lattice sys- tems with a caputo fractional time derivative.Nonlinear Anal. 135:205–222. DOI:10.1016/

j.na.2016.01.020.

[8] Wang, Z. (2008). Existence and uniqueness of solutions to stochastic volterra equations with singular kernels and non-lipschitz coefficients. Statist. Probab. Lett. 78(9):1062–1071.

DOI:10.1016/j.spl.2007.10.007.

[9] Sakthivel, R., Revathi, P., Ren, Y. (2013). Existence of solutions for nonlinear fractional stochastic differential equations.Nonlinear Anal. TMA. 81:70–86. DOI:10.1016/j.na.2012.

10.009.

[10] Anh, P.T., Doan, T.S., Huong, P.T. (2019). A variation of constant formula for caputo fractional stochastic differential equations. Statist. Probab. Lett. 145:351–358. DOI: 10.

1016/j.spl.2018.10.010.

[11] Doan, T.S., Huong, P.T., Kloeden, P.E., Vu, A.M. (2020). Euler-maruyama scheme for caputo fractional stochastic differential equations. J. Comput. Appl. Math. 380:112989.

DOI:10.1016/j.cam.2020.112989.

[12] Wang, W., Cheng, S., Guo, Z., Yan, X. (2020). A note on the continuity for caputo frac- tional stochastic differential equations.Chaos. 30(7):073106. DOI:10.1063/1.5141485.

[13] Evans, L. C. (2013). An Introduction to Stochastic Differential Equations. Providence, RI:

American Mathematical Society.

[14] Han, X., Kloeden, P.E. (2017). Random Ordinary Differential Equations and Their Numerical Solution. Singapore: Springer.

[15] Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840. New York: Springer-Verlag.

[16] Ye, H., Gao, J., Ding, Y. (2007). A generalized gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2):1075–1081. DOI: 10.1016/j.

jmaa.2006.05.061.

[17] Su, X., Li, M. (2018). The regularity of fractional stochastic evolution equations in hilbert space.Stoch. Anal. Appl. 36(4):639–653. DOI:10.1080/07362994.2018.1436973.