Abstract
We present the Cauchy-Maruyama (CM) approximation scheme and establish the existence theory of stochastic functional differential equations driven by G-Brownian motion (G-SFDEs). Several useful properties of Cauchy-Maruyama (CM) approximate solutions of G-SFDEs are given. We show that the unique solution of G-SFDEs gets convergence from Cauchy-Maruyama (CM) approximate solutions. The existence theorem for G-SFDEs is developed with the above mentioned scheme.
1. Introduction
A significant role is played by stochastic differential equations (SDEs) in a broad range of applied disciplines, including biology, economics, finance, chemistry, physics, microelectronics, and mechanics. In many applications, one assumes that the future state of the system is independent of the past states. However, under close scrutiny, it becomes apparent that a more realistic model would contain some of the past state of the system and one needs stochastic functional differential equations to formulate such systems [1]. Also see [2, 3]. The motion theory of G-Brown, corresponding to Itô’s calculus and stochastic differential equations driven by G-Brownian motion (G-SDEs), was introduced by Peng [4, 5]. Then many authors paid much attention to this theory and G-SDEs [6–10]. The existence theory for solutions of stochastic functional differential equations driven by G-Brownian motion (G-SFDEs) was introduced by Ren et al. via Picard approximation [11]. However, like the classical stochastic differential equations, the convergence of Picard approximate solutions to the unique solution of G-SFDEs under the linear growth and Lipschitz condition is still open [1]. In contrast, here we introduce the Cauchy-Maruyama approximation scheme for G-SFDEs and show that the unique solution of G-SFDEs gets convergence from CM approximate solutions , . Furthermore, using CM approximation scheme, it is shown that the G-SFDE has a unique solution.
Some mathematical preliminaries are given in subsequent section. In Section 3, the CM approximation scheme for G-SFDEs is developed. In Section 4, some properties of the CM approximate solutions are presented. In Section 5, the theory of existence for the solutions of G-SFDEs with the above stated method is established.
2. Basic Notions
Some basic definitions and notions are presented in this section [4, 5, 11–13]. Suppose that if is a basic space that is nonempty and is a space of linear real valued mappings stated on the space such as , where is any arbitrary constant and if , for every . Furthermore, for , ; that is, is a space containing the linear mappings . Here represents the space of random variables.
Definition 1. A functional is named a sublinear expectation, if for every , , , and it holds the following monotonic, subadditivity, constant preserving and positive homogeneity properties, respectively:(i) implies .(ii). (iii).(iv).
The space given by triple is named sublinear expectation space. The functional is known as a nonlinear expectation if it holds only the first two properties.
Definition 2. Assume two random vectors and , which are -dimensional and are, respectively, defined on the spaces and . These are distributed identically, represented as if
Definition 3. Suppose is a sublinear expectation space with and Then is called -distributed or G-distributed, if for every , we get for every , where and it is independent of
To define the G-Brownian motion, we suppose that represents the space of all real valued incessant paths such that having norm Consider for and the process , which is commonly known as the canonical process. Then for every rigid we get that for , , and . Take which is a sequence of -dimensional random vectors on the space . Also, for every is independent of and is -distributed. Then we represent a sublinear expectation stated on as follows. For every , and we have
Definition 4. The expectation mentioned above, that is, , is known as a G-expectation and the related process is said to be a G-Brownian motion.
The completion of is presented with the norm for by . For we have . Also, , , denotes the filtration caused due to the stated process . To define G-Itô’s integral, we consider, for every , a partition of which is a finite ordered subset where . Assume to be fixed. For the above mentioned given partition we represent Remember that . We use to denote the completion of under the norm where for , .
Definition 5. For every , G-Itô’s integral is denoted by and is defined by
Let a sequence of partition of be given by , , where .
Definition 6. The process , known as quadratic variation process of , is a continuous increasing process having and is stated by
Definition 7. Suppose is the Borel -algebra of and is the (weakly compact) combination of measures for probability that is stated on . Afterwards, the capacity attached to is given by In case of the capacity of a set to be zero then the set is called polar, that is, and a property carries quasi-surely (in short .) if it is satisfied outside a polar set.
Lemma 8 (Gronwall inequality). Let and be real numbers, for , and be a real valued continuous function on such that ; then
3. The Cauchy-Maruyama Approximation Scheme for G-SFDEs
Let , , and . Represent by the space of bounded incessant -valued mappings defined on and having norm . Suppose , , and are Borel measurable. Assume the following -dimensional G-SFDE [5]: with initial data Also, represent the process of quadratic variation of G-Brownian motion . The coefficients , , and are given mappings satisfying , , for all . The integral form of (13) with initial data (14) is given as the following: where is a given initial condition.
Definition 9. An -valued stochastic process defined on is said to be the solution of (13) having initial data (14) if it holds the properties given below:(i)the solution is continuous and it is -adapted, for all ;(ii) and ;(iii) and for every
The Cauchy-Maruyama approximation scheme for G-SFDE (15) with initial data (14) is shown as the following. For any integer , we define on as follows: for , . Obviously, can be determined by stepwise iterations over the interval Furthermore, if we define and for , , then (18) yields for . We will use the following linear growth and Lipschitz conditions, respectively. Suppose two positive numbers and such that(i)for every , (ii)for all , and ,
4. Properties of Cauchy-Maruyama Approximate Solutions
Here we show that the Cauchy-Maruyama approximate solutions satisfy some very useful properties, which are given in the form of Lemmas (10) and (12).
Lemma 10. Let the linear growth condition (20) hold. Then for all , where , , , and , , and are arbitrary positive constants.
Proof. Using the basic inequality then from (19) we have for . Applying the Burkholder-Davis-Gundy (BDG) inequalities [9] and the linear growth condition (20) and taking sublinear expectation, it is implied that Remembering the notion of , we proceed as Noting the fact that we have where and . Thus the Gronwall inequality gives Letting , we have Consequently, which is the required result (22).
Remark 11. The above lemma shows that, for all , . Similarly, as the previous lemma, one can show that
Lemma 12. For all and , where and .
Proof. Obviously, one can observe that, for and , Thus using the linear growth condition (20) and the BDG inequalities [9] and taking sublinear expectation, we obtain From (29) taking we get which is the required result (32).
Remark 13. In a similar way as the previous lemma one can prove
5. Existence of Solutions for G-SFDEs
Firstly, this section shows that the unique solution of G-SFDEs (15) gets convergence from Cauchy-Maruyama (CM) approximate solutions and then it determines that the G-SFDE (15) has a unique solution with the Cauchy-Maruyama approximation scheme. Suppose if there is a constant that is positive such as that is, the initial data is uniformly Lipschitz -continuous.
Theorem 14. Suppose that the coefficients , , and satisfy the linear growth and Lipschitz conditions (20) and (21), respectively. Let the initial data satisfy condition (37). Then where .
Proof. Using the BDG inequalities [9], the Lipschitz condition (21), and inequality (32), we have where . Then by Gronwall’s inequality Using (36) we estimate as follows: Thus substituting the value of in (40), we obtain where , which is the required result (38).
Theorem 15. Suppose that the coefficients , , and satisfy the linear growth and Lipschitz conditions (20) and (21), respectively. Let the initial data satisfy condition (37). Then defined by (18) is a Cauchy sequence in and it converges to the unique solution of the G-SFDE (15).
Proof. Firstly, it is proved that the sequence is Cauchy. Assume ; then for , By using the BDG inequalities [9], Lipschitz condition (21), and Lemma (10), we get Using (38), it yields Consequently, we have where . Thus the sequence is a Cauchy sequence in . Denoting its limits by and letting in (46) yield Now we show that satisfies the G-SFDE (15). Let ; then using the Lipschitz condition (21), Lemma 10, and the BDG inequalities [9], we get where . Thus we get as a solution of (15). To show the uniqueness, assume that the G-SFDE (15) has two solutions, say and . Then in a similar fashion as above one can prove that which gives for . Hence for all , q.s.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author appreciates and acknowledges the financial support of NUST Pakistan. The author acknowledges the careful reading and some very useful suggestions of Dr. Ali Anwar.