Existence of Solutions for G-SFDEs with Cauchy-Maruyama Approximation Scheme

and Applied Analysis 3 probability P that is stated on (Λ,B(Λ)). Afterwards, the capacity c(⋅) attached toP is given by c (D) = sup P∈P P (D) , D ∈ B (Λ) . (11) In case of the capacity of a set D to be zero then the set D is called polar, that is, c(D) = 0 and a property carries quasisurely (in short q.s.) if it is satisfied outside a polar set. Lemma 8 (Gronwall inequality). Let κ and c < d be real numbers, g(t) ≥ 0 for t ∈ [c, d], and h(t) be a real valued continuous function on [c, d] such that h(t) ≤ κ + ∫ d c g(V)h(V)dV; then h (t) ≤ κe ∫ d c g(V)dV , t ∈ [c, d] . (12) 3. The Cauchy-Maruyama Approximation Scheme for G-SFDEs Let 0 = t0 ≤ t ≤ T < ∞, a ∈ [−τ, 0], and τ > 0. Represent by BC([−τ, 0];R) the space of bounded incessant R-valued mappings φ defined on [−τ, 0] and having norm ‖φ‖ = sup a≤0 |φ(a)|. Suppose α : [t0, T] × BC([−τ, 0];R d ) → R, β : [t0, T] × BC([−τ, 0]R d ) → R, and γ : [t0, T] × BC([−τ, 0];R) → R are Borel measurable. Assume the following d-dimensional G-SFDE [5]: dY (t) = α (t, Y (t + a)) dt + β (t, Y (t + a)) d ⟨B, B⟩ (t) + γ (t, Y (t + a)) dB (t) , t ∈ [t0, T] , (13) with initial data Yt 0 = ξ = {ξ (a) : −τ ≤ a ≤ 0} is F0-measurable, BC ([−τ, 0] ;R d ) -value random variable so that ξ ∈ M G ([−τ, T] ;R d ) . (14) Also, {⟨B, B⟩(t) : t ≥ 0} represent the process of quadratic variation of G-Brownian motion {B(t) : t ≥ 0}. The coefficients α, β, and γ are given mappings satisfying α(⋅, y), β(⋅, y), γ(⋅, y) ∈ M G ([t0 − τ, T];R d ) for all y ∈ R. The integral form of (13) with initial data (14) is given as the following: Y (t)


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][7][8][9][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   (),  ≥ 1.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.

Basic Notions
Some basic definitions and notions are presented in this section [4,5,[11][12][13].Suppose that if Λ is a basic space that is nonempty and H is a space of linear real valued mappings stated on the space Λ such as  ∈ H, where  is any arbitrary constant and if  1 ,  2 , . . .,   ∈ H, ( 1 ,  2 , . . .,   ) ∈ H for every  ∈ C . Lip (R  ).Furthermore, linear mappings .Here H represents the space of random variables.

Definition 1.
A functional E : H → R is named a sublinear expectation, if for every ,  ∈ H,  ≥ 0, and  ∈ R it holds the following monotonic, subadditivity, constant preserving and positive homogeneity properties, respectively: The space given by triple (Λ, H, E) is named sublinear expectation space.The functional E 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 (Λ, H, E) and ( Λ, H, Ẽ).These are distributed identically, represented as  ∼  if Definition 3. Suppose (Λ, H, E) is a sublinear expectation space with  ∈ H and Then  is called N(0; [ 2 ,  2 ])-distributed or G-distributed, if for every ,  ≥ 0 we get for every  ∈ H, where  ∼  and it is independent of  To define the G-Brownian motion, we suppose that Λ =  0 (R + ) represents the space of all real valued incessant paths (  ) ∈R + such that  0 = 0 having norm Consider for  ∈ [0, ∞) and  ∈ Λ the process   () =   , which is commonly known as the canonical process.Then for every rigid  ∈ [0, ∞) we get that for  ≤ ,   (Λ  ) ⊆   (Λ  ), and =1 which is a sequence of -dimensional random vectors on the space (Λ, H, E).Also, for every  = 1, 2, . . .,  − 1,  +1 is independent of ( 1 ,  2 , . . .,   ) and   is N(0; [ 2 ,  2 ])-distributed.Then we represent a sublinear expectation E[⋅] stated on   (Λ) as follows.For every  = ( Definition 4. The expectation mentioned above, that is, E :   (Λ) → R, is known as a G-expectation and the related process {  ,  ≥ 0} is said to be a G-Brownian motion.
Definition 5.For every   ∈  2,0  (0, ), G-Itô's integral is denoted by (  ) and is defined by Let a sequence of partition of [0, ] be given by Definition 6.The process {⟨⟩  ,  ≥ 0}, known as quadratic variation process of {  ,  ≥ 0}, is a continuous increasing process having ⟨⟩ 0 = 0 and is stated by Definition 7. Suppose B(Λ) is the Borel -algebra of Λ and P is the (weakly compact) combination of measures for Abstract and Applied Analysis 3 probability  that is stated on (Λ, B(Λ)).Afterwards, the capacity ĉ(⋅) attached to P is given by In case of the capacity of a set  to be zero then the set  is called polar, that is, ĉ() = 0 and a property carries quasisurely (in short q.s.) if it is satisfied outside a polar set.

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).
Remark 13.In a similar way as the previous lemma one can prove