EXISTENCE OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ABSTRACT VOLTERRA OPERATORS

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1 International Journal of Differential Equations and Applications Volume 7 No. 1 23, EXISTENCE OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ABSTRACT VOLTERRA OPERATORS Zephyrinus C. Okonkwo Department of Mathematics and Computer Science Albany State University Albany, Georgia, GA 3175, USA zokonkwo@asu.alasu.edu Abstract: This paper deals with the existence of the solution process of a class of neutral stochastic functional differential equations with abstract Volterra operators defined on certain function spaces. Using fixed point techniques, existence and uniqueness of the solution process are proven. It is also noted that the method of successive approximations can be used to obtain some existence results. AMS Subject Classification: 34D5, 34F5, 6H2 Key Words: neutral stochasitc functional differential equations, Volterra operators 1. Introduction and Statement of the Problem In this paper, we study the existence and uniqueness of the solution process of neutral stochastic functional differential equations of the form d[v (x(t, ω))] = (Fx)(t, ω)dt + σ(t, x(t, ω))dz(t, ω), (1.1) with the initial condition x(, ω) = x R n. (1.2) Received: April 9, 23 c 23 Academic Publications

2 12 Z.C. Okonkwo Other auxiliary conditions can also be imposed. The underlying funciton space is E([, T ] Ω, R n ), which for every ω Ω is either the space of continuous functions C([, T ] Ω, R n ) or the space of measurable maps L p ([, T ] Ω, R n ), 1 p. For the purpose of this paper, we shall take the underlying space to be the function space L 2 ([, T ] Ω, R n ). L 2 ([, T ] Ω, R n ), T <, is the space of all product measurable random functions x : [, T ] Ω R n with square integrable sample paths. V and F are causal operators acting on the function space L 2 ([, T ] Ω, R n ) and satisfying certain regularity properties to be specified in the sequel. σ is a nonlinear map belonging to the space M 2 ([, T ] R n, R n q ) which is the space of all n q product measurable matrix-valued random functions satisfying the following condition T P(ω \ σ(t, x(t, ω)) 2 dt < ) = 1, (1.3) with the integral in (1.3) being in the ordinary Lebesque sense. Throughout this paper, we shall assume that (Ω, F, P) is a complete probability space, F is the σ-algebra of subsets of Ω; F t is the underlying filtration and P is the probability measure. Ω is the sample space, with ω Ω. Furthermore, z(t, ω) is assumed to be a normalized R q valued Wiener process, x is a Gaussian random variable, with a positive definite covariance matrix Q and independent of the Wiener process z(t, ω), see for exmaple [4]. The Volterra operator V, we have in mind, can appear in the form or in the form (V x)(t, ω) = x(t, ω) + G(x(t h, ω)), (1.4) (V x)(t, ω) = x(t, ω) + G(x(η(t), ω)), (1.5) where η(t) is a scalar real-valued function satisfying η(t) t for t [, T ], with η() =. For complete details of the properties of the operator V in (1.5), we refer the reader to Corduneanu [3]. Notice that σ(s, x(s, ω))dz(s, ω)

3 EXISTENCE OF NEUTRAL STOCHASTIC satisfies the following property for every t T : and E E{ σ(s, x(s, ω))dz(s, ω) 2 = σ(s, x(s, ω))dz(s, ω)} =, (1.6) E σ(s, x(s, ω)) 2 ds <. (1.7) Here and in the sequel, Ef will mean the mathematical expectation of f. 2. Existence and Uniqueness of the Solution Process Let us consider the system (1.1), (1.2) under the assumption that V is of the form (1.5). Futhermore, assume that (H1) G : L 2 ([, T ] Ω, R n ) L 2 ([, T ] Ω, R n ) is a contraction operator on this space; that is E G(x) G(y) 2 L 2 λe x y 2 L 2. (2.1) (H2) F : L 2 ([, T ] Ω, R n ) L 2 ([, T ] Ω, R n ) is Causal, and takes almost surely bounded sets into bounded sets with (H3) E F(x) F(y) 2 L 2 γ(t)e x y 2 L 2. (2.2) E σ(t, x(t, ω)) σ(t, y(t, ω)) 2 L 2 β(t)e x y 2 L 2. (2.3) In the sequel, we shall assume appropriately that 3λ < 1, and β(t), γ(t) L 1 for every x L 2 with t T. Theorem 2.1. Consider the neutral stochastic functional differential equation (1.1) with V given by formula (1.5) under the initial condiiton (1.2). Assume that hypotheses H1, H2, and H3 are satisfied, η being a real-valued function with η() = and η(t) t for t [, T ]. Then there exists a solution process x = x(t, ω) of the problem on [, τ] Ω [, T ] Ω with τ < T, such that x(t, ω) + G(x(η(t), ω)) is differentiable almost everywhere.

4 14 Z.C. Okonkwo Proof. Equation (1.1), (1.2) can be written in the form x(t, ω) + G(x(η(t), ω)) = G(x(η(), ω)) + x + + σ(s, x(s, ω))dz(s, ω). We assume, without loss of generality, that G(x(η(), ω)) = G(x(), ω)) = G(x ) = θ. This does not represent any restrictions to the problem. Define the operator U by (Ux)(t, ω) = x + (Fx)(s, ω)ds + (Fx)(s, ω)ds (2.4) σ(s, x(s, ω))dz(s, ω), (2.5) which makes sense for any x L 2. Equation (2.4) can be put in the form (V x)(t, ω) = (Ux)(t, ω), (2.6) for any x L 2 ([, T ] Ω, R n ). If we assume that V has an inverse on L 2 ([, T ] Ω, R n ), as well as continuous and causal, then equation (2.6) can be written in the form x(t, ω) = V 1 (Ux)(t, ω), (2.7) which is the usual form of equations with causal operators see for example Corduneanu [2]. Let us proceed in the sense of Corduneanu [3]. From hypothesis H1 and H2, with η = η(t), it is obvious that the map x(t, ω) = (V x)(t, ω) (2.8) is continuous almost everywhere for each ω Ω. This map is a homeomorphism on the space L 2, see for example Burton [1]. Notice that if we represent the right-hand side of equation (2.4) by f(t, ω), then (2.4) can be put in the form x(t, ω) + G(x(η(t), ω) = f(t, ω), (2.9)

5 EXISTENCE OF NEUTRAL STOCHASTIC i.e., (V x)(t, ω) = f(t, ω). From (2.9) we have that x(t, ω) = G(x(η(t), ω)) + f(t, ω) = (T x)(t, ω). (2.1) Under assumption H1, H2 and H3, it is obvious that T is a contraction. Hence, (2.1) has a unique solution process x(t, ω) L 2, for each f(t, ω) in L 2. This implies that V maps L 2 onto L 2 for each f L 2 and there is only one solution process x(t, ω) of (2.1). The map x(t, ω) (V x)(t, ω) has an inverse V 1 on L 2 ([, T ] Ω, R n ). Hence V 1 is continuous almost surely and (2.1) defines a homeomorphism on the space L 2. The above results therefore indicate that the product V 1 U is continuous and causal on L 2, and since U is compact, V 1 U is compact. Let D L 2 be a bounded set. Then, FD L 2 is bounded. Hence there is M > such that for any h FD, h(t) h(s) M t s. This implies the equicontinuity of the functions belonging to FD. Ascoli- Arzela criterion of compactness implies that U : L 2 L 2 is compact. The product V 1 U is continuous and compact and hence takes bounded sets to relatively compact sets. All conditons of Theorem in [2] are satisfied by the operator V 1 U which implies the existence of a solution process of equation (2.7), on some interval [, τ] [, T ]. But (2.7) is equivalent to (2.6) which in turn is equivalent to (1.1), (1.2). Uniqueness: Let x(t, ω) and y(t, ω) be two solutions of (1.1), (1.2) defined on some interval [, δ]. Then x(t, ω) y(t, ω) + G(x(η(x), ω)) G(y(η(t), ω)) = + {(Fx)(s, ω) (Fy)(s, ω)}ds {σ(s, x(s, ω)) σ(s, y(s, ω))}dz(s, ω). (2.11) Taking the mathematical expectation of both sides and using the fact that a + b + c 2 3a 2 + 3b 2 + 3c 2 we get E x(t, ω) y(t, ω) 2 3E G(x(η(t), ω)) G(y(η(t), ω)) 2 3E (Fx)(s, ω) (Fy)(s, ω) 2 ds + 3E σ(s, x(s, ω)) σ(s, y(s, ω)) 2 ds.

6 16 Z.C. Okonkwo Using our assumptions in this Theorem we obtain E x(t, ω) y(t, ω) 2 3λE x(η(t), ω) y(η(t), ω) 2 3E γ(s) x(s, ω) y(s, ω) 2 ds + 3E β(s) x(s, ω) y(s, ω) 2 ds. (2.12) We now strengthen the above inequality by invoking hypotheses H1, H2 and H3. We then obtain E x(t, ω) y(t, ω) 2 3λE x(t, ω) y(t, ω) 2 3E + 3E γ(s) x(s, ω) y(s, ω) 2 ds β(s) x(s, ω) y(s, ω) 2 ds. (2.13) Let u(t) = E x(t, ω) y(t, ω) 2, s t for t [, T ]. Inequality (2.13) can then be put in the form (1 3λ)u(t) 3 Hence for each ɛ >, we have that u(t) ɛ λ {γ(s) + β(s)}u(s)ds, t [, T ]. (2.14) {γ(s) + β(s)}u(s)ds, t [, T ]. (2.15) Let {γ(s) + β(s)}ds ζ(τ)t. From the above Gronwall-Type inequality we have u(t) ɛ exp{ 3ζ(τ)t }, t [, T ]. (2.16) (1 3λ) Since ɛ > was arbitrary, we have that u(t) = on [, T ], and hence x = y on [, T ].

7 EXISTENCE OF NEUTRAL STOCHASTIC Remark 2.2. Under certain assumptions on the operators in equation (1.1), the existence and uniqueness of (1.1), (1.2) can be obtained using the method of successive approximations, see for example Okonkwo and Ladde [4]. References [1] T.A. Burton, Integral Equations, implicit functions, and fixed points, Proc. AMS, 124 (1996), [2] C. Corduneanu, Integral Equations and Applications, Cambridge University Press (1991). [3] C. Corduneanu, Existence of solutions of neutral functional differential equations with causal operators, Journal of Differential Equations, 168, (2). [4] Z.C. Okonkwo, G.S. Ladde, Itô-type stochastic differential equations with abstract Volterra operators and their control, Dynamic Systems and Applications, 6 (1997), [5] Z.C. Okonkwo, E.G. Rowe, On neutral stochastic functional differential equations with causal operators, International J. of Appl. Math., 2, No.1 (2), [6] Z.C. Okonkwo, E.G. Rowe, A.O. Adewale, Stability of stochastic functional differential equations with causal operators, International J. of Diff. Equations and Appl., 1, No. (4) (2),

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