In this paper, we are focused upon the global uniqueness results for a stochastic integro-differential equation in Fréchet spaces. The main results are proved by using the resolvent operators combined with a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to Frigon and Granas. As an application, a controllability result with one parameter is given to illustrate the theory. appropriate function specified later and w(t), t ≥ 0 is a given K-valued Brownian motion, which will be

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... efined in Section 2. The initial data x 0 is an F 0 -adapted, H-valued random variable independent of the Wiener process w. Stochastic differential and integro-differential equations have attracted great interest due to their applications in characterizing many problems in physics, biology, mechanics and so on. Qualitative properties such as existence, uniqueness and stability for various stochastic differential and integro-differential systems have been extensively studied by many researchers, see for instance [1, 2, 3, 4, 5, 6, 7, 8, 9] and the references therein. The theory of nonlinear functional integro-differential equations with resolvent operators serves as an abstract formulation of partial integrodifferential equations which arises in many physical phenomena [10, 11, 12, 13, 14] . Just as pointed out by Ouahab in [15] , the investigation of many properties of solutions for a given equation, such as stability, oscillation, often needs to guarantee its global existence. Thus it is of great importance to establish sufficient conditions for global existence results for functional differential equations. The existence of unique global solutions for deterministic functional differential evolution equations with infinite delay in Fréchet spaces were studied by Baghli et al. [16, 17] and Benchohra et al. [18]. Motivated by the works [16, 17, 19, 18] , the main purpose of this paper is to establish the global uniqueness of solutions for the problem (1.1)-(1.2). Our approach here is based on a recent Frigon and Granas nonlinear alternative of Leray-Schauder type in Fréchet spaces [20] combined with the resolvent operators theory. The obtained result can be seen as a contribution to this emerging field. The rest of this paper is organized as follows: In section 2, we recall some basic definitions, notations, lemmas and theorems which will be needed in the sequel. In section 3, we prove the existence of the unique mild solutions for the problem (1.1)-(1.2). Section 4 is reserved for an application.