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The primary goal of this paper is to prove a near-martingale optionalstopping theorem and establish solvability and large deviations for a class ofanticipating linear stochastic differential equations. We prove the existenceand uniqueness of solutions using two approaches: (1) Ayed-Kuo differentialformula using an ansatz, and (2) a novel braiding technique by interpreting theintegral in the Skorokhod sense. We establish a Freidlin-Wentzell type largedeviations result for solution of such equations.

On near-martingales and a class of anticipating linear SDEs