In this paper, we present a sufficient condition for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. We then establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will play an important role.

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We introduce a new class of Backward Stochastic Differential Equations with weak reflections whose solution (Y, Z) satisfies the weak constraint E[Ψ(θ, Y θ )] ≥ m, for all stopping time θ taking values between 0 and a terminal time T , where Ψ is a random non-decreasing map and m a given threshold. We study the wellposedness of such equations and show that the family of minimal time t-values Y t can be aggregated by a right-continuous process. We give a nonlinear Mertens type decomposition for lower reflected g-submartingales, which to the best of our knowledge, represents a new result in the literature. Using this decomposition, we obtain a representation of the minimal time t-values process. We also show that the minimal supersolution of a such equation can be written as a stochastic control/optimal stopping game, which is shown to admit, under appropriate assumptions, a value and saddle points. From a financial point of view, this problem is related to the approximative hedging for American options.