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In this paper we are interested in a quasi-linear hyperbolic stochasticdifferential equation (HSPDE) when the vector field is merely bounded andmeasurable. Although the deterministic counterpart of such equation may beill-posed (in the sense that uniqueness or even existence might not be valid),we show for the first time that the corresponding HSPDE has a unique (Malliavindifferentiable) strong solution. Our approach for proving this result rests on:1) tools from Malliavin calculus and 2) variational techniques introduced in[Davie, Int. Math. Res. Not., Vol. 2007] non trivially extended to the case ofSDEs in the plane by using an algorithm for the selection of certainrectangles. As a by product, we also obtain the Sobolev differentiability ofthe solution with respect to its initial value. The results derived hereconstitute a significant improvement of those in the current literature on SDEson the plane and can be regarded as an analogous equivalent of the pioneeringworks by [Zvonkin, Math. URSS Sbornik, 22:129-149] and [Veretennikov, TheoryProbab. Appl., 24:354-366] in the case of one-parameter SDEs with singulardrift.

Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields