  In this paper we are interested in a quasi-linear hyperbolic stochastic
differential equation (HSPDE) when the vector field is merely bounded and
measurable. Although the deterministic counterpart of such equation may be
ill-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 (Malliavin
differentiable) 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 of
SDEs in the plane by using an algorithm for the selection of certain
rectangles. As a by product, we also obtain the Sobolev differentiability of
the solution with respect to its initial value. The results derived here
constitute a significant improvement of those in the current literature on SDEs
on the plane and can be regarded as an analogous equivalent of the pioneering
works by [Zvonkin, Math. URSS Sbornik, 22:129-149] and [Veretennikov, Theory
Probab. Appl., 24:354-366] in the case of one-parameter SDEs with singular
drift.
