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Existence, uniqueness, and $L_p$-approximation results are presented forscalar stochastic differential equations (SDEs) by considering the case where,the drift coefficient has finitely many spatial discontinuities while bothcoefficients can grow superlinearly (in the space variable). Thesediscontinuities are described by a piecewise local Lipschitz continuity and apiecewise monotone-type condition while the diffusion coefficient is assumed tobe locally Lipschitz continuous and non-degenerate at the discontinuity pointsof the drift coefficient. Moreover, the superlinear nature of the coefficientsis dictated by a suitable coercivity condition and a polynomial growth of the(local) Lipschitz constants of the coefficients. Existence and uniqueness ofstrong solutions of such SDEs are obtained. Furthermore, the classical$L_p$-error rate $1/2$, for a suitable range of values of $p$, is recovered fora tamed Euler scheme which is used for approximating these solutions. To thebest of the authors' knowledge, these are the first existence, uniqueness andapproximation results for this class of SDEs.

Existence, uniqueness and approximation of solutions of SDEs with  superlinear coefficients in the presence of discontinuities of the drift  coefficient