On well-posedness and Euler scheme for regime-switching stochastic differential equations with discontinuous drift coefficient

In this article, we study well-posedness and Euler scheme for regime-switching stochastic differential equations where the drift coefficient is piecewise Lipschitz continuous and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. The entangling of the discontinuous dynamics of the underlying Markov chain and the continuous dynamics of the solution process, along with the discontinuities in the drift coefficient, gives rise to various challenges which are resolved through a Markov chain-dependent transformation and a numerical scheme with a non-uniform discretization induced by the jump–times of the chain. To achieve mean-square convergence of order 1/2, we investigate conditional local and occupation times of the scheme near the points of discontinuity. Our approach also incorporates the case where the behaviour of discontinuity points of the drift coefficient can vary from regime to regime. Finally, we illustrate our results through numerical examples.