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We are interested in the strong convergence and almost sure stability ofEuler-Maruyama (EM) type approximations to the solutions of stochasticdifferential equations (SDEs) with non-linear and non-Lipschitziancoefficients. Motivation comes from finance and biology where many widelyapplied models do not satisfy the standard assumptions required for the strongconvergence. In addition we examine the globally almost surely asymptoticstability in this non-linear setting for EM type schemes. In particular, wepresent a stochastic counterpart of the discrete LaSalle principle from whichwe deduce stability properties for numerical methods.

Strong convergence and stability of implicit numerical methods for  stochastic differential equations with non-globally Lipschitz continuous  coefficients