On strong existence and continuous dependence for solutions of one-dimensional stochastic equations with additive Lévy noise

One-dimensional stochastic differential equations (SDEs) with additive Lévy noise are considered. Conditions for strong existence and uniqueness of a solution are obtained. In particular, if the noise is a Lévy symmetric stable process with $\alpha\in(1;2),$ then the measurability and the boundedness of a drift term is sufficient for the existence of a strong solution. We also study the continuous dependence of the strong solution on the initial value and the drift.