On degenerate stochastic equations of Itô type with jumps

The one-dimensional stochastic equation dXt=b1(Xt)dWt+b2(Xt−)dZt+a(Xt)dt,t≥0, where b1,b2,a:R→R are Borel measurable functions, W is a Brownian motion, and Z is a symmetric stable process of index 0<α<2, is considered. We prove the existence of (weak) solutions under some conditions of boundedness of coefficients when b1 can be degenerate which improves the results of Lepeltier and Marchal [Lepeltier, J.P., Marchal, B., 1976. Probléme des martingales et équations différentielles stochastiques associées á un opérateur intégro-différentiel. Ann. IHP 12 (1), 43-103] for this case. Our approach is based on Krylov's estimates for solutions X and the weak convergence arguments.