Reflected backward stochastic differential equations (RBSDE in short) were introduced by N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez [Ann. Probab. 25, No. 2, 702-737 (1997; Zbl 0899.60047)]: One adds to an ordinary BSDE a reflection term :
forcing the -part of the solution to stay above a given barrier process . In the present paper, existence and uniqueness of a solution of the RBSDE are shown in the case where the original continuous process is just càdlàg (right continuous, left limited). In particular, is allowed to have negative jumps, what means that is no more continuous neither. As an application, a link between these equations and stochastic mixed control problems is given.