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Reflected BSDE’s with discontinuous barrier and application. (English) Zbl 1015.60057

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 (K t ):

Y t =ξ+ t T f(s,Y s ,Z s )ds+K T -K t - t T Z s dB s ,

forcing the (Y t )-part of the solution (Y t ,Z t ,K t ) to stay above a given barrier process (X t ). In the present paper, existence and uniqueness of a solution of the RBSDE are shown in the case where the original continuous process (X t ) is just càdlàg (right continuous, left limited). In particular, (X t ) is allowed to have negative jumps, what means that (Y t ) is no more continuous neither. As an application, a link between these equations and stochastic mixed control problems is given.

MSC:
60H99Stochastic analysis
93E20Optimal stochastic control (systems)