On a class of backward doubly stochastic differential equations with continuous coefficients (Q2796888)

\textit{E. Pardoux} and \textit{S. Peng} [Probab. Theory Relat. Fields 98, No. 2, 209--227 (1994; Zbl 0792.60050)] have proven existence and uniqueness of solutions for backward doubly stochastic differential equations (BDSDEs) of the form NEWLINE\[NEWLINE y_t = \xi + \int_t^T f(s,y_s,z_s) \, \text{d}s + \int_t^T g(s,y_s,z_s) \, \text{d}B_s - \int_t^T z_s \, \text{d}W_s, \quad t \in [0,T] NEWLINE\]NEWLINE under Lipschitz conditions.NEWLINENEWLINELater, \textit{S. Janković} et al. [Appl. Math. Comput. 217, No. 21, 8754--8764 (2011; Zbl 1220.60034); corrigendum ibid. 218, No. 17, 9033--9034 (2012)] have proven existence and uniqueness of solutions as well as a comparison result for BDSDEs of the form NEWLINE\[NEWLINE Y_t = \xi + \int_t^T f(s,Y_s,Z_s) \, \text{d}s + \int_t^T g(s,Y_s,Z_s) \, \text{d}B_s - \int_t^T [h(s,Y_s) + Z_s] \, \text{d}W_s, \quad t \in [0,T] NEWLINE\]NEWLINE under Lipschitz, and also under non-Lipschitz conditions. Furthermore, they have studied the connection between these BDSDEs and those studied in the aforementioned paper by Pardoux and Peng.NEWLINENEWLINEThe goal of the present paper is to extend these results by proving existence of solutions for the BDSDEs considered in Janković et al. under continuity and linear growth conditions. Moreover, the author establishes the relation between these BDSDEs and those studied by Pardoux and Peng, and provides a comparison result as well as a Kneser-type theorem, which deals with the number of solutions for the BDSDE.