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Double barrier backward SDEs with continuous coefficient. (English) Zbl 0887.60065
El Karoui, Nicole (ed.) et al., Backward stochastic differential equations. Harlow: Longman. Pitman Res. Notes Math. Ser. 364, 161-175 (1997).
The authors prove an existence theorem for solution of double barrier backward stochastic differential equations. A solution of such an equation is a process $$(Y,Z,K^+, K^-)$$ such that $$L_t\leq Y_t\leq U_t$$, for all $$t\leq T$$, where $Y_t= \xi+ \int^T_t f(s, Y_s, Z_s)ds- \int^T_t Z^*_s dW_s+ (K^+_T- K^+_t)- (K^-_T- K^-_t)$ and $$E\int^T_0(|Y_s|^2+|Z_s|^2)ds<+ \infty$$. $$K^+$$ (resp. $$K^-$$) is a continuous increasing process such that $$\int^T_0(Y_s- L_s)dK^+_s= 0$$ (resp. $$\int^T_0(U_s- Y_s)dK^-_s= 0$$). The authors show the existence of a solution if $$f$$ is continuous with linear growth and if one barrier is smooth. They also give an example in which there are two solutions.
For the entire collection see [Zbl 0866.00052].

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)