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Reflected solutions of generalized anticipated BSDEs and application to reflected BSDEs with functional barrier. (English) Zbl 1272.60039
After the paper of S. Peng and Z. Yang [Ann. Probab. 37, No. 3, 877–902 (2009; Zbl 1186.60053)], Yang studied so-called generalised anticipated backward stochastic differential equations (ABSDEs), which, driven by a Brownian motion $$W=(W)_{t\in[0,T]}$$, have the form $dY_t=-f(t,(Y_r)_{r\in[t,T+h]},(Z_r)_{r\in[t,T+h]})dt+Z_tdW_t,\quad t\in [0,T],$ where $$T,C>0$$ and the given terminal conditions are $$Y_t=\xi_t,\;t\in[T,T+C]$$, and $$Z_t=\eta_t,$$ $$dt$$-a.e., $$t\in(T,T+C]$$. For all $$t\in[0,T]$$, the function $$f(t,.,.)$$ maps the elements of spaces of adapted processes into the space $$L^2({\mathcal F}_t;\mathbb{R}^m)$$ and satisfies some comprehensible Lipschitz conditions guaranteeing the existence and the uniqueness of an adapted solution $$(Y,Z)$$. The author of the present paper extends the results of Yang with the help of a fixed point theorem in a direct way to generalised ABSDEs with a lower reflecting barrier $$S$$ which is supposed to be a continuous adapted process with a square integrable positive part. An application is studied, in which the standard reflection condition $$Y_t\geq S_t,\;(Y_t-S_t)dK_t, \;t\in [0,T]$$ ($$K$$ is the adapted increasing process representing the minimal force needed to keep $$Y$$ above the barrier $$S$$) is replaced by $\operatorname{E}[\int_t^TY_s\,ds|{\mathcal F}_t]\geq S_t,\;t\in [0,T],$ $\int_0^T\big(\operatorname{E}[\int_t^TY_s\,ds|{\mathcal F}_t]-S_t\big)\operatorname{E}[K_T-K_t|{\mathcal F}_t]\,dt=0.$

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