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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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