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Anticipated backward stochastic differential equations. (English) Zbl 1186.60053
Let \(T>0\) denote a finite time horizon and \(W=(W_t)_{t\in[0,T]}\) be a \(d\)-dimensional Brownian motion. The authors consider two functions \(\delta_i:[0,T]\rightarrow \mathbb{R}_+,\, i=1,2,\) for which there is some constant \(K>0\) such that, for both of them, \(s+\delta_i(s)\leq T+K,\, s\in[0,T],\) and \(\int_t^Tg(s\delta_i(s))ds\leq C_K\int_t^{T+K}g(s)ds,\, i=1,2,\) for all non negative, integrable function \(g\).
The authors investigate in their paper the following new type of (\(m\)-dimensional) backward stochastic differential equations (BSDE):
\[ dY_t=-f(t,Y_t,Z_t,Y_{t+\delta(t)},Z_{t+\zeta(t)})dt+Z_tdW_t,\, t\in[0,T], \] where they put \(Y_t=\xi_t,\, Z_t=\eta_t,\, t\in[T,T+K],\) for given stochastic processes \(\xi\in{\mathcal S}_{\mathbb{F}}^2(T,T+K;\mathbb{R}^{m}),\, \eta\in L_{\mathbb{F}}^2(T,T+K;\mathbb{R}^{m\times d}).\) This type of BSDE generalizes the by now well known BSDE introduced by E. Pardoux and S. Peng in their pioneering paper of 1990 [Syst. Control Lett. 14, No. 1, 55–61 (1990; Zbl 0692.93064)]. Taking into account that a BSDE is solved backward, beginning from its time horizon \(T\), the above BSDE can be regarded as a translation of the concept of stochastic differential equations (SDE) with delay into the context of backward stochastic equations.
But the link between these anticipated BSDE and SDE with delay is even narrower as the authors show with their proof of duality between both types of equation. Concerning the above anticipated BSDE, the authors prove that under adequate assumptions on the driving coefficient \(f\) which generalize those for standard BSDE and include in particular the assumption that \(f(s,y,z,\xi,\eta)\) is \({\mathcal F}^W_s\)-measurable, \(s\in[0,T],\) for all square integrable random variables \(\xi,\eta,\) there exists a unique solution \((Y,Z)\). Moreover, for the one-dimensional anticipated BSDE (\(m=1\)) they establish a comparison principle and prove also a strict comparison result. Finally, their paper is completed by the study of a control problem in which the controlled state process is a linear SDE with delay, and whose cost functional is described by an anticipated BSDE.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
93E03 Stochastic systems in control theory (general)
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