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Time-dependent backward stochastic evolution equations. (English) Zbl 1139.60026
Given a time-dependent unbounded linear operator $A=(A_t)_{t\ge 0}$ on a separable Hilbert space $H$, generating a semigroup $(U_{s,t})_{s\ge t\ge 0}\subset L(H)$ which is strongly continuous in $(s,t)$, and given a cylindrical Brownian motion $W$ defined over some probability space $(\Omega,{\cal F},P),$ the author of the present paper investigates the $H$-valued backward stochastic differential equation (BSDE) $dY_t=-\{A_tY_t+f(t,Y_t,Z_t)\}dt+Z_tdW_t,\, t\in[0,T],\, Y_T=\xi\in L^2(\Omega,{\cal F}_T^W,P;H)$. Supposing that the $({\cal F}^W_t)$-progressively measurable coefficient $f(t,\omega,y,z)$ is Lipschitz in $z$ and such that $\vert f(t,y,z)-f(t,y',z)\vert\le c(\vert y-y'\vert),\, y,y'\in H,$ for some non increasing concave function $c:R^{*}_+\rightarrow R^{*}_+$ with $c(0+)=0$ and $\int_{0^+}^1c^{-1}(t)dt=+\infty,$ the author shows the existence and the uniqueness for this BSDE. Afterwards he proves that the process $Y$ has continuous trajectories. The author’s work concerns a subject which enjoys a great interest since the pioneering paper by {\it Y. Hu} and {\it S. Peng} [Stochastic Anal. Appl. 9, No. 4, 445--459 (1991; Zbl 0736.60051)], and a lot of generalizations and applications of the equation considered by Hu and Peng (the above BSDE with time-independent unbounded linear operator $A$) have been studied since then. The present paper employs standard methods for its generalization.

60H10Stochastic ordinary differential equations
60H15Stochastic partial differential equations
Full Text: EuDML