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Time-dependent backward stochastic evolution equations. (English) Zbl 1139.60026

Given a time-dependent unbounded linear operator \(A=(A_t)_{t\geq 0}\) on a separable Hilbert space \(H\), generating a semigroup \((U_{s,t})_{s\geq t\geq 0}\subset L(H)\) which is strongly continuous in \((s,t)\), and given a cylindrical Brownian motion \(W\) defined over some probability space \((\Omega,{\mathcal F},P),\) the author of the present paper investigates the \(H\)-valued backward stochastic differential equation (BSDE) \(dY_t=-\{A_tY_t+f(t,Y_t,Z_t)\}dt+Z_tdW_t,\, t\in[0,T],\, Y_T=\xi\in L^2(\Omega,{\mathcal F}_T^W,P;H)\). Supposing that the \(({\mathcal F}^W_t)\)-progressively measurable coefficient \(f(t,\omega,y,z)\) is Lipschitz in \(z\) and such that \(| f(t,y,z)-f(t,y',z)|\leq c(| y-y'|),\, y,y'\in H,\) 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 Y. Hu and 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.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Citations:

Zbl 0736.60051
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